Ann. Henri Poincar´e 3 (2002) 1 – 17 c Birkh¨ auser Verlag, Basel, 2002 1424-0637/02/010001-17 $ 1.50+0.20/0
Annales Henri Poincar´ e
On the Gap in the Spectrum of the Translations H. J. Borchers Abstract. The spectrum of the translations in local quantum field theory will be analyzed in order to show that in a positive energy representation without vacuum vector and with lowest mass m1 there is no gap in the spectrum which is larger than 2m1 . In particular in a zero mass representation there is no hole at all. These results are obtained with methods of analytic functions of several complex variables.
1 Introduction Let π be a representation of quantum field theory of local observables in the sense of Araki, Haag and Kastler [Ha92], which fulfils the spectrum condition. We assume that the representation of the translation is the minimal one, which implies that its spectrum is located on a Lorentz invariant set of the closed forward light cone V + (see [BB85] and [Bch85]). If the representation is generated from a vacuum state then Wightman’s result [Wi64] implies that the spectrum is an additive set, i.e. it fulfils S + S ⊂ S, (1.1) where S denotes the support of the spectrum. If the representation space does not contain a vacuum vector then I conjectured in [Bch96] that the spectrum fulfils the relation S + S + S ⊂ S. (1.2) In particle physics factor representations are also called super selection sectors because a superposition of vectors in two different sectors does not lead to observable consequences. Therefore, the charges, describing the different sectors, must commute with every observable. Since the time development belongs to the weak closure of the observable algebra it acts separately in every sector. In such a situation one can characterize the factor representations of the observable algebra by charge quantum numbers. Dealing with vectors carrying a charge and adding to this a particle with a non-zero charge, then the total charge changes. This implies that we obtain another sector. Therefore, one stays in the same sector only if we add particles carrying the total charge zero. Let us associate to every sector a charge quantum number Q, then the structure of the spectrum of the translation is known, provided the representation in a charged sector is generated from the vacuum sector by a charged field, in the sense of Doplicher, Haag and Roberts [DHR69a,b]. In this situation one has in the case of Abelian charges the relation SQ1 + SQ2 = SQ1 +Q2 ,
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which has been shown in [Bch65]. SQ denotes the spectrum of the sector characterized by Q. We are interested in a statement concerning only one sector. If the above picture with the charged fields is correct, then the adjoint of a charged field carries the opposite charge. Therefore, one must find to every sector a sector with opposite charge. Since the spectrum does not change by this charge conjugation one has: S−Q = SQ . More generally this relation holds for particles and anti-particles. Since a particle together with its anti-particle form a neutral pair we remain in the same sector if we add to a vector in this sector such a pair. If one takes the particle from the given sector, one obtains Eq. (1.2). In general theories one cannot expect that the representations obtained from a vacuum sector are all representations. In the presence of zero-mass particles the investigations of Doplicher and Spera [DS82], Borek [Bk82] in the non-interacting, and of Buchholz and Doplicher [BD84] in the interacting situation show, that most positive energy representations are not connected with vacuum representations. A similar situation might hold also in a gauge field theory. Therefore, Eq. (1.2) needs a separate investigation. One special result is known for arbitrary sectors [Bch86], which indicates that the conjecture is probably correct. Theorem. Let π be a covariant factor representation of a theory of local observables fulfilling the spectrum condition, and let T (a) be the unique minimal representation of the translations. Assume the spectrum of T (a) consists of two parts: (a) Isolated hyperboloids with the masses m1 < m2 · · · mi < · · · . (b) The rest starting at mc > mi where mc denotes the beginning of the continuous spectrum or the first accumulation point of the mi s. Then 3m1 ≥ mc . One would like to show that above 3m1 there is no gap in the spectrum at all. Unfortunately techniques with which one can derive such result are missing. Since we know that there can be a gap between m1 and 3m1 , such method must distinguish between the regions below 3m1 and above 3m1 . Our procedure does not do it, therefore, we only obtain that there is no gap larger than 2m1 . The technique we are using is that of analytic functions of several complex variables, in particular the edge of the wedge theorem (Brehmermann, Oehme and Taylor [BOT56]). In this case one has two analytic functions holomorphic in the forwardand backward tube respectively. These functions coincide on some real coincidence domain and the edge of the wedge theorem implies that both functions are different branches of one analytic function. The problem consists in calculating the envelope of holomorphy or of parts of it.
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In the next section the details of the problem and the technique to be used will be presented. In our situation the coincidence domain consists of three parts each of which can be handled with help of the Jost-Lehmann-Dyson representation [Dy58]. We will compute the holomorphy domains of the three parts in section 3. In the last section the influence of the different parts on each other will be discussed with help of the double cone theorem. In favorable situations there is a simpler proof of the main result. It is using the ”reentrant nose theorem”, which is a consequence of the Jost-Lehmann-Dyson representation. The method will be described at the end of section 2.
2 Representation of the problem Quantum field theory of local observables in the sense of Araki, Haag and Kastler [Ha92] is concerned with C ∗ -algebras A(O), associated with bounded open regions O ⊂ Rd , which have a common unit. These algebras shall fulfil isotony, i.e. O1 ⊂ O2 implies A(O1 ) ⊂ A(O2 ), and locality, i.e. if O1 and O2 are space-like separated, then the algebras A(O1 ) and A(O2 ) commute element-wise. In order that this statement makes sense the space Rd must be furnished with the Minkowski metric. The global algebra A is defined as the C ∗ -inductive limit of {A(O)}. We also assume that the net {A(O)} is covariant under a representation of the translation group of Rd by automorphisms, i.e. to a ∈ Rd exists an automorphism αa with αa A(O) = A(O + a). A covariant representation π of A in the representation space H is such that there exists a continuous unitary representation T (a) of the translation group of Rd implementing αa T (a)π(A)T (−a) = π(αa (A)),
A ∈ A.
In addition T (a) shall fulfil the spectrum condition, i.e. spectrum T (Rd ) is contained in the closed forward light-cone. As a consequence of the spectrum condition one can choose the representation T (a) of the translation as elements of the von Neumann algebra generated by π(A) [Bch96]. Since T (a) leaves the center of π(A)” point-wise invariant [Ar64] it is no restriction assuming that π is a factor representation. The spectrum of the translations shall have the following structure: It starts at m1 > 0 and in addition there shall be a gap in the spectrum between m2 and m3 , m2 < m3 . We want to show that these assumptions lead to a contradiction if m3 − m2 > 2m1 holds. Since the statement we are looking for is scale invariant we may assume m1 = 1. For the investigation one looks at matrix elements of the form F + (a) = (ψ, BT (a)Aψ)
(2.1)
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where ψ belongs to the representation space Hπ and A, B are operators belonging to π(A(D)). D denotes a double cone with center at the origin. One also looks at the expression (2.2) F − (a) = (ψ, T (a)AT (−a)BT (a)ψ). Locality implies that F + (a) and F − (a) coincide for a ∈ (2D) , where the prime denotes the space-like complement. We are interested in the Fourier transform of (2.1) and (2.2). The support of the Fourier transform of (2.1) is contained in the support of the spectrum S described above. In order that the Fourier transform of (2.2) has good support property we must choose ψ in a suitable manner. The best way is to choose the spectral support of ψ close to the bottom of the spectrum. Let t = (1 + , 0) then we choose the support of ψ in S ∩ (t + V − ). Because of the curvature of the hyperboloid the support of the vector is contained in the double 1 , 0). This choice allows to describe the support of the cone Dt ,t with t = ( 1+ − Fourier transform of F (a). If we denote this support by S then we get S = 2t + (V1− \ {crescent moon}), − − crescent moon = set below 2t + Hm and above 2t + Hm . 2 3
(2.3)
+ In this formula Vm+ characterizes the set {p ∈ V + ; p2 ≥ m2 }, Vm− = −Vm+ . Hm 2 2 − denotes the upper sheet of the hyperboloid p = m and Hm its lower sheet. For the future investigations we choose so small that the crescent moon domain is sufficiently big. The details will be discussed later. About the spectrum of the translations we know definitely only the values mi , i = 1, 2, 3. Whether or not there exists other gaps in the spectrum than that between m2 and m3 is not determined by the formulation of the result Thm. 4.1. Therefore, one knows only vectors with energy-momentum support in the neighbourhood of the masses mi . This motivates the choice of the vector ψ. If m1 = 0, then one has to choose for the support of ψ a small double cone such that its closure is in the interior of the forward light cone. Such vector exists because the theorem in the introduction implies that m1 = 0 is not an isolated hyperboloid. Let F (a) = F + (a)− F − (a) be the commutator function. It vanishes in (2D) . Therefore, we can write F (a) = G+ (a) − G− (a) with
supp G+ (a) ⊂ −b + V + ,
supp G− (a) ⊂ b + V − ,
(2.4)
where b is the upper tip of the double cone 2D. Looking at the Fourier transform we obtain F˜ (p) = F˜ + (p) − F˜ − (p) and
supp F˜ (p) ⊂ S ∪ S .
(2.5)
On the other hand one can write ˜ + (p) − G ˜ − (p), F˜ (p) = G
(2.6)
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˜ + (p) is the boundary value of a function holomorphic in the forward where G + tube T = {z = x + iy ∈ Cd ; x ∈ Rd , y ∈ V + }. Here V + denotes the interior ˜ − (p) is the boundary value of an of the forward light cone. Correspondingly G − + analytic function holomorphic in T = −T . Since the two functions coincide on Rd \ {S ∪ S } we have to deal with an edge of the wedge problem. If one has an edge of the wedge problem where the coincidence domain is bounded by two space-like hyper-surfaces then the domain of holomorphy can be computed with help of the Jost-Lehmann-Dyson formula [Dy58] (see also Bros, Messiah and Stora [BMS61] [Bch96]), i.e. the complement of the holomorphy domain is the union of hyperboloids not entering the coincidence domain. If we assume that the same method would also work in our situation then the coincidence domain could be enlarged provided m3 > 3 holds. But one does not know whether or not also for time-like disconnected coincidence domains a formula like the Jost-Lehmann-Dyson representation holds. Therefore, we have to use other methods of enlarging the domain of holomorphy. If the domain of holomorphy can be enlarged into S \ {2t − V1 } then one can conclude that the spectrum is smaller than assumed. The reason is the following: Let ∆ be the set Dt ,t and E(∆) the corresponding spectral projection of the translation, then we can choose an arbitrary vector in E(∆)H. Moreover, the matrix elements are formed with operators in π(A(D)). Since the sets D are arbitrary, except for the requirement that their center is the origin, any operator in ∪A(D) can be chosen. These are norm-dense in A. Therefore, the conclusion for the spectrum holds for any vector in the closure of π(A)E(∆)H. The projection F on this space belongs to π(A) . Let Z be the central carrier of F then F π(A) and Zπ(A) are isomorphic, and hence we obtain a change of ZS. In the coming investigations we are dealing with factor representations. So one has Z = 1l and the analyticity methods show directly the change of the spectrum.
2.a A simple plausibility argument If one disregards the problem of the support of the vector ψ then there is a direct argument showing that there is no gap of width larger than 2m1 . This method uses the reentrant nose theorem, which follows from the Jost-Lehmann-Dyson representation. Since in this subsection we do not present a correct proof we look at the enlargement of the domain of holomorphy, which cuts into the support of −S convoluted with the double of the support of ψ. For a correct proof one would have to interchange the gap with the crescent moon domain. At the end of this subsection an example will be given showing that this method does not work in all cases, and that one has to use the more complicated methods described in section 3 and 4. Let τ be the vector (τ, 0) and Dτ − ,τ be the double cone {τ − + V + } ∩ {τ + V − } and assume that for τ > m1 the spectral projection E(Dτ − ,τ ) of the translations does not vanish. Choosing τ then the support of F − (p) is (2τ +
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Vm−1 ) \ crescent moon. (The crescent moon part is unimportant for the coming considerations.) Therefore, for m2 < 2τ − m1 < m3
(2.7)
The support of F − (p) cuts into the gap between m2 and m3 . The gap itself is a Jost-Lehmann-Dyson domain. Therefore, one can ask for conditions such that the part of (2τ + Vm−1 ) which cuts into the gap is a reentrant nose such that part of it can be erased by holomorphic completion. This means (gap) \ (2τ + Vm−1 ) is not the complement of hyperboloids not entering this domain. If this is the case then holomorphic completion shows that (2τ + Vm−1 ) is not a stable set, contradicting the assumptions that m1 is the lower boundary of the spectrum. The critical hyperboloid is Hm1 centered at 2τ . Its upper sheet shall not cut the hyperboloid Hm3 . The upper sheets of the two hyperboloids hit each-other if m3 − 2τ > m1
or
m3 − m1 > 2τ
holds. Together with Eq. (2.7) this implies m3 − m1 > m2 + m1 ,
(2.8)
which shows that there is no gap with width larger than 2m1 . The difficulty of this approach is the existence of suitable vectors ψτ which we have to assume in the above approach. That this assumption is not always satisfied shows the following Example: Look at the situation: (i) There is a gap between m1 and 3m1 . (ii) There is a gap between 3m1 and 6m1 . Trying to apply the above method we find that ψτ must be located at the + . Therefore, 2τ is a little above 6m1 and the reentrant nose hyperboloid H3m 1 appears only if m3 > 7m1 . Therefore, the existence of the second gap can not be excluded with this method.
3 The holomorphy domain for subsets of the coincidence region The coincidence domain of our problem consists of three parts. The envelope of holomorphy of each part can be computed separately with help of the JostLehmann-Dyson representation. The union of the three domains will be the starting point for the investigation of the whole problem. The three domains in question are: 1. The complement of {V1+ ∪ (2t + V1− )}.
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2. The gap, i.e. the domain between the two upper sheets of the hyperboloids + + Hm and Hm . 2 3 3. The crescent moon shaped domain, i.e. the domain between the hyperboloids − − − 2t + Hm and 2t + Hm . This means the points below 2t + Hm and above 2 3 2 − 2t + Hm3 . We start with the first coincidence domain and notice that it is invariant under rotations. The complement of the domain of holomorphy is given by the union of d ; (p − u)2 = µ2 } the real or complex points of the hyperboloids H(µ, √ u) = {p ∈ C √ which have their center u at the surface u0 = t2 and u = eλ t2 , where e is a real unit vector in the plane p0 = 0 and 0 ≤ λ ≤ 1. The possible values of µ are determined by the condition that the real part of H(µ, u) is contained in {V1+ ∪ (2t + V1− )}. Fixing u then the possible values of µ are larger or equal than a minimal value µ0 (u). Since the zero-component of u is fixed the function µ0 (u) depends only on the norm of the space-like components of u. √ √ First let us compute the function µ0 (u). To this end set τ = t2 , τ = t2 , which implies the representation u = τ (1, λe). The hyperboloids are given by the formula (p0 − τ )2 − (p − λτ e)2 = µ2 or p0 = τ ± µ2 + (p − λτ e)2 . In order that the upper sheet of this hyperboloid belongs to V1+ there must hold τ + µ2 + (p − λτ e)2 − 1 + p2 ≥ 0. (3.1) Since we want to look at the minimum we can choose p and e parallel and set p = πe. To obtain an equation for minimum one has to differentiate (3.1) with respect to π. π π − λτ −√ . 0= 2 2 1 + π2 µ + (π − λτ ) This leads to a second degree equation for π (π − λτ )2 (1 + π 2 ) (π − λτ )2 (π −
2 λτ 1−µ2 )
with the solution π=
= π 2 (µ2 + (π − λτ )2 ), = π 2 µ2 , =
λ2 τ 2 µ2 (1−µ2 )2 ,
λτ λτ , (1 ± µ) = 2 1−µ 1∓µ
which leads to the expressions 1 + π2 µ2 + (π − λτ )2
= =
1 (1 ∓ µ)2 + λ2 τ 2 , 1∓µ µ (1 ∓ µ)2 + λ2 τ 2 . 1∓µ
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The minimal value of (3.1) shall be zero, which implies τ+
µ − 1 (1 ∓ µ)2 + λ2 τ 2 = 0. 1∓µ
Hence we have to take the − sign. This implies for the minimal µ0 (λ) µ0 (λ) = max{1 − τ 1 − λ2 , 0}.
(3.2)
Since µ has to be non-negative one must choose µ = 0 if the first expression becomes negative. This first expression is the equation of a semi-circle with radius τ. For the holomorphy domain one must find the points p + iq such that the equation (p − u + iq)2 = µ2 ≥ µ20 (3.3) cannot be fulfilled. Eq. (3.3) reads (p − u)2 − q 2 + 2i(q, p − u) = µ2 . Therefore, points of analyticity are: Either one has (q, p − u) = 0 ∀ u, or if (q, p − u) = 0 for one u then (p − u)2 < µ20 (u) + q 2 must hold.
(3.4)
Next we look at the gap as coincidence domain. This set is invariant under Lorentz transformations. Therefore, the surface for the center of the hyperboloids is also invariant, and the same holds for the minimal µ. The centers of 2 2 ) and the minimal µ is the hyperboloids are located on the surface u2 = ( m3 +m 2 m3 −m2 µ0 = . Let (p, q) be given, then exists always a u with (q, p − u) = 0. In 2 order that p + iq belongs to the domain of holomorphy (q, p − u) = 0 and (p − u)2 < µ20 + q 2
∀ u fulfilling the first eqation
(3.5)
must hold. Since (p − u) is time-like this equation can only be fulfilled if −q 2 < 2 2 ) holds. ( m3 −m 2 Finally we look at the crescent moon problem. The interesting part of the surface for the centers of the hyperboloids must be inside this coincidence domain. The maximal value of v := u, which we denote by vm is obtained if (u0 , u) belongs to both hyperboloids, i.e. if 2 = 2τ − 2 m23 + vm u0 = 2τ − m22 + vm holds. With this value for u0 we obtain for vm a second order equation 2 2 2 )(m2 + v 2 ), 4(τ − τ )2 = (m23 + vm ) + (m22 + vm ) − 2 (m23 + vm m 2 2 2 2 (4(τ − τ )2 − 2vm − (m22 + m23 ))2 = 4(m23 + vm )(m22 + vm ),
2 vm 16(τ − τ )2 − 16(τ − τ )4 + 8(τ − τ )2 (m22 + m23 ) − (m23 − m22 )2 = 0,
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which implies 2 vm =
16(τ − τ )4 + (m23 − m22 )2 − 8(τ − τ )2 (m22 + m23 ) . 16(τ − τ )2
(3.6)
2
2) This formula is valid for (τ − τ )2 < (m3 −m as should be. One concludes this 4 2 because vm has to be positive. The corresponding value of u0 is restricted to 2 2 (3.7) 2τ − m3 + v < u0 < 2τ − m22 + v 2 ,
where we have set v = u. The exact value of u0 will be determined together with the minimal value µ0 of µ by the requirement that the hyperboloid H(µ0 , u) has to touch both H − (m2 , 2t ) and H − (m3 , 2t). With π = p one obtains two equations: (3.8a) u0 + µ2 + (π − v)2 − 2τ + m22 + π 2 > 0, u0 − µ2 + (π − v)2 − 2τ + m23 + π 2 < 0. (3.8b) One finds the minimal value of (3.8a) by differentiating this equation with respect to π. π−v 2 µ + (π − v)2
+
(π − v)2 (m22 + π 2 ) =
π = 0, 2 m2 + π 2
π 2 (µ2 + (π − v)2 ).
2 This implies π = mvm . The sign of the square root has to be chosen in such a 2 ±µ manner that π ≤ v holds. The minimal value of (3.8a) is 2 )2 . (3.9a) u0 − 2τ + µ2 + v 2 ( m2µ+µ )2 + m22 + v 2 ( mm 2 +µ
The maximal value of (3.8b) is obtained by π = u0 − 2τ +
vm3 m3 −µ
which leads to
3 µ2 + v 2 ( m3µ−µ )2 + m23 + v 2 ( mm )2 . 3 −µ
(3.9b)
The minimal value of µ is obtained by setting (3.9a) resp. (3.9b) equal to zero. This implies (u0 − 2τ )2
=
= =
2 (µ2 + v 2 ( m2µ+µ )2 ) + (m22 + v 2 ( mm )2 ) + 2 +µ 2 2 2 (µ2 + v 2 ( mµ2 +µ )2 )(m22 + v 2 ( mm )2 ), 2 +µ 2 2 (m22 + µ2 )(1 + (m2v+µ)2 ) + 2 µ2 m22 (1 + (m2v+µ)2 )2 ,
(m2 + µ)2 (1 +
v2 (m2 +µ)2 )
= (m2 + µ)2 + v 2 ,
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Ann. Henri Poincar´e
(u0 − 2τ )2 − v 2 − m2 .
(3.10a)
(u0 − 2τ )2 − v 2 .
(3.10b)
In the same manner one finds µb = m3 −
Now we can determine u0 from the equation µa = µb . This leads to (m2 + m3 )2 = ((u0 − 2τ )2 − v 2 ) + ((u0 − 2τ )2 − v 2 )+ 2 ((u0 − 2τ )2 − v 2 )((u0 − 2τ )2 − v 2 ), {(m2 + m3 )2 − ((u0 − 2τ )2 − v 2 ) − ((u0 − 2τ )2 − v 2 )}2 = 4((u0 − 2τ )2 − v 2 )((u0 − 2τ )2 − v 2 ), u20 [−4(m2 + m3 )2 + 16(τ − τ )2 ] + u0 [8(m2 + m3 )2 (τ + τ ) − 32(τ + τ )(τ − τ )2 ] + (m2 + m3 )4 + 4(m2 + m3 )2 v 2 − 8(m2 + m3 )2 (τ 2 + τ 2 ) + 16(τ 2 − τ 2 )2 = 0, 0 = [u0 − (τ + τ )]2 − (τ + τ )2 − {(m2 + m3 )4 + 4(m2 + m3 )2 v 2 − 8(m2 +
1 4[(m2 +m3 )2 −4(τ −τ )2 ] m3 )2 (τ 2 + τ 2 ) + 16(τ 2
− τ 2 )2 },
and hence [u0 − (τ + τ )]2 =
(m2 +m3 )2 4[(m2 +m3 )2 −4(τ −τ )2 ] [(m2
For v = 0 we have to find u0 = (τ + τ ) − take the negative square root, i.e. u0 = (τ + τ ) −
(m2 +m3 ) 2
+ m3 )2 − 4(τ − τ )2 + 4v 2 ].
(m2 +m3 ) . 2
1+
(3.11)
This implies that one has to
4v 2 [(m2 +m3 )2 −4(τ −τ )2 ] .
(3.12)
Inserting this expression into (3.10a) we obtain µ0
= =
(τ − τ ) −
(m2 +m3 ) 2
1+
2 4v 2 [(m2 +m3 )2 −4(τ −τ )2 ]
3) )) 1 + { (m2 +m − (τ − τ 2
− v 2 − m2 ,
4v 2 2 [(m2 +m3 )2 −4(τ −τ )2 ] }
− m2 ,
which implies µ0 =
m3 −m2 2
− (τ − τ ) 1 +
4v 2 [(m2 +m3 )2 −4(τ −τ )2 ] .
(3.13)
For v larger than vm we take the surface u0 (v) = u0 (vm ), v > vm and µ0 (v) = 0.
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4 The case of a big gap In this section we want to treat the gap problem in a situation where techniques of enlarging the domain of holomorphy at real points apply. The method which shall be used is the double cone theorem discovered by Vladimirov [Vl60] and Borchers [Bch61]. We need the local version of this theorem explained in [Bch96]. Let the domain D ⊂ Cd be invariant under the map z → z ∗ . For x ∈ Rd we define the local cone Cx as follows: Cx = {y ∈ Rd ; ∃ ρ > 0
with x + iρy ∈ D}.
By the symmetry the set Cx is either Rd or it contains an even number of components. We are interested in the situation of two components. In the case where x belongs to D ∩ Rd one has Cx = Rd . Since the domain of holomorphy is open, the cone Cx is open too. If D contains real points, then they are called the coincidence ˜ = H(D)∩Rd . Asdomain K. Let H(D) be the envelope of holomorphy of D and K ˜ has a tangent hyper-plane at a point x ∈ ∂ K, ˜ sume that the coincidence domain K then the double cone theorem requires that this hyper-plane is not allowed to enter the local cone Cx . In other terms if for the original coincidence domain K at a point x ∈ ∂K the tangent hyper-plane enters the local cone Cx , then D can be enlarged through this point. In our situation the local cones Cx contain at least the open light-cones V + ∪ V − . With help of this double cone theorem we want to show the following result:
4.1
Theorem
Let π be a translational covariant factor representation of the theory of local observables fulfilling the spectrum condition for the translations. Assume that the representation of T (a) is the unique minimal one. Let the spectrum of the translations start at m1 , then there is no gap with the width larger than 2m1 . Before proving the theorem we draw the following conclusion:
4.2
Corollary
Use the assumptions of the last theorem and assume the spectrum starts at m1 = 0, then there is no gap in the spectrum. This is a simple consequence of the theorem because one has in this case 2m1 = 0. Proof of the Theorem. If m2 coincides with m1 , then the result follows from the theorem mentioned in the introduction. Therefore, it is no restriction if m2 > m1 is assumed in the proof. The result will be shown by contradiction, i.e. the assumption that there is a gap of width larger than 2m1 is not in accordance with the value of the lowest mass m1 . Since the support of the spectrum is Lorentz invariant
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[BB85],[Bch85], it is sufficient to show that one can enlarge the coincidence domain at some parts of the hyperboloid H + (m1 , 0). This will be done by computing the local cones at some points of H + (m1 , 0). Since the complement of Vm+1 ∪ (2t + Vm−1 ) and also the gap between H + (m2 , 0) and H + (m3 , 0) does not lead to an instability of H + (m1 , 0) the interesting contribution for the local cone must come from the crescent moon domain. Our domain is invariant under rotations, therefore, we can restrict our attention to the (0, 1)-plane. Since the domain of analyticity we are starting with is open it is sufficient to compute Cp ∩ R2 , where the plane in question is the (0, 1)-plane, because Cp is open in Rd . For the calculations we use again m1 = 1. We start with the local cone obtained from the complement of {V1+ ∪ (2t + − V1 )}, and afterwards we look at the enlargement of this cone influenced by the crescent moon domain. The local cone contains always the interior of the light cone, therefore, one is interested in light-like and space-like directions. In order to find them we recall first that the hyperboloids H(m, u), filling the complement of the holomorphy domain, are real. Therefore, one must have (p − u, q) = 0 if (p − u + iq) belongs to such hyperboloid. In addition one must have (p − u)2 − q 2 = m2 . So we have to look at points (p − u + iq) where one of the two conditions is violated for every of the hyperboloid filling the complement of the domain of holomorphy. The hyperboloids of interest are those which are centered in the hyper-surface described in the last section. Let p be the point of interest then there are two possibilities of finding directions of the local cone Cp . 1. The set of directions of p − u form a cone C. Then (p − u + iq) belongs to the domain of holomorphy if q is not perpendicular to any of these directions, ˆ i.e. if q or −q belongs to the interior of the dual cone C. 2. If (p − u) belongs to C and q ⊥ (p − u) then q belongs to the local cone Cp if the equations
(p − u)2 − q 2 < µ0 (u)2 ,
(p − u, q) = 0,
(4.1)
can be fulfilled. Altogether the local cone at p will consist of three parts. Two of them are consequences of the coincidence domain {complement of Vm+1 ∪ (2t + Vm−1 )} and the third part is the contribution of the crescent moon domain. (See Fig.1) We choose a point p ∈ H + (1, 0) with p1 > 0 and p21 > 1 − τ12 . This condition implies that the set p − V + does not meet the intersection of V1+ with 2t − V1+ . With this choice p − V + meets the surface for the centers of the hyperboloids in the interval [λ0 τ, τ ] where λ0 is obtained from the relation ( 1 + p21 , p1 ) − α(1, 1) = (τ, λ0 τ ),
Vol. 3, 2002
On the Gap in the Spectrum of the Translations
13
(2) (1)
tangent
(3)
Figure 1: Structure of the local cone (1) y ⊥ (p − u) ∀ u (2) y ⊥ (p − u) and (p − u)2 − y 2 < µ20 (u) (3) Contribution of the crescent moon domain. which implies 1 + p21 − α = τ, (1 − λ0 )τ
=
p1 − α = λ0 τ. 1 + p21 − p1 ,
and consequently
λ0 τ = τ + p1 − 1 + p21 . (4.2) If we take the vector u = √ τ 2 ( 1 + p21 , p1 ) in the u-hyper-surface, then we 1+p1
obtain (p − u)2 = (1 − √ τ 2 ). In this situation one has λ1 = √ p1 2 and √ 1 2 = 1+p1 1+p1 1+p1 2 1 − λ1 . This implies (p − u(λ1 ))2 = µ0 (λ1 ). (4.3) If λ0 ≤ λ < λ1 , then the second case can be fulfilled because one finds with p1 = √ λ1 2 the relation 1−λ1
(p − u(λ)) = eτ (λ1 − λ) + (p − u(λ1 )),
e2 = −1.
This implies with Eq. (3.2), (4.3) and the above value of p1 (p − u(λ))2
= (p − u(λ1 ))2 − τ 2 (λ1 − λ)2 − 2τ (λ1 − λ)(p1 − τ λ1 ) = µ0 (λ)2 − (1 + τ 2 (1 − λ2 ) − 2τ 1 − λ2 ))(1 + τ 2 (1 − λ21 ) − 2τ 1 − λ21 ) − τ 2 (λ21 + λ2 − 2λλ1 ) − 2τ (λ1 − λ)( √ λ1 2 − τ λ1 ) 1−λ1
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H. J. Borchers
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1 (λ1 −λ) = µ0 (λ)2 − 2τ ( 1 − λ21 − 1 − λ2 + λ√ ) 1−λ21 = µ0 (λ)2 − √ 2τ 2 (1 − λ1 λ − (1 − λ1 λ)2 − (λ1 − λ)2 ) 1−λ1 λ1 −λ 2 ) ) = µ0 (λ)2 − √ 2τ 2 (1 − λ1 λ − (1 − λ1 λ) 1 − ( 1−λ 1λ 1−λ1
2
< µ0 (λ) −
2 1 (λ1 −λ) √ 2τ 2 2 1−λ λ 1−λ 1
< µ0 (λ)2 .
Next we turn to the crescent moon domain. Since can be chosen arbitrarily small we take first = 0 in order to see what can be expected. In this situation the u-hyper-plane is met by every time-like direction starting from p. Looking for conditions implying that the local cone Cp contains the tangent plane at p one takes p − u0 = (1 + α)p. Since p2 = 1 and q 2 = −β 2 equation (4.1) reads (1 + α)2 + β 2 < µ20 . The quantity α is a function of p1 . In order that the tangent direction belongs to the local cone one must have µ0 > 1 + min{α(p1 )}. Later we 2 will show that α → 0 for p1 → ∞. Since µ0 = m3 −m , one obtains an enlargement 2 of the local cone for µ0 > 1. With µ0 − 1 = δ and the choice α(p1 ) < 2δ one can vary β in the interval (0, δ + 34 δ 2 ). Since (p − u) depends continuously on u one can still vary u in a small neighbourhood around u0 without violating Eq. (4.1), i.e. the local cone at p contains the tangent direction at p as an interior direction. It remains to show α(p1 ) → 0 for p1 → ∞ and to look at the variation for small 3 . With m2 +m = M , the point u0 must lie on the hyperboloid H − (M, 2t) which 2 leads to the equations −αp = 2t − ( M 2 + r2 , r), −αp1 = −r, −α 1 + p21 = 2 − M 2 + r2 ,
and hence 2=
M 2 + α2 p21 − α 1 + p21 .
This implies 4 = M 2 + α2 (1 + 2p21 ) − 4(M 2 + α2 p21 )α2 (1 + p21 ) = −α4 + α2 {2M 2 + α2 = (M 2 + 4(1 + 2p21 )) ±
2 (M 2 + α2 p21 )α2 (1 + p21 ), {(M 2 − 4) + α2 (1 + 2p21 )}2 , 8(1 + 2p21 )} = (M 2 − 4)2 , (M 2 + 4(1 + 2p21 ))2 − (M 2 − 4)2 .
Since α is bounded by 1 for all p1 we must choose the minus sign and get 1 (M 2 −4)2 2 −4)2 ≈ 2 M 2 +4(1+2p2 ) . α2 = (M 2 + 4(1 + 2p21 )) 1 − 1 − (M 2(M +4(1+2p2 ))2 1
This tends to zero for p1 → ∞ as claimed before.
1
(4.4)
Vol. 3, 2002
On the Gap in the Spectrum of the Translations
15
Finally we must look at the approximation in . For the future calculations 1 1 one sets σ = +( 1+ −1), δ = −( 1+ −1) which implies τ +τ = 2+δ, τ −τ = σ. The crescent moon domain becomes smaller for δ = 0, which simplifies the coming calculations. With these notations we obtain with (3.12) and (3.13) the following equations for α and µ0 (v). −α 1 + p21
= −
2 2 − M 2 + v 2 MM −αp1 = −v, 2 −σ 2 , 2 α 1 + p21 = 2 − M 2 + α2 p21 MM 2 −σ 2 ,
and µ0 =
m3 −m2 2
−σ
1+
α2 p21 M 2 −σ2 .
(4.5)
(4.6)
In order that the tangent at p belongs to the local cone one has to choose small enough such that m3 − m2 α2 p2 − 1 > σ 1 + M 2 −σ1 2 2 holds. To show that this is possible we have to calculate p1 as a function of α. 2 With Eq. (4.5) and with Γ = MM 2 −σ 2 = 1 − γ one obtains as equation for α 4 = M 2 + α2 (1 + p21 (1 + Γ)) − 2 (M 2 + α2 p21 Γ)α2 (1 + p21 ), 4(M 2 + α2 p21 Γ)α2 (1 + p21 ) = {(M 2 − 4) + α2 (1 + p21 (1 + Γ))}2 , 4{M 2 α2 (1 + p21 ) + α4 (p21 Γ(1 + p21 ))} = (M 2 − 4)2 + 2(M 2 − 4)α2 (1 + p21 (1 + Γ))+ α4 (1 + 2p21 (1 + Γ) + p41 (1 + Γ)2 ), p41 {−α4 γ 2 } + p21 {2α4 γ + 8α2 (1 + Γ) − 2M 2 α2 γ} + {−α4 + α2 (2M 2 + 8) − (M 2 − 4)2 } = 0, 16α2 +γ(2α4 −α2 (2M 2 −8)) (+α4 −α2 (2M 2 +8)+(M 2 −4)2 )α4 γ 2 2 1 − . (4.7) 1 − p1 = α4 γ 2 {16α2 +γ(2α4 −α2 (2M 2 −8))}2 One must choose p1 at least as big as stated in the formula. This is not too different from the value of the case = 0. With (p − u) = (1 + α)p one chooses α such that the inequality (4.1) is fulfilled. Then one can find an appropriate value for p1 using Eq. (4.7) and finally a finite value for with help of (4.6). Hence the local cone Cp contains the tangent at p, which leads to the desired contradiction. Acknowledgment. I like to thank Jacques Bros for discussions.
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References [Ar64]
H. Araki, On the algebra of all local observables, Progr. Theor. Phys. 32, 844–854 (1964).
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¨ H. J. Borchers, Uber die Vollst¨andigkeit lorentzinvarianter Felder in einer zeitartigen R¨ohre, Nuovo Cimento 19, 787–796 (1961).
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H. J. Borchers, Local Rings and the Connection of Spin with Statistics, Commun. Math. Phys. 1, 281–307 (1965).
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H. J. Borchers, Locality and covariance of the spectrum, Fizika 17, 289– 304 (1985).
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H. J. Borchers, A Remark on Antiparticles, in Lecture Notes in Physics Vol 257, 268–280 (1986).
[Bch96]
H. J. Borchers, Translation Group and Particle Representations in Quantum Field Theory, Lecture Notes in Physics m40 Springer, Heidelberg (1996).
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H. J. Borchers and D. Buchholz, The Energy-Momentum Spectrum in Local Field Theories with Broken Lorentz-Symmetry, Commun. Math. Phys. 97, 169–185 (1985).
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R. Borek, Kovariante Darstellungen Masseloser Fermi Felder, Dissertation, G¨ ottingen (1982).
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H. J. Brehmermann, R. Oehme and J. G. Taylor, Proof of dispersion relation in quantized field theories, Phys. Rev. 109, 2178–2190 (1958).
[BMS]
J. Bros, A. Messiah and R. Stora, A problem of analytic completion related to the Jost-Lehmann-Dyson formula, J. Math. Phys. 2, 639–651 (1961).
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D. Buchholz and S. Doplicher, Exotic Infrared Representations of interacting Systems, Ann. Inst. Henri Poincar´e 40, 157–184 (1994).
[DHR69a] S. Doplicher, R. Haag and J. E. Roberts, Fields, observables and gauge transformations I, Commun. Math. Phys. 13, 1 (1969). [DHR69b] S. Doplicher, R. Haag and J. E. Roberts, Fields, observables and gauge transformations II, Commun. Math. Phys. 15, 173 (1969). [DS82]
S. Doplicher and M. Spera, Representations obeying the spectrum condition, Commun. Math. Phys. 84, 505–513 (1982).
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F. J. Dyson, Integral representation of causal commutators, Phys. Rev. 110, 1460–1464 (1958).
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17
[Ha92]
R. Haag, Local Quantum Physics, Springer Verlag, Berlin-HeidelbergNew York (1992).
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V. S. Vladimirov, The construction of envelopes of holomorphy for domains of special type, Doklady Akad. Nauk SSSR 134, 251 (1960).
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A. S. Wightman, La th´eorie quantique locale et la th´eorie quantique des champs, Ann. Inst. Henri Poincar´e A, I, 403–420 (1964).
Hans J¨ urgen Borchers Institut f¨ ur Theoretische Physik Universit¨ at G¨ ottingen Bunsenstrasse 9 D-37073 G¨ ottingen Germany Communicated by Klaus Fredenhagen submitted 19/04/01, accepted 02/10/01
Ann. Henri Poincar´e 3 (2002) 19 – 27 c Birkh¨ auser Verlag, Basel, 2002 1424-0637/02/010019-9 $ 1.50+0.20/0
Annales Henri Poincar´ e
Singular Vectors of the N = 1 Superconformal Algebra K. Iohara and Y. Koga Abstract. We give two singular vector formulae of the N = 1 superconformal algebra.
1 Introduction Singular vectors of Verma modules over the Virasoro algebra play important roles both in mathematics and physics. In fact, many studies on their explicit forms have been done e.g., [Ke] etc. In particular, two formulae were obtained by B. L. Feigin and D. B. Fuchs [FeFu1], and these formulae played an essential role when they determined the structure of the fusion rings of the minimal models [FeFu2]. In this paper, we will derive similar formulae for the N = 1 superconformal algebra, i.e., the Neveu-Schwarz algebra. In our case, one technical difficulty arises, viz. factorizations in a non-commutative ring are required (see Remark 3.2 and Theorem 3.5). We note that one of our formulae, i.e. Theorem, 3.1 has been partially obtained in [BS] in a special case. Namely, they obtained their formula explicitly in the case when one of a and b is one and provided an algorithm to compute the general case via fusion procedure. Our method is based on the embedding diagram given in Proposition 2.3 and the result is given for the general cases. This paper is organized as follows: In §2, we will recall the definition of the N = 1 superconformal algebra and its Verma modules. Further, we will collect some properties of them. In §3, we will derive two singular vector formulae.
2 N = 1 Superconformal Algebra Here we recall the definition of the N = 1 superconformal algebra. Further we introduce Verma modules and state their properties.
2.1
Definitions
Let V ir 12 be the N = 1 superconformal algebra, the Neveu-Schwarz algebra, i.e., V ir 12 is the Lie superalgebra V ir 12 =
i∈Z
CLi ⊕
j∈Z+ 12
CGj ⊕ CC,
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Ann. Henri Poincar´e
and these generators satisfy the following commutation relations: deg Li = deg C = 0, deg Gi = 1, 1 [Li , Lj ] = (i − j)Li+j + (i3 − i)δi+j,0 C, 12 1 [Gi , Lj ] = (i − j)Gi+j , 2 1 1 [Gi , Gj ] = 2Li+j + (i2 − )δi+j,0 C, 3 4 [C, V ir ] = {0}. Note that V ir 12 admits the following triangular decomposition: − 0 V ir 12 = V ir+ 1 ⊕ V ir 1 ⊕ V ir 1 , 2
2
where
V ir± 1 := 2
For later use, we set
CLi ⊕
±i>0
V ir01 := CL0 ⊕ CC. 2
±i>0
+ 0 V ir≥ 1 := V ir 1 ⊕ V ir 1 . 2
2.2
CGi ,
2
2
2
Verma Modules
Next, we introduce Verma modules over V ir 12 and collect some basic properties of them. For (z, h) ∈ C2 , let Cz,h = C1z,h be the 1-dimensional representation of V ir≥ 1 defined by 2
x.1z,h = 0
if x ∈ V ir+ 1 , 2
C.1z,h = z1z,h , L0 .1z,h = h1z,h . The Verma module M (z, h) over V ir 12 with highest weight (z, h) is defined as follows: M (z, h) := U (V ir 12 ) ⊗U(V ir≥ ) Cz,h . 1 2
The values z and h are called central charge and conformal weight, respectively. For simplicity, we set |z, h := 1 ⊗ 1z,h . Since the choice of the parity of a highest weight vector is not essential in the sequel, we do not specify it. Note that M (z, h) has the following weight space decomposition: M (z, h) = M (z, h)n , M (z, h)n := {v ∈ M (z, h)|L0 .v = (h + n)v}. n∈ 12 Z≥0
A non-zero vector in {M (z, h)n }
Vir+ 1 2
is called a singular vector of level n.
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21
Here we recall two basic facts on singular vectors. The first fact is the existence of singular vectors. As a consequence of the determinant formula for M (z, h) [KW], we see the following: If there exist t ∈ C \ {0} and a, b ∈ Z>0 satisfying a ≡ b (mod 2) and 15 − 3(t + t−1 ), 2 1 1 1 h = ha,b (t) := (a2 − 1)t − (ab − 1) + (b2 − 1)t−1 , 8 4 8 z = z(t) :=
then M (z, h) has a singular vector of level 12 ab. The second fact is the uniqueness of singular vectors. Indeed, we have Proposition 2.1 ([IK1]) For any highest weight (z, h) ∈ C2 , the following hold: dim{M (z, h)n}
2.3
Vir + 1 2
≤1
(∀n ∈
1 Z≥0 ). 2
Embedding Diagrams of Verma Modules
To prove singular vector formulae, we will use embedding diagrams of Verma modules [IK1]: Here we recall them. It should be noted that non-trivial homomorphisms between Verma modules over a Lie superalgebras are not injective in general. But for the N = 1 superconformal algebra, the injectivity holds. Proposition 2.2 ([IK1]) Suppose that M (z, h) has a singular vector of level n. Then the following map is an embedding: M (z, h + n) −→ M (z, h). When z = z(t) for t ∈ Q \ {0}, we have the following embedding diagrams of Verma modules. Let p and q be integers such that z = z(± pq ), p, q ≥ 2, p − q ∈ 2Z and ( p−q 2 , q) = 1. For the integers p and q, we set ± ± ± ± ± := Kp,q (1) ∪ Kp,q (2) ∪ Kp,q (3) ∪ Kp,q (4), Kp,q ± (i) are give by where Kp,q ± (1) := {(r, s) ∈ Z2 |0 < r < q, 0 < ±s < p, rp + sq ≤ pq}, Kp,q ± (2) := {(q, s)|0 < ±s < p}, Kp,q ± (3) := {(r, −p)|0 < r < q}, Kp,q ± Kp,q (4) := {(0, ±p), (q, ±p)}.
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K. Iohara and Y. Koga
Ann. Henri Poincar´e
± For (r, s) ∈ Kp,q , we also set h(i−1)q+r,−s ± p i ≡ 1 (mod 2) q . hi := p hiq+r,s ± i ≡ 0 (mod 2) q
Then we have the following proposition: Proposition 2.3 Let us take integers p, q, r and s as above. Then there exist the following embedding diagrams of Verma modules: + For (r, s) ∈ Kp,q (1),
/ M (z, h3 ) / M (z, h2 ) / M (z, h1 ) · · · M (z, h4 ) UUU* 9 KKK KKK MMM q8 s9 q s ss K K q MM s s q K K s s M (z, h0 ). qqMMMM ssKKKK ssKKKK qqq sss sss & % % iii4 · · · M (z, h−4 ) / M (z, h−3 ) / M (z, h−2 ) / M (z, h−1 ) + (2), For (r, s) ∈ Kp,q
· · · M (z, h4 )
/ M (z, h3 )
/ M (z, h2 )
/ M (z, h3 )
/ M (z, h−2 )
/ M (z, h1 )
/ M (z, h0 ).
+ (3), For (r, s) ∈ Kp,q
· · · M (z, h−4 )
/ M (z, h1 )
/ M (z, h0 ).
+ For (r, s) = (p, q) ∈ Kp,q (4)
· · · M (z, h8 )
/ M (z, h6 )
/ M (z, h4 )
/ M (z, h2 )
/ M (z, h4 )
/ M (z, h−2 )
/ M (z, h0 ).
+ (4) For (r, s) = (0, q) ∈ Kp,q
· · · M (z, h8 )
/ M (z, h−6 )
/ M (z, h0 ).
− (1), For (r, s) ∈ Kp,q
M (z, h3 ) o · · · M (z, h4 ) o M (z, h2 ) o M (z, h1 ) jU fMMM eKKK eKKK UU q s q s s MM KKs KKsKs s q s q K M q s s M (z, h0 ). K K M q s s KK KK MM tiii xqqq ysss ysss · · · M (z, h−4 ) o M (z, h−3 ) o M (z, h−2 ) o M (z, h−1 ) − For (r, s) ∈ Kp,q (2),
· · · M (z, h4 ) o
M (z, h3 ) o
M (z, h2 ) o
M (z, h1 ) o
M (z, h0 ).
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23
− For (r, s) ∈ Kp,q (3),
· · · M (z, h−4 ) o
M (z, h3 ) o
M (z, h−2 ) o
M (z, h1 ) o
M (z, h0 ).
− For (r, s) = (p, −q) ∈ Kp,q (4),
· · · M (z, h8 ) o
M (z, h6 ) o
M (z, h4 ) o
M (z, h2 ) o
M (z, h4 ) o
M (z, h−2 ) o
M (z, h0 ).
− For (r, s) = (0, −q) ∈ Kp,q (4),
· · · M (z, h8 ) o
M (z, h−6 ) o
M (z, h0 ).
3 Singular Vector Formulae In this section, we will present our main results, singular vector formulae.
3.1
Notation
Suppose that z = z(t) and h = ha,b (t) for some a, b ∈ Z>0 such that a ≡ b (mod 2). As stated before, there exists a unique element (up to a scalar multiple) −1 ] Sa,b (t) ∈ U (V ir− 1 ) ⊗C C[t, t 2
such that Sa,b (t)|z, h is a non-zero singular vector of level 12 ab. From now on, we will give two formulae on Sa,b (t).
3.2
Loop Modules
The first formula is concerned with the representations over V ir 12 called loop modules. Although, this formula has already been considered in [IK2], we have stated there without proof. Here, we will show a way to prove this formula. Let θ be the Grassmann variable i.e. θ2 = 0. For λ, µ ∈ C and σ ∈ {0, 1}, we set σ Fλ,µ = CFi ⊕ CFi θ, i∈Z+σ 12
i∈Z+(1−σ) 12
and regard it as a V ir -module via C.Fi θγ = 0, 1 Ln .Fi θγ = {−i + µ + (n − 1)λ − nγ}Fi+n θγ , 2 Gm .Fi θγ = δγ,1 Fi+m + δγ,0 {−i + µ + (2m − 1)λ}Fi+m θ, where γ = 0, 1, n ∈ Z and m ∈ Z + 12 . σ Here we describe the action of Sa,b (t) on Fλ,µ explicitly. To do it, we set 1 if i ≡ j (mod 2), (2) δi,j = 0 if i ≡ j (mod 2).
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Ann. Henri Poincar´e
σ Then Sa,b (t) acts on F0 θγ ∈ Fλ,µ as follows:
Theorem 3.1 ([IK2]) Suppose that a, b ∈ Z>0 such that a ≡ b (mod 2). Then we have (2) γ Sa,b (t).F0 θγ = Pa,b (λ, µ, t)F− 12 ab θδab,1−γ , γ where Pa,b (λ, µ, t)2 is, up to a scalar depending on the normalization, given by
Qk,l a,b (λ, µ, t)
0≤k≤a−1 0≤l≤b−1 k−l≡γ (mod 2)
and Qk,l a,b (λ, µ, t)
1 1 1 1 1 = (µ − 2λ) − (kt 2 − lt− 2 ){(a − k)t 2 − (b − l)t− 2 } 2
1 1 1 1 1 × (µ − 2λ) − {(k + 1)t 2 − (l + 1)t− 2 }{(a − k − 1)t 2 − (b − l − 1)t− 2 } 2 2 1 1 1 + (a − 1 − 2k)t 2 − (b − 1 − 2l)t− 2 λ. 2 (1)
σ Proof. Since every weight space of the loop module Fλ,µ is of dimension 1, there γ −1 exists Φa,b (λ, µ, t) ∈ C[λ, µ, t, t ] which satisfies (2)
Sa,b (t).F0 θγ = Φγa,b (λ, µ, t)F− 12 ab θδab,1−γ . γ From now on, we verify Φγa,b (λ, µ, t) = Pa,b (λ, µ, t). To do it, we first compute the γ degree of Φa,b (t) as a polynomial in t. Following [AsFu], one obtains
degΦγa,b (t) ≤
ab + 1 , 2
where x denotes the maximal integer that does not exceed x. Let us prove γ Φγa,b (λ, µ, t) = Pa,b (λ, µ, t) by induction on the level of singular vectors. It follows from Proposition 2.3 that there exists t0 ∈ Q \ {0} such that the singular vector of level 12 ab factorize as Sa1 ,b1 (t0 )Sa2 ,b2 (t0 ) · · · Sak ,bk (t0 )|z(t0 ), ha,b (t0 ) ,
(2)
k where k > 1 and ai , bi ∈ Z>0 satisfy ai ≡ bi (mod 2) and 12 ab = i=1 21 ai bi . By the uniqueness of singular vectors and induction hypothesis, we have Φγa,b (λ, µ, t0 ) ∝
k i=1
Paγi ,bi (λ, µ, t0 ).
(3)
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Singular Vectors of the N = 1 Superconformal Algebra
25
γ Here, one can show that the right hand side of (3) coincides with Pa,b (λ, µ, t) by direct computation. On the other hand, by Proposition 2.3, we can check that the number ({t0 } such that the singular vector of level 12 ab factorize as (2) is greater than the γ γ degree of Pa,b (λ, µ, t). This implies that Φγa,b (λ, µ, t) = Pa,b (λ, µ, t) holds for any t ∈ C \ {0}. We have completed the proof.
3.3
Projections of Singular Vectors
Next we describe an explicit form of the image of Sa,b (t) under the projection − − − − − π : U (Vir− 1 ) −→ U (Vir 1 )/U (Vir 1 )[Vir 1 , [Vir 1 , Vir 1 ]]. 2
2
2
2
2
2
(4)
Remark 3.2 In [FeFu1], a similar projection was computed for the Virasoro algebra. One of the differences between their case and our case is whether the quotient space is a commutative ring or not. Indeed, the quotient space in (4) is noncommutative, and, to derive the formula, we have to consider factorizations in this non-commutative space. We introduce Xp,q , Yr ∈ U (Vir− 1 ) for p, q, r ∈ C as follows: 2
Xp,q :=
L2−1
− pG− 12 G− 32 − qG− 32 G− 12 ,
Yr := L−1 G− 12 − rG− 32 . Then we have the following lemma: Lemma 3.3
1. π(Xp,r )π(Xq,s ) = π(Xp,s )π(Xq,r ) = π(Xq,r )π(Xp,s ).
2. π(Xp,q )π(Yr ) = π(Xp,r )π(Yq ) = π(Yr )π(Xp,q ). To describe the image of π(Sa,b (t)), we introduce some notation. Let us first introduce an index set Pσa,b for σ ∈ {0, 1} and a, b ∈ Z>0 such that a − b ≡ 0 (mod 2). Set (i) (r, s) ∈ (Z>0 × Z) ∪ ({0} × Z≥σ ), (ii) 0 ≤ r ≤ a − 1 ∧ −(b − 1) ≤ s ≤ b − 1, σ , Pa,b := (r, s) (iii) a − 1 − r ≡ 0 (mod 2) ∧ b − 1 − s ≡ 0 (mod 2), (iv) a + b − (r + s) ≡ 2(1 − σ) (mod 4) and nσa,b := (Pσa,b . We remark that n0a,b − n1a,b =
0 1
if a ≡ b ≡ 0 mod 2 . if a ≡ b ≡ 1 mod 2
Moreover, let us take a sequence {(r1σ , sσ1 ), (r2σ , sσ2 ), · · · , (rnσ , sσn )}
(n := nσa,b )
of elements Pσa,b , which satisfies the following conditions:
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Ann. Henri Poincar´e
1. (riσ , sσi ) = (rjσ , sσj ) for any i, j (1 ≤ i < j ≤ n), 2. if (0, 0) ∈ Pσa,b , i.e., σ = 0, a ≡ b ≡ 1 (mod 2) and a + b ≡ 2 (mod 4), then (rnσ , sσn ) = (0, 0). Next, we define a subgroup Ga,b of the symmetric group Sn , where n := nσa,b , by Ga,b :=
{τ ∈ Sn |τ (n0a,b ) = n0a,b } Sn
if (0, 0) ∈ Pσa,b . otherwise
For i ∈ Z such that 1 ≤ i ≤ n0a,b and τ ∈ Ga,b , we define an element mτi ∈ P0a,b × (P1a,b ∪ {vac}) as follows: mτi := Further, we set
[(rτ0 −1 (i) , s0τ −1 (i) ), (ri1 , s1i )]
if 1 ≤ i ≤ n1a,b
[(rτ0 −1 (n0 ) , s0τ −1 (n0 ) ), vac] a,b a,b
if n0a,b = n1a,b ∧ i = n1a,b
.
Ωτa,b := {mτi |1 ≤ i ≤ n0a,b }.
Finally, for m ∈ Ωτa,b , we set
Sm a,b
and
1 1 1 1 X 14 (r t 2 +s t− 2 )2 , 14 (r t 2 +s t− 2 )2 := Y 1 12 − 12 2 (rt +st ) 4 1 G− 2 mτ
if m = [(r, s), (r , s )] if m = [(r, s), vac] for (r, s) = (0, 0) , if m = [(0, 0), vac] mτ
mτ
Sa,b;τ := Sa,b1 Sa,b2 · · · Sa,bn , where n := n0a,b . By Lemma 3.3, we have Lemma 3.4 Sa,b;τ does not depend on the choice of τ ∈ Ga,b . Consequently, the image of Sa,b (t) under the projection π in (4) is described as follows: Theorem 3.5 Suppose that a, b ∈ Z>0 such that a ≡ b (mod 2). Then we have π(Sa,b (t)) ∝ Sa,b;τ up to scalar multiplication depending on the normalization. Proof. The proof of this formula is essentially the same as the proof of Theorem 3.1. Hence we omit it. Acknowledgment. We would like to thank the referee for pointing out the reference [BS], where a similar result was obtained.
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References [AsFu]
A. Astashkevich and D. B. Fuchs, Asymptotics for singular vectors in Verma modules over the Virasoro algebra, Pacific J. Math. 177, no. 2, 201–209 (1997).
[BS]
L. Benoit and Y. Saint-Aubin, Fusion and the Neveu-Schwarz Singular Vectors, Int. Jour. Mod. Phys. A 9, 547–566 (1994).
[FeFu1]
B. L. Feigin and D. B. Fuchs, Verma modules over the Virasoro algebra, Lecture Notes in Math., 1060, 230–245 (1982).
[FeFu2]
B. L. Feigin and D. B. Fuchs, Cohomology of some nilpotent subalgebras of the Virasoro and Kac-Moody Lie algebras, J. Geom. Phys. 5, 209–235 (1988).
[IK1]
K. Iohara and Y. Koga, Representation Theory of Neveu-Schwarz and Ramond Algebras I: Verma Modules, preprint.
[IK2]
K. Iohara and Y. Koga, Fusion algebras for N=1 superconformal field theories through coinvariants II: N = 1 super Virasoro symmetry, J. Lie Theory 11, 305–337 (2001).
[Ke]
A. Kent, Singular vectors of the Virasoro algebra, Phys. Lett. B 273, 56–62 (1991).
[KW]
V. G. Kac and M. Wakimoto, Unitarizable highest weight representation of the Virasoro, Neveu-Schwarz and Ramond algebras, Lect. Notes in Phys., 261, 345–372, (1986).
Kenji Iohara Department of Mathematics Faculty of Science Kobe University Kobe 657-8501 Japan email:
[email protected] Communicated by Tetsuji Miwa submitted 26/01/01, accepted 08/09/01
Yoshiyuki Koga Department of Mathematics Faculty of Science and Technology Sophia University Tokyo 102-8554 Japan email:
[email protected]
Ann. Henri Poincar´e 3 (2002) 29 – 86 c Birkh¨ auser Verlag, Basel, 2002 1424-0637/02/010029-58 $ 1.50+0.20/0
Annales Henri Poincar´ e
Front Fluctuations in One Dimensional Stochastic Phase Field Equations L. Bertini∗, S. Brassesco†, P. Butt` a‡ and E. Presutti‡
Abstract. We consider a conservative system of stochastic PDE’s, namely a weakly coupled, one dimensional phase field model with additive noise. We study the fluctuations of the front proving that, in a suitable scaling limit, the front evolves according to a non–Markov process, solution of a linear stochastic equation with long memory drift.
Part I. Introduction 1 General setting, model, and results The term “sharp interface limit” denotes a scaling procedure aimed at the derivation of interfaces as geometric objects, e.g. surfaces of codimension one with bounded variation, that is, enough regular for the area measure to be well defined. Of course this makes only sense in the context of systems which undergo phase transitions and of states where different phases coexist. In the limit the other degrees of freedom are lost and we are left with the interface alone. Rigorous proofs are hard, yet a great variety of models has been successfully worked out. The mathematics involved is correspondingly rich, e.g. the theory of Γ–convergence (to study the sharp interface limit of Ginzburg–Landau like free energy functionals in relation with the equilibrium shape of the interface, as in the Wulff problem) and correspondingly the theory of Gibbsian large deviations (to study the same problems at the more microscopic level of statistical mechanics); singular limit in PDE’s, like in the Allen–Cahn, Cahn–Hilliard and phase field equations, and correspondingly, at the microscopic level, hydrodynamic limits of spin or particle systems. This paper deals with fluctuations. Here again the questions are, first, whether in a sharp interface limit the system is described by a [fluctuating] interface with closed equations of motion and, secondly, the nature of such equations. The problem greatly simplifies in one space dimension where the limit interface is represented by a point which separates the two phases (one to its left and the other one ∗ Partially
supported by Cofinanziamento MURST 1999. supported by agreement CONICIT–CNR, and Proyecto F–97 000004 (CONICIT). ‡ Partially supported by Cofinanziamento MURST 1999 and by NATO Grant PST. CLG. 976552. † Partially
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to its right). Indeed, until now, most fluctuation results are restricted to d = 1. Such a state is marginally stable (w.r.t. displacements of the interfaces, see Section 2 for a precise statement) while it becomes stable when there is a conservation law (which imposes that the mass of each phase is preserved). By the conservation law, each displacement of the interface must then necessarily come with a deformation of the profile, hence the intuition that other degrees of freedom may then enter into play. This is confirmed, but in a sense also infirmed, by our results: indeed we will see that extra degrees of freedom become relevant, but their effect can be represented in the limit by terms which depends on the previous history of the interface evolution. The system we consider is a phase field equation with additive stochastic noise, see (1.1)–(1.2) below. Without noise the interface is given by a special instantonic profile connecting the two phases; in the presence of noise, after suitable rescalings, the limit state is still represented by the same instantonic profile which is however randomly displaced. The displacement obeys an ordinary stochastic equation driven by a white noise forcing term and with a long memory drift, whose effect is to force the interface back toward its initial position, thus restoring the equilibrium of the two coexisting phases. The evolution is defined by the following stochastic equations √ 1 ∆m(t) − V (m(t)) + λh(t) dt + γ dW (a) (t) dm(t) = 2 1 d [h(t) + m(t)] = ∆h(t)dt 2
(1.1) (1.2)
where the unknowns, m(t) = m(t, x), h(t) = h(t, x), (t, x) ∈ R+ × R, are two scalar random fields. In (1.1), λ and γ are positive parameters; ∆ is the Laplacian on R and W (a) (t) is a white noise with a cutoff in the spatial covariance. This means W (a) (t) is the canonical process in the filtered probability space (Ω, F , Ft , P) where Ω := C(R+ ; S (R)) (S (R) the space of tempered distributions), F its Borel σ– algebra, Ft the canonical filtration, and P the Gaussian measure with mean zero and covariance aγ (x) := a γ β/2 x (1.3) E W (a) (t), ϕW (a) (t ), ϕ = t ∧ t ϕ, a2γ ϕ , where t ∧ t := min{t, t }, ·, · denotes the inner product in L2 (R, dx) as well as the canonical pairing between S and S . Above β > 0 and a ∈ C02 (R), i.e. a twice differentiable function with compact support. We assume the normalization a(0) = 1. Finally V (m) is the derivative of a symmetric, smooth double well potential V (m); for simplicity we assume m2 m4 − . (1.4) 4 2 Note we are omitting to write explicitly the dependence on the randomness ω ∈ Ω. This will be done throughout the whole paper without further mention. V (m) =
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In a companion paper, [2], we prove global existence and uniqueness in a space of H¨older continuous functions for the system t t √ m(t) = pt m(0) + ds pt−s [−V (m(s)) + λh(s)] + γ pt−s dW (a) (s) (1.5) 0 0 1 t ds (∆pt−s )m(s) (1.6) h(t) + m(t) = pt [h(0) + m(0)] − 2 0 where pt = et∆/2 is the heat semigroup, namely the integral operator with kernel 2
pt (x, y) =
e−(x−y) /2t √ . 2πt
(1.7)
Observe that the integral on the r.h.s. of (1.6) is well defined because m(s) is H¨ older continuous. The system (1.1)–(1.2) is defined in terms of the integral equations (1.5)–(1.6) and called “stochastic phase field equations”. General background and physical interpretation are discussed in the sequel, here we proceed by stating our main result, presented in the next theorem. We consider the initial condition m(0) = m, ¯
h(0) = 0
(1.8)
where m ¯ ξ (x) := tanh(x − ξ),
m ¯ := m ¯0
(1.9)
is a standing wave (that we call “instanton”) with “center” ξ ∈ R. We suppose that √ (1.10) λ= γ (λ) (λ) and denote by m (t), h (t) the solution of (1.1)–(1.2) with (1.10) and initial condition (1.8). Our main result is
Theorem 1.1 There exists a process x(λ) (t) in C(R+ ), adapted to Ft , such that for each τ, ε > 0
(λ) sup m (t) − m ¯ x(λ) (t) > ε = 0 . (1.11) lim P λ↓0
t≤λ−2 τ
∞
Furthermore, denoting weak convergence in C(R+ ) by =⇒ and defining, after scaling, xλ (τ ) := x(λ) λ−2 τ , we have that xλ =⇒ x as λ ↓ 0, where x(τ ) is the unique solution of τ x(s) x(τ ) = b(τ ) − 3 (1.12) ds 2π(τ − s) 0 in which b(τ ) is a one dimensional Brownian motion with diffusion coefficient D = 3/4.
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The coefficient 3 in (1.12) and the value D = 3/4 above are related to the specific choice of the potential V . Existence and properties of the process solution of (1.12) are discussed in [3]. For the physical interpretation, we start from the equation without noise and with λ = 0, namely (1.1) with λ = γ = 0. This is the well known Allen–Cahn (AC) equation with double well potential V (m), which arises as the gradient flow associated to the Ginzburg–Landau free energy functional 1 dx (1.13) F (m) = |∇m(x)|2 + V m(x) 4 R (F decreases along the solutions of the AC equation). The minimizers of F (m) are the two constant functions m+ (x) = 1 and m− (x) = −1, therefore the values of the order parameter m = ±1 correspond to pure phases and the interface for (1.13) is (up to translations) the profile m(x) which minimizes F (m) under the condition that asymptotically as x → ±∞ it converges to ±1. The associated Euler–Lagrange equation is the stationary AC equation 1 ∆m − V (m) = 0 2
(1.14)
which, imposing the above conditions at ±∞, has the instanton m ¯ of (1.9) as its unique solution (modulo translations). Therefore m ¯ ξ is the equilibrium state which has the two phases coexisting to the right and to the left of ξ, it represents the “mesoscopic interface” with ξ its “mesoscopic location” (mesoscopic instead of macroscopic because the interface is “diffuse” and the transition from one phase to the other, even though exponentially fast, is not sharp; mesoscopic instead of microscopic because the AC equation and the Ginzburg–Landau functional can be derived by a scaling procedure from particle or spin systems, i.e. from an underlying more microscopic structure). The next step is with λ > 0, but still γ = 0. Then (1.1) is coupled to (1.2) and the two together give an example of phase field equations (PFE). Here h is a thermodynamic potential conjugated to the order parameter m: if m is a magnetization density, then h is a magnetic field, our notation is inspired by such an interpretation. More commonly however, m is a relative concentration of one species in a binary alloy and h is a relative temperature. The effective potential will then depend on the relative temperature h, our choice is simply to replace V (m) by V (m) − hm. At the critical temperature, which corresponds to h = 0, the alloy can exist in two different concentrations m = ±1, but, as the temperature h changes, the equilibrium concentrations vary, one becomes stable and the other one metastable. In the presence of a given temperature profile h, the AC rate of change of the concentration density at x at time t, i.e. dm(t, x)/dt, is given by (1.1) (with γ = 0). Due to latent heat, there is a corresponding change of temperature given (in the proper units) by dh(t, x)/dt = −dm(t, x)/dt. Simultaneously, by the
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Fourier law, the temperature diffuses according to the r.h.s. of (1.2). This has a feedback in (1.1) so that (1.1) and (1.2) are coupled. In conclusion the PFE describe a change of phase including latent heat effects and, because of that, unlike for the AC equation, there is a conservation law: the integral of m + h is in fact invariant under PFE. The noise term in (1.1) takes into account some external fluctuating force and the resulting equation is known in the physical literature as model C of Hohenberg and Halperin, [14]. Since m ¯ solves (1.14), the state m = m, ¯ h = 0 is a stationary solution of PFE, which is therefore interpreted, like in AC, as the mesoscopic interface. We are then studying in Theorem 1.1 what happens to the interface when there are small (because γ → 0 in Theorem 1.1), external perturbations which produce a √ random change γdW (a) (t) of magnetization (or, in the other interpretation, of concentration), the analysis including latent heat effects. The small parameter γ in the noise term has the meaning of a ratio between mesoscopic and microscopic space units, the former referring to (1.1), the latter to some microscopic model, as for instance the Glauber + Kawasaki process introduced in [12]. A formal comparison with the microscopic model in [12] would lead to a more complex structure for the additive noise; however we stick to (1.1) (which catches the correct behavior of critical fluctuations, see [5]) to make our analysis simpler. In conclusion the scaling γ → 0 has a natural justification in terms of the microscopic origin of the noise term, the scaling of time in Theorem 1.1 is on the other hand justified a posteriori: it is the correct scaling for observing finite displacements of the interface. On the contrary, the equality (1.10) has no clear physical interpretation; it is true that a scaling with λ → 0 is widely used in the PFE literature to stress “kinetic undercooling effects”, see [11], [17], but relating λ to the noise as in (1.10) is just a matter of technical convenience. We will come back to this in the next section in the paragraph “the role of the assumptions”.
2 Heuristic analysis and outline of proofs By Theorem 1.1, with probability going to 1, the process m(λ) (t), t ≤ λ−2 τ , is always close to the manifold of instantons M = {m ¯ ξ , ξ ∈ R}, see (1.11); the theorem then identifies the motion along M, (1.12). It is then natural to describe m(λ) (t) in terms of coordinates along and transversal to M, these are the Fermi coordinates that we are going to define. Stability of instantons, Fermi coordinates. Closeness to M is a consequence of the stability of M under the AC evolution. Under AC, in fact, M attracts exponentially fast all data which are in a small neighborhood, in sup norm, · ∞ , of M: namely there are δ, c and a all positive so that if for some ξ, m − m ¯ ξ ∞ < δ, then there is a ξ for which, for all t, m(t) − m ¯ ξ ∞ ≤ ce−at , m(t) being the solution of the AC equation starting from m. This obviously fails if we add noise (thus considering (1.1) with λ = 0 and γ > 0), but the noise, in a polynomial scale, cannot drive too far away from M because of the exponential attraction of
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the deterministic part of the equation: in [10] it is shown that for any τ > 0 and ζ > 0,
lim P
γ→0
sup dist(m(t), M) ≥ γ 1/2−ζ
=0
(2.1)
t≤γ −1 τ
where “dist” denotes distance in sup norm. The analysis extends to our case with √ λ = γ and (1.1)–(1.2) coupled, as stated in (1.11) and proved in Section 8. Going back to AC without noise, observe that stability of M does not mean stability of the single instanton: let m be a small deviation from m ¯ ξ , then from what we said above it will relax under AC to some m ¯ ξ , with ξ close but not necessarily equal to ξ. In the space of all profiles m, m ¯ ξ is marginally stable along the direction M while all the other directions are stable. It is then natural to associate to each m (as above) the value ξ of the center of the limit instanton. In practice, however, it is better to work with a more “geometrical” definition. Following [10, 9], we define a center ξ of m as a real number which minimizes the ¯ x (as a function of x). Then ξ is such that L2 –norm of m − m ¯ ξ = 0. m − m ¯ ξ, m
(2.2)
The center ξ has also a dynamical interpretation. Let Lξ be the operator 1 ∆v + (1 − 3m ¯ 2ξ )v (2.3) 2 obtained by linearizing the AC equation around m ¯ ξ . It is readily seen that Lξ is ¯ ξ while the rest of self-adjoint in L2 (R, dx), it has eigenvalue 0 with eigenvector m the spectrum is on the negative axis strictly away from 0. Then, if m has center ξ, the deviation v = m − m ¯ ξ by (2.2) has no component along m ¯ ξ , hence the linearized evolution starting from v decays exponentially fast and correspondingly m converges to m ¯ ξ , so that the center of m is also the center of the instanton to which the linearized AC evolution converges. The pair {ξ, m − m ¯ ξ } is known as the Fermi coordinates of m. This notion of a center of a function plays an important role also in our proofs, so we will spend a few more words, recalling Proposition 3.2 in [9], which gives a sufficient condition for m = m(x) to have a center. Lξ v =
Proposition 2.1 There is a constant δ > 0 such that if there exists x0 ∈ R so that m− m ¯ x0 ∞ ≤ δ then the following holds for some constant C = C(δ) independent of x0 and m. (i) The function m has a unique center x and ¯ x0 ∞ |x − x0 | ≤ Cm − m
(2.4)
(ii) the center x has the expansion 3 9 ¯ x0 , m − m ¯ ,m− m x = x0 − m ¯ x0 − m ¯ x0 m ¯ x0 , m − m ¯ x0 + R(m − m ¯ x0 ), 4 16 x0 ¯ x0 3∞ . (2.5) |R(m − m ¯ x0 )| ≤ Cm − m
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Thus, by (2.1) and (1.11) and with the help of Proposition 2.1, we can talk unambiguously, with probability going to 1, of the center ξ(t) of m(t) both for the stochastic AC equation and PFE. This gives us a precise definition of the location of the interface even without sharp interface limits; we will prove convergence to (1.12) by studying the asymptotic law of ξ(t) as γ → 0. Heuristics of the AC equation and PFE with noise. We start from the simpler (and instructive) case of the AC equation with noise, i.e. (1.1) with λ = 0 but with γ > 0. This is well studied in the literature, [9, 10, 13], even though in slightly different contexts. The scaling procedure is the same as in Theorem 1.1 and it leads to the same limit law but without drift, i.e. a Brownian motion. To √ explain heuristically the result, let us regard the forcing term γdW (a) (t) as a “source of small kicks” which we decompose in a component along M and another one orthogonal to M. The latter fights against the AC drift which pushes back √ toward M, and because of the small factor γ, to a first order, we forget about orthogonal components. On the contrary the kicks along M are not contrasted and they sum up: thus we are approximating dm(t) ≈
3√ γm ¯ ξ(t) m ¯ ξ(t) , dW (a) (t), 4
m ¯ , m ¯ =
4 3
(2.6)
where ξ(t) is the center of m(t). Also m(t) ≈ m ¯ ξ(t) , hence dm(t) ≈ m ¯ ξ(t) dξ(t)
(2.7)
and, in conclusion,
3√ γm ¯ ξ(t) , dW (a) (t) (2.8) 4 namely ξ(γ −1 t) is a Brownian motion with diffusion 3/4, which is what is proved in [10]. The argument for the system (1.1)–(1.2) is similar, the only difference from the stochastic AC equation in (1.1) lies in the simple, innocent looking term λh, which is however the source of all problems. The same heuristics leading to (2.6) √ applies to (1.1) by simply adding λh(t) to the noise; writing γ = λ according to (1.10), we then get
3 ¯ ξ(t) m dm(λ) (t) ≈ λm ¯ ξ(t) , dW (a) (t) (2.9) ¯ ξ(t) , h(t)dt + m 4 dξ(t) ≈
and, using (2.7), dξ(t) ≈
3 λ m ¯ ξ(t) , h(t)dt + m ¯ ξ(t) , dW (a) (t) . 4
(2.10)
On the other hand, writing (1.2) in integral form and recalling that h(0) = 0, we have t pt−s dm(λ) (s) (2.11) h(t) = − 0
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Using again (2.7) and taking the scalar product with m ¯ ξ(t) we get from (2.11) m ¯ ξ(t) , h(t)
≈−
0
t
m ¯ ξ(t) , pt−s m ¯ ξ(s) dξ(s) .
(2.12)
The approximate system (2.10)–(2.12) (with ≈ replaced by equality) is not as easy to study as the one which approximates the stochastic AC equation, but it can be seen to give the correct result (1.12) for the limit motion of the center, [3]. To see the relation with (1.12) we make another approximation whose validity will be justified in the course of the proof of Theorem 1.1. The approximation consists in ¯ (i.e. ξ(t) → 0) in the scalar products in (2.10) and (2.12) replacing m ¯ ξ(t) by m (the reason for its validity is that the displacements of the center are finite while the field h becomes flat because the diffusion in (1.2) acts for long times). The new system is then (forgetting about the noise cutoff) 3 λm ¯ , h(t) + λdb(t) 4 t m ¯ , pt−s m ¯ dξ(s) m ¯ , h(t) = −
(2.13)
dξ(t) =
(2.14)
0
where b(t) is the Brownian motion with diffusion 3/4 of Theorem 1.1. Using (2.14) to rewrite the first term on the r.h.s. of (2.13) we get t s 3 ξ(t) = λb(t) − λ ds m ¯ , ps−s m ¯ dξ(s ) . 4 0 0 Integrating by parts, after some simple algebra, t 3 ξ(t) = λb(t) − λ ds ξ(s )m ¯ , pt−s m ¯ . 4 0
(2.15)
Approximating
¯ (y) m ¯ (x)m dy 2π(t − s ) (2.16) and recalling that m ¯ = 2, (2.15) becomes (1.12), in the above approximation, which can be made rigorous in the limit λ → 0, having set t = λ−2 τ . ¯ = m ¯ , pt−s m
dx
dy m ¯ (x)pt−s (x, y)m ¯ (y) ≈
dx
Main difficulties and outline of proof. The heuristic arguments outlined above are essentially based on a linear approximation, their validity therefore rests on a rigorous proof that the non linear effects are negligible. Since the strength of √ the noise is γ, we cannot hope to improve the a priori bounds beyond m(t) − √ m ¯ ξ(t) ∞ ≈ γ. Then the non linear terms which are, to lowest order, quadratic have order γ; since they act for a time λ−2 τ (that is γ −1 τ ), a naive estimate gives a non–negligible contribution. The fact that they are indeed negligible must then
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come from cancellations, and the true problem is to find them and take them into proper account. This is more clearly seen in the stochastic AC equation. Following [10], we split the time axis into intervals of length T , taking T long, yet very small as compared to the macroscopic time γ −1 τ , for instance T = γ −1/10 . The crucial point is an iterative procedure for which the problem reduces to the analysis of only one of these intervals, a clear advantage, because in such a “short time” the non linear effects are under control. Couplings are used for this crucial reduction. We compare in fact, in the generic interval [nT, (n + 1)T ], the true process m(t) which starts from m(nT ) and the new process m(t) ˆ which starts from m ¯ ξ(nT ) , ξ(nT ) the center of m(nT ), the coupling is simply to use for both processes the √ ¯ ξ(nT ) ∞ ≤ γ 1/2−ζ , same noise γdW (a) (t). Under the assumption that m(nT )−m see (2.1), it can be seen that, with probability going to 1 as γ → 0, m((n + 1)T ) − m((n ˆ + 1)T )∞ ≤ Cγ 1−2ζ , (C a constant). By Proposition 2.1, the displacements of the center in the interval [nT, (n + 1)T ], as computed with the two evolutions, differ proportionally to γ 1−2ζ : since the number of intervals grows proportionally to γ −1 τ /T , the sum of all these differences goes like γ 1−2ζ γ −1 τ /T , which vanishes, after choosing 2ζ < 1/10. We can then study in each interval the process which starts from an instanton. Neglecting for simplicity the cutoff on the noise (with the cutoff some extra computations are needed), then the displacement of the center in a time interval T does not depend on the initial center and it is therefore independent of the past. The displacements of the centers (each time restarting from an instanton) are thus independent variables with mean 0 (by the symmetry between right and left) and, using classical arguments on convergence to Brownian motion, in the end, we need to sum their variances; since we are already with squares, it turns out that the linear approximation is sufficiently accurate and this explains the validity of the linear approximation in the previous heuristic analysis. While the above approach works well in the stochastic AC equation, Theorem 1.1 tell us that it fails, as there are long memory effects in the limit law. More bad news: the last term in (1.12), responsible for these effects, according to the heuristic analysis of the previous paragraph, comes from the term λh(t) and since it produces a finite drift in a time λ−2 , the order of magnitude of h(t) must be ≈ λ. Therefore we need an accuracy of order λ, which is comparable with the √ deviations of m(λ) from M (recall λ = γ) that has been neglected so far. Our approach to the problem, since when we began the present work, has been “to trust” [10] and to consider the non linear terms that are left as being negligible when we linearize (1.1) around m ¯ ξ(nT ) , ξ(nT ) the center of m(λ) (nT ). It is evidently not possible to use the coupling argument of [10], yet in some maybe more complicated way, in the end we “must” see that they are not relevant. The extra term λh(t) is new w.r.t. [10] and has to be dealt anew. According to the final result and for what said before, we expect h(t) ≈ λ, but let us even suppose, pessimistically, that h(t) is of the order of unity. Its effect for a time T will then be of order λT , hence still infinitesimal. Moreover, by (2.11), h(t) can be large only if dm(λ) (s) is large, but dm(λ) (s) is under control except for the term λh(s). Due
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Ann. Henri Poincar´e
to the presence of the small factor λ this gives a virtuous feedback which allows to control the magnitude of h(t). This is done in Sections 3 and 4. We first write down an integral equation, ¯ ξ(nT ) , the superscript n recalls that we are con(3.7), for v (n) (t) = m(λ) (t) − m sidering t ∈ [nT, (n + 1)T ]. v (n) (t) is written as the sum of 8 terms, two of them, (n) (n) called Γ2 (t) and Λ3 (t) are the important ones, which give contributions to the limit, respectively the Brownian motion and the drift. All the others vanish, but at this stage this is not yet established as they depend on the unknown m(λ) (t). In Section 4 we study by iterations the integral equation (3.7) to prove bounds on v (n) (t) and on the displacements of the center. We derive in this way (1.11) and establish even sharper bounds that will be used in the successive proofs. The proof of the convergence to (1.12) is reported in the remaining sections. (n) We begin Section 5 by splitting the term Λ3 (t) into the sum of 3 other terms, see (5.1)–(5.5), which foresees the approximation done in (2.16). We then use (2.5) to deduce an equation for the displacements of the center, which are called ψn , see (5.7). The equation is (5.9), which is a sort of linear integral equation in the ψn with kernel An,k , k < n, and known data ηn : “sort of” because the ηn still depend on the unknowns v (n) (·). The elements An,k decay as (n − k)−1/2 . It is then convenient to reduce to the matrix A2 , so that we iterate once (5.9) obtaining (5.17), where the kernel is now A2n,k and the “known terms” are ηn and (Aη)n . In Section 6 we study these “known terms” which are splitted into four groups. The first one consists of truly known terms which survive in the limit (n) (they come from Γ2 (t)). The terms in the second group, which instead may de(n) pend on v (t), are all directly proved to be negligible using the a priori bounds of Section 4. For those in the third group we cannot proceed in this way, but we need to use the integral equation for v (n) (t) and only after sufficiently many iterations, we can show that they are negligible. Finally, the last group collects terms which become negligible because of stochastic cancellations. The latter are studied in Section 7, the others in Section 6. We draw the conclusions of our analysis in Section 8 where we complete the proof of Theorem 1.1. Role of assumptions. We start from the assumption (1.10) which is conceptually the most important one, the others are of a more technical nature. As already remarked, there are several studies of sharp interface limits on PFE where λ is scaled to 0. This describes an intermediate regime (called kinetic undercooling) where thermodynamical equilibrium is not fully reached. Thus our model should be regarded as kinetic undercooling in the presence of stochastic perturbations. As said, the relation between λ and γ stated in (1.10) does not have a straight physical interpretation, it is just the right way to scale (1.1)–(1.2) and have a nice limit law. One may however wonder what would happen if we took a different relation than (1.10). We have not worked out the details, but we can at least present some educated guess. If we multiply h(t) in (1.1) by a constant θ, we would then derive a limit law with such a factor multiplying the last term in (1.12). Let us then
Vol. 3, 2002
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39
consider the scaling behavior of x(t) = b(t) − 3θ
t
0
x(s) ds 2π(t − s)
(2.17)
both when θ vanishes and when it diverges. Set y(t) = θa x(θc t), then a c/2
y(t) = θ θ
b(t) − 3θθ
c/2
0
t
y(s) . ds 2π(t − s)
(2.18)
Imposing a + c/2 = 0,
1 + c/2 = 0;
a = 1,
c = −2
(2.19)
we have (2.18) equal to (1.12). Thus if we take (1.1) with λ → λ1+α and call θ = λα , we believe that our analysis extends, at least for small |α|, and that the center ξ(t) of the solution of the corresponding equation is such that λα ξ(λ−2−2α t)
(2.20)
converges in law to (1.12). Concluding remarks and perspectives. A forthcoming paper, [3], is devoted to the analysis of the limit process (1.12). This can be characterized in terms of a Brownian motion with absorption at the origin, which in turn is reduced to the well studied one dimensional Schr¨ odinger equation with Dirac’s delta potentials. A quite explicit expression for the solution of (1.12), is then available, in √ particular it shows that the displacements ξ(τ ) of the front have typical size log τ for τ large; “aging phenomena” are also present. The “cluster fluctuations” have instead the usual Brownian structure. Consider an initial state with h0 = 0 and m0 (x) the symmetric function which coincides with m ¯ −ξ (x), ξ > 0, for x ≤ 0. We interpret it as a “plus cluster” in the region (−ξ, ξ) with the minus phase outside. To make it sharp we set ξ = λ−1 -, - > 0, and √ consider the process (1.1)–(1.2) (always assuming λ = γ). Preliminary results (see also [3]) indicate that, in proper units, the coordinates ξ1 (τ ) and ξ2 (τ ) of the two centers, evolve in the limit according to a system of two stochastic equations. The system can be diagonalized into the variables ξG (τ ) := [ξ1 (τ ) + ξ2 (τ )]/2 and ξ(τ ) := ξ1 (τ ) − ξ2 (τ ). The variable ξG (τ ) has the law of a Brownian motion, ξ(τ ) is independent of ξG (τ ) and obeys an equation like (1.12). We hope to be able to accomplish the above program including the analysis of the case with several clusters (and suitable scalings), the ultimate goal being the study of interface fluctuations in many dimensions.
40
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Ann. Henri Poincar´e
Part II. Stability of fronts 3 The iterative scheme Before starting with the proofs we introduce some notation that will be used throughout the paper: Notation. C will denote a generic constant whose numerical value may change from line to line (from the statements it will appear clear on which parameters it depends). For p ∈ [1, ∞] we denote by · p the norm in Lp (R, dx). We will study the problem (1.1)–(1.10) by an iterative procedure. To this end, we divide the microscopic time–line R+ into intervals [Tn , Tn+1 ], Tn = nT , where n ∈ N and T = λ−(1∧β)/20 (β as in (1.3)). We then associate to any macroscopic time interval [0, τ ], τ > 0, the set of microscopic time intervals [Tn , Tn+1 ], with n ≤ nλ (τ ), where nλ (τ ) is the integer part of (λ2 T )−1 τ , namely nλ (τ ) = [(λ2 T )−1 τ ]. To simplify notation, we write m(t) = {m(t, x), x ∈ R} and h(t) = {h(t, x), x ∈ R} for the solution of (1.1)–(1.10) (omitting the dependence on λ). We next define, by induction on n ≥ 0, the numbers xn and the functions v (n) (t) = {v (n) (t, x), x ∈ R}, t ∈ [Tn , Tn+1 ]. They will have the property that for each t ∈ [Tk , Tk+1 ] ¯ xk v (k) (t) = m(t) − m
if v (h) (Th+1 )∞ ≤ δ for all 0 ≤ h ≤ k − 1
(3.1)
with δ as in Proposition 2.1. For n = 0 we set x0 = 0 and v (0) (t) = m(t) − m ¯ x0 , 0 ≤ t ≤ T , so that (3.1) holds. Suppose that, by the induction hypothesis, we have already defined for all k ≤ n − 1 both xk and v (k) (t) and that (3.1) holds for such k’s. If there is k ≤ n − 1 such that v (k) (Tk+1 )∞ > δ, we set xn = 0 and v (n) (t) = 0, ¯ xn−1 + v (n−1) (Tn ), which t ∈ [Tn , Tn+1 ]. Otherwise we define xn as the center of m (n−1) by Proposition 2.1 is well defined, as v (Tn )∞ ≤ δ. We then set, according to ¯ xn , t ∈ [Tn , Tn+1 ], and have (always under the assumption (3.1), v (n) (t) = m(t)− m that v (k) (Tk+1 )∞ ≤ δ for all k ≤ n − 1) ¯ xn−1 + v (n−1) (Tn ) = m ¯ xn + v (n) (Tn ), m(Tn ) = m Let
h(n) (t) = h(t),
m ¯ xn , v (n) (Tn ) = 0 . (3.2)
t ∈ [Tn , Tn+1 ],
introduce the stopping time w.r.t. the discrete filtration FTn+1
Nδ := inf k : v (k) (Tk+1 )∞ > δ (n)
(δ as in Proposition 2.1) and define gt (0) (2.3). We abbreviate gt = gt .
(3.3)
(3.4)
:= exp{tLxn }, t ∈ R+ , with Lxn as in
Vol. 3, 2002
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41
After expanding V (m(t)) in (1.8) and (1.10) about m ¯ xn , for t ∈ [Tn , Tn+1 ] as long as n ≤ Nδ , v (n) (t) and h(n) (t) are given as the solution of dv (n) (t) = Lxn v (n) (t) + λh(n) (t) − 3m ¯ xn v (n) (t)2 − v (n) (t)3 dt + λdW (a) (t)
h(n) (t) = pt−Tn h(n−1) (Tn ) − v (n) (t) + pt−Tn v (n) (Tn ) +
(3.5) t
ds Tn
∂pt−s (n) v (s) ∂s (3.6)
with initial condition v (n) (Tn ) for the first equation. Recall that v (n) (Tn ) is determined from v (n−1) (Tn ) via (3.2), and that from (3.3) we have h(n−1) (Tn ) = h(n) (Tn ). We will write in the next section v (n) (t) as a sum of 10 terms, v (n) (t) = 10 (n) i=1 Γi (t), but for the moment it is more convenient to keep some of the terms together. We are going to prove that for n < Nδ , (n)
(n)
(n)
(n)
(n)
(n)
(n)
(n)
v (n) (t) = Γ1 (t) + Γ2 (t) + Λ3 (t) + Γ4 (t) + Γ5 (t) + Γ6 (t) + Γ9 (t) + Γ10 (t) (3.7) where (n)
Γ1 (t) (n)
Γ2 (t) (n)
Λ3 (t) (n) Γ4 (t) (n)
Γ5 (t) (n) Γ6 (t) (n)
Γ9 (t) (n)
Γ10 (t)
(n)
:= gt−Tn v (n) (Tn ) (n) := λz (t) := λ n
:= −λ
k=1 t
:= λ
Tn
:= −λ
k=1 t
:= −
t
Tn
Tn t
Tn
gt−s dW (a) (s)
(3.9)
(n)
ds gt−s ps−Tk [m ¯ xk − m ¯ xk−1 ]
s
ds
Tn
∂ps−s (n) v (s ) ∂s
(n)
Tn
(n) ds gt−s
Tk
(3.12) ds
Tk−1
∂ps−s (k−1) v (s ) ∂s
(n) ¯ xn v (n) (s)2 ds gt−s m
(3.15)
−1
to (3.5) we get (n) (n) (n) (n) v (n) (t) = Γ1 (t) + Γ2 (t) + Γ9 (t) + Γ10 (t) + λ
(3.13) (3.14)
(n) ds gt−s v (n) (s)3 .
Proof of (3.7). By applying (∂t − Lxn )
(3.10) (3.11)
ds gt−s v (n) (s)
Tn n t
:= −λ := −3
t
(n)
Tn
(n) ds gt−s
(3.8) t
t Tn
(n)
ds gt−s h(n) (s) .
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L. Bertini, S. Brassesco, P. Butt` a and E. Presutti
Ann. Henri Poincar´e
We use for h(n) (s) the expression given by (3.6), and get v (n) (t) =
(n)
(n)
(n)
(n)
(n)
(n)
Γ1 (t) + Γ2 (t) + Γ4 (t) + Γ5 (t) + Γ9 (t) + Γ10 (t) t (n) ds gt−s ps−Tn h(n−1) (Tn ) + v (n) (Tn ) . +λ Tn
To complete the proof of (3.7) we thus need to show
t
λ Tn
(n) (n) (n) ds gt−s ps−Tn h(n−1) (Tn ) + v (n) (Tn ) = Λ3 (t) + Γ6 (t) .
(3.16)
From (3.2), we have ¯ xn − m ¯ xn−1 ) h(n−1) (Tn ) + v (n) (Tn ) = h(n−1) (Tn ) + v (n−1) (Tn ) − (m
(3.17)
¯ xn−1 ) reproduces We use this identity in the l.h.s. of (3.16). The difference (m ¯ xn − m (n) the last term of the sum in (3.10) (i.e. in the definition of Λ3 (t)). For the term with h(n−1) (Tn ) + v (n−1) (Tn ), we use (3.6) to write h(n−1) (Tn ) + v (n−1) (Tn )
(n−1) (n−1) (Tn−1 ) + v (Tn−1 ) + = pTn −Tn−1 h
Tn
Tn−1
ds
∂pTn −s (n−1) v (s ). ∂s
The contribution of the last integral to the l.h.s. of (3.16) gives the last term of (n) the sum in (3.13), i.e. the definition of Γ6 (t). By iteration we then get (3.16) and (3.7) is therefore proved.
4 A priori bounds We will use the representation (3.7) to prove in Proposition 4.1 below some a priori bounds on v (n) (t) and other quantities. We need first some more notation; recalling Proposition 2.1, we set ξn := −
n−1 3 m ¯ xk , v (k) (Tk+1 ) 4
(4.1)
k=0
We should think of ξn as a linear approximation to xn , since the increment ξk −ξk−1 is, according to (2.5), the linear approximation to the displacement xk − xk−1 of the center in the time interval [Tk−1 , Tk ]. We set (n) sup Vn,∗ := sup Vk , V∗ (τ ) := Vnλ (τ ),∗ (4.2) Vn := v (t) , t∈[Tn ,Tn+1 ]
∞
k≤n
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and
δ ∗ (τ ) :=
sup |xk+1 − xk |
43
(4.3)
k≤nλ (τ )
Let also Zn :=
sup s∈[Tn ,Tn+1 ]
(n) z (s)
∞
,
Zn,∗ := sup Zk k≤n
(4.4)
(1)
and Bλ,τ ⊂ Ω be the event (recall the definition of (Ω, F , Ft , P) given in Section 1) √ (1) (4.5) Bλ,τ := ω ∈ Ω : Znλ (τ ),∗ ≤ λ−ζ T . We will prove in Appendix B the following Gaussian estimate: for each τ, ζ, q > 0 there is a constant C = C(τ, ζ, q) such that for any λ > 0 (1) (4.6) P Bλ,τ ≥ 1 − Cλq . The next proposition is the key ingredient for the bound (1.11), see the beginning of §8 for the conclusion of the proof. Proposition 4.1 For each τ, ζ > 0 there is a constant C = C(τ, ζ) such that, for any λ > 0, √ √ (1) V∗ (τ ) ≤ Cλ1−ζ T , δ ∗ (τ ) ≤ Cλ1−ζ T on the set Bλ,τ . (4.7) In particular, for λ > 0 small enough, Nδ > nλ (τ )
(1)
on the set Bλ,τ .
(4.8)
Proof. Let (mλ , tλ ), tλ ∈ [Tmλ , Tmλ +1 ], mλ ≤ nλ (τ ), be the stopping times (mλ w.r.t. FTn+1 , tλ w.r.t. Ft ) so that (mλ , tλ ) = (nλ (τ ), Tnλ (τ )+1 ) if V∗ (τ ) < λT , otherwise (mλ , tλ ) = (n, t), where n is the first index such that Vn ≥ λT and t the first time in [Tn , Tn+1 ] for which v (n) (t)∞ ≥ λT . In the following we may assume λ is small enough, otherwise (4.7) holds trivially. We claim there is a constant C so that v (mλ ) (tλ )∞ ≤ CλT .
(4.9)
In fact, if tλ ∈ (Tmλ , Tmλ +1 ], (4.9) follows from the continuity of v (mλ ) (t)∞ ; otherwise, i.e. if tλ = Tmλ , v (mλ ) (Tmλ )∞ ≤ v (mλ −1) (Tmλ )∞ + m ¯ mλ − m ¯ mλ −1 ≤ λT + |xmλ − xmλ −1 | ≤ CλT from (3.2) and (2.4), since m ¯ ∞ = 1 and v (mλ −1) (Tmλ )∞ ≤ λT .
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Ann. Henri Poincar´e
We will study the evolution till time tλ , so that our next considerations are tacitly restricted to n ≤ mλ and t ≤ tλ . By (4.9), Nδ > mλ (for λ small enough) so that we can use (3.7), regarding it as an integral equation in v (n) (t). We are going to bound one after the other all eight terms on the r.h.s. of (3.7). We will use in the sequel the following bounds (see e.g. [9] and references therein). There is C < ∞ so that, for any measurable function ϕ, gt ϕ∞ ≤ Cϕ∞ .
(4.10)
Moreover, by the Perron–Frobenius Theorem, there are α > 0 and C < ∞ so that for any ϕ orthogonal to m ¯ , m ¯ , ϕ = 0, gt ϕ∞ ≤ Ce−αt ϕ∞ .
(4.11)
Then, by the last identity in (3.2), for t ∈ [Tn , Tn+1 ] (n) ≤ Ce−α(t−Tn ) v (n) (Tn )∞ Γ1 (t) ∞ ≤ Ce−α(t−Tn ) v (n−1) (Tn )∞ + m ¯ xn − m ¯ xn−1 ∞ ≤
Ce−α(t−Tn ) v (n−1) (Tn )∞ .
(4.12)
The second inequality follows from (3.2) and the last one follows from (2.4), proceeding as in the proof of (4.9). By definition (4.4) we get (n) sup (4.13) Γ2 (t) ≤ CλZn . ∞
t∈[Tn ,Tn+1 ]
(n)
We next bound Λ3 . We have 2 1 pt m ¯ z ∞ ≤ √ ≤ √ t 2πt √ ¯ z 1 = 2 for any z ∈ R. Then because pt (x, y) ≤ 1/ 2πt and m |x − x xk k k−1 | pt (m √ ¯ xk − m ¯ xk−1 ) ∞ ≤ dz pt m ¯ z ∞ ≤ xk−1 t so that, by (4.10), we get (n) Λ3 (t)
∞
≤ Cλ
n k=1
t
|xk − xk−1 | ds √ . s − Tk Tn
(4.14)
(4.15)
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45
By (2.4), we then get (n) Λ3 (t)
n √ Vk−1 √ ≤ Cλ T . n −k+1 k=1
∞
(4.16)
(n)
We will next bound Γ4 (t). We write
(n)
gt−s = pt−s −
t
ds
s
d (n) gt−s ps −s = pt−s + ds
t
s
(n)
ds gt−s [1 − 3m ¯ 2xn ]ps −s
having used (2.3) in the second equality. Then (n)
Γ4 (t)
t
= λ
s
ds
ds Tn
Tn
t
+λ
∂pt−s (n) v (s ) ∂s
t
ds
ds
Tn
s
(n) gt−s [1
−
3m ¯ 2xn ]
s
ds
Tn
∂ps −s (n) v (s ) . ∂s
By using (4.10) and (see Appendix A) ∂ pt ϕ ≤ 1 ϕ∞ ∂t t ∞ since v (n) (s )∞ ≤ Vn , we get (n) Γ4 (t)
∞
≤ CλT 2 Vn .
This is not optimal but good enough for our purposes. Analogously t (n) ds v (n) (s)∞ ≤ CλT Vn Γ5 (t) ≤ Cλ ∞
(n) Γ6 (t)
∞
≤ Cλ
n k=1
Tn
(4.18)
(4.19)
Tn
t
(4.17)
Tk
ds
ds
Tk−1
Finally, using again (4.10), we have (n) Γ9 (t) ∞ (n) Γ10 (t) ∞
n v (k−1) (s )∞ Vk−1 . ≤ CλT s − s n−k+1 k=1 (4.20)
≤
CT Vn2
(4.21)
≤
CT Vn3
(4.22)
hence, recalling the definition of (mλ , tλ ) and (4.9), (n) (n) Γ9 (t) + Γ10 (t) ≤ CλT 2 Vn ∞
∞
(4.23)
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L. Bertini, S. Brassesco, P. Butt` a and E. Presutti
Ann. Henri Poincar´e
After collecting all these bounds, we have, for t ∈ [Tn , Tn+1 ], n ≤ mλ , t ≤ tλ , (n) ≤ C e−α(t−Tn ) v (n−1) (Tn ) + λZn + λT 2 Vn v (t) ∞
∞
n n √ V Vk−1 √ k−1 . + λT +λ T n−k+1 n−k+1 k=1 k=1
(4.24)
By iterating once the above inequality we get Vn ≤ C e−αT Vn−2 + λ(Zn + Zn−1 ) + λT 2 (Vn + Vn−1 ) n n √ Vk−1 Vk−1 √ +λ T + λT . n−k+1 n−k+1 k=1 k=1 Recalling (4.4), we have Vn
≤ C e−αT Vn,∗ + λZn,∗ + λT 2 Vn,∗ n n √ Vk−1,∗ Vk−1,∗ √ +λ T + λT n−k+1 n−k+1 k=1 k=1
and since the r.h.s. is an increasing function of n, Vn,∗ ≤ C e−αT Vn,∗ + λZn,∗ + λT 2 Vn,∗ n n √ V Vk−1,∗ √ k−1,∗ + λT . +λ T n−k+1 n−k+1 k=1 k=1
(4.25)
Inequality (4.25) yields, for n ≤ mλ ≤ nλ (τ ), √ n−1 Cλ T 2 −αT √ 1 − C λT + e + λT log nλ (τ ) Vn,∗ ≤ CλZn,∗ + Vk,∗ . n−k k=0 For λ small enough the square bracket term is larger than 1/2, so that (provided we double the value of the constant C) Vn,∗ ≤ CλZn,∗ +
n−1 k=0
√ Cλ T √ Vk,∗ . n−k
(4.26)
By iteration of (4.26) we get Vn,∗ ≤ CλZn,∗ +
n−1 k=0
√ n−2 C 2 λ2 T 2 2 √ α(n, k)Vk,∗ Zk,∗ + C λ T n−k k=0
(4.27)
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where α(n, k) =
n−1
1 1 √ √ n−h h−k h=k+1
47
(4.28)
is a bounded function of n and k. We bound the second term on the r.h.s. of (4.27) by Zn,∗
n−1 k=0
√ √ C 2 λ2 T √ ≤ CZn,∗ λ2 T n ≤ CλZn,∗ n−k
since n ≤ mλ ≤ (λ2 T )−1 τ . We then get from (4.27) Vn,∗ ≤ CλZn,∗ + Cλ2 T
n−2
Vk,∗
(4.29)
k=0
from which, by Gronwall Lemma, there is C = C(τ ) such that Vn,∗ ≤ CλZn,∗ ,
for all n ≤ mλ .
(4.30)
By (4.5), choosing λ small enough, we have, for n ≤ mλ , √ (1) on the set Bλ,τ Vn,∗ ≤ Cλ1−ζ T (1)
which implies mλ = nλ (τ ) on the set Bλ,τ . Thus the first bound in (4.7) follows from (4.5) and (4.30). Since Nδ > mλ , (4.8) also follows. Since m ¯ ∞ = 1, the second estimate in (4.7) follows directly from the first one and Proposition 2.1. ¯ xn , is We are going to prove that the component of v (n) (Tn+1 ) orthogonal to m 1−ζ (n) bounded by Cλ , thus considerably improving the bound on the full v (Tn+1 ). Let 3 (n,⊥) (n) ¯ m gt := gt ¯ xn | 1 − |m 4 xn the operator whose kernel is (n,⊥)
gt
3 (n) ¯ (x)m (x, y) = gt (x, y) − m ¯ xn (y) . 4 xn
(4.31) (n)
The superscript ⊥ recalls L2 –orthogonality w.r.t. the eigenvector m ¯ xn of gt , i.e. (n) ¯ xn = m ¯ xn . It follows from (4.11) that there are constants α > 0 and C < ∞ gt m so that, for any ϕ, (n,⊥) ϕ ≤ Ce−αt ϕ∞ . (4.32) gt ∞
Let also
3 ¯ , z (n) (t)m z (n,⊥) (t) := z (n) (t) − m ¯ xn 4 xn
(4.33)
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L. Bertini, S. Brassesco, P. Butt` a and E. Presutti
Ann. Henri Poincar´e
be the component of z (n) (t) orthogonal to m ¯ xn and introduce the event (2)
Bλ,τ :=
ω∈Ω :
sup
sup
n≤nλ (τ ) t∈[Tn ,Tn+1 ]
z (n,⊥)(t)∞ ≤ λ−ζ
.
(4.34)
In Appendix B we will prove that for each τ, ζ, q > 0 there exists a constant C = C(τ, ζ, q) such that for any λ > 0 (2) P Bλ,τ ≥ 1 − Cλq . (4.35) Define now
V∗⊥ (τ ) :=
sup v (n,⊥) (Tn+1 )
n≤nλ (τ )
∞
,
3 ¯ , v (n) (Tn+1 )m v (n,⊥) (Tn+1 ) := v (n) (Tn+1 ) − m ¯ xn . 4 xn
(4.36)
Proposition 4.2 Recalling (4.5) and (4.34), set (1,2)
(1)
(2)
Bλ,τ := Bλ,τ ∩ Bλ,τ .
(4.37)
Then, for each τ, ζ > 0 there is a constant C = C(τ, ζ) such that, for any λ > 0, V∗⊥ (τ ) ≤ Cλ1−2ζ
(1,2)
on the set Bλ,τ
(4.38)
and, recalling (4.1) for the definition of ξn , sup n≤nλ (τ )+1
|ξn − xn | ≤ CT −1/2+ζ
(1,2)
on the set Bλ,τ
.
(4.39)
Proof. Let (n,⊥)
Λ3
(Tn+1 ) := =
3 (n) (n) ¯ , Λ (Tn+1 )m Λ3 (Tn+1 ) − m ¯ xn 4 xn 3 n Tn+1 (n,⊥) −λ dt gTn+1 −t pt−Tk [m ¯ xk − m ¯ xk−1 ] k=1
(4.40)
Tn
where we used (4.31). By using (4.14), (4.32), m ¯ ∞ ≤ 1, and recalling (4.3) we have n Tn+1 1 (n,⊥) ∗ (Tn+1 ) ≤ Cδ (τ )λ dt e−α(Tn+1 −t) √ Λ3 ∞ t − Tk k=1 Tn λ ≤ Cδ ∗ (τ ) √ nλ (τ ) ≤ Cλ1−ζ T −1/2 (4.41) T (1)
the last inequality being true, by (4.7), on the set Bλ,τ .
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Shorthanding by R(n) the sum of all the terms on the r.h.s. of (3.7), except (n) and Λ3 , and calling R(n,⊥) its orthogonal projection,
(n) Γ2
(n,⊥)
v (n,⊥) (Tn+1 ) = λz (n,⊥) (Tn+1 ) + Λ3
(Tn+1 ) + R(n,⊥) .
(4.42)
The last term is bounded by |R(n,⊥) | ≤ C e−αT + λT 2 + λT log nλ (τ ) V∗ (τ )
(4.43)
as it follows from (4.12) and (4.18)–(4.20), and (4.23). The bound (4.38) now follows from the definition (4.34), and equations (4.41)–(4.43). It remains to prove (4.39). Recalling that x0 = 0, from (2.5) and (4.1), xn =
n−1
(xk+1 − xk ) = ξn −
k=0
n−1 9 m ¯ xk , v (k) (Tk+1 )m ¯ xk , v (k) (Tk+1 ) 16 k=0
+
n−1
R(v (k) (Tk+1 )) (4.44)
k=0
then, for n ≤ nλ (τ ), since m ¯ 1 = 2,
¯ xn , v (n) (Tn+1 ) + |ξn − xn | ≤ nλ (τ ) 2V∗ (τ ) sup m n≤nλ (τ )
sup R(v (n) (Tn+1 ))
∞
n≤nλ (τ )
By (2.5)
sup R(v (n) (Tn+1 ))
∞
n≤nλ (τ )
≤ CV∗ (τ )3 .
.
(4.45)
(4.46)
¯ = 0, |m ¯ xn , v (n) (Tn+1 )| ≤ CV∗⊥ (τ ), (4.39) follows from (4.7), Since m ¯ , m (4.38), (4.45), and (4.46). ¯ xn , v (n) (Tn ) = 0, The bound in (4.38) holds also for v (n) (Tn ). Indeed, since m we have the following proposition. (1,2)
Proposition 4.3 Let Bλ,τ be as in (4.37). Then, for each τ, ζ > 0 there is a constant C = C(τ, ζ) such that, for any λ > 0 (1,2) on the set Bλ,τ . (4.47) sup v (n) (Tn ) ≤ Cλ1−2ζ n≤nλ (τ )
Proof. By (3.2)
∞
v (n) (Tn ) = v (n−1) (Tn ) + m ¯ xn−1 − m ¯ xn .
(4.48)
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By Taylor expansion up to fourth order, we have ¯ xn−1 m ¯ xn − m
1 ¯ −m ¯ xn−1 (xn − xn−1 ) + m (xn − xn−1 )2 2 xn−1 1 − m (xn − xn−1 )3 + an−1 (xn − xn−1 )4 ¯ 6 xn−1
=
(4.49)
where an−1 is bounded. Thus, by Proposition 2.1, we have m ¯ xn − m ¯ xn−1 =
3 m ¯ , v (n−1) (Tn )m ¯ xn−1 + αn 4 xn−1
(4.50)
where 1 1 m ¯ ¯ (xn − xn−1 )2 − m (xn − xn−1 )3 + an−1 (xn − xn−1 )4 2 xn−1 6 xn−1 9 (n−1) (n−1) (n−1) m ¯ +m ¯ xn−1 ,v (Tn )m ¯ xn−1 , v (Tn ) − R(v (Tn )) . (4.51) 16 xn−1 αn =
From (4.48) and (4.50) we get v (n) (Tn ) = v (n−1,⊥) (Tn ) − αn .
(4.52)
√ (1,2) Note that, from (4.7), on the set Bλ,τ we have supn≤nλ (τ ) |xn −xn−1 | ≤ Cλ1−ζ T ; by (4.38) and (4.46) we have |αn | ≤ Cλ2−2ζ T (a better bound is proved in Section 6). The bound (4.47) follows.
Part III. Limit motion 5
A new integral equation (1,2)
Unless otherwise stated, we will work in the set Bλ,τ which appears in Propositions 4.2 and 4.3. In particular we can use the integral representation (3.7) for all n ≤ nλ (τ ) (if λ is small enough) and the bounds of Section 4. It is now convenient (n) to decompose the term Λ3 (t) into the sum of three new terms; we thus use (4.50) to write (n) (n) (n) (5.1) Λ3 (t) = Γ3+8 (t) + Γ7 (t) where (n)
Γ3+8 (t) := (n)
Γ7 (t) :=
n
3λ m ¯ xk−1 , v (k−1) (Tk ) 4 k=1 n t (n) −λ ds gt−s ps−Tk αk −
k=1
Tn
t
Tn
(n)
ds gt−s ps−Tk m ¯ xk−1
(5.2) (5.3)
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(n)
51
(n)
and next we decompose Γ3+8 (t) = Γ3 (t) + Γ8 (t) where (n)
Γ3 (t) := −
t n−1 3λ 2 (n) m ¯ xk , v (k) (Tk+1 ) ds gt−s 4 2π(s − Tk+1 ) Tn
(5.4)
k=0
t n−1 3λ 2 (n) (k) := − m ¯ xk , v (Tk+1 ) ds gt−s ps−Tk+1 m ¯ xk − . 4 2π(s − Tk+1 ) Tn k=0 (5.5) (n) (n) Note that Γ3 (t) is obtained from Γ3+8 (t) replacing pt (x, y) by (2πt)−1/2 (recall that dx m ¯ (x) = 2). In conclusion we have (n) Γ8 (t)
v (n) (t) =
10 i=1
Let us define
(n)
Γi (t) .
3 ψn := − m ¯ , v (n) (Tn+1 ) 4 xn
(5.6)
(5.7)
so that, recalling (4.1), ξn =
n−1
ψk .
(5.8)
k=0
We then set t = Tn+1 in (5.6) and project it on m ¯ xn , getting ψn = ηn −
n−1
An,k ψk
(5.9)
3 (n) ¯ , Γ (Tn+1 ), i = 1, . . . , 10 ηn (i) = − m 4 xn i
(5.10)
k=0
where ηn :=
10 i=1 i =3
and
ηn (i),
√ 2 T 3 √ 1k
Note that ηn (3) ≡ −
n−1
An,k ψk .
(5.11)
(5.12)
k=0
Despite its appearance, (5.9) is not a linear equation in the variables ψn , because the term ηn still depends on the unknowns v (n) (t). We shall however see, using the a priori bounds of Section 4, that all contributions to ηn which contain the unknowns vanish as λ ↓ 0 (this is not exactly true, as some terms will vanish
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only after averaging over n). As already seen in Section 4 the matrix A := ((An,k )) improves after iterations. Calling ψ and η the vectors whose components are ψn and ηn , (5.9) becomes ψ = η − Aψ . (5.13) In Rnλ (τ )+1 we introduce the seminorms |y|n := sup0≤j≤n |yj | and the corresponding seminorms on matrices as M n := sup |M y|n = sup |y|n =1
n
0≤i≤n j=0
|Mi,j | .
(5.14)
By the triangular structure of A, see (5.11), we easily get the following lemma.
Lemma 5.1 For each τ > 0 there is a constant C = C(τ ) such that for any λ > 0 Cj . sup Aj n ≤ j! n≤nλ (τ )
(5.15)
Hence, from (5.13), ψ=
∞
(−A)j η = (1 + A)−1 η,
sup (1 + A)−1 n ≤ C .
(5.16)
n≤nλ (τ )
j=0
It will be convenient to consider also one iteration of (5.13), i.e. ψ = η − Aη + A2 ψ .
(5.17)
Explicitly we have (A2 )n,k =
n−1
An,j Aj,k 10≤k
(5.18)
j=k+1
We conclude the section by showing that A2 has a scaling limit. Proposition 5.2 For each τ > 0 there is a constant C = C(τ ) such that 2 sup A n,k ≤ Cλ2 T .
(5.19)
0≤k
Moreover, for each δ, τ > 0, lim
sup
λ↓0 0≤τ2 <τ1 ≤τ τ1 −τ2 >δ
2 −1 2 (λ T ) A n (τ1 ),n (τ2 ) − λ λ
9 =0. 2
(5.20)
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Proof. By (5.11)
53
n−1 1 2 √ √ A n,k ≤ Cλ2 T n − j j−k j=k+1
which proves (5.19). By (5.11) we have 9 9 (λ2 T )−1 A2 n (τ ),n (τ ) − = λ 1 λ 2 2 2π 1 )−1 nλ (τ 2 × nλ (τ1 ) − j + nλ (τ1 ) − j − 1 j=n (τ )+1 λ
2
2 −π × j − nλ (τ2 ) + j − nλ (τ2 ) − 1
.
(5.21)
1 Since π = 0 dx [x(1 − x)]−1/2 , by letting σ = τ1 − τ2 and changing the index of summation the r.h.s. of (5.21) equals 1 nλ (σ)−2 2 2 1 9 √ √ − dx 2π j=0 nλ (σ) − j + nλ (σ) − j − 1 j + j + 1 x(1 − x) 0 (5.22) which converges, as λ ↓ 0, to 0 uniformly for σ ∈ [δ, τ ], as can be easily checked. Proposition 5.2 is proved.
6 Bounds on η Recalling (5.8)–(5.10), a term η(i) (as in the previous section, we are here using vectorial notation) contributing to ψ whose seminorm is |η(i)|n = o(λ2 T ) (i.e. such that (λ2 T )−1 |η(i)|n vanishes as λ ↓ 0) does not contribute to the limiting equation for ξn since n ≤ (λ2 T )−1 τ . This is the case for some of the η(i)’s, i.e. η(1), η(7), and η(10), as we shall see. √ Clearly ηn (2) is not negligible because its typical magnitude is λ T . It will be examined in the next section, where we shall see that , summed over n it gives a finite contribution because of cancellations related to its martingale nature.The other terms, i.e. ηn (i), with i from 4 to√9, live on an intermediate stage: they are smaller than the a priori bound λ1−ζ T , yet not small enough to be directly negligible. We shall rewrite the factors v (n) (t) via (5.6), with the idea that if we get two η(i), i = 4, . . . , 10 then the corresponding terms become negligible. We will use the following notation. Definition 6.1 Let ηn (i1 , . . . , ik ), k > 1, ij ∈ {3, 4, 5, 6, 8} when j < k, ik ∈ {1, . . . , 10}, be the term which is obtained from ηn (i1 ) by replacing the v ( ) (·)– ( ) function in its expression by Γi2 (·); then, the new v ( ) (·) function which appears
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( )
is replaced by Γi3 (·) and so forth till the last one, which is not changed. Instead of an index ij may appear the symbol 3 + 8 or 4 + 5, it means that, at that stage ( ) ( ) of the iteration, we replace the v ( ) (·)–function by Γ3 (·) + Γ8 (·), or respectively ( ) ( ) by Γ4 (·) + Γ5 (·). We will also consider i1 = 9, in which case there is a product of two v– functions. Then, ηn (9|i2 , . . . , ik ; i2 , . . . , ih ) is defined by doing the branching i2 , . . . , ik (with the previous rules) for the first v function and the branching i2 , . . . , ih for the second one (the second branching may be absent). We define σn := ηn (2) + ηn (4 + 5, 2) + ηn (6, 2) + ηn (9|2; 2)
(6.1)
which will be studied in the next section where we will bound its sum over n using probabilistic arguments, σn is in fact a truly stochastic term. The difference between ηn and σn is negligible, this being the main result in this section. Let (3) ω ∈ Ω : sup An,k ηk (2) ≤ λ2−ζ T (6.2) Bλ,τ := n≤nλ (τ ) k
where ηk (2) and An,k are defined in (5.10) and (5.11) respectively. We will prove in Appendix B that for each τ, ζ, q > 0 there is C = C(τ, ζ, q) so that (3) (4) (6.4) P Bλ,τ ∩ Bλ,τ ≥ 1 − Cλq . Finally define xn,∗ := sup |xn |, k≤n
x∗ (τ ) = xnλ (τ ),∗ .
(6.5)
Proposition 6.2 Recalling (4.37), let (1,2)
(3)
(4)
Bλ,τ := Bλ,τ ∩ Bλ,τ ∩ Bλ,τ .
(6.6)
Then for each τ > 0, there is a constant C = C(τ ) so that for any λ > 0 and ζ small enough |ηn − σn | ≤ Cλ2 T 3/4 (1 + x2n+1,∗ )
on the set Bλ,τ .
(6.7)
Proof. We will call negligible a term which is bounded by the r.h.s. of (6.7). We will next examine one by one all the terms which contribute to ηn − σn and show that they are all negligible, thus proving Proposition 6.2.
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Step 1. The terms ηn (1) and ηn (10). We have that ηn (1) = 0; indeed by (3.8), (n) (n) since gt is self–adjoint and gt m ¯ xn = m ¯ xn , 3 3 (n) ¯ , g v (n) (Tn ) = − m ¯ , v (n) (Tn ) = 0 ηn (1) = − m 4 xn T 4 xn
(6.8)
by (3.2). To show ηn (10) is negligible we use the bound (4.22); since we are in the set Bλ,τ , we get, from (4.7) (n)
sup t∈[Tn ,Tn+1 ]
Γ10 (t)∞ ≤ CT 5/2 λ3−3ζ
hence
|ηn (10)| ≤ CT 5/2 λ3−3ζ . (6.9)
Step 2. The term ηn (7). By (5.3) and (4.51), (n)
Γ7 (t) = I1 + I2 + I3 + I4
(6.10)
where t n λ (n) 2 I1 := − (xk − xk−1 ) ds gt−s ps−Tk m ¯ xk−1 2 T n k=1 n " (xk − xk−1 )3 t (n) ds gt−s ps−Tk m ¯ I2 := −λ xk−1 6 T n k=1 # t (n) + (xk − xk−1 )4 ds gt−s ps−Tk ak−1 n
I3 := −
Tn
9λ m ¯ xk−1 , v (k−1) (Tk )m ¯ xk−1 , v (k−1) (Tk ) 16 k=1
I4 := λ
n t k=1
Tn
(6.11)
(6.12)
t Tn
(n)
ds gt−s ps−Tk m ¯ xk−1 (6.13)
(n) ¯ xk−1 R(v (k−1) (Tk )) . ds gt−s ps−Tk m
(6.14)
To bound I1 we first consider the term with k = n. We write, for t > Tn + 1, t t Tn +1 (n) (n) ds gt−s ps−Tn m ¯ xn−1 = + ¯ xn−1 . ds gt−s ps−Tn m Tn
Tn
Tn +1
The first integral is bounded by a constant. In the second integral, as well as in the integrals in I1 when k < n, we write ps−Tk m ¯ xk−1 = 2
∂ps−Tk m ¯ xk−1 ∂s
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and then use (4.17). We thus get recalling the definition (4.3) of δ ∗ (τ ), and using (4.10), ∗ 2 n δ (τ ) 1 ∗ 2 √ ≤C I1 ∞ ≤ Cλδ (τ ) log T + λ3 T | log λ| . n−k+1 λ T k=1
−1/2 Since m ¯ ¯ and as ak−1 is bounded, using xk−1 ∈ L1 (R, dx), pt m xk−1 ∞ ≤ Ct again (4.10) n t n ∗ 3 −1/2 ∗ 4 ds (s − Tk ) + Cλδ (τ ) (t − Tn ) I2 ∞ ≤ Cλδ (τ )
≤ C
k=1
∗
δ (τ ) √ λ T
3
Tn
λ3 T 3/2 + C
δ ∗ (τ ) √ λ T
4
k=1
λ3 T 2 .
Recalling the definition (4.2) of V∗ (τ ) and the definition (4.36) of V∗⊥ (τ ), we get √ V∗ (τ )V∗⊥ (τ ) √ I3 ∞ ≤ CλV∗ (τ )V∗⊥ (τ ) T nλ (τ ) ≤ C λ2 T . 2 λ T Finally, recalling (2.5), I4 ∞ ≤ Cλ T nλ (τ )V∗ (τ )3 ≤ C
V∗ (τ ) √ λ T
3
λ3 T 3/2 .
√ Since we are √restricting our considerations to the set where δ ∗ (τ ) ≤ λ1−ζ T , V∗ (τ ) ≤ λ1−ζ T and V∗⊥ (τ ) ≤ λ1−ζ , we get √ √ (n) Γ7 (t)∞ ≤ Cλ2−2ζ T hence |ηn (7)| ≤ Cλ2−2ζ T . (6.15) sup t∈[Tn ,Tn+1 ]
Step 3. The term ηn (8). In order to show ηn (8) is negligible (as specified at the beginning of the present proof) we cannot use directly the a priori bounds, but we use equation (5.6) to get ηn (8) =
10
ηn (8, i) + ηn (8, 3 + 8)
(6.16)
i=1 i =3,8
and show each of the terms on the r.h.s. above are negligible. We have n−1 (1) ηn (8) = An,k ψk k=0
where, recalling 2 = 1, m ¯ , Tn+1xk 3λ (1) An,k := − ds 4 Tn
(6.17)
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m ¯ xn , ps−Tk+1 m ¯ xk
−
1 1, m ¯ xk 1k
m ¯ xn , 1
Let us define (recall that m ¯ > 0)
57
(6.18)
:= λ ds dx dy − y) − ¯ xk (y) . m 2π(s − Tk+1 ) Tn (6.19) We are going to prove that for any τ > 0 there is a constant C = C(τ ) such that for any 0 ≤ k < n ≤ nλ (τ ) 1 (1) (1) 2 √ + 1k=n−1 . (6.20) An,k ≤ CBn,k ≤ Cλ 1 + xn+1,∗ T (n − k)3/2 (1) Bn,k
Tn+1
m ¯ xn (x) ps−Tk+1 (x
1
Proof of (6.20). Let xn,k := xn − xk and define, for any θ ∈ R and t > 0, e−(y−x−θ)2 /2t − 1 √ f (t, θ) = dx dy m ¯ (x)m ¯ (y) . 2πt Then, after a change of variables in (6.19), its r.h.s. becomes equal to λ f (s − Tk+1 , xn,k ).
Tn+1 Tn
ds
The case k < n − 1. After changing variables in the time integral, we get, for k < n − 1, 1 (1) (6.21) Bn,k = λT dt f T (t + n − 1 − k), xn,k . 0
Since |e
−|ξ|
− 1| ≤ |ξ| and m ¯ (x) decays exponentially to 0 as |x| → ∞, we get f (t, θ) ≤ C
1 + θ2 t3/2
(6.22)
and, from (6.21), (1) Bn,k
≤ CλT
−1/2
[1 +
x2n,k ]
1
dt 0
1 (t + n − 1 − k)3/2
which proves (6.20) when k < n − 1. The case k = n − 1. We have, after a change of variables in the time integral, T (1) Bn,n−1 = λ dt f (t, xn,n−1 ) . 0
Using the inequality f (t, θ) ≤ 4(2πt)−1/2 when t ≤ 1 and (6.22) when t > 1, we get T 1 1 + x2n,n−1 4 (1) √ + Cλ dt dt ≤ Cλ(1 + x2n,∗ ) . Bn,n−1 ≤ λ t3/2 2πt 0 1
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The proof of (6.20) is concluded. Recall now that ηn (8, i) = −
n−1 3 (1) An,k m ¯ xk , Γki (Tk+1 ), 4
i = 1, 2, 3 + 8, 4, 5, 6, 7, 9, 10.
k=0
The term ηn (8, 1) in (6.16) vanishes by the same argument as in (6.8). Since (1) we are working in the set Bλ,τ , see (4.37), we have √ n−2 1 2 √ + 1 λ1−ζ T |ηn (8, 2)| ≤ Cλ 1 + xn+1,∗ T (n − k)3/2 k=0 ≤ Cλ2−ζ 1 + x2n+1,∗
(6.23)
which is negligible for ζ small. To bound ηn (8, 3 + 8) use (6.17), (5.2), and (5.7) to write ηn (8, 3 + 8) = λ
n−1 k=0
(1) An,k
k−1
Bk,h ψh ,
Bk,h :=
h=0
Tk+1
Tk
ds m ¯ xk , ps−Th+1 m ¯ xh . √
(6.24)
We note that, by (5.7), |ψ|n ≤ CVn,∗ ; it is also easy to verify Bk,h ≤ C T (k − h)−1/2 . Plugging this bounds, together with (6.20), into (6.24), we get √ n−1 |ηn (8, 3 + 8)| ≤ Cλ2 T Vn,∗ 1 + x2n+1,∗ k=0
×
n−1
k−1
1 √ ≤ CλVn,∗ 1 + x2n+1,∗ k−h h=0 k=0
1 √ + 1k=n−1 T (n − k)3/2
1 √ + 1k=n−1 T (n − k)3/2 ≤ CλVn,∗ 1 + x2n+1,∗ (6.25)
which shows ηn (8, 3 + 8) is negligible. It remains to bound ηn (8, i), i = 4, 5, 6, 7, 9, 10. The negligibility of those terms follows directly from (6.17), (6.20), and the bounds (4.18)–(4.22), (6.15). We thus conclude √ |ηn (8)| ≤ Cλ2−ζ T 1 + x2n+1,∗ . (6.26) Step 4. The term ηn (4 + 5). We claim ηn (4 + 5) =
3λ 4
Tn+1
Tn
ds m ¯ xn , pTn+1 −s v (n) (s)
(6.27)
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Indeed s 3λ Tn+1 ∂ps−s (n) ηn (4) = − ds ds m ¯ xn , v (s ) 4 Tn ∂s Tn 3λ Tn+1 Tn+1 ∂ps−s (n) = v (s ) ds ds m ¯ xn , 4 Tn ∂s s 3λ Tn+1 = ds m ¯ xn , pTn+1 −s v (n) (s ) − ηn (5) 4 Tn which proves (6.27). In order to bound ηn (4 + 5), we use equation (5.6) to get ηn (4 + 5) =
ηn (4 + 5, 2) + ηn (4 + 5, 3, 2) + ηn (4 + 5, 3, 3) 10 + [ηn (4 + 5, i) + ηn (4 + 5, 3, i)] .
(6.28)
i=1 i =2,3
The term ηn (4 + 5, 2) is in σn . Postponing the analysis of ηn (4 + 5, 3, 2) let us first show that ηn (4 + 5, i), i = 2, 3 are negligible. The terms ηn (4 + 5, i), i = 2, 3. We have ηn (4 + 5, 1) =
3λ 4
Tn+1
Tn
(n)
ds m ¯ xn , pTn+1 −s gs−Tn v (n) (Tn )
¯ xn , see (3.2), we can apply (4.11) to deduce that since v (n) (Tn ) is orthogonal to m |ηn (4 + 5, 1)| ≤ Cλ
Tn+1
ds e−α(s−Tn ) λ1−2ζ
Tn
having used that v (n) (Tn )∞ ≤ Cλ1−2ζ , see (4.47). Thus |ηn (4 + 5, 1)| ≤ Cλ2−2ζ which for ζ small is negligible. We next show ηn (4 + 5, 8) is negligible. We have n−1
ηn (4 + 5, 8) =
3 2 λ ψk 4 k=0
$ ×
(n) m ¯ xn , pTn+1 −t gt−s
Tn+1
t
dt Tn
¯ xk ps−Tk+1 m
ds Tn
2 − 2π(s − Tk+1 ) (n)
% .
(6.29)
We now use the decomposition (4.31) for gt−s above. Recalling (6.19), the (n) term obtained by replacing gt−s (x, y) by (3/4)m ¯ xn (x)m ¯ xn (y) in (6.29) is bounded
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by Cλ
n−1 k=0
(1) Bn,k |ψk |
×
Tn+1
Tn
n−1 k=0
√ dt m ¯ xn , pTn+1 −t m ¯ xn ≤ C T λ2 Vn,∗ 1 + x2n+1,∗
1 √ + 1k=n−1 T (n − k)3/2
√ ≤ C T λ2 Vn,∗ 1 + x2n+1,∗ (n)
where we used (6.20). We next consider the case when we replace gt−s in (6.29) (n,⊥) by gt−s . Since, by (4.14) and (4.32), (n,⊥) ¯ xk ≤ Ce−α(t−s) ps−tk+1 m ¯ xk ∞ pTn+1 −t gt−s ps−Tk+1 m ∞
1 ≤ Ce−α(t−s) s − Tk+1 (n)
(n,⊥)
we can bound (6.29) (with gt−s replaced by gt−s ) as n−1
Tn+1
t
1 ds e−α(t−s) s − Tk+1 Tn Tn k=0 n−1 √ √ 1 ≤ C T λ2 Vn,∗ ≤ C λ3−ζ T + λ2−ζ T 1k=n−1 + √ n−k k=0 Cλ2
|ψk |
dt
which is negligible for ζ small enough. It remains to show ηn (4 + 5, i), i = 4, 5, 6, 7, 9, 10 is negligible. This follows from the bounds (4.18)–(4.22), (6.15), and (6.27). The terms ηn (4 + 5, 3, i). We write ηn (4 + 5, 3) =
n−1 k=0
(2)
An,k ψk
(6.30)
where (2) An,k
3λ2 = 4
Tn+1
ds Tn
$
Tn+1
ds s
(n) m ¯ xn , pTn+1 −s gs −s
2
2π(s − Tk+1 )
% 1k
By using the bounds (4.14) and (4.10), it is easy to show λ2 T 3/2 (2) 1k
(6.32)
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From (6.30) we thus get ηn (4 + 5, 3, i) =
n−1 k=0
(2)
An,k ηk (i) .
(6.33)
The first one, ηn (4 + 5, 3, 1), vanishes, by (6.8). By using the bounds (4.18)–(4.22), (6.15), (6.26), and (6.32) it is immediate to verify ηn (4 + 5, 3, i) is negligible for i = 2, 3 if ζ is chosen small enough. To bound η(4 + 5, 3, 3) we note that, by (5.12) η(4 + 5, 3, 3) = −A(2) Aψ with A = ((An,k )) as in (5.11). Hence, recalling (5.16), η(4 + 5, 3, 3) = −A(2) (1 + A)−1 Aη .
(6.34)
By (6.32), it is easy to show sup A(2) n ≤ CλT
(6.35)
n≤nλ (τ )
so that, by the bound in (5.16), |ηn (4 + 5, 3, 3)| ≤ CλT
i =3
max A ,k ηk (i) . ≤n
(6.36)
k<
By definitions (6.2) and (6.6) the term with i = 2 in the r.h.s. of (6.47) is therefore negligible. For i = 1, 7, 8, 10 (resp. i = 4, 5, 6, 9) we can use the bounds already proved (6.8), (6.15), (6.26), and (6.9) (resp. the a priori bounds (4.18)–(4.21)) together with An ≤ C to conclude that ηn (4 + 5, 3, 3) is also negligible. We are thus left with the term ηn (4 + 5, 3, 2). Recalling (6.33), we write it as ηn (4 + 5, 3, 2) =
n−1 k=0
(2,L)
An,k ηk (2) +
n−1 k=0
(2,R)
An,k ηk (2)
(6.37)
where Tn+1 Tn+1 2 3λ2 (n) ds ds m ¯ xn , pTn+1 −s gs −s 11k
An,k
:=
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Since, by (4.10) Tn+1 Tn+1 (n) ds ds m ¯ xn , pTn+1 −s gs −s 1 ≤ CT 2 Tn s n−1 (2,L) An,k ηk (2) ≤ CλT
we have
k=0
√ n−1 λ T √ ηk (2) n−k
(6.39)
k=0
(4)
which is negligible because we are in the set Bλ,τ , recall (6.3). On the other hand, by using again (4.10) we get " # 1 (2,R) (6.40) An,k ≤ Cλ2 T 3/2 1k=n−1 + (n − k − 1)3/2 whence, recalling (4.5) n−1 n−2 1 (2,R) 3−ζ 2 An,k ηk (2) ≤ Cλ T 1+ (n − k − 1)3/2 k=0
(6.41)
k=0
which is negligible. Step 5. The term ηn (6). To study this term we are going to use the same strategy as for ηn (4 + 5). The explicit expression of ηn (6) is Tk+1 n−1 3λ Tn+1 ∂ps−s (k) v (s ) ds ds m ¯ xn , ηn (6) = 4 ∂s Tn Tk
(6.42)
k=0
which we decompose as we did for ηn (4 + 5), i.e. as in (6.28) with 4 + 5 replaced by 6. The terms ηn (6, i), i = 2, 3. We have ηn (6, 1) =
Tk+1 n−1 3λ Tn+1 ∂ps−s (k) gs −Tk v (k) (Tk ) . ds ds m ¯ xn , 4 ∂s Tn Tk
(6.43)
k=0
√ (k) Since, by (3.2) and(4.7) gs −Tk v (k) (Tk )∞ ≤ Ce−α(s −Tk ) λ1−ζ T , by (4.17) the double integral above for k = n − 1 is less or equal than √ Tn+1 Tn−1 +T /2 Ce−α(s −Tn−1 ) λ1−ζ T ds ds s − [Tn−1 + T /2] Tn−1 Tn √ Tn+1 Tn −αT /2 1−ζ √ λ T Ce + ds ds ≤ Cλ1−ζ T . s−s Tn Tn−1 +T /2
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On the other hand,√for k < n − 1 the double integral in (6.43) is bounded by C(n − k + 1)−1 λ1−ζ T , so that √ |ηn (6, 1)| ≤ Cλ2−ζ T | log λ| (6.44) which is therefore negligible. We next consider ηn (6, 8); its explicit expression is Tn+1 Tk+1 s n−1 k−1 3λ2 ψh dt ds ds ηn (6, 8) = 4 T T T n k k k=0 h=0 $ % ∂pt−s (k) 2 × m ¯ xn , ¯ xh − g ps −Th+1 m . ∂t s−s 2π(s − Th+1 )
(6.45)
(n)
We now use the decomposition (4.31) for gt−s above. Since ( ) C ∂pt−s m ¯ xk ≤ ¯ xn , ∂t m (t − s)3/2 (k)
and recalling (6.19), the term obtained replacing gs−s (x, y) in (6.45) by ¯ xk (y) can be bounded by (3/4)m ¯ xk (x)m CλVn,∗
n−1 k−1 k=0 h=0
√ T (1) B ≤ Cλ3−ζ T [1 + xn+1,∗ ] (n − k)3/2 k,h
(6.46)
(k)
which is negligible. We next consider the case when gs−s in (6.45) is replaced by
(k,⊥) gs−s .
By (4.17), (4.32), and (4.14) we have ∂p 1 2 e−α(s−s ) t−s (k,⊥) gs−s ps −Th+1 m ¯ xh − ≤C ∂t t−s s − Th+1 2π(s − Th+1 ) ∞ (k)
(k,⊥)
hence the r.h.s. of (6.45) with gt−s replaced by gt−s
can be bounded by
n−1 k−1 √ Cλ2 T Vn,∗
√ 1 √ ≤ Cλ2−ζ | log λ| T (n − k) k − h k=0 h=0
which is negligible. It remains to consider the terms ηn (6, i), i ≥ 4, i = 8. We have |ηn (6, i)| ≤ Cλ| log λ|T max
sup
k≤n t∈[Tk ,Tk+1 ]
(k)
Γi (t)∞
(6.47)
which, together with (4.18) for i = 4, (4.19) for i = 5, (4.20) for i = 6, (6.15) for i = 7, (4.21) for i = 9, (6.9) for i = 10, shows that the terms ηn (6, i), i ≥ 4, i = 8 are all negligible.
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The terms ηn (6, 3, i). The term ηn (6, 3, 1) = 0. Analogously to (6.30), we write n−1
ηn (6, 3) =
k=0
(3)
An,k ψk
(6.48)
where (3)
An,k =
s Tn+1 Th n 3λ2 dt ds ds 4 Th−1 Th−1 h=k+2 Tn $ % 2 ∂pt−s (h−1) g × m ¯ xn , 1k
(6.49)
By (4.17), (4.10), and (4.14) it is easy to show λ2 T 3/2 log(n − k) (3) √ 1k
(6.50) (2)
Except for the factor log(n − k) ≤ C| log λ|, An,k has the same behaviour as An,k , compare (6.32) with (6.50). The analysis in Step 4 applies and we can therefore conclude that all the terms ηn (6, 3, i), i = 2, are negligible. We are left with ηn (6, 3, 2). As in (6.38) we define s Tn+1 Th n 2 3λ2 dt ds ds 4 2π(T − T ) T T T h+1 k+1 n h−1 h−1 h=k+2 ( ) ∂pt−s (h−1) gs−s 1 1k
(3,L)
An,k :=
(3,R) An,k
s Tn+1 Th n 3λ2 := dt ds ds 4 Th−1 Tn Th−1 h=k+2
2
(
2
− 2π(s − Tk+1 ) 2π(Th+1 − Tk+1 )
m ¯ xn ,
) ∂pt−s (h−1) gs−s 1 1k
Setting 3λ (2) Bn,h := √ 2 2πT we have
Tn+1
Tn
Th
dt
s
ds Th−1
Th−1
ds
) ( ∂pt−s (h−1) gs−s 1 m ¯ xn , ∂t
√ n−1 n h−2 (3,L) λ T (2) √ An,j ηj (2) ≤ ηj (2) Bn,h j=0 h=2 j=0 h − j
(6.52)
(6.53)
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and, using (4.10) and (4.17), λT (2) . Bn,h ≤ C n−h (4)
Thus, since we are in the set Bλ,τ , see (6.3), the l.h.s. of (6.53) is negligible. On the other hand, by using again (4.10) and (4.17) it is easy to show 1 (3,R) An,k ≤ Cλ2 | log λ| T 3/2 (n − k)3/2 n−1 n−1 (3,R) 1 3−ζ 2 A η (2) ≤ Cλ | log λ| T j n,j (n − j)3/2 j=0 j=0
hence
(6.54)
which is negligible. Step 6. The term ηn (9). The iterative scheme to estimate η(9) is the following. We first branch, using (5.6), one of the v (n) (s) which appears on the r.h.s. of (3.14) and stop the iteration for the terms η(9|i), i = 2, 3. In η(9|3) we use again the equation (5.6) and stop the iteration. We then branch the other v (n) (s) on the term η(9|2) stopping the iteration with the same rule as before. We thus get η(9)
= η(9|2; 2) + η(9|2; 3, 2) + η(9|2; 3, 3) + η(9|3, 2) + η(9|3, 3) [η(9|i) + η(9|2; i) + η(9|2; 3, i) + η(9|3, i)] . (6.55) + i =2,3
We need to show that η(9) − η(9|2; 2) is negligible. We start from η(9|i), i = 2, 3. We have 9 Tn+1 (n) ηn (9|i) = dt m ¯ xn , m ¯ xn v (n) (t)Γi (t) . (6.56) 4 Tn (n)
For i = 1, by (3.2), gt−Tn v (n) (Tn )∞ ≤ Ce−α(t−Tn ) v (n) (Tn )∞ , by (4.47) v (n) (Tn )∞ ≤ λ1−ζ , hence √ |ηn (9|1)| ≤ Cλ1−ζ T λ1−ζ (6.57) (1,2)
so that it is negligible. On the set Bλ,τ , see (4.37) and (6.6), we can use the bounds (4.7), (4.18), (4.19), (4.20), (6.15), (4.21), and (6.9), which show η(9|i) is negligible for i = 4, 5, 6, 7, 9, 10. To bound η(9|8) we note that, by (5.5), (4.7), (n) ¯ (x) > 0, and gt (x, y) ≥ 0), we have (using also m ¯ ∞ = 1, m |ηn (9|8)| ≤ λCVn,∗
n−1 k=0
|ψk |
Tn+1
dt Tn
t
1 ds s − Tk+1 Tn
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2 (n) ¯ xk (z) . dx dy dz m ¯ xn (x)gt−s (x, y) e−(y−z) /[2(s−Tk+1 )] − 1 m
× (n)
Since gt is self–adjoint in L2 (R, dx) we have Then, from (6.19) and (6.20) it follows that |ηn (9|8)| ≤ CV∗ (τ )T
n−1 k=0
(6.58)
(n)
dx m ¯ xn (x)gt−s (x, y) = m ¯ xn (y).
(1) Bn,k |ψk | ≤ CλV∗ (τ )2 T 1 + x2n+1,∗
which is negligible because of (4.7). We next consider η(9|3, i), i = 2, 3, Tn+1 t n−1 9λ ηn (9|3, i) = ηk (i) dt ds 4 Tn Tn
$
2
(n) m ¯ xn , m ¯ xn v (n) (s)gt−s
%
. 2π(s − Tk+1 ) (6.59) Note that, from (6.8) η(9|3, 1) = 0 and that, for i ≥ 4, from (5.11), k=0
n−1
|ηn (9|3, i)| ≤ CλVn,∗
k=0
|ηk (i)|
Tn+1
dt Tn
t
1 ds s − Tk+1 Tn
≤ T CVn,∗
n−1
An,k |ηk (i)| .
(6.60)
k=0
Since An ≤ C, by using again (4.7), (4.18), (4.19), and (4.20)–(6.9), we show η(9|3, i), i ≥ 4 is negligible. Next, using (5.12), (5.16) and (6.59), Tn+1 t n−1 9λ −1 (1 + A) Aη k dt ds η(9|3, 3) = − 4 Tn Tn k=0
$ ×
2 2π(s − Tk+1 )
%
(n) m ¯ xn , m ¯ xn v (n) (s)gt−s
(6.61)
whence, recalling (1 + A)−1 n ≤ C, An ≤ C, ηn (1) = 0,
√ 10 n−1 T √ |η(i)|n Vn,∗ |η(9|3, 3)|n ≤ CλT sup |(Aη(2))k | + n −k k≤n i=4 k=0 2−ζ ≤ CT λ T + λ2−2ζ T 2 1 + x2n+1,∗ Vn,∗ where we used the bounds (6.8), (6.15), (6.26), (6.9), and (4.18)–(4.21) to estimate |η(i)| together with the fact that we are on the set Bλ,τ (see (6.2) and (6.6)). Hence η(9|3, 3) is negligible. (n) Since on the set Bλ,τ Γ2 (t) has the same order as v (n) (t) (see (4.5)) we can bound η(9|2; i), η(9|2; 3, i), i = 2, 3, and η(9|2; 3, 3) as above.
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We are left with the bounds of η(9|3, 2) and η(9|2; 3, 2). We have ηn (9|3, 2) = (L) (R) ηn (9|3, 2) + ηn (9|3, 2) where ηn(L) (9|3, 2) :=
n−1 2 9λ ηk (2) 4 2π(T n+1 − Tk+1 ) k=0
Tn+1
t
dt Tn
Tn
ηn(R) (9|3, 2) := ×
Tn+1
Tn
& ' (n) ds m ¯ xn , m ¯ xn v (n) (s)gt−s 1
n−1 9λ ηk (2) 4 k=0 t
2 2 − dt ds 2π(s − Tk+1 ) 2π(Tn+1 − Tk+1 ) Tn & ' (n) × m ¯ xn , m ¯ xn v (n) (s)gt−s 1 .
(6.62)
Since, by (4.7) and (4.10), Tn+1 t & ' (n) dt ds m ¯ xn , m ¯ xn v (n) (s)gt−s 1 ≤ Cλ1−ζ T 5/2 Tn Tn √ n−1 λ T (L) 1−ζ 3/2 √ ηk (2) ηn (9|3, 2) ≤ Cλ T n−k
we have
(6.63)
k=0
(4)
which is negligible since we are in the set Bλ,τ , see (6.3). On the other hand, again by (4.7) and (4.10), Tn+1 t & ' 2 2 (n) m ¯ xn , m dt ds ¯ xn v (n) (s)gt−s 1 − Tn 2π(s − Tk+1 ) 2π(Tn+1 − Tk+1 ) Tn " # 1 ≤ Cλ1−ζ T 2 1k=n−1 + (n − k − 1)3/2 whence, by (4.13) and (4.5) n−2 (R) 2−ζ 2 ηn (9|3, 2) ≤ Cλ T k=0
1 |ηk (2)| + |ηn−1 (2)| ≤ λ3−ζ T 5/2 (n − k − 1)3/2 (6.64)
which is negligible. (2)
Since Γn (t) has the same order as v (n) (t), it follows from (6.56) that the term η(9|2; 3, 2) can be analyzed as η(9|3, 2). Proposition 6.2 is thus proved.
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7 Bounds on the stochastic terms The stochastic terms in the title are the ones left from the previous section, i.e. σn , see (6.1). As already mentioned we will obtain bounds for the sums of the components of these vectors rather than for the components themselves. We define Sn (-)
:=
n−1
- ∈ L := {2, (4 + 5, 2), (6, 2), (9|2; 2)}
ηk (-),
k=0
Sn
:=
Sn (-) =
∈L
n−1
σk
(7.1)
k=0
and call S the vector whose components are Sn . Analogously ξ is the vector whose components are ξn , see (5.8). Proposition 7.1 Recall that A is the matrix whose entries are defined in (5.11). Then ξ = S − AS + R + A2 ξ on the set Bλ,τ (7.2) where Rn :=
n−1
(ηk − σk ) −
k=0
n−1 k=0
(A[η − σ])k .
(7.3)
Moreover, for each τ > 0 there is a constant C = C(τ ) such that for any λ and ζ small enough on the set Bλ,τ . (7.4) |R|n ≤ CT −1/4 1 + x2n+1,∗ The proof is based on the following simple lemma. Lemma 7.2 Let M be a lower triangular matrix such that Mn,k = Mn ,k whenever n−1 n −k = n−k. Let also u = {uh ; h = 0, . . . , n} be a vector and define vn = h=0 uh (resp. v0 = 0). Then n−1 (M u)h = (M v)n . (7.5) h=0
Proof. The l.h.s. of (7.5) can be written as n−1 h−1
Mh,k (vk+1 − vk ) =
h=0 k=0
n−1 h−1
Mh,k vk+1 −
h=0 k=0
n−1 h−1
Mh,k vk .
h=0 k=0
We call k = k + 1 in the first sum and h = h − 1 in the second one, getting h
n−1
n−1
h=0
h=0 k =1
(M u)h =
Mh,k −1 vk −
n−2
h
h =−1 k=0
Mh +1,k vk
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We may as well extend the first sum from k = 0 because v0 = 0. We then get n−1
n−1
h=0
k=1
(M u)h =
Mn−1,k−1 vk +
h n−2
[Mh,k−1 − Mh+1,k ] vk .
h=0 k=0
Note that the second term above vanishes by the assumptions on M , and this completes the proof of (7.5). Proof of Proposition 7.1. From (5.8) and (5.17), ξn =
n−1
[η − Aη + A2 ψ]k .
k=0
Next, apply Lemma 7.2 for A and A2 . From (7.1) and (5.8), we get (7.2). Since An ≤ C, the bound (7.4) follows from Proposition 6.2. Our next task is to control the stochastic term Sn . Proposition 7.3 For any τ > 0 we have lim lim P
L→∞ λ↓0
sup |Sn (2)| > L
=0
(7.6)
- ∈ L \ {2} .
(7.7)
n≤nλ (τ )
|Sn (-)| lim P sup > T −1/4 λ↓0 1 + x n,∗ n≤nλ (τ )
= 0,
Proof of (7.6). By (3.9), (5.10), and (7.1), n−1 3λ m ¯ xk , z (k) (Tk+1 ) Sn (2) = − 4
(7.8)
k=0
which is an FTn –martingale. In Appendix B it is proved there is a C so that, for any n ≤ nλ (τ ) and t ∈ [Tn + 1, Tn+1 ], 2p sup E z (n) (t, x) p = 1, 2 . (7.9) ≤ C(t − Tn )p , x∈R
Then, by Doob’s martingale inequality, we get, for some constant C,
Cnλ (τ )λ2 T 1 P sup |Sn (2)| > L ≤ 2 E Snλ (τ ) (2)2 ≤ L L2 n≤nλ (τ ) which proves (7.6).
(7.10)
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Proof of (7.7). The case - = (4 + 5, 2). We have (recall (6.27)) Sn (4 + 5, 2) =
n−1 3λ2 Th+1 ds m ¯ xh , pTh+1 −s z (h) (s) . 4 Th h=0
This is again a FTn –martingale, so that, using again Doob’s inequality, and the bound 2 Th+1 E ds m ¯ xh , pTh+1 −s z (h) (s) ≤ CT 3 Th (which follows from (7.9)) we conclude that for each ζ > 0 there is a constant C = C(τ, ζ) such that
√ −1/4 P sup |Sn (4 + 5, 2)| > T (7.11) ≤ Cnλ (τ )λ4 T 3 T ≤ Cλ2 T 5/2 . n≤nλ (τ )
The case - = (6, 2). By (3.13) and (3.9) ) ( Tk n−1 h 3λ2 Th+1 ∂ps−s (k−1) z ds ds (s ) . Sn (6, 2) = m ¯ xh , 4 ∂s Th Tk−1
(7.12)
h=1 k=1
For each h and k above, we write, recalling that m ¯ ξ (x) = m(x ¯ − ξ), xh m ¯ xh (x) = m ¯ xk−1 (x) − dξ m ¯ ξ (x) xk−1
thus obtaining, by exchanging the sums in (7.12), Sn (6, 2) = −S˜n (6, 2) +
n−1
Xn,k
(7.13)
k=1
where Tk xh n−1 h 3λ2 Th+1 ∂ps−s (k−1) ˜ z Sn (6, 2) := ds ds dξ m ¯ ξ , (s ) (7.14) 4 ∂s Tk−1 xk−1 h=1 k=1 Th Tk 3λ2 Tn ∂ps−s (k−1) Xn,k := z ds ds m ¯ xk−1 , (s ) . (7.15) 4 Tk ∂s Tk−1 We define (n) Mk
:=
k−1 h=1
Xn,h ,
k = 1, . . . , n
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(n)
71
(n)
with M1 = 0. For each fixed n, {Mk ; k = 1, . . . , n} is an FTk−1 –martingale (n) since E(Xn,h |FTh−1 ) = 0. The quadratic variation of Mk is k 2 k−1 (n) (n) 2 Xn,h . Mh − Mh−1 = M (n) = k
h=2
h=1
We have Tk Tn Tk s ∧u 9λ4 Tn E = ds ds du du ds dx dx dy 16 Tk Tk−1 Tk Tk−1 Tk−1 ∂ps−s ∂pu−u × m ¯ (x) m ¯ (x ) gs −s (x, y) gu −s (x , y)a(λβ (y + xk−1 ))2 ∂s ∂u Tn Tk Tn Tk s ∧u 2 4 ≤ Ca∞ λ ds ds du du ds Tk Tk−1 Tk Tk−1 Tk−1 ∂pu−u ∂ps−s m ¯ m ¯ × gs −s gu −s 1∞ ∂s ∂u ∞ 1 Tn Tk Tn Tk s ∧u 1 1 2 4 ≤ Ca∞ λ ds ds du du ds )3/2 u − u (s − s Tk Tk−1 Tk Tk−1 Tk−1
2 FTk−1 Xn,k
≤ Ca2∞ λ4 | log λ|T 5/2 . Since, conditionally on FTk−1 , z k−1 is Gaussian we also have 4 FT E Xn,k ≤ Ca4∞ λ8 | log λ|2 T 5 . k−1 By the BDG inequality (see [16, §6, E. 4.1]) and the above bounds it follows that
2 n−1 4 2 (n) (n) 2 Mn E Xn,h ≤ CE M =CE ≤ C
n−1
n
4 EXn,h
h=1 2 8
h=1
+2
n−1 k−1
E
2 Xn,h E
2 FTk−1 Xn,k
k=1 h=1
≤ C n λ | log λ|2 T 5 .
(7.16)
We next use the following corollary of the Chebyshev inequality, P sup |ϑn | > δ ≤ P |ϑn |p > δ p ≤ δ −p E (|ϑn |p ) n≤N
n≤N
(7.17)
n≤N
and get, by (7.16), choosing p = 4, n−1
nλ (τ ) −1/4 P sup Xn,k > T Cn2 λ8 | log λ|2 T 5 ≤ Cλ2 T 4 . ≤T n≤nλ (τ ) k=1
n=1
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It remains to bound S˜n (6, 2). We split it into two terms by using the decomposition (4.33) for z (k−1) . We get Tk xh n−1 h 3λ2 Th+1 ∂ps−s (k−1,⊥) S˜n⊥ (6, 2) := z ds ds dξ m ¯ ξ , (s ) 4 ∂s Tk−1 xk−1 h=1 k=1 Th Tk n−1 h 3λ2 Th+1 S˜n (6, 2) := ds ds 4 Tk−1 h=1 k=1 Th xh ∂ps−s m ¯ xk−1 w(k−1) (s ) × dξ m ¯ ξ , (7.18) ∂s xk−1 where we set w(k) (t) := 34 m ¯ xk , z (k) (t). In Appendix A we prove for each γ ∈ (0, 1/2) there exists a constant C such that ( ) ϕ1 ∂pt ≤ C m ϕ , ∧ ϕ . (7.19) ¯ ∞ t3/2 ∂t tγ We thus get |S˜n⊥ (6, 2)| ≤ Cλ2 n≤nλ (τ ) 1 + xn,∗ sup
≤ Cλ2 ≤ CT
nλ (τ )−1 h Th+1
Tk
ds
h=1
k=1
nλ (τ )−1 h
h=1 −1/2
k=1
sup
Th
ds
Tk−1
√ T (h − k + 1)3/2 sup
n≤nλ (τ ) t∈[Tk−1 ,Tk ]
z
1 z (k−1,⊥) (s )∞ (s − s )3/2 sup
t∈[Tk−1 ,Tk ]
(k−1,⊥)
z (k−1,⊥) (t)∞
(t)∞ .
(7.20)
On the other hand, for each γ ∈ (0, 1/2), since m ¯ xk−1 1 ≤ C, we also have
|S˜n (6, 2)| ≤ Cλ2 n≤nλ (τ ) 1 + xn,∗ sup
≤ Cλ2
nλ (τ )−1 h Th+1 h=1
k=1
nλ (τ )−1 h h=1
≤ C T −γ
k=1
sup
Tk
ds
Th
ds
Tk−1
T 1/2−γ (h − k + 1)3/2+γ sup
n≤nλ (τ ) t∈[Tk−1 ,Tk ]
|w
1 |w(k−1) (s )| (s − s )3/2+γ sup
t∈[Tk−1 ,Tk ]
|w(k−1) (t)|
(k)
(t)| √ . T
(7.21)
Since |w(k) (t)| ≤ Cz (k−1) (t)∞ by using the Gaussian estimate (4.6) (resp. (4.35)) and (7.21) (resp. in (7.20)), the bound in (7.7) for S˜n (6, 2) now follows. n−1 The case - = (9|2; 2). We have Sn (9|2; 2) = k=0 ηk (9|2; 2) where 9λ2 Tk+1 ηk (9|2; 2) = dt m ¯ xk , m ¯ xk z (k) (t)2 . (7.22) 4 Tk
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We define γk := E ( ηk (9|2; 2)| FTk ) and then decompose Sn (9|2; 2) = Mn +
n−1
γk
(7.23)
(7.24)
k=0
where Mn is an FTn –martingale with bracket M n =
n−1
E ηk (9|2; 2)2 FTk − γk2 .
(7.25)
k=0
Note that the difference between the bracket ·n and the quadratic variation [·]n is a martingale, see [16]. We have t β 9λ2 Tk+1 2 2 dt ds dx dy m ¯ (x)m(x)g ¯ γk = t−s (x, y) a(λ (y + xk )) − 1 4 Tk Tk (7.26) where we exploited the identity 2 dx dy m ¯ (x)m(x)g ¯ (x, y) = dx m ¯ (x)m(x)g ¯ t−s 2(t−s) (x, x) = 0 which holds because x "→ gt (x, x) is an even function of x. We claim for each q > logλ−1 T = (1 ∧ β)/20 there is C = C(q) so that
−q |γk | ≤ C(λT )2 λβ−q + λβ xk,∗ + e−λ /C . (7.27) By Taylor expansion β a(λ (y + xk ))2 − 1 ≤ ≤
/ . C λβ (|y| + |xk |)1|y|≤λ−q + 1|y|>λ−q . / C λβ−q + λβ xk,∗ + 1|y|>λ−q .
(7.28)
We use the bound (7.28) in (7.26). There is C so that, for any t ∈ [0, T ] (T > 1), t t 2 ds dx dy m ¯ (x)|m(x)|g ¯ ds dx m ¯ (x)g2s (x, x) ≤ CT s (x, y) ≤ 0
0
the first two terms on the r.h.s. of (7.28) produce the first two terms on the r.h.s. of (7.27). We estimate next the contribution of the last one. Denoting by Ex,y,t the expectation w.r.t. a Brownian bridge from x to y in time t, by the Feynman–Kac formula we get t ds V (m(ω ¯ s )) ≤ e2t pt (x, y) (7.29) gt (x, y) = pt (x, y) Ex,y,t exp − 0
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whence ¯ (x) gt (x, y)2 1|y|>λ−q dx dy 1|x|≤λ−q /2 m −2q
≤
e−[λ
/(8t)]+2t
√ 2πt
dx
−2q
dy gt (x, y) m ¯ (x) = 2
e−[λ
/(8t)]+2t
√ 2πt
. (7.30)
On the other hand, since m ¯ (x) decays exponentially as x → ∞, −q ¯ (x)gt (x, y)2 1|y|>λ−q ≤ Ce−λ /C dx g2t (x, x) m ¯ (x) dx dy 1|x|>λ−q /2 m −q
e−λ /C+4t √ . ≤ C 4πt
(7.31)
Taking q > logλ−1 T , from (7.30) and (7.31) we find that the last term on the r.h.s. of (7.28) yields the last one on the r.h.s. of (7.27). From (7.27), for any λ small enough, n−1 1 sup γk ≤ CT λβ−q . (7.32) 1 + x n,∗ n≤nλ (τ ) k=0
By choosing q ∈ (logλ−1 T, β − logλ−1 T ) (recall we fixed logλ−1 T = (1 ∧ β)/20) it follows the l.h.s. of (7.32) vanishes as λ ↓ 0. We are left with the bound on the martingale part Mn . By Doob’s inequality, recalling (7.25),
λ (τ ) √ n √ −1/4 E ηk (9|2; 2)2 ≤ C T E M nλ (τ ) ≤ C T P sup |Mn | > T n≤nλ (τ )
≤
2 √ C T (λ2 T )−1 Cλ2 T 3/2
k=0
(7.33)
in the second estimate we used (recall (7.9)) Tk+1 0 2 2 E (ηk (9|2; 2) ) ≤ Cλ dt dx m ¯ (x) E z (k) (t, x)4 ≤ Cλ2 T 3/2 . Tk
From (7.32) and (7.33) (recall xn,∗ is increasing), (7.7) for - = (9|2; 2) follows.
8 Conclusion of the proof In this section we prove Theorem 1.1. As before we denote by (m(t), h(t)) the solution of (1.1)–(1.10) (omitting the dependence on λ). Let δ be as in Proposition 2.1 and define the stopping time ¯ z ∞ ≥ δ} . tδ := inf{t ≥ 0 : inf m(t) − m z∈R
(8.1)
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By Proposition 2.1, for each t ∈ [0, tδ ] the profile m(t) has a well defined center x(m(t)). We prove the theorem with x(λ) (t) := x(m(t ∧ tδ )) . Observe the process x(λ) (t) is adapted to Ft . Moreover, by Proposition 2.1 and the continuity properties of m(t), the process x(λ) (t) is continuous. Proof of (1.11). Given τ, ζ > 0, from Propositions 2.1, 4.1, and recalling (3.1) there is C = C(τ, ζ) so that, for any λ small enough, √ (1) sup sup |x(λ) (t) − xn | ≤ Cλ1−ζ T on the set Bλ,τ . tδ > λ−2 τ, n≤nλ (τ ) t∈[Tn ,Tn+1 ]
On the other hand, by (3.1), (4.2), and m ¯ ∞ ≤ 1, we have m(t) − m ¯ x(λ) (t) ∞ ≤ Vn + sup sup |x(λ) (t) − xn |
t∈[Tn ,Tn+1 ]
t∈[Tn ,Tn+1 ]
(8.2) ∀ n : Tn+1 ≤ tδ .
(8.3) From (4.6), (4.7), (8.2), and (8.3), by choosing ζ small enough (1.11) follows. To complete the proof of Theorem 1.1 we need to prove the weak convergence of the scaled process xλ (τ ) := x(λ) λ−2 τ . Let ξλ (τ ), τ ∈ R+ , be the (continuous) process obtained by linearly interpolating the values ξλ (λ2 Tn ) = ξn . From (4.6), (4.35), (4.39), and (8.2), for any τ and ε positive, lim P sup |ξλ (s) − xλ (s)| > ε = 0 . λ↓0
0≤s≤τ
It is therefore enough to prove the convergence of the process ξλ . To this end we shall use that ξn solves the equation (7.2). In Sections 7 and 8 we proved bounds on R and S, see (7.4) and Proposition 7.3 which hold with probability going to 1 as λ ↓ 0. They however depend on the unknown quantities xn,∗ . The a priori bounds of Section 4 yield xn,∗ ≤ nδ ∗ (τ ), which goes like λ−1 T −1/2+ζ , according to (4.7) and with n = nλ (τ ). With such a bound, our estimates on R and S become very bad and in any case inadequate to study the equation (7.2), for which we could only tolerate a bound on xn,∗ which diverges very weakly as λ ↓ 0. As mentioned, the bounds of Section 4 on δ ∗ (τ ) are quasi optimal, yet they yield a bound on xn,∗ which is far from correct. The point is that xn is the sum of the increments xk − xk−1 , whose size has indeed the order of δ ∗ (τ ); but there are a lot of cancellations, which make the absolute value of the sum much smaller than the sum of the absolute values. Such cancellations are lost using the a priori bounds of Section 4, we thus need to go back to the equation (7.2) itself (recalling that ξn and xn are essentially the same, see (4.39)). It seems, at this point, that we are back to a non linear problem, with the unknown ξn in both the “known terms” R and S hidden through xn . Such a non linearity is however not really dangerous,
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as in our bounds for R and S the dependence on xn,∗ is always multiplied by a function of λ which vanishes as λ ↓ 0. As we will see, this makes easy the proof of Proposition 8.1 below. Let (8.4) ξ ∗ (τ ) := sup |ξn | n≤nλ (τ )
and x∗ (τ ) be as in (6.5). Proposition 8.1 For any τ ∈ R+ , lim lim P (ξ ∗ (τ ) > L) = 0 ,
(8.5)
lim lim P (x∗ (τ ) > L) = 0 .
(8.6)
L→∞ λ↓0
L→∞ λ↓0
Proof. Since (8.6) follows from (8.5) and (4.39), we only need to prove (8.5). Let (recall (7.1)) Sn(R) := Sn (-) (8.7) ∈L\{2} (R)
and denote as usual by S (R) the vector whose components are Sn . Given τ > 0 and ζ > 0 small enough, let Bλ,τ be as in (6.6) and define Gλ,L := Aλ,L ∩ Bλ,τ where Aλ,L is the event Aλ,L :=
ω ∈ Ω : |S(2)|nλ (τ )
(R) S −1/4 n ≤ L ; sup ≤T . n≤nλ (τ ) 1 + xn,∗
(8.8)
(8.9)
From (4.6), (4.35), (6.6), (6.4), and Proposition 7.3 we have, for any τ > 0, lim lim P (Gλ,L ) = 1 .
L→∞ λ↓0
(8.10)
We are going to show that given L there is a constant c so that, for all λ small enough, in Gλ,L we have |ξn | ≤ c for all n ≤ nλ (τ ) + 1. Given any L1 > 0, let N (L1 , λ) be the first index n ≤ nλ (τ ) for which |ξn | ≥ L1 ; otherwise we set N (L1 , λ) = nλ (τ ) + 1. Then what we have to show is that there is an L1 so that for all λ small enough, N (L1 , λ) = nλ (τ ) + 1 in Gλ,L . Having fixed L1 , there is a λ1 so that for λ ≤ λ1 and n ≤ N (L1 , λ), by (4.39) (1,2) (recall Gλ,L ⊆ Bλ,τ ) |xn | ≤ |ξn | + 1 ≤ L1 + 1,
on the set Gλ,L .
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Using this, (7.4), and (8.9) we then get there is λ2 = λ2 (λ1 , L1 ) ∈ (0, λ1 ) such that for any λ ≤ λ2 and n ≤ N (L1 , λ), |R|n ≤ 1,
|S|n ≤ L + 1,
on the set Gλ,L .
(8.11)
Then, by (7.2) and (5.19), for n ≤ N (L1 , λ), |ξn | ≤ C(L + 1) + C
n−1
λ2 T |ξk |,
on the set Gλ,L .
(8.12)
k=0
By solving this inequality, we conclude that there is a c = c(C, L, τ ), independent of L1 , so that |ξn | ≤ c, for all n ≤ N (L1 , λ). By taking L1 ≥ c, we have N (L1 , λ) > nλ (τ ). Proposition 8.1 is proved. Setting Yn := Sn (2) −
n−1
An,k Sk (2)
(8.13)
k=0
from (7.4), (8.6), (8.10), and An ≤ C, we have, for each ε > 0,
lim P λ↓0
sup |S − AS + R − Y |n > ε
=0
(8.14)
n≤nλ (τ )
so that the analysis of the limiting behavior of the “known term” S − AS + R in (7.2) reduces to that of Y , which is the content of the following proposition. Proposition 8.2 Let Yλ (τ ), τ ∈ R+ , be the process obtained by linearly interpolating the values Yλ (λ2 Tn ) = Yn , n ∈ N. Then τ 3 b(s) λ↓0 Yλ (τ ) =⇒ b(τ ) − √ ds √ (8.15) τ −s 2π 0 where the convergence is in C(R+ ), and b(τ ) is a one dimensional Brownian motion with diffusion coefficient D = 3/4. Proof. We prove first that (S(2))λ (τ ) (obtained by linear interpolation from Sn (2)) converges weakly to a Brownian motion. We first prove tightness. Boundedness has already been proved, see (7.6). Since Sn (2) is a FTn –martingale, by Doob’s inequality, for each ε > 0, we have lim lim P δ↓0 λ↓0
sup
0≤σ<τ ≤T τ −σ≤δ
|(S(2))λ (τ ) − (S(2))λ (σ)| > ε
≤ lim lim δ↓0 λ↓0
C[nλ (δ) + 1]λ2 T =0. ε2
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To identify the limit we compute the bracket of Snλ (τ ) (2) as follows nλ (τ )
k=1
λ (τ ) 9λ2 n 2 E (Sk (2) − Sk−1 (2)) FTk−1 = E m ¯ xk−1 , z (k−1) (Tk )2 FTk−1 16
k=1
nλ (τ ) 9λ T dx m ¯ (x)2 a(λβ (x + xk−1 ))2 16 2
=
k=1
which converges to Dτ on the set {x∗ (τ ) ≤ L} for any L > 0. From (8.6) and routine manipulation, (S(2))λ converges weakly to a Brownian motion with diffusion D by Levy’s characterization theorem. To complete the proof, we introduce the family of linear mappings Jλ : C(R+ ) → C(R+ ) defined by nλ (τ )−1
Jλ (ϕ)(τ ) :=
Anλ (τ )−1,k−1 ϕ(λ2 T k)
k=0
when (λ2 T )−1 τ is an integer and by linear interpolation otherwise. It is easy to verify that τ 3 ϕ(s) lim Jλ (ϕ)(τ ) = J(ϕ)(τ ) := √ ds √ λ↓0 τ −s 2π 0 uniformly for ϕ in a compact set. Since J is continuous, we can apply [6, Thm. 5.3] and get 3 Jλ ((S(2))λ )(τ ) =⇒ √ 2π λ↓0
τ 0
b(s) ds √ τ −s
in C(R+ ) .
Proposition 8.2 is proved.
Conclusion of the proof of Theorem 1.1. We first show that ξλ converges by subsequences to a continuous process, and then that any limit point solves the integral equation (1.12). By the uniqueness (in law) of the latter, Theorem 1.1 follows. The boundedness of ξλ follows from (8.5). In order to prove equicontinuity, we note that from (7.2) and (8.14) we have, for each ε > 0,
n−1 2 lim P sup ξn − Yn − (A )n,k ξk > ε = 0 . λ↓0 n≤nλ (τ )
(8.16)
k=0
Setting nλ (s)−1
Ξλ (s) :=
k=0
(A2 )nλ (s),k ξk
(8.17)
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we postpone the proof that, for each ε > 0, lim lim P sup δ↓0 λ↓0
|Ξλ (s) − Ξλ (s )| > ε = 0
s,s ∈[0,τ ] |s−s |≤δ
79
(8.18)
and observe that, by (8.16), Proposition 8.2, and (8.18), for each ε > 0, lim lim P sup δ↓0 λ↓0
|ξλ (s) − ξλ (s )| > ε = 0
s,s ∈[0,τ ] |s−s |≤δ
(8.19)
which shows ξλ is tight in C(R+ ). It remains to prove (8.18). Let us fix 0 ≤ s ≤ s ≤ τ , s − s ≤ δ. We have nλ (s −δ)−1
Ξλ (s) − Ξλ (s ) =
2 (A )nλ (s),k − (A2 )nλ (s ),k ξk
k=0 nλ (s )−1
+
2 (A )nλ (s),k − (A2 )nλ (s ),k ξk
k=nλ (s −δ) nλ (s)−1
+
(A2 )nλ (s),k ξk
k=nλ (s )
so that
|Ξλ (s) − Ξλ (s )| ≤
∗
2
2 ξ (τ ) nλ (τ ) λ T + 3 ξ ∗ (τ ) nλ (δ)
sup τ1 ,τ2 ∈[0,τ ] τ1 −τ2 >δ
sup 0≤k
2 −1 2 (λ T ) (A )n (τ ),n (τ ) − λ 1 λ 2
9 2
2 (A )n,k
by using (5.19), (5.20), and (8.5) the equation (8.18) follows. We have concluded the proof that ξλ is tight. Finally, by (5.20), Proposition 8.2 and (8.16), we conclude that any limit ξ solves τ 9 τ 3 b(s) + ξ(τ ) = b(τ ) − √ ds ds ξ(s) . (8.20) 2π 0 2π(τ − s) 2 0 It is now easy to verify (1.12) and (8.20) are equivalent; indeed the latter is obtained by an iteration of the former.
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Appendices A
Some technical estimates
We have pt (x, y) = pt (x − y, 0) and # " 2 2 ∂pt (x, 0) x 1 = √ − 1 e−x /2t . ∂t π(2t)3/2 t
(A.1)
Proof of (4.17). " # 2 ∂ (pt ϕ)(x) ≤ ϕ∞ 1 dy pt (y, 0) 1 + y = 1 ϕ∞ . ∂t 2t t t Proof of (7.19). ) ( ) ( 2 ∂pt ∂ pt ϕ =− m ¯ , ϕ m ¯ , ∂t ∂x∂t " # (x − y)3 1 3(x − y) −(x−y)2 /2t = dx m ¯ (x) dy √ − ϕ(y) e t2 t π(2t)3/2 " # (x − y)3 1 3(x − y) √ = 3/2 dx m − ¯ (x) dy pt (x, y) ϕ(y) 3/2 2t t t hence, by the Young and H¨ older inequalities, (7.19) follows.
B Gaussian estimates Proof of (7.9). From (1.3) and (3.9) we have, for t ∈ [Tn + 1, Tn+1 ] E z (n) (t, x)2 |xn =
t
Tn
ds
(n)
2
dy gt−s (x, y) a(λβ y)2 ≤ C
0
t−Tn
(n)
du g2u (x, x) .
(B.1) (n) Next, since sup{gu (x, y) : x, y ∈ R; 2u ≥ 1} < ∞ (see [7, Lemma A.9]), and using also (7.29) the desired estimate holds for p = 1. Since the process z (n) conditioned on xn is Gaussian, the estimate also holds for p = 2. Proof of (4.6). Let us denote by Pn the probability P conditioned on the center xn , and by En the corresponding expectation. From (B.1) and (7.29) to treat the case Tn ≤ t ≤ Tn + 1, it follows there exists C > 0 such that, for any n ≤ nλ (τ ), sup (B.2) sup En z (n) (t, x)2 ≤ CT . t∈[Tn ,Tn+1 ] x∈R
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Proceeding as in the proof of [7, Thm. 2.3], (taking ε = 1 and µ(y) = a(λβ y)), it follows that given α ∈ (0, 1) there exists C such that, for any h, λ ∈ (0, 1], t ∈ [Tn , Tn+1 ], and x ∈ R, 2 z (n) (t, x + h) − z (n) (t, x) (B.3) ≤ CT hα . En It is not difficult to prove also H¨older continuity in t: 2 √ En z (n) (t + h, x) − z (n) (t, x) ≤ CT 2 λ−β h .
(B.4)
Indeed recall that, from (1.3), 2 (n) (n) z (t + h, x) − z (t, x) = En
t
ds Tn
h
+
2 (n) (n) dy gt−s+h (x, y) − gt−s (x, y) a(λβ y)2
ds 0
dygs(n) (x, y)2 a(λβ y)2 .
(B.5)
Using (7.29) it is easy to see that h √ ds dy gs(n) (x, y)2 a(λβ y)2 ≤ C h . 0
To bound the first term on the r.h.s. of (B.5), we use the following formula for (n) gt : t (n) ds dz pt−s (x, z)[1 − 3m ¯ 2xn (z)]gs(n) (z, y) . (B.6) gt (x, y) = pt (x, y) + 0
(n)
From the properties of the heat kernel pt , the estimate sup{gt (x, y); x, y ∈ R, t ≥ 1} < ∞, the fact that a has compact support and (7.29) for t ∈ (0, 1], one gets (B.4). Recall that zn conditioned on xn is Gaussian, and let us now define (B.7) sup sup En z (n) (t, x)2 σn2 = t∈[Tn ,Tn+1 ] x∈R
µn
=
En
sup t∈[Tn ,Tn+1 ]
z
(n)
(t)∞
.
(B.8)
Borell’s inequality (see for instance [1, Thm. 2.1]) tells us that, if µn < √ −ζ T λ , then
√ −ζ 2 √ T − µ ) (λ n z (n) (t)∞ > λ−ζ T ≤ 4 exp − sup . (B.9) Pn 2σn2 t∈[Tn ,Tn+1 ]
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To estimate µn , we use (see [1, Cor. 4.15]), ∞ µn ≤ K dr log N (r),
Ann. Henri Poincar´e
(B.10)
0
where K is a universal constant and N (r) is the minimal number of balls of radius r needed to cover [Tn , Tn+1 ] × R when considering in that set the metric 3 2 . d (t, x), (s, y) = En z (n) (t, x) − z (n) (s, y) Let us then estimate N (r). Recall that, from (B.2), we only need to consider √ r ≤ CT , since for larger r, N (r) = 1 and the integrand in (B.10) equals zero. For each r > 0, consider the sets A1 = [Tn , Tn+1 ] × [−(λ2β r)−1 , (λ2β r)−1 ] and A2 = ([Tn , Tn+1 ] × R) \ A1 . Let us estimate first the number of balls needed to cover A2 . From now on, to avoid introducing new constants in the notation we suppose that the support of a is included in [−1, 1], and that√λ is sufficiently small. From (7.29), and since Tn+1 − Tn = T ≤ λ−β/20 and r ≤ CT , it is easy to see that if (t, x) ∈ A2 t ≤ En z (n) (t, x)2 ds dy e4(t−s) pt−s (x, y)2 a(λβ y)2 Tn 2 t (λ2β r)−1 − λ−β 1 4T ≤ Ce ds √ exp − 2(t − s) t−s Tn ≤
exp [−λ−β /(2r2 )] ≤
r2 . 4
(B.11)
From (B.11), it follows that we may cover A2 with just one ball. From (B.3) and (B.4), it follows that A1 may be covered by (λ7β r9 )−1 balls of radius r. Then, from (B.10) it follows that √CT 0 K dr log N (r) ≤ K dr log (1 + (λ7β r9 )−1 ) 0 0 ∞ ∞ log (1 + u9 ) 9 log u −7β/9 −7β/9 du ≤ Kλ du Kλ 2 −7β/9 −7β/9 λ√ λ√ u u2 CT CT √ C(β) T | log λ| (B.12)
µn
≤ ≤ ≤
∞
where we have done the change of variables u = λ−7β/9 r−1 , and C(β) is a constant that depends on β. Then, we may apply (B.9), to obtain
√ Pn z (n) (t)∞ > λ−ζ T ≤ 4 exp (−Cλ−ζ/2 ) . (B.13) sup t∈[Tn ,Tn+1 ]
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Finally, recall that, given xn , the random variables sup t∈[Tn ,Tn+1 ]
83
z (n) (t)∞ ; n ≤ nλτ
are independent, and the estimate (B.13) does not depend on n (in particular, it does not depend on xn ). Then nλ (τ ) (1) P Bλ,τ ≥ 1 − 4 exp (−Cλ−ζ/2 )
(B.14)
and (4.6) follows. Proof of (4.35). The estimate (4.35) follows by the same procedure used to prove (4.6). We use the notation introduced in that proof. Recall that z
n,⊥
t
(t) = Tn
and define 2 σ(n,⊥)
=
sup
n,⊥ gt−s dW (a) (s)
(B.15)
sup En z (n,⊥) (t, x)2
(B.16)
t∈[Tn ,Tn+1 ] x∈R
µ(n,⊥)
En
=
sup t∈[Tn ,Tn+1 ]
z
(n,⊥)
(t)∞
.
(B.17)
Then, = En z (n,⊥)(t, x)2 ≤ =
2 3 (n) ¯ xn (x)m gt−s (x, y) − m ¯ xn (y) a(λβ y)2 4 Tn 2 ∞ 3 ¯ (x)m a∞ ds dy gt−s (x, y) − m ¯ (y) 4 0 ∞ 3 2 ¯ (x) ≤ C a∞ ds g2t (x, x) − m (B.18) 4 0
t
ds
dy
(see [8] for the last inequality). It is easy to see that the estimates (B.3) and (B.4) also hold for z (n,⊥) . Moreover, ¯ (x)2 (B.19) En z (n,⊥) (t, x)2 ≤ CEn z (n) (t, x)2 + Ctm (see (4.33)), hence from (B.11) and recalling that m ¯ is exponentially small as |x| → ∞, it follows that, for λ small enough, r2 En z (n,⊥) (t, x)2 ≤ 4
for (t, x) ∈ A2 .
(B.20)
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Then N (r) ≤ 1 + (λ7β r9 )−1 , so proceeding as in (B.12), but now with T = 1, we obtain µn,⊥ ≤ C| log λ| (B.21) and the proof finishes as that of (4.6). Proof of (6.4). Note that, conditionally on the xk−1 ’s, the random variables {ηk (2); k ≤ nλ (τ )} are independent and Gaussian with variance bounded by CT . Therefore (6.4) is easily deduced from the following elementary lemma. LemmaB.1 Let {ωh , h = 1, . . . , N } be mean zero i.i.d. random variables and set n Yn := h=1 (n − h + 1)−1/2 ωh . Assume that for each p ∈ [1, ∞) we have E|ωi |p < ∞. Then for each ζ > 0 and q < ∞ there exists a constant C = C(ζ, q) such that ζ P sup |Yn | > N (B.22) ≤ CN −q . n≤N
k (n) Proof. Let Yk := h=1 (n − h + 1)−1/2 ωh , k = 1, . . . , n, which is a martingale with quadratic variation [Y (n) ]k =
k h=1
1 ω2 n−h+1 h
hence, by the BDG inequality (see [16, §6, E. 4.1]), for each p ∈ [1, ∞) there exists C = C(p) such that
p/2 1 2 [Y ]n ≤ CE ω E (|Yn | ) = E ≤ CE n−h+1 h h=1
p/2 n n−1 p/2 1 1 p 2/p (E|ωh | ) ≤C ≤C ≤ C (log n)p/2 . n−h+1 k+1
p
|Yn(n) |p
(n)
p/2
h=1
n
k=0
By using (7.17) we thus get P
sup |Yn | > N −ζ
n≤N
≤ N −ζp
N
p/2
E (|Yn |p ) ≤ CN 1−ζp (log N )
n=1
taking p large enough the lemma follows.
Acknowledgments. S. Brassesco acknowledges the very kind hospitality of the Department of Mathematics of the University of Rome Tor Vergata. Part of this work was done while P. Butt` a was visiting that Department with a Fellowship in Mathematics of the “Istituto Nazionale di Alta Matematica Francesco Severi”.
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85
References [1] R. J. Adler, An Introduction to Continuity, Extrema and Related Topics for General Gaussian Processes, Lecture Notes Monograph series 12, Institute of Mathematical Statistics 1990. [2] L. Bertini, S. Brassesco, P. Butt` a and E. Presutti, Stochastic phase field equations: existence and uniqueness, Ann. Henri Poincar´e 3, 87–98 (2002). [3] L. Bertini, P. Butt` a, E. Presutti and E. Saada, Interface fluctuations in a conserved system: derivation and long time behavior, Preprint. [4] L. Bertini, P. Butt` a and B. R¨ udiger, Interface dynamics and Stefan problem from a microscopic conservative model, Rend. Mat. Appl. (7) 19, 547–581 (1999). [5] L. Bertini, E. Presutti, B. R¨ udiger and E. Saada, Dynamical fluctuations at the critical point: convergence to a nonlinear stochastic PDE, Teor. Veroyatnost. i Primenen 38, no. 4, 689–741 (1993); Translation in Theory Probab. Appl. 38, no. 4, 586–629 (1993). [6] P. Billingsley, Convergence of Probability Measures, New York: Wiley 1968. [7] S. Brassesco, Stability of the Instanton under small random perturbations, Stoch. Proc. Appl. 54, 309–330 (1994). [8] S. Brassesco and P. Butt`a, Interface fluctuations for the D=1 Stochastic Ginzburg–Landau equation with non–symmetric reaction term, J. Statist. Phys. 93, 1111–1142 (1998). [9] S. Brassesco, P. Butt` a, A. De Masi and E. Presutti, Interface fluctuations and couplings in the d = 1 Ginzburg–Landau equation with noise, J. Theoret. Probab. 11, 25–80 (1998). [10] S. Brassesco, A. De Masi and E. Presutti, Brownian fluctuations of the interface in the d = 1 Ginzburg–Landau equation with noise, Annal. Inst. H. Poincar´e 31, 81–118 (1995). [11] G. Caginalp and X. Chen, Convergence of the phase field model to its sharp interface limits, European J. Appl. Math. 9, 417–445 (1998). [12] A. De Masi, P. Ferrari and J. L. Lebowitz, Reaction–diffusion equations for interacting particle systems, J. Statist. Phys. 44, 589–644 (1986). [13] T. Funaki, The scaling limit for a stochastic PDE and the separation of phases, Prob. Theory Relat. Fields 102, 221–288 (1995) . [14] P. C. Hohenberg and B. I. Halperin, Theory of dynamic critical phenomena, Rev. Mod. Phys. 49, 435–479 (1977).
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[15] P. L. Lions and P. E. Souganidis, Fully nonlinear stochastic partial differential equations: non–smooth equations and applications, C. R. Acad. Sci. Paris S´er. I Math. 327, 735–741 (1998). [16] M. M´etivier, Semimartingales: A Course on Stochastic Processes, Berlin: de Gruyter 1982. [17] H. M. Soner, Convergence of the phase–field equations to the Mullins–Sekerka problem with kinetic undercooling, Arch. Rational Mech. Anal. 131, 139–197 (1995). [18] F. G. Tricomi, Integral Equations, New York: Dover 1985. Lorenzo Bertini Dipartimento di Matematica Universit` a di Roma La Sapienza Piazzale Aldo Moro 2 00185 Roma Italy email:
[email protected] Stella Brassesco Departamento de Matem´aticas Instituto Venezolano de Investigaciones Cient´ıficas Apartado Postal 21827 Caracas 1020–A Venezuela email:
[email protected] Paolo Butt` a Dipartimento di Matematica Universit` a di Roma La Sapienza Piazzale Aldo Moro 2 00185 Roma Italy email:
[email protected] Errico Presutti Dipartimento di Matematica Universit` a di Roma Tor Vergata Via della Ricerca Scientifica 00133 Roma Italy email:
[email protected] Communicated by Jean-Pierre Eckmann submitted 11/01/01, accepted 13/06/01
Ann. Henri Poincar´e 3 (2002) 87 – 98 c Birkh¨ auser Verlag, Basel, 2002 1424-0637/02/010087-12 $ 1.50+0.20/0
Annales Henri Poincar´ e
Stochastic Phase Field Equations: Existence and Uniqueness L. Bertini∗, S. Brassesco, P. Butt` a† and E. Presutti†
Abstract. We consider a conservative system of stochastic PDE’s, namely a one dimensional phase field model perturbed by an additive space–time white noise. We prove a global existence and uniqueness result in a space of continuous functions on R+ × R. This result is obtained by extending previous results of Doering [3] on the stochastic Allen–Cahn equation.
1 Introduction and results We consider a phase field system with additive stochastic noise, which is formally written as 1 ∂t m(t) = ∆m(t) − V (m(t)) + λh(t) + aη(t) 2 (1.1) 1 ∂t [h(t) + m(t)] = ∆h(t) 2 where (t, x) ∈ R+ × R, m(t) = m(t, x), h(t) = h(t, x) are two scalar random fields (we omit to write explicitly the dependence on the randomness), λ is a positive parameter, ∆ is the Laplacian on R, V (m) = m4 /4 − m2 /2 is a double well potential, a = a(x) is a bounded and continuous function, and η(t) = η(t, x) is a space–time white noise, i.e. E (η(t, x) η(s, y)) = δ(t − s) δ(x − y). In particular a translationally covariant noise is obtained for a = 1. The deterministic system obtained by setting a = 0 in (1.1), usually referred to as phase field equations, describes the kinetic of phase segregation when the presence of the latent heat is taken into account. The first equation describes in fact the evolution of the order parameter m which is coupled to the external field h (which can be thought as the excess temperature measured from the melting temperature); h is however itself a dynamic variable which diffuses and, via the coupling λ, introduces a feedback into m whose effect is to slow down the phase segregation process. We stress that q = m + h is locally conserved as apparent from the second equation in (1.1). Scaling limits of the deterministic phase field equations as λ → 0 have been considered in [2] and [6]. ∗ Partially
† Partially
976552.
supported by Cofinanziamento MURST 1999. supported by Cofinanziamento MURST 1999 and by NATO Grant PST.CLG.
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Due to the phenomenological character of the phase field model, the introduction of a random forcing term appears natural and makes possible to discuss statistical properties of solutions. We also mention that the stochastic system (1.1) is very similar to the so–called, in the physical literature on critical phenomena, model C of Hohenberg and Halperin [5]. The specific choice of an additive white noise only in the first equation has been done to keep the model as simple as possible: the random forcing term is non–conservative, whereas the conservation law is still linear and not perturbed. Moreover, for λ small, m and h are weakly coupled so that we may refer to (1.1) as a weakly conservative system. This simplifying feature might help in developing a mathematical theory for phase segregation in conservative models. Indeed, in the companion paper [1], front fluctuations for (1.1) are analyzed in a suitable scaling limit as λ → 0 and a = O(λ). The need of an existence and uniqueness result for the stochastic system (1.1) in [1] is the main motivation for the present paper. Referring to [1] for a more exhaustive discussion on the stochastic phase field equations, we next state precisely our results. For α > 0 and γ ∈ (0, 1], let us define the following norms on C(R): ϕCα (R)
:= sup e−α|x| |ϕ(x)|
ϕCαγ (R)
:= ϕCα (R) + sup e−α(|x|+|y|)
x∈R
x=y
|ϕ(x) − ϕ(y)| |x − y|γ
and also the following ones on C(R+ × R) f Cα (R+ ×R)
:=
f Cαγ (R+ ×R)
:=
2
sup e−α
t/2
f (t)Cα(R)
t/2
f (t)Cαγ (R)
t∈R+ 2
sup e−α t∈R+
we shall denote by Cα (R), . . . , the corresponding Banach spaces. Let pt = et∆/2 be the heat semigroup, namely the integral operator with kernel pt (x, y) = (2πt)−1/2 exp{−(x − y)2 /2t}. We introduce the process Z(t) = t Z(t, x) given by Z(t) = 0 ds pt−s [aη(s)]. Then Z is the mean zero Gaussian process with covariance t∧s E (Z(t, x)Z(s, y)) = Γ(t, s; x, y) := du dz pt−u (x−z)ps−u (y−z)a(z)2 (1.2) 0
where t ∧ s := min{t, s}. In the next lemma we state some properties of the paths of Z, which follow, by standard Gaussian arguments, from the properties of the covariance Γ, see e.g. [3] and references therein. Lemma 1.1 For each α > 0 and γ ∈ (0, 1/2) we have that Z ∈ Cαγ (R+ × R), P–a.s. We shall denote by Ft the filtration given by Ft := σ{Z(s, x); (s, x) ∈ [0, t] × R}. In the following we consider a fixed realization of Z ∈ Cαγ (R+ × R).
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Let q := m + h and U (m) := V (m) − λm; we formulate the problem (1.1) in the integral form t ds pt−s [−U (m(s)) + λq(s)] + Z(t) m(t) = pt m(0) + 0
1 q(t) = pt q(0) − 2
0
(1.3) t
ds ∆pt−s m(s)
note that the integral on the r.h.s. of the second equation is well defined (the result is a continuous function) provided x → m(s, x) is H¨older continuous. Our main result is the following existence and uniqueness result for the system (1.3) on the space of H¨ older continuous functions. Theorem 1.2 Let m(0), q(0) ∈ Cαγ (R) for any α > 0 and γ ∈ (0, 1/2). Then there exists a unique Ft –adapted process (m, q) ∈ Cαγ (R+ × R) × Cαγ (R+ × R) for any α > 0 and γ ∈ (0, 1/2) which solves (1.3). ˜ on Cα (R+ × R) given by We introduce the linear operators G, G t G[F ] (t) := ds pt−s F (s) 0 t 1 ˜ ] (t) := G[F ds (t − s) ∆pt−s F (s) 2 0
(1.4)
and set R(t) := pt m(0)+ λt pt q(0)+ Z(t). By plugging the second equation in (1.3) into the first one we get that m solves the problem ˜ m = R − G[U (m)] − λ G[m] .
(1.5)
By standard estimates on the heat kernel, it is easy to verify that for each α > 0 and γ ∈ (0, 1] there exist an α > 0 and a constant C = C(α, γ) such that Gf Cαγ (R+ ×R) ≤ Cf Cα (R+ ×R) , ˜ C γ (R ×R) ≤ Cf C (R ×R) . Gf + + α α Furthermore, for each α > 0 and γ ∈ (0, 1), there exists an α > 0, γ ∈ (0, 1) and a constant C = C(α, γ) such that t ds ∆pt−s f (s) ≤ Cf C γ (R ×R) . 0
γ Cα (R+ ×R)
α
+
Therefore Theorem 1.2 can be easily deduced from the following existence and uniqueness result for the problem (1.5) on the space of continuous functions. Theorem 1.3 Let m(0), q(0) ∈ Cα (R) for any α > 0. Then there exists a unique Ft –adapted process m ∈ Cα (R+ × R) for any α > 0 which solves (1.5).
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In the rest of the paper we prove Theorem 1.3. A uniqueness and existence result for equation 1.5 with λ = 0 is given in [3]. See also [4] for the analogous result in a bounded domain. We shall follow closely the proof in [3] for the one dimensional case, referring to that paper for some technical Lemmata. The term ˜ coming from λh in (1.1), is the source of the difficulties. Since we cannot λG, estimate the Lp norm of h in terms of the Lp norm m, the necessary a priori bounds are not a straightforward extension of that in [3]. To overcame this problem we shall estimate an appropriate negative Sobolev norm of h in terms of the Lp norm of m, see Lemmata 2.4 and 2.5 below. We finally remark that the quartic double well potential, V (m) = m4 /4 − 2 m /2 has been chosen only for notation simplicity; the proof works for any polynomial of even degree with positive leading coefficient. Since the parameter λ will be kept fixed throughout all the paper we omit to indicate the dependence on it. We shall denote by C a generic positive constant whose numerical value may change from line to line.
2 Finite volume approximations ∞ Let CK (R) be the space of infinitely differentiable functions on R with compact ∞ support and introduce L = {Λ ∈ CK (R) : 0 ≤ Λ ≤ 1}. For Λ ∈ L, we introduce the following finite volume approximations of the problem (1.5)
˜ mΛ = ΛR − G[ΛU (mΛ )] − λ G[Λm Λ] .
(2.1)
In this section we establish a global existence result for (2.1) together with some bounds uniform for Λ ∈ L. To this end we need to introduce some more notation. For α > 0, let %(x) := e−α|x| . We shall denote by % also the measure %(x)dx on R. We introduce the following finite measures on R+ × R, omitting the dependence on α > 0 from the notation 2
µ(dt, dx) µT (dt, dx)
:= :=
e−α t/2 %(x) dx dt χ[0,T ] (t) µ(dt, dx)
µΛ (dt, dx) µT,Λ (dt, dx)
:= :=
Λ(x) µ(dt, dx) χ[0,T ] (t) Λ(x) µ(dt, dx)
where χ[0,T ] denotes the characteristic function of [0, T ]. For ν a measure and f a function use the notation ν(f ) = dν f . For p ∈ (1, ∞), we introduce the Sobolev space H1p (%) obtained by completing ∞ CK (R) with respect to the norm ϕpH p () := ∇ϕpLp () + ϕpLp () 1
where ∇ denotes the derivative with respect to x. Since H1p (%) ⊂ Lp (%), for p, q ∈ q (1, ∞) such that p−1 + q −1 = 1 we introduce the dual space H−1 (%) by completing
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Lq (%) with respect to the norm q +H−1 () :=
sup
%(+ ϕ) .
ϕ : ϕ H p () ≤1 1
For f = f (t, x) we shall use the notation f pH p
−1 (µT )
T
:= 0
2
dt e−α
t/2
f (t)pH p
−1 ()
omitting to write T on the l.h.s. if T = ∞. From Lemma 1.1 it follows that for each α > 0 and p ∈ [1, ∞) we have Z ∈ Lp (µ). Furthermore for each T > 0 and Λ ∈ L we have sup sup |Λ(x)Z(t, x)| < ∞.
t∈[0,T ] x∈R
Moreover, by the assumptions on the initial data, the same holds for R(t) = pt m(0) + λt pt q(0) + Z(t). ˜ has a We next state a local existence result for the problem (2.1). Since G kernel which can be estimated as the one of G, the proof of the next result, which is based on Picard iterations, is the same as [3, Prop. 1] and we omit it. Lemma 2.1 For each Λ ∈ L and each realization of Z in Cα (R+ × R) there is a time T0 > 0 such that there exists a unique Ft –adapted continuous and bounded solution of (2.1) on [0, T0 ] × R. To show the existence of a global solution, it is enough to prove that if mΛ is a continuous solution of (2.1) on [0, T ∗) × R then sup sup |mΛ (t, x)| < ∞ .
(2.2)
t∈[0,T ∗ ) x∈R
The key ingredient for proving (2.2) is the following a priori bound on the Lp (µT ∗ ,Λ ) norm of solutions. Proposition 2.2 Let mΛ be a continuous solution of (2.1) on [0, T ∗ ) × R. For each α > 0 and p ∈ [1, ∞) there exists a constant C = C α, p, RLp(µ) < ∞, independent of T ∗ > 0 and of Λ ∈ L, such that mΛ Lp (µT ∗ ,Λ ) ≤ C .
(2.3)
The proof of the proposition is split in several Lemmata, the first one, which is proven integrating by parts, is [3, Lemma 7].
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Lemma 2.3 If both f (t, x) and (∂t − 12 ∆)f are continuous on (0, T ) × R, f (0) = 0 and |f |2n+2 and |∇f |2n+2 are in L1 (µT ), then, for any n = 0, 1, . . . , 1 2n+1 µT f ∂t − ∆ f ≥ (2n + 1)µT f 2n |∇f |2 . 2
(2.4)
Let mΛ be a continuous solution of (2.1). We define uΛ := mΛ − ΛR, 1 ˜ qΛ := ∂t − ∆ G[Λm Λ] . 2
(2.5)
From (2.1), if mΛ is continuous and bounded on [0, T ] × R, then uΛ ∈ C ∞ ˜ Moreover, for t ∈ [0, T ], ((0, T ] × R) by the regularizing properties of G and G. uΛ (t, x) (together with its derivative) is exponentially decaying as x → ∞. Lemma 2.4 Let mΛ be a continuous solution of (2.1) on [0, T ∗ ) × R; then for each n = 0, 1, . . . and β > 0 there exists a constant C = C(n, β), independent of T ∗ > 0 and Λ ∈ L, such that 2n+2 µT ∗ ,Λ e−(2n+2)βt u2n+1 U (mΛ ) ≤ Ce−βt qΛ H 2n+2 Λ (µ −1
T∗)
.
(2.6)
Proof. We apply Lemma 2.3 with f = e−βt uΛ and T < T ∗ . From (2.1) we get 1 ∂t − ∆ e−βt uΛ = −βe−βt uΛ − e−βt [ΛU (mΛ ) + λqΛ ] 2 hence, by (2.4)
2n+2 2n+1 2 µT e−(2n+2)βt (2n + 1)u2n + uΛ ΛU (mΛ ) Λ (∇uΛ ) + βuΛ 2n+1 (2.7) qΛ . ≤ −λµT e−(2n+2)βt uΛ Let p, q ∈ (1, ∞) such that p−1 + q −1 = 1; then for each γ > 0 there exists a constant C = C(γ, p) such that for any a, b ∈ R |a b| ≤ γ|a|p + C|b|q
(2.8)
q (%), we have therefore, by the duality between H1p (%) and H−1
|%(f g)| ≤ γf pH p () + CgqH q 1
−1 ()
.
(2.9)
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By applying (2.9) we get T 2n+1 2 q dt e−α t/2−(2n+2)βt % uΛ (t)qΛ (t) µT e−(2n+2)βt u2n+1 Λ ≤ Λ 0 T
2n+2 2n+2 2n+2 2 ≤ dt e−α t/2−(2n+2)βt γ(2n + 1) 2n+1 % uΛ (t)2n 2n+1 (∇uΛ (t)) 2n+1 0 2n+2 +γ% u2n+2 (t) + Cq (t) Λ Λ H 2n+2 ()
T
2
2n+2
−1
2
γ(2n + 1) 2n+1 c1 (n)% uΛ (t)2n (∇uΛ (t)) + uΛ (t)2n+2 0 2n+2 2n+2 +γ% uΛ (t) + CqΛ (t)H 2n+2 () −1 2n+2 2 = γ(2n + 1) 2n+1 c1 (n)µT e−(2n+2)βt u2n Λ (∇uΛ )
2n+2 2n+2 + Ce−(2n+2)βt qΛ 2n+2 + γ (2n + 1) 2n+1 c1 (n) + 1 µT e−(2n+2)βt uΛ H 2n+2 (µ ) ≤
dt e−α
t/2−(2n+2)βt
−1
T
(2.10) where we used, in the third step, H¨ older inequality in the form (as follows from (2.8)) 2n+1
2n+1 %(f g) ≤ c1 (n) % |f | n+1 + % |g| n 2n+2
2n+2
with f = [uΛ (t)n ∇uΛ (t)] 2n+1 and g = uΛ (t)n 2n+1 . Choosing γ = γ(n, λ, β) small enough and taking T ↑ T ∗ the lemma now follows from (2.7) and (2.10). Lemma 2.5 For each p ∈ (1, ∞) there exists a constant C = C(p) such that, for any α, β, T > 0 and F ∈ Lp (µT ), −βt 1 ˜ 1 e e−βt F p G[F ] ∆ − ≤ C 1 + . (2.11) ∂ t L (µT ) 2 α H p (µT ) −1
Proof. We can write
˜ ](t) = G[F 0
t
ds pt−s
s
0
1 ds ∆ps−s F (s ) 2
(2.12)
so that (in distribution sense)
1 ˜ ] (t) = ∂t − ∆ G[F 2 We thus get ∂t − 1 ∆ G[F ˜ ] (t) p 2 H
−1 ()
1 ≤ 2
t
0
0
t
1 ds ∆pt−s F (s) . 2
ds ∆pt−s F (s)H p
−1 ()
(2.13)
.
(2.14)
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∞ For q −1 + p−1 = 1 and ϕ ∈ CK (R) we now have
(ϕ, ∆pt−s F (s)) = (%ϕ, ∆pt−s F (s))L2 (dx) ϕ ∇% = − % ∇ϕ + , ∇pt−s F (s) % L2 (dx) ≤ ∇ϕLq () + ϕLq () sup |∇ log %(x)| ∇pt−s F (s)Lp ()
(2.15)
x∈R
since supx∈R |∇ log %(x)| = α we conclude ∆pt−s F (s)H p
−1 ()
≤ (1 + α) ∇pt−s F (s)Lp () .
(2.16)
As exp{−α|x|} ≤ exp{−α|y|} exp{α|x − y|}, −α|x|/p ≤ dy e−α|y|/p |F (s, y)| |∇pt−s (x − y)| eα|x−y|/p , (2.17) |∇pt−s F (s) (x)| e by applying Young’s inequality for convolutions we then obtain ∇pt−s F (s)Lp () ≤ F (s)Lp () dx |∇pt−s (x)| eα|x|/p . We define ψα,p (t) :=
dx |∇pt (x)| e
α|x|/p
2 2α exp ≤√ + p 2πt
α2 t 2p2
(2.18)
.
(2.19)
Then, from (2.14), (2.16) and (2.18), 2 1 ˜ − α2pt −βt χ[0,T ] (t) e ∂t − 2 ∆ G[F ] (t) p H−1 () 2 α s 1+α ≤ ds χ[0,T ] (s)e− 2p −βs F (s)Lp () χ[0,T ] (t − s) 2 × e−
α2 (t−s) −β(t−s) 2p
ψα,p (t − s) .
Again by Young’s inequality for convolutions we get −βt 1 ˜ e ∂t − ∆ G[F ] = p 2 H−1 (µT ) 1/p p T 2 1 ˜ − α2 t −βpt dte ∂t − 2 ∆ G[F ](t) p 0 H−1 () 1/p T T 2 2 1+α p − α2pt − α2 t −βpt ≤ dt e ψα,p (t) dt e F (t)Lp () . 2 0 0
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By using the estimate in (2.19) and recalling p > 1, it is easy to show there is a constant C = C(p) > 0 so that α2 t 1+α T 1 dt e− 2p ψα,p (t) ≤ C 1 + . 2 α 0
The lemma is proved.
3
Proof of Proposition 2.2. Recalling (2.5) and that U (m) = m − (1 − λ)m, by expanding the l.h.s. of (2.6), using Lemma 2.5, and Λ2 ≤ Λ, we get there exists a constant C = C(α, n, β) such that µT ∗ ,Λ e−(2n+2)βt u2n+4 Λ
≤ CµT ∗ ,Λ e−(2n+2)βt |uΛ |2n+1 u2Λ |ΛR| + |uΛ |(ΛR)2 + |ΛR|3 2n+2 . (2.20) + |uΛ |2n+1 |ΛR| + |ΛR|2n+2 +uΛ Let
1/(2n+4) 2n+4 M := µT ∗ ,Λ e−(2n+2)βt uΛ
by using repeatedly H¨ older inequality in (2.20) we get
M 2n+4 ≤ C M 2n+3 ΛRL2n+4(µ) + M 2n+2 1 + ΛR2L2n+4(µ)
+ M 2n+1 ΛRL2n+4(µ) 1 + ΛR2L2n+4(µ) + ΛR2n+2 L2n+4(µ) we then conclude that M is bounded by some constant C = C(α, n, β, ΛRL2n+4(µ) ). Recalling mΛ = uΛ + ΛR, by using triangular and Cauchy–Schwartz inequalities, we have mΛ Lp (µΛ,T ∗ ) ≤ ΛRLp(µΛ,T ∗ ) + eβt L2p (µΛ,T ∗ ) e−βt uΛ L2p (µΛ,T ∗ ) . Since, for each p ∈ [1, ∞), ΛRLp(µΛ,T ∗ ) ≤ RLp(µ) < ∞, by choosing β = β(α, p) small enough and n = n(p) large enough, the proposition follows. Proof of (2.2). Let Y := sup{|x| : Λ(x) > 0}; writing explicitly the kernels in the integral equation (2.1), and using Young’s inequality for convolutions, it follows sup sup |mΛ (t, x)| ≤
t∈[0.T ∗ ) x∈R
+e
α2 4
T ∗+ α 2Y
×
T∗
+λ
dt 0
sup sup |Λ(x)R(t, x)|
t∈[0,T ∗ ) x∈R
T∗
dt 0
2
12
dz pt (z)
dz pt (z)2
2
z 1 + 2t 2
2
µT ∗ ,Λ (dt, dx) U (mΛ (t, y))
2 12
µT ∗ ,Λ (dt, dx) mΛ (t, y)2
12
12 (2.21)
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where we used also that Λ2 ≤ Λ. By using that, as follows from Proposition 2.2, U (mΛ ) is in L2 (µT ∗ ,Λ ), and the properties of the heat kernel, it is easy to see that the r.h.s. of (2.21) is bounded. The following lemma states an a priori uniform bound for the solutions of (2.1), which by (2.2) are defined for any t ≥ 0. Lemma 2.6 Let mΛ be a continuous solution of(2.1) on R+ × R. Then for each α > 0, p ∈ [1, ∞) there exists a constant C = C α, p, RLp(µ) < ∞ independent of Λ ∈ L such that mΛ Lp (µ) ≤ C. Proof. Since the continuous solution of equation (2.1) exist globally in time, the inequality (2.3) may be extended for T ∗ ↑ ∞; we get ΛmΛ Lp (µ) ≤ mΛ Lp (µΛ ) ≤ C < ∞ .
(2.22)
˜ are bounded operators in Lp (µ) (see [3, Note that for p > 1 both G and G ˜ From the integral equation Lemma 9] for G and the same proof also works for G). (2.1), the bound (2.22), and R ∈ Lp (µ) the lemma follows.
3 Infinite volume equation In this section we conclude the proof of Theorem 1.3 by removing the truncation Λ in (2.1); following [3], we prove first that, provided β is chosen large enough, exp (−βt) Λ mΛ converges in Lp (µ). Lemma 3.1 Let mΛ be a solution of (2.1). For each p ∈ [1, ∞) there exists a positive constant kp < ∞ such that, for any α > 0 and β ≥ kp (1 + α−p ), {exp (−βt) Λ mΛ , Λ ∈ L} is Cauchy in Lp (µ) as Λ ↑ 1. Proof. Recalling (2.5), if we consider f = e−βt [uΛ − uΛ¯ ], from equation (2.1),we have
1 ¯ (mΛ¯ ) − λe−βt [qΛ − qΛ¯ ] . ∂t − ∆ f = −βf − e−βt ΛU (mΛ ) − ΛU 2 By the same computations as in Lemma 2.4 with f as above, (see equations (2.7) and (2.10)), but choosing γ = γ(n, λ) independent of β and taking T ↑ ∞, there exists a constant c1 = c1 (n) independent of α, β such that 2n+1
¯ (mΛ¯ ) µ e−(2n+2)βt [uΛ − uΛ¯ ] ΛU (mΛ ) − ΛU . + (β − 1)µ e−(2n+2)βt [uΛ − uΛ¯ ]2n+2 ≤ c1 e−βt (qΛ − qΛ¯ ) 2n+2 H 2n+2 (µ) −1
(3.1)
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By using Lemma 2.5 there is a constant c2 = c2 (n) such that e−βt (qΛ − qΛ¯ ) 2n+2 2n+2 H−1 (µ)
¯ Λ¯ 2n+2 . (3.2) ≤ c2 1 + α−(2n+2) µ e−(2n+2)βt ΛmΛ − Λm On the other hand, since for any a, b ∈ R we have (a3 − b3 )(a − b) ≥ (a − b)4 /4, there exists a constant c3 = c3 (λ) such that
¯ (mΛ¯ ) (uΛ − uΛ¯ ) ΛU (mΛ ) − ΛU ¯ ¯ 3¯ − (1 − λ) (uΛ − uΛ¯ ) ΛmΛ − Λm ¯ Λ¯ = mΛ − mΛ¯ − (Λ − Λ)R Λm3Λ − Λm Λ
1 ¯ Λ¯ 4 − c3 (uΛ − uΛ¯ )2 ΛmΛ − Λm ≥ 4
¯ m2 (1 + m2 ) + m2¯ (1 + m2¯ ) + R2 (1 + R2 ) − c3 |1 − Λ| + |1 − Λ| Λ Λ Λ Λ
(3.3)
By plugging the bounds (3.2) and (3.3) into (3.1) and using H¨ older inequality, we find there exists a c4 = c4 (n, α, λ) such that 1 −(2n+2)βt ¯ Λ¯ 2n+4 µ e ΛmΛ − Λm 4 2n+2 + β − 1 − c3 − c1 c2 1 + α−(2n+2) µ e−(2n+2)βt (uΛ − uΛ¯ ) ¯ L2 (µ) ≤ c4 1 − ΛL2 (µ) + 1 − Λ 2n+4 2n+4 2n+4 × 1 + mΛ L4n+8 (µ) + mΛ¯ L4n+8 (µ) + RL4n+8 (µ) . Given p ≥ 2, let n = n(p) = [p/2] − 1 and kp such that 1 + c3 + c1 c2 (1 + α−(2n+2) ) ≤ kp (1 + α−p ) for any α > 0. The lemma now follows from Lemma 2.6 and R ∈ Lp (µ) for any p ∈ [1, ∞). The proof of Theorem 1.3 can now be completed as in [3], we shall just sketch the argument. Proof of Theorem 1.3. To prove existence of a continuous solution of (1.5), we first note that, for each p ∈ [1, ∞) and α so large that α2 > kp (1 + α−p ), mΛ is Cauchy in Lp (µ). This follows from Lemmata 2.6, 3.1, and H¨older inequality; moreover the ˜ map L2 (µ) into Cα (R+ × R), see limit m satisfies equation (1.5). Since G and G [3, Lemma 12], we also have m ∈ Cα (R+ × R). In order to show m ∈ Cα (R+ × R) for any α > 0, we note that, by Lemma 2.6, mΛ is uniformly bounded in Lp (µ); we can thus find a weakly convergent subsequence mΛk → m . On the other hand, by Lemma 3.1, e−βt Λk mΛk converges strongly in Lp (µ) for β ≥ kp (1 + α−p ), hence m = m µ–a.s. Since m ∈ Lp (µ) for any α > 0 and p ∈ [1, ∞), by the same argument as above, we get m ∈ Cα (R+ ×R) and m = m . To prove uniqueness, let m1 and m2 be two continuous solutions of (1.5). By applying Lemma 2.3 to the function f = e−βt [m1 − m2 ] and repeating the same computations as in Lemma 3.1 it is easy to show m1 = m2 .
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Acknowledgments. L. Bertini acknowledges the very kind hospitality of Departamento de Matem´aticas, Instituto Venezolano de Investigaciones Cient´ıficas.
References [1] L. Bertini, S. Brassesco, P. Butt` a and E. Presutti, Front fluctuations in one dimensional stochastic phase field equations, Ann. Henri Poincar´e 3, 29–86 (2002). [2] G. Caginalp and X. Chen, Convergence of the phase field model to its sharp interface limits, European J. Appl. Math. 9, 417–445 (1998). [3] C. R. Doering, Nonlinear parabolic stochastic differential equations with additive colored noise on Rd × R+ : a regulated stochastic quantization, Comm. Math. Phys. 109, 537–561 (1987). [4] W. G. Faris and G. Jona–Lasinio, Large fluctuations for a nonlinear heat equation with noise, J. Phys. A 15, 3025–3055 (1982). [5] P. C. Hohenberg and B. I. Halperin, Theory of dynamic critical phenomena, Rev. Mod. Phys. 49, 435–479 (1977). [6] H. M. Soner, Convergence of the phase–field equations to the Mullins–Sekerka problem with kinetic undercooling, Arch. Rational Mech. Anal. 131, 139–197 (1995). Lorenzo Bertini Dipartimento di Matematica Universit` a di Roma La Sapienza Piazzale Aldo Moro 2 00185 Roma Italy email:
[email protected]
Stella Brassesco Departamento de Matem´ aticas Instituto Venezolano de Investigaciones Cient´ıficas Apartado Postal 21827 Caracas 1020–A Venezuela email:
[email protected]
Paolo Butt` a Dipartimento di Matematica Universit` a di Roma La Sapienza Piazzale Aldo Moro 2 00185 Roma Italy email:
[email protected]
Errico Presutti Dipartimento di Matematica Universit` a di Roma Tor Vergata Via della Ricerca Scientifica 00133 Roma Italy email:
[email protected]
Communicated by Jean-Pierre Eckmann submitted 30/01/01, accepted 13/06/01
Ann. Henri Poincar´e 3 (2002) 99 – 105 c Birkh¨ auser Verlag, Basel, 2002 1424-0637/02/010099-7 $ 1.50+0.20/0
Annales Henri Poincar´ e
Some High-Complexity Hamiltonians with Purely Singular Continuous Spectrum D. Damanik and B. Solomyak Abstract. We give examples of discrete one-dimensional Schr¨ odinger operators with potential having high combinatorial complexity that have purely singular continuous spectrum. This is accomplished by constructing minimal, palindromic subshifts with entropy arbitrarily close to maximal entropy. For a dense Gδ set of elements in such a subshift, purely singular continuous spectrum can then be established using a criterion of Hof, Knill, and Simon.
1 Introduction In this paper we discuss discrete one-dimensional Schr¨odinger operators (Hφ)(n) = φ(n + 1) + φ(n − 1) + V (n)φ(n)
(1)
in 2 (Z) with potential V taking finitely many values. It is known that for these operators, any spectral type can occur. Concretely, periodic potentials lead to purely absolutely continuous spectrum [12], Sturmian potentials (to be defined below) lead to purely singular continuous spectrum [4], and random potentials lead almost surely to pure point spectrum [2]. A point of view that has been proposed and investigated recently [3] is to link spectral properties of the operators to combinatorial properties of the potentials, particularly their combinatorial complexity. Apparently, the spectra become more and more singular as combinatorial complexity is increased. For example, referring to the above classes, periodic potentials have minimal complexity, Sturmian potentials have minimal complexity among the aperiodic potentials, and, on the other extreme, random potentials have (almost surely) maximal complexity. For potentials of intermediate complexity (i.e., potentials intermediate between Sturmian and random), there are almost no definite results on the operator side, in particular for potentials with positive entropy. Our purpose here therefore is to exhibit concrete examples of potentials with high combinatorial complexity that lead to purely singular continuous spectrum. This indicates that a possible correspondence between complexity and spectral type might be quite complicated and we discuss this issue at the end of the paper. Let us be more specific. Let A ⊆ R be a finite set and consider some V ∈ AZ . A finite string w = a1 . . . ak of elements of A is called a word and its length is |w| = k. A word a1 . . . ak is called a subword of V if there is some m such that
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V (m + i) = ai , 1 ≤ i ≤ k. Let f (n) denote the number of different subwords of V having length n. The function f : N → N is called the complexity function of V . A classical result by Morse and Hedlund states that f is bounded if and only if V is periodic [7]. Moreover, any unbounded f satisfies the universal lower bound f (n) ≥ n + 1 for every n. There exist V ’s with f (n) = n + 1 for every n and these sequences are called Sturmian (if they satisfy the additional requirement that every subword occurs infinitely often). Sturmian sequences are therefore of minimal complexity among the aperiodic sequences. On the other hand, if we generate a random sequence in AZ (by i.i.d. random variables giving non-zero weight to every element of A), we get f (n) = |A|n with probability one. One defines the entropy of V by h(V ) = limn→∞ n1 ln f (n). By a subadditivity argument one can show that the limit exists. The maximal entropy in AZ is therefore ln |A|. Our goal is the following: Given A with |A| ≥ 2 and any number h < ln |A|, we will construct a potential V with entropy h(V ) > h such that the operator (1) has purely singular continuous spectrum. The organization is as follows. In Section 2 we use a construction that is similar to the one in Furstenberg [5] which gives a minimal, palindromic subshift Ω ⊂ AZ with high entropy. We then show in Section 3 that for an uncountable set of V ’s in Ω (which, in fact, is a dense Gδ ), the operator (1) has purely singular continuous spectrum. This will follow easily from Section 2, Hof, Knill and Simon [8], Kotani [10], and Last and Simon [11].
2 Construction of minimal, palindromic subshifts with large entropy In this section we construct subshifts Ω ⊆ AZ which are minimal, contain arbitrarily long palindromes, and have large entropy. Our approach is similar to the one used by Furstenberg in [5] who constructs minimal subshifts with positive entropy by generating suitable Toeplitz sequences. Let us put the topology of pointwise convergence on AZ and define the shift transformation T : AZ → AZ by T x(n) = x(n + 1). Any closed, T -invariant subset Ω ⊆ AZ is called a subshift. A subshift Ω is called minimal if the T -orbit of every V ∈ Ω is dense in Ω. It is relatively easy to see that if Ω is minimal, every V ∈ Ω has the same set of subwords. In this case, one may associate a complexity function f : N → N with the subshift because the complexity function is the same for each one of its elements. Similarly, we can then define the entropy of a minimal subshift as above by taking h(Ω) = limn→∞ n1 ln f (n). We shall be interested in minimal subshifts that contain palindromes. A palindrome is a word w = a1 . . . ak that is invariant under reversal, that is, w = ak . . . a1 . We define the palindrome complexity function p : N → N0 := N ∪ {0} for a sequence (resp., a minimal subshift) by p(n) = number of palindromes of length n occurring in V (resp., the elements of the subshift). For general background on palindrome complexity, we refer the reader to [1] (and references therein).
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We call a minimal subshift palindromic if it contains arbitrarily long palindromes (equivalently, lim supn→∞ p(n) > 0). This notion was introduced by Hof, Knill and Simon [8] who also established a connection between palindromicity and spectral theory. Our main result is the following: Theorem 1 Given any A ⊆ R with k elements, where k ≥ 2, and any number h < ln k, there exists a minimal, palindromic subshift Ω ⊆ AZ with entropy larger than h. Proof. The strategy is as follows. We construct a Toeplitz sequence T by starting with the “empty sequence” and successively filling the “holes” with periodic sequences (also having holes) of rapidly increasing periods. This ensures that the limit sequence, provided that it exists, is uniformly recurrent, that is, every subword occurs infinitely often and with bounded gaps. This implies that its T -orbit closure in AZ is a minimal subshift, see, e.g., [6]. We therefore have to take care of two issues: In the generation process, we have to introduce sufficiently many palindromes (to get palindromicity) and sufficiently many subwords (to get large entropy). The first issue is dealt with by “symmetrizing” Furstenberg’s construction [5], the second issue only requires a slightly more elaborate line of argument. We first decompose Z into a suitable disjoint union of arithmetic progressions: Let c0 = 0 and let {dn } be an increasing sequence of positive integers satisfying dn |dn+1 for every n. Let J0 = (c0 + d0 Z), and then for every n ≥ 0, n cn+1 = min m ≥ 0 : m ∈ Ji i=0
and Jn+1 = (cn+1 + dn+1 Z) ∪ (−cn+1 + dn+1 Z). It is clear that Z is the disjoint union of the sets Jn , n ≥ 0. We will choose a sequence {sn } ∈ AN0 and then define the sequence V ∈ AZ by V (m) = sn if m ∈ Jn . From the construction it follows that V is uniformly recurrent and hence we obtain a minimal subshift Ω by taking its T -orbit closure. It also follows that V contains arbitrarily long palindromes since it is symmetric about the origin: V (m) = V (−m) for every m. Thus, Ω is palindromic. To arrange for entropy larger than h, we have to choose the sequences {dn } and {sn } suitably. Given any ε > 0, we can choose a rapidly increasing sequence {dn } sufficient to ensure cn < 1 + ε. (2) lim sup n→∞ n ε −1 Indeed, we can make sure that ∞ m=0 dm < 4 . Since [−cn + 1, cn − 1] ∩ Z ⊂
n−1 m=0
Jm
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2cn − 1 +1 , dm
(2cn − 1)ε +n 4
which implies (2) for ε > 0 sufficiently small. For every m, (2) implies that there are infinitely many n such that cn+m − cn+1 < (1 + ε)m.
(3)
From this point, we can proceed very similarly to Furstenberg. For the reader’s convenience, we provide a sketch of the subsequent arguments. Let Qm = {n ∈ N : (3) holds }. It is possible to choose a sequence {sn } such that for every m and every word a1 . . . am ∈ Am , there exists r ∈ Qm such that sr+1 . . . sr+m = a1 . . . am . Furstenberg [5] used that this holds for a random sequence in AZ . There is also a constructive way: We can just inductively define the sequence {sn } by prescribing the places in Qm for all the words in Am so that no overlaps occur, neither between words on the same m-level nor for words of different levels. It helps of course that we have in each step only finitely many words, so that there is always enough (infinite) space left to work with. With this sequence {sn } we define V as described above. We get that for every word a1 . . . am ∈ Am , there exists r such that cr+m −cr+1 < (1+ε)m and V (cr+j ) = aj , 1 ≤ j ≤ m. Thus there exists n such that V (n+1)V (n+2) . . . V (n+(1+ε)m) contains the symbols a1 , . . . , am (possibly with gaps) in precisely that order. Since every fixed block V (n + 1)V (n + 2) . . . V (n + (1 + ε)m) can contain at most (1+ε)m ordered m-tuples, we deduce that for every m, V must contain at least m |A|m (1+ε)m m
subwords of length (1 + ε)m. It follows from Stirling’s formula that, for λ > 1, m λm λλ ≤ const . m (λ − 1)λ−1 Thus, choosing ε sufficiently small, we can conclude the proof.
3 Application to Schr¨ odinger operators Given a minimal subshift (Ω, T ), certain combinatorial properties of Ω are linked to spectral properties of (1) with V ∈ Ω. Recall that f (n) counts the number of subwords of length n that occur in elements of Ω and p(n) counts the number of palindromes of length n that occur in elements of Ω.
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Proposition 3.1 (Absence of absolutely continuous spectrum) If the complexity function obeys lim supn→∞ f (n) = ∞, then for every V ∈ Ω, the operator H in (1) has empty absolutely continuous spectrum. Remark. If lim supn→∞ f (n) < ∞, then every V ∈ Ω is periodic and hence the operator H has purely absolutely continuous spectrum. Proof. It follows from the assumption and Morse-Hedlund that every V ∈ Ω is aperiodic. Moreover, since (Ω, T ) must have a T -invariant measure, it follows that it also has an ergodic measure µ. Kotani [10] then implies that for µ-almost every V ∈ Ω, the operator H has empty absolutely continuous spectrum. Finally, it was shown by Last-Simon [11] that minimality implies independence of σac (H) of V ∈ Ω. Putting this together, we get the assertion. Proposition 3.2 (Absence of point spectrum) If the palindrome complexity function obeys lim supn→∞ p(n) > 0, then for a dense Gδ -set of V ’s in Ω, the operator H in (1) has empty point spectrum. Proof. This was shown by Hof, Knill and Simon in [8]. We remark that it is possible to construct uncountably many elements V ∈ Ω for which (1) has empty point spectrum. Combining Theorem 1, Proposition 3.1, and Proposition 3.2, we obtain the following theorem: Theorem 2 Given any A ⊆ R with k elements, where k ≥ 2, and any number h < ln k, there exists a closed, T -invariant set Ω ⊆ AZ such that 1. (Ω, T ) is minimal, 2. (Ω, T ) has entropy larger than h, 3. for a dense Gδ set of V ∈ Ω, the operator H in (1) has purely singular continuous spectrum. Let us summarize our results: For each finite alphabet with at least two elements, it is possible to construct uniformly recurrent two-sided sequences with entropy arbitrarily close to the maximal possible value such that the associated Schr¨ odinger operator has purely singular continuous spectrum. This phenomenon should be seen in the general context of the interplay between combinatorial complexity and spectral properties of Schr¨ odinger operators. It turns out that sufficiently high complexity along with uniform recurrence does not necessarily imply pure point spectrum. One may ask, however, the following: Given a uniformly recurrent sequence V , consider the associated subshift Ω and some ergodic measure µ on Ω. Is it true then that if V has high complexity, µalmost every element of Ω generates an operator with pure point spectrum? If this turns out to be true, one will have an interesting result as a byproduct – the
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spectral type for potentials from some minimal subshift over a finite alphabet need not be constant! Such a non-constancy result is known in the uniformly almost periodic case; see Jitomirskaya and Simon [9] who prove this for the family of almost Mathieu operators. However, in the finite alphabet case such an example has not yet been found.
References [1] J.-P. Allouche, M. Baake, J. Cassaigne and D. Damanik, Palindrome complexity, preprint (2000), to appear in Theoret. Comput. Sci. [2] R. Carmona, A. Klein and F. Martinelli, Anderson localization for Bernoulli and other singular potentials, Commun. Math. Phys. 108 (1987), 41–66 [3] D. Damanik, Gordon-type arguments in the spectral theory of one-dimensional quasicrystals, in Directions in Mathematical Quasicrystals, Eds. M. Baake and R. V. Moody, CRM Monograph Series 13, AMS, Providence, RI (2000), 277–304 [4] D. Damanik, R. Killip, and D. Lenz, Uniform spectral properties of onedimensional quasicrystals, III. α-continuity, Commun. Math. Phys. 212 (2000), 191–204 [5] H. Furstenberg, Disjointness in Ergodic Theory, minimal sets, and a problem in diophantine approximation, Math. Systems Theory 1 (1967), 1–49 [6] H. Furstenberg, Recurrence in Ergodic Theory and Combinatorial Number Theory, Princeton University Press, 1981. [7] G. A. Hedlund and M. Morse, Symbolic dynamics, Amer. J. Math. 60 (1938), 815–866 [8] A. Hof, O. Knill and B. Simon, Singular continuous spectrum for palindromic Schr¨ odinger operators, Commun. Math. Phys. 174 (1995), 149–159 [9] S. Jitomirskaya and B. Simon, Operators with singular continuous spectrum: III. Almost periodic Schr¨ odinger operators, Commun. Math. Phys. 165 (1994), 201–205 [10] S. Kotani, Jacobi matrices with random potentials taking finitely many values, Rev. Math. Phys. 1 (1989), 129–133 [11] Y. Last and B. Simon, Eigenfunctions, transfer matrices, and absolutely continuous spectrum of one-dimensional Schr¨ odinger operators, Invent. Math. 135 (1999), 329–367 [12] G. Teschl, Jacobi Operators and Completely Integrable Nonlinear Lattices, Mathematical Surveys and Monographs 72, AMS, Providence (2000)
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David Damanik Department of Mathematics 253–37, California Institute of Technology Pasadena, CA 91125 USA email:
[email protected] Boris Solomyak Department of Mathematics University of Washington Seattle, WA 98195-4350 USA email:
[email protected] Communicated by Jean Bellissard submitted 05/01/01, accepted 11/09/01
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Ann. Henri Poincar´e 3 (2002) 107 – 170 c Birkh¨ auser Verlag, Basel, 2002 1424-0637/02/010107-64 $ 1.50+0.20/0
Annales Henri Poincar´ e
Asymptotic Completeness for Rayleigh Scattering J. Fr¨ ohlich, M. Griesemer and B. Schlein
Abstract. It is expected that the state of an atom or molecule, initially put into an excited state with an energy below the ionization threshold, relaxes to a ground state by spontaneous emission of photons which propagate to spatial infinity. In this paper, this picture is established for a large class of models of non-relativistic atoms and molecules coupled to the quantized radiation field, but with the simplifying feature that an (arbitrarily tiny, but positive) infrared cutoff is imposed on the interaction Hamiltonian. This result relies on a proof of asymptotic completeness for Rayleigh scattering of light on an atom. We establish asymptotic completeness of Rayleigh scattering for a class of model Hamiltonians with the features that the atomic Hamiltonian has point spectrum coexisting with absolutely continuous spectrum, and that either an infrared cutoff is imposed on the interaction Hamiltonian or photons are treated as massive particles. We show that, for models of massless photons, the spectrum of the Hamiltonian strictly below the ionization threshold is purely continuous, except for the ground state energy.
1 Introduction Ever since the inception of the quantum theory of atoms interacting with the quantized radiation field, theoreticians have expected that when an atom (with an infinitely heavy nucleus) in a state where all electrons are bound to the nucleus is targeted by a finite number of photons in such a way that the total energy of the composed system remains below the ionization threshold of the atom the following physical processes unfold: First, some of the electrons in the shells of the atom are lifted into an excited state by absorbing incoming photons; but, since the total energy is below the ionization threshold, they remain bound to the nucleus. As time goes on, the excited state relaxes to a ground state of the atom by spontaneous emission of photons, which propagate essentially freely to spatial infinity. Thus, asymptotically, the state of the total system, atom plus quantized radiation field, describes an atom in its ground state and a cloud of photons escaping to infinity with the velocity of light. Relaxation of an excited initial state to a ground state by emission of outgoing radiation is the simplest example of an “irreversible process”, accompanied by information loss at infinity, occuring in an open quantum system with infinitely many degrees of freedom. It would seem worthwhile to attempt to understand this process mathematically precisely; (see Sect. 10).
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The picture described above suggests that the scattering operator describing Rayleigh scattering of light off an atom with a static nucleus, i.e., the scattering operator restricted to the subspace of states with energies below the ionization threshold of the atom, is unitary; (see Sect. 9 and 10). If true - one says that “asymptotic completeness” (AC) is valid for Rayleigh scattering. In attempting to establish this picture mathematically, one faces the problem that, in the scattering of light at an atom, an arbitrarily large number of soft photons of arbitrarily small total energy can, in priciple, be produced (in processes of high order in the feinstructure constant). Perturbative calculations of scattering amplitudes suggest, however, that in the analysis of Rayleigh scattering of light at an atom with a static nucleus, one does not encouter a genuine infrared catastrophe of the kind first described by Bloch and Nordsieck. Yet, the mathematical problems connected with controlling very large numbers of very soft photons in a mathematically rigorous, non-perturbative way are quite substantial and have not been fully mastered, yet. In order to simplify matters to a manageable size, we propose to study Rayleigh scattering and the phenomenon of relaxation to a ground state for models of massive photons and for models of massless photons with an infrared cutoff. In this paper, results in this direction are proven. In order to avoid inessential technical complications, we consider models of “scalar photons”, bosons. But our analysis can be extended to the quantum electrodynamics of non-relativistic electrons (bound to a static nucleus) interacting with the quantized electromagnetic field, provided we work within the dipole approximation and impose an (arbitrarily tiny) infrared cutoff on the electron-field interaction. We plan to study more difficult scattering problems in similar models and their consequences for “irreversible phenomena” in future papers. Next, we describe our main results in more detail. To avoid starting with a list of assumptions, we formulate our results for a concrete, simple model, which is physically relevant and captures the main features of the problems we propose to solve. For precise assumptions and other models see Section 3. Consider N non-relativistic electrons subject to a potential V , which may also include two–body interactions. The electrons are linearly coupled to a quantized field of relativistic bosons. The Hamilton operator of this system is H = K ⊗ 1 + 1 ⊗ dΓ(ω) + φ(G) and acts on the Hilbert space H = Hel ⊗F, where Hel = ∧N L2 (R3 ; C2 ) is the antisymmetric tensor product of N copies of L2 (R3 ; C2 ), and F is the bosonic Fock space over L2 (R3 ). The operator K describes the time evolution of the electrons without radiation and is given by K = −∆ + V , where ∆ denotes the Laplacian on R3N . (In our units, and the electron mass are equal to one.) Electron spin will be neglected henceforth.
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We assume that V− , the negative part of V , is infinitesimally form–bounded with respect to −∆, that K is essentially self-adjoint on C0∞ (R3N ), and that inf σess (K) > inf σ(K), i.e., that K has bound states. In particular, inf σ(K) is an isolated eigenvalue of K. The operator dΓ(ω) describes the energy of free bosons. Formally dk ω(k)a∗ (k)a(k) dΓ(ω) = R3
where a(k) and a∗ (k) are the usual annihilation- and creation operators (operator– √ valued distributions) depending on the wave vector k, and ω(k) = k 2 + m2 is the energy of a relativistic particle with momentum k and mass m ≥ 0. The operator φ(G) = dk Gx (k)a(k) + Gx (k)a∗ (k) describes the interaction between electrons and bosons. In this introduction, we −ik·xi e κ(k), and κ(k) ∈ C0∞ (R3 ). If m = 0 we cut off the choose Gx (k) = N i=1 infrared modes from the interaction by assuming that κ(k) = 0, for |k| small. Thus, in any case (1) inf ω(k) > 0. k∈supp κ
Without this assumption, it is presently not known how to control the number of soft bosons produced in the course of the time evolution. Let Σ denote the ionization threshold. This is the smallest energy the system can reach when one or several electrons have been moved to infinity. States with energy below Σ are exponentially localized w.r. to the electron coordinates. More precisely
eα|x| E∆ (H) < ∞ if ∆ ⊂ (−∞, Σ) and sup ∆ + α2 < Σ. Clearly Σ ≥ inf σ(H), and, for one-electron atoms and if the coupling is weak enough, it is known that Σ > inf σ(H) [GLL00, BFS98]. Of course, one expects Σ > inf σ(H) for all neutral atoms and molecules. Note that Σ = ∞ if V (x) → ∞, for |x| → ∞, i.e., when σ(K) is discrete. This situation is included in our analysis, but one of our main points is to prove results on Rayleigh scattering when Σ < ∞. For states ϕ ∈ H with energy below Σ, that is ϕ = E(−∞,Σ) (H)ϕ, one expects that ϕt = e−iHt ϕ is well approximated, in the distant future, by a linear combination of states of the form a∗ (h1,t ) . . . a∗ (hn,t )e−iEt ϕb
(2)
where ϕb is a bound state of H, Hϕb = Eϕb , hj,t = e−iωt hj , j = 1, . . . , n, are one– particle wave functions of freely propagating bosons, and a∗ (h) = h(k)a∗ (k) dk, a(h) := (a∗ (h))∗ . This property is called asymptotic completeness (AC) for
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Rayleigh scattering. It asserts, in particular, that the asymptotic dynamics of escaping photons is well approximated by their free dynamics. This requires that the strength of the interaction between ballistically moving bosons and electrons decays at an integrable rate. In our model of non-relativistic electrons bound to static nuclei this is true, thanks to the spatial localization of the electrons. For massless bosons, it follows more generally from the fact that the propagation velocity of electrons is strictly smaller than the velocity of the bosons, i.e., the velocity of light, [FGS00]. To give a mathematically more precise formulation of AC, let us introduce asymptotic creation operators a∗+ (h). Let ϕ = Eλ (H)ϕ, hj ∈ L2 (R3 ), j = 1, . . . , n and Mj = sup{ω(k)|hj (k) = 0}. Then a∗+ (h1 ) . . . a∗+ (hn )ϕ = lim eiHt a∗ (h1,t ) . . . a∗ (hn,t )e−iHt ϕ t→∞
(3)
exists if λ+
n
Mi < Σ.
(4)
i=1
Asymptotic completeness of Rayleigh scattering is the statement that linear combinations of states of the form (3), (4), with ϕ ∈ Hpp (H), are dense in E(−∞,Σ) (H)H. Assuming (1), we prove AC for all m ≥ 0, with an infrared cutoff imposed when m = 0. If m = 0 one can show more: If the Pauli principle for the electrons is neglected then H has a unique ground state ϕ0 [BFS98, GLL00] but no other stationary states with energy below inf σess (K) − ε, ε > 0, provided g is sufficiently small and the life times of all excited states of Hg=0 , as computed by Fermi’s Golden Rule, are finite. This follows from results of Bach et al. [BFSS99] together with an argument given in the present paper, which excludes eigenvalues close to inf σ(H). As a consequence, states of the form inf σ(H) + M (hi ) ≤ inf σess (K) − ε a∗+ (h1 ) . . . a∗+ (hn )ϕ0 , are dense in Einf σess (K)−ε (H)H, for some ε > 0 depending on the coupling constant. Moreover, we show that every state ψt ∈ Einf σess (K)−ε (H)H eventually relaxes to the ground state ϕ0 , in the following sense: Let A denote the C ∗ algebra generated by the Weyl operators eiφ(h) where φ(h) = a(h) + a∗ (h), and h ∈ S(R3 ), the Schwartz space of test functions. By taking sums of tensor products of operators in A with arbitrary bounded operators acting on the N –electron Hilbert ˜ space one obtains a C ∗ algebra A. “Relaxation of ψt to the ground state ϕ0 ” is the statement that lim ψt , Aψt = ϕ0 , Aϕ0 ψ, ψ,
t→∞
(5)
˜ and for all ψ ∈ Einf σ (K)−ε (H)H. This is our second for all operators A ∈ A, ess main result. It essentially follows from asymptotic completeness and, of course, from the absence of eigenvalues in (inf σ(H), inf σess (K) − ε].
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Asymptotic Completeness for massive bosons was previously established by Derezi´ nski and G´erard, for confined electrons (i.e., Σ = ∞) and under a somewhat unphysical short-range assumption [DG99]. From this important paper, and from [DG00], we have learned how to translate techniques from N -body quantum theory to quantum field theory. Before [DG99], Arai had established AC in the standard model of non-relativistic QED in the dipole–approximation and with V (x) = x2 , a model which is explicitly soluble [Ara83]. Later, Spohn extended this result, using the Dyson series, to include potentials which are small perturbations of x2 [Spo97]. Our proof of asymptotic completeness adapts methods and techniques from the scattering theory of N -particle Schr¨ odinger operators to the present situation. In particular, we use a Mourre estimate and propagation estimates, and we rely on localization techniques in bosonic configuration space. As in the more recent papers on N -body quantum scattering, we derive AC from the fact that the mean square diameter dΓ(y 2 )ψt of a given state ψt , with y the position operator in bosonic configuration space, diverges like t2 if ψ is away, in energy, from thresholds and eigenvalues. Correspondingly, a central object in our proof is an asymptotic observable W that measures the square of the asymptotic velocities of the escaping photons. That is, W = lim dΓ(y 2 /2t2 )t t→∞
= lim
t→∞
d dΓ(y 2 /2t)t . dt
(6)
Thanks to the ballistic escape property mentioned above, W is positive and thus invertible, on suitable spectral subspaces. We construct a Deift-Simon wave operator W+ with the property that W+ W −1 is a right-inverse of an extended wave operator on a dense subspace of Hcont (H) ∩ E(−∞,Σ) (H)H, where Hcont (H) is the orthogonal complement of all eigenvectors. The proof is completed with an inductive argument explained further below. Let us temporarily assume that the interaction φ(G) vanishes, in order to explain the ideas underlying the construction of W and W+ in their purest form. The main observation is that D2
1 y2 = (∇ω − y)2 ≥ 0, 2t t
(7)
where D denotes the Heisenberg derivative [iω, .] + ∂/∂t. As a consequence the time derivative of the expression (6), whose limit is W , is non-negative. Since d/dtdΓ(y 2 /2t)t is bounded uniformly in time, it follows that d2 /dt2 dΓ(y 2 /2t)t = dΓ(D2 [y 2 /2t])t is integrable. This propagation estimate, with small modifications to accommodate the interaction, proves existence of W and suffices to establish existence of W as a strong limit. Existence of W+ requires, in addition, some geometry in bosonic configuration space. Once these asymptotic operators are constructed, AC follows by induction in the number of energy intervals of length m, the smallest energy of a boson.
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Assuming that AC holds on the spectral subspace of H corresponding to energies below min(Σ, m(n − 1)), we prove AC for energies below min(Σ, mn). Roughly speaking, the positivity of W on suitable spectral subspaces E∆ (H)H, where ∆ ⊂ (−∞, min(Σ, mn)), allows us to show that at least one boson of a given state ϕ ∈ E∆ (H)H escapes to infinity. It thus carries away an energy of at least m. The energy distribution of the remaining system is contained in (−∞, m(n − 1)), where asymptotic completeness holds by assumption, and hence ϕt , for large t, is of the form (2). Obviously the positivity of the boson mass, or condition (1), in the case of more general dispersion relations, is absolutely crucial in this argument. Our strategy and the constructions of W and W+ are strongly inspired by ideas and constructions developed by Graf and Schenker for N -body quantum scattering theory [GS97]. The Mourre estimate we use is essentially the one of Derezi´ nsky and G´erard [DG99]. We follow closely the notation of [DG99]; but, otherwise, there are only few similarities between our approach to AC and the one in [DG99]. Our paper is organized as follows. In Sect. 2, we consider the quantum theory of the bosons. We briefly review the standard formalism of second quantization and introduce some basic notions that are useful in scattering theory. In Sect. 3, we describe the physical systems and define the models studied in this paper. We formulate some basic assumptions on the Hamiltonians generating the dynamics in these models which will be important to gain mathematical control over Rayleigh scattering. We describe some concrete examples of models. In Sect. 4, we review and prove results on spectral properties of the Hamiltonians of our models. In particular, we recapitulate a theorem on the existence of a ground state and on the location of the essential spectrum. We state a Mourre estimate and properties of the set of thresholds and eigenvalues. In Sect. 5, we construct the Møller wave operators of our models on spectral subspaces corresponding to bound electrons, using a variant of Cook’s argument; see also [FGS00] for more detailed results. Sect. 6 contains our basic propagation estimates needed for the construction of an asymptotic observable and of a Deift-Simon wave operator. An asymptotic observable, W , is constructed in Sect. 7 and shown to be selfadjoint, positive on appropriate spectral subspaces, and to commute with the Hamiltonian H. In Sect. 8, a Deift-Simon wave operator is constructed and shown to invert, with respect to W , an extended variant of the Møller wave operator on spectral subspaces where W is positive. By combining the results of previous sections, asymptotic completeness for Rayleigh scattering is established in Sect. 9 with the help of an inductive argument in the number of asymptotic bosons. In Sect. 10, models of massless bosons with an infrared cutoff are analyzed, and the phenomenon of relaxation to a ground state is exhibited. A novel positivecommutator estimate is proven which, together with results in [BFSS99], excludes
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the existence of point spectrum above the ground state energy (and below the ionization threshold). Our most difficult and innovative results appear in Sects. 6 through 10. Various technical arguments are deferred to appendices, the most important ones being Appendices E through G. Acknowledgements. We thank V. Bach, Chr. G´erard, G.-M. Graf and I. M. Sigal for many useful discussions of problems related to those studied in this paper.
2 Fock Space and Second Quantization The natural Hilbert space of states of the radiation field is the Fock space. Let h be a complex Hilbert space and let ⊗ns h denote the n-fold symmetric tensor product of h. The bosonic Fock space over h F = F (h) = ⊕n≥0 ⊗ns h is the space of sequences ϕ = (ϕn )n≥0 , with ϕ0 ∈ C, ϕn ∈ ⊗ns h, and with the scalar product given by (ϕn , ψn ), ϕ, ψ := n≥0
where (ϕn , ψn ) denotes the inner product in ⊗n h. The vector Ω = (1, 0, . . .) ∈ F is called the vacuum. By F0 ⊂ F we denote the dense subspace of vectors ϕ for which ϕn = 0, for all but finitely many n. The number operator N on F is defined by (N ϕ)n = nϕn .
2.1
Creation- and Annihilation Operators
The creation operator a∗ (h), h ∈ h, on F is defined by √ a∗ (h)ϕ = n S(h ⊗ ϕ), for ϕ ∈ ⊗n−1 h, s and extended by linearity to F0 . Here S ∈ B(⊗n h) denotes the orthogonal projection onto the subspace ⊗ns h ⊂ ⊗n h. The annihilation operator a(h) is the adjoint of a∗ (h) restricted to F0 . Creation- and annihilation operators satisfy the canonical commutation relations (CCR) [a(g), a∗ (h)] = (g, h),
[a# (g), a# (h)] = 0.
In particular [a(h), a∗ (h)] = h 2 , which implies that the graph norms associated with the closable operators a(h) and a∗ (h) are equivalent. It follows that the closures of a(h) and a∗ (h) have the same domain. On this common domain we define 1 (8) φ(h) = √ (a(h) + a∗ (h)). 2
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The creation- and annihilation operators, and thus φ(h), are bounded relative to the square root of the number operator:
a# (h)(N + 1)−1/2 ≤ h
(N + 1)−1/2 a# (h) ≤ h .
(9)
More generally, for any p ∈ R and any integer n
(N + 1)p a# (h1 ) . . . a# (hn )(N + 1)−p−n/2 ≤ Cn,p h1 · . . . · hn . This follows from a∗ (h)N = (N − 1)a∗ (h), a(h)N = (N + 1)a(h), and from (9).
2.2
The Functor Γ
Let h1 and h2 be two Hilbert spaces and let b ∈ B(h1 , h2 ). We define Γ(b) : F (h1 ) → F (h2 ) Γ(b)|` ⊗ns h1 = b ⊗ . . . ⊗ b. In general Γ(b) is unbounded but if b ≤ 1 then Γ(b) ≤ 1. From the definition of a∗ (h) it easily follows that Γ(b)a∗ (h) = a∗ (bh)Γ(b), Γ(b)a(b∗ h) = a(h)Γ(b),
h ∈ h1 . h ∈ h2 .
(10) (11)
where (11) is derived by taking the adjoint of (10). If b∗ b = 1 on h1 then these equations imply that Γ(b)a(h) = a(bh)Γ(b) Γ(b)φ(h) = φ(bh)Γ(b)
2.3
h ∈ h1 h ∈ h1 .
(12) (13)
The Operator dΓ(b)
Let b be an operator on h. Then dΓ(b) : F (h) → F (h) n dΓ(b)|` ⊗ns h = (1 ⊗ . . . b ⊗ . . . 1). i=1
For example N = dΓ(1). From the definition of a∗ (h) we get [dΓ(b), a∗ (h)] = a∗ (bh) [dΓ(b), a(h)] =−a(b∗ h), where the second equation follows from the first one by taking the adjoint. If b = b∗ then i[dΓ(b), φ(h)] = φ(ibh). (14) Note that dΓ(b)(N + 1)−1 ≤ b .
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The Operator dΓ(a, b)
Suppose a, b ∈ B(h1 , h2 ). Then dΓ(a, b) : F (h1 ) → F (h2 ) n dΓ(a, b)|` ⊗ns h = (a . . . a ). ⊗ . . . a ⊗b ⊗ a ⊗ j=1
j−1
n−j
For a, b ∈ B(h) this definition is motivated by Γ(a)dΓ(b) = dΓ(a, ab),
and
[Γ(a), dΓ(b)] = dΓ(a, [a, b]).
If a ≤ 1 then dΓ(a, b)(N + 1)−1 ≤ b .
2.5
The Tensor Product of two Fock Spaces
Let h1 and h2 be two Hilbert spaces. We define a linear operator U : F (h1 ⊕ h2 ) → F (h1 ) ⊗ F(h2 ) by UΩ = Ω ⊗ Ω U a (h) = [a∗ (h(0) ) ⊗ 1 + 1 ⊗ a∗ (h(∞) )]U,
(15)
∗
for h = (h(0) , h(∞) ) ∈ h1 ⊕h2 . This defines U on finite linear combinations of vectors of the form a∗ (h1 ) . . . a∗ (hn )Ω. From the CCRs it follows that U is isometric. Its closure is isometric and onto, hence unitary. It follows that U a(h) = [a(h(0) ) ⊗ 1 + 1 ⊗ a(h(∞) )]U.
(16)
Furthermore we note that U dΓ(b) = [dΓ(b0 ) ⊗ 1 + 1 ⊗ dΓ(b∞ )]U
if
b=
b0 0
0 b∞
(17)
For example U N = (N0 + N∞ )U where N0 = N ⊗ 1 and N∞ = 1 ⊗ N . Let Fn = ⊗ns h and let Pn be the projection from F = ⊕n≥0 Fn onto Fn . Then the tensor product F ⊗ F is norm-isomorphic to ⊕n≥0 ⊕nk=0 Fn−k ⊗ Fk , the corresponding isomorphism being given by ϕ → (ϕn,k )n≥0, k=0..n where ϕn,k = (Pn−k ⊗ Pk )ϕ. In this representation of F ⊗ F and with pi (h(0) , h(∞) ) = h(i) , U becomes U |` ⊗ns (h ⊕ h) =
n 1/2 n k=0
k
p0 ⊗ . . . ⊗ p0 ⊗ p∞ ⊗ . . . ⊗ p∞ .
n−k factors
k factors
(18)
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Factorizing Fock Space in a Tensor Product
Suppose j0 and j∞ are linear operators on h and j : h → h ⊕ h is defined by ∗ h2 and consequently j ∗ j = jh = (j0 h, j∞ h), h ∈ h. Then j ∗ (h1 , h2 ) = j0∗ h1 + j∞ ∗ ∗ j0 j0 + j∞ j∞ . On the level of Fock spaces, Γ(j) : F (h) → F (h ⊕ h), and we define ˘ Γ(j) = U Γ(j) : F → F ⊗ F. ˘ ∗ Γ(j) ˘ It follows that Γ(j) = Γ(j ∗ j) which is the identity if j ∗ j = 1. Henceforth ∗ j j = 1 is tacitly assumed in this subsection. From (10) through (13), (15) and (16) it follows that # ˘ ˘ Γ(j)a (h) = [a# (j0 h) ⊗ 1 + 1 ⊗ a# (j∞ h)]Γ(j) ˘ ˘ Γ(j)φ(h) = [φ(j0 h) ⊗ 1 + 1 ⊗ φ(j∞ h)]Γ(j).
(19) (20)
Furthermore, if ω = ω ⊕ ω on h ⊕ h, then by (17) ˘ Γ(j)dΓ(ω) = U Γ(j)dΓ(ω) = U dΓ(ω)Γ(j) − U dΓ(j, ω j − jω) ˘ ˘ ω j − jω) = [dΓ(ω) ⊗ 1 + 1 ⊗ dΓ(ω)]Γ(j) − dΓ(j,
(21)
˘ b) = U dΓ(a, b) was introduced. In particular Γ(j)N ˘ where the notation dΓ(a, = ˘ Finally we remark that, by (18), (N0 + N∞ )Γ(j). ˘ ` ⊗ns h = Γ(j)|
n 1/2 n k=0
2.7
k
j0 ⊗ . . . ⊗ j0 ⊗ j∞ ⊗ . . . ⊗ j∞ .
n−k factors
(22)
k factors
The “Scattering Identification”
An important role will be played by the scattering identification I : F ⊗ F → F defined by I(ϕ ⊗ Ω) = ϕ I(ϕ ⊗ a (h1 ) · · · a∗ (hn )Ω) = a∗ (h1 ) · · · a∗ (hn )ϕ, ∗
ϕ ∈ F0 ,
and extended by linearity to F0 ⊗ F0 . (Note that this definition is symmetric with respect to the two factors in the tensor product.) There is a second characterization of I which will often be used. Let ι : h ⊗ h → h be defined by ι(h(0) , h(∞) ) = h(0) + h(∞) . Then (23) I = Γ(ι)U ∗ with U as above. To see this consider states of the form ψ = a∗ (h1 ) · · · a∗ (hm )Ω ⊗ a∗ (g1 ) · · · a∗ (gn )Ω m n
= [a∗ (hi ) ⊗ 1] [1 ⊗ a∗ (gj )] Ω ⊗ Ω i=1
j=1
(24)
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in F0 ⊗ F0 . By equations (15) and (10) Γ(ι)U ∗ [a∗ (hi ) ⊗ 1] = Γ(ι)a∗ (hi , 0)U ∗ = a∗ (hi )Γ(ι)U ∗ and similarly Γ(ι)U ∗ [1 ⊗ a∗ (gj )] = a∗ (gj )Γ(ι)U ∗ . Furthermore Γ(ι)U ∗ Ω ⊗ Ω = Ω = I(Ω ⊗ Ω). This shows that Γ(ι)U ∗ ψ = Iψ for ψ given by (24) which proves equation (23). √ Since ι = 2, the operator I is unbounded. Lemma 1 For each positive integer k, the operator I(N + 1)−k ⊗ χ(N ≤ k) is bounded. Let j : h → h ⊕ h be defined by jh = (j0 h, j∞ h) where j0 , j∞ ∈ B(h). If ˘ is a right inverse of I, that is, j0 + j∞ = 1, then Γ(j) ˘ I Γ(j) = 1. ˘ Indeed I Γ(j) = Γ(ι)U ∗ U Γ(j) = Γ(ιj) = Γ(1) = 1.
3 Definition of the System and Basic Assumptions 3.1
The Electron System
The dynamics of the electron system (atom) is given by a self-adjoint operator K on L2 (X), where X is a measure space. Typically X = (Rn , dn x) or X = {1, . . . , n} equipped with the counting measure, in which case L2 (X) = Cn . We assume that K is bounded from below and that inf σess (K) > inf σ(K).
(H0)
In other words, inf σ(K) is an isolated eigenvalue of K. We use |x| to denote the norm of x ∈ X if X is a euclidean space. Otherwise |x| := 0.
3.2
The Radiation Field
Pure states of the radiation field are described by vectors in the bosonic Fock space F (h) over h = L2 (Rd , dk). Their time evolution is generated by the Hamiltonian dΓ(ω), where ω denotes multiplication with a real-valued function ω(k) on Rd . For easy reference, we summarize all further properties of ω in the following assumption. m := inf ω(k) > 0, and ω(k) → ∞ as |k| → ∞, ω ∈ C ∞ (Rd ), and ∂ α ω is bounded for all α = 0, (H1) ∇ω(k) = 0 if k = 0. Remarks. (i) The last condition ensures positivity of the Mourre √ constant away from thresholds. (ii) Typical examples we have in mind are ω(k) = k 2 + m2 and
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smooth dispersions ω that differ from |k| only for small |k|, where ω is chosen such that inf k ω(k) > 0 (see Section 10). (iii) Bosons with non-zero spin or helicity, with h = L2 (Rd , dk) ⊗ Cs , can also be handled. We then k as a pair interpret (k, λ) ∈ Rd × {1, . . . , s} and define integration over k as sλ=1 dd k. Throughout this paper, y denotes the position operator in h, i.e., y = i∇k .
3.3
The Composed System
The dynamics of the composed system of matter (electron system) interacting with the radiation field is given by the Hamilton operator H = K ⊗ 1 + 1 ⊗ dΓ(ω) + φ(G) = H0 + φ(G), where
⊕
φ(Gx ) dx
φ(G) = X
and φ has been defined in (8). It acts on the Hilbert space H = L2 (X) ⊗ F ∼ = ⊕ F dx. For each x ∈ X, G ∈ h, and we assume that x X sup Gx < ∞. x
−1/2
and hence that φ(G)(H0 + i)−1/2 are bounded It follows that φ(G)(N + 1) operators. This implies that φ(G) is infinitesimal w.r.to H0 , which shows that H is self-adjoint on D(H0 ) and bounded from below. All further assumptions are listed below and will be cited upon use. (H2) Exponential decay: There exists an ionization threshold Σ > inf σ(H) with the property that
eα|x|E∆ (H) < ∞, for some α > 0, and for any given closed interval ∆ contained in (−∞, Σ). (H3) Fall-off of the form factor: For arbitrary α > 0, sup e−α|x| (1 + y 2 )Gx < ∞. x
(H4) Dispersion of the form-factor: If h ∈ C0∞ (Rd \{0}) then, for arbitrary α > 0, sup e−α|x| |(Gx , ht )| dt < ∞. x
Here ht = e
−iωt
h.
(H5) Short range condition: There exists a µ > 1 such that sup e−α|x| χ(|y| ≥ R)y n Gx ≤ Cα R−µ+n x
for all
R≥0
for all α > 0 and n ∈ {0, 1, 2}. Further hypotheses (IR), (H6), and (H7) are introduced in Section 10.2.
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Remarks. (i) These hypotheses are satisfied and, with the exception of (H2), easily verified for many concrete systems (see the next section and Sect. 10.3). (ii) Hypothesis (H2) says that the particle system is localized near the origin for small energies. This is the central physical assumption in this paper. Since (H2) can often be derived from (H0) (see [BFS98, GLL00]) it is legitimate to impose (H2). Assumption (H2) makes (H0) obsolete in all of this paper with the exception of Section 10. (iii) In (H5), χ(|y| ≥ R) stands for multiplication with the characteristic function of the set {y : |y| ≥ R}. More generally we use χ(A < c) and χ(A ≤ c) to denote the spectral projections E(−∞,c) (A) and E(−∞,c] (A) of a given self-adjoint operator A. (iv) Hypothesis (H3) follows from (H5). ˜ =H⊗F We conclude this section by defining an extended Hilbert space H and an extended Hamilton operator ˜ = H ⊗ 1 + 1 ⊗ dΓ(ω), H
(25)
which describes a time evolution where the photons in the auxiliary Fock space do not interact with the particles. With the help of (25) and the scattering identification I the time evolution (2) will be described as a unitary time evolution on the extended Hilbert space in Section 5. Moreover the extended Hamilton operator is ˜ we define invaluable in the spectral analysis of H (see Section 4). In analogy to H ˜ H0 = H0 ⊗ 1 + 1 ⊗ dΓ(ω).
3.4
Examples of Concrete Physical Systems
In this section some concrete systems are discussed for which the hypotheses (H0) through (H5) are all satisfied. See also Section 10.3 where we explain how the standard model of non-relativistic QED fits into a slightly expanded version of the general framework introduced above. The example discussed in the introduction, where K is an arbitrary atomic Schr¨ odinger operator, that is, K = −∆ + V on L2 (R3N ), √ satisfying hypothesis (H0), ω(k) = k 2 + m2 , and Gx (k) = g
N
e−ik·xj κ(k),
κ ∈ C0∞ (R3 ),
j=1
satisfies hypotheses (H1) through (H5), for g sufficiently small. In fact, (H2) is proven in [BFS98], with Σ = inf σess (H) − g supx dk |Gx (k)|2 /ω(k), (H3) follows from y 2 = −∆k , (H4) from |(Gx , ht )| ≤ Ct−3/2 , see e.g. [RS79], and (H5) even holds in the strong form sup e−α|x| χ(|y| ≥ R)y n Gx ≤ Cn,µ R−µ x
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for arbitrary n, µ ∈ N. To see this, let f (y) = (1 + |y|)µ κ ˆ (y) and put N = 1 for simplicity (this means that we consider the case with only one electron). Then −2α|x| n 2 −2α|x|
χ(|y| ≥ R)y Gx = e |y n κ ˆ (x − y)|2 dy e |y|≥R −2α|x| =e (1 + |x − y|)−2µ |y n f (x − y)|2 dy. |y|≥R
This decays exponentially as R → ∞ for |x| ≥ R/2, and if |x| < R/2 it is bounded by (1 + R/2)−2µ supx e−2α|x| χ(|y| ≥ R)y n f (x − ·) 2 which is of order O(R−2µ ), because κ ∈ C0∞ (R3 ) by assumption and thus f is rapidly decreasing. The spin-boson model, where K is a hermitian n × n matrix on Cn = L2 (X), with X = {1, . . . , n}, also fits into the our general framework. Suppose ω is as above and Gx ∈ S(R3 ) for all x ∈ X. Then hypotheses (H0) through (H5) are satisfied with the convention that |x| ≡ 0. Hypotheses (H0) and (H2) are trivial, (H4) is seen as in the first example above, and (H3) and (H5) follow from the fact that Gx ∈ S(R3 ), for x = 1, . . . , n.
4 The Spectrum of Pauli-Fierz Hamiltonians 4.1
Essential Spectrum and Existence of a Ground State
Theorem 2 Assume (H1) and (H2) (Sect. 3), and let E = inf σ(H). Then inf σess (H) ≥ min{Σ, E + m}. In particular, inf σ(H) is an isolated eigenvalue of H. Proof. Given λ ∈ σess (H) with λ < Σ we need to show that λ ≥ E + m. Let ∆ be an open interval in R containing λ with sup ∆ < Σ and let (ϕn )n≥0 ⊂ E∆ (H)H with (H − λ)ϕn → 0, ϕn = 1 and ϕn 8 0. Then λ = lim ϕn , Hϕn n→∞
and we estimate the r.h.s from below. Let j0,R , j∞,R ∈ C ∞ (R3 ) be a partition of 2 2 unity defined as in Lemma 32 with j0,R + j∞,R = 1. Pick α > 0 according to (H2) α|x| such that e E∆ (H) is bounded. Then supn eα|x| ϕn < ∞ by assumption on ϕn and hence by Lemma 32 and Lemma 31 ˘ R )Hϕn ˘ R )∗ Γ(j ϕn , Hϕn = ϕn , Γ(j ˜ Γ(j ˘ R )ϕn + o(R0 ) ˘ R )∗ H = ϕn , Γ(j
(26)
uniformly in n. From H ≥ E and dΓ(ω) ≥ m − mχ(N = 0), it is clear that ˜ ≥ (E + m) − mχ(N∞ = 0). H
(27)
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˘ R )∗ χ(N∞ = 0)Γ(j ˘ R ) = Γ(j 2 ) and since Since Γ(j 0,R 2 2 E∆ (H)Γ(j0,R )E∆ (H) = (E∆ (H)eα|x| ) × (e−α|x| Γ(j0,R )E∆ (H))
is compact by Lemma 34 in Appendix E, the equation (26) combined with (27) implies that λ = lim ϕn , Hϕn ≥ E + m + o(R0 ). n→∞
Letting R → ∞ this proves the theorem.
4.2
The Mourre Estimate
Next we establish a type of Mourre theorem with conjugate operator A = dΓ(a) and 1 a = [iω, y 2 /2] = (∇ω · y + y · ∇ω). 2 That is we prove positivity of i[H, A] on spectral subspaces of H away from thresholds and eigenvalues, and, as in N -body quantum theory, we obtain important spectral properties of H as a byproduct. Here the thresholds are the elements of τ := σpp (H) + Nm, (0 ∈ N). The Mourre inequality will allow us to show that dΓ(y 2 )ψt ≥ ct2 ,
as |t| → ∞,
(28)
where c > 0 for states separated in energy from thresholds and eigenvalues. This together with the above mentioned spectral properties suffices to derive AC. As (28) can only be true if the particles are spatially confined, our Mourre estimate only holds for energies below Σ. On a suitable dense subspace, i[H, A] = dΓ(|∇ω|2 ) − φ(iaG). We use this equation to define the operator i[H, A] on ∪µ<Σ Eµ (H)H. Note that φ(−iaG)ϕ makes sense for ϕ ∈ Ran Eµ (H), µ < Σ, thanks to the exponential decay and the boundedness of e−α|x| φ(−iaG)(N + 1)−1/2 . The following virial theorem is an important ingredient in the proof of Theorem 4. Furthermore, in the case of massless bosons and IR-cutoff interaction (see Section 10) it will allow us to prove absence of eigenvalues close to, but different from the ground state energy. Lemma 3 (Virial Theorem) Assume hypotheses (H1), (H2), and (H3). If Hϕ = Eϕ and E < Σ then ϕ, i[H, A]ϕ = 0. The proof of this lemma is deferred to Appendix E.
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The following theorem is the main result of this section. Theorem 4 Assume (H1), (H2) and (H5). Then (i) For each λ ∈ (−∞, Σ)\τ there exists an open interval ∆ constant Cλ , and a compact operator E such that
λ, a positive
E∆ (H)[iH, A]E∆ (H) ≥ Cλ E∆ (H) + E. (ii) Non-threshold eigenvalues in (−∞, Σ) have finite multiplicity and can accumulate only at thresholds. Furthermore τ ∩ (−∞, Σ] is closed and countable. (iii) If λ ∈ (−∞, Σ)\τ is not an eigenvalue, then there exists an open interval ∆ λ and a positive constant Cλ such that E∆ (H)[iH, A]E∆ (H) ≥ Cλ E∆ (H). The proof of this theorem follows the lines of the proof in [DG99] with only minor modifications due to the presence of continuous spectrum in the particle Hamiltonian K. For the sake of completeness we have included a proof of Theorem (4) in this paper, but it is deferred to Appendix E.
5 The Wave Operator ˜ = H ⊗ 1 + 1 ⊗ dΓ(ω) on H ˜ = H ⊗ F, and let Recall from Section 3 that H PB (H) in (29) denote the orthogonal projector onto Hpp (H), the closure of the space spanned by all eigenvectors of H. The purpose of this section is to establish existence of the wave operator ˜
Ω+ = s − lim eiHt Ie−iHt PB (H) ⊗ 1 t→∞
(29)
˜ corresponding to compact intervals ∆ ⊂ (−∞, Σ). on spectral subspaces of H Furthermore we will see that Ω+ is isometric if restricted to vectors in Hpp (H)⊗F. The existence of (29) will essentially follow from the existence of asymptotic field operators (30) a'+ (h)ϕ = lim eiHt a' (ht )e−iHt ϕ t→∞
and the existence of products of such operators, which is established in the next theorem. Theorem 5 Assume hypotheses (H1), (H2) and (H4) are satisfied, and let f, h ∈ L2 (Rd ). i) If ϕ = Eη (H)ϕ for some η < Σ then the limit a'+ (h)ϕ = lim eiHt a' (ht )e−iHt ϕ t→∞
exists. Here ht = e−iωt h.
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ii) The canonical commutation relations [a+ (g), a∗+ (h)] = (g, h)
[a'+ (h), a'+ (g)] = 0,
and
hold true, in form–sense, on χ(H < η)H for all η < Σ. iii) Let m = inf{ω(k) : h(k) = 0} and M = sup{ω(k) : h(k) = 0}. Then a∗+ (h) Ran χ(H ≤ E) ⊂ Ran χ(H ≤ E + M ) a+ (h) Ran χ(H ≤ E) ⊂ Ran χ(H ≤ E − m). iv) Suppose ϕ = Eη (H)ϕ, hi ∈ L2 (Rd ; C) for i = 1, . . . , n and let Mi = sup{ω(k) : hi (k) = 0}. Then a∗+ (h1 ) . . . a∗+ (hn )ϕ = lim eiHt a∗ (h1,t ) . . . a∗ (hn,t )e−iHt ϕ provided that η +
t→∞
n i=1
Mi < Σ.
v) Suppose η < Σ and ϕ ∈ Eη (H)Hpp (H). Then a+ (h)ϕ = 0
for all
h ∈ L2 (Rd ).
√ Remark. For relativistic massive bosons, that is for ω(k) = k 2 + m2 with m > 0, as well as in the case of relativistic electrons and massless bosons, the asymptotic field operators actually exist on a dense subspace of H irrespective of Σ (see [FGS00]). Proof. i) Assume first that h ∈ C0∞ (Rd \{0}). By Cook’s argument it suffices to show that ∞
(Gx , ht )ϕt dt < ∞. (31) 1
This follows from the assumptions (H2) and (H4). For the proofs of ii), iii) and iv) we refer to [FGS00]. v) It suffices to show that a+ (h)ϕ = 0 if Hϕ = Eϕ. Statement v) then follows from the boundedness of a+ (h)Eη (H). Since ht 8 0 weakly as t → ∞ we have s − limt→∞ a(ht )(H + i)−1/2 = 0. Hence a+ (h)ϕ = lim eiHt e−iEt (E + i)1/2 a(ht )(H + i)−1/2 ϕ = 0. t→∞
Next we prove existence of the extended wave operator ˜ ˜ + := s − lim eiHt Ie−iHt Ω t→∞
˜ Since Ω ˜ + agrees with Ω+ on vectors in on a suitable spectral subspace of H. Hpp (H) ⊗ F, this will immediately imply existence of Ω+ .
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Lemma 6 Assume the hypotheses of the theorem above are satisfied. ∗ ∗ a) Suppose n ψ = ϕ ⊗ a (h1 ) . . . a (hn )Ω where ϕ = Eλ (H)ϕ and λ + i=1 Mi < Σ, where Mi is defined as in Theorem 5, iv). Then ˜ ˜ + ψ := limt→∞ eiHt Ie−iHt ψ exists and Ω
˜ + ψ = a∗ (h1 ) . . . a∗ (hn )ϕ. Ω + + ˜ H ˜ for all µ < Σ. ˜ + exists on Eµ (H) b) Ω Proof. Statement a) follows from ˜
e−iHt ϕ ⊗ a∗ (h1 ) . . . a∗ (hn )Ω = e−iHt ϕ ⊗ a∗ (h1,t ) . . . a∗ (hn,t )Ω, the definition of I and Theorem 5, iv). ˜ ˜ is bounded uniformly in t, it suffices to prove (b) Since eiHt Ie−iHt Eµ (H) ˜ H. ˜ By Lemma 30 finite linear com˜ + on a dense subspace of Eµ (H) existence of Ω binations of vectors of the form described in part a) span such a subspace and hence b) follows from a). The following theorem is the main result of this section. Theorem 7 Assume (H1), (H2) and (H4). Then ˜
Ω+ = s − lim eiHt Ie−iHt PB (H) ⊗ 1 t→∞
˜ H, ˜ Ω+ = 1 and hence Ω+ has a unique extension, also exists on ∪µ<Σ Eµ (H) ˜ H. ˜ Ω+ is isometric on (PB (H) ⊗ 1)EΣ (H) ˜ H ˜ and hence denoted by Ω+ , to EΣ (H) Ran Ω+ is closed. Furthermore ˜
eiHt Ω+ = Ω+ eiHt . ˜ H ˜ follows from Lemma 6 b). By Proof. The existence of Ω+ on ∪µ<Σ Eµ (H) Lemma 6 a), the CCR in Theorem 5 ii), and by part v) of that theorem, Ω+ is ˜ H. ˜ So Ω+ is a partial isometry and hence isometric on (PB (H) ⊗ 1) ∪µ<Σ Eµ (H)
Ω+ = 1. All these properties carry over to the closure of Ω+ .
6 Propagation Estimates This section establishes the propagation estimates needed later on to construct the asymptotic observable W (see Eq. (6)) and the Deift Simon wave operator W+ . To begin with, we define a smooth convex function S(y, t) which modifies y 2 /2t for y in a neighborhood of size tδ , δ ∈ (0, 1), of the origin. The operator W will be constructed using S(y, t) in place of y 2 /2t. This will not affect W but allows us to prove its existence.
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Here and henceforth we use the following notation for the various Heisenberg derivatives. Suppose A is an operator in H. Then we define ∂A , ∂t ∂A . D0 A := i[H0 , A] + ∂t Similarly, the Heisenberg derivative d of an operator a on the one boson sector h is defined by ∂a da = i[ω, a] + . ∂t Furthermore, if A(R) is a family of operators on H, and R ∈ R+ , we write DA := i[H, A] +
A(R) = O(Rm )N p
6.1
if
A(R)(N + 1)−p = O(Rm ).
Construction of S(y, t)
Pick m ∈ C0∞ (R+ ) with m ≥ 0, supp(m) ⊂ [1, 2] and dσ m(σ) = 1. Set S0 (y) = dσ m(σ)χ(y 2 /2 > σ) (y 2 /2 − σ). Then S0 is smooth and
S0 (y) =
where b = −
0 y 2 /2 + b
(32)
y 2 /2 ≤ 1 y 2 /2 ≥ 2
dσ m(σ)σ. It follows that S0 (y) = y 2 /2 + a(y) + b
(33)
C0∞ (R3 ).
This formula will allow us to apply Lemma 27 in Appendix A where a ∈ to [iω, S0 ]. It is important that S0 is convex, which is easy to see from the definition. Next we define a scaled version of S0 by S(y, t) = t−1+2δ S0 (y/tδ )
(34)
where 0 < δ < 1. Lemma 8 For all integers n ≥ 0 and all α y2 n α D ∂ S− = O t−(n+1)+δ(2−|α|) . 2t Here D = ∂/∂t or D = ∂/∂t + (y/t) · ∇ and ∂ α is any spacial derivative of order |α|. In particular ∇S =
y + O(t−1+δ ), t
These estimates hold uniformly in y ∈ Rd .
y2 ∂S = − 2 + O(t−2+2δ ). ∂t 2t
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Proof. By definition of S(y, t) and (33), S(y, t) − y 2 /2t = t−1+2δ a(yt−δ ) + t−1+2δ b. The second term clearly enjoys the desired property. For the first term we have ∂ α [t−1+2δ a(yt−δ )] = t−1+δ(2−|α|) (∂ α a)(yt−δ ) where the right hand side is of the form tc h(yt−δ ) with h ∈ C0∞ (R3 ). Both ∂/∂t(tc h) and D(tc h) are again of this form with c replaced by c − 1. This proves the lemma. Lemma 9 Suppose ω ∈ C ∞ (Rd ) and ∂ α ω is bounded for all α = 0. Then dS =
1 ∂S (∇ω · ∇S + ∇S · ∇ω) + + O(t−1 ), 2 ∂t
and for any smooth vector field v(y, t) and Dv = ∂/∂t + v · ∇, d(dS) =(∇ω − v) · S
(∇ω − v) + (Dv ∇S) · (∇ω − v) + (∇ω − v) · (Dv ∇S)
(35) (36)
+ Dv2 S − (Dv v) · ∇S + O(t−1−δ ).
(37)
Proof. The first part follows from (33) and Lemma 27. By definition of the Heisenberg derivative, d(dS) is given by d(dS) = [iω, [iω, S]] + 2[iω, ∂S/∂t] +
∂2S . ∂t2
By Lemma 8, (33), and Lemma 27 ∂S ∂S +∇ · ∇ω + O(t−2 ) ∂t ∂t [iω, [iω, S]] = ∇ωS
∇ω + O(t−1−δ ).
2[iω, ∂S/∂t] = ∇ω · ∇
To prove the second equation an explicit formula for [iω, S] − ∇ω · ∇S is also needed (see the proof of Lemma 27). Since for every smooth vector field v(y, t) ∂S ∂2S ∂S +∇ · ∇ω + 2 ∂t ∂t ∂t =(∇ω − v) · S
(∇ω − v) + (Dv ∇S) · (∇ω − v) + (∇ω − v) · (Dv ∇S)
∇ω · S
∇ω + ∇ω · ∇
+ Dv2 S − (Dv v) · ∇S
(38)
the lemma follows.
6.2
Propagation Estimates
We are now ready to state and prove our propagation estimates. Note that these are basic propagation estimates which are well known in other contexts (see [GS97]).
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Proposition 10 Assume (H1) and (H3), let χ = χ ¯ ∈ C0∞ (R), and suppose α|x| e χ(H) is a bounded operator for some α > 0. Let χ = χ(H) and v = y/t. Then, if λ > 0 is large enough, there exists a constant C such that ∞ 1 dt χψt , χ(λ2 ≤ dΓ(v 2 ) ≤ 2λ2 )χψt ≤ C ψ 2 t 1 for all ψ ∈ H. Here ψt = e−iHt ψ. ˜ and with H and dΓ(v 2 ) Remark. This propagation estimate equally holds on H 2 2 ˜ replaced by H and dΓ(v ) ⊗ 1 + 1 ⊗ dΓ(v ). This is needed for the proof of the remark to Proposition 11. ˜ Proof. h ∈ C0∞ (1/2, 3) with h(r) = 1 on [1, 2], 0 ≤ h ≤ 1 and set h(s) = s 2Pick
˜ ˜ ˜ ˜ ds h (s ). Note that h(s) = h(3) for s ≥ 3. Hence g(s) = h(s) − h(3) for 0 s ≥ 0 and g(−s) = g(s) define a C0∞ -function g on R. The operator B = 2 ˜ −χh(dΓ(v /λ2 ))χ is our propagation observable. Since B is bounded the theorem will follow if we show that DB := [iH, B] +
C ∂B ≥ χ(H) χ(1 ≤ dΓ(v 2 /λ2 ) ≤ 2) χ(H) + integrable terms ∂t t
for some C > 0 if λ is large enough. Henceforth we use the abbreviations h, 2 ˜ ˜ and g to denote the operators h(dΓ(v 2 /λ2 )), h(dΓ(v /λ2 )) and g(dΓ(v 2 /λ2 )), h respectively. Clearly 1 ∂B 2 = χh2 dΓ(v 2 /λ2 ) χ ≥ χh2 χ. ∂t t t
(39)
Next ˜ = χ[idΓ(ω), ˜h]χ + χ[iφ(G), h]χ. ˜ −[iH, B] = χ[iH, h]χ ˜ = [iφ(G), g], Consider first the second term on the right hand side. By [iφ(G), h] the Helffer–Sj¨ ostrand functional calculus (see Appendix A.2), and by (H3) 1 1 ˜
e−α|x|[iφ(G), h](N + 1)− 2 ≤ C e−α|x|φ(−iv 2 G)(N + 1)− 2
C 1 ≤ 2 2 sup e−α|x| φ(iy 2 Gx )(N + 1)− 2 = O(t−2 ). λ t x (40)
˜ is integrable. By Lemma 28 Hence χ[iφ(G), h]χ ˜ = [idΓ(ω), g] [idΓ(ω), h] 1 = g dΓ(∇ω · v/λ + v/λ · ∇ω) + O(t−2 )N λt 1 = hdΓ(∇ω · v/λ + v/λ · ∇ω)h + O(t−2 )N λt C ≤ h(N + 1)h + O(t−2 )N . λt
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Since [χ, h] = O(t−1 ) it follows that ˜ ≤ χ[idΓ(ω), h]χ
C 2 h + O(t−2 ). λt
For C/λ < 1 this in conjunction with (39) proves the proposition. Proposition 11 Assume (H1) and (H5), let χ = χ ¯ ∈ C0∞ (R), and suppose α|x| e χ(H) is a bounded operator for some α > 0. Let f ∈ C0∞ (R), with 0 ≤ f ≤ 1, f (x) = 0 for x ≥ 2, and f (x) = 1 for 0 ≤ x ≤ 1. Denote χ = χ(H) and f = f (dΓ(v 2 /λ2 )), where v = y/t and λ ∈ R. Then, for λ large enough ∞ dt ψt , χf dΓ ((∇ω − v) · S
(∇ω − v)) f χψt ≤ C ψ 2 . 1
˜ and with H, dΓ(v 2 ) and dΓ(P ) Remark. This proposition equally holds on H
˜ dΓ(v 2 ) ⊗ 1 + 1 ⊗ dΓ(v 2 ) and (here P = (∇ω − v) · S (∇ω − v)), replaced by H, dΓ(P ) ⊗ 1 + 1 ⊗ dΓ(P ). This is needed for the proof of Theorem 15. Proof. Let γ(t) = ψt , χf dΓ(dS)f χψt . From ±dS ≤ const(v 2 + 1) it follows that dΓ(dS)f (N +1)−1 is a bounded operator and thus that sup|t|≥1 |γ(t)| < ∞ because of the cut-off χ. Next we show that γ (t) ≥ψt , χf dΓ((∇ω − v) · S
(∇ω − v))f χψt (41) + ψt , B(t)ψt , where B(t) is a bounded operator such that dt|ψt , B(t)ψt | ≤ C ψ 2 . By the Leibnitz rule γ (t) = ψt , χ(Df )dΓ(dS)f χψt + h.c. + ψt , χf (DdΓ(dS))f χψt .
(42)
Only the last term will contribute to (41). DdΓ(dS) = D0 dΓ(dS) + [iφ(G), dΓ(dS)] = dΓ(d(dS)) + φ(−idS G) where d(dS) is given by Lemma 9. Since v = y/t the terms in (37) are of order O(t−1−ε ) where ε = min(δ, 1 − δ). For the terms of (36) we have ± [(Dv ∇S) · (∇ω − v) + (∇ω − v) · (Dv ∇S)] ≤ t2−δ (Dv ∇S)2 + t−2+δ (∇ω − v)2 = O(t−2+δ )(1 + v 2 ) which, thanks to the cutoffs f and χ, gives an integrable contribution to γ (t). This shows that ψt , χf dΓ(d2 S)f χψt ≥ψt , χf dΓ((∇ω − v) · S
(∇ω − v))f χψt + O(t−1−ε ) ψ 2
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To estimate the contribution due to φ(−idS G) use
χf φ(−idSG)f χ ≤ const e−α|x| φ(−idSG)(N + 1)−1/2 . and dS = ∇S · ∇ω − i
2 2 (∂rs S)(∂rs ω) +
r,s
(43)
∂S + O(t−1−δ ). ∂t
This shows that dS = χ(|y| ≥ t )dS + O(t−δ−1 ) and in conjunction with Lemma 8 and (H5) it follows that (43) is integrable. To estimate the contribution in (42) due to Df note that Df = D0 f + [iφ(G), f ]. δ
The second term gives a contribution of order O(t−2 ). This is seen in the same way as (40). Next choose g ∈ C0∞ (R) with supp(g) ⊂ (1, 2) and gf = f and denote g = g(dΓ(v 2 /λ2 )). Then χ(D0 f )dΓ(dS)f χ = χg(D0 f )dΓ(dS)gf χ + O(t−2 ) and hence c |ψt , χ(D0 f )dΓ(dS)f χψt | ≤ g(N + 1)χψt 2 + O(t−2 ) t which is integrable by Proposition 10.
7 The Asymptotic Observable In this section existence of the asymptotic observable W is proved. An auxiliary version Wλ of W will involve a space cutoff at |y| = λt in the bosonic configuration space. W is then obtained in the limit λ → ∞. To define the space cutoff we pick, once and for all, a function f ∈ C0∞ (R) with 0 ≤ f ≤ 1, f (x) = 0 for x ≥ 2 and f (x) = 1 for 0 ≤ x ≤ 1. The space cutoff is the operator f [dΓ(v 2 /λ2 )] or 1 ⊗ f [dΓ(v 2 /λ2 )] on F or H respectively. Here v = y/t and λ ∈ R. For brevity these operators will also be denoted by f if there is no danger of confusion. Since (dS)2 ≤ const(v 4 + 1), which follows from Lemma 9 and Lemma 27, and since f ∈ C0∞ (R), the operator dΓ(dS)f (N + 1)−1 is bounded. Theorem 12 Assume Hypotheses (H1), (H2), and (H5). If χ ∈ C0∞ (R) with supp χ ⊂ (−∞, Σ), then Wλ = s − lim eiHt χf dΓ(dS)f χe−iHt t→∞
exists, is self-adjoint, and commutes with H.
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Proof. To prove existence of Wλ ϕ we use Cook’s argument, i.e. we show that ∞ d dt ψt , χf dΓ(dS)f χϕt ≤ C ψ . dt 1 As in the proof of Proposition 11 one shows that d ψt , χf dΓ(dS)f χϕt = ψt , χf dΓ[(∇ω − v) · S
(∇ω − v)]f χϕt dt + ψt , B(t)ϕt ,
(44)
∞ where B(t) is a bounded t-dependent observable such that 1 dt|ψt , B(t)ϕt | ≤ C ψ
ϕ . Since S
≥ 0 the first term on the right side defines a non-negative sesquilinear form in ψ and ϕ and hence we can apply Schwarz |ψt , χf dΓ[(∇ω − v) · S
(∇ω − v)]f χϕt | ≤ |ψt , χf dΓ[(∇ω − v) · S
(∇ω − v)]f χψt |
1/2
× |ϕt , χf dΓ[(∇ω − v) · S
(∇ω − v)]f χϕt |
1/2
.
This, together with Proposition 11 after an application of H¨ older’s inequality, shows that also the first term in (44) is integrable. This proves existence of Wλ ϕ. Clearly Wλ is bounded and symmetric. To prove that Wλ commutes with H it suffices to show that e−iHt Wλ eiHt = Wλ for all t ∈ R. This follows from −iHt e−iHs Wλ eiHs ϕ − Wλ ϕ = lim eiHt χ[f dΓ(dS)f ]t+s ϕ t χe t→∞
−1 because χ[f dΓ(dS)f ]t+s ). Indeed t χ = O(t
∂ ∂f ∂f ∂ (f dΓ(dS)f ) = f dΓ(dS)f + dΓ(dS)f + f dΓ(dS) ∂t ∂t ∂t ∂t where ∂f /∂t = O(t−1 ) and χf dΓ(d(∂S/∂t))f χ = O(t−1 ). The latter follows from (33) and Lemma 27. In the next step we remove the space cutoff f . This will allow us to prove positivity of W = limλ→∞ Wλ away from thresholds and eigenvalues. Proposition 13 Under the assumptions of Theorem 12, the limit W = limλ→∞ Wλ exists in operator norm sense, and W is given by ϕ, W ψ = lim ϕ, eiHt χdΓ(dS)χe−iHt ψ t→∞
for all ϕ, ψ ∈ D(dΓ(y 2 )) ∩ D(N ). W commutes with H.
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Proof. We pick ϕ, ψ ∈ D(dΓ(y 2 )) ∩ D(N ) and we consider the difference |ϕt , χdΓ(dS)χψt −ϕt , χf dΓ(dS)f χψt | ≤ |ϕt , χ(1 − f )dΓ(dS)f χψt | + |ϕt , χdΓ(dS)(1 − f )χψt | ≤ (1 − f )χϕt
dΓ(dS)f χψt + (1 − f )χψt
dΓ(dS)χϕt . (45) Since (1 − f (s))2 ≤ s2 , we have
(1 − f )χϕt ≤
1 C
dΓ(y 2 /t2 )χϕt ≤ 2 (1/t2 dΓ(y 2 + 1)ϕ + ϕ ), 2 λ λ
(46)
and analogously for ϕ replaced by ψ. The second inequality in (46) follows from Lemma 36 in Appendix F. To handle the factor dΓ(dS)χϕt on the r.h.s. of (45) use that, by Lemma 8 and Lemma 9, dS =
1 y2 (∇ω · y + y · ∇ω) − 2 + O(t−1+δ ). 2t 2t
(47)
Part (iv) of Lemma 37 (with y replaced by y/t) and Lemma 36 in Appendix F lead then to
dΓ(dS)χϕt ≤ C (1/t2 dΓ(y 2 + 1)ϕ + ϕ ). (48) Insering (46) and (48) into (45) and using dΓ(dS)f χψt ≤ C ψ we find |ϕt , χdΓ(dS)χψt − ϕt , χf dΓ(dS)f χψt | ≤
C (1/t2 dΓ(y 2 + 1)ϕ + ϕ ) λ2 × (1/t2 dΓ(y 2 + 1)ψ + ψ ),
for arbitrary ϕ, ψ ∈ D(dΓ(y 2 ))∩D(N ), and for all t ≥ 1. This shows that W exists as a weak limit and that |ϕ, (W − Wλ )ψ| ≤
C
ϕ ψ , λ
which proves that W also exists as a norm limit. Finally, that W commutes with H follows from the fact that Wλ commutes with H for each λ. Using the Mourre inequality from Theorem 4, we next prove positivity of W away from thresholds and eigenvalues. Note that this is the only place where the Mourre inequality is used. Proposition 14 Assume hypotheses (H1), (H2) and (H5) are satisfied. Assume, moreover, that the energy cutoff χ in the definition of W satisfies supp χ ⊂ (−∞, Σ)\S, where S = σpp (H) + m · (N ∪ {0}) is the set of all eigenvalues and thresholds of H. Then W ≥ d χ2 , for some d > 0. In particular, if ∆ ⊂ (−∞, Σ)\S and χ|`∆ = 1, then W ≥ d on RanE∆ (H).
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Proof. By the compactness of σ(H) ∩ (−∞, Σ)\S, and since W commutes with H, it is enough if we prove that, for each x ∈ (−∞, Σ)\S, W |`RanEUx (H) = EUx (H)W EUx (H) ≥ dx EUx (H)χ2 ,
(49)
where Ux is an arbitrarily small neighborhood of x, and dx > 0. If we choose Ux sufficiently small then, by the Mourre estimate (Theorem 4), we can find a function χ ˜ ∈ C0∞ (R) such that χ ˜ = 1 on Ux and ˜ χ(H)[iH, ˜ A]χ(H) ˜ ≥ 2dx χ(H),
(50)
for some dx > 0 (χ ˜ is a smoothed characteristic function of Ux ). Here A = dΓ(a) = ˜ we have EUx (H)χ(H) ˜ = EUx (H), and dΓ([iω, y 2 /2]). Because of the choice of χ thus Eq. (49) follows if we prove that χW ˜ χ ˜ ≥ dx χχ ˜ 2,
(51)
where χ ˜ = χ(H). ˜ To this end we choose ψ ∈ D(dΓ(y 2 )) ∩ D(N ) and let ψt = −iHt e ψ. Then, by Lemma 35, χψ ˜ ∈ D(dΓ(y 2 )) ∩ D(N ) and thus, by Proposition 13, we have ˜ χ ˜ χψt ψ, χW ˜ χψ ˜ = lim ψt , χ χdΓ(dS) t→∞
d ψt , χ χdΓ(S) ˜ χ ˜ χψt , t→∞ dt
(52)
= lim
where the second equality holds because, by Hypothesis (H5), χ[iφ(G), ˜ dΓ(S)]χ ˜= −χφ(iSG) ˜ χ ˜ = o(t−1+δ ). From (52) it now follows that 1 ψt , χ χdΓ(S) ˜ χ ˜ χψt t 1 2 = lim 2 ψt , χ χdΓ(y ˜ /2)χ ˜ χψt , t→∞ t
ψ, χW ˜ χψ ˜ = lim
t→∞
(53)
where, in the second equality, we used the definition of the function S(y). Now we have t 2 2 ˜ /2)χe ˜ −iHt = χdΓ(y ˜ /2)χ ˜+ ds eiHs χ[iH, ˜ dΓ(y 2 /2)]χe ˜ −iHs eiHt χdΓ(y 0 t y2 2 G)χe ˜ −iHs = χdΓ(y ˜ /2)χ ˜− ds eiHs χφ(i ˜ 2 0 t iHs (54) + ds e χA ˜ χe ˜ −iHs 0
t y2 2 G)χe ˜ −iHs = χdΓ(y ˜ /2)χ ˜− ds eiHs χφ(i ˜ 2 0 t s + tχA ˜ χ ˜+ ds dr eiHr χ[iH, ˜ A]χe ˜ −iHr . 0
0
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Note that the operator φ(iy 2 G)χ ˜ is bounded. Moreover the expectation values of 2 χdΓ(y ˜ )χ ˜ and of χA ˜ χ ˜ in the state χψ are finite, because ψ ∈ D(dΓ(y 2 )) ∩ D(N ), ±A ≤ C dΓ(y 2 + 1) and because of Lemma 35 (see Appendix F). Thus, after division by t2 , only the last term in (54) gives a non-vanishing contribution to (53) in the limit t → ∞. By (50) s 1 t ds dr ψr , χ χ[iH, ˜ A]χ ˜ χψr ≥ dx ψ, χ2 χψ, ˜ ψ, χW ˜ χψ ˜ = lim 2 t→∞ t 0 0 which proves Eq. (51) because D(dΓ(y 2 )) ∩ D(N ) is dense in H.
8 Inverting the Wave Operator The Deift-Simon wave operator W+ , to be constructed in this section, inverts the ˜ + with respect to Wλ in the sense that extended wave operator Ω ˜
˜ + W+ = lim eiHt Ie−iHt W+ . Wλ = Ω t→∞
(55)
On spectral subspaces where Wλ is positive and thus invertible, W+ Wλ−1 is then ˜ + . Formally, and when space and energy cutoffs are in fact a right inverse of Ω ignored, then W+ is given by −iHt ˜ ˘ ˘ (56) W+ = s − lim eiHt dΓ(j t , djt )dΓ(S) + Γ(jt )dΓ(dS) e t→∞
˘ t ) = 1 and where jt = (j0,t , j∞,t ) and j0,t + j∞,t = 1. By the last identity, I Γ(j ˘ ˘ IdΓ(jt , djt ) = D0 [I Γ(jt )] = 0. Hence (55) is obvious at least on this formal level. The functions j0,t and j∞,t are constructed as follows. Let j0 , j∞ ∈ C ∞ (Rd ) where j0 (y) = 1 for |y| < 1, j0 (y) = 0 for |y| > 2, and let j∞ = 1 − j0 . Next set j',t (y) = j' (y/ut) where u > 0 is a fixed parameter. By construction of jt and W+ , Eq. (56), photons with velocity u or larger are mapped to the second Fock space where their interaction with the electrons is turned off. First we prove existence of W+ in Theorem 15 and then we prove (55). Theorem 15 together with the Mourre estimate, Theorem 4, is the heart of our proof of AC. Recall from Section 7 that DA, D0 A and da denote Heisenberg derivatives of operators A and a on H and h respectively. If B is an operator on the extended ˜ and if C maps H to H, ˜ then we set Hilbert space H ˜ B] + DB := i[H,
∂B ∂t
˜ := iHC ˜ − CiH + ∂C . DC ∂t ˜ 0 are defined in a similar way using H0 and H ˜ 0 rather The derivatives D0 , and D ˜ than H and H. Finally the Heisenberg derivative db of an operator b mapping the
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Ann. Henri Poincar´e
one-boson sector h to h ⊕ h (that is, bh = (b0 h, b∞ h) with b0,∞ being operators on h) is defined by ∂b db0 ω 0 = db = i b − b iω + . db∞ 0 ω ∂t The function f ∈ C0∞ (R) and the space cutoff denoted by the same letter are defined as in the previous section, and f˜ = f [dΓ(v 2 /λ2 ) ⊗ 1 + 1 ⊗ dΓ(v 2 /λ2 )] is ˜ the space cutoff on H. The next theorem is the main result of this section. Theorem 15 Assume Hypotheses (H1), (H2), and (H5). If χ ∈ C0∞ (R) with supp χ ⊂ (−∞, Σ), then ˜ ˘ t )dΓ(S) f χe−iHt ˜ 0 Γ(j W+ = s − lim eiHt χ ˜f˜D t→∞
˜ Furthermore exists. Here χ = χ(H) and χ ˜ = χ(H). ˜
e−iHt W+ = W+ e−iHt . Proof. By Cook’s argument we need to show that, for all ϕ ∈ H, there exists a constant C such that ∞ d dt ψ, W+ (t)ϕ ≤ C ψ
dt 1 for all ψ ∈ H, where ˜
W+ (t) = eiHt χ ˜f˜Qf χe−iHt ˘ t )dΓ(S) = dΓ(j ˜ 0 Γ(j ˘ t , djt )dΓ(S) + Γ(j ˘ t )dΓ(dS). Q=D In form sense d ˜ ˜ f˜Qf ]χe−iHt W+ (t) = eiHt χ ˜D[ dt ˜ f˜Qf ] = (Df˜)Qf + f˜(DQ)f ˜ D[ + f˜Q(Df ). The contributions due to Df˜ and Df are dealt with as in Proposition 11 and are ˜ is the sum integrable due to Proposition 10. The operator DQ ˜ =D ˜ 0 Q + i(φ(G) ⊗ 1)Q − Qiφ(G) DQ where the last two terms give a contribution of order t−µ because of Hypothesis (H5). To show this write ˘ t )dΓ(S)idΓ(ω) ˘ t )dΓ(S) − Γ(j Q =i [dΓ(ω) ⊗ 1 + 1 ⊗ dΓ(ω)] Γ(j ∂ ˘ − Γ(j t )dΓ(S) ∂t
(57)
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and commute iφ(G) from the right through the terms in (57) using (14) and (20) (before differentiating with respect to t). This is a lengthy computation which leads to ˘ t )dΓ(S) ˜ 0 Γ(j i(φ(G) ⊗ 1)Q − Qiφ(G) =i [φ(j∞ G) ⊗ 1 − 1 ⊗ φ(j∞ G)] D ˘ t) − i [φ(d(j0 S)iG) ⊗ 1 + 1 ⊗ φ(d(j∞ S)iG)] Γ(j ˘ t) ˜ 0 Γ(j − i [φ(j0 SiG) ⊗ 1 + 1 ⊗ φ(j∞ SiG)] D ˘ t )dΓ(S). − [φ(d(j0 )G) ⊗ 1 + 1 ⊗ φ(d(j∞ )G)] Γ(j Using Hypothesis (H5), Eq. (33) and Lemma 27 one shows that each of these terms is of order t−µ . We demonstrate this for the last term. By Lemma 27 t(dj∞ )G = t ∇ω · ∇j∞ − i (∂rs ω)(∂rs j∞ ) + ∂t j∞ G + O(t−2 )G. (58) r,s
Since derivatives of ω are bounded, and since derivatives of j∞ are of order t−1 and live on {|y| ≥ ut}, Equation (58) in conjunction with Hypothesis (H5) implies −α|x| φ(t(dj∞ )G)(N + 1)−1 = O(t−µ ). e On the other hand dΓ(S/t)f (N + 1)−1 is bounded. Hence ˘ t )dΓ(S)f χ χ ˜f˜ 1 ⊗ φ((dj∞ )G)Γ(j =χe ˜ α|x| f˜ 1 ⊗ e−α|x| φ(t(dj∞ )G)(N∞ + 1)−1 (N∞ + 1)(N0 + N∞ + 1)−1 ˘ t )dΓ(S/t)f (N + 1)−1 (N + 1)2 χ × Γ(j is of order t−µ . ˜ 0 Q we work in the representation F ⊗F = To handle the contribution due to D ˜ 0 Q maps ⊕n≥0 ⊕nk=0 Fn−k ⊗ Fk where Fk is the k-boson subspace of F . Since D Fn to ⊕nk=0 Fn−k ⊗ Fk one has ˜ 0 Qf χϕt = ˜f˜D ψt , χ
∞ n
˜ 0 Qβt,n αt,nk , Pn−k ⊗ Pk D
(59)
n=0 k=0
where αt,nk = Pn−k ⊗ Pk f˜χψ ˜ t , βt,n = Pn f χϕt and Pn is the orthogonal projection F → Fn . Note that χ ˜ = χ ˜ χ(N∞ ≤ n∞ ) for some n∞ large enough, and hence ˜ 0 Qβt,n | for each αt,nk = 0 for k > n∞ . Next we estimate |αt,nk , Pn−k ⊗ Pk D given n and k separately. To this end we identify Fn−k ⊗ Fk with a subspace of L(Rdn ). Then, on Fn , the operator ˘ t )dΓ(S) = Jnk Sn =: Snk Pn−k ⊗ Pk Γ(j
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acts by multiplication with a function Jnk Sn where Sn (y, t) = by (22) 1/2 n j0 ⊗ . . . ⊗ j0 ⊗ j∞ ⊗ . . . ⊗ j∞ . Jnk =
k n−k factors
Ann. Henri Poincar´e
n i=1
S(yi , t) and (60)
k factors
˜ 0 Q is given by In terms of Snk the operator Pn−k ⊗ Pk D ˜ 0 Q = D02 Pn−k ⊗ Pk Γ(j ˘ t )dΓ(S) = D02 Snk Pn−k ⊗ Pk D
(61)
where D0 Snk = [iΩ, Snk ] + ∂Snk /∂t and Ω(k1 , . . . , kn ) = ni=1 ω(ki ). Let V = (y1 , . . . , yn )/t ∈ Rdn and let DV = ∂/∂t + V · ∇ denote the material derivative w.r. to V . In the appendix we show that
D02 Snk =(∇Ω − V ) · Snk (∇Ω − V )
+ (DV ∇Snk ) · (∇Ω − V ) + (∇Ω − V ) · (DV ∇Snk ) 1/2 2 2 n O(t−1−δ ) + DV Snk + n k for |y|/t ≤ 2λ (Lemma 40) and that 1/2 n n m α 2 yi /2t = nm+1 O(t−(1+m)+2(δ−|α|) ) Snk − Jnk DV ∂ k i=1
(62)
(63)
(Lemma 39). These are analogs of Lemma 9 and Lemma 8. The binomial factor in (62) and (63) stems from (60) and will be estimated by nk . From Eq. (63) we get for the last term of (62) n yi2 2 2 DV (Jnk Sn ) = Jnk DV Sn − = nk/2+3 O(t−3+2δ ) , (64) 2t i=1 where we used that DV f = 0 for any function f (y, t) which only depends on V and that DV2 (y 2 /2t) = 0. For the second and third term in Eq. (62), Eq. (63) shows that ± (DV ∇Snk ) · (∇Ω − V ) + (∇Ω − V ) · (DV ∇Snk ) ≤ t2−δ (DV ∇Snk )2 + t−2+δ (∇Ω − V )2 = O(t−2+δ )(n + V 2 )nk+4
(65)
Combining (59) and (61) with (62), (64) and (65) we get ∞ n
˜ 0 Qf χϕt ≤ ˜f˜D |αt,nk , (∇Ω − V ) · Snk (∇Ω − V )βt,n | ψt , χ n=0 k=0 ∞ n
+
n=0 k=0
(66)
αt,nk βt,n n
k+4
O(t
−1−ε
)
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where ε = min(1 − δ, δ) > 0. Here nk+1 ≤ nn∞ +1 because αt,nk = 0 for k > n∞ . H¨older’s inequality in the form ∞ n
Ank Bn ≤
n=0 k=0
∞ n
1/2 A2nk
n=0 k=0
∞
1/2 (n + 1)Bn2
(67)
n=0
where Ank , Bn ≥ 0, will be used frequently in the following. It shows that the second term in (66) is bounded from above by f˜χψ ˜ t (N +1)n∞ +5/2 f χϕt O(t−1−ε ) which is integrable. To deal with the first term in (66) we use that
±Snk (y, t)
1/2 n ≤ const × n Sn
(y, t), k 2
for |y| ≤ 2λt and ut1−δ ≥ 2 by Lemma 38. This allows us to estimate the contribu
tion due to Snk χ(|V | ≤ 2λ). The contribution due to Snk χ(|V | > 2λ) is bounded −2 4 by O(t )(n + 1) αt,nk βt,n thanks to the space cutoff f . Together with (66), the Schwarz inequality, and αt,nk = 0 for k > n∞ , this implies that
|αt,nk , (∇Ω − V )·Snk (∇Ω − V )βt,n |
≤ |αt,nk , (∇Ω − V ) · Sn
(∇Ω − V )αt,nk |
1/2
× |βt,n , (∇Ω − V ) · Sn
(∇Ω − V )βt,n |
1/2
nn∞ /2+2
+ O(t−2 )(n + 1)4 αt,nk βt,n
1−δ ≥ 2. Now insert this bound into (66), use that (∇Ω − V ) · Sn
(∇Ω − V ) = for nut
i=1 (∇ω(ki ) − yi /t) · S (∇ω(ki ) − yi /t), sum over k and n, and apply (67) to see that ∞ n
|αt,nk , (∇Ω − V ) · Snk (∇Ω − V )βt,n |
n=0 k=0
12 ≤ f˜χψ ˜ t , dΓ (∇ω − v)S
(∇ω − v) ⊗ 1 + 1 ⊗dΓ (∇ω − v)S
(∇ω − v) f˜χψ ˜ t 1 × f χϕt , N (n∞ +5)/2 dΓ (∇ω − v) · S
(∇ω − v) N (n∞ +5)/2 f χϕt 2 + O(t−2 ) f˜χψ ˜ t (N + 1)9/2 f χϕt . This is integrable w. r. to t by Proposition 11. ˜ To prove the last statement it suffices to show that e−iHs W+ eiHs ϕ = W+ ϕ for all s ∈ R. This follows from ˜ ˜ −iHt e−iHs W+ eiHs ϕ − W+ ϕ = lim eiHt χ[ ˜ f˜Qf ]t+s ϕ t χe t→∞
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if we prove that χ∂/∂t[ ˜ f˜Qf ]χ = O(t−1 ). The contributions due to ∂t f˜ and ∂t f are easily seen to be of order t−1 . As for ∂t Q note that, by Lemma 39, 1/2 ∂ 2 Snk 2 n Pn−k ⊗ Pk ∂t Q = D0 ∂t Snk = ∇Ω · ∇(∂t Snk ) + +n O(t−2 ) k ∂t2 1/2 n = n2 O(t−1 ) k ˜ for |y/t| ≤ 2λ. Use this, (67), and that k ≤ n∞ thanks to the energy cutoff χ. ˜ + W+ as we show in the next lemma. Some By construction of W+ , Wλ = Ω minor technical difficulties in its proof are due to the presence of the cutoffs and due to the unboundedness of I. Lemma 16 Suppose Wλ and W+ are defined as in Theorem 12 and Theorem 15. Then, under the assumptions of these theorems, ˜ ˜ + W+ . Wλ = s − lim eiHt Ie−iHt W+ = Ω t→∞
Proof. By definition of W+ we have, for all ϕ ∈ H, ˜
˜ 0 [Γ(j ˘ t )dΓ(S)]f χϕt + o(1), Ie−iHt W+ ϕ = I χ ˜f˜D
for t → ∞,
(68)
˜
where ϕt = e−iHt ϕ. Note here that e−iHt W+ ϕ is in the domain of I, for all t ∈ R, ˜ + ϕ, for any χ ∈ C ∞ (R), with χ = 1 on supp χ. From because W+ ϕ = χ (H)W 0 (68) it follows now, if we expand the free Heisenberg derivative, that ˜ ˘ t , djt )dΓ(S)f χϕt + I χ ˘ t )dΓ(dS)f χϕt + o(1) Ie−iHt W+ ϕ = I χ ˜f˜dΓ(j ˜f˜Γ(j ˘ t )f dΓ(dS)f χϕt + o(1). ˘ t , djt )dΓ(S)f χϕt + I χ ˜Γ(j = f I χd ˜ Γ(j
(69)
To prove the second equality commute f˜ to the left in the first term, using that, by Lemma 28, [χ, ˜ f˜] = O(t−1 ), and to the right in the second term, using that ˘ t ) = Γ(j ˘ t )f . Choose now χ1 ∈ C0∞ (R), with χ1 = 1 on supp χ, and supp χ1 ⊂ f˜Γ(j (−∞, Σ), and set χ1 = χ1 (H). Then χ1 χ = χ, and thus, by (69), ˜
˘ t , djt )dΓ(S)f χ1 χϕt + I χ ˘ t )f dΓ(dS)f χ1 χϕt + o(1) ˜ dΓ(j ˜ Γ(j Ie−iHt W+ ϕ = f I χ ˘ t , djt )χ1 dΓ(S)f χϕt + I χ ˘ t )χ1 f dΓ(dS)f χϕt + o(1), = f Iχ ˜ dΓ(j ˜ Γ(j (70) where, in the second equality, we used that [f dΓ(S)f, χ1 ]χ = O(t−1 ), for t → ∞, which easily follows, expanding χ1 in a Helffer-Sj¨ ostrand integral, by Hypothesis (H3), by Lemma 28 and because, by assumption, eα|x| χ is a bounded operator, for some α > 0. Below we will show that ˘ t ) χ1 = χ + o (1) and Iχ ˜ Γ(j ˘ t , djt ) χ1 = o (t−1 ). Iχ ˜ dΓ(j
(71) (72)
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Inserting these two equations in (70) it follows, since dΓ(S/t)f χ is uniformly bounded in t, ˜
Ie−iHt W+ ϕ = χf dΓ(dS)f χe−iHt ϕ + o(1) = e−iHt Wλ ϕ + o(1), which proves the lemma. It only remains to prove (71) and (72). We begin proving ˘ t ) = 1H (because, by construction of jt , j0,t + (71). To this end we note that I Γ(j j∞,t = 1) and thus, for any n ∈ N, ˘ t )χ + IE(n,∞) (N∞ )Γ(j ˘ t )χ, ˘ t )χ = IE[0,n] (N∞ )Γ(j χ = I Γ(j
(73)
where E∆ denotes the characteristic function of the set ∆. Now we claim that the norm of the second term on the r.h.s. of the last equation can be made arbitrarily small by choosing n sufficiently large. This follows because ˘ t )χ = IE(n,∞) (N∞ )Γ(j ˘ t )E(n,∞) (N )χ, IE(n,∞) (N∞ )Γ(j ˘ t ) ≤ 1 for all n ∈ N, that which implies, since IE(n,∞) (N∞ )Γ(j ˘ t )χ ≤ IE(n,∞) (N∞ )Γ(j ˘ t ) E(n,∞) (N )(N + 1)−1 (N + 1)χ
IE(n,∞) (N∞ )Γ(j C . ≤ n+2 (74) On the other hand the first term on the r.h.s. of (73) can be written as ˘ t )χχ1 ˘ t )χ = I(N0 + 1)−n E[0,n] (N∞ )(N0 + 1)n Γ(j IE[0,n] (N∞ )Γ(j ˘ t )χ1 + o(1), = I(N0 + 1)−n E[0,n] (N∞ )(N0 + 1)n χ ˜Γ(j
(75)
where the second equality follows because of Lemma 32, and because I(N0 + 1)−n E[0,n] (N∞ ) is a bounded operator (see Lemma 1). Now, for n ∈ N sufficiently large, E[0,n] (N∞ )χ ˜ = χ. ˜ This remark, together with (73), (74) and (75) shows ˘ t )χ1 < ε, for t sufficiently large. This proves (71). that, for any ε > 0, χ − I χ ˜Γ(j ˘ t , djt ) = D0 (I Γ(j ˘ t )) = D0 (1H ) = 0, Eq.(72) follows in a very similar way by IdΓ(j −1 ˘ using that IE(n,∞) (N∞ )dΓ(jt , djt )(N + 1) ≤ const and applying Lemma 33 (see Appendix D). So far the positive parameter u in the construction of W+ was arbitrary. The next lemma now shows that, by choosing u small, we can neglect the possibility to find zero “escaping photons” in states in the range of W+ . This will be important in the proof of asymptotic completeness, Theorem 19. Lemma 17 Assume Hypotheses (H1), (H2) and (H5) hold, and let the Deift-Simon wave operator W+ be defined as in Theorem 15. Then
(1 ⊗ E{0} (N ))W+ ≤ 2 u2 (N + 1)1/2 χ(H) 2 .
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Proof. For ϕ ∈ H and ψ ∈ H⊗F we define ϕt = e−iHt ϕ respectively ψt = e−iHt ψ. Then by definition of W+ , we have ˜ ˘ t )dΓ(S) f e−iHt χϕ ˜ 0 Γ(j ˜f˜D ψ, W+ ϕ = lim ψ, eiHt χ t→∞ (76) ˜ Γ(j ˘ t )dΓ(S) f χϕt . = lim ψt , χ ˜f˜D t→∞
The last equality follows because, by assumption, the operator eα|x| χ is bounded for some α > 0 and because the norm of the operator ˘ t )dΓ(S) − Γ(j ˘ t )dΓ(S)iφ(G) f (N + 1)−2 (77) e−α|x| (iφ(G) ⊗ 1)Γ(j tends to 0, as t → ∞. To see this write the operator in (77) as ˘ t )iφ(G) dΓ(S)f (N + 1)−2 ˘ t ) − Γ(j e−α|x| (iφ(G) ⊗ 1)Γ(j ˘ t )e−α|x| [iφ(G), dΓ(S)]f (N + 1)−2 . + Γ(j
(78)
Now the operator in the first line equals, by (20), − ie−α|x| {φ((j0,t − 1)G) ⊗ 1 + 1 ⊗ φ(j∞,t G)} (N0 + N∞ + 1)−1 ˘ t ) dΓ(S)f (N + 1)−1 , × Γ(j and tends to 0 as t → ∞. This follows because dΓ(S)f (N + 1)−1 = O(t), while, by ˘ t ) is of order O(t−µ ), for some µ > 1. Hypothesis (H5), the factor on the left of Γ(j To handle the operator in the second line of (78) use that [iφ(G), dΓ(S)] = φ(−iSG), and that, by (H5), sup e−α|x| φ(−iSGx )(N + 1)−1 = sup e−α|x| φ(χ(|y| ≥ tδ )iSGx )(N + 1)−1
x
x
= O(t−1+δ ). This implies that also the term in the second line of (78) tends to zero as t → ∞. From (76) it follows now, because terms involving the Heisenberg derivative of f , or f˜ give, by Lemma 28, a vanishing contribution in the limit t → ∞, that d ˘ t )dΓ(S)f χϕt = lim 1 ψt , χ ˘ t )dΓ(S)f χϕt ψt , χ ˜f˜Γ(j ˜f˜Γ(j t→∞ t dt 1 ˘ ∗ = lim Γ(j ˜ t , f dΓ(y 2 /2t)f χϕt , t ) χψ t→∞ t (79)
ψ, W+ ϕ = lim
t→∞
where, in the last equality, we used Lemma 8 to replace S by y 2 /2t. Consider in particular the case ψ = (1 ⊗ E{0} (N ))ψ; then there exists some α ∈ H with ψ = α ⊗ Ω, and ˘ t )∗ χψ ˜ t = I (Γ(j0,t ) ⊗ Γ(j∞,t )) ((χαt ) ⊗ Ω) = Γ(j0,t )χαt . Γ(j
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For such ψ = α ⊗ Ω, (79) reads ψ, W+ ϕ = lim
t→∞
1 αt , χf Γ(j0,t )dΓ(y 2 )f χϕt . 2t2
(80)
Now we note that |αt , χf Γ(j0,t )dΓ(y 2 )f χϕt | = |αt , χf dΓ(j0,t , j0,t y 2 )f χϕt | ≤ (N + 1)1/2 χαt (N + 1)1/2 χϕt j0,t y 2
≤ 4u t α ϕ (N + 1) 2 2
1/2
(81)
2
χ ,
where, in the last step, we used that j0,t y 2 ≤ sup|y|≤2u t y 2 ≤ 4t2 u2 . Since
α = ψ , it follows from (81) and (80) that |ψ, (1 ⊗ E{0} (N ))W+ ϕ| ≤ 2u2 (N + 1)1/2 χ(H) 2 ψ ϕ .
9 Asymptotic Completeness As explained in the introduction we prove asymptotic completeness by induction in the energy measured in units of m. The first step is the following, essentially trivial lemma. The idea is that AC on Ran Eη (H) as explained in the introduction ˜ in Eq. (2), implies the same property for Ie−iHt on Ran Eη (H) ⊗ F, the photons from F merely contributing to the asymptotically free radiation. Lemma 18 Assume that hypotheses (H1), (H2) and (H4) are satisfied, and let the ˜ + and Ω+ be defined as in Lemma 6 and in Theorem 7, respecwave operators Ω ˜ tively. Suppose Ran Ω+ ⊃ Eη (H)H and µ < Σ. Then for every ϕ ∈ Ran Eµ (H) ˜ there exists a vector ψ ∈ Ran Eµ (H) such that ˜ + (Eη (H) ⊗ 1)ϕ = Ω+ ψ. Ω ˜ H ˜ then ψ ∈ E∆ (H) ˜ H. ˜ If ϕ ∈ E∆ (H) ˜ H ˜ can be approximated by a sequence Proof. By Lemma 30 every given ϕ ∈ Eµ (H) ˜ ˜ of vectors ϕn ∈ Eµ (H)H which are finite linear combinations of vectors of the form (82) γ = α ⊗ a∗ (h1 ) . . . a∗ (hn )Ω, n where α = Eλ (H)α and λ + i=1 Mi < µ, with Mi = sup{|k| : hi (k) = 0}. Let ˜ be of the form (82). Then γ∈H ˜
˜
Ie−iHt (Eη (H) ⊗ 1) γ = Ie−iHt Eη (H)α ⊗ a∗ (h1 ) . . . a∗ (hn )Ω = a∗ (h1,t ) . . . a∗ (hn,t ) e−iHt Eη (H)α.
(83)
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˜ and we may assume β = Eη (H)β ˜ By assumption Eη (H)α = Ω+ β for some β ∈ H by the intertwining relation for Ω+ . From (83) it follows that ˜
Ie−iHt (Eη (H) ⊗ 1) γ = a∗ (h1,t ) . . . a∗ (hn,t )e−iHt Ω+ β ˜
= a∗ (h1,t ) . . . a∗ (hn,t )Ie−iHt (PB ⊗ 1) β ˜ + a∗ (h1,t ) . . . a∗ (hn,t ) e−iHt Ω+ β − Ie−iHt (PB ⊗ 1) β , (84) where PB denotes the orthogonal projector onto Hpp (H). After inserting a factor (N + 1)−n/2 (N + 1)n/2 in the second factor on the r.h.s. of last equation, just in front of the braces, we get ˜
˜
Ie−iHt (Eη (H) ⊗ 1) γ−Ie−iHt (PB ⊗ 1) (1 ⊗ a∗ (h1 ) . . . a∗ (hn ))β
≤ a∗ (h1,t ) . . . a∗ (hn,t )(N + 1)−n/2
˜ × (N + 1)n/2 e−iHt Ω+ β − Ie−iHt (PB ⊗ 1)β . The first factor on the r.h.s. is bounded by a finite constant, uniformly in t. The second factor converges to zero as t → ∞. To see this use that it stays bounded for ˜ all integers n, a fact which follows from the boundedness of (H + i)n/2 Ω+ Eµ (H) n/2 ∗ ∗ ˜ ˜ ˜ and of (N + 1) I Eµ (H). Since (1 ⊗ a (h1 ) . . . a (hn ))β ∈ Eµ (H)H it follows that ˜ + (Eη (H) ⊗ 1)γ = Ω+ (1 ⊗ a∗ (h1 ) . . . a∗ (hn ))β. Ω Hence to each ϕn , as defined at the very beginning, there exists a vector ψn ∈ ˜ such that Ω ˜ + (Eη (H) ⊗ 1)ϕn = Ω+ ψn . The left hand side converges to Eµ (H)H ˜ Ω+ (Eη (H) ⊗ 1)ϕ as n → ∞, and hence the right side converges as well. Since Ω+ is isometric on Hpp (H) ⊗ F it follows that (PB ⊗ 1)ψn is Cauchy and hence has ˜ Thus Ω ˜ + (Eη (H) ⊗ 1)ϕ = Ω+ ψ which proves the first part a limit ψ ∈ Eµ (H)H. ˜ + and of the lemma. The second part follows from the intertwining relations for Ω Ω+ . Theorem 19 (Asymptotic Completeness) Assume hypotheses (H1) through (H5) are satisfied. Then Ran Ω+ ⊃ E(−∞,Σ) (H)H. Proof. The proof is by induction. We show that Ran(Ω+ ) ⊃ E(−∞,km) (H)E(−∞,Σ) (H)H
(85)
holds for all integers k with (k − 1)m ≤ Σ. For k < 0 and |k| large enough (85) is trivially correct because H is bounded from below. Hence we may assume (85) holds for k = n − 1. To prove it for k = n it suffices to show that Ran Ω+ ⊃ E∆ (H)H
(86)
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for compact intervals ∆ ⊂ (−∞, nm) ∩ (−∞, Σ)\S because, by Theorem 4, the union of such subspaces is dense in E(−∞,nm) (H)E(−∞,Σ) (H)Hpp (H)⊥ , because, clearly, RanΩ+ ⊃ E(−∞,Σ) (H)Hpp (H) and because RanΩ+ is closed. Now choose χ ∈ C0∞ (R) such that χ = 1 on ∆ and supp χ ⊂ (−∞, Σ)\S. Let λ, u > 0 and define Wλ and W+ in terms of χ, λ and u as in Theorem 12 and 15. Define moreover the asymptotic observable W = limλ→∞ Wλ as in Proposition 13. By Proposition 14, the operator W : E∆ (H)H → E∆ (H)H is onto and hence for every given ϕ = E∆ (H)ϕ there exists a ψ ∈ Ran E∆ (H) such that ϕ = W ψ. Given ε > 0 we pick u, λ small, respectively large enough so that ˜ + χ(N∞ = 0)W+ ψ < ε,
Ω
(W − Wλ )ψ < ε,
(87)
˜ Then by ˜ + E∆ (H). by using Lemma 17, Proposition 13, and the boundedness of Ω Lemma 16 ˜ + W+ ψ = Ω ˜ + χ(N∞ > 0)W+ ψ + Ω ˜ + χ(N∞ = 0)W+ ψ. Wλ ψ = Ω
(88)
The vector χ(N∞ > 0)W+ ψ has at least one boson in the outer Fock space and thus an energy of at most (n − 1)m in the inner one. More precisely χ(N∞ > 0)W+ ψ = χ(H < (n − 1)m) ⊗ χ(N > 0)W+ ψ. ˜ H ˜ such that Hence by induction hypothesis and Lemma 18 there exist γ ∈ E∆ (H) ˜ + χ(N∞ > 0)W+ ψ = Ω+ γ. Ω This equality together with (87) and (88) shows that ˜ + χ(N∞ > 0)W+ ϕ < 2ε
ϕ − Ω+ γ = W ψ − Ω which proves the theorem.
10 Massless Photons This section is devoted to the case where the bosons are massless photons, but the soft modes do not interact with the particles (electrons). That is H = K ⊗ 1 + 1 ⊗ dΓ(|k|) + φ(G) = H0 + φ(G)
(89)
and (IR)
Gx (k) = 0
if
|k| < m
for some m > 0. As before, we assume that Gx ∈ h for each x ∈ X and that supx Gx < ∞. Then φ(G)(H0 + 1)−1/2 is again bounded and hence H is selfadjoint on D(H0 ). The key idea is to compare H with the modified Hamiltonian Hmod = K ⊗ 1 + 1 ⊗ dΓ(ω) + φ(G) where K and G are as above but the dispersion ω(k) = |k| is modified for |k| < m. We choose ω in such a way that (H1) is satisfied (with m/2 instead of m) and ω(k) = |k| for |k| ≥ m.
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˜ = Asymptotic completeness for H is essentially a corollary of Theorem 19. Let H H ⊗ 1 + 1 ⊗ dΓ(|k|) and let Ω+ and Ω+,mod be defined in terms of H, Hmod , and ˜ mod = Hmod ⊗ 1 + 1 ⊗ dΓ(ω). H Theorem 20 (Asymptotic Completeness) Assume (IR), (H2), and (H5) for the ˜ H ˜ and system defined by (89). Then the wave operator Ω+ exists on E(−∞,Σ) (H) Ran(Ω+ ) ⊃ E(−∞,Σ) (H)H. Proof. We split the Fock space into two Fock spaces, one with interacting photons the other one with non-interacting photons. Henceforth the subindices i and s refer to interacting and soft respectively. Let Ki = {k ∈ Rd : |k| ≥ m}
hi = L2 (Ki )
(90)
Ks = {k ∈ R : |k| < m}
hs = L (Ks ).
(91)
d
2
Then h = hi ⊕ hs and correspondingly F (h) is isomorphic to F (hi ) ⊗ F(hs ) with an isomorphism U as given in Section 2.5. By assumption on G and ω and by (17) U Hmod U ∗ = Hi ⊗ 1 + 1 ⊗ dΓ(ωs ) U HU ∗ = Hi ⊗ 1 + 1 ⊗ dΓ(|k|s ) with respect to the factorization H = (L2 (X) ⊗ F(hi )) ⊗ F(hs ). Here Hi = K ⊗ 1 + 1 ⊗ dΓ(|k|) + φ(G) on Hi := L2 (X) ⊗ F(hi ). It follows that Hmod and H have the same eigenvectors, they are of the form U ∗ (ϕi ⊗ Ωs ) where ϕi is an eigenvector of Hi . Hence PB (H) = PB (Hmod ). Furthermore ˜
˜
eiHt Ie−iHt = eiHmod t e−idΓ(ω−|k|)t Iei[dΓ(ω−|k|)⊗1+1⊗dΓ(ω−|k|)]t e−iHmod t ˜
= eiHmod t Ie−iHmod t because Hmod and dΓ(ω − |k|) commute. This shows that Ω+ = Ω+,mod and hence ˜ mod ≤ µ)H ˜ and that Ran Ω+ ⊃ χ(Hmod ≤ µ)H for all that Ω+ exists on χ(H µ < Σ, by Theorem 19. To reformulate these results in terms of Hi we consider U ⊗ U as a map from ˜ mod ≤ F ⊗ F to Fi ⊗ Fi ⊗ Fs ⊗ Fs . Then U Ω+ (U ∗ ⊗ U ∗ ) exists on (U ⊗ U )χ(H ˜ ˜ ˜ ˜ ˜ µ)H ⊃ χ(Hi ≤ µ)Hi ⊗ Ωs ⊗ Ωs , where Hi = Hi ⊗ 1 + 1 ⊗ dΓ(|k|) on Hi = Hi ⊗ Fi , and Ran U Ω+ (U ∗ ⊗ U ∗ ) ⊃ U χ(Hmod ≤ µ)H ⊃ χ(Hi ≤ µ)Hi ⊗ Ωs . Furthermore U Ω+ (U ∗ ⊗ U ∗ ) = Ωi ⊗ (Is (χ(Ns = 0) ⊗ 1s )) ˜
where Ωi = s − limt→∞ eiHi t Ii e−iHi t (PB (Hi ) ⊗ 1). In fact U I(U ∗ ⊗ U ∗ ) = Ii ⊗ Is and U PB (H)U ∗ = PB (Hi ) ⊗ χ(Ns = 0). It follows that U Ω+ (U ∗ ⊗ U ∗ ) exists ˜ i ≤ µ)H ˜ i ⊗ Fs ⊗ Fs ⊃ (U ⊗ U )χ(H ˜ ≤ µ)H ˜ and that its range contains on χ(H ˜ ≤ µ)H ˜ and by the χ(Hi ≤ µ)Hi ⊗ Fs ⊃ U χ(H ≤ µ)H. Hence Ω+ exists on χ(H ˜ ˜ intertwining relation for Ω+ , the range of Ω+ |`χ(H ≤ µ)H contains χ(H ≤ µ)H. The theorem now follows because µ < Σ was arbitrary and Ω+ = 1.
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Relaxation to the Ground State
With the help of AC established in the last section we next show that states below the ionization threshold relax to the ground state in the sense (5) under the dynamics generated by the Hamilton operator (89). We begin by summarizing results due to Bach et al. [BFSS99], [BFS98] on the point spectrum of H that are needed in this section. No infrared cutoff is assumed in the following discussion. Consider the Hamilton operator Hg = K ⊗ 1 + 1 ⊗ dΓ(|k|) + gφ(G) where K = −∆ + V on L2 (Rn ) and V is operator-bounded w.r.t. −∆ with relative bound zero. This assumption allows for typical N -body Schr¨ odinger operators [HS00]. We assume that 1 2 + 1 dk < ∞ (92) sup |Gx (k)| |k| x to ensure self adjointness of Hg on D(Hg=0 ). Following [BFSS99] we furthermore assume that |(k · ∇k )2 Gx (k)|2 <∞ (H6) sup(1 + |x|2 )−M/2 dk |k| x for some M ≥ 0. All excited bound states of Hg will be unstable if their life time as given by Fermi’s Golden Rule is finite. To state this condition we have to introduce some notation. Suppose E0 < E1 < · · · < inf σess (K) are the isolated eigenvalues of K with finite multiplicity. Let mj be the multiplicity of the eigenvalue Ej , and let ϕj,l ∈ L2 (Rn ), for l = 1, . . . mj be an orthonormal base of the eigenspace of K to the eigenvalue Ej . Then for each 0 ≤ i < j and for each k ∈ Rd we define the mi × mj matrix Aij (k) := ϕi,r , Gx (k)ϕj,s . Now, for each j ≥ 0 we define the mj × mj matrix Γj = (93) A∗ij (k)Aij (k)δ(ω(k) − Ej + Ei )dk . i:i<j
The eigenvalues of this matrix are then the resonance widths in second order perturbation theory corresponding to the eigenvalues Ej . To show that Hg has no eigenvalues in neighborhoods of the eigenvalues of the unperturbed Hamiltonian H0 , we therefore need the following assumption: (H7) Fermi Golden Rule. For each j ≥ 1 we have Γj > 0. The following theorem summarizes results from [BFSS99] and [BFS98].
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Theorem 21 i) (Exponential decay, [BFS98]) Suppose µ < inf σess (K) and ε > 0. Then there exists a constant M = M (ε) such that
eα|x|χ(Hg < µ) < M for all α, g with inf σess (K) − µ − α2 − g 2 supx dk |Gx (k)|2 /|k| > ε. ii) (Existence and uniqueness of the ground state, [BFS98, GLL00]) If inf σ(K) < inf σess (K) − g 2 supx dk |Gx (k)|2 /|k| and Gx (−k) = Gx (k) then Eg := inf σ(Hg ) is a non-degenerate eigenvalue of Hg . Moreover if ψg is an eigenvector of Hg corresponding to the eigenvalue Eg , then
Pψg − Pϕ0 ⊗Ω → 0,
as
g → 0,
(94)
where Pψg and Pϕ0 ⊗Ω denote the orthogonal projections onto the spaces spanned by the ground states ψg and ϕ0 ⊗ Ω, respectively. iii) (Absence of excited eigenstates, [BFSS99]) Assume (H6) and (H7). Set ∆ = [E0 + ε, µ], for fixed ε > 0 and µ < inf σess (K). Then σpp (Hg ) ∩ ∆ = ∅ for g > 0 sufficiently small. If the infrared cutoff (IR) is imposed, then assumption (92) simplifies to supx Gx < ∞ and all results of the above theorem then hold for g sufficiently small. Note that m must be small in order for (H7) to hold, because transitions between energy levels Ej with separation less than m are suppressed by the infrared cutoff. Next we prove absence of eigenvalues above and close to Eg for g small, and assuming (IR). As in [BFSS99] we argue by contradiction and prove a virial theorem as well as the positivity of [iH, A] on a spectral interval (Eg , E1 − ε] and for a suitable conjugate operator A. Lemma 22 (Virial Theorem) Assume (IR) and (H3). If Hϕ = Eϕ and E < inf σess (K) − g 2 supx dk |Gx (k)|2 /|k| then ϕ ∈ D(N ) and ϕ, (N − gφ(iaG)) ϕ = 0. Proof. With the notation of the proof of Theorem 20 the eigenvector ϕ is of the form U ∗ (ϕi ⊗ Ωs ). It follows that ϕ is an eigenvector of Hmod and thus in D(Hmod ) ⊂ D(N ). By Theorem 21, part i), eα|x| ϕ ∈ H for some α > 0 and hence Lemma 3 applies to ϕ and Hmod . This shows that ϕ, [dΓ(|∇ω|2 ) − gφ(iaG)]ϕ = 0 which proves the theorem because, by the form of ϕ, dΓ(|∇ω|2 )ϕ = N ϕ.
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Theorem 23 (Positive commutator) Assume (IR), (H0), and (H3). Set ∆ = (Eg , E1 − ε], for some fixed ε > 0 (here E1 is the first point in the spectrum of K above inf σ(K)). Then there is a constant C > 0 such that E∆ (Hg ) (N − g φ(iaG)) E∆ (Hg ) ≥ CE∆ (Hg ), for all g > 0 sufficiently small. In particular, by Lemma 22, it follows that Hg has no eigenvalues in ∆, if g > 0 is small enough. Proof. Using N ≥ 1 − 1 ⊗ PΩ we get E∆ (Hg ) (N − g φ(iaG)) E∆ (Hg ) ≥ E∆ (Hg ) (1 − 1 ⊗ PΩ − g φ(iaG)) E∆ (Hg ) = E∆ (Hg ) − E∆ (Hg ) (1 ⊗ PΩ ) E∆ (Hg ) − gE∆ (Hg )φ(iaG)E∆ (Hg ) −α|x| α|x| 1/2 ≥ E∆ (Hg ) 1 − g sup e
iaGx E∆ (Hg )e
(Ni + 1) E∆ (Hg )
x
− E∆ (Hg ) (1 ⊗ PΩ ) E∆ (Hg ) (95)
where Ni = |k|>m dk a∗ (k)a(k) is the operator counting the number of interacting bosons (which is bounded w.r.t. Hg ), and where PΩ is the orthogonal projector onto Ω. By Hypothesis (H3), and because aGx ≤ const ( yGx + Gx ), the number in the parenthesis in the first term on the r.h.s. of the last equation is larger than C, for any C < 1, if g > 0 is small enough. It remains to show that the last term in (95) converges to 0 as g → 0. To do this we split it into two parts, according to E∆ (Hg ) (1 ⊗ PΩ ) E∆ (Hg ) = E∆ (Hg ) E{E0 } (K) ⊗ PΩ E∆ (Hg ) (96) + E∆ (Hg ) E[E1 ,∞) (K) ⊗ PΩ E∆ (Hg ), where K is the particle Hamiltonian, and E0 and E1 are its ground state energy and its first exited eigenvalue. Since Pψg E∆ (Hg ) = 0 the first term in Eq. (96) can be written as E∆ (Hg ) E{E0 } (K) ⊗ PΩ E∆ (Hg ) = E∆ (Hg ) Pϕ0 ⊗Ω − Pψg E∆ (Hg ), (97) which converges to zero by Theorem 21, part ii). Consider now the second term on the r.h.s. of (96). Choose χ ∈ C∞ 0 (R), such that 0 ≤ χ ≤ 1, χ(s) = 0 if s > E1 − ε/2, and χ(s) = 1 if s ∈ ∆ (this is a smooth version of the characteristic function E∆ ). Then we have on the one hand χ(Hg )E∆ (Hg ) = E∆ (Hg ) and on the other hand E[E1 ,∞) (K) ⊗ PΩ χ(H0 ) = 0. Thus we get E[E1 ,∞) (K) ⊗ PΩ E∆ (Hg ) = E[E1 ,∞) (K) ⊗ PΩ (χ(Hg ) − χ(H0 )) E∆ (Hg ). (98)
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Now if χ ˜ is an almost analytic extension of χ, in the sense of the Helffer–Sj¨ostrand functional calculus (see Appendix A.2), then we have g χ(Hg ) − χ(H0 ) = − ˜ − Hg )−1 φ(G)(z − H0 )−1 . dxdy ∂z¯χ(z π This implies, since χ has a compact support, that
χ(Hg ) − χ(H 0 ) ≤ Cg, for some constant C > 0, and thus, by (98), that E[E1 ,∞) (K) ⊗ PΩ E∆ (Hg ) → 0 as g → 0. This completes the proof of the theorem. The last theorem, together with Theorem 21, proves the following corollary. Corollary 24 Assume Hypotheses (IR), (H3), (H6), and (H7). If µ < inf σess (K), then σpp (Hg ) ∩ (Eg , µ) = ∅, for all g > 0 sufficiently small. With the help of this corollary and Theorem 20 we next prove relaxation to the ground state in the sense of the following theorem. To define the algebra of observables let A denote the C ∗ algebra generated by all Weyl operators W (h) = exp(iφ(h)), with h ∈ L2 (Rd , dk). By taking tensor products of operators in A with bounded operators acting on the Hilbert space Hel = L2 (Rn , dx) of the electrons ˜ one obtains a C ∗ algebra, which we denote by A. Theorem 25 (Relaxation to the ground state) Assume Hypotheses (IR), (H0), and (H3) through (H7). Choose µ < inf σess (K). Then, for sufficiently small values of the coupling constant g > 0, the Hamiltonian Hg exhibits the property of relaxation to the ground state for states with energy less than µ. This means that, if g > 0 is sufficiently small, then, for all A ∈ A˜ and for all ψ ∈ Ran(H ≤ µ), we have lim ψt , Aψt = ψg , Aψg ψ, ψ,
t→∞
(99)
where ψt = e−iHt ψ and ψg denotes the ground state of Hg . Proof. Since the C ∗ algebra A is generated by the Weyl-operators W (h) = eiφ(h) , and because products of Weyl-operators are again Weyl-operators (up to some unimportant phase) it is enough if we prove (99) for A = B ⊗ W (h), where B is a bounded operator on Hel and h ∈ S(Rd ). By Theorem 20 we know that the system we are considering is asymptotically complete. On the other hand we know, from Corollary 24, that the ground state ψg is the only eigenstate of Hg , which lies in the range of the spectral projection χ(Hg ≤ µ). These two results imply that each ψ ∈ Ranχ(Hg ≤ µ) can be written as limit of a sequence of finite linear combinations of states like a∗+ (f1 ) . . . a∗+ (fm )ψg , with fi ∈ S(Rd ). Since we are dealing only with bounded operators, it follows that it is enough to prove (99) for A = B ⊗ W (h) and for ψ=
N i=1
ci a∗+ (f1i )a∗+ (f2i ) . . . a∗+ (fni i )ψg .
(100)
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In this case we have lim e−iHt ψ, Ae−iHt ψ
t→∞
N
=
c¯i cj lim e−iHt
i,j=1 N
=
ni l=1
t→∞
c¯i cj lim t→∞
i,j=1
ni l=1
a∗+ (fli )ψg , Ae−iHt nj m=1
i a∗ (fl,t )ψg , A
nj m=1
j a∗+ (fm )ψg
(101)
j a∗ (fm,t )ψg ,
where we used the definition of the asymptotic fields a∗+ (h). Notice now that the ground state ψg of Hg is in the domain of arbitrary powers of the field-Hamiltonian dΓ(|k|). Moreover we know that the Weyl operators leave D(dΓ(|k|)n ) invariant. This follows by the commutation relations [W (h), dΓ(|k|)] = −φ(i|k|h)W (h) + 1/2 Re (|k|h , h) W (h). These remarks imply that we can rewrite the limit in the r.h.s. of the last equation as lim
t→∞
ni l=1
i a∗ (fl,t )ψg , A
= lim ψg , t→∞
nj m=1 ni l=1
= lim A∗ ψg ,
j a∗ (fm,t )ψg
i a(fl,t ) (B ⊗ W (h))
nj m=1
j a∗ (fm,t )ψg
nj m=1
j a∗ (fm,t )ψg (102) nj ni i ∗ j + lim ψg , B ⊗ m=1 a (fm,t )ψg . l=1 a(fl,t ), W (h)
t→∞
ni l=1
i a(fl,t )
t→∞
If we expand the commutator in the last equation, we get a sum of ni terms. Each i , h)L2 . Now, since we have assumed that of these terms contains a contraction (fl,t i d fl , hr ∈ S(R ), we have i (fl,t , hr ) = dkf i (k)hr (k)ei|k|t → 0 as t → ∞, l and thus the second term on the r.h.s. of (102) vanishes. To handle the first term n on the r.h.s. of (102) we use that, by Lemma 26, limt→∞ l=1 a(fl,t )ψg = 0. Assuming ni ≥ nj , this implies that the first term on the r.h.s. of (102) vanishes if ni > nj , and that lim
t→∞
ni
l=1
i a(fl,t )
nj
m=1
j a∗ (fm,t )ψg = lim ψg ψg , t→∞
i a(fl,t )
ni i l=1 a+ (fl ) ni ∗ i l=1 a+ (fl )ψg ,
= ψg ψg , = ψg
ni l=1
nj m=1
j a∗ (fm,t )ψg
nj ∗ j m=1 a+ (fm )ψg nj ∗ j m=1 a+ (fm )ψg ,
for all ni ≥ nj . Using the antisymmetry of the inner product it is analogously proven that the r.h.s. of (102) vanishes if ni < nj .
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Thus, for arbitrary ni , nj , nj ni
j i )ψg ,A a∗ (fm,t )ψg lim a∗ (fl,t
t→∞
m=1
l=1
= A∗ ψg , ψg
ni l=1
a∗+ (fli )ψg ,
nj m=1
j a∗+ (fm )ψg .
The theorem now follows if we insert this result into (101) and compute the sum over i, j. Lemma 26 Suppose N n > 0 and ϕ ∈ D((Hg + i)n/2 ). Then, if h1 , . . . , hn ∈ L2 (Rd , dk) with hi 2ω = dk |hi (k)|2 (1 + 1/|k|) < ∞, we have lim
t→∞ n i=1
Proof. Notice that
that
n
a(hj,t )ϕ ≤ (
j=1
n
a∗ (hi )(dΓ(|k|) + 1)−n/2 ≤ C
n
a(hj,t ) −
j=1
+
a(hj,t )ϕ = 0.
(103)
j=1
n
n i=1
hi ω . This implies
˜ j,t ))(H + i)−n/2
(H + i)n/2 ϕ
a(h
j=1
n
˜ j,t )ϕ
a(h
j=1
≤
n ˜ j,t ) . . . a(h ˜ n,t ) (H + i)−n/2
a(h1,t ) . . . a(hj,t − h j=1
× (H + i)
n/2
ϕ +
n
˜ j,t )ϕ
a(h
j=1
≤C
n
˜ j ω . . . hn ω +
h1 ω . . . hj − h
j=1
n
˜ j,t )ϕ , a(h
j=1
where we used that (dΓ(|k|) + 1)n/2 (Hg + i)−n/2 < ∞ (see [FGS00]). Because of the last equation it is enough to prove (103) when hj ∈ C0∞ (Rd \{0}). In this case we have M = minj inf{|k| : hj (k) = 0} > 0, and there exists f ∈ C ∞ (Rd ) with f (k) = 0 if |k| < M/2, and f (k) = 1 if |k| ≥ M . Then, on the one hand, dΓ(f ) is bounded w.r.t. Hg (and higher powers of dΓ(f ) are bounded w.r.t. corresponding powers of Hg ). This implies that ϕ ∈ D((dΓ(f ) + 1)n/2 ). On the other hand n −n/2 . Thus, with ψ = (dΓ(f ) + 1)+n/2 ϕ, j=1 a(hj,t ) is bounded w.r.t. (dΓ(f ) + 1) we have n n
a(hj,t )ϕ = a(hj,t )(dΓ(f ) + 1)−n/2 ψ. j=1
j=1
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Since nj=1 a(hj,t )(dΓ(f ) + 1)−n/2 is uniformly bounded in t, it is enough if we show that the r.h.s. of the last equation converges to 0, as t → ∞ for ψ = α ⊗ a∗ (f1 ) . . . a∗ (fm )Ω. To this end we write n
j=1
a(hj,t )(dΓ(f ) + 1)− 2
n
m
a∗ (fi )Ω = (dΓ(f ) + 1 + n)− 2
n
i=1
= (dΓ(f ) + 1 + n)
a(hj,t )
j=1
−n 2
n
n
a(hj,t ) ,
j=1
m
m
a∗ (fi )Ω
i=1
a∗ (fi ) Ω.
i=1
Expanding the commutator, we find a sum of terms containing n contractions (hj,t , fi )L2 . The lemma now follows, because all these contractions converge to 0 as t → ∞.
10.3
QED in Dipole Approximation
As mentioned in the introduction, our methods can be extended to prove AC, as well as our further main results, for atoms described by “non-relativistic QED” in the dipole approximation. In this section we briefly explain how this is accomplished. For an introduction to the standard model of non-relativistic QED and for the justification of the dipole approximation we refer to [BFS98]. Here we merely show how this model fits into our general framework. We consider a non-relativistic electron interacting with the quantized radiation field in Coulomb gauge. (The generalization to N electrons is straightforward.) States of this system are described by vectors in the Hilbert space HT = Hat ⊗ FT , where Hat = L2 (R3 , dx), and FT is the bosonic Fock space over hT = {h ∈ L2 (R3 ; C3 )| k · h(k) = 0 a.e.}, the space of transversal photons. The Hamilton operator is HT = K ⊗ 1 + 1 ⊗ Hf + φ(G)
on HT ,
where K = −∆ + V is assumed to satisfy Hypothesis (H0) and Hf = dΓ(|k|). To describe QED in the dipole approximation, we set Gx (k) = κ(k)g(x)P (k)x
(104)
where x ∈ R3 is the position of the electron, and P (k), for k = 0, is the orthogonal projection onto the plane orthogonal to k ∈ R3 . That is, Pij (k) = δij − ki kj /k 2 . As before we assume κ ∈ C0∞ (R3 ), and κ(k) = 0 if |k| < m, for some m > 0. The factor g ∈ C0∞ (R3 ) is a space cutoff necessary to make Gx bounded as a function of x. From a physical point of view this simplification is legitimate, because the electrons are exponentially localized near the origin (see also [BFS98]). Note that Gx ∈ hT and hence φ(Gx ) is one operator in FT = F (hT ), not a triple of operators as Gx (k) ∈ C3 might suggest.
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The above description of QED in Coulomb gauge is equivalent to the more traditional one in which a choice of polarization vectors ελ (k) ⊥ k, λ = 1, 2, is made and photons are described by functions in L2 (R3 ; C2 ). In fact, using 2 3 2 the polarization vectors an isomorphism L (R ; C ) → hT can be defined by h(k, λ) → λ=1,2 ελ (k)h(k, λ). Under this isomorphism the coupling function Gx (λ, k) = κ(k)g(x)ε∗λ (k) · x, used in the dipol approximation, is mapped to (104). The advantage of our description is that k → Gx (k), unlike Gx (λ, k), is (twice) differentiable as required by Hypothesis (H5). The problem with Gx (λ, k) is that the polarization vectors ελ (k) cannot even be chosen so that they are continuous as functions of k on the unite sphere (an easy application of a famous theorem due to H. Hopf). However the Hilbert space HT is not invariant under the action of many operators which are used in the proof of AC. To circumvent this problem we enlarge the one-boson space by adding hL = {h ∈ L2 (R3 ; C3 )|P (k)h(k) = 0 a.e.}, the space of longitudinal photons. These longitudinal photons do not interact with the electrons and will drop out again at the end. Let H = Hat ⊗ F where F is the bosonic Fock space over h = L2 (R3 ; C3 ) = hT ⊕ hL and let H = K ⊗ 1 + 1 ⊗ Hf + φ(G) on H, where K and G are defined as above. This Hamilton operator is of the form (89) and hypotheses (IR), (H5), and (H2) are satisfied by assumption on κ and K (a proof of (H2) is contained in [GLL00] and [BFS98], see also Theorem 21). Theorem 20 thus tells us that the wave operator ˜
Ω+ = s − lim eiHt Ie−iHt PB (H) ⊗ 1 t→∞
˜ < Σ)H ˜ and that AC holds in the form exists on χ(H ˜ < Σ)H ˜ = χ(H < Σ)H. Ω+ χ(H Using this and the fact that longitudinal photons do not interact with the electron it is now easy to establish AC for HT . To this end recall form Section 2.6 that F is isomorphic to FT ⊗ FL where FL = F (hL ). Hence we may identify F with FT ⊗ FL and F ⊗ F with FT ⊗ FT ⊗ FL ⊗ FL . Then H = HT ⊗ 1L + 1T ⊗ Hf,L on H = HT ⊗ HL , and ˜ = H ⊗ 1 + 1 ⊗ Hf H ˜ T ⊗ (1L ⊗ 1L ) + (1T ⊗ 1T ) ⊗ (Hf,L ⊗ 1L + 1L ⊗ Hf,L ) =H ˜ = H⊗F = H ˜ T ⊗ FL ⊗ FL, where H ˜ T = HT ⊗ FT . Furthermore I = IT ⊗ IL on H as an operator from F ⊗ F = FT ⊗ FT ⊗ FL ⊗ FL to F = FT ⊗ FL , where IT
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and IL are defined in a way analogous to I. Let PL denote the projection onto the vacuum ΩL of FL . Then PB (H) = PB (HT ) ⊗ PL and from eiHf,L t IL e−i(Hf,L ⊗1L +1L ⊗Hf,L )t = IL we get Ω+ = ΩT+ ⊗ ΩL +
˜ < Σ)H ˜ on χ(H
where is the wave operator defined in terms of HT , and ΩL + = IL (PL ⊗ 1L ). It follows that ˜ < Σ) H ˜ T ⊗ ΩL ⊗ ΩL = ΩT χ(H ˜ T < Σ)H ˜ T ⊗ ΩL , (105) Ω+ χ(H + ΩT+
˜ T < Σ)H ˜ T . On the other hand, which show in particular that ΩT+ exists on χ(H using AC for H, the left side of (105) can be written as ˜ < Σ) H ˜ T ⊗ ΩL ⊗ ΩL = (1 ⊗ PL )Ω+ χ(H ˜ < Σ)H ˜ Ω+ χ(H (106) = (1 ⊗ PL )χ(H < Σ)H = χ(HT < Σ)HT ⊗ ΩL which in conjunction with (105) shows that ˜ T < Σ)H ˜ T = χ(HT < Σ)HT . ΩT+ χ(H That is, AC in the sense of Theorem 20 holds for HT .
A
Pseudo Differential Calculus and Functional Calculus
This appendix collects our main tools for computing commutators.
A.1 Pseudo Differential Calculus on Fock Space Lemma 27 Suppose f ∈ S(Rd ), g ∈ C n (Rd ) and sup|α|=n ∂ α g ∞ < ∞. Let p = −i∇. Then i[g(p), f (x)] = i
1≤|α|≤n−1
= (−i)
(−i)|α| α (∂ f )(x)(∂ α g)(p) + R1,n α!
1≤|α|≤n−1
i|α| α (∂ g)(p)(∂ α f )(x) + R2,n α!
where
Rj,n ≤ Cn sup ∂ α g ∞ |α|=n
dk |k|n |fˆ(k)|.
In particular, and most importantly, if n = 2 in the limit ε → 0 i[g(p), f (εx)] = ε∇g(p) · ∇f (εx) + O(ε2 ) = ε∇f (εx) · ∇g(p) + O(ε2 ).
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dk eikx fˆ(k). The first equation follows from g(p)eikx − eikx g(p) = eikx e−ikx g(p)eikx − g(p)
Proof. Let f (x) =
= eikx [g(p + k) − g(p)]
(107)
and Taylor’s formula g(p + k) − g(p) =
(∂ α g)(p)
1≤|α|≤n−1
1
+n
kα α!
dt (1 − t)n−1
0
(∂ α g)(p + tk)k α /α!.
|α|=n
To obtain the second equation write g(p)eikx − eikx g(p) = −[g(p − k) − g(p)]eikx instead of (107). Lemma 28 Suppose ω is in C ∞ (Rd ) and has bounded derivatives. If f ∈ C0∞ (R), then 1 [idΓ(ω), f ] = f dΓ(∇ω · y/t + y/t · ∇ω) + O(t−2 )N t 2 where f = f (dΓ(v )), v = y/t and f = f (dΓ(v 2 )). Proof. The operators dΓ(ω) and f commute with N , hence it suffices to prove the equation on ⊗ns h. On this subspace n n 2 i[dΓ(ω), f ] = vi ) . ω(kj ), f ( j=1
i=1
To evaluate [ω(kj ), f ( ni=1 vi2 )] for given fixed j ∈ {1, . . . , n} consider f ( vi2 ) as a function of yj only, and apply Lemma 27. It follows that n 1 2 ω(kj ), f ( vi ) = 2 f 2yj · ∇ω(kj ) + Rj,t t i=1 where
Rj,t ≤ t−2 sup ∂ α ω ∞ |α|=2
and fˆj (k) = (2π)−d
|k|2 |fˆj (k)|dk
n
e−ik·yj f yj2 +
yl2 dyj .
l=1,l=j
It is easy to see that |fˆj (k)| ≤
Cp (1 + |k|)p
for all
p∈N
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n l=1,l=j
155
yl2 . It follows that Rj,t ≤
n Cn 1 yj /t · ∇ω(kj ) ≤ 2 .
[idΓ(ω), f ] − f 2 t t j=1
(108)
The same equation holds with ∇ω(kj ) · yj /t instead of yj /t · ∇ω(kj ), as follows from [∇ω(kj ), yj /t] = ∆ω(kj )/t = O(t−1 ). In conjunction with (108) this proves the lemma.
A.2 Helffer-Sj¨ ostrand Functional Calculus Suppose f ∈ C0∞ (R; C) and A is a self-adjoint operator. A convenient representation for f (A), which is often used in this paper, is then given by f (A) = −
1 π
dxdy
∂ f˜ (z) (z − A)−1 , ∂ z¯
z = x + iy,
which holds for any extension f˜ ∈ C0∞ (R2 ; C) of f with |∂z¯f˜| ≤ C|y|, f˜(z) = f (z)
and
1 ∂ f˜ (z) = ∂ z¯ 2
∂f ∂f +i ∂x ∂y
(z) = 0
for all
z ∈ R.
(109) Such a function f˜ is called an almost analytic extension of f . A simple example is given by f˜(z) = (f (x) + iyf (x)) χ(z) where χ ∈ C0∞ (R2 ) and χ = 1 on some complex neighborhood of supp f . Sometimes we need faster decay of |∂z¯f˜| as |y| → 0 in the form |∂z¯f˜| ≤ C|y|n . In that case we work with the almost analytic extension n k (iy) χ(z) f˜(z) = f (k) (x) k! k=0
where χ is as above. We call this an almost analytic extension of order n. For more details and extensions of this functional calculus the reader is referred to [HS00] or [Dav95].
˜ < c)H ˜ B Representation of States in χ(H ˜ < c) proved in this section is used in The representation of states in Ran χ(H Lemma 18 and Theorem 20. Lemma 29 Suppose ω(k) ≥ 0 is continuous an let c > 0. Then the space of lin∗ ∗ 2 d ear n combinations of vectors of the form a (h1 ) . . . a (hn )Ω with hi ∈ L (R ) and i=1 sup{ω(k) : k ∈ supp(hi )} < c is dense in χ(dΓ(ω) < c)F .
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Proof. It suffices to prove the assertion in each n-boson sector Fn separately and without Bose-Einstein symmetrization. That is, given n ∈ N, it suffices to show that the space Dc of finite linear combinations of vectors h1 ⊗ . . . ⊗ hn with n n i=1 sup{ω(k) : k ∈ supp(hi )} < c is dense in χS(c) (⊗ h) where χS(c) is the characteristic function of the set * ) n sup{ω(k) : k ∈ supp(hi )} < c . S(c) = (k1 , . . . , kn ) ∈ Rnd : i=1
Let ψ ∈ χS(c) (⊗ h). By standard arguments one constructs functions ψε ∈ C0∞ (Rnd ), ε > 0, with ψε = χS(c−ε) ψε and ψε − ψ → 0 as ε → 0. Now pick f ∈ C0∞ (Rd ) with supp(f ) ⊂ B(0, 1) and Rd f dk = 1, and let fδ (k) = δ −d f (k/δ), where δ > 0. Let Fδ (k1 , . . . , kn ) = ni=1 fδ (ki ). Then Fδ ∗ψε −ψ → 0 as ε, δ → 0, and, by construction of Fδ and ψε , the function k → Fδ (k − k )ψε (k ) belongs to Dc for every given fixed k ∈ supp(ψε ) if δ ≤ δ(ε). Hence the Riemann sums of Fδ(ε) ∗ ψε (k) = Fδ(ε) (k − k )ψε (k )dk
n
Rnd
belong to Dc as well. Since these Riemann sums converge to Fδ(ε) ∗ ψε uniformly in k, by the uniform continuity of Fδ(ε) and ψε , it follows that Fδ(ε) ∗ ψε ∈ D c and hence that ψ = limε→0 Fδ(ε) ∗ ψε ∈ Dc . Lemma 30 Suppose ω(k) = |k| or ω satisfies (H1), and let c > 0. Then the set of all linear combinations of vectors of the form ϕ ⊗ a∗ (h1 ) . . . a∗ (hn )Ω,
λ+
N
Mi < c
(110)
i=1
where ϕ = χ(H < λ)ϕ for some λ < c, n ∈ N and Mi = sup{ω(k) : hi (k) = 0}, is ˜ < c)H. ˜ dense in χ(H ˜ < c)ϕ ⊗ a∗ (h1 ) . . . a∗ (hn )Ω with ϕ ∈ H, Proof. Obviously the set of vectors χ(H ˜ ˜ Thus it suffices to approximate hi ∈ h and n arbitrary is dense in χ(H < c)H. such vectors by vectors of the form (110). In the sense of a strong Stieltjes integral ˜ χ(H < c) = dEλ (H) ⊗ χ(dΓ(ω) < c − λ). This means that ˜ < c)ϕ ⊗ a∗ (h1 ) . . . a∗ (hn )Ω χ(H E∆i (H)ϕ ⊗ χ(dΓ(ω) < c − λi )a∗ (h1 ) . . . a∗ (hn )Ω = lim sup |∆i |→0
i
where ∆i = (λi−1 , λi ]. The lemma now follows from Lemma 29 applied to χ(dΓ(ω) < c − λi )a∗ (h1 ) . . . a∗ (hn )Ω.
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Number–Energy Estimates
Thanks to the positivity of the boson mass, assumption (H1), one has the operator inequality N ≤ aH + b, (111) for some constants a and b. The purpose of this Section is to prove that also higher powers of N are bounded with respect to the same powers of H. This easily follows from (111) if the commutator [N, H] is zero, that is, for vanishing interaction. Otherwise it follow, as we will see, from the boundedness of adkN (H)(H + i)−1 for all k. Lemma 31 Assume the hypotheses (H1) and (H2) and suppose m ∈ N ∪ {0}. i) Then uniformly in z, for z in a compact subset of C,
(N + 1)−m (z − H)−1 (N + 1)m+1 = O(| Im z|−m ). ii) (N + 1)m (H + i)−m is a bounded operator. iii) If χ ∈ C ∞ (R) with supp χ ⊂ (−∞, Σ), then N m eα|x| χ(H) is a bounded operator, provided α > 0 is small enough. Proof. i) This is proved by induction in m. For m = 0 it suffices to show that N (H + i)−1 is bounded, because (H + i)(H − z)−1 = O(| Im(z)|−1 ) for z in a compact subset of C. The operator N (H0 + i)−1 is bounded because of the positivity of the boson mass, assumption (H1). Since φ(G) is infinitesimally small with respect to H0 , it follows that H0 (H + i)−1 is bounded and hence so is N (H + i)−1 . Next assume that the assertion in i) holds for any non–negative integer less than a given m ∈ N. To prove it for m we commute (z−H)−1 with (N +1)m+1 and get (N +1)−m (z − H)−1 (N + 1)m+1 = (N + 1)(z − H)−1 + (N + 1)−m (z − H)−1 [H, (N + 1)m+1 ](z − H)−1 = (N + 1)(z − H)−1 + (N + 1)−m (z − H)−1 m+1 m + 1 × (−1)l (N + 1)m+1−l adlN (H)(z − H)−1 l l=1 m+1 m + 1 −1 −1 = (N + 1)(z − H) + (N + 1) l l=1
× (N + 1)−(m−1) (z − H)−1 (N + 1)m+1−l il φ(il G)(z − H)−1 .
O(| Im z|−m+1 )
By induction hypothesis and because φ(il G)(z−H)−1 is of order O(| Im z|−1 ), for z in a compact set in C, it follows that the r.h.s. of the last equation is of order O(| Im z|−m ).
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ii) Follows directly from i) if we put z = −i and write (N + 1)m (H + i)−m = (N + 1)m (H + i)−1 (N + 1)−m+1 × (N + 1)m−1 (H + i)−1 (N + 1)−m+2 . . . . . . (N + 1)(H + 1)−1 . iii) We begin to write N m eα|x| χ(H) = N m eα|x| (H + i)−m e−α|x| e+α|x| χ(H)(H + i)m = N m (i + e+α|x| He−α|x| )−m e+α|x| χ(H)(H + i)m = N m (i + H + δHα )−m e+α|x| χ(H)(H + i)m , where δHα = 2iαp·x/|x|− 2α/|x|− α2 , and α > 0 is so small that δHα (H + i)−1 < 1 (this ensures that (i + H + δHα ) = (1 + δHα (H + i)−1 )(H + i) is invertible, with a bounded inverse given by (H + i)−1 (1 + δHα (H + i)−1 )−1 ). The last equation implies iii), since N m (i + H + δHα )−m is a bounded operator. This can be shown in the same way we showed that N m (H + i)−m is a bounded operator, using that N commutes with δHα .
D Commutator Estimates 2 Let j0 , j∞ ∈ C ∞ (Rd ) be real-valued with j02 + j∞ ≤ 1, j0 (y) = 1 for |y| ≤ 1 and j0 (y) = 0 for |y| ≥ 2. Given R > 0 set j#,R (y) = j# (y/R) and let jR : h → h ⊕ h denote the operator defined by jR h = (j0,R h, j∞,R h).
Lemma 32 Assume hypotheses (H1), (H2), (H5). Suppose α > 0, m ∈ N∪{0}, and that jR is as above. Suppose also that χ, χ ∈ C0∞ (R), with supp χ ⊂ (−∞, Σ). Then, for R → ∞, ˘ R )H − H ˜ Γ(j ˘ R ) (N + 1)−m−1 = O(R−1 ), i) e−α|x| (N0 + N∞ + 1)m Γ(j ˜ Γ(j ˘ R ) − Γ(j ˘ R )χ(H) χ (H) = O(R−1 ). ii) (N0 + N∞ + 1)m χ(H) Proof. i) From the intertwining relations (20) and (21) we have that ˘ R )H − H ˜ Γ(j ˘ R ) = − dΓ(j ˘ R , adω (jR )) Γ(j ˘ R ). + [φ((j0,R − 1)G) ⊗ 1 + 1 ⊗ φ(j∞,R G)]Γ(j By Lemma 27 adω (jR ) = O(R−1 ). Hence ˘ R , adω (jR ))(N + 1)−m−1 = O(R−1 ). (N0 + N∞ + 1)m dΓ(j
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To see that the other two terms lead to contributions of order O(R−1 ) write (N0 +N∞ + 1)m [φ((j0,R − 1)G) ⊗ 1 + 1 ⊗ φ(j∞,R G)] = [φ((j0,R − 1)G) ⊗ 1 + 1 ⊗ φ(j∞,R G)](N0 + N∞ + 1)m m m + (−i)l [φ(il (j0,R − 1)G) ⊗ 1 + 1 ⊗ φ(il j∞,R G)](N0 + N∞ + 1)m−l , l l=1
˘ R ) = Γ(j ˘ R )(N + 1)j and use Hypothesis (H5). then use (N0 + N∞ + 1)j Γ(j ii) Let χ ˜ be an almost analytic extension of χ of order m, as defined in Sect. A.2. Then we have ˜ Γ(j ˘ R ) − Γ(j ˘ R )χ(H))χ (H) = − 1 dxdy ∂z¯χ ˜ (N0 + N∞ + 1)m (χ(H) π ˜ −1 e−α|x| (H ˜ Γ(j ˘ R ) − Γ(j ˘ R )H) × (N0 + N∞ + 1)m (z − H) × eα|x| χ (H)(z − H)−1 . Then the statement follows by i) because ˜ −1 (N0 + N∞ + 1)−m+1 = O(| Im z|−m ), (N0 + N∞ + 1)m (z − H)
(112)
m α|x|
and because (N + 1) e ciently small.
χ (H) is a bounded operator, provided α > 0 is suffi-
Lemma 33 Assume hypotheses (H1), (H2), (H5). Suppose α > 0, m ∈ N ∪ {0}. Let jR be as above and set djR = [iω, jR ] + ∂jR /∂R. Suppose also that χ, χ ∈ C0∞ (R), with supp χ ⊂ (−∞, Σ). Then ˘ R , djR )H − Hd ˜ Γ(j ˘ R , djR ) (N + 1)−m−2 = i) e−α|x| (N0 + N∞ + 1)m dΓ(j O(R−2 ),
˜ Γ(j ˘ R , djR ) χ (H) = O(R−2 ). ˘ R , djR )χ(H) − χ(H)d ii) (N0 + N∞ + 1)m dΓ(j
Proof. We begin proving part i). To this end note that ˘ R ,djR )H − Hd ˜ Γ(j ˘ R , djR ) dΓ(j ˘ R , djR )dΓ(ω) − (dΓ(ω) ⊗ 1 + 1 ⊗ dΓ(ω))dΓ(j ˘ R , djR ) = dΓ(j
(113)
˘ R , djR ). ˘ R , djR )φ(G) − (φ(G) ⊗ 1)dΓ(j + dΓ(j Consider the term on the first line on the r.h.s. of the last equation. We have ˘ R , djR )|` ⊗n h ˘ R , djR )dΓ(ω)−(dΓ(ω) ⊗ 1 + 1 ⊗ dΓ(ω))dΓ(j dΓ(j s =U (jR ⊗ · · · ⊗ djR ⊗ · · · ⊗ [jR , ω] ⊗ · · · ⊗ jR ) i=j
+
n i=1
(jR ⊗ · · · ⊗ [djR , ω] ⊗ · · · ⊗ jR ) .
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Each one of the n2 terms on the r.h.s. of the last equation is of order O(R−2 ), by Lemma 27. Hence ˘ R , djR )dΓ(ω) (N0 + N∞ +1)m dΓ(j ˘ R , djR ) (N + 1)−m−2 = O(R−2 ). −(dΓ(ω) ⊗ 1 + 1 ⊗ dΓ(ω))dΓ(j Consider now the term on the second line on the r.h.s. of (113). This is given by ˘ R , djR )φ(G)−(φ(G) ⊗ 1)dΓ(j ˘ R , djR ) = (φ((j0,R − 1)G) ⊗ 1 + 1 ⊗ φ(j∞,R G)) dΓ(j ˘ R ). ˘ R , djR ) + (φ(dj0,R G) ⊗ 1 + 1 ⊗ φ(dj∞,R G)) Γ(j × dΓ(j By Hypothesis (H5) it follows, in the same way as in the proof of Lemma 32, that this term gives a contribution of order O(R−µ−1 ). This completes the proof of part i). Part ii) of the Lemma follows in the same way as in Lemma 32.
E
Mourre Estimate
This section contains the proofs of Lemma 3 (the Virial Theorem) and of Theorem 4 (the Mourre Estimate). Recall that A stands for dΓ(a) with a = 1/2(∇ω · y + y · ∇ω). Proof of Lemma 3 (Virial Theorem). First we prove the virial theorem for a regularized variant, Aε , of A = Aε=0 defined on D(H), and then we let ε → 0. Let aε = 1/2(∇ω · yε + yε · ∇ω) where yε = y(1 + εy 2 )−1 and let Aε = dΓ(aε ). Then iHϕ, Aε ϕ − iAε ϕ, Hϕ = ϕ, dΓ(i[ω, aε ]) − φ(iaε G) ϕ (114) for all ϕ ∈ D(K) ⊗ F0 , which is a core for H. Since all operators in (114) are H-bounded this equation extends to D(H). If ϕ is an eigenvector H then the left side of (114) vanishes because Aε ⊂ A∗ε . Thus it suffices to prove that the right side converges to ϕ, i[H, A]ϕ, as ε → 0, for ϕ ∈ Eµ (H)H, µ < Σ. We show that and
φ(iaε G)ϕ → φ(iaG)ϕ,
(115)
dΓ(i[ω, aε ])ϕ → dΓ(|∇ω| )ϕ
(116)
2
as ε → 0. Eq. (115) follows from aε G−aG → 0 and supx e−α|x| φ((a−aε )G)(N + 1)−1/2 < ∞ by Lebesgue’s dominated convergence theorem if α > 0 is chosen in such a way that eα|x| (N + 1)1/2 ϕ belongs to H. Here we used (H2) and (H3). To prove (116) note that i[ω, yε ] → ∇ω strongly and hence that i[ω, aε ] → |∇ω|2 strongly. Since moreover supε i[ω, aε ] < ∞ this proves (116) for all ϕ ∈ D(N ). The property proved in the following lemma has a well known analog, called local compactness, in the theory of Schr¨ odinger operators. We used it in the proof of Theorem 2 and will need it again in the subsequent proof of the Mourre estimate, Thm. 4.
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Lemma 34 Assume (H1) and suppose f ∈ L∞ (X) and g ∈ L∞ (Rd , dy) with
g ∞ ≤ 1. If f (x) → 0, as |x| → ∞, and g(y) → 0, as |y| → ∞ in Rd , then f (x) ⊗ Γ(g)(H + i)−1/2
is compact.
This lemma follows from the fact that p2 and dΓ(ω) are form bounded w.r. to H and from the positivity of the boson mass. Proof. From m > 0 it follows that χ(N > n)Γ(g)(dΓ(ω) + 1)−1/4 → 0 as n → ∞. Furthermore Γ(g)(dΓ(ω) + 1)−1/4 |` ⊗ns h = g(y1 ) . . . g(yn )[ω(k1 ) + . . . + ω(kn ) + 1]−1/4 which is compact. It follows that Γ(g)(dΓ(ω) + 1)−1/4 is a compact operator on F . Since f (x)(p2 + 1)−1/4 is compact on L2 (X) and (p2 + 1)1/4 ⊗ (dΓ(ω) + 1)1/4 (H + i)−1/2 is a bounded operator, the operator f ⊗ Γ(g)(H + i)−1/2 is compact. The proof of Theorem 4 is by induction in the number of energy steps of size m, the minimal energy of a free boson. Assuming that the theorem holds for λ < min(Σ, (n − 1)m) we prove it for λ < min(Σ, nm). To this end we need a partition of unity in the bosonic configuration space. Let j0 , j∞ ∈ C ∞ (Rd ), with 2 j02 + j∞ = 1, and with 0 ≤ j0 ≤ 1, j0 (y) = 1 if |y| ≤ 1 and j0 (y) = 0 if |y| ≥ 2. Given R > 0 we set j#,R (y) = j# (y/R). Each boson h ∈ h will be split into the two parts j0,R h and j∞,R localized near the origin and near infinity respectively. Recall ˘ R ) does the corresponding localization in Fock space. In from Section 2.6 that Γ(j ˘ R ) through H or any bounded function of H Appendix D we saw how to move Γ(j (see Lemma 32). Since [iH, A] has a structure similar to H, with ω and G replaced by |∇ω|2 and by iaG, respectively, it is easy to show that an analogue of Lemma 32 also holds for [iH, A]. In particular we use that ˘ R )[iH, A] − ([iH, A] ⊗ 1 + 1 ⊗ dΓ(|∇ω|2 ))Γ(j ˘ R ) (N + 1)−1 = o(R0 ), e−α|x| Γ(j (117) as R → ∞. Proof of Theorem 4 (Mourre Estimate). First, we introduce the Mourre constants d(λ) :=
Ω⊥
inf
dΓ(|∇ω(k)|2 )
σpp (H)+dΓ(ω(k))=λ
˜ := d(λ)
inf
σpp (H)+dΓ(ω(k))=λ
dΓ(|∇ω(k)|2 ),
where the superscript Ω⊥ in the definition of d means that we exclude the vacuum sector to compute the infimum. Note that d vanishes only on thresholds, while d˜ vanishes on thresholds and on eigenvalues of H. We introduce, moreover, the ˜ For κ > 0, we set ∆κ = [λ − κ, λ + κ], smeared out versions of the functions d, d. λ
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˜ and then we define dκ (λ) = inf µ∈∆κλ d(µ) and d˜κ (λ) = inf µ∈∆κλ d(µ). By definition of these functions we have the inequality Ω⊥ inf d˜κ (λ − dΓ(ω(k))) + dΓ(|∇ω(k)|2 ) ≥ dκ (λ).
(118)
For all n ∈ N we will show the following statements. H1 (n): Let ε > 0 and λ ∈ [E0 , E0 + nm) ∩ (−∞, Σ). Then there exists an open interval ∆ λ and a compact operator E such that E∆ (H)[iH, A]E∆ (H) ≥ (d(λ)− ε)E∆ (H) + E. H2 (n): Let ε > 0 and λ ∈ [E0 , E0 + nm) ∩ (−∞, Σ). Then there exists an open ˜ − ε)E∆ (H). interval ∆ λ such that E∆ (H)[iH, A]E∆ (H) ≥ (d(λ) H3 (n): Let κ, ε0 , ε > 0. Then there exists δ > 0 such that for all λ ∈ [E0 , E0 + nm − ε0 ] ∩ (−∞, Σ), one has E∆κλ (H)[iH, A]E∆κλ (H) ≥ (d˜κ (λ) − ε)E∆κλ (H). S1 (n): τ is a closed and countable set in [E0 , E0 + nm) ∩ (−∞, Σ). S2 (n): For any closed intervall I ⊂ [E0 , E0 + nm) ∩ (−∞, Σ), with I ∩ τ = ∅, one has dim Ran EI∩σpp (H) < ∞. Note here that all the claims of the theorem follow from these statements, if we prove them for any n ∈ N. Actually, the statements give no new information if n is so large that E0 + nm > Σ. We will prove these statements by induction in n. Since H1 (1) and S1 (1) are obvious, all the statements, for any n ∈ N, follow if we prove the implications: H1 (n) ⇒ H2 (n), H2 (n) ⇒ H3 (n), H1 (n) ⇒ S2 (n), S2 (n − 1) ⇒ S1 (n), and S1 (n) ∧ H3 (n − 1) ⇒ H1 (n). Now the implication S2 (n − 1) ⇒ S1 (n) is trivial. Moreover the implications H1 (n) ⇒ H2 (n), H2 (n) ⇒ H3 (n) and H1 (n) ⇒ S2 (n) are proved by standard abstract arguments, which are typical in the proof of any Mourre inequality. It only remains to prove that S1 (n) together with H3 (n − 1) imply the statement H1 (n). To this end we fix λ ∈ [E0 , E0 + nm) ∩ (−∞, Σ) and ε > 0. Choose now χ ∈ C0∞ (R) with supp χ ⊂ [λ−δ, λ+δ]∩(−∞, Σ), where δ is some positive constant which will be fixed later on. Then we have χ(H)[iH, A]χ(H) ˘ R )∗ E{0} (N∞ )Γ(j ˘ R )χ(H)[iH, A]χ(H) = Γ(j ˘ R )χ(H)[iH, A]χ(H) ˘ R )∗ E[1;∞) (N∞ )Γ(j + Γ(j (119) + , ∗ 2 ˘ ˜ = Γ(qR )χ(H)[iH, A]χ(H) + Γ(jR ) χ(H) [iH, A] ⊗ 1 + 1 ⊗ dΓ(|∇ω| ) ˘ R ) + o(R0 ), ˜ [1;∞) (N∞ )Γ(j × χ(H)E ˘ R ) to the right. The first where we used Lemma 32 and Eq. (117) to commute Γ(j term on the r.h.s. of (119) is compact, by Lemma 34. To handle the second term we want to diagonalize 1 ⊗ dΓ(ω) and 1 ⊗ dΓ(|∇ω|2 ) on the range of E[1;∞) (N∞ ). Using the induction hypothesis S1 (n) we find κ > 0 such that dκ (λ) ≥ d(λ) − ε/3.
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Then, by H3 (n − 1), we know that if δ > 0 is small enough we have + , E∆δλ (H + dΓ(ω(k))) [iH, A] ⊗ 1 + 1 ⊗ dΓ(|∇ω(k)|2 ) E∆δλ (H + dΓ(ω(k))) ε ×E[1;∞) (N∞ ) ≥ E∆δλ (H + dΓ(ω(k))) d˜κ (λ − dΓ(ω(k))) + dΓ(|∇ω(k)|2 ) − 3 ε ×E[1;∞) (N∞ ) ≥ (dκ (λ) − )E∆δλ (H + dΓ(ω(k)))E[1;∞) (N∞ ) 3 2J ≥ (d(λ) − )E∆δλ (H + dΓ(ω(k)))E[1;∞) (N∞ ), 3 where, in the second inequality we used (118). It follows, since supp χ ⊂ ∆δλ , that , + ˜ [1;∞) ≥ (d(λ) − 2J )χ2 (H)E ˜ [1;∞) (N∞ ). ˜ [iH, A] ⊗ 1 + 1 ⊗ dΓ(|∇ω|2 ) χ(H)E χ(H) 3 Now we insert this in the second term on the r.h.s. of (119) and then we com˘ R ) back to the left. Using that mute, using again Lemma 32 and Eq. (117), Γ(j ˘ R )∗ E[1,∞) (N∞ )Γ(j ˘ R ) = 1 − Γ(j 2 ) and that, by Lemma 34, Γ(j 2 )χ(H) is a Γ(j 0,R 0,R compact operator, we find, from (119), χ(H)[iH, A]χ(H) ≥ (d(λ) −
2ε 2 )χ (H) + E + o(R0 ). 3 δ/2
The statement H1 (n) then follows if we choose χ such that χ = 1 on ∆λ , if we multiply (119) from the right and from the left with E∆δ/2 (H) and if we choose R λ sufficiently large.
F Invariance of Domains In this section the invariance of the domain of dΓ(y 2 ) with respect to the action of χ(H) for smooth functions χ is proven. Moreover we prove in Lemma 36 that the norm of dΓ(y 2 /t2 )χ(H)e−iHt ϕ remains uniformly bounded for all t ≥ 1, if ϕ ∈ D(dΓ(y 2 + 1)). These results are used in Proposition 13 to prove the existence of the asymptotic observable W . Lemma 35 Assume Hypotheses (H1), (H2), (H3) are satisfied. Suppose moreover that ϕ ∈ D(dΓ(y 2 )) ∩ D(N ) and that χ ∈ C ∞ (R) with supp χ ⊂ (−∞, Σ). Then we have
dΓ(y 2 )χ(H)ϕ ≤ C( dΓ(y 2 + 1)ϕ + ϕ ). Proof. Find χ1 ∈ C0∞ (R) with χχ1 = χ, and supp χ1 ⊂ (−∞, Σ). Put χ = χ(H) and χ1 = χ1 (H). Then we have dΓ(y 2 )χ = dΓ(y 2 )χχ1 = χdΓ(y 2 )χ1 + [dΓ(y 2 ), χ]χ1 .
(120)
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Now expand the χ in the commutator in an integral, according to the Helffer– Sj¨ ostrand functional calculus (see Appendix A.2). We get 2i 2 [dΓ(y ), χ]χ1 = − ˜ (z − H)−1 dΓ(a)(z − H)−1 χ1 dxdy ∂z¯χ π (121) i dxdy ∂z¯χ + ˜ (z − H)−1 φ(iy 2 G)χ1 (z − H)−1 , π where χ ˜ is an almost analytic extension of χ, in the sense of the Helffer-Sj¨ostrand functional calculus, and a = [iω(k), y 2 /2] = 1/2(∇ω · y + y · ∇ω). The second term on the r.h.s. of the last equation is bounded, because, by hypotheses (H2) and (H3), φ(y 2 G)χ1 < ∞. To handle the first term on the r.h.s. of (121) we commute the factor dΓ(a) to the right of the resolvent (z − H)−1 . Using [dΓ(a), H] = idΓ(|∇ω|2 ) − iφ(iaG), we see that the contributions arising from this commutator are bounded, because |∇ω| < const and because, using again (H2) and (H3), φ(iaG)χ < ∞. Thus we have, from (120) dΓ(y 2 )χ = χdΓ(y 2 )χ1 + C χ (H)dΓ(a)χ1 + bounded,
(122)
where χ is the first derivative of χ. Now we commute the two operators dΓ(y 2 ) and dΓ(a) in the two terms on the r.h.s. of the last equation to the right of χ1 . For example , for the term χdΓ(y 2 )χ1 , we find χdΓ(y 2 )χ1 = χχ1 dΓ(y 2 ) + χ[dΓ(y 2 ), χ1 ] 1 = χdΓ(y 2 ) − ˜1 χ(z − H)−1 [dΓ(y 2 ), H](z − H)−1 dxdy ∂z¯χ π (123) 2i = χdΓ(y 2 ) − 2 ˜1 χ(z − H)−1 dΓ(a)(z − H)−1 dxdy ∂z¯χ πt i + ˜1 (z − H)−1 χφ(iy 2 G)(z − H)−1 . dxdy ∂z¯χ π The third term on the r.h.s. of the last equation is bounded by (H2) and (H3). To handle the second term we commute dΓ(a) to the right. We get 2i ˜1 χ(z − H)−1 dΓ(a)(z − H)−1 − dxdy ∂z¯χ π 2i = − ˜1 χ(z − H)−2 dΓ(a) dxdy ∂z¯χ π 2 + ˜1 (z − H)−2 χdΓ(|∇ω|2 )(z − H)−1 dxdy ∂z¯χ π 2 − ˜1 (z − H)−2 χφ(iaG)(z − H)−1 . dxdy ∂z¯χ π The first term on the r.h.s. of the last equation is proportional to χχ 1 dΓ(a), where χ 1 is the first derivative of χ1 , and thus vanishes, since χ1 is constant on supp χ.
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The other two terms on the r.h.s. of the last equation are bounded. It follows, by (123), that χdΓ(y 2 )χ1 = χdΓ(y 2 ) + bounded. Similarly we find, that χ dΓ(a)χ1 = χ dΓ(a) + bounded. These two results imply, by (122), that + ,
dΓ(y 2 )χϕ ≤ C dΓ(y 2 )ϕ + dΓ(a)ϕ + ϕ . (124) The Lemma now follows using Lemma 37, part (iv), to estimate the second term on the r.h.s. of the last equation. Lemma 36 Assume hypotheses (H1), (H2) and (H3) are satisfied. Suppose moreover that ϕ ∈ D(dΓ(y 2 )) ∩ D(N ) and that χ ∈ C ∞ (R) with supp χ ⊂ (−∞, Σ). Then we have
dΓ(y 2 )e−iHt χ(H)ϕ ≤ C( dΓ(y 2 + 1)ϕ + t2 ϕ ), for all t ≥ 1. Proof. We begin by noting, that eiHt dΓ(y 2 )e−iHt χ(H) − dΓ(y 2 )χ(H) = =2
t
ds eiHs [iH, dΓ(y 2 )] χ(H)e−iHs
0 t
ds e
iHs
dΓ(a) χ(H)e
−iHs
0
−
t
ds eiHs φ(iy 2 G) χ(H)e−iHs ,
0
(125) where a = [iω, y 2 /2] = 1/2(∇ω · y + y · ∇ω). The second integral on the r.h.s. of the last equation is bounded, with norm of order t, because, by (H2) and (H3),
φ(iy 2 G)χ < ∞. To handle the first integral on the r.h.s. of the last equation use the expansion t 2 ds eiHs dΓ(a)χ(H)e−iHs 0 t s = 2t dΓ(a)χ(H) + 2 ds dr eiHr [iH, dΓ(a)] χ(H)e−iHr 0
0
t
s
= 2t dΓ(a)χ(H) + 2 ds dr eiHr dΓ(|∇ω|2 ) χ(H)e−iHr 0 0 t s −2 ds dr eiHr φ(iaG) χ(H)e−iHr . 0
0
Since |∇ω| is bounded and φ(iaG)χ(H) < ∞, both the integrals on the r.h.s. of the last equation are bounded and of order t2 . This implies, by (125), that
dΓ(y 2 )e−iHt χ(H)ϕ ≤ dΓ(y 2 )χ(H)ϕ + 2t dΓ(a)χ(H)ϕ + Ct2 ϕ
= dΓ(y 2 )χ(H)ϕ + 2t2 dΓ(a/t)χ(H)ϕ + Ct2 ϕ . (126)
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By Lemma 37, part (iv), with y replaced by y/t, we have
dΓ(a/t)χϕ ≤ dΓ(y 2 /t2 + 1)χϕ ≤ dΓ(y 2 /t2 )χϕ + C ϕ . The Lemma now follows inserting the last equation into (126) and applying Lemma 35. Lemma 37
(i) For any operator A A2 + (A∗ )2 ≤ AA∗ + A∗ A.
(ii) If a = 1/2(y · ∇ω + ∇ω · y) and ω satisfies (H1) then a2 ≤ const(y 2 + 1). (iii) If a is a symmetric operator in h then dΓ(a)2 ≤ N dΓ(a2 ) . (iv) With a as in (ii) we have dΓ(a)2 ≤ const dΓ(y 2 + 1)2 . Proof.
(i) This follows from 0 ≤ (iA − iA∗ )2 = −A2 − (A∗ )2 + AA∗ + A∗ A.
(ii) By (i), (∇ω · y + y · ∇ω)2 ≤ 2[(∇ω · y)(y · ∇ω) + (y · ∇ω)(∇ω · y)] where (y · ∇ω)(∇ω · y) ≤ ∇ω 2∞ y 2 (∇ω · y)(y · ∇ω) ≤ 2( ∇ω 2∞ + ∆ω 2∞ )(y 2 + 1)
(iii) It suffices to prove this for states ϕnin Ran χ(N = n). such vecFor n n tors ϕn , dΓ(a)2 ϕn = dΓ(a)ϕn 2 ≤ ( i=1 ai ϕn )2 ≤ n i=1 ai ϕn 2 = ϕn , N dΓ(a2 )ϕn , where ai denotes the operator a acting on the i-th boson. (iv) This follows from (ii), (iii), and N ≤ dΓ(y 2 + 1).
G
Some Technical Parts of Theorem 15
Lemma 38 There exists a constant C = C(λ, u) such that 1/2 n ±(Jnk Sn ) (y, t) ≤ Cn Sn
(y, t) k
2
for |y| ≤ 2λt and ut1−δ ≥ 2. Here y = (y1 , . . . , yn ), yi ∈ Rd .
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Proof. We drop n the combinatorial factor in the definition of Jnk for convenience. Let S0 (y) = i=1 S0 (yi ). From (34) and the scaling factor 1/ut in the arguments of Jnk we get tSn
(y, t) = S0
(y/tδ ) and
t(Jnk Sn )
(y, t) = Jnk (z) · S0
(w) + α−2 Jnk (z)S0 (w)
+ α−1 ∇Jnk (z) ⊗ ∇S0 (w) + α−1 ∇S0 (w) ⊗ ∇Jnk (z)
(127)
where w = yt−δ , z = y/ut, α = ut1−δ and hence w = αz, α ≥ 2 and |z| ≤ 2λ/u. Proving the lemma thus amounts to showing that the r.h.s. of (127) (and its negative) are bounded by const × n2 S0
(w). The first term in (127) enjoys the bound. To estimate the other three terms we need some “N -body geometry”. For each a ∈ {0, ∞}n , let 1a : Rnd → Rnd denotes the orthogonal projection onto the subspace {w ∈ Rnd |wi = 0 if ai = 0} and let wa = 1a w. Set
χa (σ, w) =
χ(ws2 /2 > σ).
s:as =∞
r:ar =0
Then
χ(wr2 /2 ≤ σ)
χa ≡ 1. Hence, by (32) and since χa (σ, w) χ(wi2 /2 > σ) = χa (σ, w)δai ,∞ , (128) S0 (w) = dσ m(σ)S0 (σ, w) a
S0 (σ, w) =
n
χ(wi2 /2 > σ)(wi2 /2 − σ) =
χa (σ, w)(wa2 /2 − σa )
(129)
a
i=1
where σa = σ · #{i : ai = ∞}. Furthermore ∇S0 (σ, w) =
χa (σ, w)wa
(130)
χ(σ, w)1a
(131)
a
S0
(σ, w) ≥
a
in the sense that ∇S0 (w) =
dσ m(σ)
S0
(w) ≥
χa (σ, w)wa and
dσ m(σ)
χa (σ, w)1a .
By (128) it suffices to estimate the last three terms of (127) for S0 (w) replaced by S0 (σ, w) uniformly in σ. If wi2 /2 ≤ 1 and α ≥ 2 then |zi | ≤ 1 and hence ∂ β jε (zi ) = 0 for all derivatives of non-zero order β and ε ∈ {0, ∞}. Hence for w such that χa (σ, w) = 0
Jnk (z) = 1a Jnk (z)1a
(132)
∇Jnk (z) = 1a ∇Jnk (z).
(133)
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From (129), (132) and (131) we get
(z)S0 (σ, w) ≤ a−2 w2 /2 Jnk (z)
±a−2 Jnk
Ann. Henri Poincar´e
χa (w)1a
a
≤ Cn2 S0
(σ, w). The last two terms in (127) are estimated similarly using (130), (133) and (131). Lemma 39
1/2 y2 m+1 n D ∂ O(t−(1+m)+δ(2−|α|) ) Snk (y, t) − Jnk (y, t) =n k t m α
Proof. The difference Snk (y, t) − Jnk (y, t)y /2t = Jnk (y, t)t 2
−1+2δ
n
[a(yi t−δ ) + b]
(134)
i=1
is of the form t−c f (y/t)h(yt−δ ) where y = (y1 , . . . , yn ) and f, h ∈ C ∞ ∩ L∞ (Rnd ). Under differentiation (134) becomes a sum of terms of the same form where in each term the exponent of t is decreased by at least δ or 1 for differentiation w.r. to y or t (or D) respectively. This accounts for the power in t. From the explicit form the right hand side of (134) and (60) we see that every derivative w.r. to yi (at most) doubles the number of terms, while every derivative w.r. to t multiplies it by n + 1. Lemma 40 For |y|/t ≤ 2λ we have
(∇Ω − V ) D02 Snk =(∇Ω − V ) · Snk + (D∇Snk ) · (∇Ω − V ) + (∇Ω − V ) · (D∇Snk ) 1/2 n + DV2 Snk + n2 O(t−1−δ ) k
where V = y/t ∈ Rnd .
1/2 since it appears in all terms on both sides of the Proof. We drop the factor nk equation. By definition of D0 , ∂ 2 Snk . (135) ∂t2 = Jnk Sn and the Leibnitz rule
D02 Snk = [iΩ, [iΩ, Snk ]] + 2[iΩ, ∂Snk /∂t] + To evaluate the double commutator we use Snk and get for |y/t| ≤ 2λ
[iΩ, [iΩ, Snk ]] =[iΩ, [iΩ, Jnk ]]Sn + 2[iΩ, Jnk ][iΩ, Sn ] + Jnk [iΩ, [iΩ, Sn ]]
Sn + ∇Jnk ⊗ ∇Sn + ∇Sn ⊗ ∇Jnk + Jnk Sn
] ∇Ω (136) =∇Ω · [Jnk + n2 O(t−1−δ )
=∇Ω · Snk ∇Ω + n2 O(t−1−δ )
(137)
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where we used [iΩ, Jnk ] = ∇Ω·∇Jnk +nO(t−2 ), [iΩ, Sn ] = ∇Ω·∇Sn +nO(t−1 ) (and the same with the order in the dot products reversed) as well as [iΩ, [iΩ, Sn ]] =
∇Ω · Sn
∇Ω + nO(t−1−δ ) (see Lemma9), [iΩ, [iΩ, Jnk ]] = ∇Ω · Jnk ∇Ω + n2 O(t−3 ) α 1−|α| ) for |y/t| ≤ 2λ and |α| ≤ 2. In a similar way one shows and |∂ Sn | = O(t that [iΩ, ∂Snk /∂t] = ∇Ω · ∇ =∇
∂Snk + n2 O(t−2 ) ∂t
∂Snk · ∇Ω + n2 O(t−2 ). ∂t
(138)
The lemma now follows from (135), (137) and (138) as Lemma 9 did from analogous equations.
References [Ara83]
A. Arai, Rigorous theory of spectra and radiation for a model in quantum electrodynamics, J. Math. Phys. 24(7), 1896–1910 (1983).
[BFS98]
V. Bach, J. Fr¨ ohlich and I. M. Sigal, Quantum electrodynamics of confined nonrelativistic particles, Adv. Math. 137(2), 299–395 (1998).
[BFSS99] V. Bach, J. Fr¨ ohlich, I. M. Sigal and A. Soffer, Positive commutators and spectrum of Pauli–Fierz Hamiltonian of atoms and molecules, Comm. Math. Phys. 207(3), 557–587 (1999). [Dav95]
E. B. Davies, The functional calculus, J. London Math. Soc. (2), 52(1), 166–176 (1995).
[DG99]
J. Derezi´ nski and C. G´erard, Asymptotic completeness in quantum field theory Massive Pauli-Fierz Hamiltonians, Rev. Math. Phys. 11(4), 383– 450 (1999).
[DG00]
J. Derezi´ nski and C. G´erard, Spectral scattering theory of spatially cutoff P (φ)2 Hamiltonians, Comm. Math. Phys. 213(1), 39–125 (2000).
[FGS00] J. Fr¨ ohlich, M. Griesemer and B. Schlein, Asymptotic electromagnetic fields in models of quantum-mechanical matter interacting with the quantized radiation field, Adv. Math., to appear. Los Alamos arXiv math-ph/0009033. [GLL00] M. Griesemer, E. H. Lieb and M. Loss, states in non–relativistic QED, Invent. math. 145(3), 557–595 (2001). [GS97]
G. M. Graf and D. Schenker, Classical action and quantum N -body asymptotic completeness, In Multiparticle quantum scattering with applications to nuclear, atomic and molecular physics (Minneapolis, MN, 1995), pages 103–119. Springer, New York (1997).
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[HS00]
W. Hunziker and I. M. Sigal, The quantum N –body problem, J. Math. Phys. 41(6), 3448–3510 (2000).
[RS79]
M. Reed and B. Simon, Methods of modern mathematical physics: Scattering Theory. Volume 3, Academic Press (1979).
[Spo97]
H. Spohn, Asymptotic completeness for Rayleigh scattering, J. Math. Phys. 38(5), 2281–2296 (1997).
J. Fr¨ ohlich and B. Schlein Theoretical Physics ETH–H¨ onggerberg CH–8093 Z¨ urich Switzerland email:
[email protected] email:
[email protected] M. Griesemer Department of Mathematics University of Alabama at Birmingham Birmingham, AL 35294 USA email:
[email protected] Communicated by Gian Michele Graf submitted 09/04/01, accepted 09/10/01
Ann. Henri Poincar´e 3 (2002) 171 – 201 c Birkh¨ auser Verlag, Basel, 2002 1424-0637/02/010171-31 $ 1.50+0.20/0
Annales Henri Poincar´ e
Self-Adjointness of the Pauli-Fierz Hamiltonian for Arbitrary Values of Coupling Constants F. Hiroshima Abstract. The Pauli-Fierz Hamiltonian describes a system of N electrons minimally coupled to a quantized radiation field. The electrons have spin and an ultraviolet cutoff is imposed on the quantized radiation field. For arbitrary values of coupling constants, self-adjointness and essential self-adjointness of the Pauli-Fierz Hamiltonian are proven by means of a functional integral.
1 Introduction and main results The Pauli-Fierz Hamiltonian [33] governs a system of N -electrons minimally coupled with a quantized radiation field. The N -electrons are assumed to have spin and the quantized radiation field is smeared by an ultraviolet cutoff. The purpose of this paper is to establish self-adjointness and essential self-adjointness of the Pauli-Fierz Hamiltonian for arbitrary values of coupling constants. For sufficiently small values of coupling constants self-adjointness of the PauliFierz Hamiltonian on an explicit domain, say D, has previously been established. In [21] we have shown essential self-adjointness on D for arbitrary values of coupling constants, for N = 1 under certain integrable conditions on the ultraviolet cutoff. In this paper we show self-adjointness on the same domain D for all N ≥ 1 under some weaker conditions on the ultraviolet cutoff.
1.1
The Pauli-Fierz Hamiltonian
Suppose that the N -electrons move in the d-dimensional space, where d ≥ 3. Let L2 := L2 (Rd ). The Hilbert space of state vectors is defined by N [d/2] ⊗ L2 ⊗ F, Htotal := C2
(1.1)
where [k] denotes the integer part of k. Here F is the symmetric Fock space over Cd−1 ⊗ L2 , i.e., F :=
∞ n=0
⊗ns Cd−1 ⊗ L2 ,
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where ⊗ns denotes the symmetric tensor product with ⊗0s [Cd−1 ⊗ L2 ] := C. Under the identification Cd−1 ⊗ L2 ∼ = L2 (Rd × {1, 2, ..., d − 1}) we have F∼ =
∞
⊗ns L2 (Rd × {1, 2, ..., d − 1}).
n=0
In what follows we identify (1.1) as [d/2] [d/2] Htotal ∼ ) ⊗ L2 (RdN ) ⊗ F ∼ ) ⊗ L2 (RdN ; F ). = (⊗N C2 = (⊗N C2
(1.2)
The vacuum vector in F is defined by Ω := {1, 0, 0, 0, ...}. The smeared annihilation and creation operators are denoted by ar (f ) and a†r (f ), f ∈ L2 , r = 1, ..., d − 1, respectively. They are linear in f and satisfy the canonical commutation relations: [as (f ), a†r (g)] = (f¯, g)L2 δrs ,
[ar (f ), as (g)] = 0,
[a†r (f ), a†s (g)] = 0
on the finite particle subspace (0) Ffin := ∪∞ , Ψ(1) , Ψ(2) , ...} ∈ F Ψ(n) = 0, n ≥ m , m=0 {Ψ where (f, g)K denotes the scalar product on K. Note that (Ψ, a†r (f )Φ)F = (ar (f¯)Ψ, Φ)F , and we formally write as ar (f ) = ar (k)f (k)dk,
a†r (f ) =
Ψ, Φ ∈ Ffin ,
a†r (k)f (k)dk.
We write the norm on K as · K . Unless confusion may arise we write simply (f, g) and f for (f, g)K and f K, respectively. Note that in our notation (f, g) is linear in g and anti-linear in f . Under the identification (1.2) the Pauli-Fierz Hamiltonian is given by 2 1 j j σµ ⊗ pµ + ajexµ (xj ) ⊗ 1 − eAµ (xj ) +1⊗V ⊗1+1⊗1⊗Hf , (1.3) HPF := 2 where the summation over repeated indices is automatically understood unless otherwise stated. Here we adopt the unit: c = = 1, e denotes the electron charge regarding as the coupling constant, and pj := (pj1 , · · · , pjd ) and xj := (xj1 , · · · , xjd ) are the-jth electron momentum and its canonical position operators in L2 , respectively. Let σ1 , ..., σd denote 2[d/2] × 2[d/2] matrices satisfying the anti-commutation relations: {σµ , σν } = 2δµν 1, µ, ν = 1, ..., d. Then σ j := (σ1j , · · · , σdj ) is given by σµj := 1 ⊗ · · · ⊗
the jth σµ
N
⊗ · · · ⊗ 1,
µ = 1, ..., d,
j = 1, ..., N,
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where σµ appears as the jth factor. V denotes an external potential. We assume that there exists a constant 0 ≤ aV < 1, and bV ≥ 0 such that
V f ≤ aV ∆f + bV f ,
f ∈ D(∆),
(1.4)
where D(T ) denotes the domain of T . ajex : Rd → R denotes an external vector potential assumed to be (1) ajex ∈ C 1 (Rd ),
(2) ajex ∞ < ∞,
(3) ∂µj ajex ∞ < ∞,
(1.5)
where · ∞ denotes the sup norm. The free Hamiltonian in F is defined by Hrad := ω(k)a†r (k)ar (k)dk with the dispersion relation ω(k) := |k|. ∼ L2 (RdN ; F ), the transverse vector poUnder the identification, L (R ) ⊗ F = tential quantized by the Coulomb gauge is defined by 1 ˆ ˆ + ar (k)eikx λ(k) dk, Aµ (x) := √ erµ (k) a†r (k)e−ikx λ(−k) 2 2
dN
where er = (er1 , ..., erd ) denotes polarization vectors satisfying er (k) · es (k) = δrs ˆ ∈ L2 is an ultraviolet cutoff and er (k) · k = 0 for almost everywhere k ∈ Rd . λ function, and physically reasonable choice is ρˆ(k) ˆ λ(k) = , (2π)d ω(k) where eρ denotes the electron charge density; ρ(x)dx = 1, Rd
ˆ ∈ L2 , µ = 1, ..., d, and fˆ is the Fourier transform of f . Note that in the case of kµ λ divA(x) = 0 on some dense domain. We assume that ˆ ˆ λ(−k) = λ(k) which ensures that HPF is a symmetric operator in Htotal . We abbreviate 1 ⊗ 1 ⊗ X, 1 ⊗ X ⊗ 1, X ⊗ 1 ⊗ 1 by X unless confusions may arise. Thus we write (1.3) as HPF =
N 2 1 j p + ajex (xj ) − eA(xj ) + V + Hf + Hspin . 2 j=1
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The spin term is given by Hspin := − where
bµν ex j
1 2
j σµj σνj eBµν (xj ) − bµν (x ) , ex j
1≤µ<ν≤d
denotes the external field, bexj := i∇j ∧ ajex ,
and Bµν the quantized magnetic field, j j 1 ˆ ˆ k ∧ er (k) e−ikx λ(−k) − eikx λ(k) dk. B(xj ) := i∇j ∧ A(xj ) = √ 2 A typical example is as follows: d = 3, (σ1 , σ2 , σ3 ) is the 2 × 2 Pauli matrices: 0 1 0 −i 1 0 σ1 := , σ2 := , σ3 := , 1 0 i 0 0 −1 and V the Coulomb potential smeared by the electron charge eρ, e2 ρ(xi − k)ρ(xj − k ) Vsmear (x1 , · · · , xN ) := − dkdk , | 4π |k − k 3 3 R ×R 1≤i<j≤N
or the Coulomb potential in M static nuclei: VCoulomb (x1 , · · · , xN , y 1 , · · · , y M ) := −
e2 4π
1≤i<j≤N
|e|Zj 1 + , i j |x − x | j=1 i=1 |xj − y i | N
M
where y 1 , ..., y M are constants in Rd , Z1 , ..., ZM positive constants. Over the past few decades a considerable number of studies have been made on the Pauli-Fierz type models (including spin-boson models and the Nelson model [31]) by many authors. In particular spectral analysis of the Pauli-Fierz type models has been investigated in e.g., [2]-[13], [15, 16] [19]-[29], [32, 36, 37]. Ground states are defined as the vectors associated with the bottom of the spectrum of a self-adjoint operator. Existence of ground states of the Pauli-Fierz type Hamiltonian was proven for weak couplings in [2, 4, 19, 20]. In particular, Bach-Fr¨ ohlich-Sigal [5] proved it without infrared cutoffs. For arbitrary values of coupling constants, existence of ground states has been shown by Spohn [37] for the Nelson Hamiltonian, by G´erard [15] for rather general models, and by GriesemerLieb-Loss [16] for the Pauli-Fierz Hamiltonian. The multiplicity of ground states has been established by Bach-Fr¨ohlich-Sigal [4] for the Nelson Hamiltonian, by Hiroshima [20] for a spinless Pauli-Fierz Hamiltonian, and by Hiroshima-Spohn [23] for the Pauli-Fierz Hamiltonian including spin. Resonances have been investigated by H¨ ubner-Spohn [26], Bach-Fr¨ ohlich-Sigal [3, 4], Bach-Fr¨ ohlich-Sigal-Soffer [6], and Jakˇsi´c-Pillet [27], moreover scattering theory by H. Krohn [24], H¨ ubner-Spohn [25], Derezi´ nski-G´erard [8], and Fr¨ ohlich-Griesemer-Schlein [13].
Vol. 3, 2002
1.2
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The Friedrichs extension
To our best knowledge a general way to define a self-adjoint extension of HPF for all values of coupling constants is to take the Friedrichs extension since HPF consists of the quadratic part and a potential V . Let
Kµj := σµj pjµ + ajexµ − eAµ (xj ) . We define the form QPF (Φ, Ψ) by QPF (Φ, Ψ) =
1 j 1/2 1/2 (K Φ, Kµj Ψ) + (Hrad Φ, Hrad Ψ) + (|V |1/2 Φ, |V |1/2 Ψ) 2 µ
with the form domain 1/2
d j 1/2 Q := ∩N ). j=1 ∩µ=1 D(Kµ ) ∩ D(Hrad ) ∩ D(|V |
ˆ ∈ L2 . Since Q is dense, the Friedrichs extension of HPF exists for λ ˆ ∈ L2 . Then the Friedrichs extension of HPF exists. Proposition 1.1 Let λ ˆ is needed to define the Friedrichs It must be noted that no extra conditions on λ extension. As is well known, this extension, however, do not give information of the domain, and there still remains the possibility that several distinct self-adjoint extensions exist. In general, the dynamics of a system is determined by the one-parameter unitary time-evolution generated by the Hamiltonian. A densely defined Hamiltonian, however, may have several distinct self-adjoint extensions. Then we need to choose a particular self-adjoint extension to define a dynamics. In other words, distinct self-adjoint extensions determine distinct time-evolutions. It will be useful to keep this aspect in mind as we examine spectral properties of the Pauli-Fierz Hamiltonian derived by the Friedrichs extension.
1.3
Main results
In [21] we have shown that for arbitrary values of coupling constants HPF is essentially self-adjoint on D(∆) ∩ D(Hrad ) in the case when (1) N = 1 and (2) ˆ ∈ L2 . In this paper we prove self-adjointness on the same domain ˆ √ω, ω 2 λ λ/ ˆ √ω, ω λ ˆ ∈ L2 . Before D(∆) ∩ D(Hrad ) under the assumption (1) N ≥ 1 and (2) λ/ moving on to the main theorems it is desirable to describe why we pay attention ˆ to conditions on λ. 1.3.1 Domain and cutoffs ˆ We shall mention the relaWe shall now look carefully into the assumption on λ. ˆ tionship between D(HPF ) and the ultraviolet cutoff λ.
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ˆ kµ λ ˆ ∈ L2 , µ = 1, ..., d. Then, for Ψ ∈ C ∞ (RdN )⊗[F fin ∩ D(Hrad )] (⊗ Let λ, 0 denotes algebraic tensor product), it holds that Ψ ∈ D(pj · A(xj )) ∩ D(A(xj ) · pj ). By the Coulomb gauge, we have [pjµ , Aµ (xj )]Ψ =
N
divA(xj )Ψ = 0.
j=1
ˆ ωλ ˆ ∈ L2 , then Hence, in particular, if λ, N 2 1 j 1 e2 p − eA(xj ) = p2 − eA(xj ) · pj + A2 (xj ) 2 j=1 2 2
(1.6)
fin ∩D(Hrad )]. Put the right-hand side of (1.6) as K. Clearly holds on C0∞ (RdN )⊗[F j 2 1 N both 2 j=1 p − eA(xj ) and K are closable. Let us denote their closed exten2 N sions by the same symbols 12 j=1 pj − eA(xj ) and K, respectively, and let N be the number operator given by N := a†r (k)ar (k)dk. From the inequalities:
a†r (f )Ψ ≤ f ( N 1/2Ψ + Ψ ),
ar (f )Ψ ≤ f
N 1/2Ψ ,
ar (f )as (g)Ψ ≤ 8 f
g ( N Ψ + Ψ ), ˆ ωλ ˆ ∈ L2 , then where ar = ar or a†r , it follows that, if λ, ˆ 2 ( ∆Ψ + N Ψ + Ψ )
KΨ ≤ C λ
(1.7)
2 N with some constant C. Hence 12 j=1 pj − eA(xj ) is well defined on D(∆) ∩ D(N ). In addition to this, from the inequalities: √ 1/2
a†r (f )Ψ ≤ f / ω
Hrad Ψ + f
Ψ ,
(1.8)
√ 1/2 (1.9)
ar (f )Ψ ≤ f / ω
Hrad Ψ , √ √ √
ar (f )as (g)Ψ ≤ ( f / ω + f )( g/ ω + g + ωg + ωg ) ×( Hrad Ψ + Ψ ),
(1.10)
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ˆ √ω, ω λ ˆ ∈ L2 , then it follows that, if λ/
KΨ ≤ C ( ∆Ψ + Hrad Ψ + Ψ ),
(1.11)
ˆ √ω and ω λ . ˆ Thus we see that where C is a constant depending only on λ/ N 1 j j 2 is well defined on D(∆) ∩ D(Hrad ). j=1 p − eA(x ) 2 Let us summarize the main points that have been mentioned above. ˆ ωλ ˆ ∈ L2 , HPF is well defined on D(∆) ∩ Proposition 1.2 (1) In the case of λ, √ ˆ ˆ ∈ L2 , HPF is well defined on D(Hrad ) ∩ D(N ). (2) In the case of λ/ ω, ω λ D(∆) ∩ D(Hrad ). The Pauli-Fierz Hamiltonian can be split out as HPF = −∆/2 + Hrad + eHint + V + H(a), where Hint :=
1 j −p · A(xj ) − A(xj ) · pj + eA(xj )2 + Hspin , 2
and H(a) :=
1 j j p · aex + ajex · pj + (ajex )2 − eajex · A(xj ). 2
ˆ ∈ L2 . From (1.5), (1.8), (1.9), and (1.10), it follows that, ˆ √ω, ω λ Let λ/
Hint Ψ ≤ a (−∆/2 + Hrad )Ψ + b Ψ with some constant a and b, and for arbitrary , > 0
H(a)Ψ ≤ , (−∆/2 + Hrad )1/2 Ψ + b Ψ
(1.12)
with some constant b for Ψ ∈ D(−∆/2 + Hrad ) = D(∆) ∩ D(Hrad ). Then, by (1.4), we obtain
(eHint + V + H(a))Ψ ≤ (|e|a + aV + ,) (−∆/2 + Hrad )Ψ + (b + bV + b ) Ψ . ˆ √ω, ω λ ˆ ∈ L2 , for e such that Then in the case of λ/ |e|a + aV + , < 1, HPF is self-adjoint on D(∆) ∩ D(Hrad ) by the Kato-Rellich theorem. This was established in e.g., [4, 5, 6, 19, 20, 32].
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1.3.2 Main results Several articles also have been devoted to the study of self-adjointness of the Pauli-Fierz type Hamiltonians. It is noteworthy that Arai [1] proves that the PauliFierz Hamiltonian in the dipole approximation is self-adjoint for arbitrary values of coupling constants by means of the Nelson commutator theorem [34, Theorem X.36]. In contrast, self-adjointness of the full Pauli-Fierz Hamiltonian for all values of coupling constants is not obvious. From the argument of Section 1.3.1 we may, therefore, reasonably conclude that it is worth while to prove √ self-adjointness for arbitrary values of coupling ˆ ∈ L2 instead of λ/ ˆ √ω, ω 2 λ ˆ ω, ω λ ˆ ∈ L2 imposed constants under the condition λ/ in [21]. Furthemore by Proposition 1.2 (1) it is interesting to consider essential ˆ ωλ ˆ ∈ L2 but λ/ ˆ √ω ∈ L2 . Our main results in this self-adjointness of HPF for λ, paper are as follows: ˆ √ω, ω λ ˆ ∈ L2 . Then HPF is self-adjoint on D(∆) ∩ Theorem 1.3 Suppose that λ/ D(Hrad ) and bounded from below. Moreover HPF is essentially self-adjoint on any core of −∆/2 + Hrad . ˆ √ω, ω λ ˆ ∈ L2 . Then there exists a constant C such that Corollary 1.4 Let λ/
∆Ψ + Hrad Ψ ≤ C( HPF Ψ + Ψ ),
Ψ ∈ D(∆) ∩ D(Hrad ).
Proof. From Theorem 1.3 and the closed graph theorem,
(−∆/2 + Hrad )Ψ ≤ C ( HPF Ψ + Ψ ),
Ψ ∈ D(∆) ∩ D(Hrad ),
follows with some constant C . Thus the corollary holds true.
Next we consider a more √ subtle case. As we have mentioned in the previous ˆ ω ∈ L2 , HPF is a priori not defined on D(∆) ∩ subsection, for the case of λ/ n D(Hrad ) but well defined on D(∆) ∩ D(Hrad ) ∩ D(N ). Let C ∞ (T ) := ∩∞ n=1 D(T ). ˆ ωλ ˆ ∈ L2 . Then HPF is essentially self-adjoint on Theorem 1.5 Suppose that λ, ∞ ∞ C (∆) ∩ D(Hrad ) ∩ C (N ) and bounded from below. Finally also it is of mathematical interest to consider essential self-adjointness of N 2 1 j p − eA(xj ) + V. 2 j=1
(1.13)
Our method is also available to (1.13). ˆ ωλ ˆ ∈ L2 . Then 1 N pj − eA(xj ) 2 + V is essentially selfTheorem 1.6 Let λ, j=1 2 adjoint on C ∞ (∆) ∩ C ∞ (N ) and bounded from below.
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179
The outline of our strategy is from Step 1 to Step 4. Step 1: Let N 2 1 j H := p − eA(xj ) + Hrad , 2 j=1 then HPF = H + V + Hspin + H(a). For sufficiently small coupling constants, the matrix elements of the heat semigroup e−tH can be expressed by a functional integral. This type of functional integrals were first studied in [30], and there have been numerous related works e.g., [14, 17, 10]. We see that for arbitrary values of coupling constants the functional integral itself is well defined and yields a heat semigroup generated by a self-adjoint operator H. Step 2: we prove that ⊃ HD(∆)∩D(H ) . H rad ˆ √ω, ω 2 λ ˆ ∈ L2 . In This statement was checked in [18] only for the case when λ/ √ 2 ˆ ˆ this paper we relax this assumption to λ/ ω, ω λ ∈ L . b Step 3: we show that D(∆) ∩ D(Hrad ) is invariant under e−tH , i.e., b
e−tH {D(∆) ∩ D(Hrad )} ⊂ D(∆) ∩ D(Hrad ), is essentially self-adjoint on D(∆)∩D(Hrad ) by [34, Theorem which implies that H X.49]. Step 4: We prove that + E)−1 (−∆ + Hrad )(H is closed on D(∆)∩D(Hrad ). is bounded for every E > 0. Thus we conclude that H Hence H is self-adjoint on D(∆) ∩ D(Hrad ) and bounded from below, and H is also self-adjoint on D(∆) ∩ D(Hrad ) and bounded from below. For arbitrary , > 0 there exists c > 0 such that, for Ψ ∈ D(∆) ∩ D(Hrad ),
Hspin Ψ ≤ , (−∆/2 + Hrad )Ψ + c Ψ , and (1.4) derives
V Ψ ≤ aV HΨ + bV Ψ with some 0 ≤ aV < 1 and 0 ≤ bV . This, together with (1.12) and Corollary 1.4, implies that HPF = H + V + Hspin + H(a) is self-adjoint on D(∆) ∩ D(Hrad ). The paper is organized as follows: In Section 2 we construct a self-adjoint extension in terms of a functional integral. In Section 3 we show some relative bounds by using the functional integral representation. In Section 4 we prove the main theorems. Section 5 is devoted to an appendix.
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Ann. Henri Poincar´e
2 A self-adjoint extension of HPF 2.1
A short review of a probabilistic description
For a moment let us look at H=
N 2 1 j p − eA(xj ) + Hrad 2 j=1
acting in H = L2 (RdN ) ⊗ F. It is useful to take a Schr¨odinger representation of F to investigate the semigroup generated by H. Let W := ⊕d L2 and W0 := ⊕d L2 (Rd+1 ). q and q0 denote bilinear forms defined by 1 dµν (k)fˆµ (k)ˆ gν (k)dk, f, g ∈ W, q(f, g) := 2 Rd q0 (f, g) := dµν (k)fˆµ (k, k0 )ˆ gν (k, k0 )dkdk0 , f, g ∈ W0 , Rd+1
where dµν (k) :=
erµ (k)erν (k)
= δµν
kµ kν − = |k|2
k k ⊗ 1− , |k| |k| µν
k ∈ Rd ,
is the transverse projection. Let (Q, µ) and (Q0 , µ0 ) be probability spaces associated with the Gaussian random variables of zero mean (φ(f ), f ∈ W ) and (φ0 (g), g ∈ W0 ), respectively. Their covariances are given by 1 φ(f )φ(g)µ(dφ) = q(f, g), f, g ∈ W, 2 Q and
φ0 (f )φ0 (g)µ(dφ0 ) = Q0
1 q0 (f, g), 2
f, g ∈ W0 ,
respectively. The finite particle subspace L2fin(Q) is defined by L2fin (Q) := L.H.{:φ(f1 ) · · · φ(fn ):, 1|fj ∈ W, j = 1, ..., n, n ≥ 1}, where L.H.{· · · } is the linear hull of the vectors in {· · · } and :X: denotes the Wick product of X. Let ω : W → W be a self-adjoint operator defined by ω := ⊕d ω(−i∇).
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181
Define Hf and Nf by Hf :φ(f1 ) · · · φ(fn ):=
n
:φ(f1 ) · · · φ(ˆ ω fj ) · · · φ(fn ):,
j=1
Hf 1 = 0, and Nf :φ(f1 ) · · · φ(fn ):= n :φ(f1 ) · · · φ(fn ):, Nf 1 = 0. Let Aµ (λ) := φ(⊕dν=1 δµν λ),
µ = 1, ..., d.
The Hilbert space HS is given by HS := L2 (RdN ) ⊗ L2 (Q), and the following identification HS ∼ = L2 (RdN ; L2 (Q)) is used without notice. We define a symmetric operator HS on HS by HS := HA + Hf , where
1 j (p − eA(λ(· − xj )))2 . 2 j=1 N
HA :=
We define a linear operator T of F to L2 (Q) by ˆ 1 ) · · · Aµ (λ ˆ n ): Ω =:Aµ (λ1 ) · · · Aµ (λn ):, • T :Aµ1 (λ n 1 n • T Ω = 1. It is seen that T can be extended to a unitary operator of F to L2 (Q). We denote the extension as the same symbol T . In particular T : D(N l ) −→ D(Nfl ), l ) −→ D(Hfl ), T : D(Hrad
for arbitrary l ≥ 0. Moreover T implements the equivalences Aµ (x) ∼ = Aµ (λ(· − x)) for each x ∈ Rd , N ∼ = Nf , and Hrad ∼ = Hf . Thus we proved the following proposition.
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Ann. Henri Poincar´e
ˆ √ω, ω λ ˆ ∈ L2 . Then Proposition 2.1 (1)Let λ/ T HT −1D(∆)∩D(Hf ) = HS D(∆)∩D(Hf ) . ˆ ωλ ˆ ∈ L2 . Then (2) Let λ, T HT −1D(∆)∩D(Hf )∩D(Nf ) = HS D(∆)∩D(Hf )∩D(Nf ) . ˆ ωλ ˆ ∈ L2 . Then (3)Let λ, T
N 2 1 j p − eA(xj ) T −1 D(∆)∩D(Nf ) = HA D(∆)∩D(Nf ) . 2 j=1
Then, instead of H, we shall study self-adjointness of HS for the time being.
2.2
A self-adjoint extension
Let {b(T )}T ≥0 = {bjµ (T )}µ=1,...,d,j=1,...,N,T ≥0 denote the dN -dimensional Brownian motion starting at the origin on a probability space (P, db). We set XT := x + b(T ), XTjµ = xjµ + bjµ (T ), M := RdN × P , and dX := dx ⊗ db. We see that the map Λ : t → λ(· − t) ˆ ˆ ∈ L2 . is an L2 -valued function on Rd . Note that (λ(· − t))(k) = e−ik·t λ(k). Let ω n λ n d 2 Then Λ ∈ Cb (R ; L ), the set of n-times continuously differentiable bounded (up ˆ ωλ ˆ ∈ L2 . Then the L2 to n-times derivatives) L2 -valued functions on Rd . Let λ, valued stochastic integral, t λ(· − Xs )dbjµ (s) ∈ L2 (P ) ⊗ L2 , 0
ˆ ω2λ ˆ ∈ L2 , it is proven that is well defined. Note that, in the case λ, 2n
1 j j jµ jµ λ(· − Xtk/2 Xtk/2 s − lim n ) + λ(· − Xt(k−1)/2n ) n − Xt(k−1)/2n n→∞ 2 k=1
t
λ(· − Xs )dbjµ (s)
= 0
in L2 (P )⊗L2 . Let Ξt : L2 (Q) → L2 (Q0 ) be the second quantization of the isometry ⊕d ξt : W → W0 , which is defined by −itk0 ω(k) e fˆ(k), f ∈ L2 . (ξ t f )(k, k0 ) = √ π ω(k)2 + |k0 |2
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183
Then Ξt also is an isometry and satisfies Ξ∗s Ξt = e−|s−t|Hf .
(2.1)
Since ξs λ is strongly continuous in s in L2 (Rd+1 ), the L2 (Rd+1 )-valued stochastic integral, t
ξs λ(· − Xs )dbjµ (s) ∈ L2 (P ) ⊗ L2 (Rd+1 ), 0
is well defined. Let
Jt0 := Jt0 (X) = e−ieφ(L(t)) ,
where L(t) := L(X, t) :=
N
λ(· − Xs )dbjµ (s). 0
j=1
Moreover let
t
⊕dµ=1
Jt := Jt (X) = Ξ∗0 e−ieφ0 (K(t)) Ξt ,
where K(t) := K(X, t) :=
N
⊕dµ=1
ξs λ(· − Xs )dbjµ (s). 0
j=1
We define, for F, G ∈ H,
t
0 IF,G
(F0 , Jt0 Gt ) dX ,
:= M
and
(F0 , Jt Gt ) dX ,
IF,G := M
where F0 := F (X0 ), Gt = G(Xt ). The key lemma of this paper is as follows. ˆ ωλ ˆ ∈ L2 . Then there exists a nonnegative self-adjoint operLemma 2.2 (1) Let λ, ator HA such that b 0 IF,G = (F, e−tHA G), F, G ∈ HS , A ) ⊃ D(∆) ∩ D(Nf ) with and D(H A ⊃ HA D(∆)∩D(N ) . H f ˆ √ω ∈ L2 . Then D(H A ) ⊃ D(∆) ∩ D(Hf ) with In addition, assume λ/ A ⊃ HA D(∆)∩D(H ) . H f ˆ ωλ ˆ ∈ L2 . Then there exists a nonnegative self-adjoint operator H S such (2) Let λ, that b IF,G = (F, e−tHS G), F, G ∈ HS ,
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S ) ⊃ D(∆) ∩ D(Hf ) ∩ D(Nf ) with and D(H S ⊃ HS D(∆)∩D(H )∩D(N ) . H f f ˆ √ω ∈ L2 . Then D(H S ) ⊃ D(∆) ∩ D(Hf ) with In addition, assume λ/ S ⊃ HS D(∆)∩D(H ) . H f Proof. See Appendix.
3 Relative bounds and invariant domains We first state a fundamental fact. Lemma 3.1 Let K be a Hilbert space. (1) Let T be a densely defined closed operator in K and g ∈ K. Let D be a core of T . Moreover suppose (3.1) |(T f, g)| ≤ Rg f for all f ∈ D with some constant Rg . Then g ∈ D(T ∗ ) with T ∗ g ≤ Rg . (2) Let S be a densely defined closed operator and U a densely defined linear operator in K. Let DS be a core of S and DU the domain of U . Suppose that |(U f, Sg)| ≤ R f
g
(3.2)
for all f ∈ DU and g ∈ DS with some constant R. Then U ∗ S has unique bounded operator extension U ∗ S with U ∗ S ≤ R. In particular, if S is a bounded operator, then U ∗ S is a bounded operator.
3.1
Relative bounds and invariant domains for ∆
Let Hd := −∆/2 + Hf . ˆ ωλ ˆ ∈ L2 , E > 0 and t > 0. Then both ∆k e−tHbS , k ≥ 0, and Lemma 3.2 Let λ, S + E)−1 are bounded operators. In particular, e−tHbS leaves C ∞ (∆) invariant. ∆(H Proof. We fix k ≥ 0. From the functional integral representation (5.19) it follows that b (3.3) |(F, e−tHS G)| ≤ (|F |, e−tHd |G|). 2fin (Q) ∩ C ∞ (Hf )]. Let FR denote the real part of F Take F ∈ D := C0∞ (RdN )⊗[L and FI its imaginary part, i.e., F = FR + iFI . Define subsets of RdN × Q by dN Q+ × Q|(∆k FR )(x, φ) > 0}, R := {(x, φ) ∈ R dN Q− × Q|(∆k FR )(x, φ) < 0}, R := {(x, φ) ∈ R dN × Q|(∆k FI )(x, φ) > 0}, Q+ I := {(x, φ) ∈ R dN Q− × Q|(∆k FI )(x, φ) < 0}. I := {(x, φ) ∈ R
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Self-adjointness of the Pauli-Fierz Hamiltonian
Define also F1 :=
FR (x, φ), 0,
−FR (x, φ), 0, FI (x, φ), F3 := 0, −FI (x, φ), F4 := 0,
185
(x, φ) ∈ Q+ R, (x, φ) ∈ Q+ R, (x, φ) ∈ Q− R, (x, φ) ∈ Q− R,
F2 :=
(x, φ) ∈ Q+ I , (x, φ) ∈ Q+ I , (x, φ) ∈ Q− I , (x, φ) ∈ Q− I .
Note that Fi ∈ D(∆k ) and ∆k Fi > 0, i = 1, 2, 3, 4. Since |∆k FR | = ∆k F1 + ∆k F2 and |∆k FI | = ∆k F3 + ∆k F4 , we have b
b
b
|(∆k F, e−tHS G)| ≤ |(∆k FR , e−tHS G)| + |(∆k FI , e−tHS G)| ≤ |(|∆k FR |, e−tHd |G|)| + |(|∆k FI |, e−tHd |G|)| =
4
(∆k Fi , e−tHd |G|) ≤
i=1
4
Fi
G .
i=1
Since F1 2 + F2 2 = FR 2 , F3 2 + F4 2 = FI 2 , and FR 2 + FI 2 = F 2 , 4 we have i=1 Fi ≤ 2 F . Hence b
|(∆k F, e−tHS G)| ≤ 2 F
G .
(3.4)
b
Since D is a core of ∆k , the boundedness of ∆k e−tHS follows from Lemma 3.1. By means of (3.3) we have S + E)−1 G)| ≤ (|F |, (Hd + E)−1 |G|). |(F, (H S + E)−1 HS ⊂ D(∆) and Note that ∆(Hd + E)−1 is bounded. Thus, (H S + E)−1 G)| ≤ 2 F
G |(∆F, (H S + E)−1 is bounded by Lemma 3.1. follows in a similar way as (3.4). Hence ∆(H Thus we get the desired results. ˆ ωλ ˆ ∈ L2 . Then the same statements as Lemma 3.2 hold with Lemma 3.3 Let λ, S replaced by H A . In particular, e−tHbA leaves C ∞ (∆) invariant. H Proof. We have
b
|(F, e−tHA G)| ≤ (|F |, e−t(−∆/2) |G|). Hence, the same proof as of Lemma 3.2 yields the desired results.
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3.2
Ann. Henri Poincar´e
Relative bounds and invariant domains for Hf and Nf
We identify W0 = ⊕d L2 (Rd+1 ) ∼ = L2 (R) ⊗ W . Let ω 0 := 1 ⊗ ω : W0 → W0 . Let E refer to the expectation value with respect to db. The following is useful for b estimating matrix elements of e−tHS . k/2 ˆ ∈ L2 . Then K(t) ∈ D( ˆ ωλ ˆ ∈ L2 and ω k/2 λ ω0 ) and Proposition 3.4 Let λ,
(2m)!
k/2 ˆ 2m tm dm N 2m ω k/2 λ E ω0 K(t) 2m W0 ≤ L2 . 2m
Proof. See [21, Theorem 4. 6]. We define Hf0 : L2 (Q0 ) → L2 (Q0 ) by Hf0 :φ0 (f1 ) · · · φ0 (fn ):=
n
:φ0 (f1 ) · · · φ0 ( ω0 fj ) · · · φ0 (fn ):,
j=1
Hf0 1 = 0. Then
Ξ∗t Hf0 = Hf Ξ∗t .
(3.5)
Let π0 (f ) := i[Hf0 , φ0 (f )]. Thus we have [φ0 (f ), π0 (g)] = iq0 ( ω0 f¯, g).
(3.6)
ˆ ωλ ˆ ∈ L2 . Then it is easily seen that Jt maps D(Hf ) onto itself. From (3.5) Let λ, and (3.6) it follows that e2 ∗ −ieφ0 (K(t)) [Hf , Jt ] = Ξ0 e ω0 K(t), K(t)) Ξt (3.7) −eπ0 (K(t)) + q0 ( 2 holds on D(Hf ). ˆ ωλ ˆ ∈ L2 . Then for Ψ ∈ D(Hf ) we have e−tHbS Ψ ∈ D(Hf ), and Lemma 3.5 Let λ, there exists a constant C such that √ b
[e−tHS , Hf ]Ψ ≤ C(t + t) (Hf + 1)1/2 Ψ . (3.8) Proof. Let F, G ∈ D(Hf ). Note that (Hf F )(x) = Hf (F (x)) for each x ∈ RdN . Then bS −tH (Hf F, e G) = (F0 , Hf Jt Gt ) dX M
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Self-adjointness of the Pauli-Fierz Hamiltonian
187
(F0 , [Hf , Jt ]Gt ) dX +
=
(F0 , Jt Hf Gt ) dX
M
M
b
(F0 , [Hf , Jt ]Gt ) dX + (F, e−tHS Hf G).
= M
We have by (3.7)
(F0 , [Hf , Jt ]Gt ) dX M
e2 ∗ −ieφ0 (K(t)) = ω0 K(t), K(t)) Ξt Gt dX . F0 , Ξ0 e −eπ0 (K(t)) + q0 ( 2 M
Note that
π0 (K(t))Ξt Ψ ≤ θ (Hf + I)1/2 Ψ with
d−1 1/2 ω0 K(t) ). ω0 K(t) + θ := √ (2 2
Moreover 1/2
ω0 K(t), K(t))| ≤ ω0 K(t) 2 := η. |q0 ( Using Proposition 3.4, we have 1/2
ω0 K(t) 2 + E ωK(t) 2 ) E(θ2 ) ≤ (d − 1)2 (4E ˆ 2) ˆ 2 + tN 2 d ω λ ≤ (d − 1)2 (4tN 2 d ω 1/2 λ ˆ 2 ), ˆ 2 + ω λ ≤ 4tN 2 d(d − 1)2 ( ω 1/2 λ and ˆ 4. E(η 4 ) ≤ 6t2 N 4 d2 ω 1/2 λ By the Schwartz inequality we have (F0 , [Hf , Jt ]Gt ) dX M
1/2
≤ |e|
1/2
F0 2 θ2 dX
(Hf + I)1/2 Gt 2 dX
M
M 2
e + 2
1/2
F0 η dX M
Rd
≤
2
M
It then follows that 1/2 2 2
F0 θ dX ≤ M
1/2
Gt dX
2 4
1/2 (Eθ
2
) F (x) 2L2 (Q) dx
ˆ + ω λ ) F ˆ 4tN 2 d(d − 1)2 ( ω 1/2 λ
,
.
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F. Hiroshima
and
1/2
F0 η dX 2 4
M
≤
≤
Ann. Henri Poincar´e
1/2 (Eη
Rd
4
) F (x) 2L2 (Q) dx
√ ˆ 2 F . 6t2 N 4 d2 ω 1/2 λ
√ (F0 , [Hf , Jt ]Gt ) dX ≤ C( t + t) F
(Hf + I)1/2 G
Thus
(3.9)
M
with C := max
1√ 4 2 ˆ 2 ˆ + ω λ ), ˆ 4N 2 d(d − 1)2 ( ω 1/2 λ 6N d ω λ . 2
Hence we proved that √ b |(Hf F, e−tHS G)| ≤ C( t + t) + 1 (Hf + 1)G
F . b
Then e−tHS G ∈ D(Hf ) by Lemma 3.1. Moreover, since bS −tH , Hf ]G) = (F0 , [Hf , Jt ]Gt ) dX , (F, [e M
(3.8) follows from (3.9).
ˆ ωλ ˆ ∈ L2 . Let Ψ ∈ D(Hf ) and E > 0. Then, for all k > 0, Lemma 3.6 Let λ, S + E)−k Ψ ∈ D(Hf ) and (H ∞ b S + E)−k Ψ = 1 Hf (H t−1+k Hf e−tHS e−tE Ψdt, (3.10) Γ(k) 0 ∞ where Γ(k) := 0 e−x xk−1 dx is the usual Gamma function. Proof. Note that S + E)−k Ψ = (H
1 Γ(k)
∞
b
e−tHS e−tE t−1+k Ψdt
0
in the strong sense. We have b
b
b
Hf e−tHS Ψ ≤ [e−tHS , Hf ]Ψ + e−tHS Hf Ψ √ ≤ C(1 + t + t) (Hf + 1)Ψ with some constant C. Let Φ ∈ D(Hf ). It is obtained that ∞ 1 b −k (Hf Φ, (HS + E) Ψ) = (Hf Φ, e−tHS e−tE t−1+k Ψ)dt Γ(k) 0
(3.11)
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Self-adjointness of the Pauli-Fierz Hamiltonian
=
1 Γ(k)
∞
189
b
(Φ, Hf e−tHS e−tE t−1+k Ψ)dt.
0
By virtue of (3.11), S + E)−k Ψ)| ≤ |(Hf Φ, (H ≤
1 Γ(k)
∞
C(1 +
1 Γ(k)
∞
b
Φ
Hf e−tHS e−tE t−1+k Ψ dt
0
√ t + t)e−tE t−1+k dt Φ
(Hf + 1)Ψ .
0
S + E)−k Ψ ∈ D(Hf ) by Lemma 3.1 and (3.10) holds by the strong Hence (H ∞ b integrability of 0 Hf e−tHS e−tE t−1+k Ψdt. Then the lemma follows. ˆ ωλ ˆ ∈ L2 . Then H 1/2 (H S + E)−1/2 is a bounded operator. Lemma 3.7 Let λ, f S G) for G ∈ D(∆) ∩ D(Hf ) ∩ D(N ), Proof. Since (G, Hf G) ≤ (G, HS G) = (G, H we have 1/2
Hf
S + E)−1/2 G ≤ G , (H
G ∈ D(∆) ∩ D(Hf ) ∩ D(N ).
Thus the lemma follows, since D(∆) ∩ D(Hf ) ∩ D(N ) is dense.
ˆ ωλ ˆ ∈ L2 . Then Hf (H S + E)−1 is a bounded operator. Lemma 3.8 Let λ, S + E)−1/2 Ψ ∈ D(Hf ) Proof. Let Ψ ∈ D(Hf ). From Lemma 3.6, it follows that (H −1 and (HS + E) Ψ ∈ D(Hf ). Then we have S + E)−1 Ψ = Hf (H S + E)−1/2 Ψ + [(H S + E)−1/2 , Hf ](H S + E)−1/2 Ψ, S + E)−1/2 Hf (H (H and
S + E)−1/2 , Hf ](H S + E)−1/2 Ψ = [(H ∞ 1 1 b S + E)−1/2 Ψdt √ [e−tHS , Hf ]e−tE (H Γ(1/2) 0 t
holds. We have
S + E)−1/2 , Hf ](H S + E)−1/2 Ψ
[(H ∞√ t+t C S + E)−1/2 Ψ e−tE dt √ (Hf + I)1/2 (H ≤ Γ(1/2) 0 t ∞ √ C S + E)−1/2
Ψ . ≤ (1 + t)e−tE dt (Hf + I)1/2 (H Γ(1/2) 0
(3.12)
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S + E)−1/2 Hf (H S + E)−1/2 is a bounded operator. Hence, by By Lemma 3.7, (H virtue of (3.12), we have S + E)−1 Ψ ≤ C Ψ
Hf (H with some constant C . Since D(Hf ) is dense, the lemma follows.
Next we show a relative bound of Nf . The procedure is similar to that of Hf . We define Nf0 : L2 (Q0 ) → L2 (Q0 ) by Nf0 :φ0 (f1 ) · · · φ0 (fn ):= n :φ0 (f1 ) · · · φ0 (fn ): . Nf0 1 = 0. Then
Ξ∗t Nf0 = Nf Ξ∗t .
Let
(3.13)
π0 (f ) := i[Nf0 , φ0 (f )].
Thus we have
[φ0 (f ), π0 (g)] = iq0 (f¯, g).
For arbitrary l ≥ 0, Jt maps Nfl Jt
=
Ξ∗0 e−ieφ0 (K(t))
D(Nfl )
(3.14)
onto itself and
l e2 0 Ξt −eπ0 (K(t)) + q0 (K(t), K(t)) + Nf 2
(3.15)
holds on D(Nfl ). ˆ ωλ ˆ ∈ L2 . Then both of e−tHbS and e−tHbA leave C ∞ (Nf ) invariLemma 3.9 Let λ, ant. Proof. Let F, G ∈ C ∞ (Nf ). Then b
(Nfl F, e−tHS G) =
M
(F0 , Nfl Jt Gt ) dX .
By (3.15), we easily obtain
Nfl Jt Gt ≤ C1 (X) (Nf + 1)l Gt . Here C1 (X) = C1 (X, t) is a constant depending on K(t) n , n = 0, 1, 2, ..., l, and satisfies by (3.4), E(C1 (X)2 ) ≤ P (t), where P (t) = al tl + · · · + a1 t + a0 with some aj ≥ 0. Hence b
|(Nfl F, e−tHS G)| ≤ P (t) F
(Nf + 1)l G .
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b
Thus e−tHS G ∈ D(Nfl ). Since l is arbitrary, b
e−tHS C ∞ (Nf ) ⊂ C ∞ (Nf ). Similarly we have b
(Nfl G, e−tHA G) =
M
(Nfl F0 , e−ieφ(L(t)) Gt ) dX =
M
(F0 , Nfl e−ieφ(L(t)) Gt ) dX .
We also see that
Nfl e−ieφ(L(t)) Gt ≤ C2 (X) (Nf + 1)l Gt (X) with E(C2 (X)2 ) ≤ Q(t), where Q(t) = bl tl + · · · + b1 t + b0 with some bj ≥ 0. Hence b
|(Nfl F, e−tHA G)| ≤ Q(t) F
(Nf + 1)l G . b
Thus the lemma follows in the same way as with e−tHS .
4 Proofs of the main theorems ˆ √ω, ω λ ˆ ∈ L2 . Then HS is self-adjoint on D(∆) ∩ D(Hf ). Lemma 4.1 Let λ/ S ) ⊃ D(∆) ∩ D(Hf ). Since e−tHb S leaves D(∆) ∩ D(Hf ) Proof. Note that D(H S . By Lemmas 3.2 and 3.8, we have invariant, D(∆) ∩ D(Hf ) is a core of H S Ψ + Ψ ),
∆Ψ + Hf Ψ ≤ C( H
S ), Ψ ∈ D(H
(4.1)
with some constant C, and by (1.8)-(1.10) S Ψ = HS Ψ ≤ C ( ∆Ψ + Hf Ψ ),
H
Ψ ∈ D(∆) ∩ D(Hf ),
S is closed on D(∆) ∩ D(Hf ). Thus H S is with some positive constant C . Hence H self-adjoint on D(∆) ∩ D(Hf ). By Lemma 2.2, we have HS D(∆)∩D(Hf ) ⊂ HS . Thus HS is self-adjoint on D(∆) ∩ D(Hf ). ˆ √ω, ω λ ˆ ∈ L2 . From the proof of Lemma 4.1 we conclude that Remark 4.2 Let λ/ S . HS = H
(4.2)
ˆ ωλ ˆ ∈ L2 . Then HS is essentially self-adjoint on C ∞ (∆) ∩ Lemma 4.3 Let λ, ∞ D(Hf ) ∩ C (Nf ).
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b
S ) ⊃ D(∆) ∩ D(Nf ) ∩ D(Hf ). Since e−tHS leaves C ∞ (∆) ∩ Proof. Note that D(H ∞ S is essentially self-adjoint on C ∞ (∆) ∩ D(Hf ) ∩ D(Hf ) ∩ C (Nf ) invariant, H S ⊃ HS D(∆)∩D(N )∩D(H ) . Thus HS is essentially selfC ∞ (Nf ). By Lemma 2.2, H f f ∞ adjoint on C (∆) ∩ D(Hf ) ∩ C ∞ (Nf ). ˆ ωλ ˆ ∈ L2 . Then HA is essentially self-adjoint on C ∞ (∆) ∩ Lemma 4.4 Let λ, ∞ C (Nf ). A ) ⊃ D(∆) ∩ D(N ). Since e−tHb A leaves C ∞ (∆) ∩ C ∞ (Nf ) Proof. Note that D(H A ⊃ A is essentially self-adjoint on C ∞ (∆)∩C ∞ (Nf ). By Lemma 2.2, H invariant, H HA D(∆)∩D(Nf ) . Thus HA is essentially self-adjoint on C ∞ (∆) ∩ C ∞ (Nf ). ˆ ωλ ˆ ∈ L2 . Then Hspin + H(a) in the Schr¨ Lemma 4.5 (1) Let λ, odinger represenˆ ˆ tation is infinitesimally small with respect to HS . (2) Let λ, ω λ ∈ L2 . Then V is relatively bounded with respect to both HA and HS with the relative bound aV < 1. Proof. Since for arbitrary , > 0 we have
(Hspin + H(a))Ψ ≤ , (−∆/2 + Hf )Ψ + b Ψ with some constant b , (4.1) and (4.2) imply (1). (2) follows from [21, Lemma 5.10]. A proof of Theorem 1.3 By Lemmas 4.1 and 4.5, HS +V +Hspin +H(a) is self-adjoint on D(∆)∩D(Hf ) and bounded from below. By Proposition 2.1, HPF is self-adjoint on D(∆) ∩ D(Hrad ). The essential self-adjointness of HPF follows from Corollary 1.4. A proof of Theorem 1.5 By Lemmas 4.3 and 4.5, HS + V + Hspin + H(a) is essentially self-adjoint on C ∞ (∆) ∩ D(Hf ) ∩ C ∞ (N ). Since the unitary operator T in Proposition 2.1 maps T −1 : C ∞ (∆)∩D(Hf )∩C ∞ (Nf ) → C ∞ (∆)∩D(Hrad )∩C ∞ (N ), HPF is essentially self-adjoint on C ∞ (∆) ∩ D(Hrad ) ∩ C ∞ (N ). A proof of Theorem 1.6 By Lemmas 4.4 and 4.5, HA + V is essentially self-adjoint on C ∞ (∆) ∩ C ∞ (Nf ). Since the unitary operator T in Proposition 2.1 maps T −1 : C ∞ (∆) ∩ C ∞ (Nf ) → j j 2 ∞ C ∞ (∆)∩C ∞ (N ), 12 N j=1 p − eA(x ) +V is essentially self-adjoint on C (∆)∩ ∞ C (N ).
5 Appendix ˆ √ω, ω λ ˆ ∈ L2 and In this Appendix we prove Lemma 2.2. For the case when λ/ 2ˆ 2 ω λ ∈ L , Lemma 2.2 has been proven in [21]. We want to remove the assumption ˆ ∈ L2 . We have ω2λ IF,G = (F, Rt G).
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Here Rt G := E (Jt G(Xt )) .
(5.1)
Using Markov properties of Ξt [35] and the Brownian motion b(t), we can directly prove that Rt has the semigroup property, i.e., Rt+s G = Rt Rs G. Thus Rt defines the semigroup in HS . It is not easy, however, to see directly from (5.1) that its self-adjoint generator is a certain extension of HS . So let us devote a little more space to proving Lemma 2.2. We define a contractive symmetric operator Qs as follows. For F ∈ HS , let (Q0 F )(x) := F (x), and j j j j (Qs F )(x) := ps (|x − y|)e(−ie/2)Aµ (λ(·−x )+λ(·−y ))(xµ −yµ ) F (y)dy, s > 0, RdN
(5.2) 2 ˆ ω2λ ˆ∈ for almost every x ∈ Rd , where ps (T ) := (2πs)−dN/2 e−T /(2s) , T ∈ R. Let λ, 2fin (Q) and G ∈ HS , a direct calculation shows that L2 . For F ∈ C0∞ (RdN )⊗L g(s) := (Qs F, G) is differentiable in s > 0 and lims→0+ g (s) = −(HA F, G). Hence lim
n→∞
(Qt/n F, G) − (F, G) = −(HA F, G), t/n
t > 0.
ˆ ∈ L2 and F, G ∈ HS . Put fn (s) := (Qs/2n )2n . Then Lemma 5.1 Let λ (F, fn (s)fm (t)G)HS = (F (X0 ), e−ieφ(Lmn (s,t)) G(Xs+t ))L2 (Q) dX .
(5.3)
M
Here Lmn (s, t) :=
N
jµ ⊕dµ=1 lmn (s, t),
j=1 jµ lmn (s, t)
1 2
:=
2m
j j jµ jµ λ(· − Xsk/2 Xsk/2 m +t ) + λ(· − Xs(k−1)/2m +t ) m +t − Xs(k−1)/2m +t
k=1 2n
1 j j jµ jµ + λ(· − Xtk/2 Xtk/2 n ) + λ(· − Xt(k−1)/2n ) n − Xt(k−1)/2n 2 k=1
ˆ ω2λ ˆ ∈ L2 , then In particular, if λ, (F (X0 ), e−ieφ(L(s+t)) G(Xs+t )) dX . lim (F, fn (s)fm (t)G) = m,n→∞
(5.4)
M
Proof. See [18, the proof of Lemma 4.6].
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F. Hiroshima
Ann. Henri Poincar´e
Lemma 5.2 Let fn ∈ L2 (P ) ⊗ W . Assume that lim E fn − f 2W = 0. n→∞
Then s − limn→∞ eiφ(fn ) = eiφ(f ) on L2 (P ) ⊗ L2 (Q). Proof. It is enough to prove the weak convergence of eiφ(fn ) G for G in a dense domain in L2 (P ) ⊗ L2 (Q). Note that
1/2
eiφ(fn ) − 1 Φ L2 (Q) ≤ C fn W ( Nf Φ L2 (Q) + Φ L2 (Q) ) 1/2
for Φ ∈ D(Nf
) with some constant C. Thus it follows that, for
F ∈ L2 (P ) ⊗ L2 (Q)
G ∈ L∞ (P )⊗D(N f
1/2
and
),
F, (eiφ(fn ) − eiφ(f ) )G 2 2 L (P )⊗L (Q)
iφ(fn ) iφ(f )) = db F (b), e G(b) −e 2 L (Q) P
1/2 db F (b)
fn (b) − f (b) W Nf G(b) + G(b) ≤ P
1/2 1/2 sup Nf G(b) L2 (Q) + G(b) L2 (Q) . ≤ F K E fn − f 2W b∈Rd
Hence the lemma follows.
ˆ ωλ ˆ ∈ L2 . Then there exists a nonnegative self-adjoint operator Lemma 5.3 Let λ, A in HS such that, for F, G ∈ HS , H b
0 IF,G = (F, e−tHA G)HS .
(5.5)
ˆ ∈ L2 . Define ˆ ω2λ Proof. Let λ, St G := E Jt0 G(Xt ) . Then St ≤ 1. From (5.4) it follows that 0 . lim (F, fn (t)G) = (F, St G) = IF,G
n→∞
Thus it is seen that (F, St Ss G) =
lim (F, fn (t)fm (s)G)
m,n→∞
(5.6)
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195
(F (X0 ), e−ieφ(L(t+s)) G(Xt+s )) dX = (F, St+s G).
= M
Hence t, s ≥ 0.
St Ss = St+s ,
(5.7)
Since fn (t), n ≥ 0, is symmetric, we have St∗ = St ,
t ≥ 0.
(5.8)
ˆ ωλ ˆ ∈ L2 and ω 2 λ ˆn be such that λ ˆn , ω λ ˆn, ω2λ ˆ ∈ L2 . Let λ ˆ n ∈ L2 Next we let λ, 2 ˆ ˆ ˆ ˆ and that λn → λ and ω λn → ω λ strongly in L as n → ∞. Let St (n) and Ln (t) ˆ replaced by λ ˆ n , respectively. We have defined by St and L(t) with λ ˆ−λ ˆn 2 = 0. lim E L(t) − Ln (t) 2W = lim tdN λ (5.9) n→∞
n→∞
By Lemma 5.2 and the Lebesgue dominated convergence theorem, we obtain s − lim St (n) = St .
(5.10)
St Sr = s − lim St (n)Sr (n) = s − lim St+r (n) = St+r
(5.11)
s − lim St = 1
(5.12)
n→∞
We easily check that n→∞
n→∞
by (5.7), t→∞
by the definition of St , and finally St∗ = s − lim St (n)∗ = s − lim St (n) = St n→∞
n→∞
(5.13)
by (5.8). Then, (5.11), (5.12) and (5.13) imply that {St }t≥0 is a symmetric strongly continuous one-parameter semigroup. Thus there exists a nonnegative self-adjoint A such that operator H b e−tHA = St , t ≥ 0. Thus we get the desired results.
A . Next we investigate the domain of H ˆ ωλ ˆ ∈ L2 . Then H A ⊃ HA D(∆)∩D(N ) . (1) Let λ, f √ ˆ ∈ L2 . Then H ˆ ω, ω λ A ⊃ HA D(∆)∩D(H ) . (2) Let λ/ f
Lemma 5.4
ˆ ω2λ ˆ ∈ L2 . We have, by (5.2), 2fin (Q), F ∈ HS , and λ, Proof. Let G ∈ C0∞ (RdN )⊗L n
−1 2 1 −tHbA 1 Qt/2n − 1 j n e − 1 G, F = lim G, (Q ) F t/2 n→∞ t 2n t/2n j=0
196
F. Hiroshima
1
= lim
n→∞
0
Qt/2n − 1 G, (Qt/2n )[ns] F t/2n
ds =
1
Ann. Henri Poincar´e
b
−(HA G, e−tsHA F )ds.
(5.14)
0
Since we show that, by (5.14) and ˆ
HA G ≤ C λ ( ∆G + Nf G + G ),
(5.15)
ˆ ∈ L2 . equation (5.14) extends to G ∈ D(∆) ∩ D(Nf ) and F ∈ HS . Suppose ω 2 λ 2ˆ 2 ˆ ˆ ˆ ˆ ˆ Let λn be such that λn , ω λn , ω λn ∈ L and that λn → λ strongly as n → ∞. ˆ replaced by λ ˆ n . Thus (5.18) A (n) and HA (n) are defined by H A and HA with λ H A and HA , respectively. Then, by the A (n) and HA (n) instead of H holds with H definition of HA , b
b
s − lim e−tHA (n) = s − lim St (n) = St = e−tHA .
(5.16)
ˆ n − λ ( ∆F ˆ
(HA (n) − HA )F ≤ C λ
+ Nf F + F ),
(5.17)
n→∞
n→∞
Since it is seen that HA (n)F converges to HA F strongly as n → ∞, Hence, by a limiting argument 1
1 −tHbA b e − 1 G, F = −(HA G, e−tsHA F )ds, (5.18) t 0 ˆ with λ, ˆ ωλ ˆ ∈ L2 and for G ∈ D(∆) ∩ D(Nf ), F ∈ HS . Thus taking holds for λ ˆ ωλ ˆ ∈ L2 , then t → ∞ on both sides of (5.18), we see that, if λ, A ⊃ HA D(∆)∩D(N ) . H f ˆ ∈ L2 and ω 2 λ ˆ ∈ L2 . We have ˆ √ω, ω λ Next we let λ/
HA G ≤ C1 ( ∆G + Hf G + G ), ˆ √ω , λ , ˆ and ω λ . ˆ Hence (5.18) extends to G ∈ where C1 depends only on λ/ 2ˆ 2 ˆ ˆ n /√ω, λ ˆn , ω λ ˆn, ω2λ ˆ n ∈ L2 D(∆)∩D(Hf ). Suppose ω λ ∈ L . Let λn be such that λ √ √ ˆ n / ω → λ/ ˆ ω, λ ˆn → λ ˆ and ω λ ˆn → ωλ ˆ strongly as n → ∞. H A (n) and that λ and HA (n) are defined in the same way as above. It holds that
(HA (n) − HA )G ≤ C2 (n)( ∆G + Hf G + G ), ˆ n − λ)/ ˆ √ω , λ ˆ n − λ ˆ and ω(λ ˆ n − λ) ˆ with where C2 (n) depends on only (λ limn→∞ C2 (n) = 0. Then (5.18) holds for G ∈ D(∆) ∩ D(Hf ) and F ∈ HS . The ˆ √ω, ω λ ˆ ∈ L2 , then limiting argument similar to (5.16) leads to that, if λ/ A ⊃ HA D(∆)∩D(H ) . H f Thus the lemma follows. A ) ∩ D(Hf ) is dense, H S := H A + ˙ Hf is well defined. Since D(H
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197
ˆ ωλ ˆ ∈ L2 . Then H S ⊃ HS D(∆)∩D(N )∩D(H ) . (1) Let λ, f f √ 2 ˆ ∈ L . Then H ˆ ω, ω λ S ⊃ HS D(∆)∩D(H ) . (2) Let λ/ f
Lemma 5.5
Proof. It follows from Lemma 5.4. Next we construct a functional integral representation of e ˆ ω2λ ˆ ∈ L2 . Then Lemma 5.6 Let λ, b
(F, e−tHS G)HS = IF,G ,
bS −tH
.
F, G ∈ HS .
(5.19)
Proof. We show an outline of the proof. See [18] for details. By the Trotter-Kato product formula and (2.1) we have b
b
(F, e−tHS G) = lim (F, (e−tHSA /2 e−tHf /2 )2 G) n
n
n
n→∞
b
b
b
= lim (F, Ξ∗0 (Ξ0 e−tHSA /2 Ξ0 )(Ξ∗t/2n e−tHSA /2 Ξt/2n ) · · · (Ξt e−tHSA /2 Ξ∗t )Ξt G). n
n
n
n→∞
Using (5.2) and the Markov property [35] of Ξt , the right hand side above is = lim (Ξ0 F0 , e−ieφ0 (Kn (t)) Ξt Gt ), n→∞
where Kn (t) =
N
⊕dµ=1
j=1
n −1 (i+1)t/2n 2
it/2n
i=0
ξit/2n λ(· − Xs )dbjµ (s).
Hence we get (5.19). ˆ ωλ ˆ ∈ L2 but ω 2 λ ˆ ∈ L2 . We want to extend (5.19) for the case when λ, ˆ ωλ ˆ ∈ L2 . Then (5.19) holds true. Lemma 5.7 Let λ, Proof. We put
n n n 2 b Pn := e−tHA /2 e−tHf /2
and
n n n 2 b Pn (m) := e−tHA (m)/2 e−tHf /2 .
ˆ replaced by λ ˆm . A with λ A (m) is defined by H Here H bA (m) bA −tH −tH From s − limm→∞ e =e it follows that s − limm→∞ Pn (m) = Pn . It is easily proven that (Ξ0 F0 , e−ieφ0 (Knm (t)) Ξt Gt ) dX . (F, Pn (m)G) = M
Here
N 2 −1
n
Knm (t) :=
⊕dµ=1
j=1 i=0
Let
N 2 −1 j=1 i=0
ti/2n
n
Kn (t) :=
t(i+1)/2n
⊕dµ=1
ξti/2n λm (· − Xs )dbjµ (s).
t(i+1)/2n ti/2n
ξti/2n λ(· − Xs )dbjµ (s).
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Since ξs is an isometry, we have lim E Knm (t) − Kn (t) 2W0
m→∞ −1 N d 2
n
= lim
m→∞
E
−1 N d 2
n
m→∞
E
n 2 −1
m→∞
i=0
ξti/2n (λm (· − Xs ) − λ(· − Xs )) dbjµ (s) W0
2
t(i+1)/2n
(λm (· − Xs ) − λ(· − Xs )) dbjµ (s)
ti/2n
j=1 µ=1 i=0
= lim dN
2
t(i+1)/2n
ti/2n
j=1 µ=1 i=0
= lim
W
t ˆ ˆ 2 = lim tdN λ ˆ m − λ ˆ 2 = 0.
λm − λ m→∞ 2n
Hence, by Lemma 5.2, we have lim (F, Pn (m)G) = (F, Pn G) =
m→∞
(Ξ0 F0 , e−ieφ0 (Kn (t)) Ξt Gt ) dX .
(5.20)
M
Moreover by the triangle inequality, E Kn (t) − K(t) 2W0 ≤
N d
n 2 −1
#
N d
$
t(i+1)/2n
E
2
ξti/2n λ(· − Xs )dbjµ (s) − ξti/2n λ(· − Xti/2n )δbjµ (i)
ti/2n
i=0
+
E
j=1 µ=1
W0 n 2 −1
j=1 µ=1
ξti/2n λ(· − Xti/2n )δbjµ (i) −
i=0
2
t
,
ξs λ(· − Xs )dbjν (s) 0
W0
(5.21) where δbjµ (i) := bjµ (t(i + 1)/2n ) − bjµ (ti/2n ). The first term on the right-hand side of (5.21) is N d lim E n→∞
j=1 µ=1
n 2 −1
i=0
#
t(i+1)/2n ti/2n
$
2
= 0.
λ(· − Xs )dbjµ (s) − λ(· − Xti/2n )δbjµ (t, n)
Noting that ˆ (1 − e−|t−s|ω )λ)| ˆ ≤ |t − s| λ
ω ˆ ˆ ˆ − ξt λ ˆ 2 2 = |(λ, λ ,
ξs λ L
W
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ˆ ωλ ˆ ∈ L2 , we see that the second term also converges to zero as n → ∞. Thus, if λ, then lim E Kn (t) − K(t) 2W0 = 0. n→∞
Taking n → ∞ on the both sides of (5.20), we obtain (5.19).
A proof of Lemma 2.2 (1) follows from Lemmas 5.3 and 5.4. (2) follows from Lemmas 5.5 and 5.7.
Acknowledgments. I thank T. Ichinose for his encouragement and H. Spohn for his hospitality of Technische Universit¨ at M¨ unchen. I am grateful to J. L˝ orinczi for a careful reading of my manuscript. I also thank Graduiertenkolleg “Mathematik in ihrer Wechselbeziehung zur Physik” for financial support.
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[12] J. Fr¨ ohlich, Existence of dressed one electron states in a class of persistent models, Fortschritte der Physik 22, 159–198 (1974). [13] J. Fr¨ ohlich, M. Griesemer and B. Schlein, Asymptotic electromagnetic fields in a mode of quantum-mechanical matter interacting with the quantum radiation field, mp-arc 00–374, preprint(2000). [14] J. Fr¨ ohlich and Y. M. Park, Correlation inequalities and thermodynamic limit for classical and quantum continuous systems II. Bose-Einstein and FermiDirac statistics, J. Stat. Phys. 23, 701–753 (1980). [15] C. G´erard, On the existence of ground states for massless Pauli-Fierz Hamiltonians, Ann. H. Poincar´e 1, 443–459 (2000). [16] M. Griesemer, E. Lieb and M. Loss, Ground states in non-relativistic quantum electrodynamics, preprint (2000). [17] L. Gross, The free euclidean Proca and electromagnetic fields, Functional integral and its application, Proceedings of the international conference, the Cumberland Lodge, Windsor Great Park, London in 1974. A. M. Arthurs (editor), Clarendon Press, Oxford (1975). [18] F. Hiroshima, Functional integral representation of a model in quantum electrodynamics, Rev. Math. Phys. 9, 489–530 (1997). [19] F. Hiroshima, Ground states and spectrum of non-relativistic quantum electrodynamics, Trans. of Amer. Math. Soc. 353, 4497–4528 (2001). [20] F. Hiroshima, Ground states of a model in nonrelativistic quantum electrodynamics I, J. Math. Phys. 40, 6209-6222 (1999), II, J. Math. Phys. 41, 661–674 (2000). [21] F. Hiroshima, Essential self-adjointness of translation-invariant quantum field models for arbitrary coupling constants, Commun. Math. Phys. 211, 585–613 (2000). [22] F. Hiroshima and H. Spohn, Enhanced binding through coupling to a quantum field, Ann. H. Poincar´e 2, 1159–1187 (2001). [23] F. Hiroshima and H. Spohn, Ground state degeneracy of the Pauli-Fierz model including spin. mp-arc 01–407, preprint (2001). [24] R. Høegh-Krohn, Asymptotic fields in some models of quantum field theory I, J. Math. Phys. 9, 2075–2080 (1968), II J. Math. Phys. 10, 639–643 (1969), III J. Math. Phys. 11, 185–189 (1969). [25] M. H¨ ubner and H. Spohn, Radiative decay: nonperturbative approaches, Rev. Math. Phys. 7, 363–387 (1995). [26] M. H¨ ubner and H. Spohn, Spectral properties of the spin-boson Hamiltonian, Ann. Inst. Henri Poincar´e 62, 289–323 (1995).
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[27] V. Jakˇsi´c and C. A. Pillet, On a model for quantum friction I. Fermi’s golden rule and dynamics at zero temperature, Ann. Inst. H. Poincar´e, 62, 47–68 (1995). [28] E. Lieb and M. Loss, Self-energy of electrons in non-perturbative QED, preprint (1999). [29] J. L˝ orinczi, R. Minlos and H. Spohn, The infrared behaviour in Nelson’s model of a quantum particle coupled to a massless scalar field, preprint (2001). [30] E. Nelson, Schr¨ odinger particles interacting with a quantized scalar field, Analysis in Function Space, Proceedings U. S. A. (1963). [31] E. Nelson, Interaction of nonrelativistic particles with a quantized scalar field, J. Math. Phys. 5 (1964), 1190–1197. [32] T. Okamoto and K. Yajima, Complex scaling technique in non-relativistic massive QED, Ann. Inst. H. Poincar´e 42, 311–327 (1985). [33] W. Pauli and M. Fierz, Zur Theorie der Emission langwelliger Lichtquanten, Nuovo Cimento 15, 167–188 (1938). [34] M. Reed and B. Simon, Methods of Modern Mathematical Physics II, Academic Press (1975). [35] B. Simon, The P (φ)2 Euclidean (Quantum) Field Theory, Princeton Univ. Press (1974). [36] H. Spohn, Ground state(s) of the spin-boson Hamiltonian, Commun. Math. Phys. 123, 277–304 (1989). [37] H. Spohn, Ground state of quantum particle coupled to a scalar boson field, Lett. Math. Phys. 44, 9–16 (1998). Fumio Hiroshima Zentrum der Mathematik Technische Universit¨ at M¨ unchen Gabelsberger straße 49 M¨ unchen Germany Permanent address: Department of Mathematics and Physics Setsunan University Ikeda-naka-machi 17-8 572-8508, Osaka Japan email:
[email protected] Communicated by Bernard Helffer submitted 17/07/01, accepted 19/11/01
Ann. Henri Poincar´e 3 (2002) 203 – 267 c Birkh¨ auser Verlag, Basel, 2002 1424-0637/02/020203-65 $ 1.50+0.20/0
Annales Henri Poincar´ e
Phase Diagram of Horizontally Invariant Gibbs States for Lattice Models P. Holick´ y, R. Koteck´ y and M. Zahradn´ık Abstract. We study interfaces between two coexisting stable phases for a general class of lattice models. In particular, we are dealing with the situation where several different interface configurations may enter the competition for the ideal interface between two fixed stable phases. A general method for constructing the phase diagram is presented. Namely, we give a prescription determining which of the phases and which of the interfaces are stable at a given temperature and for given values of parameters in the Hamiltonian. The stability here means that typical configurations of the limiting Gibbs state constructed with the corresponding interface boundary conditions differ only on a set consisting of finite components (“islands”) from the corresponding ideal interface.
0 Introduction Before stating our main result in its full generality in the next section, we shall explain the main idea for a particular model. Namely, we shall consider interfaces between two stable phases for the three-dimensional Blume-Capel model. To every site i ∈ Z3 a spin x(i) is attached attaining the values x(i) ∈ {−1, 0, +1}. The Hamiltonian in a finite volume Λ ⊂ Z3 with boundary conditions z is given as (x(i) − x(j))2 + J (x(i) − z(j))2 − λ x(i)2 − h x(i). HΛ (x|z) = J i,j i,j∈Λ
i,j i∈Λ,j ∈Λ /
i∈Λ
i∈Λ
First two sums are over pairs of nearest neighbours, J > 0 is fixed. It is easy to see that the phase diagram in the (λ, h)-plane and at vanishing temperature consists of three regions of ground states of constant spins, x ≡ +1, x ≡ −1, and x ≡ 0, separated by half-lines h = 0, λ ≥ 0; h = λ, λ ≤ 0; and h = −λ, λ ≤ 0. With the help of Pirogov-Sinai theory [PS] one can show that the phase diagram at small temperatures is a smooth deformation of this zero temperature phase diagram [BS] as indicated in Fig. 1. Notice that the region of the phase 0 is expanding as the temperature grows. This is easy to understand by observing that while the phases + and − have only one type of lowest energy excitation, namely, flipping a single spin to the value 0, the phase 0 allows two excitations of this order, flipping to +1 or −1, which gives the phase 0 an advantage at non-vanishing temperatures when excitations contribute to the free energy. The coexistence of all three phases occurs at the line (λ = λ0 (T ), h = 0), where λ0 (0) = 0 and the function λ0 (T ) is growing in T .
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T
λt (T )
λt (0) = 2J
λ
h Fig. 1. Phase diagram of Blume-Capel model. The line λt (T ) corresponds to coexistence of two different interface patterns on the boundary between stable plus and minus phases. Let us consider now, for h = 0, λ > 0, an interface between plus phase in the upper half-space and minus phase in the lower half-space. Depending on the value of λ, different arrangements of spins on the interface yield the minimal energy. Namely, if λ ≥ 2J, the most convenient way is to switch directly from plus to minus spins — the configuration y (I) , y (I) (i) = 1 whenever i3 ≥ 0 and y (I) (i) = −1 whenever i3 < 0, is a ground state. Indeed, it is not difficult to see that the difference of the energy of any configuration x that differs from y (I) on finite number of sites and that of the configuration y (I) itself, is nonnegative, HΛ (x|y (I) ) ≥ HΛ (y (I) |y (I) ). On the other side, if λ ≤ 2J, it is favourable to separate pluses and minuses by a layer of spins 0; the ground state is the configuration y (II) , y (II) (i) = 1 whenever i3 > 0, y (II) (i) = 0 for i3 = 0, and y (II) (i) = −1 whenever i3 < 0. We will see that this behaviour subsists at small temperatures. Namely, there exists a smooth transition function λt (T ) emanating from the point λt (0) = 2J at T = 0 such that, if λ ≥ λt (T ), the boundary conditions y (I) yield (at temperature T and in the thermodynamic limit) the Gibbs state whose typical configurations differ from y (I) only on small islands — the state corresponding to y (I) is stable. Similarly, for λ0 (T ) + δ < λ ≤ λt (T ) the state corresponding to y (II) is stable. At the coexistence line λ = λt (T ) both states are stable. Notice that the transition is
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of the first order type — the variable that exhibits a discontinuity while passing through the transition line is, for example, the density of spin 0 at the interface. We are actually concerned here with the phenomenon of “prewetting” of the microscopic ± interface by zero spins. Notice that we excluded a small neighbourhood of the coexistence line λ0 (T ). It is expected that for λ very close to λ0 (λ − λ0 ∼ O(e−β ) as β → ∞) the layer of zero spins spreads over several lattice sites, with its thickness growing due to “entropic repulsion” as λ λ0 (it was this type of wetting that was discussed for λ = λ0 in [BL]). Our aim in this paper is to study this type of surface phenomena in a general case. Namely, we are interested in situations where, as in the example above, several different interface configurations may enter the competition for the ideal interface between two fixed stable phases. The simpler case with a single ground state interface in the considered region of parameters (as is the case, for example, for the standard Ising model) is well understood [HKZ] as a straightforward, though rather technically involved, generalization of the standard Dobrushin treatment [D 72]. Simplifying slightly, the main idea is to rewrite the finite volume Gibbs state with interface boundary conditions in terms of the probability distribution of the interface contour separating the regions of two coexisting stable phases. To prove the existence of the interface Gibbs state in the thermodynamic limit, one shows that typical interface contours differ only locally from the ideal ground state interface configuration. To this end one splits the interface contour into regions of ideal interface at different heights — the ceilings — separated by walls. The crucial observation that there is a one-to-one correspondence between interface contours and collections of compatible walls, and that the walls are distributed in an independent way, allows one to use a generalization of the Peierls argument, with walls playing the role of contours, to prove that the probability of walls is dumped and to conclude the existence of interface Gibbs state. The most important fact that was skipped out in the simplified description above, is that the interface is actually surrounded, from above and below, by two different — possibly asymmetric — coexisting phases. Their influence is taken into account by rewriting the corresponding partition functions, with the help of the Pirogov-Sinai theory, in terms of cluster expansions. After separating a suitable normalizing factor, we expand the cluster terms intersecting the interface. As a final result, one has to deal with an interface decorated by clusters. The problem in the general case is that presence of various competing interfaces leads to existence of different types of ceilings. To treat the probability distribution of collection of walls one has to cope with the matching conditions on families of walls. Namely, each wall is “labeled” on any connected component of its boundary, where it is attached to surrounding ceilings, by the type of the corresponding ceiling. For the collection of walls stemming from an interface, these labels on boundaries of different walls attached to the same ceiling must coincide. But this is the starting point of the standard Pirogov-Sinai analysis of probabilities of collections of labeled contours. One only has to be slightly more careful when applying these ideas.
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One point to consider is the nonlinear dependence on the original Hamiltonian (and its parameters) of the decorating cluster terms originated from the coexisting phases surrounding the interface. These have to be attached to the considered walls and one thus has to be prepared to deal with a nontrivial dependence of wall weights on parameters of the original Hamiltonian. Second point is that the decorations may actually stick out of the considered finite volume and as a result one is dealing with infinitely many wall-cluster aggregates intersecting the given finite volume. One thus has to be careful when applying standard cluster expansions to treat correctly these situations having also in mind that the weight factors of terms that are not entirely contained in a given volume may actually depend on it yielding a fixed limiting weight factor only when volume expands to infinity. Last but not least, the weight factors of the wall-cluster aggregates might be actually negative, due to presence of cluster terms whose sign is not determined. There are different strategies how to deal with these problems. Here, we have chosen the most conservative one. Namely, we rewrite the interface partition function (and the corresponding Gibbs state) in such a way that we can use directly (essentially) standard Pirogov-Sinai theory with the role of contours played by “shadows” of wall-cluster aggregates projected to the plane of ideal interface. To get positive weights allowing in the final account, for example, to estimate the probability of external shadows in the standard manner, we add a suitable cluster sum into the exponent contributions to the interface partition function, absorbing it in the same time into a small change of weight of interfaces. The cluster contributions of this added sum can be easily chosen in such a way that the positivity of combined cluster terms is assured. There are at least two alternative approaches. First, one may base all the discussion on an extension of Pirogov-Sinai theory to complex parameters [BI]. The first steps in this direction (dealing only with one type of interface and generalizing thus [HKZ]) are done in [BCF 1]. The study in [BCF 1] is motivated by the investigation of interfaces in quantum lattice models [BCF 2]. Other approach, proposed recently by two of us [HZ], is to develop a new alternative to PirogovSinai theory based on the idea of “expanding away”, one by one, all contours and walls without ever passing through intermediate contour models with their cluster expansions as it is the case in the standard Pirogov-Sinai theory. This method is conceptually promising and we expect that it will allow a treatment of a great variety of models with different types of interface, wetting, and other “stratified” states (i.e. consisting of several interfaces). The paper [HZ] develops the theory in a very general situation and it has to be supplied by a detailed study of the dependence of stability of resulting interfaces on parameters of the original Hamiltonian to draw the phase diagram at nonvanishing temperatures. A great deal of the present paper treats this particular problem in the situation of an interface with different ceilings and has thus its importance even though it is based on a rather standard Pirogov-Sinai approach and does not evoke the approach from [HZ].
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The paper is organized as follows. In Section 1, we set the problem, introduce the class of models to be treated with assumptions (Peierls conditions) that assure a dominance by a particular class of ideal interfaces (ceilings). The main result (Theorem 1) is presented in a general form stressing the smooth dependence of the resulting full phase diagram on the temperature (including the description of regions where particular interfaces become stable — cf. Fig. 1). It follows from Basic Lemma whose proof is postponed, after various preparatory steps in Sections 3 and 4, to Section 4.4. Section 1 contains, in addition to Theorem 1 with its proof following from Basic Lemma, also the characterization of the ground state phase diagram including interface ground states (Proposition 1.1.3) as well as its completeness under the condition of removing of degeneracy (Corollary 1.1.4) with their proofs in Section 1.3. Section 2 is devoted to a brief reformulation and a slight extension of the Pirogov-Sinai theory. First, we summarize the results concerning contour models in a form needed for our purposes. This part includes contour models with boundary dependence (models whose contour weights depend slightly on the boundary of the considered volume and are translation invariant only for contours far from the boundary). The corresponding results are of an independent interest, given the fact that one often obtains a reformulation in terms of such a contour model. Next, we introduce labeled contour models and summarize the Pirogov-Sinai theory in Theorem 2 (characterization of stable phases) proved as a consequence of Proposition 2.2.1 (properties of stable phases) that is proved in Section 2.3. Again, also the situation of labeled contour models with boundary dependence is considered (Corollary 2.2.2). A standard application is the description of periodic Gibbs states in terms of contour models. This is needed in the further treatment and it is presented in Section 3 to set the notation. Finally, in Section 4 we reformulate the Gibbs states with interfaces in terms of labeled contour models with role of contours played by the shadows of walls decorated by the clusters of the stable phases above and below the interface. This is done in a series of steps that yield an expression for the weights of shadows that are sufficiently dumped to allow once more the application of Theorem 2.
1 Setting and the main result 1.1
Setting; ground state phase diagram
We shall consider classical lattice models on a ν-dimensional lattice Zν (ν ≥ 3) with a finite set S of spin values attached to each lattice site i ∈ Zν . The configuration ν space will be denoted by X(= S Z ), the space of restrictions xΛ = (x(i) : i ∈ Λ) of configurations x = (x(i) : i ∈ Zν ) ∈ X to Λ ⊂ Zν by XΛ (= S Λ ). We endow the lattice Zν with the ∞ -metric (ρ(i, j) = maxk=1,...,ν |ik − jk |). Connected (Rconnected) set A ⊂ Zν is then defined as a set whose any two sites i, j ∈ A can be
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joined by a sequence of sites i = i1 , i2 , . . . , j = ik from A such that ρ(i , i +1 ) ≤ 1 ( ρ(i , i +1 ) ≤ R) for all = 1, 2, . . . , k − 1. Every translation invariant Hamiltonian H = (UA ; A ⊂ Zν , diamA < R) of range R with interactions UA : XA → R can be identified with H = (U[A] ), where [A] runs over all (finitely many) equivalence classes of subsets of Zν defined by shifts of an A ⊂ Zν with diamA < R. So the translation invariant Hamiltonians of range R form a finite-dimensional vector space of Hamiltonians denoted by H(R). More precisely, it can be identified with the space of all vectors (UA (xA ); i(A) = 0, diamA < R, xA ∈ XA ), where xA are ordered in a fixed way. Here i(A) ∈ A is a site chosen in A in a fixed canonical way (say, the first site in A in a fixed lexicographic order). Throughout we use · to denote the euclidean norm on H(R), · ∞ the maximum norm on it, and dim H(R) to denote its dimension. Using HΛ (x|z) to denote the Hamiltonian in Λ with boundary conditions z ∈ X, UA (xA ) HΛ (x|z) = A∩Λ=∅ c
with x = z in Λ , we introduce the partition function Z(Λ|z; H) = exp{−HΛ (x|z)}. x=zinΛc
It will be useful to define also Z(Y, Λ|z; H) =
exp{−HΛ (x|z)}
xΛ ∈Y x=zinΛc
for any Y ⊂ XΛ . The Gibbs state in a finite volume Λ ⊂ Zν under a boundary condition x ∈ X with Hamiltonian H is the probability µ(·, Λ|z; H) on X defined by µ({x}, Λ|z; H) =
Z({xΛ }, Λ|z; H) Z(Λ|z; H)
whenever x = z in Λc . The set of all Gibbs states in (possibly infinite) V ⊂ Zν under a boundary condition x ∈ XZν \V with Hamiltonian H introduced by means of the DLR equations (see [HKZ], Section 2.1, for discussion of the situation with V Zν ) will be denoted by G(V |x; H). For V = Zν the boundary condition x is necessarily empty and we get the standard definition of Gibbs states (G(Zν | ∅, H) = G(H)). The inverse temperature β does not appear here as an independent parameter; it is incorporated into the constants of the Hamiltonian H. (See also Remark 4 in Section 1.2.) It is well known that the set of all Gibbs states G(H) is the closed convex hull of all possible weak limits limΛn Zν µ(·, Λn |z; H) of finite volume Gibbs states.
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A configuration x ∈ X is called a ground configuration of a Hamiltonian H = (UA ) if “the energy of every finite perturbation x ˜ of x (˜ x differs from x at a finite number of lattice sites) is not smaller than the energy of x”: (UA (˜ xA ) − UA (xA )) ≥ 0. (1.1) H(˜ x; x) = A
If the above difference is positive for each finite perturbation x˜ = x, we refer to x as to a ground state (isolated ground configuration). We use g(H) (resp. g per (H)) to denote the set of all ground configurations corresponding to H (resp. periodic ground configurations). Our aim will be to discuss phase diagrams including a class of Gibbs states with an (“horizontal”) interface. Before taking into account any excitations, we shall describe the phase diagram at vanishing temperature—the ground state phase diagram. We thus suppose that a set G of configurations is given (to play the role of possible ground states). We shall restrict ourselves to horizontally periodic configurations, i.e. we suppose that every x ∈ G is periodic with respect to translations in the first (ν − 1) coordinates. We use Gper to denote the subset of all x ∈ G that are also vertically periodic, i.e. periodic with respect to the ν-th coordinate, and we put Ghor = G \ Gper . Further we assume that each x ∈ Ghor is identical to two configurations y1 , y2 ∈ Gper above and below certain heights, respectively. Namely, for each x ∈ Ghor there exist two states y1 , y2 ∈ Gper and a pair of constants t1 (x), t2 (x) such that x(i) = y1 (i) once iν ≥ t1 (x) and x(i) = y2 (i) once iν ≤ t2 (x). We may assume that t1 (x) and t2 (x) are chosen as the minimal, resp. maximal, constant with this property. We suppose also that the set G is finite up to vertical translations, i.e. there exists a finite subset of G such that any configuration x ∈ G is a vertical translation of a configuration from the considered finite set. In particular, Gper is finite, Gper = {x1 , . . . , xr }, and there exist a finite constant t (maximal thickness of interfaces) such that t ≥ t1 (x) − t2 (x) for all pairs t1 (x), t2 (x) above1 . To control the suppression of excitations with respect to configurations from G, we rely on an extended Peierls condition. To introduce it, let us first define “the specific energy at the site i” by UA (xA ) (H) (1.2) Ei (x) = |A| A i
for each configuration x ∈ X. Here |A| refers to the number of sites in A. The no (H) (H) tation EΛ (x) = i∈Λ Ei (x) will be also used. For every periodic configuration x, x ∈ X per , we also define the specific energy ex (H) of x by (H) 1 ex (H) = lim Ei (x) (1.3) ν n→∞ |Vn | ν i∈Vn
1 This
fact implies that the interface introduced below is necessarily connected.
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(with Vnν denoting a cube consisting of nν lattice sites). For any cube Vnν whose side (H) is a multiple of the periodicity of x ∈ X per, one clearly has EVnν (x) = |Vnν |ex (H), (H) i.e. ex (H) = |V1ν | i∈V ν Ei (x). n n It is useful to introduce an averaged specific energy2 at site i for any configuration x by the right hand side of the last equation. Namely, let p be a common multiple of periods of all configurations from Gper as well as horizontal periods of all configurations from Ghor , for any i ∈ Zν let Vp (i) be the cube Vp (i) = j ∈ Zν ; j = i + k, k = (k1 , . . . , kν ) ∈ {0, 1, . . . , p − 1}ν , and for any configuration x ∈ X let (H)
Ei
(x) =
1 (H) Ej (x). |Vp (i)| j∈Vp (i)
It is easy to verify that for any Λ, the sum a local boundary term, i∈Λ
(H)
Ei
(H)
(x) = EΛ (x) +
j∈Λc
(H)
Ej
(x)
i∈Λ
(H)
Ei
(H)
(x) differs from EΛ (x) by
|Vp∗ (j) ∩ Λ| (H) |Vp∗ (j) ∩ Λc | − . E (x) j |Vp∗ (j)| |Vp∗ (j)| j∈Λ
Here, Vp∗ (j) = {i ∈ Zν ; j ∈ Vp (i)} = i ∈ Zν ; i = j − k, k ∈ {0, 1, . . . , p − 1}ν . The explicit form of the boundary term is not very relevant; an important fact is, however, that whenever x and x ˜ differ only on a finite set, then (H) (H) (H) (H) x) − EΛ (x) = E i (˜ x) − E i (x) (1.4) EΛ (˜ i∈Λ
i∈Λ
once Λ is sufficiently large. More exactly, the equality holds if x(i) = x ˜(i) implies (H)
that d(i, Λc ) > R + p. Clearly, E i (x) = ex (H) for any x ∈ Gper . Let, now, an integer d ≥ R be chosen so that it is surpassing both periodicity p as well as interface maximal thickness t, d > max{R − 1, p, t}. Consider the set of all elementary cubes consisting of dν lattice sites. A bad cube of a configuration x ∈ X is an elementary cube D for which xD differs from yD for every y ∈ Gper . Notice that the choice of d ensures that the only configurations with no bad cube are those from Gper . The boundary B(x) of x is the union of all bad cubes of x. ˜ is its finite perturbation (differing from x on a finite set of If x ∈ Gper and x 2 The reader who is ready to sacrifice the subtlety of the case of periodic but not necessarily translation invariant configurations in Gper can skip the present paragraph, suppose that all (H) (H) x ∈ Gper are translation invariant, and replace everywhere E i (x) by Ei (x). Actually, as explained after the formulation of Basic Lemma in Section 1.3 below, by introducing “block spins” one can rewrite the model (possibly lowering the upper bound on allowed temperatures) in such a way that all configurations in Gper become translation invariant as well as all configurations in Ghor become horizontally translation invariant.
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lattice sites), then, necessarily, B(˜ x) is finite. Any connected component Γ of B(˜ x) is calleda contour (of x ˜) and we use ∂(˜ x) to denote the set of all contours of x˜, B(˜ x) = Γ∈∂(˜x) Γ. Notice that the configuration x ˜ coincides with one of the states x). The only x ∈ Gper that coincides with x ∈ Gper on every component of Z \ B(˜ x ˜ on d-boxes that intersect both Γ and the only infinite component of Zν \ Γ, is called the external boundary condition of the contour Γ of x. Finally, we use e0 (H) for minx∈Gper ex (H) and denote g0per (H) = {x ∈ Gper ; ex (H) = e0 (H)}. The standard Peierls condition3 , with ρ > 0 and with respect to Gper , that is used in the Pirogov-Sinai theory, can be formulated as the bound (H) (per) E i (xΓ ) − ex (H) ≥ ρ|Γ| (P ) i∈Γ
for any x ∈ Gper , any contour Γ, and any configuration xΓ such that Γ is its only contour with external boundary condition x. Notice that if all configurations in Gper are actually translation invariant, E i can be simply replaced by Ei , yielding Peierls condition in the form (H)
EΓ (xΓ ) − ex (H)|Γ| ≥ ρ|Γ|.
(P(per) )
Let us remark that one can normalize the condition with respect to the minimum e0 (H) instead of ex (H) [BK]. This is a useful trick that enables, with some additional care in relevant estimates, to get a uniform validity of the theory far away from lines of coexistence. However, we will restrict our considerations to a small neighbourhood of a fixed Hamiltonian anyway and will thus abstain from this extension. Turning now to the configurations containing interfaces, we first introduce the notion of a wall. Namely, consider a configuration x ∈ Ghor and its excitation x ˜ differing from x on a finite set of lattice sites. Let I(˜ x) denote the infinite connected component of B(˜ x) (notice that B(˜ x) has only one infinite component (cf. footnote 1)) and let I(˜ x), an interface, be the pair (I(˜ x), x ˜I(˜x) ). Notice that I(˜ x) is splitting Zν \ I(˜ x) into two infinite components. Denoting, for any i ∈ Zν , by C(i) the column {(i1 , . . . , iν−1 , n); n ∈ Z} and by Cd (i) its d-neighbourhood, we use C(˜ x) to denote the set of those sites i of I(˜ x) for which there exists a configuration y ∈ G such that I(˜ x) ∩ Cd (i) = I(y) ∩ Cd (i) and x ˜ = y on it. The set C(i) ∩ I(y), for such i ∈ C(˜ x), is called a y-ceiling column. x) \ Further, a pair w = (W, x˜W ), where W is a connected component of I(˜ C(˜ x), is a wall of I(˜ x). We denote by W(I(˜ x)) the collection of all walls of I(˜ x). Whenever w = (W, x˜W ) is a wall, there exists a configuration yw ∈ G and its perturbation xw such that w is the only wall of xw , W(I(xw )) = {w}. For any wall 3 The
word standard here corresponds to the fact that here we are dealing only with “contours immersed in periodic configurations” in contrast with “walls of an interface” as will be the case below. On the other hand, we are dealing here with a Peierls condition valid uniformly on a neighbourhood of Hamiltonians — this type of the Peierls condition is sometimes [EFS] called “extended Peierls condition”.
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w, we use IW = I(yw )∩( i∈W C(i)). Notice that this definition is consistent—the right hand side above is well defined only from geometry of W . Indeed, any wall has an “outside rim” from which the set I(yw ) can be uniquely read of. A Hamiltonian H = (UA ) is said to fulfill the Peierls condition (P) with respect to the set G (and with constant ρ > 0) if it satisfies (P
(per)
) as well as
(H) (H) E i (xw ) − e0 (H) − E i (yw ) − e0 (H) ≥ ρ|W | i∈W
(P
(hor)
)
i∈IW
for any wall w and the corresponding configuration yw ∈ Ghor and its excitation xw . Again, if all configurations in Gper are translation invariant, the condition takes simpler form (H) (H) EW (xw ) − e0 (H)|W | − EIW (yw ) − e0 (H)|IW | ≥ ρ|W |.
(P(hor) )
The symbol (P) denotes that both (P(per) ) and (P(hor) ) are satisfied. The following lemma is related to Lemma 2.1 from [S]. Lemma 1.1.1 Let H satisfy the Peierls condition with respect to a non-empty finite set Gper ⊂ X per. Then all periodic ground configurations of H are ground states, their set g per (H) coincides with the set g0per (H), and g0per (H) minimizes specific energy over all X per , g0per (H) = {x ∈ X per; ex (H) = e0 (H)}. Proof. 1. We first show that each element x of Gper with ex (H) = e0 (H), is a ground state of H. This, in particular, implies g0per(H) ⊂ g per (H). Let y differ from x on a non-empty finite subset of Zν . Let Λ ⊂ Zν be finite and such that x(i) = y(i) implies that dist(i, Λc ) > p + R. (per)
By (1.4), (P (H)
), and the choice of x and Λ, we get (H)
H(y; x) = EΛ (y) − EΛ (x) = (H) (H) (H) (y) − E i (x) = i∈Λ E i (y) − ex (H) = i∈Λ E i (H) (H) = i∈B(y) E i (y) − ex (H) + i∈Λ\B(y) (E i (y) − ex (H)) > ρ|B(y)| > 0. (H)
Notice that for i ∈ / B(y) the values E i (y) equal to some of ez (H) with z ∈ Gper and so they are greater or equal to ex (H) = e0 (H). Hence the claim of the first step is verified. 2. Let ey (H) > e0 (H) for some periodic configuration y. Then we may consider a configuration y˜ that is equal to y outside of some large cube Λ ⊂ Zν and coincides with x in Λ, where x ∈ g per (H). It is obvious that H(˜ y; y) < 0 if Λ is large enough due to the inequality ey (H) > e0 (H) = ex (H). So y is not a ground configuration. 3. Finally, let y be a periodic configuration and y ∈ / Gper . Let us consider x x configurations yΛ such that yΛ equals to y on a sufficiently large cube Λ ⊂ Zν and
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to x ∈ g0per (H) on Λc . We may notice that by the finite range of the potential ey (H) − ex (H) = limν Λ→Z
x H(yΛ ; x) , |Λ|
where the limit is e.g. over any sequence of cubes Λ with diameters tending to infinity. We get, similarly as in step 1, for sufficiently large neighbourhood V of Λ that (H) x (H) x x H(yΛ ; x) = EV (yΛ ) − EV (x) > ρ|B(yΛ )|. x There is c > 0 independent of Λ such that |B(yΛ )| > c|Λ| as y is periodic. Summing up what we observed, we get that ey (H) > ex (H) implying the inclusion {x ∈ X per; ex (H) = e0 (H)} ⊂ Gper and finishing thus the proof.
Our next aim is to study all horizontally periodic ground configurations of H. hor In analogy with ex (H) for x ∈ X per, we consider a configuration x ∈ G∩Xy,z per hor with y, z ∈ g0 (H), y = z, where Xy,z is defined as the set of all horizontally hor if periodic configurations asymptotically coinciding with y and z (i.e. x ∈ Xy,z x(i) = y(i) whenever iν ≥ t1 and x(i) = z(i) whenever iν ≤ t2 for some t1 , t2 ∈ Z), and define (H) 1 (H) ex (H) = lim Ei (x) − Ei (y) . (1.5) ν−1 n→∞ |Vn | ν−1 i:(i1 ,...,iν−1 )∈Vn
Notice that only finite number of terms in the sum does not vanish and that the configurations y and z can be read off from x and we do not introduce them explicitly into the notation. Notice also that (H) ex (H) = E i (x) − e0 (H) , i∈C(0)
where the sum is again only formally infinite. Notice, that even though we keep the same notation as in (1.3), no confusion can arise since these two definitions concern disjoint classes of configurations x; periodic ones in (1.3) and horizontally periodic ones above. Similar notion to e0 (H), that according to Lemma 1.1.1 equals min{ex (H); x ∈ X per}, is hor ey,z 0 (H) = min{ex (H); x ∈ G ∩ Xy,z }. We also use the notation hor g0y,z (H) = {x ∈ Ghor ∩ Xy,z ; ex (H) = ey,z 0 (H)}.
Finally, we say that a set G is admissible if it is non-empty and, up to vertical translations, finite set of horizontally periodic configurations such that for each hor hor for some y, z ∈ Gper , y = z, and that G ∩ Xy,z = ∅ for x ∈ Ghor it is x ∈ Xy,z per each y, z ∈ G , y = z.
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Lemma 1.1.2 Let G be admissible and let H satisfy the Peierls condition (P) with respect to G and some ρ > 0. Then a horizontally periodic configuration x is a ground state, equivalently a ground configuration, of H if and only if either x ∈ g0per (H) hor or x ∈ Xy,z for some y, z ∈ g0per (H), y = z, and ex (H) = ey,z 0 (H). Moreover, such configuration x automatically belongs to G. Proof. According to Lemma 1.1.1, it suffices to consider only non-periodic configurations. 1. We first show that each x ∈ g0y,z (H) for some y, z ∈ g per (H), y = z, is a ground state. Let x ˜ differ from x on a non-empty finite subset of Zν and let Λ ⊂ Zν be a finite cube such that x ˜(i) = x(i) implies that dist(i, Λc ) > R + p. Let xI be the unique configuration with B(xI ) consisting only of the infinite component of B(˜ x). Then H(˜ x; x) = H(˜ x; xI ) + H(xI ; x). (per)
Using (P ), we get that H(˜ x; xI ) ≥ 0, or even that H(˜ x; xI ) > 0 if xI = x (and I I thus x ˜ = x ) when also H(x ; x) = 0. (H) (H) If xI = x, we rewrite H(xI ; x) = i∈Λ E i (xI ) − E i (x) , by adding and subtracting the term e0 (H)|Λ|, as the sum over disjoint columns C of the terms of the form (H) (H) E i (xI ) − e0 (H) − E i (x) − e0 (H) . i∈C∩Λ
i∈C∩Λ
For any x ¯-ceiling column C this term equals to ex¯ (H)−ey,z 0 (H) that is nonnegative by the definition of ey,z 0 (H). Noticing also that for a column C that intersects a (H)
wall w of xI every contribution E i (xI ) − e0 (H) with i ∈ C ∩ W c is nonnegative, and using the bound eyw (H) ≥ ex (H) = ey,z 0 (H), and the transcription of the definition of ex (H) after (1.5) above, we are left with (H) (H) H(xI ; x) ≥ E i (xI ) − e0 (H) − E i (yw ) − e0 (H) > 0 w
i∈W
i∈IW hor
by the Peierls condition (P ). 2a. Let x be a horizontally periodic ground configuration that is not periodic. If, in the upper half space, there are only finitely many horizontal layers of width d with x equal to some element of g per(H) on them, then we easily obtain a per contradiction with (P ) by considering cubes Λ that are large enough and take x on Λ and some x0 ∈ g per (H) on Λc . Notice that the volume of bad cubes in Λ
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is of the order of the volume of Λ. So there must be infinitely many such layers of a configuration y ∈ g per(H) in the upper half-space and infinitely many layers of z ∈ g per(H) (possibly equal to y) in the lower half-space . Further, whenever there are two disjoint layers on which x equals to the same element y of g per (H), it necessarily equals y also in-between of these two layers. Indeed, let us suppose that x is not equal to y in-between. Considering parallelepipeds Λ that have the upper mentioned layer just above its top and the lower mentioned layer just below its bottom and changing x on the d-neighbourhood of the boundary of Λ to y may increase the energy at most by a contribution of the order of the size of the side-wise boundary of Λ (excluding its top and bottom), whereas changing additionally x to y everywhere inside Λ causes a decrease of the per energy by a contribution of the order of the size of the top of Λ (applying (P and the fact that between those two layers there must be a layer containing, due to horizontal periodicity of x, a periodic horizontal grid of bad cubes). Since the size of the top, or bottom, of Λ are asymptotically larger than that of the rest of the boundary of Λ, the energy for Λ large enough decreases and so x is not a ground hor for some y, z ∈ g per(H), configuration. This contradiction shows that x ∈ Xy,z y = z. per hor (H), y = z, x ∈ Xy,z , then x is not 2b. If ex (H) > ey,z 0 (H) for y, z ∈ g hor a ground configuration because replacing it by some element x ¯ of Xy,z for which y,z ex¯ (H) = e0 on sufficiently large cube, with its top in the upper region for both configurations and its bottom in the lower region for both configurations x and x¯, we increase the energy. hor hor , we get that ex (H) > ey,z ) investi3. If x ∈ / G and x ∈ Xy,z 0 (H) from (P 1 w gating the limit of ex (H) = |V ν−1 H(x ; w) for some sequence of cubes V tending n Vn | n
y,z c hor to Zν . Here xw Vn is equal to w on Vn for some w ∈ G ∩ Xy,z with ew (H) = e0 (H) and to x on Vn .
Our aim now is to describe “the phase diagram at zero temperature” — the ground state phase diagram. Consider a Hamiltonian H0 with the set G0 of all horizontally periodic ground states of H0 , G0 = g(H0 ). To describe the ground state phase diagram in a (sufficiently small) neighbourhood of H0 in H(R) means to specify, for every subset G ⊂ G0 , the set of all Hamiltonians H for which G is the set of all horizontally periodic ground configurations, G = g(H). In fact we are describing the phase diagram for H from some cone containing a neighbourhood of H0 . Proposition 1.1.3 (ground state phase diagram) Suppose that H0 satisfies the Peierls condition (P) with a constant ρ0 > 0 and with respect to an admissible set G0 of horizontally periodic ground states, G0 = g(H0 ). ¯ : H ¯ − H0 < Then there exists ε > 0 such that for each H ∈ Kε (H0 ) = {β H εH0 , β > 1} the set g(H) of all horizontally periodic ground configurations is contained in G0 , each x ∈ g(H) is a ground state and either •
x ∈ Gper ¯ ∈ Gper ¯ (H) : x 0 and ex (H) = min{ex 0 }
(1.6)
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•
Ann. Henri Poincar´e
hor x ∈ G0 ∩ Xy,z for some y, z ∈ g per(H), y = z, and hor ex (H) = min{ex¯ (H) : x¯ ∈ G0 ∩ Xy,z }
(1.7)
Moreover, for each 0 < ε¯ < ε there exists a constant ρε¯ such that limε¯→0+ ρε¯ = ρ0 H and each H ∈ Kε¯(H0 ) satisfies the Peierls condition (P) with the constant ρε¯ H . 0 Proof. To show the validity of Peierls condition for the Hamiltonian H with respect 0 to G0 and a suitable ρε¯, it suffices to compare it with the condition for H H H close to H0 . Hence the characterization from Lemmas 1.1.1 and 1.1.2 is valid implying the claims. Remark. Notice that only non-empty admissible sets g(H) ⊂ g(H0 ) will appear. To enable the realization of all admissible G ⊂ G0 in a neighbourhood of H0 (“full ground state phase diagram”) it is necessary to assume that a condition of removing of degeneracy is fulfilled. To formulate it and to enable a global study of the phase diagram at a non-vanishing temperature it is convenient to extend the functional H → ex (H) defined by (1.5) only for those Hamiltonians H for which y, z ∈ g per (H). This can be done for example by defining 1 ex (H) = lim [Ei (x) − min ey (H)] (1.8) ν−1 n→∞ |Vn y∈Gper | 0 i∈I(x) (i1 ,...,iν−1 )∈Vnν−1
hor hor for every horizontally periodic x ∈ Xy,z with y, z ∈ g per (H). Using |G ∩ Xy,z |∼ver hor to denote the number of elements of G ∩ Xy,z taken up to vertical translations, we formulate the condition of removing of degeneracy as the following assumption.
(RD) An affine subspace H0 ⊂ H(R) removes degeneracy of G0 = g(H0 ), an admissible set of horizontally periodic configurations, if the set of hor N0 = |g0per (H0 )| − 1 + (|g0 (H0 ) ∩ Xy,z |∼ver − 1) per y,z∈g0 (H0 ) y=z
linear functionals {ex (H) − ex0 (H), x ∈ g0per(H0 ), x = x0 } ∪ hor , x = xy,z , y, z ∈ g0per(H0 ), y = z} ∪{ex (H) − exy,z (H), x ∈ g0 (H0 ) ∩ Xy,z hor with arbitrarily chosen x0 ∈ Gper and xy,z ∈ G0 ∩ Xy,z is a set of linearly independent functionals on H0 . Notice that choosing H0 = H(R) with R large enough, degeneracy is always removed. (It is sufficient to choose R larger than the smallest periods of configurations in G0 as well as “thickness” of I(x) for all x ∈ G0 ∩ X hor .) If we considered G consisting of translation invariant configurations, already single site potentials (“external fields”) would be enough to remove the degeneracy. Let us also remark that one might consider a more general “manifold of parameters” H0 ⊂ H(R).
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Corollary 1.1.4 (completeness of the ground state phase diagram) Supposing the validity of (RD) (with H0 = H(R)) in addition to the assumptions of Proposition 1.1.3 and denoting Hgr (G) = {H : G = g(H)}, one has Hgr (G) ∩ Kε (H0 ) = ∅ for each admissible G ⊂ g0 (H0 ) and Hgr (G1 ) ∩ Kε (H0 )⊂ = Hgr (G2 ) ∩ Kε (H0 ) whenever G1 ⊃ G . Actually the set H (G) is the intersection of the corresponding gr = 2 number of hyperplanes and open half-spaces in H(R) whose boundaries are yielded by equalities contained in (1.6) and (1.7).
1.2
The main result
Our aim is to show that the phase diagram including all horizontally periodic states is a small distortion of the ground state phase diagram described in Proposition 1.1.3 and Corollary 1.1.4 above. A Gibbs state µ of a Hamiltonian H ∈ H(R) is said to be a perturbation of a (ground) configuration x ∈ G0 if for µ-almost all configurations x ˜ there exists a connected subset M ⊂ Zν such that x ˜ differs from x only outside M , x ˜M = xM , the R-components of its complement Zν \ M are finite and, if x ∈ G0 ∩ X hor , also the set M0 = {i ∈ I(x)|M ⊃ I(x) ∩ C(i)} is connected. Theorem 1 Let H0 be a translation invariant Hamiltonian, H0 ∈ H(R), fulfilling the Peierls condition (P) with respect to an admissible set of ground states G0 = g(H0 ) and a sufficiently large ρ0 . Let further an affine subspace H0 ⊂ H(R) containing H0 remove degeneracy of G0 (condition (RD)). Then there exist constants ε > 0, c > 0, and a one-to-one mapping T from Kε (H0 ) onto a subset of H(R) so that (i) T (H0 ∩ Kε (H0 )) ⊂ H0 ; (ii) for any x ∈ G, with G ⊂ g(H0 ) admissible, there exists a Gibbs state that is a perturbation of x whenever H ∈ T (Hgr (G) ∩ Kε (H0 ))(=: H(G)); (iii) T (H) − H ≤ e
H −(ρ0 −c) H 0
for each H ∈ Kε (H0 );
(iv) (T (H1 ) − H1 ) − (T (H2 ) − H2 ) ≤ e for all H1 , H2 ∈ Kε (H0 ).
H H
−(ρ0 −c) min( H1 , H2 ) 0
0
H1 − H2
Remarks. 1. We expect that the phase diagram is complete in the sense that the only horizontally periodic Gibbs states for H ∈ H(G) are those corresponding to x ∈ G. Even though we have not a proof of this fact in a general situation, it should follow from the completeness of the ground state phase diagram by a method similar to that of [Z]. 2. The Gibbs states from (ii) satisfy an exponential decay of correlations. Also, explicit integral formulas describing them can be written using the integration with respect to measures on families of external contours and on families of walls which describe the interface of the states corresponding to x ∈ Ghor . We shall only describe the probabilities of external contours in Section 3 and of “shadows” in
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Section 4 and we abstain from a discussion how to reconstruct all the probability using them. It could be done similarly like in [HKZ] Section 6. 3. The affine space H0 may be given in the form H0 = {Hα = H0 + N k=1 αk Hk }. It is customary to choose Hamiltonians Hk ∈ H(R) in a suitable way so that their number N may be taken as the minimal possible (N = N0 from (RD)). One could replace H0 by an N -dimensional smooth manifold defined by a mapping H(α), α ∈ RN (with N ≤ N0 ), from a neighbourhood V of a point α(0) ∈ RN into H(R) with H0 = H(α(0) ). The condition of removing of degeneracy then would be stated as the condition of maximality of the rank of the N0 × N matrix with the entries
∂ ex (H(α)) − ex0 (H(α)) , x ∈ g per(H0 ), x = x0 , and ∂αk α=α(0)
∂ , ex (H(α)) − exy,z (H(α)) ∂αk α=α(0) hor , x = xy,z , y, z ∈ g per(H0 ), y = z. x ∈ g(H0 ) ∩ Xy,z We shall not pursue the case of a general manifold H0 any further. 4. Usually one considers pairs, a Hamiltonian H and a temperature T , and assigns them the Gibbs states defined by T1 H (we put Boltzmann constant k = 1). In Theorem 1 we included the temperature into the Hamiltonian to avoid an overparametrization (multiplying both, the Hamiltonian and the temperature, by the same factor, we get exactly the same set of Gibbs states as originally). One can reformulate Theorem 1 exposing explicitly the temperature. Namely, one may introduce the temperature-depending mapping TT (H) = T · T (
H ). T
It describes the phase diagram at the slice of constant temperature T : the Gibbs states yielded by T1 TT (H) correspond (by (ii)) to the ground states of H T (the same as the ground states of H). Thus, for a fixed temperature T , one actually considers H0 as a space of parameters and the mapping TT shows how one should deform the ground state phase diagram to get the phase diagram at the temperature T . The condition (iii) shows that limT →0 TT = id, the identical mapping, and that the limit is attained exponentially fast. Notice that iv) yields a Lipschitz condition for the mapping (T, H) → (T, TT (H)). We derive Theorem 1 from the following lemma on the existence of a function characterizing the presence of stable states whose proof is the main content of the present paper and is presented in Sections 3, 4 and 5. We use here and in what follows ∂α+ ¯ to denote the directional one-sided ¯ (α) + derivative in the direction α ¯ , i.e. ∂α¯ f (α) = limt→0+ f (α+tα)−f . Let us recall t that the hamiltonian H = (U[A] ; A ⊂ Zν , diamA < R) is an element of the finitedimensional space H(R).
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Basic Lemma. Let the assumptions of Theorem 1 be fulfilled (with ρ0 sufficiently large). Then there exist ε > 0 and mappings hx : Kε (H0 ) → R (H → hx (H)) for each x ∈ G0 such that (i) a) for H ∈ Kε (H0 ) and x ∈ Gper such that hx (H) = minx˜∈Gper hx˜ (H) (such x 0 is then called stable), there exists an extremal Gibbs state µ ∈ G(H) that is a perturbation of x; hor b) for H ∈ Kε (H0 ) and x ∈ G0 ∩ Xy,z , where y, z ∈ Gper 0 , y = z, are stable per elements of G0 and hx (H) = minx˜∈G0 ∩Xy,z hor hx (H) (x is stable), there exists ˜ an extremal Gibbs state µ ∈ G(H) that is a perturbation of x;
(ii) there exists c > 0 (independent of ρ0 ) such that a)
|hx (H) − ex (H)| ≤ e
−(ρ0 −c)
H H0
(1.9)
for each H ∈ Kε (H0 ) and b)
+ + ∂H ¯ hx (H) − ∂H ¯ ex (H) ≤ e
−(ρ0 −c)
H H0
¯ H
(1.10)
¯ ∈ H(R). for any H ∈ Kε (H0 ), H Remark (passing to horizontally translation invariant setting). Since we suppose that G0 is finite up to vertical translations, we may and shall suppose that all are actually translation invariant and all elements of Ghor are elements of Gper 0 0 horizontally translation invariant by considering a modified model. Namely, we can choose some partition of Zd into a grid of cubes with edges of a length that is some (e.g. the smallest possible) common multiple of the periods of all concerned periods. The set of “spins” attained at such a cube B consist then of all configurations on B with values in S. This changes the number of “spins” in dependence on the periods only. In the proofs that follow in Sections 3, 4 and 5, we use this observation and suppose that G0 consists of translation invariant and horizontally translation invariant configurations. Proof of Theorem 1. Denoting the parameter ε of the cone from Basic Lemma by ε˜, the assertion (i) implies that it is enough to find a suitable mapping T (H) solving the equations hx (T (H)) − hx0 (T (H)) = ex (H) − ex0 (H) for x, x0 ∈ Gper 0 , x = x0 , and hx (T (H)) − hxy,z (T (H)) = ex (H) − exy,z (H) for y, z ∈ Gper 0 , y = z and (1.11) hor , x = xy,z x, xy,z ∈ G0 ∩ Xy,z for all H ∈ Kε (H0 ) for some 0 < ε < ε˜. We see from Basic Lemma (i) immediately that, if x is a horizontally invariant ground configuration of H, then x is stable with respect to T (H). The number of different equations in (1.11) is N0 (see (RD) above). To get a unique solution T (H) satisfying (i) of the theorem we add the following
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equations. Namely, consider a decomposition H(R) = H(0) ⊕ H(1) into two orthogonal subspaces such that dim H(0) = N0 , H(0) + H0 ⊂ H0 , and H(0) + H0 removes degeneracy of G0 . Thus every H ∈ H(R) can be decomposed uniquely as H = H0 + H (0) + H (1) , H (0) ∈ H(0) , H (1) ∈ H(1) . Denoting H (1) = Π(1) (H) we consider the additional equations Π(1) (T (H)) = Π(1) (H).
(1.12)
The right hand sides of equations (1.11), (1.12) define an invertible linear mapping L from H(R) onto RN0 × H(1) while the left-hand sides define a mapping F that differs only slightly from L on Kε˜(H0 ) according to Basic Lemma (ii). More precisely, we put L(H) = (ex (H) − ex0 (H); x ∈ Gper 0 \ {x0 }), per (1) (ex (H) − exy,z (H); y = z, y, z ∈ Gper (H) 0 , x ∈ G0 ∩ Xy,z \ {xy,z }), Π and
F (H) = (hx (H) − hx0 (H); x ∈ Gper 0 \ {x0 }),
per (1) (hx (H) − hxy,z (H); y = z, y, z ∈ Gper , x ∈ G ∩ X \ {x }), Π (H) . 0 y,z y,z 0
If we express L in suitable orthogonal coordinates, we get by (RD) that the absolute value of the jacobian of L, |jL (H)|, is strictly positive for all H ∈ H(R). ˜ = T (H) fulfilling (1.11) and (1.12), i.e. fulfilling Our task thus is to find H ˜ = L(H) for all H ∈ Kε (H0 ) for some ε. Equivalently, we want the equation F (H) to solve the equation ˜ = H − L−1 P (H), ˜ H (1.13) ˜ = F (H) ˜ − L(H) ˜ is a mapping fulfilling the bound where P (H) ˜ H
˜ ∈ Kε˜(H0 ) ˜ ≤ e−(ρ0 −c) H0 for H P (H) due to Basic Lemma (ii)a), and ˜ H ˜ H
˜ 1 ) − P (H ˜ 2 ) ≤ e−(ρ0 −c) min( H10 , H20 ) H ˜1 − H ˜ 2 for H ˜ 1, H ˜ 2 ∈ Kε˜(H0 ) P (H due to Basic Lemma (ii)b). ˜ of the mapping Q : H ˜ → H − L−1 P (H) ˜ in (1.13) To find the fixed point H for a fixed H ∈ Kε (H0 ), we use the Banach contraction principle. We have ˜ 2 ) ≤ L−1 e−(ρ0 −c)(1−ε) H ˜1 − H ˜ 2 ˜ 1 ) − Q(H Q(H ˜ 1, H ˜ 2 ∈ Kε (H0 ). for H
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Since the constant L−1 e(ρ0 −c) (1−ε) is smaller than one if ρ0 is sufficiently large (we may and shall suppose that ε ≤ 12 ), it suffices to show that Q maps ˜ | H ˜ − H ≤ ε˜ } into itself for H ∈ K ε˜ (H0 ). This allows us to take U ε˜ (H) = {H 2 2 2 ε = 2ε˜ . ˜ − H ≤ ε˜ . Then Let H 2 ˜ − Q(H) ≤ L−1 e−(ρ0 −c)(1−˜ε) Q(H)
ε˜ ε˜ ≤ . 2 2
Hence, for sufficiently large ρ0 (chosen independently of H ∈ Kε (H0 )), we get that Q is a contraction on Uε (H). The unique solution T (H) in Uε (H) fulfills the assertions (i) and (ii) of Theorem 1 according to (i) of Basic Lemma. Using (ii) of Basic Lemma, we get the asked properties (iii) and (iv) of Theorem 1 easily.
2 Labeled contour models We present here a brief reformulation of the essential part of Pirogov-Sinai theory and its slight extension to a form needed for our application. The task is to grasp some control over a description of “labeled contour models” that arise in the study of (“physical”) Gibbs states. The characteristic feature of the Pirogov-Sinai theory is a reformulation in terms of “contour models”. The following presentation follows essentially [Z], but it brings some improvements (see especially Theorem 2 and Proposition 2.2.1; compare also the paper [BK] using some ideas from a preliminary version of the present paper).
2.1
Contour models
We use here the word contour simply for any finite connected (in the sense of nearest neighbours) non-empty subset of Zν (ν ≥ 2). Given any contour Γ ⊂ Zν , we define its exterior, ExtΓ, to be the only infinite connected component of Zν \ Γ. The interior, IntΓ, of Γ is the union of the other (finite) connected components of Z \ Γ. We denote V (Γ) = Γ ∪ IntΓ. A set ∂ of contours is compatible if any pair of distinct elements of ∂ has a disconnected union. A contour model is given by introducing a contour functional4 (“contour weights”) Ψ which maps the set of contours to [0, ∞). Considering a set L of contours, we define the contour model partition function in L by Ψ(Γ) , (2.1) Z(L; Ψ) = ∂⊂L Γ∈∂
with the sum taken over all compatible families ∂ of contours from L. Notice that this definition makes sense not only for any finite set L, but supposing that the 4 We restrict our attention here to the case of real-valued contour functionals that will arise in our context.
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sum and products converge, also for some infinite L. In dealing with general sets L we follow the abstract setting from [KP]. Given a volume Λ ⊂ Zν , we shall be concerned with two particular cases of sets of contours, namely the set L(Λ) of all contours in Λ, L(Λ) = {Γ : dist(V (Γ), Λc ) > 1}, and the set LΛ of all contours intersecting Λ, LΛ = {Γ : V (Γ) ∩ Λ = ∅}. For a finite L we can also introduce the contour model probability distribution in L by Ψ(Γ) µ({∂}, L; Ψ) = Γ∈∂ (2.2) Z(L; Ψ) for any compatible ∂ in L. As usual, the empty product is put equal to one. For an infinite L we can introduce the compact space of families of contours from L as a closed subspace of {0, 1}L with the product topology, and consider the weak limit of measures (2.2). It is useful to introduce a special symbol, e.g. ρ(∂, L; Ψ), for the “correlations” ¯ L; Ψ). µ({∂¯ : ∂ ⊂ ∂}, We may now summarize the main facts concerning contour models with a contour functional Ψ satisfying the inequality Ψ(Γ) ≤ exp(−τ |Γ|) for any contour Γ,
(2.3)
i.e. with Ψ being a τ -functional. Most of the assertions of the following proposition as well as their proofs can be found, for example, in [S, Se, Br]. In view of an application to “volumes” of the form LΛ we rely on an abstract version of contour models and cluster expansions as presented in [KP, D 96]. In particular, we define the distancedist(∂, L) from a set ∂ of compatible contours to a set L of contours as minC Γ∈C |Γ|, where the minimum is taken over all clusters whose union is sets of contours C (i.e. Γ Γ = ∅. connected) such that C ∩ L = ∅ and Γ∈∂ Γ∈C Let us introduce also the notion of external contours. Namely, a contour Γ ∈ ∂ ¯ for each Γ ¯ ∈ ∂, Γ ¯ = Γ. If ∂ is a family of is an external contour of ∂ if Γ ⊂ Ext Γ ¯ for some external contours such that every Γ ∈ ∂ is either external or Γ ⊂ Int Γ ¯ contour Γ of ∂, then we say that external contours exist. We use ϑ(∂) to denote the set of all external contours. Proposition 2.1.1 (contour models) There exist constants τcl ≡ τcluster (ν) and ccl ≡ ccluster(ν) (both depending only on the dimension ν of the lattice) such that, whenever L is an arbitrary set of contours and Ψ is a τ -functional with τ ≥ τcl , then the weak limit µ(·, L; Ψ) = lim µ(·, K; Ψ) K L
over the system of finite subsets ordered by inclusion exists, and : a) For µ-almost all families ∂ there exists the set ϑ(∂) of external contours. b) Whenever ∂ is a compatible family in L, we have ρ(∂, L; Ψ) ≤ exp(−(τ − 1) |Γ|). Γ∈∂
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c) Whenever ∂ is a compatible family of contours in the intersection of sets L1 , L2 , we have |ρ(∂, L1 ; Ψ) − ρ(∂, L2 ; Ψ)| |Γ|) exp(−(τ − ccl ) dist(∂, L1 L2 )). ≤ exp(−(τ − 1) Γ∈∂
(Here L1 L2 is the symmetric difference L1 L2 = (L1 \ L2 ) ∪ (L2 \ L1 ).) Hence also, for any finite non-empty set L ⊂ L1 ∩ L2 and any mapping ϕ of families of contours to real numbers, living5 on L, with |ϕ(∂)| ≤ ϕ, one has
ϕ(∂)µ(d∂, L1 ; Ψ) − ϕ(∂)µ(d∂, L2 ; Ψ)
≤ ϕ exp(−(τ − ccl ) dist(L, L1 L2 )). d) Let us suppose further that Ψ is translation invariant. Then the limiting “pressure” 1 p(Ψ) = limν log Z(L(Λ); Ψ) Λ Z |Λ| (with the limit in the van Hove sense) exists. The partition function Z(L(Λ); Ψ) satisfies the approximation log Z(L(Λ); Ψ) = |Λ| p(Ψ) + ε |∂Λ| with |ε| ≤ exp(−(τ − ccl )), and |p(Ψ)| ≤ exp(−(τ − ccl )). e) Let us suppose now that a family of translation invariant τ (α) -functionals Ψ(α) , τ (α) ≥ τcl , is given, depending on a parameter α from an open set Ω ⊂ Rn and suppose that the one-sided derivative in the direction α ¯ fulfills for some α ∈ Ω, α ¯ ∈ Rn , the bound
+ (α)
∂α¯ Ψ (Γ) ≤ exp(−τ (α) |Γ|)||¯ α|| (α) for every contour Γ. Then the derivative ∂α+ ) satisfies the bound ¯ p(Ψ
+
∂α¯ p(Ψ(α) ) ≤ exp(−(τ (α) − ccl ))||¯ α||. 5 We say that ϕ lives on L if ϕ(∂ ) = ϕ(∂ ) whenever the set of all contours from ∂ that are 1 2 1 contained in L coincides with the set of all contours from ∂2 contained in L.
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Proof. The assertions a)-c) are standard and their proof may be based on the cluster expansion as presented in the statement a) of Proposition 2.1.3. The statement d) follows from a rather straightforward extension of standard proofs of cluster expansion (Proposition 2.1.3 b)). The constant ccl (ν) can be derived from the constant c# = c# (ν) determining the number of “contours of given length”,
{Γ | Γ ! 0, |Γ| = k} ≤ ec# (ν)k .
(2.4)
Namely, it can be shown (see [KP]) that ccl (ν) = c# (ν) + a +
log(1 + a) ∼ c# (ν) + 1.58 log a
(2.5)
√
with a = 5−1 2 . One often meets a situation where the contour functional ΨΛ (Γ) depends on the volume Λ once the contour Γ crosses the boundary of Λ. Also in this case one has a full control of the limiting contour model. Proposition 2.1.2 (contour models with dependence on the boundary) Let V ⊂ Zν be arbitrary and let a sequence Vn of sets converging to V be given, Vn " V . Let for any Vn a τ -functional ΨVn be given so that ΨVm (Γ) = ΨVn (Γ) whenever Γ∈ / LVm Vn 6 and suppose that τ ≥ τcl with τcl from Proposition 2.1.1. Then the limit ΨV = lim ΨVn n→∞
is uniquely determined and the limiting measure µ(·, LV ; ΨV ) = lim µ(·, LVn ; ΨVn ) n→∞
exists. It coincides with the measure µ(·, LV ; ΨV ) = lim µ(·, K; ΨV ) K LV
from the preceding proposition. Further, for any (possibly infinite) V, V1 , V2 ⊂ Zν , one has: a) Whenever ∂ is a compatible family in LV , we have ρ(∂, LV ; ΨV ) ≤ exp(−(τ − 1)
|Γ|).
Γ∈∂ 6 It
means also that ΨΛ (Γ) coincides with Ψ
Z(Γ) for every contour Γ in Λ. ν
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b) Whenever ∂ is a compatible family of contours in the intersection of sets LV1 , LV2 , we have |ρ(∂, LV1 ; ΨV1 ) − ρ(∂, LV2 ; ΨV2 )| |Γ|) exp(−(τ − ccl ) dist(∂, LV1 LV2 )) ≤ exp(−(τ − 1) Γ∈∂
with ccl = ccl (ν) from Proposition 2.1.1. For any finite non-empty set Λ ⊂ V1 ∩ V2 and any mapping ϕ, living on LΛ , of families of contours to real numbers, with |ϕ(∂)| ≤ ϕ, one has
ϕ(∂)µ(d∂, LV1 ; ΨV1 ) − ϕ(∂)µ(d∂, LV2 ; ΨV2 )
≤ ϕ exp(−(τ − ccl ) dist(Λ, V1 V2 )). Suppose, further, that Ψ ≡ ΨZν is translation invariant. Then c) The limiting “free energy” (or “pressure”) p(Ψ) = limν Λ Z
1 log Z(LΛ ; ΨΛ ) |Λ|
(with the limit in the van Hove sense) exists with p(Ψ) the same as in c) of the preceding proposition7 . The partition function Z(LΛ ; ΨΛ ) satisfies also the approximation log Z(LΛ ; ΨΛ ) = |Λ| p(Ψ) + ε |∂Λ| with |ε| ≤ exp(−(τ − ccl )). Proof. Due to the conditions on the functionals ΨVn , the value ΨVn (Γ) stays constant, for every Γ, once n is sufficiently large. Taking into account the possibility to verify the convergence of measures by proving the convergence of correlations ρ(∂, LVn ; ΨVn ) for finite families ∂ and observing that it may be approximated, from a cluster expansion, by restricting the contour model on a finite number of contours, the statements of Proposition 2.1.2 follow from cluster expansions (Proposition 2.1.3 below) and Proposition 2.1.1 (cf. [HKZ] Proposition B.2). Let us also briefly summarize few standard facts about the cluster expansion in a form suitable for our purposes [KP, D 96]. A proof of b) in Proposition 2.1.3 appears in several papers [BK, DKS]. Propositions 2.1.1 and 2.1.2 follow easily from Proposition 2.1.3 below. The results do not serve for the proofs of the above propositions only; we use the explicit form of the expansion in an essential way in Section 4 when studying the probability of interfaces. 7 Notice that it means in particular that the statement 1 c) is valid also with L replaced by Λ sets L(Λ) , where L(Λ) ⊂ L(Λ) ⊂ LΛ .
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Proposition 2.1.3 (cluster expansion) There exist constants τcl ≡ τcl (ν) and ccl ≡ ccl (ν) such that the following statements hold true. a) Whenever Ψ is a τ -functional with τ ≥ τcl , then there exists a (“cluster”) functional ΨT that maps all non-empty clusters of contours (i.e. sets C of contours with connected support C = Γ∈C Γ) to R such that log Z(L; Ψ) =
ΨT (C)
(2.6)
C⊂L
for every finite set L of contours. The functional ΨT satisfies, for every i ∈ Zν , the bound
ΨT (C) exp (τ − ccl ) |Γ| ≤ 1, (2.7) C:C i
Γ∈C
with the sum taken over all clusters with support C = Γ∈C Γ containing a fixed site i ∈ Zν . The value ΨT (C) depends only on values of Ψ(Γ) with Γ ∈ C, and it is translation invariant once the functional Ψ is translation invariant. b) Suppose further that τ (α) - functionals Ψ(α) , τ (α) ≥ τcl , depend on a parameter α from an open set Ω ⊂ Rn and satisfy for every contour Γ, α ∈ Ω, and any α ¯ ∈ Rn , the inequality
+ (α)
∂ Ψ (Γ) ≤ exp(−τ (α) |Γ|)||¯ α||. (2.8) α ¯ Then, for every i ∈ Zν and all α ∈ Ω, one has
∂ + Ψ(α)T (C) exp (τ (α) − ccl ) |Γ| ≤ ||¯ α||. α ¯ C:C i
(2.9)
Γ∈C
Remark. Assuming, instead of (2.8), the existence of the Fr´echet derivative and a bound on its norm, we actually get the existence of the Fr´echet derivative also for Ψ(α)T , with a corresponding bound. Resuming over all clusters with coinciding support and denoting Ψ T (C) = ΨT (C), we will get a formulation that is particularly suitable for our C:
S
Γ∈C
Γ=C
implementations. Corollary 2.1.4 Let τcl and ccl be the constants from Proposition 2.1.3. If Ψ is a τ -functional with τ ≥ τcl , then log Z(L(Λ); Ψ) = Ψ T (C) (2.10) C⊂Λ(0)
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for every finite Λ ⊂ Zν . The sum above runs over all connected subsets C of the set Λ(0) = {i ∈ Λ dist(i, V c ) > 1}. The functional Ψ T (C) satisfies the bound
Ψ T (C) exp (τ − ccl ) |C| ≤ 1. (2.11) C i
Let, moreover, Ψ be translation invariant. Then the limits p(Ψ) = limν Λ Z
1 1 log Z(L(Λ); Ψ) = limν log Z(LΛ ; Ψ) Λ Z |Λ| |Λ|
(the limit in the van Hove sense) exist and satisfy the bounds
p(Ψ) ≤ exp{−(τ − ccl )} and
+
∂α¯ p(Ψ) ≤ exp{−(τ − ccl )}||¯ α||.
(2.12) (2.13)
The function p(Ψ) is explicitly given by p(Ψ) =
Ψ T (C) . |C|
(2.14)
C 0
Remark. Notice, that the assumption of the translation invariance of Ψ in Propositions 2.1.1 c), 2.1.2 c), and the above corollary is actually not necessary. It is enough to introduce pi (Ψ) =
ΨT (C) Ψ T (C) = |C| |C|
C:C i
and to replace the terms |Λ| p(Ψ) by
C i
pi (Ψ).
i∈Λ
For periodic Ψ the limit p(Ψ) is obtained as a mean of pi (Ψ) over the cell of periodicity.
2.2
Labeled contour models
We consider a finite set Q = {1, . . . , r} of “labels” and call γ = (Γ, λ) a labeled contour if its support Γ ≡ Γ(γ) ⊂ Zν is a finite non-empty connected set (a contour), and λ = λ(γ) assigns to each connected component of the boundary ∂Γ of Γ some q ∈ Q. A labeled contour γ is called a q-contour if the label assigned to its external boundary (the boundary of Γ∪Int Γ) is q. A family x of labeled contours is said to be compatible and matching if their supports are compatible and their labels match (i.e., considering a connected component, say C, of Zν \ {Γ(γ) : γ ∈ x}, all
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connected components of the boundaries of Γ’s adjacent to C are labeled by the same label q; we say that C is a q-component ). Compatible and matching families x of labeled contours play a role of configurations for labeled contour models 8 . Whenever x is such that for the family ∂ = {Γ(γ) : γ ∈ x} of its supports, there exists the set ϑ(∂) of external contours (in the sense of the definition preceding Proposition 2.1.2), we introduce the set ϑ(x) (ϑq (x)) of external labeled contours of the family x as the subset of those γ = (Γ, λ) ∈ x whose support Γ is an external (q−)contour of ∂. We say that x is included in V ⊂ Zν if dist(V (Γ(γ)), V c ) > 1 for γ ∈ x. Whenever Λ ⊂ Zν , we consider the set X(Λ) of compatible matching families x of labeled contours that are included in Λ. We introduce the set XqΛ as the set of all compatible matching families of labeled contours that intersect Λ and are such that external contours exist and all external contours as well asq all = contours not belonging to X(Λ) are q-contours. Let us also denote X Λ q XΛ . If x ∈ XΛ , the union of m-components of Λ \ {Γ(γ) : γ ∈ x} together with supports of all m-contours of x is denoted by Λm (x). Clearly, the number of labeled contours γ, for which supp γ = Γ, is bounded by |S||Γ| . A labeled contour functional Φ (“an exponential of a contour Hamiltonian”) maps the set of labeled contours into [0, ∞). We also assume that a vector ϕ = (ϕ1 , . . . , ϕr ) ∈ Rr (of “specific energies of some translation invariant configurations”) is given. We consider a labeled contour model with a boundary condition q ∈ Q by introducing, for any Λ ⊂ Zν and any X(Λ) , X(Λ) ⊂ X(Λ) ⊂ XΛ , the (labeled contour model) partition functions exp − ϕm |Λm (x)| Φ(γ). (2.15) Z(X(Λ) |q; Φ, ϕ) = x∈X(Λ)
γ∈x
m∈Q
Similarly we introduce the (labeled contour model) probability distribution ϕm |Λm (x)| Φ(γ). µ({x}, X(Λ) | q; Φ, ϕ) = Z −1 (X(Λ) |q; Φ, ϕ) exp − m∈Q
γ∈x
(2.16) Again, the notation for partition functions and probability distributions above is distinguishing them from those for a lattice spin model (c.f. Section 1.1) only by using different variables. When stressing the fact that we are dealing with partition functions and probability distributions of a labeled contour model, we will use the notation Z cont and µcont . Remarks. 1. For any Λ ⊂ Zν , the sets X(Λ) (resp. XΛ ) may be embedded into ν the product space {1, . . . , r}Z by assigning to every x that configuration from ν {1, . . . , r}Z which attains the value q at all lattice sites in Λq (x). On the sets X(Λ) ν (resp. XΛ ) we consider the topology inherited from the compact space {1, . . . , r}Z . 8 We use the same notation as for spin configurations of classical lattice models in view of the existence of a natural identification of a class of lattice configurations with a given compatible and matching family of labeled contours (see Section 3).
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Let us notice that for an infinite Λ, even if x = lim xn with xn such that ϑ(xn ) consists of q-contours and in the same time ∂(xn ) converges, in the topology of families of contours used in Section 2.1, to a compatible family of contours ∂ such that external contours ϑ(∂) exists, it might be that ϑ(x) = ϑq (x) with q = q. 2. Notice that there are small formal differences in our formulation and that of [S]. First, our definition of contours does not specify configurations on the support. We also found useful to consider directly the “weights of contours”, thus the ˜ in [S, Z]. functional Φ above corresponds to exp(−Φ) In the standard Pirogov-Sinai theory, the functional Φ˜ as well as the r-tuple ϕ = (ϕ1 , . . . , ϕr ) are linear in the Hamiltonian H (they are given in terms of explicit formulae — see e.g. (1.4) and (1.6) from [Z]). However, in the present application, we shall meet a more general situation. 3. Readers accustomed to standard Pirogov-Sinai formulation may also notice that the partition function Z(X(Λ)|q; Φ, ϕ) corresponds, roughly speaking, to the (relative) diluted partition function multiplied by the factor exp(ϕq |Λ|) (since the standard approach is based on relative Hamiltonian with respect to the energy corresponding to the external boundary condition). The crucial part of the Pirogov-Sinai theory is formulated in Theorem 2 below. First, we need a definition and some more notation. Definition. A phase q is called cs -stable in Λ (with respect to Φ and ϕ) if Z(X(Λ)|λ; Φ, ϕ) ≤ exp(cs |∂Λ|), Z(X(Λ)|q; Φ, ϕ) for every labeling λ (i.e. a label λ(b) ∈ Q for every connected component b of the boundary ∂Λ, chosen in a compatible way). A phase q is said to be cs -stable (with respect to Φ and ϕ) if it is cs -stable in Λ for every non-empty finite Λ. The main aim of the Pirogov-Sinai theory is to provide a characterization of stable phases showing, in the same time, that any stable phase gives rise to a distinct Gibbs state that is a perturbation of the corresponding ground configuration. We split these claims into two statements. First, in Theorem 2, we characterize the stability (see (S)) in terms of certain functions hq (Φ, ϕ) that are close to external fields ϕq . Proposition 2.2.1 then describes the properties of the stable phases. Theorem 2 (characterization of stable phases). Let, for every α from an open set Ω ⊂ Rn of parameters and any Λ ⊂ Zν , a nonnegative labeled contour functional (α) ΦΛ and a vector ϕ(α) of “specific energies” be given (with labels from a finite set (α) (α) Q) such that ΦΛ1 (γ) = ΦΛ2 (γ) whenever supp γ ⊂ Λ1 ∩ Λ2 . We denote Φ(α) = (α)
ΦZν . We suppose that a continuous function τ (α) is given so that for every labeled contour γ, and all α ∈ Ω, we have (1)
(α)
ΦΛ (γ) ≤ exp(−τ (α) |Γ(γ)|).
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Then, there exist a constant τ = τlabeled(cs , ν, |S|) and functions hq (Φ(α) , ϕ(α) ) characterizing cs -stability9 whenever τ (α) ≥ τ . Namely, the phase q is cs -stable with respect to Φ(α) and ϕ(α) if and only if hq (Φ(α) , ϕ(α) ) = min(hm (Φ(α) , ϕ(α) )).
(S)
m
The functions hq (Φ(α) , ϕ(α) ) can be chosen in such a way that, denoting h(Φ , ϕ(α) ) = minm (hm (Φ(α) , ϕ(α) )), one has
Z(Λ|q; Φ(α) , ϕ(α) ) ≥ exp −hq (Φ(α) , ϕ(α) ) |Λ| − ε |∂Λ| (α)
and
Z(Λ|q; Φ(α) , ϕ(α) ) ≤ exp −h(Φ(α) , ϕ(α) ) |Λ| + ε |∂Λ|
with ε = e−(τ −ccl ) + e−(τ −ccl −c# −log |S|−1) . Moreover, there exists a constant c = clabeled(cs , ν, |S|) such that
≤ exp −(τ (α) − c − ccl )
hq (Φ(α) , ϕ(α) ) − ϕ(α) q (α)
(α)
for q = 1, . . . , r and α ∈ Ω. Supposing, moreover, for any α ∈ Ω and α ¯ ∈ Rn , the bounds on the (one(α) (α) sided) directional derivatives of ΦΛ and ϕq ,
+ (α)
∂ Φ (γ) ≤ exp −τ (α) |Γ(γ)| ||¯ (2) α|| and α ¯ Λ
+ (α)
∂ ϕ ≤ M ||¯ α|| for some M > 0, (3) α ¯
q
there exists constants τ = τ (cs , M, ν, |S|) and c = c (cs , ν, |S|) (possibly larger than those above) such that for τ (α) > τ we have:
+
≤ exp −(τ (α) − c − ccl ) ||¯
∂α¯ hq (Φ(α) , ϕ(α) ) − ϕ(α) α|| q and, denoting h(Φ(α) , ϕ(α) ) = minm (hm (Φ(α) , ϕ(α) )), also
+
∂α¯ h(Φ(α) , ϕ(α) ) ≤ M + exp −(τ (α) − c − ccl ) ||¯ α|| for α ∈ Ω and α ¯ ∈ Rn . Remark. Notice, in particular, that if Ω is convex, the functions hq (Φ(α) , ϕ(α) ) − (α) ϕq are, as functions of α, Lipschitz with the constant exp −(τ (α) − c − ccl ) . The theorem follows from an explicit construction of functions hq (Φ(α) , ϕ(α) ) ¯ introduced below. The expression in terms in terms of contour functionals Ψ and Ψ of those functionals also yields the properties of stable phases. 9 Stability implies a good control of the corresponding states. These implications, as well as an explicit construction of the characterizing functions hq (Φ(α) , ϕ(α) ), are presented in Proposition 2.2.1 and Corollary 2.2.2.
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Proposition 2.2.1 (properties of stable phases) Under assumption (1) of Theorem 2: a) Existence of contour functionals Ψ For every finite Λ, α ∈ Ω and q = 1, . . . , r, (α) there exist uniquely determined nonnegative contour functionals Ψq,Λ such that (Λ) ; Ψ(α) Z(X(Λ) |q; Φ(α) , ϕ(α) ) = exp −ϕ(α) q |Λ| Z(L q )
(2.17)
for any L(Λ) , L(Λ) ⊂ L(Λ) ⊂ LΛ with X(Λ) = {x ∈ XΛ ; γ ∈ X implies Γ(γ) ∈ L(Λ) }, and (α)
(α)
µ({x : ϑ(x) = ϑ}, X(Λ) | q; ΦΛ , ϕ(α) ) = µ({∂ : ϑ(∂) = ϑ}, L(Λ) ; Ψq,Λ ) (α)
(2.18)
(α)
for any collection ϑ of external q-contours. Moreover, Ψq,Λ1 (Γ) = Ψq,Λ2 (Γ) when(α)
ever10 V (Γ) ⊂ Λ1 ∩ Λ2 . The functional Ψq,Λ satisfies, for any Λ, the bound
(α)
Ψ (Γ) ≤ exp −(τ (α) − c ) |Γ| , q,Λ
whenever Γ is such that q is cs -stable in every component of Int Γ (with respect to Φ(α) and ϕ(α) ) and c ≥ c# (ν) + cs + log |S|. (α) ¯ Taking Ψ(α) = Ψ ν , nonnegative contour b) Existence of contour τ -functionals Ψ q q,Z
(α)
¯ q exist, such that, for every contour Γ, we have functionals Ψ (α) (α) ¯ • Ψq (Γ) ≤ Ψq (Γ), (α) (α) ¯ •Ψ whenever q is cs -stable in Int Γ,
q (Γ) =
Ψq (Γ)
¯ (α) • Ψq (Γ) ≤ exp −(τ (α) − c ) |Γ| with c ≥ c# + cs + log |S|. Supposing that the functional Φ(α) is translation invariant (in an obvious way ¯ (α) with respect to shifts in Zν ), the functional Ψ can be chosen to be translation q invariant. Further on, there exists τ = τ (cs , ν, |S|) such that for τ ≥ τ we have ¯ (α) The functions hq (Φ(α) , ϕ(α) )(≡ c) Description of hq (Φ(α) , ϕ(α) ) in terms of Ψ ¯ (α) , ϕ(α) )) defined by hq (Ψ ¯ (α) ), hq (Φ(α) , ϕ(α) ) = ϕq − p(Ψ q (α)
¯ q ) defined in Corollary 2.1.4, characterize the stability (i.e., satisfy the with p(Ψ equivalence (S)). Further on, whenever α is such that q is cs -stable with respect to Φ(α) , ϕ(α) , one has hq (Φ(α) , ϕ(α) ) = − lim 10 Supposing
1 log Z(X(Λ)|q; Φ(α) , ϕ(α) ) = h(Φ(α) , ϕ(α) ). |Λ| (α)
(α)
equality ΦΛ1 (γ) = ΦΛ2 (γ) for supp γ ∩(Λ1 Λ2 ) = ∅, we get Ψq,Λ1 (Γ) = Ψq,Λ2 (Γ) whenever dist(V (Γ), Λ1 Λ2 ) > 1.
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(α)
Supposing, moreover, the smoothness of ΦΛ and ϕq (the conditions (2) and (3) of Theorem 2), there exist a constant τ = τ (cs , M, ν, |S|) (possibly larger then τ (cs , ν, |S|) above) such that for τ (α) > τ we have : a ) Smoothness of contour functionals Ψ(α) For any Λ,
+ (α)
∂α¯ Ψ (Γ) ≤ exp −(τ (α) − c ) |Γ| ||¯ α|| q,Λ with c ≥ cs + c# + log |S|+ 1 + ln 2, whenever Γ (and α) are such that q is cs -stable in every component of Int Γ (with respect to Φ(α) and ϕ(α) ). ¯ (α) There exist functionals Ψ ¯ (α) satisfying b ) Smoothness of contour functionals Ψ q b), with c ≥ 2cs + c# + log |S| + 1, and
+ (α)
∂α¯ Ψ ¯ q (Γ) ≤ exp −(τ (α) − c ) |Γ| ||¯ α|| with c ≥ 4cs + c# + 2 + 2 log(|S| + 1), for every Γ and every α ∈ Ω. (α) ¯ (α) Observation. Notice that, by b) and c), hq (Φ, ϕ) = minm (hm (Φ, ϕ)) iff Ψ = Ψq . q Further, the notion of cs -stability of a phase q actually does not depend on cs . Hence, in the following, we will just say that a phase q is stable. (However, changes of cs may lead to changes of τ .)
Remarks. 1. Clearly, there exists a choice of sufficiently large c (cs , ν, |S|) such that it can
(α)
(α) ¯ q (Γ) from a) be used simultaneously in the bounds on Ψq (Γ) and Ψ
+ (α)
+ (α)
¯ and b) as well as in the bounds on ∂α¯ Ψq (Γ) and ∂α¯ Ψq (Γ) from a ) and b ) . 2. The limit in c) is over finite volumes in the van Hove sense and defines the quantity corresponding to “pressure” in the physical model based on a Hamiltonian at a particular temperature. (α)
3. Studying the contour model probabilities µ(·, Λ; Ψq ) for stable q and their limits for simply connected Λ tending to infinite simply connected volumes, we can describe labeled contour model probabilities in an infinite volume under boundary condition q by conditioning over external contours (cf. Proposition 2.1.2). 4. It suffices to suppose (1) (with sufficiently large τ (α) ), to get the existence (α) ¯ (α) = (Ψ ¯ (α) of Ψ(α) = (Ψq ), Ψ q ) fulfilling a), b), and c). However, we were not able to follow exactly the method described in [Z] or [BI] to prove the smoothness b ) (and hence also a )). Therefore, we extend the assertion of Theorem 1 from [Z] in Lemma 2.3.1 to get a), b), and c) for a wider class ¯ (α) fulfilling b ) among ¯ (α) . We shall later find a functional Ψ of functionals Ψ them (Lemmas 2.3.1 and 2.3.2 and the final part of the proof of Theorem 2). 5. If we take H ∈ H(R) for the parameter α and Kε (H0 ) for the set Ω, the H (see Proposition 1.1.3). function τ (H) can be chosen as τ (H) := ρε H 0
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6. It is enough to suppose the estimates from (1) and (2) for the sum is over q-contours γ = (Γ, λ) with Γ fixed.
γ
233
(α)
ΦΛ (γ), where
7. The first point in b) can be weakened to ¯ (α) (Γ) ≤ econst|Γ| Ψ(α) (Γ), Ψ q q and the second point can be weakened by introducing a stronger notion of stability (see [BK]). We postpone the proof of Proposition 2.2.1 to Section 2.3 below. Theorem 2 then easily follows. Proof of Theorem 2. The only claims that are not directly included in Proposition 2.2.1 are the bounds on derivatives of hq (Φ(α) , ϕ(α) ) and h(Φ(α) , ϕ(α) ). Using b ) ¯ (α) of Proposition 2.2.1, the bound (2.13) on p(Ψ q ), and the assumption (3), we get
+
∂ hq (Φ(α) , ϕ(α) ) ≤ M + exp −(τ (α) − c − ccl ) ||¯ α||. α ¯
(α) To estimate ∂α+ , ϕ(α) ) , we first notice that if, for a fixed α, there exists ¯ h(Φ q ∈ Q such that h(Φ(α) , ϕ(α) ) = hq (Φ(α) , ϕ(α) ) and h(Φ(α) , ϕ(α) ) < hm (Φ(α) , ϕ(α) ), m = q, the claim immediately follows from the bound above. If h(Φ(α) , ϕ(α) ) = ¯ ⊂ Q, and h(Φ(α) , ϕ(α) ) < hq (Φ(α) , ϕ(α) ), q ∈ Q \ Q, ¯ we get, hq (Φ(α) , ϕ(α) ), q ∈ Q for a fixed direction α ¯, (α) (α) ∂α+ , ϕ(α) ) = min ∂α+ , ϕ(α) ). ¯ h(Φ ¯ hq (Φ ¯ q∈Q
¯ the directional Indeed, if the right hand side is attained for several q’s in Q, derivative on the left hand side equals any one of them, since they are equal anyway. We will use the results of Theorem 2 and Proposition 2.2.1 together with the results on contour models to describe the limit Gibbs states obtained under special boundary conditions as it is customary in the standard Pirogov-Sinai theory. However, in our application, the partition function of the “physical” model in Λ is described in terms of a labeled contour model with contours that may reach out of the volume Λ and with contour functional that (for such contours) depends on Λ. This is the reason why we formulate the following result on the description of labeled contour models under a stable boundary condition in a slightly more general situation than it is customary. Let thus a translation invariant functional Φ and a vector ϕ be as above, and suppose that sequences of finite sets Λn " V and functionals ΦΛn are given so that • ΦΛm (γ) = ΦΛn (γ) whenever m, n, and γ with support Γ are such that {γ} ∈ XΛn ∩ XΛm and dist(V (Γ), Λm Λn ) > 1, • 0 ≤ ΦΛn (γ) ≤ exp{−τ |Γ(γ)|} for all n and all γ.
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We study partition functions Z(X(Λ) |q; ΦΛ , ϕ) and the probabilities µ(·, X(Λ) | q; ΦΛ , ϕ) for X(Λ) ⊂ X(Λ) ⊂ XΛ . Corollary 2.2.2 For τ sufficiently large and a phase q stable with respect to Φ, ϕ (as in Theorem 2), there exists, for a (possibly infinite) simply connected set V ⊂ Zν and any Y ⊂ XV , a unique probability measure µ(·, Y | q; ΦV , ϕ) defined as a limit of µ(·, Y ∩ XΛ | q; ΦΛ , ϕ) with finite Λ converging to V . For almost all x ∈ XV there are external q-contours — the set ϑ(x) is defined and consists of q-contours. Moreover, for every bounded continuous function f : XV → R, there exist (τ − c − cs )-functionals Ψq,V such that f (x)µ(dx, XV |q; ϑ(x) = ϑ(∂))µ(d∂, LV |Ψq,V ). f (x)µ(dx, XV |q; ΦV , ϕ) = (2.19) Here µ(dx, XV |q; ϑ(x) = ϑ(∂)) is naturally defined as the probability µ(dx, XV |q; ΦV , ϕ) under the condition ϑ(x) = ϑ(∂). The measures µ(d∂, LV |Ψq,V ) fulfill the assertions of Proposition 2.1.2. (a) (a) Let Ei (V ), i ∈ V , a ∈ R, be the set Ei (V ) = {x ∈ XV ; ∃γ ∈ x such that (a) V (Γ) ! i, |Γ| ≥ a}. There exists a constant C such that µ(Ei (V ), XV |q; ΦV , ϕ) ≤ Ce−τ a for any i ∈ V and a ∈ R. Proof. For any finite Λ we define Ψq,Λ by Ψq,Λ (Γ) =
q {γ}∈XΛ γ: supp γ=Γ
ΦΛ (γ)
Z(X(IntΛ Γ)|λ(γ); Φ, ϕ) . Z(X(IntΛ Γ)|q; Φ, ϕ)
(2.20)
By λ(γ) in the numerator we indicate that the partition function is considered with the boundary condition induced by λ(γ). Here IntΛ (Γ) is the union of those components of Int Γ whose distance from Zν \Λ is at least 1. In principle, also those components that are not fully contained in Λ are contributing, in a multiplicative way, to the numerator as well as denominator above. However, since for any γ in the sum above the label of the boundary of any such component is q, the concerned contributions to the numerator and denominator cancel. By induction, one can first verify that (α) (α) (Λ) ; Ψq,V ). Z(X(Λ) |q; ΦV , ϕ(α) ) = exp −ϕ(α) q |Λ| Z(L The functionals Ψq,Λ clearly satisfy the condition (about the independence on Λ for contours sufficiently far from the boundary) from Proposition 2.1.2 and the limiting functional Ψq,V as well as the measure µ(d∂, LV ; Ψq,V ) are well defined. To prove (2.19) we follow the proof of Proposition 3.4 from [HKZ]. The equality (2.19) holds for every finite Λ. Supposing that f is a cylindric function living on Λ, the right hand side of (2.19) can be approximated, up to a set of small measure, say E, by a cylindrical continuous function living on a large finite Λ(E).
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The equality (2.19) for such a continuous bounded function then follows from Proposition 2.1.2 b). (a) The bound on probability of Ei (V ) is a standard implication of Proposition 2.1.2 a) applied to the contour functional Ψq,V .
2.3
Proof of Proposition 2.2.1
Since the assertion of Proposition 2.2.1 is an improvement of the results from [S] and [Z], we present here only the necessary changes or complements to the proofs from [Z]. In the same way as in Section 2.2, we often omit the superscript (α) if it cannot cause any misunderstanding. Sometimes we write Z(Λ|q) instead of Z(X(Λ)|q; Φ(a) , ϕ(α) ) and Z(Λ; Ψ) instead of Z(L(Λ); Ψ). 1. Proof of a) Recall that the equality (2.18) is fulfilled iff Ψ is defined by (2.20) (cf. proof of Corollary 2.2.2). The estimate a) follows easilyfrom the definition of stability of q in Int Γ and the bound exp (c# + log |S|) |Γ| on the number of γ’s in (2.20). ¯ explicitly as in [Z], 2. Implication b) ⇒ c) Our strategy is not to define Ψ ¯ that ensure, in particular, the conclusion c) of but to isolate those properties of Ψ ¯ (α) satisfying the assumpProposition 2.2.1. Then we look for a suitably smooth Ψ tions of the following lemma whose assertion (iv) is essentially identical to c). A ¯ (α) is given in Lemma 2.3.2. To satisfy the assumptions of the particular choice of Ψ following Lemma 2.3.1, one has to assume that τ − cs − c# − log |S| ≥ τ˜0 (cs , ν, |S|) with τ˜0 (cs , ν, |S|) determined in course of the proof of Lemma 2.3.1. Thus τ needed for validity of Proposition 2.2.1 c) can be taken as τ = τ˜0 (cs , ν, |S|) + cs + c# + log |S|. ˜ q are translation-invariant (non-negative) τ -functionals for q = 1, . . . , r, If Ψ ˜ r ), we define the following quantities: ˜ ˜ 1, . . . , Ψ and Ψ = (Ψ ˜ ϕ) = ϕq − p(Ψ ˜ q ), hq (Ψ, ˜ ϕ) = min hq (Ψ, ˜ ϕ), h(Ψ, q
˜ ϕ) = hq (Ψ, ˜ ϕ) − h(Ψ, ˜ ϕ). aq (Ψ, ˜ = {Ψ ˜ q } is a contour Lemma 2.3.1 There exists τ˜0 ≡ τ˜0 (cs , ν, |S|) such that if Ψ functional satisfying ˜ •Ψ ˜ q (Γ) ≤ Ψq (Γ), b) ˜ •Ψ q is cs -stable in Int Γ,
q (Γ) =
Ψq (Γ) whenever
˜
• Ψq (Γ) ≤ exp −˜ τ |Γ| and if τ˜ ≥ τ˜0 , then, denoting ε˜0 ≡ ε˜0 (˜ τ ) := ε(˜ τ ) + ε(˜ τ − c# − log |S| − 1) with ε(τ ) = e−(τ −ccl) , the following holds:
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(i) If q is not cs -stable in Λ, then ˜ ϕ) |Λ| > (cs − 2˜ aq (Ψ, ε0 ) |∂Λ| .
˜ ϕ) |Λ| − ε˜0 |∂Λ| . (ii) Z(Λ|q; Φ, ϕ) ≥ exp −hq (Ψ,
˜ ϕ) |Λ| + ε˜0 |∂Λ| . (iii) Z(Λ|q; Φ, ϕ) ≤ exp −h(Ψ, ˜ ϕ) = 0. Whenever q is cs -stable, (iv) q is cs -stable iff aq (Ψ, ˜ ϕ) = − lim 1 log Z(Λ|q; Φ, ϕ) hq (Ψ, |Λ|
1 = min lim − log Z(Λ|λ; Φ, ϕ) (= h(Φ, ϕ)), λ |Λ| with the minimum taken over all multi-indices λ. Proof. Assuming (i)–(iii) for all proper subsets of Λ, one can prove (ii) and (iii) following the inductive proof of Theorem 1.7 in [Z]. To be more precise we give some commentary to it. The proof of (ii) remains τ ) = e−˜τ +ccl here). In the proof of (iii) we unchanged (it suffices to take ε˜0 ∼ ε(˜ 11 get the formula [Z, (1.44)] and we intend to apply [Z, Main Lemma, (2.13)] to the functional ε0 (˜ τ ) − h + ϕq ) |Γ| ≡ τ¯ |Γ| Ξ(Γ) = (˜ τ − c# − log |S| − 2˜ for a fixed q. Observe now that the equation τ ∗ = τ¯ − ε(τ ∗ ) has, for every τ¯ ≥ τcl , a solution τ ∗ such that ε(τ ∗ ) → 0 for τ¯ → ∞. For the auxiliary functional Ξ∗ (Γ) ≡ τ ∗ |Γ| := Ξ(Γ) − ε(τ ∗ ) |Γ| of [Z, (2.10)] we need τ ∗ ≥ τcl , i.e., ε0 (˜ τ ) − h + ϕq ≥ τcl + ε(τ ∗ ) τ¯ = τ˜ − c# − log |S| − 2˜ to be able to use the results of Section 2.1. Therefore we choose τ˜0 large enough to ensure that for some δ > 0 we have 2ε(˜ τ0 − c# − log |S| − 2δ) < δ, and
ϕq ≥ h(˜ τ0 ) − δ,
τ0 − c − 2δ), ε(τ ∗ ) < δ. τ˜0 ≥ τcl + c# + log |S| + 2δ + ε(˜
11 We only will have a factor 2˜ ε0 instead of 3ccl in (1.44). This can be traced back to the fact that, in the induction hypothesis used for (1.38), the role of 2ccl is played by ε˜0 .
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Then, for τ˜ > τ˜0 , one has τ˜ − c# − log |S| − 2ε(˜ τ ) − h + ϕq ≥ τ˜ − c# − log |S| − 2δ ≥ τcl + ε(˜ τ − c# − log |S| − 2δ) ≥ τcl + ε(τ ∗ ) because τ ∗ ≥ τ˜ − c − 2δ. With a particular choice of δ, say δ = 1/3, the choice of τ˜0 depends on ν and |S| only. Hence, we bound the right hand side of [Z, (1.44)] by ˜ ϕ)) exp{−h |Λ| + ε(˜ τ ) |∂Λ|}Z q (Λ, Ξ, aq (Ψ, ˜ ϕ)) the partition function from [Z] Main Lemma, (2.12) (dewith Z q (Λ, Ξ, aq (Ψ, noted by Z(Λ, H) there) defined with the functional Ξ above and the parameter ˜ ϕ) defined before the present lemma. This yields [Z] (2.13) in the form aq (Ψ, ˜ ϕ)) ≤ eε(τ Z q (Λ, Ξ, aq (Ψ, or, equivalently,
Z(Λ; Ξ∗ ) ≤ ep(Ξ
∗
∗
)|∂Λ|
)|Λ|+ε(τ ∗ )|∂Λ|
,
.
The bound (iii) follows using [Z, Main Lemma] and the inequality ε(τ ∗ ) ≤ ε(˜ τ− c# − log |S| − 1). The only difference between [Z, Theorem 1] and our Lemma 2.3.1 (i) – (iii) concerns now the derivation of (i) and stems from our definition of cs -stable sets. Indeed, if q is not cs -stable in Λ, there exists a multi-index λ such that Z(Λ|λ) > exp cs |∂Λ| . Z(Λ|q) From (ii) and (iii) we have ˜ ϕ)|Λ| − ε˜0 |∂Λ| , and Z(Λ|˜ ˜ ϕ)|Λ| + ε˜0 |∂Λ| Z(Λ|q) ≥ exp −hq (Ψ, q) ≤ exp −h(Ψ, and thus (i) follows. ˜ ϕ) = 0. Then q is stable by (i) once τ˜ To prove (iv), suppose first that aq (Ψ, is so large that cs − 2˜ ε0 ≥ 0. ˜ q (Γ) for every Γ by Let, on the other side, q be cs -stable. Then Ψq (Γ) = Ψ ˜ Thus the condition b). ˜ q ) = exp ϕq |Λ| Z(Λ|q; Φ, ϕ) Z(Λ; Ψq ) = Z(Λ; Ψ and
˜ q ) = hq (Ψ, ˜ ϕ) = − lim log Z(Λ|q) . ϕq − p(Ψ (2.21) |Λ| ˜ ϕ) = 0, we know already that q0 is stable and Taking now any q0 such that aq0 (Ψ, thus Z(Λ|q0 ) ≤ exp cs |∂Λ| . exp(−cs |∂Λ|) ≤ Z(Λ|q) ˜ ϕ) = hq (Ψ, ˜ ϕ) and aq (Ψ, ˜ ϕ) = 0. Hence hq0 (Ψ,
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Further, if q is cs -stable, we have Z(Λ|λ) ≤ exp cs |∂Λ| Z(Λ|q) for any λ. Thus, taking into account (2.21), we get log Z(Λ|λ) log Z(Λ|q) lim − ≥ lim − = h(Φ, ϕ). |Λ| |Λ| Hence
1 log Z(Λ|λ; Φ, ϕ) . h(Φ, ϕ) = min lim − λ |Λ|
˜ of Lemma 2.3.1. ˜ satisfying b) 3. The choice of Ψ Lemma 2.3.2 Let c˜ ≥ cs . The functionals Z(Int Γ|λ(γ)) Z(Int Γ|λ(γ)) exp c˜|∂ Int Γ| ˜ Ψq (Γ) = , Φ(γ) min , ˜ Z(Int Γ|q) maxλ˜ Z(Int Γ|λ) γ
(2.22)
with the sum taken over all labeled q-contours with Γ = Γ(γ) and with labeling ˜ satisfy λ(γ), and with the maximum in the second term taken over all labelings λ, ˜ the assumptions b) of Lemma 2.3.1 above with τ˜ = τ − c˜ − c# − log |S|. Proof. It follows clearly from (2.22) and the definition of stability.
Remark. 1. In particular, combining Lemma 2.3.2 with Lemma 2.3.1 (for τ˜ = τ − c˜ − c# − log |S| ≥ τ˜0 ), we can conclude that • there exists a cs -stable q0 for every α; • for any cs -stable q one has − lim
log Z(Λ|q) log Z(Λ|λ) = min −lim ; λ |Λ| |Λ|
• Z(Λ|q) ≤ exp −h(Φ, ϕ) |Λ| + ε˜0 |∂Λ| for every q; • Z(Λ|q) ≥ exp −h(Φ, ϕ) |Λ| − ε˜0 |∂Λ| for every cs -stable q. ˜ 2. Once the existence of a stable q0 is established, we notice that replacing λ ˜ q (Γ) satisfying assumptions of Lemma 2.3.1 by a constant q˜ in (2.22) leads to a Ψ with τ˜ = τ − 2cs − c# − log |S| (taking c˜ and thus τ˜ sufficiently large, we have E˜0 ≤ cs ). ˜ for Ψ, ¯ we proved b) and c) from Proposition 2.2.1. The task now 3. Taking Ψ ¯ so that it also is, assuming conditions (2) and (3) from Theorem 2, to choose Ψ satisfies b ) (cf. the step 5 below).
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4. Estimates of derivatives of partition functions and proof of a ). Lemma 2.3.3 For sufficiently large τ ≡ τ (cs , M, ν, |S|), τ (α) > τ , and taking ε˜0 ≡ ε˜0 (τ (α) ) as in Lemma 1, one has12
+ (α)
∂ Z (Λ | λ) ≤ M |Λ|(1 + ε˜0 ) exp −h(α) |Λ| + ε˜0 |∂Λ| ||¯ α||, (2.23) α ¯ whenever Λ is a union of simply connected finite sets and λ is an arbitrary labeling. Moreover,
Z (α) (Λ | λ)
+
α||, (2.24)
∂α¯
≤ exp |∂Λ| ||¯ (α) Z (Λ | q) if q is cs -stable in Λ with respect to Φ(α) , ϕ(α) . Proof. The first inequality can be proven in a rather straightforward way, by induction in |Λ| (cf. also [BK] (Lemma 2.2)). Namely, considering the derivative of S (α) e−ϕq |Λ\ Int γ| Φ(α) (γ)Z (α) (Int γ | λ(γ)), Z (α) (Λ | q) = ϑ
γ∈ϑ
with the sum over collections ϑ of external q-contours in Λ (including the empty one), we get (taking, without loss of generality, ||¯ α|| = 1)
+ (α)
∂α¯ Z (Λ | q) ≤ ≤ M |Λ|Z (α) (Λ | q) +
∂ + Φ(α) (γ)Z (α) (Int γ | λ(γ)) e−ϕ(α) q |γ| Z (α) (Ext γ | q) ≤ α ¯ γ
≤ exp{−h
(α)
(α) 1 + M | Int γ|(1 + ε˜0 ) e−τ |γ|+3˜ε0 |γ| ≤ |Λ| + ε˜0 |∂Λ|} M |Λ| + γ
≤ M |Λ|(1 + ε˜0 ) exp{−h(α) |Λ| + ε˜0 |∂Λ|}. (α)
The first term corresponds to the derivative of ϕq using assumption (3) of Theorem 2. In the second term, we summed over contours not affected by the derivative, yielding Z (α) (Ext γ | q) with Ext γ = Λ \ (Int γ ∪ γ). To get the second inequality ˜ from Lemma 2) to bound Z (α) (Λ | q), we use Lemma 1 (iii) (with the help of Ψ (α) (α) Z (Int γ | λ(γ)), and Z (Ext γ | q), the assumption (2) of Theorem 2 to bound (α) (α) (γ)|, induction hypothesis (2.23) to bound |∂α+ (Int γ | λ(γ))|, and the |∂α+ ¯Φ ¯Z (α) (α) fact that ϕq ≥ h − ε˜0 . To get the last inequality, we use the bound
(α) 1 + M | Int γ|(1 + ε˜0 ) e−τ |γ|+3˜ε0 |γ| ≤ ε˜0 ,
γ:γ∪Int γ i
where the sum is taken over all γ encircling a fixed site i. 12 We
explicitly indicate here the dependence on α.
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To get (2.24), we use (2.23) and Lemma 1 (ii) and (iii) (with existence of ˜ assured by Lemma 2) and get needed Ψ
Z (α) (Λ | λ)
+
ε0 |∂Λ|
∂α¯ (α)
≤ M |Λ|(1 + ε˜0 ) exp 2˜ Z (Λ | q) ε0 |∂Λ| ≤ exp |∂Λ| +M |Λ|(1 + ε˜0 ) exp 4˜
for τ (α) sufficiently large (and thus ε˜0 sufficiently small).
Proof of a ) of Proposition 2.2.1. We differentiate the right hand side of (2.20). (α)
(α)
(X(IntΛ Γ)|λ(γ);Φ,ϕ) Evaluating the derivative of the product ΦΛ (γ) Z Z (α) , we use the (X(IntΛ Γ)|q;Φ,ϕ) assumption (2) of Theorem 2, the cs -stability of q in IntΛ Γ, assumption (1) of Theorem 2, and the bound (2.24) for the set IntΛ Γ, to get the bound (again, ||¯ α|| = 1)
Z (α) (X(IntΛ Γ)|λ(γ); Φ, ϕ)
+ (α)
∂α¯ ΦΛ (γ)
≤ Z (α) (X(IntΛ Γ)|q; Φ, ϕ)
≤ e−τ
(α)
|Γ| cs |Γ|
e
+ e−τ
(α)
|Γ| |Γ|
e
≤ e−(τ
(α)
−cs −1−ln 2)|Γ|
that yields (a ) with c = cs + 1 + ln 2 + c# + log |S|.
5. Definition of Ψ satisfying b), and simultaneously b’), of Proposition 2.2.1 ˜ q from (2.22) is the use of According to 3. and 4., the only problem with Ψ ˜ non-smooth. Therefore we introduce a the min in its definition which makes Ψ “smooth version of min” first and then we apply it to define Ψq by a natural modification of (2.22). Lemma 2.3.4 For any η > 0, there is a function minη : Rr → R, r ∈ N, such that (i) minη (u) ≤ min(u) for u = (u1 , . . . , ur ) ∈ Rr ; (ii) minη (u) = ui whenever ui ≤ min{uj | j = i} − η; (iii) minη ∈ C ∞ (Rr ) and it is 1-Lipschitz; (iv)
∂ ∂ui
minη (u) = 0 if ui ≥ min(u) + 2η.
Proof. We may define minη as the convolution of min and a nonnegative function ϕη ∈ C ∞ (Rr ) that is symmetric (i.e. of the form ψη (u)), fulfills ϕη (u)du = 1 Rr
and ϕη (u) = 0 if u ≥ η2 . Namely, we put minη = ϕη ∗ min. Then (i) holds because ϕη ∗ min(u) = ϕη (u − v) min(v)dv ≤ ϕη (u − v)vi dv = ϕη (v)(ui − vi )dv = Rr
Rr
Rr
ϕη (v)dv −
= ui Rr
The last integral is zero because of the symmetry of ϕη .
ϕη (v)vi dv = ui . Rr
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To verify (ii), notice that ϕη (v) min(u − v)dv = ϕη ∗ min(u) = v≤ η2
and
241
ϕη (v)(ui − vi )dv = ui . v≤ η2
The condition (iii) follows immediately from the fact that min is 1-Lipschitz ϕη (v)dv = 1 with ϕη (v) ≥ 0.
Rr
Finally, we verify (iv). Let |˜ ui − ui | < η and u ˜j = uj for j = i. Then minη (˜ u) = ϕη (v) min(˜ u − v)dv = ϕη (v) min(u − v)dv = minη (u). Here v< η2
v< η2
˜i − vi ≥ min(u) + we used that u˜i ≥ min(u) + η and u holds and Lemma 4 is proved.
η 2
≥ min(u − v). So (iv)
Proof of b) and b ) of Proposition 2.2.1. For η > 0 and any contour γ, we introduce the shorthand
Z(Λ|λ(γ)) exp (cs + η)|∂Λ| Z(Λ|λ(γ)) , log ; q˜ = q , M(γ, η) = minη|∂Λ| log Z(Λ|q) Z(Λ|˜ q) where Λ = Int Γ; q, q˜ ∈ {1, . . . , r}. Put now (α) Ψq (Γ) = Φ(α) (γ) exp M(γ, η) ,
(2.25)
γ
where the sum is over q-contours with support Γ. (α) First we show that the functional Ψq fulfills the three assumptions of ˜b) of Lemma 1 (with τ˜ = τ − c , c = c# + log |S| + cs + ε˜0 + η + 1) and thus b) of Proposition 2.2.1 (with c = c# + log |S| + 2cs + 1) . (α)
(α)
Namely, the first point (Ψq ≤ Ψq ) follows from (2.20), (2.25) and the property (i) of minη (cf. Lemma 4). (α)
(α)
The second point (Ψq (Γ) = Ψq (Γ) if q is cs -stable in Λ) needs moreover the definition of cs -stability of q in Λ and the property (ii) of minη (cf. Lemma 4). (α) To show the third point (|Ψq (Γ)| ≤ exp −(τ (α) − c )|Γ| ), we use the fact (α) that, by Lemma 2, there is a cs -stable q0 . So |Ψq (Γ)| ≤ exp −(τ (α) − c# − (α) log |S| − cs − ε˜0 − η))|Γ| by the definition of Ψq , the assumption (1) of Theorem 2, Lemma 4 (i), and the definition of cs -stability of q in Λ. Using the choice η = 1, we get b) with c = c# + log |S| + 2cs + 1. (α)
Now, we shall show that Ψq fulfills the estimate (b ). First, notice that the following auxiliary estimate holds due to Lemma 4 (i),
∂
exp minη (log t1 , · · · , log tr )
∂ti
∂ 1
≤ exp minη (log t1 , · · · , log tr ) · minη (u) · ≤ 1 ∂ui ti
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for t1 , . . . , tr > 0. We differentiate ∂α+ ¯
Φ(α) (γ) exp M(γ, η)
γ
=
γ
(α) (α) ∂α+ (γ) exp M(γ, η) + Φ (γ)∂α+ ¯Φ ¯ exp M(γ, η) . (2.26) γ
(α) We are going to use (2) of Theorem 2 to estimate ∂α+ (γ) and (1) of Theorem ¯Φ (α) 2 to estimate Φ (γ). Using the existence of a cs -stable q0 we get further that
exp M(γ, η) ≤
Z(Λ|λ(γ)) exp (cs + η)|∂Λ| Z(Λ|λ(γ)) , log ; q˜ = q ≤ ≤ exp min log Z(Λ|q) Z(Λ|˜ q) ≤ exp (2cs + η)|∂Λ| . To establish the estimate (b ) it remains to estimate the derivative
+
∂α¯ exp M(γ, η) ≤ exp M(γ, η) ∂α+ ¯ M(γ, η) ≤ ∂ ∂ minη|∂Λ| (log t1 , . . . , log tr )ti =tqi (α) · + (tqj (α)) , ≤ exp M(γ, η) ∂tj ∂α¯ j
Z(Λ|λ(γ)) Z(Λ|q)
Z(Λ|λ(γ)) exp (cs +η)|∂Λ| Z(Λ|˜ q)
and tq˜(α) = where we use the notation tq (α) = for q˜ = q. We consider the product of the two terms of the last bound. The first one is bounded by exp (2cs + η)|Γ| as we already noticed. The otherone may be reduced to the sum over j’s such that tqj (α) < min(t) · exp 2η|∂Λ| by Lemma 4 (iv). Now we may use our auxiliary estimate of the partial derivatives over tj ’s. Since we confined ourselves to those special j’s, we may use Lemma 3 to estimate the partial tq (α) and tq˜(α) derivatives of for those q and q˜ for which tq (α) ≤ exp cs |Γ| and tq˜(α) ≤ exp (2cs + η)|Γ| . Inserting the above bounds into (2.26), we finally get
(α)
+ (α) exp (2cs + η)|Γ|
∂α¯ Ψq (Γ) ≤ ec# |Γ| e−τ +|S| exp (2cs + η)|Γ| exp (2cs + η)|Γ| ||¯ α||. Hence, taking again η = 1, we get (b ) with c = c# + 4cs + 2 + 2 log(|S| + 1) .
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3 Periodic Gibbs states In this short section we recall how the standard Pirogov-Sinai theory leads to a description of Gibbs states of classical lattice models in terms of contour models. Indeed, after reformulating a model with Hamiltonian H in terms of a labeled contour model, we can apply results of Section 2.2. = Recall that we suppose that all periodic configurations from Gper 0 {x1 , . . . , xr } are actually translation invariant. This does not mean a loss of generality as explained in the remark following Basic Lemma 1.2. Considering, in the standard way (in the present context, cf. [[HKZ, Lemma 3.1]), the “diluted” partition functions (H) Z d (Λ|xq ; H) = exp{−EΛ (x)}, x=xq in (Λc )R (H)
with (Λc )R = {i ∈ Zν ; dist(i, Λc ) ≤ R+1}, we introduce ϕ(H) = (ϕq ) and Φ(H) in such a way that we can replace the diluted sums Z d (Λ|xq ; H) by the corresponding labeled contour model partition functions Z cont (X(Λ)|q; Φ(H) , ϕ(H) ) discussed in ˙ Section 2.2 (cf(2.15)); we use here the superscript “cont” to stress that we are concerned with partition functions (resp., probability distributions) of a labeled contour model. Namely, let us introduce the labeled contour model (with labels Q = = {x1 , x2 , . . . , xr }) with {1, . . . , r}, where we denote Gper 0
and Φ(H) (γ) =
ϕ(H) = exq (H) q
(3.1)
(H) (H) exp −EΓ (x) + EΓ (xq ) ,
(3.2)
x∼γ
whenever γ is a q-contour. The sum is over configurations on Γ that can be ex(H) tended to a configuration having γ as its contour (for definitions of EA (x) and ex (H) see (1.2) and (1.3)). Then, one can easily verify that Z d (Λ|xq ; H) = Z cont (X(Λ)|q; Φ(H) , ϕ(H) )
(3.3)
for every finite Λ. Moreover, the state µ({x}, Λ|xq ; H) can be linked with the labeled contour model probability distribution (2.16), µ({x}, Λ|xq ; H) = µcont ({z}, X(Λ) | q; Φ(H) , ϕ(H) ). x∈XΛ x∼z
Here the sum is over all spin configurations x ∈ XΛ such that the set of their labeled contours is just z (it means that one is summing only over configurations on γ∈z supp Γ with boundaries fixed by labeling of contours γ ∈ z).
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Next, we verify that the assumptions of Theorem 2 for the labeled model described by Φ(H) and ϕ(H) are satisfied. Recall that the Hamiltonian H0 satisfies the Peierls condition (P(per) ) with respect to the set G0 and constant ρ0 . We can choose ε so that each H ∈ Kε (H0 ) satisfies, according to Proposition 1.1.3, the Peierls H , where limε→0 ρε = ρ0 , condition (P(per) ) with a sufficiently large constant ρε H 0 if ρ0 is large enough. Now, we consider the open set Kε (H0 ) ⊂ H(R) to play the role of the set Ω from Theorem 2 (thence the Hamiltonians H stand for the parameters α of Theorem 2). The functional Φ(H) (γ) defined by (3.2) satisfies, for any q-contour γ, the condition (1) of Theorem 2. Indeed, using (P(per) ) and the estimate |S||Γ| on the H
−(ρ
−log |S|)|Γ|
cardinality of the set {x; x ∼ γ} we get Φ(H) (γ) ≤ e ε H0 . To verify the condition (2), we consider the derivative
+ (H) (H) + (H) (γ)| = ∂H exp − (Ei (x) − Ei (xq ) . |∂H ¯Φ ¯ x∼γ
(3.4)
i∈Γ
Referring to (1.2), notice that ∂UA ∂(zA ) ( 0
0
i∈A
UA (xA ) |A| )
≤
i∈A,[A]=[A0 ]
1 |A|
≤ 1,
whenever A0 is such that the canonically fixed site i(A0 ) equals 0 (cf. the discussion at the beginning of Section 1.1) and a configuration zA0 ∈ S A0 is fixed. Therefore, taking again into account that the cardinality of {x; x ∼ γ} is at most |S||Γ| and (H) (H) −ρ H |Γ| ¯ = (U ¯A0 (zA0 )|i(A0 ) = 0), the that e−(EΓ (x)−EΓ (xq )) ≤ e ε H0 , we get, for H bound + (H) (γ)| ≤ e|Γ| log |S| e |∂H ¯Φ
H |Γ| 0
−ρε H
(H)
+ (|∂H ¯ Ei
(H)
+ (x)| + |∂H ¯ Ei
(xq )|) =
i∈Γ
=e
U (x ) ∂
A A U A0 (zA0 ) + ∂UA0 (zA0 ) |A| i∈Γ A0 ,zA0 i∈A
∂ UA ((xq )A )
+ U A0 (zA0 ) ≤ ∂UA0 (zA0 ) |A|
H |Γ|(log |S|−ρε H ) 0
×
i∈A
≤e
H +log |S|) 0
|Γ|(−ρε H
≤e
¯ |Γ|2 dim H(R)H
H |Γ|(−ρε H +log |S|+1+log 2+log dim H(R)) 0
¯ H.
Hence, (H)
+ (H) |∂H (γ)| ≤ e−τΦ ¯Φ
with (H)
τΦ
= ρε
|Γ|
¯ H,
H − log |S| − 1 − log 2 − log dim H(R). H0
(3.5) (3.6)
Thus Φ(H) satisfies the conditions (1) and (2) from Theorem 2 and Proposition (H) (H) 2.2.1 with the constant τΦ playing the role of τ (α) . Further, ϕq , in the place
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(α)
of ϕq , satisfies the condition (3) from Theorem 2 with M = 1. Recalling the notation ε˜0 , ccl , and c (cf. Lemma 2.3.1, Proposition 2.1.3, and Theorem 2), and applying Theorem 2, the equality (3.3), and Lemma 2.3.1, we obtain Proposition 3.1 Under assumptions of Theorem 1: a) For every H ∈ Kε (H0 ) as above, defining Φ(H) and ϕ(H) by (3.1) and (3.2), we obtain a labeled contour model that satisfies (1) – (3) with the respective (H) constants τ (H) = τΦ and M = 1. Moreover, Z d (Λ|xq ; H) = Z cont (X(Λ)|q; Φ(H) , ϕ(H) ) for every finite Λ. b) For every x ∈ Gper 0 , there exists a function hx (·) on Kε (H0 ) such that there exists an extremal Gibbs state µ ∈ G(H) that is a perturbation of x whenever hx˜ (H). Moreover, hx (H) = h(H) = minx˜∈Gper 0 (H)
−τh |hx (H) − ϕ(H) , x |≤e (H)
+ (H) −τh |∂H , ¯ (hx (H) − ϕx )| ≤ e
(3.7) (3.8)
and, finally, (H)
+ −τh ¯ )H |∂H ¯ h(H)| ≤ (1 + e
for every x ∈
Gper 0
and any H ∈ Kε (H0 ). Here, and in what follows, (H)
τh Further, for every x ∈
(H)
= τΦ
− ccl − c .
Z d (Λ|x; H) ≥ exp −hx |Λ| − ε˜0 |∂Λ| Gper 0 .
(3.9)
(3.10) (3.11)
If xq is stable, then
Z d (Λ|xq ; H) ≤ exp −h|Λ| + ε˜0 |∂Λ|
(H)
and there exists a (τΦ
(3.12)
− c )-functional Ψq such that
Z d (Λ|xq ; H) = e−exq (H)|Λ| Z(L(Λ); Ψq )
(3.13)
for every finite Λ.
4 Gibbs states with interfaces The aim of the present section is to start from the situation of Theorem 1 and to rewrite the partition functions Z(Y, Λ|y; H), y ∈ Ghor 0 , in terms of labeled contour models that can be treated by the methods of Section 2.2.1. In fact, we are doing this only for a class of sets Y of configurations with the aim to estimate the probability of a particular interface. Throughout this section we suppose that the assumptions of Theorem 1 (i.e. also of Basic Lemma) are satisfied. In particular,
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ρ0 is assumed to be sufficiently large, with the exact bounds specified in the course of the exposition in the present section. We shall proceed in several steps that culminate in Proposition 4.4 below that yields an expression of probabilities of some sets of configurations from X in terms of corresponding interfaces which, in their turn, are related to certain labeled contour model. The probability measure that we have in mind is constructed, following Dobrushin [D 72], by a suitable limit starting from Gibbs states in finite volumes with boundary conditions y ∈ Ghor 0 .
4.1
Interfaces in finite volumes
To consider Gibbs states in Basic Lemma for non-periodic elements of G0 we fix a particular H ∈ Kε (H0 ), two different configurations from Gper 0 , say xp , xq with p, q ∈ {1, . . . , r}, and a configuration y ∈ G0 ∩ Xxhor to play the role of boundary p ,xq conditions. The Hamiltonian H being fixed, we shall often omit a reference to it from the notation. Following [HKZ] we say that a configuration x has a y-interface if its boundary B(x) has a unique infinite component I(x), I(x) \ I(y) has only finite components, and Zν \ I(x) has exactly two infinite R-components. We say that the sites in one of them are lying above and those in the other one below the interface. Further, we use I(x) to denote the pair (I(x), xI(x) ) and say that I is a y-interface if there exists a configuration x that has a y-interface I = I(x). We begin with a study of the partition function Z(Λ|y; H) in a fixed finite ν volume Λ ⊂ Zν . If x is any configuration (i.e. x ∈ S Z , ν ≥ 2) that equals y in Λc , then x has an interface ([HKZ], Lemma 4.1). We use J (y, Λ) to denote the set of interfaces of configurations x considered above. Our aim is to study the probability of interfaces and thus we begin with rewriting, for a fixed I ∈ J (y, Λ), the partition function Z(I, Λ|y; H) defined as Z(I, Λ|y; H) = Z({x : I(x) = I}, Λ|y; H) in five steps. (H)
Step 1 (sum over interfaces) Notice first that the energy EI∩Λ (x) does not depend (H) on xI c and will be denoted by EI∩Λ (I). The volume ΛR (= {i ∈ Zν : d(i, Λ) ≤ R + 1}) is, by means of an interface I ∈ J (y, Λ) with the support I, split up into several parts: I ∩ ΛR , the R-components Λm (I), m ∈ {p, q}, of ΛR \ I containing the sites lying above or below I, respectively, and, finally, the remaining finitely many R-components of ΛR \ I denoted by Intj I, j = 1, 2, . . . . The configuration x equals to some xq(j) ∈ Gper = {x1 , x2 , . . . , xr }, in IR ∩ Intj I if I(x) = I. 0 We recall that, given the Hamiltonian H, the “physical” partition function in Λ under the boundary condition z ∈ X is expressed by exp{−HΛ (x|z)}, Z(Λ|z; H) = where HΛ (x|z) =
A∩Λ=∅
x=z in Λc
UA (xA ) with x = z in Λc .
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Using now directly concerned definitions, we get Z(I, Λ|y; H), Z(Λ|y; H) = I∈J (y,Λ)
where |A ∩ ΛR | = UA (y) Z(I, Λ|y; H) exp − |A| A⊂Λc (H) = exp{−EI∩ΛR (I)} Z d (Λm (I)|xm ; H) Z d (Intj I|xq(j) ; H). (4.1) j
m∈{p,q}
The equalities above correspond to Lemma 4.2 from [HKZ] that differs only by the usage of the relative diluted partition function Θ instead of Z d . Step 2 (cluster expansion) From now on we suppose that xp and xq are stable with respect to the considered Hamiltonian H and so we may apply the PirogovSinai theory as formulated in Section 2. We express the diluted partition functions (H) (H) with boundary conditions xp , xq in terms of contour functionals Ψp and Ψq (H)T and thus, referring to (3.13) and (2.10), in terms of cluster functionals Ψp and (H)T Ψq . Namely, |A ∩ ΛR | d Z(I, Λ|y; H) exp − UA (y) Z (Intj I|xq(j) ; H)× = |A| j A⊂Λc (H) Ψ(H)T (C) − exm (H) |Λm (I)| . (4.2) × exp −EI∩ΛR (I) + m m∈{p,q} C⊂Λm (I) (H)
(H)
Here, the contour functionals Ψp and Ψq satisfy, according to Proposition (H) 2.2.1, the bounds a) and a ) with the sufficiently large constant τΦ − c . Further, (H)T (H)T and Ψq satisfy, in view of Proposition 2.1.3 (if the cluster functionals Ψp (H) (H) τΦ − c ≥ τcl ), the bounds (2.7) and (2.9) with the constant τh defined in Proposition 3.1 above. Step 3 (extraction of bulk terms) We are going to extract a bulk term independently of I using the fact that (see (S) in Theorem 2, Proposition 2.2.1 c), (2.14) of Corollary 2.1.4 and (3.1)) ϕ(H) − p
Ψ(H)T (C) p C i
|C|
= ϕ(H) − q
Ψ(H)T (C) q C i
|C|
.
Denoting this expression by h(H) (notice that, since the phases xp and xq are stable, one has h(H) = − limν Λ Z
1 1 log Z d (Λ|xq¯; H) = − limν log Z(Λ|z; H) Λ Z |Λ| |Λ|
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for any q¯ ∈ {1, . . . , r} and z ∈ X (the first equality, for stable q¯, follows by (3.13), (3.1), (2.10) and Proposition 2.2.1 c); the other one is standard), we get |A ∩ ΛR | = Z(I, Λ|y; H) exp h(H) |ΛR |} exp{− UA (y) |A| A⊂Λc
= exp − EI∩ΛR (I) − h(H) |I ∩ ΛR | log Z d (Intj I|xq(j) ; H) + h(H) |Intj I| × exp j
× exp −
Ψ(H)T (C) m
m∈{p,q} CΛm (I)
|C ∩ Λm (I)| . (4.3) |C|
Step 4 (extraction of surface terms) The next step is to extract a surface term that does not depend on I. If C ⊂ Zν and m ∈ {p, q}, we write C ∗ m whenever there exist i, j ∈ C such that |i − j| = 1, i ∈ ∂ΛR ∩ ∂Λm (I(y)), j ∈ (ΛR )c . We put Λ χΛ m (C) = 1 if C ∗ m and χm (C) = 0 otherwise. Using this notation, we have |A ∩ ΛR | |C ∩ ΛR | + Z(I,Λ|y;H)exp h(H)|ΛR |− UA (y) Ψ(H)T (C) m |A| |C| A⊂Λc m∈{p,q} C∗m log Z d (Intj I|xq(j) ; H)+h(H) |Intj I| = exp − EI∩ΛR (I)−h(H) |I ∩ ΛR | + × exp
m∈{p,q} C∩I=∅
j
|C ∩ ΛR | |C ∩ Λm (I)| Ψ(H)T (C) χΛ − . (4.4) m m (C) |C| |C|
The above equality corresponds to Lemma 4.3 in [HKZ]. Notice that the factor on the left-hand side of (4.4) does not depend on I. |C∩ΛR | |C∩Λm (I)| Step 5 (positivity of cluster terms) The terms Ψm (C)(χΛ ) m (C) |C| − |C| are not necessarily positive, a feature that would be useful in further application of our version of the Pirogov-Sinai theory. However, the terms depending on C may be turned into explicitly positive by adding a suitable sum in the exponent and absorbing it into a small change of weight of interfaces in the same time. For a reasonable choice of the added sum (to secure the positivity of cluster terms and to (H) (H) allow the proof of Lemma 4.3 below) we shall use the bounds |Ψm (Γ)| ≤ e−τh |Γ| (H)
(H)T
(H) and Ψm (C) ≤ e−τh |C| with τh from Proposition 3.1 (see also Step 2). Hence (H)T
|A ∩ ΛR | UA (y) Z(I, Λ|y; H) exp h(H) |ΛR | − |A| A⊂Λc (H) |C ∩ ΛR | −τh |C| = + 2 × exp Ψ(H)T (C) e m |C| C∗p m∈{p,q} C∗m
C∗q
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(H) = exp − EI∩ΛR (I)−h(H) |I ∩ ΛR | + log Z d (Intj I|xq(j) ; H)+h(H) |Intj I| j
|C ∩ ΛR | |C ∩ Λm (I)| Ψ(H)T − ) (C)(χΛ m m (C) |C| |C| m∈{p,q} C∩I=∅ (H) Λ × exp 2 e−τh |C| (|C ∩ I ∩ ΛR | + χΛ (C) · χ (C)) p q
× exp
C∩I=∅
(H) e−τh |C| |C ∩ I ∩ ΛR | . (4.5) × exp −2 C∩I=∅
We added the terms 2
e
(H) −τh |C|
and added and subtracted 2
(H)
e−τh
|C|
C∩I=∅
C∗q C∗p
(H) κ(τh ) |I
|C ∩ I ∩ ΛR |. The idea is that the latter equals (H) (H) e−τh |C| κ(τh ) = 2
∩ ΛR | with (4.6)
C:0∈C
and may be absorbed into a controllable change of energy, while the former does not depend on I and may be extracted as a “border of surface term”. In the same |C∩ΛR | m (I)| time, whenever the term χΛ − |C∩Λ is non-vanishing (in any case, m (C) |C| |C| Λ its absolute value is bounded by 1), the term |C ∩ I ∩ ΛR | + χΛ p (C) · χq (C) is at Λ least 1. Indeed, if χm (C) = 0 and in the same time C ∩Λm (I) = ∅, then necessarily C ∩ I ∩ ΛR = ∅ (we took into account that C ∩ I = ∅). If, say, χΛ p (C) = 1 and in Λ the same time χq (C) = 0 and C ∩ ΛR = C ∩ Λp (I), then C ∩ I ∩ ΛR = ∅. Finally, Λ the claim is trivial if χΛ p (C) = χq (C) = 1. As a result, |C ∩ ΛR | |C ∩ Λm (I)| Λ − + (C) χ (C) Ψ(H)T m m |C| |C| (H) Λ + e−τh |C| |C ∩ I ∩ ΛR | + χΛ (C) · χ (C) ≥ 0 (4.7) p q
(H)T
(H) since Ψm (C) ≤ e−τh |C| . ˜ Λ|y, H) to denote the left-hand side of (4.5), we stress that the Using Z(I, ˜ ratio Z/Z does not depend on I and thus the probabilities of interfaces defined by Z˜ or Z do not differ. To evaluate those probabilities, we recall the notion of walls in order to rewrite (4.5) in terms of them. A pair w = (W, xW ), where W is a connected component of I(x) \ C(x), is a wall of I(x). We denote by W(I(x)) the collection of all walls of I(x). For any w ∈ W(I(x)) we put (H)
E (H) (w) = EW (x).
(4.8)
Let us recall also that given a wall w, we use xw ∈ X to denote the configuration for which I(xw ) = B(xw ) and w is the only wall of I(xw ), and yw to denote
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the unique element of G0 which differs from xw in at most finite number of sites. Recall that the set G0 ∩ Xxhor consists of configurations that are vertical shifts of p ,xq finitely many “representatives” xp,q;1 , . . . , xp,q;np,q . Let s(w) be such that xp,q;s(w) is a vertical shift of yw . We use π to denote the projection of Zν onto Zν−1 defined by π(i1 , . . . , in−1 , in ) = (i1 , . . . , in−1 ), and recall that IW = I(yw ) ∩ π −1 (W ). Notice further that each set Intj I is surrounded by the support W of a single wall w of I and use j ◦ W to denote that this is the case. Let Ip,q;s be the support of the interface of xp,q;s , s = 1, 2, . . . , np,q and Bp,q;s (I) be the set {i ∈ BR = π(ΛR ) : π −1 (i) ∩ I is a vertical shift of the “xp,q;s -ceiling column” }∪ {π(W ); s(w) = s} for I ∈ J (y, Λ). Let Tp,q;s be the number of sites in π −1 (0) ∩ Ip,q;s , i.e. the “thickness” of Ip,q;s , and T be the maximum of all Tp,q;s ’s over p, q ∈ {1, . . . , r}, s ∈ (H) (H) {1, . . . , np,q }. Let us recall that 2 e−τh |C| |C ∩ I ∩ ΛR | = κ(τh )|I ∩ ΛR |. C∩I=∅
˜ (H) (w) to denote the “modified energy of the wall w”, Further, we use E ˜ (H) (w) = E (H) (w) − h(H) |W | + κ(τ (H) ) |W | − E h − (log Z d (Intj I|xq(j) ; H) + h(H) |Intj I|), (4.9) j◦W (H)
and ϕp,q;s to denote the “modified specific energy of an s-ceiling between xp above and xq below”, (H) (H) ) − h(H) Tp,q;s . ϕ(H) p,q;s = Eπ −1 (0)∩Ip,q;s (xp,q;s ) + κ(τh
(4.10)
(H)
We also use ΦΛ (w; p, q) to denote the “modified weight corresponding to a wall w of an interface separating xp and xq ”, the wall functional (H) (H) ˜ (w) − ϕ(H) ΦΛ (w; p, q) = exp − E p,q;s(w) |π(W )|
(4.11)
and, finally, we define the cluster functional
|C ∩ ΛR | |C ∩ Λm (I)| (H) ΦΛ,I (C; p, q) = exp − (C) χΛ Ψ(H)T + m m (C) |C| |C| m∈{p,q} (H) Λ (C)χ (C) − 1. (4.12) + e−τh |C| |C ∩ I ∩ ΛR | + χΛ p q
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Using this notation, we rewrite (4.5) as ˜ Λ|y; H) = Z(I,
np,q (H) ΦΛ (w; p, q)
w∈W(I)
s=1
×
exp{−ϕ(H) p,q;s |Bp,q;s (I)|}×
(H) 1 + ΦΛ,I (C; p, q) , (4.13)
C:C∩I=∅,C∩BR =∅
and notice that by (4.7) we have (H)
ΦΛ,I (C; p, q) ≥ 0.
4.2
(4.14)
Interfaces in cylinders with finite base
˜ ˜ |y; H) = Our next aim is to show that Z(V I∈J (y;V ) Z(I, V |y; H) can be defined −1 for an infinite cylinder V = π (B) with a finite base B(= π(V )) ⊂ Zν−1 and to ˜ V |y; H). study Z(I, Step 6 (a wall bound) The main aim of this step is to prove that, for any finite volume Λ ⊂ Zν and any interface I of a configuration which equals y in Λc , ˜ Λ|y; H) ≤ ecI |BR |(H+1) exp −(ρε H − cI ) Z(I, H0
|W | ,
(4.15)
w∈W(I)
for some “interface” constant cI , once ρε is sufficiently large. (H) To estimate the wall functional ΦΛ (w; p, q) defined by (4.11), we bound first (cf. (4.9) and (4.10)) (H)
(H)
E (H) (w) − h(H) |W | + κ(τh ) |W | − ϕp,q;s(w) |π(W )| = (H) (H) = EW (xw ) − e0 (H)|W | − EIW (yw ) − e0 (H)|IW | + (H)
+ (e0 (H) − h(H))(|W | − |IW |) + κ(τh
)(|W | − |IW |).
(4.16)
H First of all we use Peierls condition (Phor ) with the constant ρε H (see 0 Proposition 1.1.3) to the first parenthesised part on the right-hand side and we get
(H) H (H) |W |. EW (xw ) − e0 (H)|W | − EIW (yw ) − e0 (H)|IW | > ρε H0
(4.17)
By (3.7) we get
(H)
(e0 (H) − h(H))(|W | − |IW |) ≤ e−τh |W |(1 + T ).
(4.18)
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(H)
We estimate the artificial term κ(τh ), used to achieve the positivity of using its definition (4.6) and the bound (2.4), by
(H) ΦΛ,I (C; p, q),
(H)
0 ≤ κ(τh
)≤2
∞
(H)
e−(τh
−c# )k
(H)
≤ e−(τh
−c# −log 3)
(4.19)
k=1 (H)
once τh
is sufficiently large. Hence (H)
κ(τh
(H)
)(|W | − |IW |) ≥ −e−(τh
−c# −log 3)
|W |T.
(4.20)
Further, due to (3.12) we have the bound (H) ε˜0 (τh )|∂ Intj I| ≤ log Z d (Intj I|xq(j) ; H) + h(H)| Intj I| ≤ j◦W
j◦W
≤
(H) 2ν ε˜0 (τh )|W |
(H)
≤ e−(τh
−cZ )
|W | (4.21)
for a constant cZ . Here we used that (H)
) ≤ 2e−(τh
(H)
≤ e−(τh −cε ) , for a suitable constant cε , by Lemma 2.3.1. Here j◦W means any sum over a subset of {j; j ◦ W }. Substituting the just derived inequalities (4.17), (4.18), (4.20) into (4.16), we get (H)
ε˜0 (τh
(H)
E (H) (w) − h(H) |W | + κ(τh ρε
−ccl −c# −log |S|−1)
(H)
) |W | − ϕp,q;s(w) |π(W )| ≥
(H) (H) H |W | − e−τh |W |(1 + T ) − e−(τh −c# −log 3) |W |T. (4.22) H0
Using further (4.21) and having in mind (4.11) with (4.9) and (4.10), we conclude that (H)
ΦΛ (w; p, q) ≤ (H) (H) (H) H |W |+e−τh |W |(1+T )+e−(τh −c# −log 3) |W |T +e−(τh −cZ ) |W | ≤ exp −ρε H0 H − cw )|W | , (4.23) ≤ exp −(ρε H0 where cw is a positive constant which can be chosen arbitrarily small if taking (H) τh , i.e. ρε , sufficiently large in the same time. Now we estimate the contribution of the specific ceiling energies to the ˜ Λ|y; H). Due to (4.10), (4.19), and the inequality |h(H)| ≤ logarithm of Z(I, (H) −τh H + e following from (3.7), we get (H)
−(τh |ϕ(H) p,q;s | ≤ t(H + e
−c# −log 3)
(H)
+ H + e−τh ) ≤ cϕ (H + 1),
(4.24)
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for some real constant cϕ . It follows that p,q
ϕ(H) p,q;s |Bp,q;s (I)| ≤
n
s=1
max
s=1,...,np,q
(H)
ϕp,q;s π(ΛR ) ≤ cϕ ( H + 1) BR . (4.25) (H)
Finally, we get bounds for the nonnegative cluster functionals ΦΛ,I (C; p, q). (H)
Using (4.14) from Step 5, (4.12), the fact that |Ψm (C)| ≤ e−τh |C| , and the inequalities 0 ≤ ex − 1 ≤ 2x for x nonnegative and small enough, we get (H)T
(H)
0 ≤ ΦΛ,I (C; p, q) ≤ 4(|C| + 2)e−τh (H)
|C|
(H)
≤ e−(τh
−cC )|C|
,
(4.26)
where cC is a real constant. So we can estimate the third product from (4.13) by
(H) 1 + ΦΛ,I (C; p, q) ≤
C:C∩I=∅,C∩BR =∅
(H) 1 + e−(τh −cC )|C| ≤
i∈I∩ΛR C:dist(C,i)≤|C|
≤ exp cπ (|I ∩ ΛR |) ≤ exp cπ (|BR | +
|W |)
(4.27)
w∈W(I)
for some constant cπ . Here, we used (H) 1 + e−(τh −cC )|C| ≤ exp C:dist(C,i)≤|C|
(H)
e−(τh
−cC )|C|
C:dist(C,i)≤|C|
(H) ≤ exp |C|ν e−(τh −cC )|C| . (4.28) C:i∈C
Applying the estimates (4.23), (4.25), and (4.27) to (4.13), we get ˜ Λ|y; H) ≤ Z(I, H ≤ exp{− − cw )|W |} exp{cϕ (H + 1)|BR |} exp{cπ |BR | (ρε H0 w∈W(I)
+
w∈W(I)
H − cI ) cπ |W |} ≤ ecI |BR |(H+1) exp −(ρε H0
|W |
w∈W(I)
with a suitable constant cI , getting thus (4.15). Step 7 (partition functions in infinite cylinder sets) Let us use χV to denote, for V = π −1 (B), the analogue of χΛ . Further, we introduce the analogs of the wall and (H) (H) cluster functionals ΦΛ (w; p, q) and ΦΛ,I (C; p, q), defined by (4.11) and (4.12). We denote the corresponding functionals, obtained by substituting V for Λ, by (H) (H) ΦV (w; p, q) and ΦV,I (C; p, q).
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To avoid overloading the functionals with an additional label, we slightly abuse the notation observing that the functionals defined above can be easily distinguished by the number of their indices and arguments from general labeled (H) contour functional of Section 2.2 as well as from the functionals ΦB (A; p, q) and (H) ΦB (σ; p, q), introduced in the next section, that yield the labeled aggregate and shadow models. In correspondence to Lemma 4.7 in [HKZ], and due to the estimate (4.26), we may use the fact that (H) (H) ΦV,I (Ck ; p, q) 1 + ΦV,I (C; p, q) = C:C∩I=∅,C∩VR =∅
{Ck } finite Ck ∩I=∅,Ck ∩VR =∅
k
to obtain the following expression for the limit partition function (H) ˜ Λ|y; H) = ˜ V |y; H) = lim Z(I, Z(I, ΦV (w; p, q)× Λ V
{Ck } finite Ck ∩I=∅,Ck ∩VR =∅
w∈W(I)
np,q
×
exp{−ϕ(H) p,q;s |Bp,q;s (I)|}
s=1
(H)
ΦV,I (Ck ; p, q). (4.29)
k
Thus the probabilities of any interface I ∈ J (y, Λ) defined by µ {x ∈ X : I(x) = I}, Λ|y; H = converge to the probability of I ∈ J (y, V ) = µ {x ∈ X : I(x) = I}, V |y; H =
˜ Λ|y; H) Z(I, ˜ ¯ ¯ I∈J (y,Λ) Z(I, Λ|y; H)
Λ⊂V
J (y, Λ) defined by
˜ V |y; H) Z(I, . ˜ ¯ ¯ I∈J (y,V ) Z(I, V |y; H)
(4.30)
The sum in the denominator above converges since, according to (4.15), it H −(ρε H −cI )|W | 0 can be bounded by ecI |BR |(H+1) i∈B , where W stands i∈W e for connected subsets of Zν (possible supports of walls shifted vertically to intersect B). Here we have used the important fact that the interfaces from J (y, V ) are uniquely determined by their walls even if we “forget” their vertical position. This observation goes back to “(y, V )-admissible families of standard walls ([HKZ] Lemma 2.2)” and originates in [D 72]. (H) Now we shall recall or derive some estimates on the functionals ΦV (w; p, q) (H) and ΦV,I (C; p, q) and their derivatives as well as a bound on the derivative of (H)
ϕp,q;s needed later. Lemma 4.3 Let xp , xq ∈ G0 be two stable translation invariant configurations at H ∈ Kε (H0 ). Let s ∈ {1, . . . , np,q } be arbitrary.
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(H)
Then ΦV (w; p, q) is translation invariant, ΦV,I (C; p, q) is translation invariant “inside V ”, and the following bounds are fulfilled. (H) (H) (a) 0 < ΦV (w; p, q) ≤ e−τI |W | ; (H)
(b) 0 ≤ ΦV,I (C; p, q) ≤ e−τI (H)
|C|
;
(H) + (H) (a ) |∂H ΦV (w; p, q)| ≤ H exp{−τI |W |}; (H) + (H) ΦV,I (C; p, q)| ≤ H exp{−τI |C|}; (b ) |∂H + (H) (c) |∂H ϕp,q;s | ≤ HMs . Ms(≡ Mshadow ) and cI are suitable constants, Here ρε log H0 − cI H.
(H)
and τI
(H)
= τh
−
Proof. (a) Taking the limit Λ " V in (4.23), we get (H) H (H) − cw )|W | ≤ e−(τh −cw )|W | 0 ≤ ΦV (w; p, q) ≤ exp −(ρε H0 by (3.6) and (3.10). (b) Taking the limit Λ " V in (4.26) we get (H) (H) 0 ≤ ΦV,I (C; p, q) ≤ exp −(τh − cC )|C| . (H) (a ) By the definitions (4.9) of E˜ (H) (w) and (4.11) of ΦΛ (w; p, q), with Λ replaced by V , we have (H)
+ ∂H ΦV (w; p, q) =
(H) (H) + exp −E (H) (w) + (h(H) − κ(τh ))|W | + ϕp,q;s(w) |π(W )| × = ∂H
Z d (Intj I|xq(j) ; H) = exp{−h(H)| Intj I|} j◦W (H) (H) (H) + = ΦV (w; p, q)∂H −E (H) (w) + (h(H) − κ(τh ))|W | + ϕp,q;s(w) |π(W )| + (H) (H) + exp −E (H) (w) + (h(H) − κ(τh ))|W | + ϕp,q;s(w) |π(W )| × Z d (Int I|x ; H) Z d (Intj I|xq(j) ; H) j q(j) + ∂H . (4.31) × I|} exp{−h(H)| Int exp{−h(H)| Intj I|} j j =j j◦W
j ◦W
To get the sought estimate (a ), we begin by estimating the first summand in (4.31). Due to (a), it suffices to show that (H) (H) + −E (H) (w) + (h(H) − κ(τh ))|W | + ϕp,q;s(w) |π(W )| | ≤ C (1) |W | H |∂H (H)
with a suitable constant C (1) that can be chosen independent of τh
(H)
for large τh
.
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Indeed, this follows from (3.9), and the following bounds:
+ (H)
(H)
∂ ¯ E (w) ≤ ∂ +¯ Ei (x) ≤ dim H(R) H W H H
(4.32)
i∈W (H)
by linearity of Ei
(x) in H (see (4.8) to understand the first inequality); (H)
+ ∂H ¯ τh
= ρε
¯ H H0
(4.33)
which follows from (3.10) and (3.6); (H)
+ |∂H ¯ κ(τh
)| ≤ 2
(H)
e−τh
|C|
|C|ρε
¯ H H0
0∈C
(H)
¯ −(τh ≤ He
−log
ρε H0
−cκ )
¯ const, ≤ H
(4.34)
with a suitable constant cκ , follows from (4.6) and (4.33); + (H) ¯ |∂H ¯ ϕp,q;s | ≤ Ms H
(4.35)
follows from (4.10) using an analogy with (4.32), (3.9), and (4.34). To estimate the other summand in (4.31), we use (4.22), (4.21) and, moreover,
Z d (Int I|x ; H) j q(j)
+
¯ exp{cZ W }.
∂H¯
≤ H exp{−h(H)| Intj I|}
(4.36)
To show the last bound, we use (2.23) of Lemma 2.3.3, (3.12), (3.9), and the ν bound | Intj I| ≤ |∂ Intj I| ν−1 ≤ e|∂ Intj I| to get
Z d (Int I|x ; H) |(∂ + Z d (Int I|x ; H)) exp{−h(H)| Int I|}| j j q(j) j ¯ q(j)
+
H +
∂H¯
≤ exp{−h(H)| Intj I|} exp{−2h(H)| Intj I|} + |Z d (Intj I|xq(j) ; H)) exp{−h(H)| Intj I|} | Intj I| ∂H ¯ h(H)| ≤ exp{−2h(H)| Intj I|} (H) ¯ Intj I| exp ε˜0 |∂(Intj I)| Ms (1 + ε˜0 ) + (1 + e−τh ) ≤ He ¯ cZ |W | , ≤ H| (4.37)
+
where cZ is a suitable constant. We used the obvious inequality | Intj I| ≤ |∂(Intj I)|ν .
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(b ) From the definition (4.12) we have
|C ∩ VR | |C ∩ Vm (I)|
+ (H) Ψ(H)T − )+ (C)(χVm (C) |∂H ¯ ΦV,I (C; p, q)| ≤ exp m |C| |C| m∈{p,q} (H)
+ e−τh |C| (|C ∩ I ∩ VR | + χVp (C)χVq (C)) × (H) + (H)T + (H) × |∂H (C)| + (|C| + 1) exp{−τh |C|}|∂H ||C| ≤ ¯ Ψm ¯ τh m∈{p,q}
¯ exp{−(τ (H) − log ≤ H h
ρε − cC )|C|} H0
(4.38)
with some constant cC . + (H)T (C)|, Here we used the bound (2.9) from Proposition 2.1.3 to estimate |∂H ¯ Ψm as well as the bound (4.33) above. (c) The needed estimate was already proved in (4.35) above.
4.3
Aggregate and labeled shadow models
We shall now project the walls, as well as clusters decorating the interface, to Zν−1 (⊂ Zν ), by applying the projection π. In that way, using the equalities (4.29) and (4.30), we introduce a shadow model, in terms of which we shall grasp the characteristic features of interfaces. The series of preceding steps will be concluded by Proposition 4.4 that describes the properties of the labeled shadow model as well as the relations to the probabilities of interfaces via an intermediary aggregate model. Given an interface I = (I, xI ) ∈ J (y, V ) (where V = π −1 (B) as in Section 4.2 and y ∈ Xxhor ) and a finite family {Ck } of clusters Ck , Ck ∩I = ∅, Ck ∩VR = ∅ (we p ,xq say that “{Ck } decorates I at V ”), and using W(I) to denote the collection of walls of I, we first consider connectedcomponents, to be called shadows of (I, {Ck }), of the projection π( w∈W(I) W ∪ k Ck ). If (W, C) is a collection of walls of I, W ⊂ W(I), and clusters from {Ck }, C ⊂ {Ck }, such that Σ = π( w∈W W ∪ C∈C C) is a shadow, we introduce an aggregate A of (I, {Ck }) as the “piece of interface above the shadow” together with the corresponding clusters. Namely, A = (I A , C), where I A = (IA , xIA ) with IA = π −1 (Σ) ∩ I. The shadow Σ is called the support of A. We identify aggregates that can be moved one into another by a vertical translation (by shifting both, the corresponding walls and clusters). Notice, that the relative position of concerned clusters with respect to the set IA is, in a given aggregate, always fixed. An aggregate A = (I A , C) with support Σ is naturally labeled since for any i ∈ ∂Σ, the column π −1 (i) is an “s-column” of xI with some s ∈ {1, . . . , np,q }. It is clear that this s is identical for all i’s from each component of ∂Σ. We denote this labeling by λ(A). As a result, the shadow Σ corresponding to an aggregate A may be labeled by λ(A) as described above. A shadow Σ endowed with such a labeling λ = λ(A)
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will be called a labeled shadow (of (I, {Ck })). We write σ = (Σ, λ), λ(σ) = λ, and supp σ = Σ. Notice that the labeled shadows of a pair (I, {Ck }), as above, form compatible matching families in the sense of Section 2.2, whereas, strictly speaking, labeled aggregates do not fit into that scheme, neither as a system of contours in Zν nor in Zν−1 . To include them would need a slight generalization of the setting from Section 2.2. Labeled shadows are “labeled contours” in the sense of Section 2.2. Considwith fixed p, q ∈ Q, p = q, ering now generally a configuration xp,q;s ∈ G0 ∩ Xxhor p ,xq such that xp , xq are stable, and s ∈ (1, . . . , np,q ) (in the role of the triplet y, xp , xq from the preceding subsection), we introduce the weight of an aggregate A = (I A , C) compatible with V and y by (H)
ΦB (A; p, q) =
(H)
ΦV (w; p, q)
w∈W(I A )
(H)
ΦV,I (C; p, q).
(4.39)
C∈C
Summing over all aggregates A corresponding to a fixed labeled shadow σ (the same support and the labeling λ(σ) = λ(A) corresponding to xA as above), we get the shadow weight (H)
ΦB (σ; p, q) =
(H)
ΦB (A; p, q).
(4.40)
A:supp A=supp σ λ(A)=λ(σ) (H)
(H)
n
p,q Finally, we use ϕp,q to denote the vector (ϕp,q;s )s=1 . It is defined as in (4.10) with stable xp , xq . Preparing for a direct application of the Pirogov-Sinai theory from Section 2 to the model defined by labeled shadows, notice first that, even though we took with a volume B ⊂ Zν−1 only interfaces from J (xp,q;s , V ) with the walls inside of V = π −1 (B), the clusters Ck may stick out of V and, correspondingly, the aggregates may not be contained in V and the shadows may not be contained in B. Anticipating this feature, we actually considered, in Section 2, the generalization allowing for volume depending contour weights. Thus, for any finite B ⊂ Zν−1 , we introduce the generalized ensemble aggr XB (p, q) of all compatible and matching families of labeled aggregates with walls in V = π −1 (B) and with clusters intersecting V and the ensemble Xshad B (p, q) of all aggr families of shadows that correspond to families from Xaggr B (p, q). The set XB (p, q) (Xshad B (p, q)) is actually the union, over s ∈ (1, . . . , np,q ), of all sets characterized by the external labels of all external aggregates (shadows) being fixed to equal s — i.e. families of aggregates (shadows) consistent with boundary conditions xp,q;s . The set Xshad B (p, q) will play the role of the abstract set XΛ from Section 2.2. Recall that, in accordance with the notation from the first paragraph of shad Section 2.2, for any A ∈ Xaggr B (p, q) (S ∈ XB (p, q)), we use Bs (A) (Bs (S)) to denote the set of sites in B that are outside of the supports of all aggregates A ∈ A
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(shadows σ ∈ S) and are labeled by the label s of the corresponding ceiling. Now, we define Z aggr Xaggr Z aggr {A}, Xaggr B (p, q)|s0 ; H = B (p, q)|s0 ; H , A∈Xaggr (p,q) B ext. label s0
where Z
aggr
np,q (H)
aggr
{A}, XB (p, q)|s0 ; H = ΦB (A; p, q) exp − ϕ(H) p,q;s Bs (A) A∈A
s=1
(4.41) if A ∈ Xaggr (p, q) with external label s ; otherwise it is not defined (or, rather, it 0 B is put to equal 0). Similarly, Z shad Xshad Z shad {S}, Xshad B (p, q)|s0 ; H = B (p, q)|s0 ; H S∈Xshad (p,q) B ext. label s0
with np,q
(H)
(p, q)|s ; H = Φ (σ; p, q) exp − ϕ(H) Z shad {S}, Xshad 0 B p,q;s Bs (S) , B σ∈S
s=1
(4.42) whenever S ∈ Xshad B (p, q) with external label s0 ; otherwise it is put to equal 0. shad Notice that Xaggr B (p, q) (respectively, XB (p, q)) contains an infinite number of configurations of aggregates (shadows). Nevertheless, the convergence of (4.41) and (4.42) is an easy consequence of the bounds (a) and (b) from Lemma 4.3. The probability of a compatible matching family of labeled aggregates A ∈ Xaggr B (p, q) with external label s0 is, correspondingly, defined by Z aggr {A}, Xaggr aggr B (p, q)|s0 ; H aggr . µ {A}, XB (p, q)|s0 ; H = (4.43) Z aggr Xaggr (p, q)|s ; H 0 B Similarly, for S ∈ Xshad B (p, q) with external label s0 we introduce Z shad {S}, Xshad B (p, q)|s0 ; H shad shad . µ {S}, XB (p, q)|s0 ; H = Z shad Xshad (p, q)|s ; H 0 B
(4.44)
These measures can be used to evaluate the probability of perturbations of the (H) interface of xp,q;s0 , once a sufficient decay of functional ΦB (σ; p, q) in dependence on the size of | supp σ| is guaranteed as we shall see in Section 4.4. Proposition 4.4 (interfaces in terms of aggregates and shadows) Let xp , xq be two translation invariant ground states of H0 that are stable for some H ∈ Kε (H0 ) for ε small enough, ρ0 large enough, and let B ⊂ Zd−1 be finite. Then the following claims hold.
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(i) For any s ∈ {1, . . . , np,q }: aggr aggr Z shad Xshad XB (p, q)|s; H B (p, q)|s; H = Z ˜ π −1 (B)|xp,q;s ; H). Z(I, = I
Further, aggr µshad {S}, Xshad µ {A}, Xaggr B (p, q)|s; H = B (p, q)|s; H , A
where the sum runs through A ∈ Xaggr B (p, q) such that supp A = supp σ for all A ∈ A and the labels corresponding to mathcalA and A coincide, λ(A) = λ(S). Finally, using W(A) to denote the set of all walls corresponding to the collection of aggregates A, we have µ {x ∈ X : W(I(x)) = W}, π −1 (B)|xp,q;s ; H = µaggr {A}, Xaggr B (p, q)|s; H A:W(A)=W
for any admissible collection of walls W. (H)
(H)
(H)
n
p,q (ii) The functional ΦB (σ; p, q) and the vector ϕp,q = (ϕp,q;s )s=1 define a labeled contour model that fulfills all the assumptions of Theorem 2 and Proposition 2.2.1. Namely,
(H)
• ΦB (σ; p, q) is nonnegative, • ΦB (σ; p, q) = ΦB (σ ; p, q) = ΦZν−1 (σ; p, q) whenever σ and σ are such that supp σ, supp σ ⊂ B and σ is a translation of σ (“the functional (H) ΦB (·; p, q) is, inside B, translation invariant and independent of B”). Also, (H) (H) there exists a constant τs differing from τI by a fixed constant depending only on ν and |S| and a constant Ms , such that the following bounds hold: (H)
(H)
(H)
(H)
(H)
(1) ΦB (σ; p, q) ≤ exp{−τs |supp σ|} for any labeled shadow σ consistent with boundary conditions xp,q,s in B; (H) + (H) ¯ (2) |∂H | supp σ|}; ¯ ΦB (σ; p, q) ≤ H exp{−τs + (H) ¯ (3) |∂H ¯ ϕp,q;s | ≤ Ms H.
Proof. (i) It follows from (4.39) – (4.44) with the use of (4.29) and (4.30). (H) (ii) The functional ΦB (A; p, q) is nonnegative by its definition (4.39) and by (4.14).
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The translation invariance inside B follows from the definitions (4.39), (4.11), (4.12), from the translation invariance of the functionals involved and the invariance inside B of χVm (C), and by inspecting cardinalities of the sets involved in (4.12) for sets C fully contained in V . (H)
The corresponding claims for ΦB (σ; p, q) follow from (4.40). (1) We use first Lemma 4.3(a) and (b) to get, by (4.39) and (4.40), that (H)
0 ≤ ΦB (σ; p, q) ≤
(H)
exp{−τI
(H)
|W | − τI
w∈W(I)
(I,C)
|C|},
C∈C
where the sum is over aggregates that correspond each to a pair of an interface I ∈ I(xp,q;s , V ) and of a finite family C such that C ∩ VR = ∅ and C ∩ I = ∅, for all C ∈ C, and π( C∪ W ) = Σ = supp σ as above (i.e. over (I, C) with C∈C
w∈W(I)
the only aggregate A = (I A , C)). |C| + Since, obviously, A = (W(I), C) ≡ C∈C
w∈W(I)
|W | ≥ | supp σ|, we
have (H)
ΦB (σ; p, q) ≤ e−(τI (H)
−ζ)| supp σ|
exp{−ζ(
(I,C)
w∈W(I)
|W | +
|C|)},
C∈C
(H)
for any ζ > 0, ζ < τI . We shall show that, for some sufficiently large ζ, in dependence on ν and |S| only, the last sum is at most ec| supp σ| for suitable c which also depends on |S| and ν only. We begin with an observation. Namely, introducing the notion of an extent of an aggregate A as ext A = IA ∪ C, we notice that it is a connected set of cardinality at most |W |+ C∈C w∈W(I A ) (1 + T ) |C| ≤ (1 + T )A. In the same time we may suppose, referring to the C∈C
identification of aggregates under vertical translations, that each ext A contains an element i of supp σ. As a consequence, there are at most ecs A aggregates A with i ∈ ext A and given A. Namely, every such A can be identified with a connected path over ext A of the length at most 2ν(T + 1)A, each point of which is moreover equipped with a label saying whether this point of the path belongs to a wall, to a cluster, or to the remaining part W∪ C ext A \ w∈W(I A )
C∈C
of the interface I A . The label will also say what spin at this point is attained if it is a point of a wall. The number of needed labels is thus depending on ν and |S|
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only. As a result, we get (H)
ΦB (σ; p, q) ≤ e−(τI (H)
−ζ)| supp σ|
exp{−ζ(
(H)
≤e
−ζ)| supp σ|
(H) −(τI −ζ)| supp σ|
|W | +
w∈W(I)
A=(I,C)
≤ e−(τI
|C|)} ≤
C∈C
e−ζA ≤
i∈supp σ A:i∈ext A
(H)
e−(ζ−cs )k ≤ e−(τI
−ζ−1+log(1−e−cs ))| supp σ|
i∈supp σ k≥| supp σ|
≤ e−τs
(H)
| supp σ|
,
where ζ ≥ 2cs . (2) By differentiating (4.40) and using (4.39) as well as the estimates (a), (b), (a ), (b ) of Lemma 4.3, we obtain (H)
+ |∂H ¯ ΦB (σ; p, q)| ≤
+
A=(I,C) w∈W(I)
(H)
¯ exp{−τ H I
¯ exp{−τ (H) H I
¯ w∈W(I)
¯ − |w|
¯ w∈W(I)
A=(I,C) C∈C (H)
¯ −(τI ≤ He
(H)
¯ − τI |w|
(H) τI
−ζ−1)| supp σ|
¯ |C|}+
¯ C∈C
¯ ≤ |C|}
¯ C∈C
exp{(−ζ + 1)A}.
A=(I,C)
Here the sum
above is taken over all aggregates A with supp A = supp σ
A=(I,C)
and λ(A) = λ(σ). Now, we use the same argument as in (1) with ζ − 1 instead of ζ, so that for ζ − 1 ≥ 2cs we get + −τs |∂H ¯ ΦB (σ; p, q)| ≤ e
(H)
(H)
| supp σ|
.
(3) This is the estimate of Lemma 4.3(c) above.
4.4
Gibbs states with interfaces. Proof of Basic Lemma
To conclude the proof of Basic Lemma, it remains to define, for any y ∈ Ghor 0 , the functions hy : Kε (H0 ) → R such that the statements (i) (b) and (ii) of Basic Lemma hold. As anticipated in Section 4.1, we fix a particular H ∈ Kε (H0 ), two different configurations from Gper 0 , say xp , xq with p, q ∈ {1, . . . , r}, and a configuration y ∈ G0 ∩ Xxhor to be identified with a triplet (p, q, s), s ∈ {1, . . . , np,q }. The p ,xq (H)
(H)
(H)
corresponding functionals ΦB (A; p, q), ΦB (σ; p, q), and vectors ϕp,q describing the aggregate and shadow models in the preceding Section 4.4 were defined (or considered) only if xp and xq were stable and distinct. We may however notice that (H) ϕp,q;s are by (4.10) actually well-defined for general pairs of (distinct) xp and xq
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(H)
from Gper 0 . To define ΦB (A; p, q) and ΦB (σ; p, q), we then use formulas (4.39) (H) and (4.40) referring to (4.11) and (4.12). While ΦV (w; p, q) again is well-defined (H) even if xp , xq are arbitrary elements of Gper 0 , we need to define ΦV,I (C; p, q) so that the bounds (1), (2), and (3) of Proposition 4.4 (ii) hold true. This can be easily achieved if we simply replace Ψp and Ψq in (4.12), and further on, by Ψp and Ψq , respectively, where Ψp and Ψq are the auxiliary functionals from Proposition 2.2.1 (b ), so that we may apply all the cluster expansion technique to them to get (ii) of Proposition 4.4 for all p, q’s. This is the starting point for a second use of the Pirogov-Sinai strategy. This (and fixed p, q) playing the role of reference time with different y ∈ G0 ∩ Xxhor p ,xq states and thus the label in Theorem 2 and Proposition 2.2.1 taking values s with s ∈ {1, . . . , np,q }. The function hy (H) from Basic Lemma is, for y corresponding (H) to the triplet (p, q, s), defined to be equal to the function hs (Φ(H) (·; p, q), ϕp,q ) whose existence is assured by Theorem 2. The claim (ii) of Basic Lemma is then an immediate consequence of Theorem 2 (approximating hy (H) (and its derivative) (H) (H) in terms of ϕp,q;s ) and the equation (4.10) relating ϕp,q;s to ey (H) (cf. (3.7), (3.9), and (4.34)). (Of course, if either xp or xq is not stable, we lose the equality (4.13) and as a result we cannot claim anything like (i) of Proposition 4.4.) It remains to prove the statement (i) (b) of Basic Lemma. Therefore we assume that both xp and xq are stable, and that y = xp,q;s is stable for the (H) (H) shadow model determined by ΦB (σ; p, q) and ϕp,q . We are going to derive the properties of the Gibbs measure µ(·|y, H) obtained from measures µ(·, Vn = π −1 (Bn )|y, H) as a weak limit with Bn growing up to Zν−1 , which were investigated in Step 7 above. Such measures are Gibbs states as they are weak limits of µ(·, Λn |y, H) for volumes Λn = Bn × [−kn , kn ], with kn growing to infinity sufficiently quickly (cf. [D 72, HKZ (3.2)]). To show that almost every configuration is a perturbation of y, it suffices to verify that for every ε and any Λ ⊂ Zν , there exist constants a, b, d > 0 so that, defining X(Λ, a, b) = x ∈ X; w ∈ W(I(x)), V (W ) ∩ Λ = ∅ ⇒ diamW ≤ a and γ a contour of x, V (Γ) ∩ Λ = ∅ ⇒ diamΓ ≤ b , with V (W ) denoting the union of W and all finite components of Zν \ W , we have µ X(Λ, a, b), π −1 (B)|y; H > 1 − ε
(4.45)
whenever B ⊂ Zν−1 is such that dist(π(Λ), B c ) ≥ d. To this end we first show that it is unprobable that shadows intersecting a given finite volume in Zν−1 are large and similarly for corresponding aggregates in Zν . Finally we estimate the probability of existence of large contours intersecting a fixed finite volume in Zν .
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(c)
Let thus EM be the set of all shadow configurations containing a large shadow intersecting or surrounding a fixed finite set M ⊂ Zν−1 . Namely, for any such M and c > 0 we introduce (c) EM = S ∈ Xshad B (p, q); there exists σ ∈ S such that V (Σ) ∩ M = ∅ and diamΣ > c , with Σ = supp σ and V (Σ) (for any finite set Σ ⊂ Zν−1 ) denoting the union of Σ with all finite components of Zν−1 \ Σ. Using the main results of Section 2 (Theorem 2, Proposition 2.2.1, and Corollary 2.2.2) to the shadow model, we may show, in a standard way, that for a given finite M ⊂ Zν−1 and a positive ε, constants d = d (M, ε) and c = c(M, ε) may be found such that (c) 1 µshad EM , Xshad B (p, q)|s; H ≤ ε 2
(4.46)
for all B ⊂ Zν−1 such that dist(M, B c ) ≥ d . For a fixed configuration S of shadows of diameter at most c and with V (Σ)∩ M = ∅ for every σ ∈ S, consider the event consisting of those configurations of walls and clusters (configurations of aggregates) whose set of shadows intersecting M is fixed and equals S. The conditional probability of each of these events, given such a fixed S, is independent of B for B sufficiently large. Since there are countably many such events, and due to (4.46), we may find a = a(M, ε) such that, supposing dist(M, B c ) > d , we get 1 µaggr (A(M, a), Xaggr B (p, q)|s; H) > 1 − ε, 2 with A(M, a) = {A ∈ Xaggr B (p, q); [w ∈ W(A) and V (W ) ∩ π −1 (M ) = ∅] ⇒ diamW ≤ a}. Comparing µaggr and µ (Proposition 4.4 (i)), we get the bound 1 µ X(M, a), π −1 (B)|y; H > 1 − ε 2
(4.47)
for
X(M, a) = x ∈ X; w ∈ W(I(x)) and V (W ) ∩ π −1 (M ) = ∅ ⇒ diamW ≤ a .
Given a finite Λ ⊂ Zν with projection π(Λ) = M , xΛ is fully determined if we know the contours γ with V (Γ) ∩ Λ = ∅ and walls w with V (W ) ∩ Λ = ∅. Let I be a given interface compatible with the boundary condition y on Λc . If γ is a contour such that V (Γ) ∩ Λ = ∅ and, in the same time, V (Γ) ⊂ Int W = V (W ) \ W
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for some wall, then necessarily V (W ) ∩ Λ = ∅. Thus it suffices to consider contours that are either above or below I, Γ ⊂ ( j Intj I ∪ I)c . Introducing X(I, Λ, b) = {x ∈ X; I(x) = I and [γ is a labeled contour of x, Γ ⊂ ( Intj I ∪ I)c , V (Γ) ∩ Λ = ∅] ⇒ diamΓ ≤ b}, j
we get
1 µ X(I, Λ, b), π −1 (B)|y, I; H > 1 − ε, (4.48) 2 independently of I for B ⊂ Zν−1 with dist(M, B c ) > d for some b = b(M, ε) and d = d(M, ε) > d . Here we consider the conditional probability under the condition that the interface with support I is present. The bounds (4.47) and (4.48) together yield (4.45). Remark. Notice that having (4.45) for Λ’s that are singletons, we get that almost all configurations with respect to the limit Gibbs state are perturbations of y. In fact, we may prove an explicit expression for the limit probability µ(·|y; H) by means of the limit conditional probability “µ(·, π −1 |I)” and the limit probability on families of walls “ limν−1 µI (·, π −1 (Bn )|y; H)” similarly as (2.1) was proved Bn Z
in [HKZ, Section 6.2.2]. However, it is a little bit more complicated in our more general situation. The probabilities µI can be expressed using their projections to families of external walls. Also we don’t have the exponential estimates as (a) and (b) of (i) and (ii) in [HKZ, Theorem 2]. However, one may check that the above estimate (4.45) is sufficient to carry on the proof of the respective identity.
References [BCF 1] C. Borgs, J. Chayes and J. Fr¨ ohlich, Dobrushin states for classical spin systems with complex interactions, Jour. Stat. Phys. 89, 895 (1997). [BCF 2] C. Borgs, J. Chayes and J. Fr¨ ohlich, Dobrushin states in quantum lattice systems Jour. Stat. Phys. 89, 591 (1997). [BI]
C. Borgs and J. Imbrie, A Unified Approach to Phase Diagrams in Field Theory and Statistical Mechanics, Commun. Math. Phys. 123, 305–328 (1989).
[BK]
C. Borgs and R. Koteck´ y, A Rigorous Theory of Finite-Size Scaling at First-Order Phase Transitions, Journ. Stat. Phys. 61, 79–119 (1990).
[BL]
J. Bricmont and J. L. Lebowitz, Wetting in Potts and Blume-Capel Models, Journ. Stat. Phys. 46, 1015–1029 (1987).
[BS]
J. Bricmont and J. Slawny, Phase transitions in systems with a finite number of dominant ground states, Journ. Stat. Phys. 54, 89–161 (1989).
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[Br]
D. C. Brydges, A short course on cluster expansions, Critical phenomena, Random Systems, Gauge Theories, Les Houches 1984, North-Holland, Amsterdam, 129–183 (1986).
[D 96]
R. L. Dobrushin, Estimates of Semi-invariants for the Ising Model at Low Temperatures, Amer. Math. Soc. Transl. (2) 177, 59–81 (1996).
[D 72]
R. L. Dobrushin, Gibbs State Describing Coexistence of Phases for a Three-Dimensional Ising Model, Teor. Ver. Pril. 17, 619–639 (1972), English transl. in Theor. Probability Appl. 17, 582–600 (1972).
[DKS]
R. L. Dobrushin, R. Koteck´ y and S. Shlosman, The Wulff construction: a global shape from local interactions AMS, Translations of Mathematical Monographs 104, Providence, Rhode Island (1992).
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A.C.D. van Enter, R. Fern´ andez and A. Sokal, Regularity properties and pathologies of position-space renormalization-group transformations: Scope and limitations of Gibbsian theory, J. Stat. Phys. 72, 879–1167 (1993).
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P. Holick´ y, R. Koteck´ y and M. Zahradn´ık, Rigid interfaces for lattice models at low temperatures, Journ. Stat. Phys. 50, 755–812 (1988).
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P. Holick´ y and M. Zahradn´ık, Stratified low-temperature phases of stratified spin models. A general Pirogov-Sinai approach, preprint.
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R. Koteck´ y and D. Preiss, Cluster Expansion for Abstract Polymer Models, Commun. Math. Phys. 103, 491–498 (1986).
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S. Pirogov and Ya. G. Sinai, Phase diagrams of classical lattice systems, Theor. Mat. Phys., 25, 1185–1192 (1975), 26, 39–49 (1976).
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E. Seiler, Gauge theories as a problem of constructive quantum field theory and statistical mechanics Lecture notes in physics, Vol. 159, Springer Verlag Berlin, Heidelberg, New York (1982).
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Ya. G. Sinai, Theory of phase transitions: Rigorous results, Pergamon Press, Oxford-New York-etc (1982).
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P. Holick´ y Department of Mathematical Analysis Charles University Sokolovsk´a 83, Praha 8 Czech Republic email:
[email protected]
Vol. 3, 2002
Phase Diagram of Horizontally Invariant Gibbs States for Lattice Models
Roman Koteck´ y Center for Theoretical Study Charles University Jilsk´ a 1, 110 00 Praha 1 Czech Republic email:
[email protected] Miloˇs Zahradn´ık Department of Mathematical Analysis Charles University Sokolovsk´a 83, Praha 8 Czech Republic email:
[email protected] Communicated by Jennifer Chayes submitted 12/12/00, accepted 16/11/01
To access this journal online: http://www.birkhauser.ch
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Annales Henri Poincar´ e
The Infrared Behaviour in Nelson’s Model of a Quantum Particle Coupled to a Massless Scalar Field J. L˝ orinczi, R. A. Minlos and H. Spohn Abstract. We prove that Nelson’s massless scalar field model is infrared divergent in three dimensions. In particular, the Nelson Hamiltonian has no ground state in Fock space and thus it is not unitarily equivalent with the Hamiltonian obtained from Euclidean quantization. In contrast, for dimensions higher than three the Nelson Hamiltonian has a unique ground state in Fock space and the two Hamiltonians are unitarily equivalent. We also show that the Euclidean Hamiltonian has no spectral gap.
1 Introduction A quantum particle coupled to a free Bose field can be described by the Hamiltonian 1 − ∆ + V (q) ⊗ 1 + 1 ⊗ ω(k)a∗ (k)a(k)dk (1.1) HgN = 2 Rd 1 1 + ˆ(k)eik·q ⊗ a(k) + ˆ(−k)e−ik·q ⊗ a∗ (k) dk. d/2 2ω(k) (2π) Rd In the first term we have the Schr¨ odinger operator Hp = −(1/2)∆ + V for the quantum particle. V is assumed to be a confining potential growing at infinity, hence Hp has a purely discrete spectrum. The second term is the energy of the Bose field with dispersion relation ω(k) ≥ 0. For a shorthand we set Hf = Rd ω(k)a∗ (k)a(k)dk. a∗ (k), a(k) are the Bose creation and annihilation operators satisfying the canonical commutation relations [a(k), a∗ (k )] = δ(k − k ), [a(k), a(k )] = 0 = [a∗ (k), a∗ (k )]. The last term in (1.1) describes a translation invariant interaction between the particle and the field, which is linear in the field operators. is the charge distribution and mollifies the coupling between particle and field. We assume to be smooth, radial and of rapid decay. ˆ denotes the Fourier transform of . We use the normalization Rd (x)dx = e ≥ 0; e then cali2 d ⊗ Fsym with Fsym brates the strength of the interaction. HgN acts on L 2(R , dq) 2 d the symmetric Fock space over L (R , dk). If Rd |ˆ (k)| (ω(k)−1 + ω(k)−2 )dk < ∞, then HgN is bounded from below and the interaction is infinitesimally Hf -bounded. Therefore HgN is a self-adjoint operator with domain D(Hp ⊗ 1) ∩ D(1 ⊗ Hf ). introduces an ultraviolet cut-off, which may be removed by letting (x) → eδ(x) or, equivalently, ˆ(k) → (2π)−d/2 e. In his famous paper [17] Nelson proved that in
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the case ω(k) = k 2 + m2b , mb > 0, at the expense of renormalizing the energy, HgN stays well-defined in the point charge limit. Here we will keep the ultraviolet cut-off and will study the infrared behaviour, which is of interest only for massless bosons, mb = 0, i.e. ω(k) = |k|. (1.2) We call HgN with dispersion relation (1.2) the massless Nelson model and denote the corresponding Hamiltonian by HN . For d = 1, 2 the Nelson Hamiltonian is not bounded from below, thus we require d ≥ 3. In fact, we will mostly concentrate on dimension d = 3 for which, as proved below, the massless Nelson model is infrared divergent. Before stating our project specifically let us explain in general terms the phenomenon of infrared divergence in the context of Hamiltonians of the type (1.1). It is convenient somewhat to generalize and start from HG = K ⊗ 1 + 1 ⊗ Hf + (v(k) ⊗ a∗ (k) + v(k)∗ ⊗ a(k)) dk. (1.3) Rd
K is a self-adjoint operator acting on the particle space K and it is assumed to have a purely discrete spectrum. k → v(k) is a function of operators acting on K such that HG is bounded from below. Clearly, HgN is the special case where K = L2 (Rd , dq), K = Hp , v(k) = (2π)−d/2 e−ik·q ˆ(−k)/(2ω(k))1/2 . The Hamiltonian HG acts on K ⊗ Fsym . We say that HG is infrared divergent if it has no ground state in K ⊗ Fsym . The physical ground state has then an infinite number of very low energy bosons forcing a representation of the canonical commutation relations different from the Fock representation. HG should be thought of only as a formal object and one has first to construct the Hilbert space containing the physical ground state, as well as the corresponding Hamiltonian. Ideally, one would like to decide whether the model is infrared divergent or not directly in terms of K and v in (1.3). At present, this issue is only partially understood. G´erard proved [9] that if, beside some conditions ensuring the selfadjointness of H, Rd
ω(k)−2 v(k)(K + 1)−1 2 dk < ∞
(1.4)
holds, then HG has a ground state in K ⊗ Fsym . In the particular case (1.1), if |ˆ (k)|2 ω(k) + ω(k)−2 + ω(k)−3 dk < ∞, (1.5) Rd
then HgN has a unique ground state in L2 (Rd , dx)⊗Fsym , see [24]. Thus (1.5) must be contravened for HgN to get infrared divergent. We note that for the massless Nelson model in three dimensions indeed |ˆ (k)|2 dk = ∞. (1.6) 3 R3 |k|
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On the other hand, by (1.5) the massless Nelson model is infrared convergent for d ≥ 4. The integrals in (1.4), resp. (1.5), becoming infinite is only a necessary condition for infrared divergence to occur. To conjecture sufficient conditions we have to rely more closely on model systems. The best understood model is the spin-boson Hamiltonian where K = C2 , K = σx , v(k) = λ(k)σz , and σx , σz are the Pauli matrices. Some evidence comes also from the generalized spin-boson models studied J in [1], where v(k) = j=1 Bj λj (k) with Bj = Bj∗ acting on K. These examples suggest that, in addition to (1.4) turning infinite, v(k) must have non-vanishing ground state expectation (Ψ, v(k)Ψ)K for small k, with Ψ ∈ K the ground state of K. If this happens, the model is divergent for all couplings e = 0. If, however, v(0) has zero ground state expectation, then cancellations may occur making the model stay infrared convergent in spite of the integral in (1.4) being infinite. This is the case for the spin-boson Hamiltonian with v(k) = αλ(k)(σz + m) and 2 |λ| /ω 2 dk = ∞. If m = 0, the model is infrared divergent for all α = 0. If d R m = 0, cancellations appear and for sufficiently small α the model is infrared convergent while for large α it becomes divergent [24]. The same phenomenon takes place in the dipole approximation of the massless Nelson model in d = 1 where K = L2 (R, dq), K = Hp , v(k) = q · ˆ(k) [19]. If V (q) = V (−q), the onset of infrared divergence depends on how strong the coupling is; otherwise the model is infrared divergent no matter how large the (non-zero) coupling is. For the three dimensional Nelson model’s Hamiltonian we have e (Ψ, v(k)Ψ) = + O(1), 2|k|
k → 0.
(1.7)
Thus we expect, and will actually prove, that HN is infrared divergent for every coupling e = 0. Fr¨ ohlich [8] considered the massless Nelson model for V = 0 at fixed total momentum p in which case the Hamiltonian acting on Fsym is H(p) =
1 2
p−
ka∗ (k)a(k)dk
2
R3
Hf +
1 (2π)3/2
+ R3
1 2|k|
(ˆ (k)a(k) + ˆ(−k)a∗ (k))dk.
(1.8)
With an infrared cut-off and for sufficiently small |p|, H(p) has a unique ground state in Fsym . The removing of the infrared cut-off yields a physical ground state lying outside Fock space, and a non-Fock representation of the canonical commutation relations. In [8] a C ∗ -algebraic approach was developed. In contrast, in our work we follow the route of constructive quantum field theory by introducing a suitable probability measure on the space of Brownian paths. In terms of this path measure we can define both the physical ground state and the Hamiltonian governing the excitations relative to the ground state. While the explicit construction
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will be done in full detail in what follows, we now explain some of the particular features of the path measure associated with HN . In the Euclidean picture we have the trajectories t → qt of the particle, resp. t → ξt (x), x ∈ Rd , of the field appearing as basic ingredients. For assigning them a statistical weight we define the formal Euclidean action for the particle by 1 2 q˙ + V (qt ) dt, Sp ({qt }) = (1.9) 2 t R and by
Sf ({ξt }) =
R
Rd
1 (∂t ξt (x))2 + (∇ξt (x))2 dxdt 2
(1.10)
for the field. From (1.1) we deduce that the particle-field interaction restricted to the time interval [−T, T ] is given by Sint,T ({qt , ξt }) =
T −T
Rd
(x − qt )ξ(t, x)dxdt =
T
−T
(ξt ∗ )(qt )dt
(1.11)
with ∗ denoting convolution. Thus, heuristically, the full Euclidean path measure is 1 dqt e−Sp ({qt }) dξt (x) e−Sf ({ξt }) e−Sint,T ({qt ,ξt }) . (1.12) Z d+1 t∈R
(t,x)∈R
The first factor above provides an a priori weight for the paths qt . The corresponding path measure is denoted by N 0 , see Section 2.1 below. The second factor provides an a priori weight for the paths ξt . Clearly, this is a Gaussian measure, denoted by G in Section 2.2 below. In the spirit of the Feynman-Kac formula we define a semigroup through the quadratic form (F¯ , e−tH G)(q, ξ) = EN 0 ×G [e−
Rt 0
(ξt ∗)(qs )ds
F¯ (q0 , ξ0 )G(qt , ξt )],
(1.13)
t ≥ 0, where F, G are suitable functions. e−tH is a contractive semigroup on L2 (Rd , dN0 ) ⊗ L2 (S , dG) (here dN0 and dG are the t = 0 distributions under N 0 and G, respectively), moreover it is unitarily equivalent with e−tHN , t ≥ 0, on L2 (Rd , dq) ⊗ Fsym . To construct the physical ground state and its associated Hilbert space we have to give sense to the path measure (1.12) in the infinite time limit T → ∞. With respect to the limit measure (whenever it exists), the process t → qt , ξt (x) is jointly a time-reversible stationary Markov process. Therefore this stochastic process has a canonically defined semigroup Tt , t ≥ 0, self-adjoint on the Hilbert space weighted by the t = 0 stationary measure. Thus Tt has a self-adjoint generator so that Tt = e−tHeuc , t ≥ 0, which by definition is the Euclidean Hamiltonian Heuc . Clearly, by construction, Heuc ≥ 0 and Heuc 1 = 0, i.e. Heuc has an eigenvalue at the bottom of its spectrum. If HN is infrared convergent, then Heuc and H are unitarily equivalent and one may work with either of them on an equal
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footing. However, in the infrared divergent case they are not unitarily equivalent. Since Heuc accommodates the ground state, it should be regarded as the proper Hamiltonian of the system. Once path measures are introduced, one might wonder how infrared divergence is reflected in their properties. As explained in more detail below, within the context of the Nelson model a sharp criterion can be found: If HN is infrared divergent, then the t = 0 distribution of the path measure for the interacting system is singular with respect to the t = 0 distribution of the path measure for the non-interacting system. A short outline of the paper is the following. In Section 2 we recall the basic processes leading up to Section 3 where the definition of the joint particle-field path measure appearing in (1.12) is given. Based on the cluster expansion in [13], under suitable assumptions on V and for sufficiently small e, we establish the existence of the infinite time path measure. Thereby Heuc will be well defined. In Section 4 we discuss infrared divergence in terms of the path measure for the interacting system and prove that the massless Nelson model in d = 3 is infrared divergent. In Section 5 we prove that for d ≥ 4 it is however infrared convergent. Finally, in Section 6 we show that Heuc has no spectral gap. We remark that the quantized Maxwell field is less infrared divergent since by minimal coupling the Bose field couples to the momentum and not to the position as in (1.1). As proved in [3, 11], the Pauli-Fierz operator has a ground state in Fock space also in three dimensions, which suggests that in the case of the Maxwell field Heuc and H are unitarily equivalent.
2 Path measures for the particle and the field 2.1
Euclidean quantization of the particle
On the Hilbert space L2 (Rd , dq) we define the particle Hamiltonian 1 Hp = − ∆ + V (q). 2
(2.1)
The external potential V : Rd → R is a multiplication operator. We consider two classes of potentials. (P1) V is bounded from below, continuous, and having the asymptotics V (q) = C|q|2α + o(|q|2α )
(2.2)
for large |q|, with some constant C > 0 and exponent α > 1. (P2) V is of Kato-class bounded from below as V (q) ≥ C|q|2α with some constant C > 0 and exponent α > 0.
(2.3)
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For details on the Kato class we refer to [5]. In either case above Hp has discrete spectrum and a unique, strictly positive ground state ψ0 lying at its bottom Ep , i.e. Hp ψ0 = Ep ψ0 . Associated with Hp we consider the stationary P (φ)1 -process qt with time t = 0 distribution (invariant measure) dN0 = ψ02 (q)dq,
(2.4)
as defined through the stochastic differential equation dqt = ∇ ln ψ0 (qt ) dt + dBt
(2.5)
where dBt denotes Brownian motion. The path measure of this process will be denoted by N 0 ; for more details see [5] and references therein. (Here and henceforth we use calligraphic letters for path measures and straight characters for time t = 0 distributions.) Lemma 2.1 N 0 -almost all paths Q = {qt } are continuous and for any T > 0 they satisfy (2.6) |qt − qs | ≤ C|t − s|1/8 ∀s, t ∈ [−T, T ], where C = C(Q, T ) > 0 is independent of s, t. Moreover, N 0 -almost surely |qt | ≤ C1 (ln(|t| + 1))1/(α+1) + C2 , ∀t ∈ R,
(2.7)
where C1 > 0 is a constant dependent only on the dimension of the space, C2 = C2 (Q), and α is the exponent appearing in (P1), resp. (P2).
Proof. See [5]. The process qt is time-reversible. Its stochastic semigroup is generated by 1 Hpeuc f = − ∆f − ∇ ln ψ0 · ∇f. 2
(2.8)
Hpeuc is self-adjoint on the Hilbert space L2 (Rd , dN0 ). The map f → f /ψ0
(2.9)
from L2 (Rd , dx) to L2 (Rd , dN0 ) is unitary and transforms Hp − Ep into Hpeuc .
2.2
Euclidean quantization of the free field
We introduce the massless free field in d + 1 dimensions, d ≥ 3. For any f ∈ S(Rd+1 ), i.e. the Schwartz space over Rd+1 , we denote ξ(f ) = f (t, x)ξt (x)dtdx. (2.10) Rd+1
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According to the action (1.10) the quantum field {ξ(f ), f ∈ S(Rd+1 )} is described by the generalized Gaussian measure G on S (Rd+1 ) with zero mean and covariance covG (ξ(f1 ), ξ(f2 ))
= EG [ξ(f1 )ξ(f2 )] = (−∂t2 − ∆)−1 f1 , f2 ˆ f1 (k0 , k)fˆ2∗ (k0 , k) = dkdk0 . k02 + k 2 R Rd
(2.11)
Here fˆ1 , fˆ2 ∈ S(Rd+1 ) denote the Fourier transforms of f1 , f2 ∈ S(Rd+1 ). The scalar product is taken in the space L2 (Rd+1 , dtdx). G is a probability measure on the space S (Rd+1 ) taken together with its associated Borel σ-field [7, 15]. The particle trajectories are a stochastic process qt taking values in Rd . Following this example, it will be convenient to regard the field configurations ξt as a stochastic process taking values in a suitable Hilbert space to be specified below. To carry out this construction first note that the covariance (2.11) can be extended to test functions of the form ft (x , t ) = ϕ(x )δ(t − t) ϕ
(2.12)
with ϕ ∈ S(Rd ). Thus the random process ϕ ξt (ϕ) = ξ(ft ), ϕ ∈ S(Rd ), t ∈ R,
is well-defined. From (2.11) and (2.13) it follows that ˆ ∗2 (k) −|k||s−t| ϕ ˆ 1 (k)ϕ 1 e dk. EG [ξs (ϕ1 )ξt (ϕ2 )] = 4 Rd |k|
(2.13)
(2.14)
Clearly, ξt (ϕ) is a stationary Gaussian process. Its invariant (t = 0) measure G defined on S (Rd ) is again Gaussian with mean 0 and covariance given by (2.14) at s = t. Moreover, the process ξt (ϕ) is time-reversible [10]. Lemma 2.2 {ξt , t ∈ R} is a Markov process.
Proof. See [7, 16].
Next we construct a Hilbert space BD such that ξt ∈ BD and t → ξt is normcontinuous with probability 1. Let D be the positive self-adjoint operator in the space L2 (Rd , dx) given by a jointly continuous symmetric kernel ˆ )dk ˆ D(k, k )ξ(k (2.15) (Dξ)(k) = Rd
with ker D = {0}. We introduce the real Hilbert space BD ⊂ S (Rd ) with norm ˆ 1 )ξˆ∗ (k2 )dk1 dk2 D(k1 , k2 )ξ(k (2.16) ξ 2BD = Rd ×Rd
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ˆ and ξˆ∗ (k) = ξ(−k). Clearly,
1 EG ξt 2BD = EG ξ0 2BD = 4 Hence, if
Rd
Rd
D(k, k) dk. |k|
D(k, k) dk < ∞, |k|
(2.17)
(2.18)
the measure G is concentrated on the space BD and the random process ξt takes its values from this set; for more details see [10]. By an easy calculation we obtain that 2 D(k, k) 1 4 −|k||t−s| (1 − e )dk + (2.19) EG ( ξt − ξs BD ) = 4 |k| Rd 1 D(k1 , k2 )2 (1 − e−|k1 ||s−t| )(1 − e−|k2 ||s−t| )dk1 dk2 . 2 Rd Rd |k1 ||k2 | Thus if in addition to (2.18) we assume that D(k, k)dk < ∞ and D(k1 , k2 )2 dk1 dk2 < ∞,
(2.20)
then, using that |k|−1 (1 − e−|k||s−t| ) ≤ |s − t|, we obtain
EG ξt − ξs 4 ≤ C |t − s|2 .
(2.21)
Rd
From (2.21) and Kolmogorov’s criterion of continuity of random processes (see [10, 21]) we conclude Lemma 2.3 Under conditions (2.18) and (2.20) almost all paths of the process ξt are continuous in the metric of BD . Moreover, for any finite interval T > 0 these paths have the property that ξs − ξt BD ≤ C |t − s|1/8 , ∀s, t ∈ [−T, T ]
(2.22)
where C = C({ξt }, T ) is independent of s, t. Lemma 2.3 states that with probability 1 the process ξt is realized on C(R, BD ). We denote the corresponding Gaussian path measure again by G. Since the staeuc tionary Markov process ξt is reversible, its stochastic semigroup e−tHf acting on L2 (BD , dG) is self-adjoint and Hfeuc is the generator of this semigroup. Lemma 2.4 There exists a unitary map Fsym → L2 (BD , dG) transforming the free field Fock space Hamiltonian Hf into Hfeuc . This map can be constructed by using the Itˆo-Wick transformation, see [22].
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For concreteness we give an explicit form of the kernel D(k1 , k2 ) which satisfies (2.18) and (2.20) above. Consider the operator ¯ = (−∆k + |k|2 )−(d+1) . D
(2.23)
The eigenvalues of this operator are λn =
1 , (n1 + ... + nd + d/2)d+1
(2.24)
¯ is with multi-index n = (n1 , ..., nd ) of integer entries ni ≥ 0, i = 1, ..., d. Hence D ¯ a nuclear and positive operator, and its kernel D(k1 , k2 ) is jointly continuous [4]. Let |k|1 denote the function |k|, if |k| < 1, |k|1 = 1, otherwise. We set ¯ 1 , k2 )|k2 |1 . D(k1 , k2 ) = |k1 |1 D(k
(2.26)
It is then easily seen that this kernel satisfies conditions (2.18) and (2.20). For later use we also note that the level sets KC = {ξ ∈ BD : (D−1/2 ξ, ξ)BD ≤ C}
(2.27)
are compact in BD .
3 Path measure for the interacting system In the previous section we defined the stochastic processes qt and ξt governing the particle, resp. the free field trajectories. The joint field-particle process on the space C(R, Rd × BD ) is then given by the path measure P 0 = N 0 × G. In order to implement the interaction as in (1.12) we take P 0 as reference process and modify it with a density given by the exponential of Sint , compare with (1.11),
T 1 d PT = exp − (ξt ∗ )(qt )dt d P 0 , (3.1) ZT −T where
ZT =
exp −
T
−T
(ξt ∗ )(qt )dt d P 0
(3.2)
is the normalizing partition function. Since G is a Gaussian measure, by an easy computation we obtain
T 1 T |ˆ (k)|2 −|k||t−s| ZT = exp cos(k · (qt − qs )) e dtds dk dN 0 . 8 −T −T |k| d R (3.3) This integral is well-defined for every T > 0.
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We are interested whether the sequence of measures PT has a limit as T → ∞. The following notion of convergence will be used. Let M be a metric space, and C(R, M ) the space of continuous paths {Xt } with values in M . For any interval IT = [−T, T ] ⊂ R let MT ⊂ C be a sub-σ field of the Borel σ-field C generated by the evaluations {Xt : t ∈ IT }, i.e. MT = σ(Xt , t ∈ IT ). We say that a sequence of probability measures {µn } on C(R, M ) converges locally weakly to the probability measure µ if for any 0 ≤ T < ∞ the restrictions µn |MT converge weakly to the measure µ|MT , see [10]. We have then the following result. Theorem 3.1 Let d ≥ 3, 0 < e ≤ e∗ , with sufficiently small e∗ > 0, and V satisfy (P1). Then there exists the local weak limit PT → P as T → ∞. P is a probability measure of a stationary reversible Markov process on the path space C(R, Rd ×BD ). Moreover, the stationary distribution P of P can be obtained as the weak limit P = limT →∞ PT , where PT is the distribution of {q0 , ξ0 } under PT . Proof. For any fixed path Q = {qt : t ∈ R} ∈ C(R, Rd ) consider the conditional distribution Q (3.4) PT = PT ( · |{qt } = Q) viewed as a probability measure on C(R, BD ). Then the distribution PT on C(R, Rd × BD ) can be written as Q (3.5) PT (S × A) = PT (A)dNT (Q), ∀T > 0 S
where A ⊂ C(R, BD ) and S ⊂ C(R, Rd ), and NT is the Q-marginal of PT . Since the process ξt is Gaussian and the coupling is linear in it, this marginal can be computed explicitly and is given by
T T 1 exp − W (qs − qt , s − t)dsdt dN 0 , (3.6) dNT = ZT −T −T where W (q, t) = −
1 8
Rd
|ˆ (k)|2 cos(k · q) e−|k||t| dk. |k|
(3.7)
NT is a Gibbs measure for the finite interval [−T, T ]. The reversible diffusion process N 0 is the reference measure, and W is the (effective) pair interaction potential. We denote the distribution of q0 under NT by NT . We use identity (3.5) to show the existence of the T → ∞ limit of PT . We will do this by examining separately the limits of PTQ and NT , starting with the former measure. PTQ is a Gaussian measure with covariance given by (2.11) and mean ϕ ϕ(x)gTt (x; Q)dx, ϕ ∈ S(Rd ), EP Q [ξt ( )] = (3.8) T
Rd
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where gTt (x; Q)
= =
T ˆ(k) 1 cos(k · (x − qτ )) e−|k||t−τ | dkdτ (3.9) − (2π)d/2 −T Rd 4|k| 1 eik·x gˆTt (k; Q)dk (2π)d/2 Rd
and gˆTt (k; Q)
ˆ(k) =− 4|k|
T
e−ik·qτ e−|k||t−τ | dτ.
(3.10)
−T
This function has the properties |ˆ gTt (k; Q)| ≤
|ˆ (k)| 2|k|2
(3.11)
|ˆ gTt (k; Q) − gˆTs (k; Q)| ≤
C|ˆ (k)| |t − s| 2|k|
(3.12)
for any s, t ∈ R, |s − t| < 1, and with some constant C > 0. Furthermore, we have that for any t and Q lim gˆTt (k; Q) = gˆt (k; Q) (3.13) T →∞
exists in the norm topology of BD , uniformly with respect to Q and t in every bounded interval. The T → ∞ limit of (3.9) is given explicitly by ∞ ˆ(k) 1 cos(k · (x − qτ )) e−|k||t−τ | dkdτ. g t (x; Q) = − (3.14) d/2 (2π) −∞ Rd 4|k| From (3.12) it follows that gˆTt (k; Q) as a function of t belongs to C(R, BD ) and sup ˆ gTt ( · ; Q) BD < ∞
(3.15)
t
uniformly in Q and T ≤ ∞. Moreover, as follows from (3.11), for arbitrary Q, T , and t, gˆTt (·, Q) ∈ KC for some suitable C > 0, with KC defined by (2.27). Thus by (3.12) and (3.15) the set of restrictions gTt (·, Q), |t| ≤ R to any interval [−R, R], ∀Q ∈ C(R, Rd ), ∀T ≤ ∞, is a compact set in C([−R, R], BD ). Hence it follows that the restrictions PTQ |MR form a weakly compact set of measures. The characteristic functional of the Gaussian measure PTQ is iξ(f ) χQ Q [e ] T (f ) = EPT
∞
∞
fˆ(s, k)fˆ∗ (t, k) −|k||t−s| e dt ds dk = exp −1/8 |k| −∞ −∞ Rd ∞ dt gˆTt (k; Q)fˆ(t, k)dk . × exp i −∞
Rd
×
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Q Q From (3.13) it follows that χQ T (f ) → χ (f ) as T → ∞, where χ (f ) is the Q characteristic functional of the Gaussian measure P with covariance (2.14) and mean ϕ ϕ(x)g t (x; Q)dx. EP Q [ft ] = (3.16) Rd
Thus from the convergence of characteristic functionals χQ T and the weak comQ pactness of the restrictions PT |MR the weak local convergence limT →∞ PTQ = P Q follows for any Q ∈ C(R, Rd ), see [10]. In order to complete our argument we need to control the NT → N limit. This problem has been investigated in [13] by a cluster expansion technique and some of the results relevant in our context are quoted in Theorem 3.2 below. In particular, the local weak limit NT → N exists, as stated under (1) in Theorem 3.2. This then completes the proof of the first claim. Since PT has the Markov property and is reversible, the limit distribution P has the same properties. Stationarity of P follows by point (2) of Theorem 3.2 and the obvious equality d P θs Q (ξt−s ) = d P Q (ξt )
(3.17)
θs being the time-shift acting on paths. The existence of the weak limit P = limT →∞ PT follows by the argument above by borrowing statement (3) of Theorem 3.2 below. This completes the proof of the theorem. We conclude this section by stating some facts on N and related measures. Results on the existence and uniqueness of Gibbs measures relative to Brownian motion are available for some time [19], and a more general framework has been considered more recently in [13]. In the following theorem we list some results taken from [13] to be used below adapted for the case of Nelson’s model. For details we refer to this paper, in particular for results on the uniqueness of N which we do not discuss here. Theorem 3.2 For d ≥ 3, sufficiently small e > 0, and potentials satisfying (P1) the following properties hold: 1. There exists the local weak limit NT → N as T → ∞. 2. The measure N is invariant with respect to time shifts and time reflections. 3. The t = 0 distributions NT , resp. N, of the measures NT , resp. of N , are mutually absolutely continuous with respect to N0 , i.e. there exists a constant c > 0 such that 1 dNT ≤ (q0 ) ≤ c, (3.18) c dN0 uniformly in T and q0 . Moreover, dN dNT (q0 ) → (q0 ) as T → ∞ dN0 dN0 pointwise.
(3.19)
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4. For N0 -almost all q the conditional distribution NT ( · |q0 = q) converges locally weakly to N ( · |q0 = q). 5. N -almost all paths Q = {qt } ∈ C(R, Rd ) have the property that for all t ∈ R |qt | ≤ C1 (ln(|t| + 1))1/(α+1) + C2 (Q)
(3.20)
where C1 > 0 is a constant, and C2 is a function of the path. 6. For any bounded functions F1 , F2 on Rd the following estimate holds on their covariance: EN [F1 (qs )F2 (qt )] − EN [F1 (qs )]EN [F2 (qt )] ≤ C
sup |F1 | sup |F2 | |s − t|γ + 1
(3.21)
with some γ > 0 and a suitable prefactor C > 0.
4 The infrared divergent case : d = 3 After having introduced the path measures for the interacting system, we turn now to compare the Fock quantization and the Euclidean quantization of the interacting system. The first one is constructed through the Feynman-Kac formula, EP 0 [F (q0 , ξ0 )G(qt , ξt )e−
Rt 0
(ξs ∗)(qs )ds
] = (F¯ , e−tH G)H0 , t ≥ 0,
(4.1)
which defines the semigroup exp(−tH) on H0 = L2 (Rd × BD , dP0 ). Its generator, the Hamiltonian H, is unitarily equivalent with HN . For the Euclidean quantization we start with the semigroup Tt associated with the time reversible Markov stochastic process {qt , ξt } distributed according to P as defined through EP [F (q0 , ξ0 )G(qt , ξt )] = (F¯ , Tt G)H ,
t ≥ 0,
(4.2)
on the Hilbert space H = L2 (Rd × BD , dP). Tt is a symmetric contractive semigroup. Hence there exists a self-adjoint semibounded operator Heuc generating it, i.e. Tt = exp(−tHeuc ), which by definition will be viewed as the Hamiltonian of the system obtained in Euclidean quantization. Note that the constant function 1 is the ground state of Heuc . Since Tt is positivity improving [3, 20], this ground state is unique in H. In this and the following section we investigate whether the Hamiltonians H and Heuc are unitarily equivalent. We use throughout this paper the following Definition 4.1 The Hamiltonian H is called infrared divergent if it has no ground state in H0 . Let {Pλ } be the family of spectral projections for the self-adjoint operator H bounded from below, and denote dσ1 (λ) = d(1, Pλ 1)H0 , the spectral measure for 1 ∈ H0 . Then E0 ≤ E0 := inf supp dσ1 (λ). (4.3)
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E0 is the bottom of the spectrum of H. We also define the approximate ground state e−T (H−E0 ) 1 (4.4) ΨT = −T (H−E ) . 0 1 e Theorem 4.1 Suppose V satisfies condition (P2). Then H is infrared divergent if and only if limT →∞ (1, ΨT ) = 0. This theorem results directly from Lemma 4.2 We have the following two cases: (i) If lim supT →∞ (1, ΨT ) > 0, then σ1 ({E0 }) > 0 and the limit lim ΨT := Ψ.
(4.5)
T →∞
exists. Moreover, Ψ > 0, E0 = E0 , and Ψ is the unique ground state of H at eigenvalue E0 . (ii) If lim supT →∞ (1, ΨT ) = 0, then σ1 ({E0 }) = 0. Moreover, E0 = E0 and H has no ground state in H0 . Proof. (i) By the spectral theorem we have ∞ −(λ−E )T 0 e dσ1 (λ) E0 (1, ΨT ) = 1/2 . ∞ −2(λ−E )T 0 e dσ (λ) 1 E
(4.6)
0
Take some a > E0 . Then ∞ a −(λ−E )T ∞ −(λ−E )T 0 0 dσ1 (λ) dσ1 (λ) + a e−(λ−E0 )T dσ1 (λ) E0 e E0 e . 1/2 ≤ 1/2 ∞ −2(λ−E )T a −2(λ−E )T 0 0 e dσ1 (λ) e dσ1 (λ) E E 0
0
(4.7) Schwarz’s inequality gives
a
E0
e
−(λ−E0 )T
dσ1 (λ) ≤
Hence (4.7) further becomes ∞ −(λ−E )T 0 dσ1 (λ) E0 e 1/2 ∞ −2(λ−E )T 0 e dσ (λ) 1 E
1/2
a E0
≤
e
−2(λ−E0 )T
dσ1 (λ)
1/2 (σ1 ([E0 , a]))
1/2
(σ1 ([E0 , a]))
=
1/2
(σ1 ([E0 , a]))
(4.8)
+ a E0
0
.
+ a E0
e−(a−E0 )T
1/2 e−2(λ−E0 )T dσ1 (λ) 1
1/2 . e−2(λ−a)T dσ1 (λ)
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Since E0 = inf supp dσ1 , the denominator diverges for T → ∞ and 0 < lim sup(1, ΨT ) ≤ (σ1 ([E0 , a])) T →∞
1/2
.
(4.9)
Taking a → E0 , this implies σ1 ({E0 }) > 0. Again, by the spectral theorem the limit (4.5) exists and Ψ ≥ 0, since e−tH is positivity preserving [3]. Ψ is an eigenfunction of H at eigenvalue E0 . Since e−tH is positivity improving [3], Ψ > 0 and therefore E0 = E0 by the uniqueness part of the Perron-Frobenius theorem [20]. (ii) By contraposition take σ1 ({E0 }) > 0. Then for T → ∞the numerator of (4.6) converges to σ1 ({E0 }) and the denominator converges to σ1 ({E0 }), which contradicts limT →∞ (1, ΨT ) = 0. We finally prove that E0 = E0 . We have that inf supp dσ1 (λ) = − lim
T →∞
1 log(1, e−T H 1) = E0 . T
(4.10)
By an easy computation, for the set of functions 0 < c− ≤ f ≤ c+ < ∞,
(4.11)
with some constants c− , c+ , we also have inf supp dσf (λ) = − lim
T →∞
1 log(f, e−T H f ) = E0 . T
(4.12)
The linear span of the set of functions defined by (4.11) is dense in H. Indeed, take f as specific combinations of indicator functions, f = 1A + >1Ac with arbitrary sets
∅, A ⊂ H and some > > 0; this set is dense in H. If E0 > E0 then P{λ<E0 } H = and the linear span of the set (4.11) would not be dense in H. Thus E0 = E0 . Note that as our proof shows, actually also the limit limT →∞ (1, ΨT ) exists. Theorem 4.3 For d = 3, (P1)-potentials and 0 < e ≤ e∗ H has no ground state in H0 (and hence HN has no ground state in L2 (Rd , dq) ⊗ Fsym ). In particular, H and Heuc are not unitarily equivalent. Note that the d = 3 case (1.6) holds. We expect that the restriction to weak coupling is of technical nature only. Lemma 4.4 For d = 3 and 0 < e ≤ e∗ P is singular with respect to P0 . Hence limT →∞ (1, ΨT ) = 0. Proof. We first show that P is singular with respect to P0 . Clearly, it suffices to find a function F such that it is almost surely strictly positive with respect to the free measure P0 and EP [F ] = 0. We choose s(x)ξ(x)dx (4.13) F (q, ξ) = exp R3
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with s satisfying s 2 :=
R3
Ann. Henri Poincar´e
|ˆ s(k)|2 dk < ∞. |k|
(4.14)
The function F is defined and positive for G-almost all ξ ∈ BD (see [12]). We have the identity (4.15) EPT [F ] = ENT EPQ [F ] T
PQ T
According to our standard notation is the time t = 0 distribution of PTQ . PQ T is absolutely continuous with respect to G for any Q, and its density is given by dPQ T = exp ξ(x)mT (x; Q)dx − 2 |ˆ gT0 (k; Q)|2 |k|dk (4.16) dG R3 R3 with mT (x; Q) =
1 (2π)3/2
R3
m ˆ T (k; Q)eik·x dk
and gT0 (k; Q) = − ˆ(k) m ˆ T (k; Q) = 4|k|ˆ Thus we obtain
EPQ [F ] = exp T
R3
T
e−ik·qτ e−|k||τ | dτ.
(4.17)
(4.18)
−T
sˆ(k)ˆ gT0 (k; Q)dk
1 2 + s . 8
(4.19)
Since ˆ(0) = e > 0, there exists some k ∗ > 0 such that ˆ(k) > 0 whenever 0 < |k| < k ∗ . We choose then s such that its Fourier transform is 1 ∞∗ e−|k||t| ζ dt, if |k| < k ∗ (k) ˆ T ln t(ln ln t) sˆ(k) = 0, otherwise with a suitable constant T ∗ such that ln T ∗ > 1, and exponent 0 < ζ < 1. First we have to show that (4.14) is satisfied, i.e. k∗ ∞ ∞ |ˆ s(k)|2 e−|k|(s+t) dsdt dk ≤ C kdk ζ |k| ∗ T ∗ ln t ln s(ln ln t ln ln s) R3 0 ∞ ∞ T 1 dtds = C 2 ln t ln s(ln ln t)ζ (ln ln s)ζ (t + s) ∗ ∗ T T t/s dt ds = 2C 2 t ln t(ln ln t)ζ s ln s(ln ln s)ζ (1 + t/s) ∗ T ≤s≤t<∞ ∞ ∞ ew dw dv = 2C w )2 ζ (v + w)(ln(v + w))ζ (1 + e v(ln v) ∗ 0 ∞ ∞u dv −w ≤ 2C e dw <∞ (4.21) 2 (ln v)2ζ v ∗ 0 u
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with u = ln t, v = ln s, w = u − v, u∗ = ln T ∗ above. Now we write ∞ dk T −i(k,qs ) −|k||s| e−|k||t| 1 0 sˆ(k)ˆ gT (k; Q)dk = − e e ds dt ζ 2 |k|
0 there is a constant C = C(ε) such that N C(R, R3 ) SC ≤ ε where SC = {Q ∈ C(R, R3 ) : |qt | ≤ C0 (ln(|t| + 1))1/(α+1) + C}.
(4.23)
Then for Q ∈ SC we estimate T ∞ dt ds ≥ 2 2 ) ln t(ln ln t)ζ (|q | + (|s| + |t|) s −T T∗ ∞ T dt ds 1/(α+1) + C)2 + (|s| + |t|)2 ) ln t(ln ln t)ζ −T T ∗ ((C0 (ln(|s| + 1)) The right hand side of this last expression goes to infinity as T → ∞. Hence, for any δ > 0 we can find some T¯ = T¯ (δ, C) such that EPQ [F ] ≤ δ, ∀Q ∈ SC , T > T¯.
(4.24)
T
Moreover, it follows from (4.21) and (4.23) that there is a constant B such that EPQ [F ] ≤ B for all Q and T . Hence, by combining (4.19), (4.23) and (4.24), we T have EPT [F ] = ENT EPQ [F ] → 0, as T → ∞. (4.25) T
Consider now the function FC¯ =
F (ξ),
if F (ξ) ≤ C¯
¯ C,
otherwise.
Clearly, for T → ∞, EPT [FC¯ ] → 0. On the other hand, EPT [FC¯ ] → EP [FC¯ ] with ¯ This implies then that EP [FC¯ ] = 0 for any C¯ and hence T → ∞, for any C. EP [F ] = 0.
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We show now the second assertion of the lemma. Indeed by using Schwarz’s inequality and F given by (4.13) above, we get (ΨT , 1) =
ΨT dP0 =
ΨT F 1/2 F −1/2 dP0 ≤
Ψ2T F dP0
1/2
F −1 dP0
1/2 .
(4.27) As by the first assertion of the lemma the first integral at the right hand side above converges to zero as T → ∞, and since the second integral is bounded, the claim follows. This completes the proof of the lemma. Proof of Theorem 4.3. straightforward by combining Lemma 4.4 with Theorem 4.1. In order to define H and Heuc we first had to construct the Gibbs measure N for the interacting system which required e to be sufficiently small. In addition, as a consequence of the construction, we used in (4.23) that the fluctuations of qt are logarithmic. If we set the more modest goal of only showing that H has no ground state in H0 , then here is an alternative proof avoiding all restriction on the coupling constant, and moreover valid for a more general class of potentials. Theorem 4.5 Suppose ≥ 0 and V is of class (P2). Then for d = 3 H has no ground state in H0 . Proof. For any finite T > 0 the measure PT is absolutely continuous with respect to P0 , and ZT− (q, ξ)ZT+ (q, ξ) dPT (q, ξ) = (4.28) dP0 ZT where RT ZT+ (q, ξ) = EP 0 [e− 0 (ξt ∗)(qt )dt |q0 = q, ξ0 = ξ]. (4.29) T 0 − − ZT is defined by changing 0 for −T in the exponent, and obviously ZT = ZT+ . By applying the Feynman-Kac formula we obtain ZT+ (q, ξ) = (e−T H 1)(q, ξ)
(4.30)
ZT = e−T H 1 2
(4.31)
and 2
0
in the norm of L (R × BD , dP ). Thus 2 −T H 1)(q, ξ) (e dPT (q, ξ) = . dP0 e−T H 1 2 d
(4.32)
Now by (4.28), (4.29) and (4.30) we have that (1, ΨT ) =
EP0 [ZT+ (q, ξ)] EP0 [ZT− (q, ξ)] √ √ = ZT ZT
(4.33)
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Furthermore
RT EP0 [ZT+ (q, ξ)] = EN0 EN+0 EG [e− 0 (ξt ∗)(qt )dt ]|q0 = q
287
(4.34)
where N+0 is the distribution of paths in the forward direction Q+ = {qt : t ≥ 0}. A similar expression holds for EP0 [ZT+ (q, ξ)] taken with respect to N−0 for paths in the backward direction Q− = {qt : t ≤ 0}. Using the Markov property of N 0 and Schwarz inequality, we obtain by (4.33) R0 RT 1 (1, ΨT )2 ≤ EN 0 EG [e− −T (ξt ∗)(qt )dt ] EG [e− 0 (ξt ∗)(qt )dt ] . (4.35) ZT Moreover, since RT RT RT EG e− 0 (ξt ∗)(qt )dt = e− 0 0 W (qt −qs ,t−s)dtds ,
(4.36)
and by a similar expression for the [−T, 0] interval, we get RT RT R0 R0 1 EN 0 e− 0 0 W (qt −qs ,t−s)dsdt e− −T −T W (qt −qs ,t−s)dsdt (1, ΨT )2 ≤ ZT RT RT RR 1 − W (qt −qs ,t−s)dsdt+ W (qt −qs ,t−s)dtds ∆T = EN 0 e −T −T ZT RR W (qt −qs ,t−s)dsdt = ENT e ∆T (4.37) Here ∆T = (−T, 0) × (0, T ) ∪ (0, T ) × (−T, 0). In position space the interaction potential reads π (x) (y) W (q, t) = − dx dy < 0. 2 R3 (q + x − y)2 + t2 3 R
(4.38)
To show that the left hand side of (4.33) converges to zero, we first restrict to the set (4.39) AT := {|qt | ≤ T λ , ∀|t| ≤ T } with some λ < 1. By using the estimate (qt − qs + x − y)2 + (t − s)2 ≤ 8T 2λ + 2(x − y)2 + (t − s)2 , we obtain
(4.40)
(x) (y) (qt − qs + x − y)2 + (t − s)2 ∆T R3 R3 T T 1 ≥2 dt ds dx dy (x) (y) 2λ 8T + 2(x − y)2 + (t + s)2 3 3 0 0 R R 8T 2λ + 2(x − y)2 + T 2 = dx dy (x) (y) log . (4.41) 8T 2λ + 2(x − y)2 R3 R3 dtds
dx
dy
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The right hand side in (4.41) goes to infinity as T → ∞, since λ < 1. Hence we have shown that lim ENT exp W (qt − qs , t − s)dsdt 1AT = 0. (4.42) T →∞
∆T
We still need to prove that the limit is zero also on the complement of AT . We use again that W (q, t) < 0 and estimate W (qt − qs , t − s)dsdt (1 − 1AT ) ≤ ENT [(1 − 1AT )] . ENT exp ∆T
(4.43) Writing out the expectation with respect to NT and by using Schwarz inequality, we obtain RT RT EN 0 e− −T −T W (qt −qs ,t−s)dtds (1 − 1AT ) RT RT (4.44) ENT [(1 − 1AT )] = EN 0 e− −T −T W (qt −qs ,t−s)dtds 1/2 RT RT EN 0 e−2 −T −T W (qt −qs ,t−s)dtds 1/2 RT RT ≤ (EN 0 [(1 − 1AT )]) − −T −T W (qt −qs ,t−s)dtds EN 0 e Since 0 < −2
T
−T
T
−T
W (qt − qs , |t − s|)dsdt < T
R3
|ˆ (k)|2 dk |k|2
(4.45)
and R3 dk|ˆ |2 /|k|2 < ∞, the first factor above is bounded by exp(aT ) with some a > 0. To complete the proof we have to show that this exponential growth is balanced by the second factor. The estimate
EN 0 [(1 − 1AT )] = N 0
{qt :
sup t∈[−T,T ]
|qt | ≥ T λ }
≤ f (T )e−cT
λ(2α+1)
(4.46)
is a slight modification of Lemma 5.2 in [13] based on Varadhan’s lemma taken together with the bound exp(−c|x|2α+1 ) on the decay of ψ0 . Here c > 0, f (T ) is a polynomially growing correction, and α > 0 is the exponent appearing in (2.3). By choosing 1/(2α + 1) < λ < 1, the right hand side of (4.46) goes to zero with T → ∞. We conclude this section by showing that even though for d = 3 there is no ground state in Fock space, the particle stays well localized in the physical ground state. (Further properties of the ground state are discussed in [6].) Denote ˜ T (q) = dNT (q) Λ dN0
(4.47)
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and set
˜ T (q)ψ0 (q)2 , ΛT (q) = Λ
289
(4.48)
i.e., the particle density with respect to Lebesgue measure in the approximate odinger operator (2.1)). ground state (ψ0 is the ground state of the Schr¨ Theorem 4.6 Suppose the potential V satisfies (P1). Then for sufficiently small couplings 0 < e ≤ e∗ there exist some constants c2 > c1 > 0 such that the limit Λ(q) = limT →∞ ΛT (q) exists and c1 ψ02 (q) ≤ Λ(q) ≤ c2 ψ02 (q)
(4.49)
for every q ∈ Rd . Proof. The pointwise existence of limT →∞ dNT /dN0 = dN/dN0 (see (3) in Theorem 3.2) and the equality ˜ ˜ T (q) = dN (q) Λ(q) := lim Λ (4.50) T →∞ dN0
combined with estimate (3.18) give the result.
5 The infrared convergent case : d ≥ 4 Theorem 5.1 Suppose d ≥ 4 and that V satisfies (P1). Then for sufficiently weak couplings 0 < e < e∗ H has a unique strictly positive ground state Ψ in H0 at eigenvalue E0 (and hence HN has a unique ground state in L2 (Rd , dq) ⊗ Fsym ). Moreover, P is absolutely continuous with respect to P0 and dP dPT = lim = lim Ψ2T = Ψ2 . T →∞ dP0 T →∞ dP0
(5.1)
Furthermore, there exists a unitary map Γ : H → H0 such that Γ−1 (H − E0 )Γ = Heuc . Note that for d ≥ 4 we have (1.5). Proof. Using the representation for dPQ T /dG given by (4.16) note that there is a constant C > 0 such that |m ˆ T (k; Q)| ≤ C
|ˆ (k)| |k|
(5.2)
uniformly in Q and T , and hence it follows that the functional Rd ξ(x)mT (x; Q)dx is well defined for G-almost all ξ ∈ BD . Similarly, we have |ˆ (k)|2 0 2 |ˆ gT (k; Q)| |k|dk ≤ C dk < ∞ (5.3) 3 Rd Rd |k|
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uniformly in Q and T . Also, the density dPQ 1 = exp ξ(x)m(x; Q)dx − |ˆ g 0 (k; Q)|2 |k|dk dG 4 Rd Rd exists, where
(5.4)
m(k; ˆ Q)eik·x dk,
m(x; Q) = Rd
m(k; ˆ Q) = |k|ˆ g 0 (k; Q). We write now the Radon-Nikodym derivative dPT /dP0 in another way, that is dPQ dPT dNT T (ξ)|q0 = q (q, ξ) = ENT (q) (5.5) dP0 dG dN0 for every 0 < T < ∞. By Fubini’s theorem for any q ∈ Rd and 0 < T ≤ ∞ there is a set Ω = Ω(q, T ) ⊂ BD , P0 (Ω) = 1, such that for ξ ∈ Ω the conditional 0 expectation ENT [dPQ T /dP (ξ)|q0 = q] exists and hence (5.5) is well defined. Next, from (5.5) we see 1/2 1/2 1/2 dPQ dPT dNT T (ξ)|q0 = q (q, ξ) (q) dGdN0 . = ENT EP0 dP0 dG dN0 (5.6) We apply Jensen’s inequality and use (4.16) to obtain 1/2
dPQ T (ξ)|q0 = q ≥ ENT dP0 1 exp ξ(x)ENT [mT (x; Q)|q0 = q] dx− 2 Rd
0 1 gT (k; Q)|2 | q0 = q |k|dk . ENT |ˆ 8 Rd By using once again Jensen’s inequality and the fact that the field has zero mean under G, we find 1/2 dPT EP0 (q, ξ) ≥ (5.7) dP0 1/2
0 dNT 1 2 ENT |ˆ (q) dN0 (q). gT (k; Q)| |q0 = q |k|dk exp − 8 R3 dN0 Estimate (3.11) gives then
0 |ˆ (k)|2 ENT |ˆ gT (k; Q)|2 |q0 = q ≤ C . |k|4
(5.8)
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Since by (3.18) dNT /dN0 is bounded from below uniformly in T and q, we arrive at 1/2 dPT |ˆ (k)|2 EP0 (q, ξ) dk ≥ C3 > 0. (5.9) ≥ C1 exp −C2 3 dP0 Rd |k| By (5.9) it then follows that the left hand side above cannot converge to zero and, as it follows from (4.32), (4.33) and Lemma 4.2 (i), there exists a function Ψ ∈ H0 , coinciding with the ground state of H, such that dPT dP = Ψ2 = . 0 T →∞ dP dP0 lim
(5.10)
We conclude by showing that there is a unitary map Γ : H → H0 , F → F Ψ transforming H − E0 into Heuc . Indeed, by using (4.1) and (4.2) we have (F¯ , Tt G)H
= = = = = =
EP [F (q0 , ξ0 )G(qt , ξt )] R T +t 1 lim EP 0 [e− −T (ξs ∗)(qs )ds F (q0 , ξ0 )G(qt , ξt )] T →∞ Z2T +t (e−T H 1, F e−tH Ge−T H 1)H0 lim T →∞ (1, e−(2T +t)H 1)H0 Ψ(q0 , ξ0 )F (q0 , ξ0 )[e−tH GΨ](q0 , ξ0 )dP0 e−tE0 −t(H−E0 ) (ΓF , e ΓG)H0 −1 −t(H−E0 ) ¯ (F , Γ e ΓG)H .
This completes the proof of the theorem. The proof above requires a uniform lower bound on dNT /dN0 , which is at present available only under the strong assumptions (P1) and 0 < e ≤ e∗ . In [24] a uniform lower bound on (ΨT , e−τ Hp ⊗ PΩ ΨT ) has been obtained by using, as here, a functional integral representation (PΩ is the projection onto 1 ∈ L2 (BD , dG)). In our context this technique proves the existence of the ground state of HN for d ≥ 4 with assumption (P2) on the potential and with no restriction on e.
6 Lack of spectral gap for Heuc ˆ ⊂ H of functions having the property that EP [f ] = 0. Consider the subspace H Obviously, this subspace is invariant under the semigroup generated by Heuc . We ˆ by H ˆ euc . denote the restriction of this operator to H ˆ euc = 0. Theorem 6.1 For any d ≥ 3 we have inf Spec H Proof. Consider the functional on H Fh (ξ) = ξ(x)h(x)dx := ξ(h), h ∈ S(Rd ). Rd
(6.1)
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ˆ and Clearly, ξ(h) − EP (ξ(h)) := F˜ (h) is an element of H, ∞ ˆ (e−Heuc t F˜h , F˜h )H = e−tλ dσh (λ) := Gh (t)
(6.2)
0
with dσh (λ) = d(F˜h , Pˆλ F˜h ) ˆ euc . Lemma 6.2 below implies that and {Pˆλ } the spectral family of H
(6.3)
inf supp dσh (λ) = 0,
and this on its turn implies inf sup Pˆλ = 0. Lemma 6.2 The following asymptotics holds for t → ∞, 2 ˆ 1 C|h(0)| + o d−1 Gh (t) = d−1 t +1 t +1
(6.4)
with some constant C > 0. Proof. ˆ (e−Heuc t F˜h , F˜h )H
= covP (ξt (h), ξ0 (h)) = EN [EP Q [(ξt (h) − EP Q [ξ0 (h)])(ξ0 (h) − EP Q [ξ0 (h)])]] + EN [(EP Q [ξt (h)] − EP [ξt (h)])(EPQ [ξ0 (h)] − EP [ξ0 (h)])]
Here we used the identity Eµ [(F − Eµ [F ])(G − Eµ [G])] = +
Eµ [(F − Eµ [F |F])(G − Eµ [G|F])] (6.5) Eµ [(Eµ [F |F] − Eµ [F ])(Eµ [G|F] − Eµ [G])]
valid for an arbitrary probability measure µ, where Eµ ( · |F) is conditional average with respect to some σ-field F , and F and G are given random variables. Since for any fixed trajectory Q the conditional covariance of ξt (x) is the same as before, we get e−|k||t| ˆ EP Q [(ξt (h) − EP Q [ξt (h)])(ξ0 (h) − EP Q [ξ0 (h)])] = |h(k)|2 dk. (6.6) |k| Rd ˆ Then by choosing a sufficiently rapidly decreasing smooth function h with h(0) = h(x)dx = 0, we see that the integral (6.6) behaves asymptotically like 2 d−1 ˆ /t + o(1/td−1 ). const|h(0)| We turn next to the second term in (6.5). We have 1 ∞ ˆ(k) −ik·qτ −|k||t−τ | ˆ e e dτ dk (6.7) h(k) EP Q [ξt (h)] = − 4 −∞ Rd |k| and by stationarity EP [ξt (h)] = −
1 4
dk Rd
∞
ˆ(k) −|k||t−τ | −ik·q ˆ h(k)E e ] dτ. N [e |k| −∞
(6.8)
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Hence EP Q [ξt (h)] − EP [ξt (h)] = 1 ∞ ˆ(k) −|k||t−τ | −ik·qτ ˆ e dτ (e − EN [e−ik·q ])dk. − h(k) 4 −∞ |k| d R Furthermore (EP Q [ξt (h)] − EP [ξt (h)]) (EP Q [ξ0 (h)] − EP [ξ0 (h)]) = ∞ ∞ (k2 ) −|k||t−τ1 | −|k2 ||τ2 | 1 ˆ 2 ) ˆ(k1 )ˆ ˆ 1 )h(k e dτ1 dτ2 dk1 dk2 h(k e × 16 −∞ −∞ |k1 ||k2 | Rd Rd × e−ik1 ·qτ1 − EN [e−ik1 ·q ] e−ik2 ·qτ2 − EN [e−ik2 ·q ] Making use of (3.21) on the decay of correlations for N we get that (6.9) EN [(EP Q [ξt (h)] − EP [ξt (h)])(EP Q [ξ0 (h)] − EP [ξ0 (h)])] ∞ ∞ −|k1 ||t−τ1 |−|k2 ||τ2 | (k2 ) e ˆ 2 ) ˆ(k1 )ˆ ˆ 1 )h(k ≤ C¯ dk1 dk2 dτ1 dτ2 h(k |k ||k |τ1 − τ2 |γ + 1 d d 1 2| −∞ −∞ R R ∞ ∞ dτ1 dτ2 ˆ 2 ¯ ≤ C(sup |h|) d−1 + 1)(|τ |d−1 + 1)(|τ − τ |γ + 1) (|t − τ | 1 2 1 2 −∞ −∞ with some C¯ > 0. Denote L1 (t) =
1 |t|d−1
+1
, L2 (t) =
1 +1
|t|γ
(6.10)
The right hand side of (6.9) above is the double convolution L1 ∗ L1 ∗ L2 . Then we get by Fourier transformation ˆ 2 (y), ˆ 1 (y)2 L (L1 ∗ L1 ∗ L2 )(y) = L
y ∈ R.
(6.11)
By general results on the asymptotics of Fourier transforms [2] we have for small y ˆ 1 (y) ∼ |y|d−2 L ˆ 2 (y) ∼ |y|γ−1 L ˆ 2 (y) ∼ |y|2d+γ−5 . The asymptotics of the integral at the right hand ˆ 1 (y)2 L Thus L side of (6.9) becomes then 1/t2d+γ−4. Note that for d ≥ 3 and γ > 0, 2d + γ − 4 > d − 1, and thus the second term in (6.5) is o(1/td−1 ). This completes the proof. Acknowledgments. J.L. is supported by Deutsche Forschungsgemeinschaft within Schwerpunktprogramm “Interagierende stochastische Systeme von hoher Komplexit¨ at”. R.A.M. thanks Zentrum Mathematik of Technische Universit¨ at M¨ unchen for warm hospitality and financial support. He also thanks the Russian Fundamental Research Foundation (grants 99-01-00284 and 00-01-00271) and CRDF (grant NRM 1-2085) for financial support.
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References [1] A. Arai, M. Hirokawa and F. Hiroshima, On the absence of eigenvectors of Hamiltonians in a class of massless quantum field models without infrared cutoff, J. Funct. Anal. 168, 470-497 (1999). [2] V.I. Arnold, S. Gussein-Zahde and A. Varchenko, Singularities of Differentiable Maps, vol. 2, Nauka, Moscow (1984) (in Russian). [3] V. Bach, J. Fr¨ ohlich and M. Sigal, Quantum electrodynamics of confined nonrelativistic particles, Adv. Math. 137, 299-395 (1998). [4] F. Berezin and M. Shubin, Schr¨ odinger Equation, Nauka, Moscow, 1983 (in Russian). [5] V. Betz and J. L˝ orinczi, A Gibbsian description of P (φ)1 -processes, submitted for publication. [6] V. Betz, F. Hiroshima, J. L˝ orinczi, R.A. Minlos and H. Spohn, Ground state properties of a particle coupled to a scalar quantum field, to appear Rev. Math. Phys. (2001). [7] R.L. Dobrushin and R.A. Minlos, An investigation of the properties of generalized Gaussian random fields, Selecta Math. Sov. 1 (3) (1984). [8] J. Fr¨ ohlich, On the infrared problem in a model of scalar electrons and massless scalar bosons, Ann. Inst. Henri Poincar´e 19, 1–103 (1973). [9] C. G´erard, On the existence of ground states for massless Pauli-Fierz Hamiltonians, Ann. H. Poincar´e 1, 443–459 (2000). [10] I. Gikhman and A. Skorokhod, The Theory of Random Processes, Springer, vol 1. [11] M. Griesemer, E.H. Lieb and M. Loss, Ground states in non-relativistic quantum electrodynamics, preprint (2000). [12] M.A. Lifshitz, Gaussian random processes, Publ. TBiMS, Kiev (1995) (in Russian; English translation forthcoming at Kluwer Academic Publishers). [13] J. L˝ orinczi and R.A. Minlos, Gibbs measures for Brownian paths under the effect of an external and a small pair potential, J. Stat. Phys. 105, 607–649 (2001). [14] V.A. Malyshev and R.A. Minlos, Gibbs Random Fields, Kluwer Academic Publishers, 1991. [15] R.A. Minlos, The generalized random processes and their continuation to a measure, Proc. Moscow Math. Soc. 8, 497–518 (1959).
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[16] G. Molchan, The characterization of Gaussian fields with Markov property, Doklady Acad. Sci. 197 (4), 1971 (in Russian). [17] E. Nelson, Schr¨ odinger particles interacting with a quantized scalar field, in: Proc. Conf. on the Theory and Applications of Analysis in Function Space, W.T. Martin and I. Segal, eds., MIT, 1964, p. 87. [18] E. Nelson, Interaction of nonrelativistic particles with a quantized scalar field, J. Math. Phys. 5, 1190–1197 (1964). [19] H. Osada and H. Spohn, Gibbs measures relative to Brownian motion, Ann. Probab. 27, 1183–1207 (1999). [20] M. Reed and B. Simon, Methods of Modern Mathematical Physics, vol. IV, Academic Press, 1978. [21] A. Shiryaev, Random Processes, Publ. Moscow State Univ., 1972 (in Russian). [22] B. Simon, The P (φ)2 Euclidean Quantum Field Theory, Princeton University Press, 1974. [23] H. Spohn, Ground state(s) of the spin-boson Hamiltonian, Commun. Math. Phys. 123, 277–304 (1989). [24] H. Spohn, Ground state of a quantum particle coupled to a scalar Bose field, Lett. Math. Phys. 44, 9–16 (1998). J´ ozsef L˝orinczi and Herbert Spohn Zentrum Mathematik Technische Universit¨ at M¨ unchen 80290 M¨ unchen Germany email: [email protected] email: [email protected] Robert A. Minlos Dobrushin Mathematics Laboratory Institute for Information Transmission Problems Bolshoy Karetny per. 19 101447 Moscow Russia email: [email protected] Communicated by Vincent Rivasseau submitted 29/01/01, accepted 13/11/01
Ann. Henri Poincar´e 3 (2002) 297 – 330 c Birkh¨ auser Verlag, Basel, 2002 1424-0637/02/020297-34 $ 1.50+0.20/0
Annales Henri Poincar´ e
Generalized Dirac Equations in Implicit Form for Constrained Mechanical Systems F. Barone and R. Grassini
Abstract. The dynamics of constrained mechanical systems is framed in the differential-geometric context of Dirac manifolds and Dirac equations in implicit form. The aim is to show that Lagrangian and Hamiltonian dynamics admit a common mathematical structure taking its place in the above context.
1 Introduction (i) This paper is essentially an amplification of our previous note [3], concerning the differential-geometric formulation of Lagrangian dynamics for (autonomous) constrained mechanical systems. We broaden the scope of the above note by setting it in the context of Dirac manifolds and Dirac equations [7, 29], with the goal of deepening our understanding of both the Lagrangian and the Hamiltonian side of dynamics and their equivalence [4]. Mechanical systems described by (nonregular) Lagrangians and (nonlinear) constraints are widely treated in literature (see the nonexhaustive list of references below) and offer an amazing variety of philosophies and techniques: coordinate or geometric or mixed formalisms; dynamic laws conceived as algebraic equations (whose unknowns are vector fields) or differential equations (whose unknowns are parametrized curves) or something in the middle; Lagrangian or Hamiltonian approaches, scarcely ever lying on some basic mechanical principle. Our approach exhibits the following features. The treatment, aiming at a global and intrinsic formulation of dynamics, is coherently coordinate-free in all of its deductive developments (the coordinate expression of the final results is added for the sake of completeness). The laws of dynamics are conceived as differential equations in implicit form on manifolds, i.e. submanifolds of tangent bundles [23, 28], which are the direct geometric interpretation of the ordinary differential equations of elementary Analysis. All of our equations arise from a basic principle, which – generalizing an idea by Tulczyjew [28] – embodies a geometrized version of d’Alembert’s principle (core of Lagrangian dynamics) and Legendre transformation (springboard to Hamiltonian dynamics).
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Transition from the Lagrangian to the Hamiltonian side of dynamics proceeds along a path which faithfully recovers – in geometric terms – the line of thought of Dirac’s approach to Hamiltonian dynamics [13, 14]. (ii) The paper is organized as follows. In section 2 we give a list of the main geometric tools used in the sequel – notations, smooth bundles, tangent derivations [28], geometry of tangent bundles [8, 11, 15, 16] and geometry of iterated bundles [28]. In section 3, we briefly recall our approach [3] to a general kind of mechanical systems, described by a smooth manifold Q and a geometric structure Θ, the latter being the generator of a differential equation T (on cotangent bundle T ∗ Q) equivalent –modulo a suitable Legendre transformation– to a differential equation E (on tangent bundle T Q) which provides the above mentioned version of d’Alembert’s principle, i.e. the ‘Lagrangian’ side of dynamics. The main point here is the fact that E will prove to have the structure of an implicit equation extracted – through a typical second-order condition – from a generalized Dirac equation D on T Q (belonging to a kind of equations which generalize the Dirac equations – or Hamilton equations on Dirac manifolds – discussed in [29]). In section 4 we give a further insight into the dynamics of (Q, Θ) on T ∗ Q, by reconstructing – in geometric terms – Dirac’s approach to Hamiltonian dynamics [13, 14]. We start indeed from the Lagrangian side E of dynamics and operate on it (under suitable hypotheses) via Legendre transformation, obtaining the ‘Hamiltonian’ side of dynamics in the form of a differential equation H – fully equivalent to T – expressed in terms of the canonical symplectic structure on T ∗ Q. The main point here is again the fact that H will prove to have the structure of an implicit equation extracted – through a new kind of second-order condition – from a generalized Dirac equation D1 on T ∗ Q. In section 5 we specialize (Q, Θ) – Θ encompassing classical ingredients such as a (nonregular) Lagrangian function L and a (nonpotential) force field F , as well as (nonlinear) constraints and constraint reactions – so as to recover, from equation H, the canonical formalism ( in an expanded version) of Hamilton-Dirac dynamics [13, 14, 24, 25, 28]. In section 6 we present classical and relativistic particle dynamics as concrete examples of our general theory. In particular, the singularity of a relativistic Lagrangian will be shown to give rise to typical cases of Dirac equations which fail – on their own – to be equivalent to the laws of dynamics. In section 7 we conclude with some brief remarks, where the focal points of the work are underlined and looked at in perspective for further researches.
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2 Preliminaries Here is a list of the main geometric tools used in this paper. (i) For any smooth manifold M , we shall adopt the following notations. T M and T ∗ M are the tangent and cotangent bundles of M , whose bundle projections onto M are denoted τM : T M → M and πM : T ∗ M → M , respectively. χ(M ) is the Lie algebra of the vector fields on M . ΛM is the exterior graded algebra of M (in particular, Λ0 M ⊂ ΛM is the ring of the real-valued smooth functions on M ). For any smooth mapping ψ : N → M between manifolds N and M , T ψ : T N → T M is the vector bundle morphism called tangent mapping of ψ (whose restriction to any fibre Tp N is denoted Tp ψ) and ψ ∗ : ΛM → ΛN is the pull-back of the exterior algebra of M into that of N by ψ. ϑM ∈ Λ1 T ∗ M is the Liouville 1-form on T ∗ M , defined by ϑM : T ∗ M → T ∗ T ∗ M : ξ → ϑM (ξ) := ξ ◦ Tξ πM . From ϑM one draws – through the exterior derivative d of ΛT ∗M – the canonical symplectic 2-form ωM := −dϑM ∈ Λ2 T ∗ M . (ii) If ψ : N → M is a smooth bundle over M (i.e. a submersion onto M ), then V ψ := x ∈ T N | T ψ(x) = 0 is a vector bundle over N , and so is its annihilator V o ψ := ξ ∈ T ∗ N | ξ | x = 0 , ∀x ∈ V ψ s.t. τN (x) = πN (ξ) (where | denotes the usual pairing between forms and vectors). It is easy to check that there exists a unique bundle morphism ψ : V o ψ → T ∗ M satisfying – for all p ∈ N and ξ ∈ Vpo ψ – ξ = ψ (ξ) ◦ Tp ψ ∗ and that each restricion ψ |V o ψ : Vpo ψ → Tψ(p) M is a linear isomorphism. p
The critical subset of a 1-form ϑ ∈ Λ1 N relative to ψ, is defined by Σ := p ∈ N | ϑ(p) ∈ Vpo ψ .
Through ψ , ϑ|Σ trasforms into ϑ¯ := ψ ◦ ϑ|Σ : Σ → T ∗ M
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which is a section of πM along ψ|Σ , i.e. πM ◦ ϑ¯ = ψ|Σ , and satisfies – at each p∈Σ– ¯ ◦ Tp ψ . ϑ(p) = ϑ(p) (iii) The basic tangent derivations of ΛM (see [28]) are the following. iT : ΛM → ΛT M is the tangent-derivation (of degree -1) which vanishes on Λo M and acts on Λ1 M by θ ∈ Λ1 M → iT θ ∈ Λo T M with iT θ : T M → R : x → (iT θ)(x) := ix θ where ix θ := θ τM (x) | x . iT then acts on Λ2 M by Ω ∈ Λ2 M → iT Ω ∈ Λ1 T M with iT Ω : T M → T ∗ T M : x → (iT Ω)(x) := ix Ω ◦ Tx τM where ix Ω = Ω τM (x) | x, · . Remark that the critical subset of iT Ω ∈ Λ1 T M relative to τM is the whole T M and, at each x ∈ T M , (2.1) τM ◦ iT Ω (x) = ix Ω . From iT one also draws – through the exterior derivatives d of ΛM and ΛT M – the tangent derivation (of degree zero) given by dT := iT d + diT . (iv) In the geometry of the tangent bundle associated with a smooth manifold Q (see [8, 11, 15, 16]), the key role is played by the vertical lifting ν : T Q ×Q T Q → T T Q, whose restriction νv to the fibre {v} × Tq Q ≡ Tq Q over any v ∈ T Q (with q := τQ (v)) is the canonical isomorphism νv : Tq Q → Vv τQ of Tq Q = τQ−1 (q) onto its own tangent space Tv (Tq Q) = Ker Tv τQ = Vv τQ . On the one hand, ν transforms the diagonal mapping diagT Q : T Q → T Q ×Q T Q : v → (v, v) into the dilation vector field ∆ := ν ◦ diagT Q : T Q → T T Q : v → νv (v) . On the other hand, ν transforms the couple of fibrations (τT Q , T τQ ) : T T Q → T Q ×Q T Q : x → τT Q (x), T τQ (x) into the vertical vector bundle endomorphism S := ν ◦ (τT Q , T τQ ) : T T Q → T T Q : x → ντT Q (x) T τQ (x) . The vertical tangent bundle V τQ will then be described as the set of all vectors x ∈ T T Q satisfying S(x) = 0 .
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Similarly, the semibasic cotangent bundle V o τQ can be described as the set of all covectors ξ ∈ T ∗ T Q satisfying iS ξ := ξ ◦ S|Tv T Q = 0 with v := πT Q (ξ) . Remark that the adjoint vector bundle endomorphism iS : T ∗ T Q → T ∗ T Q also defines a derivation (of zero degree) of ΛT Q vanishing on Λo T Q, from which one draws – through the exterior derivative d of ΛT Q – the derivation (of degree 1) of ΛT Q given by dS := iS d − diS . Finally, the second tangent bundle T 2 Q, defined as the set of all vectors x ∈ T T Qsatisfying T τQ (x) = τT Q (x) , can as well be characterized by condition S(x) = ∆ τT Q (x) . The above affine bundle over T Q arises from the following considerations. For any smooth curve γ : I → Q, defined on an open interval I of the real d d : I → T Q (where du ∈ χ(R) line R, the tangent lifting is given by γ˙ := T γ ◦ du I is the canonical vector field associated with u := id ). As τ ◦ γ ˙ = γ, the second Q R d tangent lifting γ¨ := T γ˙ ◦ du I : I → T T Q satisfies T τQ ◦ γ¨ = γ˙ = τT Q ◦ γ¨ , i.e. Im γ¨ ⊂ T 2 Q . So T 2 Q is where all of the γ¨ ’s live. (v) In the geometry of the iterated bundles associated with Q (see [28]), the key role is played by the following two canonical diffeomorphisms. First, we recall that there exists a unique diffeomorphism α : T T ∗Q → T ∗ T Q satisfying πT Q ◦ α = T πQ and dT ϑQ = α∗ ϑT Q . Remark that, for any v ∈ T Q and θv ∈ Tv∗ T Q , τT ∗ Q α−1 (θv ) = θv ◦ νv .
(2.2)
In particular, for any 0-form L ∈ Λo M on an open submanifold M ⊂ T Q , F L := τT ∗ Q ◦ α−1 ◦ dL : M → T ∗ Q is the fibre derivative of L . Next, we recall that, from the canonical symplectic 2-form ωQ ∈ Λ2 T ∗ Q, one draws the diffeomorphism β : T T ∗Q → T ∗ T ∗ Q : z → iz ωQ which satisfies πT ∗ Q ◦ β = τT ∗ Q . Remark that, for any p ∈ T ∗ Q and hp ∈ Tp∗ T ∗ Q , T πQ β −1 (hp ) = hp ◦ νp
(2.3)
(where νp : Tq∗ Q → Vp πQ is the canonical isomorphism of Tq∗ Q onto its own tangent space at p ∈ Tq∗ Q). In particular, for any 0-form H ∈ Λo W on an open submanifold W ⊂ T ∗ Q , F H := T πQ ◦ β −1 ◦ dH : W → T Q is the fibre derivative of H.
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3 Generalized Dirac equation on tangent bundle In this section, we shall be concerned with an ‘abstract’ kind of constrained mechanical system, described by a smooth manifold Q (the configuration space of the system) and a suitable 1-form Θ (which will later be specialized so as to embody classical mechanical ingredients). The dynamics of (Q, Θ), already studied in [3] on tangent bundle T Q, will briefly be recalled and, by doing so, a key role will be shown to be played by a type of implicit differential equation generalizing the Dirac equations introduced in [29]. (i) Let us define the system (Q, Θ). Q is a smooth manifold. Θ := ρ∗ θ + Φ is a 1-form on the total space Y of a smooth bundle ρ : Y → M over an open submanifold M of T Q. The terms appearing in Θ are as follows. θ is a 1-form on M . Φ is a 1-form on Y s.t. Σ (critical subset of Φ relative to ρ) is nonempty and := ρ ◦ Φ| (section of πM along ρ| ) is semibasic. Φ Σ Σ The above terms will furtherly be specialized in section 5. Let us now define the dynamics of (Q, Θ). As the critical subset of Θ relative to ρ does not differ from Σ, one can consider the composite : Σ → T ∗M ⊂ T ∗T Q := ρ ◦ Θ| = θ ◦ ρ| + Φ Θ Σ Σ which Tulczyjew’s canonical diffeomorphism α−1 : T ∗ T Q → T T ∗Q transforms into : Σ → T T ∗Q . K := α−1 ◦ Θ The announced dynamics is the implicit differential equation on T ∗ Q given by
T := Im K ⊂ T T ∗Q .
The integral curves k’s of T – living in momentum phase space T ∗ Q and characterized by Im k˙ ⊂ T – will be called the momentum trajectories of (Q, Θ). The base integral curves γ’s of T – living in configuration space Q and characterized by γ := πQ ◦ k for some integral curve k – will be called the motions of (Q, Θ). A remarkable feature of the above dynamics is its second-order-like behaviour, consisting in the fact that the correspondence k → γ := πQ ◦ k between integral curves and base integral curves is invertible, the inverse being given by the Legendre lifting γ → k := L ◦ γ˙ , where L := τT ∗ Q ◦ α−1 ◦ θ : M → T ∗ Q (satisfying πQ ◦ L = τQ |M ) is the Legendre morphism defined by θ.
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As a consequence, L turns out to establish a one-to-one correspondence c → k := L ◦ c between the integral curves of E and the integral curves of T , E being the implicit differential equation on M given by E := x ∈ T M | ∃ y ∈ Σ : T τQ (x) = ρ(y) , τM (x) = ρ(y) , x = η τM (x) − Φ(y) with and
η := d i∆ θ − θ ∈ Λ1 M
: T M → T ∗ M : x → x := ix ω musical morphism of ω := −d iS θ ∈ Λ2 M
(E and T clearly admit the same base integral curves). The problem of integrating T is then reduced, through L , to the problem of integrating E . By the very definition, E satisfies both the property E ⊂D where D = x ∈ T M | ∃ y ∈ Σ : τM (x) = ρ(y) , x = η τM (x) − Φ(y)
(3.1)
is a kind of equation we shall examine in (ii), and – owing to the additional condition T τQ (x) = ρ(y) – the property E ⊂ T 2Q which makes E exhibit the typical second-order behaviour, consisting in the fact that the correspondence c → γ := τQ ◦ c between integral curves and base integral curves is invertible, the inverse being given by the tangent lifting γ → c := γ˙ . E will then be extracted from D by adding the second-order condition T τQ (x) = ρ(y) to those characterizing D , thus obtaining E = D ∩ T 2 Q. As a consequence, D is not generally equivalent to E, for it may admit more integral curves than E does (not all of the integral curves of D will then correspond to motions of (Q, Θ)). The problem of integrating E will, in principle, be solved by determining the integral curves c’s of D, characterized by c=ρ◦ς
,
◦ς c˙ = η ◦ c − Φ
for some curve ς in Σ, and then sorting out those which satisfy the second-order condition T τQ ◦ c˙ = ρ ◦ ς (i.e., putting γ := τQ ◦ c , γ˙ = c).
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(ii) The geometric structure underlying equation D will now be examined. First recall that a Dirac manifold is a couple (M, Ω), where M is a smooth manifold and Ω is a (generalized) Dirac structure, i.e. Ω ⊂ T M ×M T ∗ M (see [7]). Then remark that, by means of Ω, any subset S ⊂ T ∗ M naturally determines the implicit equation {x ∈ T M | ∃ χ ∈ S : (x, χ) ∈ Ω} ⊂ T M . In particular, if Ψ is a 1-form on the total space of a bundle ρ over M (with ⊂ T ∗M := ρ ◦ Ψ| : Σ → T ∗ M , then Im Ψ nonempty critical subset Σ) and Ψ Σ yields x ∈ T M | ∃ y ∈ Σ : x, Ψ(y) ∈ Ω ⊂ T M , which will be called the generalized Dirac equation generated by Ψ. When Ω := {(x, χ) ∈ T M ×M T ∗ M | x = χ} is the Dirac structure defined by the musical morphism of a presymplectic (i.e. closed) 2-form ω on M , the −1 ⊂ T M. generalized Dirac equation generated by Ψ is the inverse image (Im Ψ) Now turn back to the presymplectic manifold (M, ω) introduced in (i) and put Ψ := ρ∗ η − Φ ∈ Λ1 Y (the critical subset of Ψ relative to ρ is the same as that of Φ, i.e. Σ = ∅). , it follows that – for any x ∈ T M – one has x ∈ = η ◦ ρ| − Φ From Ψ Σ −1 iff there exists a y ∈ Σ satisfying x = η ρ(y) − Φ(y), (Im Ψ) which implies τM (x) = πM ( x) = ρ(y). From (3.1) it then follows that D=
−1
, (Im Ψ)
(3.2) ∗
i.e. D is the generalized Dirac equation on (M, ω) generated by Ψ := ρ η − Φ. Remark that, if Φ vanishes identically, the above equation reduces to the generalized Dirac equation generated by η, i.e. −1 Do := (Im η) = x ∈ T M | x = η τM (x) , which, if η is exact, falls into the range of Dirac equations (or implicit Hamilton equations on Dirac manifolds) discussed in [29] and turns out to be an explicit Hamilton equation Do = Im η iff ω is symplectic (i.e. admits an inverse morphism ) – in that case θ will be said to be a regular 1-form. Also remark that, if Φ does not vanish identically, all of the integral curves of D lie on C := ρ(Σ) ⊂ M . If C = M , it is meaningful to compare D and Do . = 0 (i.e. Σ = Y coincides with the subset Σo where Φ vanishes ). D is Let Φ −1 then a part of Do , namely D = τM (C)∩Do , the role of Φ being that of introducing a kinematic constraint C with the aim of selecting – from the integral curves of Do – those lying on C (moreover, regularity implies D = E, i.e. D incorporates the second-order condition). = 0 (i.e. Σo = Σ). D then differs from Do in that Φ carries not Let Φ only a kinematic constraint C bound to contain all of the integral curves, but altering Do in an effective way (under the additional also a constraint reaction Φ condition of iS θ being dS -closed, i.e. dS iS θ = 0, regularity still implies D = E).
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4 Generalized Dirac equation on cotangent bundle We shall still deal with the general kind of system (Q, Θ) considered in section 3. The aim will now be to develop (under suitable hypotheses) the geometric study of its dynamics on cotangent bundle T ∗ Q by following – from the methodological point of view – Dirac’s approach to Hamiltonian dynamics [13, 14] (i) The dynamics of (Q, Θ) will be reconsidered under the following hypotheses. (1) M1 := Im L is an embedded submanifold of T ∗ Q; (2) L1 : M → M1 – defined by ι1 ◦L1 = L with ι1 : M1 -→ T ∗ Q – is a submersion; (3) η C = (L∗1 η1 )C for some η1 ∈ Λ1 M1 . We start from equation E. Let x ∈ T M ∩ T 2 Q. One has x ∈ E iff and
τM (x) = ρ(y)
(4.1)
x = η τM (x) − Φ(y)
(4.2)
for some y ∈ Σ. As τM (x) = τT Q (x) = T τQ (x) = T πQ T L(x) = T πQ T L1 (x) , condition (4.1) reads (4.3) T πQ T L1 (x) = ρ(y) . Moreover, as τM1 T L1 (x) = L1 τM (x) , condition (4.1) implies τM1 T L1 (x) = ρ1 (y) where ρ1 := L1 ◦ ρ : Y → M1 is a smooth bundle over M1 satisfying πQ ◦ ρ1 = τQ ◦ ρ . Now, from iS θ = L∗ ϑQ and then ω = L∗ ωQ = L∗1 (ι∗1 ωQ ) = L∗1 ω1 with ω1 := ι∗1 ωQ ∈ Λ2 M1 , it follows that
x = 1 (T L1 (x) ◦ TτM (x) L1
(where 1 denotes the musical morphism of ω1 ). Similarly, from η C = (L∗1 η1 )C , it follows that
η τM (x) = η1 τM1 T L1 (x) ◦ TτM (x) L1 .
(4.4)
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is semibasic, we can consider the composite Φ := τ ◦ Φ :Σ→ Finally, as Φ Q T Q and then define the section of πT ∗ Q along ρ1 |Σ given by ∗
ϕ : Σ → T ∗ T ∗ Q : y → ϕ(y) := Φ(y) ◦ Tρ1 (y) πQ , obtaining – for all y ∈ Σ − − Φ(y) = Φ(y) ◦ Tρ(y) τQ = Φ(y) ◦ Tρ(y) (πQ ◦ ι1 ◦ L1 ) = Φ(y) ◦ Tρ1 (y) πQ ◦ Tρ1 (y) ι1 ◦ Tρ(y) L1 = Φ(y) ◦ (Tρ1 (y) πQ )T ◦ Tρ(y) L1 ρ1 (y) M1 = ϕ(y) T ◦ Tρ(y) L1 . M1 ρ1 (y)
the critical subset of Φ relative Remark that, owing to the semibasicity of Φ, to ρ1 does not differ from Σ and then, if we put 1 := ρ1 ◦ Φ| : Σ → T ∗ M1 , Φ Σ we obtain – for all y ∈ Σ – 1 (y) = ϕ(y) Φ T
ρ1 (y) M1
since Φ(y) = Φ(y) ◦ Ty ρ = ϕ(y) T
ρ1 (y) M1
◦ T y ρ1 .
As L1 is a submersion, from the above expression of x , η τM (x) and Φ(y) it follows that conditions (4.1,2) imply
1 1 (y) . T L1 (x) = η1 τM1 T L1 (x) − Φ (4.5) So conditions (4.1,2) imply – and are clearly implied by – conditions (4.3,4,5). Therefore x ∈ E iff T L1 (x) ∈ H, with H := z ∈ T M1 | ∃ y ∈ Σ : T πQ (z) = ρ(y), τM1 (z) = ρ1 (y), 1 1 (y) . z = η1 τM1 (z) − Φ That amounts to saying E = (T L1 )−1 (H) ∩ T 2 Q . By the very definition, H satisfies both the property H ⊂ D1
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where D1 := z ∈ T M1 | ∃ y ∈ Σ : τM1 (z) = ρ1 (y) ,
1
1 (y) z = η1 τM1 (z) − Φ
is the generalized Dirac equation on (M1 , ω1 ) generated by Ψ1 := ρ∗1 η1 − Φ (see (3.1,2)), and – owing to the additional condition T πQ (z) = ρ(y) – the property H ⊂ T2 with
T2 := z ∈ T T ∗Q | T πQ (z) ∈ M , τT ∗ Q (z) = L T πQ (z)
which makes H exhibit the same second-order-like behaviour as that of T . H will then be extracted from D1 by adding the second-order condition T πQ (z) = ρ(y) to those characterizing D1 . As a consequence, D1 is not generally equivalent to H, for it may admit more integral curves than H does (not all of the integral curves of D1 will then be momentum trajectories of (Q, Θ)). The problem of integrating H will, in principle, be solved by determining the integral curves k’s of D1 , characterized by k = ρ1 ◦ ς
,
1
1 ◦ ς k˙ = η1 ◦ k − Φ
for some curve ς in Σ, and then sorting out those which satisfy the second-order condition T πQ ◦ k˙ = ρ ◦ ς (i.e. – putting γ := πQ ◦ k – γ˙ = ρ ◦ ς , whence k = L ◦ γ). ˙ The integral curves of H are then the Legendre liftings of the corresponding base integral curves, which in turn do not differ from those of E = (T L)−1 (H)∩T 2 Q (all of them being characterized by Im γ˙ ⊂ M , Im (L ◦ γ) ˙ · = Im (T L ◦ γ¨ ) ⊂ H). So the integral curves of H are the same as those of T , i.e. H is equivalent to T . (ii) D1 can be given an alternative formulation, making direct use of the canonical symplectic structure ωQ of T ∗ Q. Assume that (4) η1 = ι∗1 h for some h ∈ Λ1 W , with W ⊂ T ∗ Q open submanifold containing M1 . Let z ∈ T T ∗ Q. Recall that z ∈ D1 iff z ∈ T M1 and there exists a y ∈ Σ s.t. p := τT ∗ Q (z) = ρ1 (y) 1 z = η1 (p) − ϕ 1 (y)T M . p
1
(4.6)
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From ω1 := ι∗1 ωQ and η1 = ι∗1 h , it follows that 1 z = β z Tp M1 (β being the musical morphism of ωQ ) and η1 (p) = h(p)Tp M1 . As a consequence, condition (4.6) reads β z − h(p) + ϕ(y) =0 Tp M1 which means β
z − h(p) + ϕ(y) ∈ (Tp M1 )o
or z − Xh (p) + Xϕˆ (y) ∈ β
−1
(Tp M1 )o ,
is a section of τT ∗ Q where Xh := β −1 ◦ h is a vector field on W and Xϕˆ := β −1 ◦ ϕ along ρ1 |Σ . So we obtain D1 = T M1 ∩ z ∈ T T ∗Q | ∃ y ∈ Σ : p := τT ∗ Q (z) = L ρ(y) , (4.7) −1 z − Xh (p) + Xϕˆ (y) ∈ β (Tp M1 )o . We then extract H from D1 by adding T πQ (z) = ρ(y), that is, H = T M1 ∩ z ∈ T T ∗Q | ∃ y ∈ Σ : T πQ (z) = ρ(y) , p := τT ∗ Q (z) = L ρ(y) , z − Xh (p) + Xϕˆ (y) ∈
β −1
o
(Tp M1 )
(4.8) .
In the sequel, the above equations will be discussed by distinguishing the case of a regular 1-form θ (whose Legendre morphism L is a local diffeomorphism) from the case of a singular (i.e. nonregular) one. (iii) We shall first consider a system (Q, Θ = ρ∗ θ + Φ) where θ satisfies (1)-(4), with (1) reinforced by the following hypothesis (implying the singularity of θ): (5) M1 = φ−1 (µ), where φ = (φ1 , . . . , φl ) : W → Rl (with l < dim W ) is a submersion at every point of M1 . From (5) it follows that – at any p ∈ M1 – Tp M1 = Ker dφ(p)
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(where dφ(p) = dφ1 (p), . . . , dφl (p) : Tp T ∗ Q → Rl is the differential of φ at p) and then (Tp M1 )o
= =
Span dφ(p) λ dφ(p) λ∈Rl
(with λ dφ := λi dφi := λ1 dφ1 + . . . + λl dφl ), whence β −1 (Tp M1 )o = λ Xφ (p) λ∈Rl (with λ Xφ := λi Xφi and Xφi := β −1 ◦ dφi ). So, in the present case, equations (4.7,8) take the form (with Lagrange multipliers λ ∈ Rl ) D1 = T M1 ∩ z ∈ T T ∗Q | ∃ (λ, y) ∈ Rl × Σ : (4.9) p := τT ∗ Q (z) = L ρ(y) , z = Xh (p) + λ Xφ (p) − Xϕˆ (y) and
H = T M1 ∩ z ∈ T T ∗Q | ∃ (λ, y) ∈ Rl × Σ : T πQ (z) = ρ(y) , p := τT ∗ Q (z) = L ρ(y) , z = Xh (p) + λ Xφ (p) − Xϕˆ (y) .
(4.10)
As to the operation of extracting H from D1 , we want to signal a special situation, in which such an operation leads to the elimination of the unknown Lagrange multipliers λ ∈ Rl , replaced by suitable values of some known functions on M . With reference to the 1-forms h ∈ Λ1 W and η = L∗ h ∈ Λ1 M , consider the following bundle morphisms F h := T πQ ◦ Xh : W → T Q : p → F h(p) = h(p) ◦ νp and
F η := τT ∗ Q ◦ α−1 ◦ η : M → T ∗ Q : v → F η(v) = η(v) ◦ νv
(see (2.2,3)). Then consider the vertical vector field on M given by Γh : M → V τQ : v → Γh (v) := νv F h L(v) , as well as the dilation vector field ∆ : T Q → V τQ : v → ∆(v) := νv (v) . Let iS θ be dS -closed (i.e. dS iS θ = 0).
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Under such a hypothesis, check that – for all v ∈ M – Tv L Γh (v) = νL(v) F η(v) = Tv L ∆(v) and then ∆(v) − Γh (v) ∈ Ker Tv L .
(4.11)
Under the above hypothesis, also the vector fields Γφ = (Γφ1 , . . . , Γφl ) defined on M by
Γφi : M → V τQ : v → Γφi (v) := νv F φi L(v) (with F φi := T πQ ◦ Xφi ) satisfy – for all v ∈ M – Γφi (v) ∈ Ker Tv L (since Tv L Γφi (v) = νL(v) F L∗ φi (v) and L∗ φi = µi ). Let us now reinforce condition (5) by requiring F φ(p) = F φ1 (p), . . . , F φl (p) to be a linearly independent system of vectors, for all p ∈M1 . The above implies that Γφ (v) = Γφ1 (v), . . . , Γφl (v) is a basis of Ker Tv L, for all v ∈ M . As a consequence, there exists a unique l-tuple J = (J1 , . . . , Jl ) of real-valued functions on M s.t. ∆(v) − Γh (v) = J(v)Γφ (v)
(4.12)
νv v − Fh L(v) = νv J(v)F φ L(v) v = F h L(v) + J(v)F φ L(v)
(4.13)
for all v ∈ M . Finally, let z ∈ H. Owing to (4.10), one has T πQ (z) = ρ(y), p := τT ∗ Q (z) = L T πQ (z) and z = Xh (p) + λ Xφ (p) − Xϕˆ (y), with (λ, y) ∈ Rl × Σ . The first and the second condition, in view of (4.13), imply T πQ (z) = F h(p) + J T πQ (z) F φ(p) . The third condition implies T πQ (z) = =
T πQ Xh (p) + λ T πQ Xφ (p) F h(p) + λ F φ(p)
= ϕ(y)◦ν (since T πQ Xϕˆ (y) = (T πQ ◦β −1 ◦ ϕ)(y) ρ1 (y) = Φ(y)◦Tρ1 (y) πQ ◦νρ1 (y) = 0).
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As F φ(p) is linearly independent, we deduce λ = J T πQ (z)
311
(4.14)
which shows the announced elimination in H of the unknown Lagrange multipliers. (iv) We shall now consider a system (Q, Θ = ρ∗ θ + Φ) where θ is a regular 1-form satisfying condition (3). Regularity, i.e. the fact of L being a local diffeomorphism, implies that M1 is an open submanifold of T ∗ Q and L1 is a local diffeomorphism as well. As a consequence, conditions (1) and (2) are automatically satisfied, and so is condition (4) with W = M1 and h = η1 . Equation D1 then takes the form given by (4.7). −1 −1 More precisely, as β (Tp M1 )o = β (Tp T ∗ Q)o is the null subspace of Tp T ∗ Q (for all p ∈ M1 ), one has D1 = z ∈ T T ∗Q | ∃ y ∈ Σ : p := τT ∗ Q (z) = L ρ(y) , z = Xh (p) − Xϕˆ (y) .
(4.15)
Now one should extract H from D1 by adding T πQ (z) = ρ(y) . However note that, if iS θ is dS -closed, D1 incorporates the above second-order condition, since – for any z ∈ D1 – one has T πQ (z) = T πQ Xh (p) − T πQ Xϕˆ (y) = T πQ Xh (p)
= F h L ρ(y) =
ρ(y)
(the last being due to the fact that (4.11) now reads ∆(v) − Γh (v) = 0, equality i.e. F h L(v) = v, for all v ∈ M ). So, under the above hypothesis of dS -closedness, one has H = D1 .
(4.16)
= 0. The same result can be obtained under the hypothesis Φ In such a case, indeed, from (4.15) it follows that D1 = Im (Xh L(C) ) .
Moreover, from T 2 Q∩(T L)−1 (H) = E = D = Im η|C , it follows that Im (T L◦ η|C ) ⊂ H .
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As H ⊂ D1 ⊂ Im Xh , we deduce T L ◦ η C = Xh ◦ τT ∗ Q ◦ T L ◦ η C = Xh ◦ L ◦ τM ◦ η C = Xh ◦ LC which in turn implies D1 = Im (Xh L(C) ) = Im (Xh ◦ LC ) = Im (T L ◦ η C ) ⊂ H whence (4.16). In particular, if Φ = 0, one has H = D1 = Im Xh
(4.17)
and the above equation is the integrable part of T , i.e. the region of T swept by the tangent liftings of its integral curves (since an explicit equation like H = Im Xh ⊂ T M1 is entirely covered by the tangent liftings of its own integral curves, which are the same as those of T ). The above result can furtherly be strengthened when θ is hyperregular (i.e. L is an injective local diffeomorphism). Note that a hyperregular 1-form automatically fulfils conditions (1)-(4) and therefore naturally falls into the range of the present theory. In such a case, if Φ = 0, as well as H = D1 = Im Xh ⊂ T , one has T ⊂ Im Xh . Indeed, any z ∈ T = Im (α−1 ◦ θ) ⊂ T2 satisfies T πQ (z) ∈ M and p := τT ∗ Q (z) = L T πQ (z) , z = (α−1 ◦ θ) T πQ (z) ; similarly, Xh (p) ∈ T satisfies T πQ Xh (p) ∈ M and
p = τT ∗ Q Xh (p) = L T πQ Xh (p) , Xh (p) = (α−1 ◦ θ) T πQ Xh (p) ; hence
L T πQ (z) = L T πQ Xh (p) T πQ (z) = T πQ Xh (p)
(α−1 ◦ θ) T πQ (z) = (α−1 ◦ θ) T πQ Xh (p) z = Xh (p) .
So, if θ is hyperregular and Φ = 0, one has H = D1 = Im Xh = T .
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5 Canonical formalism We shall now specialize (Q, Θ) so as to recover – from section 4 – the canonical formalism (extended to an enlarged mechanical context) of Hamilton-Dirac dynamics [13, 14, 24, 25, 28]. (i) Let Q be a smooth manifold and Θ := ρ∗ θ + Φ a 1-form on the total space Y := M × Rp of a product bundle ρ : M × Rp → M over an open submanifold M of T Q, with θ and Φ satisfying the following requirements. θ is a 1-form on M s.t. iS θ is dS -exact, i.e. iS θ = dS L , L being a given 0-form on M (called Lagrangian function). Such a condition also reads θ = dL + F with F semibasic 1-form on M (called covector force field). Note that, as F is semibasic, the Legendre morphism defined by θ is just the fibre derivative of L, i.e. L = F L (as a consequence, θ is regular iff L is regular, i.e. F L is a local diffeomorphism). Moreover, one has ω = −ddS L and η = dE − F , with E := ∆L − L . Φ is a 1-form on M × Rp defined – at any (v, π) ∈ M × Rp – by Φ(v, π) := πA(v), B(v) ∈ Tv∗ M × Rp = Tv∗ M × Tπ∗ Rp , where A = (A1 , . . . , Ap ) is a p-tuple of semibasic 1-forms on M (and then πA := πj Aj := π1 A1 + . . . + πp Ap ) and B = (B 1 , . . . , B p ) is a p-tuple of 0-forms on M (in the case A = 0 and B = 0, (Q, Θ) –also denoted (Q, L, F )– will be called a Klein-Godbillon system [16, 20]). The critical subset of Φ relative to ρ is Σ = B −1 (0) × Rp , whence the kinematic constraint C = B −1 (0) – which will be assumed to be nonempty. Moreover π) = πA(v) ∈ V o τQ at each v ∈ C, the constraint reactions are given by Φ(v, v ∗ := τ ◦ Φ and A j := or, equivalently, by Φ(v, π) = π A(v) ∈ Tτ (v) Q (where Φ Q Q
τQ ◦ Aj ). The ‘Lagrangian dynamics’ of the above system (Q, Θ) is expressed by the second-order equation E = D ∩ T 2 Q, where the generalized Dirac equation D – given by (3.1) – takes the form (with multipliers π ∈ Rp ) D = x ∈ T M | B τM (x) = 0 , ∃ π ∈ Rp : x = dE τM (x) − F τM (x) − πA τM (x) (for a Klein-Godbillon system with regular Lagrangian, D = E reduces to the explicit Lagrange equation – without multipliers – given in [16]). (ii) The ‘Hamiltonian dynamics’ of the above (Q, Θ) will be deduced from section 4 under the following hypotheses.
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Firstly, Legendre morphism L = F L will be assumed to satisfy conditions (1) and (2). Secondly, energy E = ∆L − L will be assumed to be projectable by L, i.e. E = L∗ H , H being a real-valued smooth function, called Hamiltonian function, defined on some open submanifold W of T ∗ Q containing M1 := Im L . Thirdly, covector force field F will be assumed – at least on C – to be projctable by L, i.e. F C = (L∗ f )C , f being a 1-form on W (which, in view of the semibasicity of F , will satisfy, for all v ∈ C, f L(v) = F (v) ◦ TL(v) πQ , with F := τQ ◦ F ). Conditions (3) and (4) will then be satisfied with h := dH − f . The dynamics of (Q, Θ) is therefore governed by equation H, extracted from D1 , in the form (4.7,8). The fields Xh and Xϕˆ therein appearing will now be evaluated. As to Xh , recall that Xh := β −1 ◦ h = β −1 ◦ dH − β −1 ◦ f and then (putting XH := β −1 ◦ dH, Xf := β −1 ◦ f ) Xh = XH − Xf . As to Xϕˆ , recall that – at any y = (v, π) ∈ Σ = B −1 (0) × Rp – one has and ϕ(y) = Φ(y) ◦ Tρ1 (y) πQ = π A(v) ◦ TL(v) πQ = πϕ(v), Xϕˆ (y) = β −1 ◦ ϕ(y) 1 p j j (v) ◦ TL(v) πQ ; then, putting Xϕ := with ϕ := (ϕ , . . . , ϕ ) and ϕ (v) := A (Xϕ1 , . . . , Xϕp ) and Xϕj := β −1 ◦ ϕj , we obtain Xϕˆ (y) = πXϕ (v) . As a consequence equation (4.7) reads D1 = T M1 ∩ z ∈ T T ∗ Q | ∃ (v, π) ∈ T Q × Rp : v ∈ M , B(v) = 0 , p := τT ∗ Q (z) = L(v) , z − (XH − Xf )(p) + π Xϕ (v) ∈ β
−1
(Tp M1 )o ,
whereas equation (4.8) – i.e. the dynamics of (Q, Θ) – reads H = T M1 ∩ z ∈ T T ∗Q | T πQ (z) ∈ M , B T πQ (z) = 0 , p := τT ∗ Q (z) = L T πQ (z) , −1 ∃ π ∈ Rp : z − (XH − Xf )(p) + π Xϕ T πQ (z) ∈ β (Tp M1 )o .
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Remark that the last of conditions characterizing D1 can as well be expressed – by putting H1 := ι∗1 H – in the form β z + f (p) + π ϕ(v) = dH1 (p) Tp M1
and then – in the case of a Klein-Godbillon system (Q, L, F ) with F = 0 – D1 turns out to be equivalent to D1 := z ∈ T T ∗ Q | p := τT ∗ Q (z) ∈ M1 , = dH1 (p) ( βz) Tp M1
which is the Dirac equation formulated in [24, 25, 28]. In the sequel, we shall distinguish the regular from the singular case. (iii) First we assume hypothesis (5) – implying the singularity of L. In such a case, equations D1 and H – given by (4.9,10) – read
and
D1 = T M1 ∩ z ∈ T T ∗Q | ∃ (λ, v, π) ∈ Rl × T Q × Rp : v ∈ M , B(v) = 0 , p := τT ∗ Q (z) = L(v) , z = (XH − Xf )(p) + λ Xφ (p) − π Xϕ (v)
(5.1)
H = T M1 ∩ z ∈ T T ∗ Q | T πQ (z) ∈ M, B T πQ (z) = 0, p := τT ∗ Q (z) = L T πQ (z) , ∃(λ, π) ∈ Rl × Rp : z = (XH − Xf )(p) + λ Xφ (p) − π Xϕ T πQ (z) .
(5.2)
For a Klein-Godbillon system (Q, L, F ), the above equations take the form D1 = T M1 ∩ z ∈ T T ∗Q | p := τT ∗ Q (z) , φ(p) = µ ,
∃ λ ∈ Rl : z = (XH − Xf )(p) + λ Xφ (p)
and
H = T M1 ∩ z ∈ T T ∗Q | T πQ (z) ∈ M , p := τT ∗ Q (z) = L T πQ (z) ,
∃ λ ∈ R : z = (XH − Xf )(p) + λ Xφ (p) ,
(5.3)
(5.4)
l
whence H = D1 ∩ T2 . Let k : I → T ∗ Q : t → k(t) be a smooth curve in T ∗ Q, and put γ := πQ ◦ k . k is an integral curve of (5.1), iff there exist curves Λ : I → Rl : t → λ(t) , c : I → T Q : t → c(t) , Π : I → Rp : t → π(t)
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such that
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Im c ⊂ M , B ◦ c = 0 , k = L ◦ c, ˙ k = (XH − Xf ) ◦ k + Λ(Xφ ◦ k) − Π(Xϕ ◦ c) .
(5.5)
Such a k will be an integral curve of (5.2) – i.e. a momentum trajectory of (Q, Θ) – iff it satisfies the above conditions with c = γ˙ .
(5.6)
In the case of (Q, L, F ), k is an integral curve of (5.3), iff
φ◦k =µ, k˙ = (XH − Xf ) ◦ k + Λ(Xφ ◦ k)
(5.7)
for some time-dependent Lagrage multipliers Λ . Such a k will be an integral curve of (5.4) – i.e. a momentum trajectory of (Q, L, F ) – iff it satisfies the above requirements with the first one replaced by the stronger Legendre condition Im γ˙ ⊂ M , k = L ◦ γ˙
(5.8)
Recall that, if F φ is linearly independent on M1 , then –owing to (4.14)– the Lagrange multipliers λ appearing in equations (5.2,4) are replaced by the values J T πQ (z) . As a consequence, the time-dependent Lagrange multipliers Λ appearing in the corresponding expressions of conditions (5.5,7), will be replaced by the functions J ◦ γ˙ . (iv) Now we assume that L is a regular Lagrangian. The dynamics of (Q, Θ) is then governed by equation (4.15,16) which now reads H = D1 = z ∈ T T ∗Q | T πQ (z) ∈ M , B T πQ (z) = 0 , p := τT ∗ Q (z) = L T πQ (z) , ∃ π ∈ Rp : z = (XH − Xf )(p) − π Xϕ T πQ (z) . For a Klein-Godbillon system (Q, L, F ), the dynamics – given, under the hypothesis of regularity, by (4.17) – is an explicit equation on the open submanifold M1 ⊂ T ∗ Q, namely H = D1 = Im (XH − Xf )
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A smooth curve k in T ∗ Q (with projection γ := πQ ◦ k) is then a momentum trajectory of (Q, Θ), iff Im γ˙ ⊂ M , B ◦ γ˙ = 0 , k = L ◦ γ˙ , (5.9) ˙ ˙ k = (XH − Xf ) ◦ k − Π(Xϕ ◦ γ) for some time-dependent multipliers Π . In the case of (Q, L, F ), k is a momentum trajectory, iff Im k ⊂ M1 , k˙ = (XH − Xf ) ◦ k .
(5.10)
(v) For a coordinate formulation of the theory, choose an admissible chart (U, ξ) on Q s.t. T U ∩ M = ∅ (the corresponding charts on T Q and T ∗ Q will be denoted by ξo and ξ1 , respectively). Condition (5.5) – characterizing the integral curves of D1 under hypothesis (5) – are locally expressed by
ξo−1 q(t), v(t) ∈ M , B j ξo−1 q(t), v(t) = 0 ∂L q(t), v(t) h ∂v ∂H ∂φi q(t), p(t) + λi (t) q(t), p(t) q˙h (t) = ∂ph ∂ph ∂H p˙ h (t) = − h q(t), p(t) + Fh q(t), v(t) ∂q ∂φi − λi (t) h q(t), p(t) + πj (t)Ajh q(t), v(t) ∂q (where q(t), v(t) := ξo c(t) and q(t), p(t) := ξ1 k(t) ). The momentum trajectories of the system will locally be characterized by the above equations therein putting, in view of (5.6), ph (t) =
v(t) = q(t) ˙ . Remark that, for a Klein-Godbillon system with F = 0, the integral curves of D1 are locally characterizzed, owing to (5.7), by constraint equations
φi ξ1−1 q(t), p(t) = µi coupled with Hamilton-Dirac scalar equations [13, 14] ∂H ∂φi q(t), p(t) + λi (t) q(t), p(t) ∂ph ∂ph ∂H ∂φi p˙h (t) = − h q(t), p(t) − λi (t) h q(t), p(t) ∂q ∂q q˙h (t) =
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whereas the momentum trajectories will locally be characterized, in view of (5.8), by Legendre equations ∂L ˙ ⊂ M , ph (t) = h q(t), q(t) ˙ ξo−1 q(t), q(t) ∂v coupled with Hamilton-Dirac scalar equations. ∂φi Recall that, if the rank of ∂p is l on M1 , then, in the scalar equations h locally characterizing the momentum trajectories, time-dependent Lagrange mul ˙ . tipliers λi (t) are replaced – according to [4] – by functions Ji q(t), q(t) If L is a regular Lagrangian, then – owing to (5.9,10) – all the terms containing Lagrange multipliers λi (t) disappear from the above scalar equations and, in the case of a Klein-Godbillon system, Legendre equations become redundant too, being sufficient to require – in view of (5.10) – ξ1−1 q(t), p(t) ⊂ M1 .
6 Examples Classical and relativistic particle dynamics will now be framed in our geometric construction by furthertly specializing the mathematical model illustrated in section 5. (i) We start from classical dynamics. Let Q := Rn be the space of all the configurations of n particles in a frame = carrying of reference R (3-dimensional affine space, modelled on a vector space R an Euclidean inner product “ · ”). For each i = 1, . . . , n , ri : Q → R will denote the i-th projection of Rn onto = will be defined by putting, for any q ∈ Q , r | R and ri : T Q → R i Tq Q := dq ri : n = → Tr (q) R ≡ R = (tangent mapping of ri at q or i-th projection of R =n Tq Q ≡ R i = Then dri and dr will denote the maps which take each q ∈ Q to dq ri onto R). i and each v ∈ T Q to dv ri , respectively. Now let Θ := ρ∗ θ + Φ, with θ := dL + F , be a 1-form on Y := T Q × Rp defined as follows. L is the kinetic energy in R of the masses mi > 0 associated with the particles, defined by quadratic form 1 mi ri · ri : T Q → R . 2 i=1 n
L :=
= acting in R on F is the virtual work of the vector force fields F=i : T Q → R the particles, defined by bundle morphism F :=
n
F=i · (dri ◦ τQ ) : T Q → T ∗ Q .
i=1
Φ corresponds to a couple (A, B) of the kind introduced in section 5(i).
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From the Lagrangian point of view, the dynamics of (Q, Θ) is expressed by equation E = D ∩ T 2 Q, where D is the generalized Dirac equation given in section 5(i). So a smooth curve γ : I → Q is a motion of (Q, Θ), iff it satisfies – for all t ∈ I – condition γ¨ (t) ∈ E , which – as is shown in [3] – takes the form of a constraint equation B γ(t) ˙ =0 (6.1) followed by d’Alembert’s principle n d2 (ri ◦ γ) = Fi γ(t) ˙ − mi (t) · δγ(t) ri = 0 , ˙ dt2 i=1 where δγ(t) ri := dγ(t) ri KerA( . ˙ ˜ γ(t)) ˙
2 2 2 1 ◦γ) = n , F= := (F=1 , . . . , F=n ) : , . . . , mn d (rdtn2◦γ) : I → R If we put m ddtγ2 := m1 d (r dt2
⊥ = n and KerA˜ γ(t) ˙ := orthogonal complement of Ker A˜ γ(t) ˙ ⊂ TQ → R n n = = Tγ(t) Q ≡ R in R , d’Alembert’s principle can be given the geometric form ⊥ d2 γ = Φ(t) := m 2 (t) − F= γ(t) ˙ ∈ Ker A˜ γ(t) ˙ . dt
(6.2)
If A = 0 and B = 0, then (Q, Θ) – i.e. (Q, L, F ) – corresponds to a system of 2 free particles, whose motions in R are determined by Newton’s law m ddtγ2 = F= ◦ γ˙ . If A = 0 and B = 0, then (Q, Θ) corresponds to a system of free particles, among whose motions in R – determined by Newton’s law – one wants to select those satisfying condition (6.1). If A = dS B with B = 0, then (Q, Θ) corresponds to a system of constrained particles, forced to obay a constraint condition (6.1) on position-velocity in R and then acted upon, along any motion γ, by an additional vector force field or = constraint reaction Φ(t), which –after d’Alembert (6.2)– keeps orthogonal to the ˜ spaces Ker A(v) = Ker F B(v) of admissible virtual displacements at v ∈ C = B −1 (0) . In the latter situation, a standard case is that of nonholonomic or affine functional constraints B := iT a + τQ∗ b (a and b being p-tuple of 1-forms and 0-forms on Q, respectively). In such a case, the admissible virtual displacements at v ∈ C (with q := τQ (v)) sweep the space ˜ Ker A(v) = Ker a(q) and then the constraint reactions at v take their values in ⊥ ˜ Ker A(v) = Span=a(q) (where – for any j = 1, . . . , p – =aj (q) · = aj (q)). In particular, exact 1-forms a = dg (with g = (g 1 , . . . , g p ) : Q → Rp of constant rank) and vanishing 0-forms b = 0, define the integrable functional constraints B := dT g
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which – through (6.1) – only allow of motions lying on the fibres {Qπ := g −1 (π) , π ∈ Rp } of g (C is indeed the union of the tangent bundles of such fibres). In such a case, the set of all the admissible virtual displacements coincides with C, ˜ = Ker dg(q) = since – for any v ∈ C (with q := τQ (v)) – one has Ker A(v) Tq Qπ (where π := g(q)), and then the constraints reactions at v take their values ⊥ ˜ = Span (grad g)q = (Tq Qπ )⊥ (where – for any j = 1, . . . , p – in Ker A(v) j j (grad g )q · = dg (q)). Another standard case, is that of holonomic functional constraints B := (τQ∗ g , dT g) (with g = (g 1 , . . . , g m ) : Q → Rm of constant rank on Qo := g −1 (0)), which – through (6.1) – allow of motions lying on just one fibre of g, namely Qo (C is indeed the tangent bundle of Qo ). In such a case, the set of all the admissible virtual displacements still coincides with C, since – for any v ∈ C (with q := τQ (v) ∈ Qo ) ˜ – one has Ker A(v) = Ker dg(q) = Tq Qo and then the constraint reactions at v ⊥ ˜ take their values in Ker A(v) = Span (grad g)q = (Tq Qo )⊥ . Remark that the above holonomic constraints can – in a sense – be cancelled by replacing (Q, Θ) with the Klein-Godbillon system (Qo , Lo , Fo ), defined by putting Lo := j ∗ L and Fo := j ∗ F with j : T Qo -→ T Q, since both systems admit the same motions. From the Hamiltonian point of view, the dynamics of (Q, Θ) is expressed – owing to the hyperregularity of L – by the generalized Dirac equation D1 given in section 5(iv), with H := (L−1 )∗ L and f := (L−1 )∗ F . As a consequence, a smooth curve k : I → T ∗ Q is a momentum trajectory of (Q, Θ), iff it satisfies conditions (5.9), owing to which k is required to be the Legendre lifting of a motion γ : I → Q of (Q, Θ), i.e. k = L ◦ γ˙ . In order to evaluate such a momentum trajectory, let us consider an arbitrary vector field ζ ∈ χ(Q) and its tangent lifting Z ∈ χ(T Q). As Z and ζ are τQ -related to each other, we have k | ζ ◦ γ = = =
L ◦ γ˙ | ζ ◦ γ dL ◦ γ˙ | SZ ◦ γ ˙ n mi (ri ◦ γ) ˙ · (dri ◦ γ) ˙ | SZ ◦ γ ˙ i=1
=
n d mi (ri ◦ γ) · (dri ◦ γ) | ζ ◦ γ dt i=1
whence (owing to the arbitrariness of ζ) k=
n i=1
mi
d (ri ◦ γ) · (dri ◦ γ) . dt
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The above result reads
where m
dγ := dt
321
dγ k= m · dt
d d =n m1 (r1 ◦ γ), . . . , mn (rn ◦ γ) : I → R dt dt
is the n-tuple of linear momenta of the masses m = (m1 , . . . , mn ) along γ . Finally remark that, in presence of holonomic constraints, the operation of replacing (Q, Θ) with (Qo , Lo , Fo ) implies the replacement of any momentum trajectory k = L ◦ γ˙ with the momentum trajectory ko = ˙ the latter being Lo ◦ γ, related to the former – at each time t – by ko (t) = k(t)T Q . γ(t)
o
(ii) Let us now turn to relativistic particle dynamics. Let Q be a 4-dimensional smooth manifold (space-time), equipped with a Lorentz metric tensor g (gravitational field) and a 2-form F (electromagnetic field). The causal character of a Lorentz metric tensor g : T Q → T ∗ Q (symmetric vector bundle isomorphism of signature +, −, −, −) allows one to distinguish in T Q the time-like vectors (characterized by g(v) | v > 0) and, in particular, under the hypothesis of existence of a time-like vector field ζ on Q (time-orientability), the future-pointing time-like vectors, sweeping the open submanifold of T Q given by M := v ∈ T Q | g(v) | v > 0 , g(v) | ζ τQ (v) < 0 . A future-pointing, time-like, smooth curve γ in Q (i.e. one satisfying Im γ˙ ⊂ M ), together with all of its orientation-preserving reparametrizations, determines an oriented orbit Im γ, which is meant to be the world line of a material particle. Each one of the above world lines can be parametrized by proper time, so as to obtain a (uniquely determined) smooth curve γ satisfying Im γ˙ ⊂ M with g ◦ γ˙ | γ ˙ = 1 and such a curve is said to be the life history of a material particle. In order to determine the possible life histories and world lines of a material particle (m, e) of proper mass m > 0 and electric charge e ∈ R, living in the gravitational and electromagnetic fields (g, F), we shall adopt the following kind of mechanical system (Q, Θ = ρ∗ θ + Φ). ρ : Y := M × R → M is the natural projection of M × R onto M . θ := dL + F ∈ Λ1 M is given by the differential of relativistic Lagrangian √ L := m 2K, arising from quadratic form K : M → R : v →
1 g(v) | v , 2
plus Lorentz electromagnetic force field F := −e iT FM .
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Φ ∈ Λ1 Y corresponds to a couple (A, B) with √ A = 0 , B = 2K − 1 . From the Lagrangian point of view, the dynamics of (Q, Θ) is expressed by E = D∩T 2 Q , where D denotes the generalized Dirac equation given in section 5(i). As m 1 dK ∧ dS K ω := −ddS L = √ ωK + 2K 2K (with ωK := −ddS K) and
E := ∆L − L = 0 ,
one obtains (see [3]) D = x ∈ T M | v := τM (x) ∈ C , (λ, µ) ∈ R2 : x = λ∆(v) + µXK (v) − where
C := B −1 (0) = {v ∈ M | |v| :=
1 XF (v) m
g(v) | v = 1}
is an embedded submanifold of T Q, XK ∈ χ(M ) is the Riemannian spray uniquely determined by iXK ωK = dK and XF ∈ χ(M ) is the vertical field uniquely determined by iXF ωK = F . Hence E = x ∈ T M | v := τM (x) ∈ C , ∃ λ ∈ R : x = X(v) + λ∆(v) where X ∈ χ(M ) is defined by X := XK −
1 XF . m
√ := E ∩ T C, which – owing to ∆B = 2K As τM (E) = C, E is equivalent to D and XB = 0 , i.e. X|C ∈ χ(C) – turns out to be an explicit equation on C, namely = Im (X ) . D C A smooth curve γ in Q is then a motion of (Q, Θ), iff it is a base integral i.e. curve of D, Im γ˙ ⊂ C , γ¨ = X ◦ γ˙ . Note that the second of the above conditions is also expressed by iγ¨ ωK = 1 ◦ γ¨ = 1 F ◦ γ˙ (where [K] : T 2 Q ∩ τ −1 (M ) → V o τQ : dK − m F ◦ γ˙ , i.e., [K] TQ m := τ ◦ [K] and F := τ ◦ F ). Equivalently, x → dK τT Q (x) − ix ωK , [K] Q Q passing from covector to vector formalism through g−1 , we obtain m∇γ˙ γ˙ = F¯ ◦ γ, ˙ where ∇ is the Levi-Civita connection of g (whence, as one can check in terms
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◦ γ¨ ) and F¯ := g−1 ◦ F (whence, owing to (2.1), of components, ∇γ˙ γ˙ = g−1 ◦ [K] −1 F¯ ◦ γ˙ = −e g ◦ iγ˙ F). So γ is a motion of (Q, Θ), iff it satisfies g ◦ γ˙ | γ ˙ =1 ,
m∇γ˙ γ˙ = −e g−1 ◦ iγ˙ F
which are the standard laws (see, e.g., [26] p.295 and 155) characterizing the possible life histories of a particle (m, e) living in (g, F). From the Hamiltonian point of view, one is interested in the momentum trajectories of (Q, Θ), which – as is known – project down onto the motions and are in turn determined by the motions through Legendre lifting γ → L ◦ γ, ˙ where m m L = FL = √ FK = √ g|M . 2K 2K Now remark that the mass-shell M1 := L(M ) = L(C) = p ∈ W | |p| := p | g−1 (p) = m is an embedded submanifold of T ∗ Q contained in the open subset W := g(M ), since 2 m M1 = φ−1 2 with φ := K ◦ g−1 |W (whence the singularity of L), and the smooth mapping L1 : M → M1 induced by L onto M1 is a submersion, since Ker Tv L = Span ∆(v) for all v ∈ M (as one could infer, e.g., from the study of the affine bundle structure of equation (6.3) in the sequel). Moreover, one trivially has E = L∗ H with H : W → R given by H =0. Finally, one can easily check that F C = (L∗ f )C with f ∈ Λ1 W defined, at each p ∈ W , by f (p) := −e iu(p) F ◦ Tp πQ where u(p) :=
1 −1 g (p) . m
As a consequence, the differential equation of the momentum trajectories will be deduced from the generalized Dirac equation (5.1), which reads D1 = T M1 ∩ z ∈ T T ∗Q | p := τT ∗ Q (z) ∈ L(C) = M1 , ∃ λ ∈ R : z = λ Xφ (p) − Xf (p) .
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From D1 we extract the equation of the momentum trajectories by means of (5.2), where – as F φ(p) = 0 for all p ∈ M1 – Lagrange multiplier λ is to be replaced by (4.14). In order to evaluate the function J : W → R appearing in (4.14) – and defined by (4.12) – we remark that, at any v ∈ M ,
Γφ (v) : = νv F φ L(v)
= νv g−1 L(v) m νv (v) = |v| m √ = ∆ (v) 2K
Γh (v) := νv F h L(v) = 0
and since
F h L(v)
= h L(v) ◦ νL(v) = −f (p) ◦ νp , p := L(v) = e iu(p) F ◦ Tp πQ ◦ νp = 0.
So, from
m ∆ Γφ = √ 2K
and Γh = 0 , we obtain – in view of (4.12) – J=
1 √ 2K . m
So, owing to (4.14), equation (5.2) reads H = T M1 ∩ z ∈ T T ∗ Q | T πQ (z) ∈ C , p := τT ∗ Q (z) = L T πQ (z) , 1 Xφ − Xf (p) . z= m Note that the vector field X1 :=
1 Xφ − Xf ∈ χ(W ) m
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satisfies X1 φ = −Xf φ = − dφ | Xf = f | Xφ = 0 since, for any p ∈ W , f (p) | Xφ (p)
= −e iu(p) F | Tp πQ Xφ (p) = −e iu(p) F | Fφ(p) = −e iu(p) F | g−1 (p)
= −e m iu(p) F | u(p) = 0. As a consequence, X1 is tangent to M1 , i.e. X1 M1 ∈ χ(M1 ) , and then on M1 it defines the explicit equation
1 := Im X1 D . M1 1 . From the above expression of H, it follows that H ⊂ D On the other hand, one also has D1 ⊂ H, since – for any p ∈ M1 – z := X1 (p) satisfies T πQ (z) = =
1 T πQ ◦ Xφ (p) − T πQ ◦ Xf (p) m 1 −1 g (p) ∈ C m
(remark that T πQ ◦ Xf (p) = f (p) ◦ νp = −e iu(p) F ◦ Tp πQ ◦ νp = 0) and then 1
g−1 (p) = p = τT ∗ Q (z) . L T πQ (z) = L m Therefore H turns out to be an explicit equation in M1 , namely 1 . H=D A smooth curve k : I → T ∗ Q is then a momentum trajectory of (Q, Θ), i.e. 1 , iff it satisfies an integral curve of H = D Im k ⊂ M1
,
k˙ = X1 ◦ k
which are therefore the conditions characterizing the momentum trajectories corresponding to the possible life histories of the particle. (iii) We shall now consider the above relativistic system (Q, Θ) with B = 0, i.e. the Klein-Godbillon system (Q, L, F ). For a neutral particle e = 0 – which is the case we shall be dealing with – one has F = 0 and then the system will be referred to as (Q, L).
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From the Lagrangian point of view, the dynamics of (Q, L) can be deduced from the generalized Dirac equation D given in section 5(i), which now reads D = x ∈ T M | ∃ (λ, µ) ∈ R2 : x = λ∆ τM (x) + µXK τM (x) . Remark explicitly that not all of the base integral curves of D are motions of the system (consider, e.g., an integral curve c of ∆M ; as Im ∆M ⊂ D, c is an integral curve of D as well, but the corresponding base integral curve γ := τQ ◦ c is not a motion of the system since its tangent lifting γ˙ = T τQ ◦ c˙ = T τQ ◦ ∆ ◦ c = 0 does not even lie in M ). The differential equation of the motions will indeed be extracted from D via intersection with T 2 Q , thus obtaining (6.3) E = x ∈ T M | ∃ λ ∈ R : x = XK τM (x) + λ∆ τM (x) . The base integral curves of E will all be deduced from the base integral curves := Im(XK | ) ∈ χ(C) by eliminating their proper time parametrization. of D C Let σ : Io → Q be a future-pointing, time-like, smooth curve. A diffeomorphism s : Io → I with derivative s > 0, yields a new futurepointing, time-like, smooth curve γ := σ ◦ s−1 : I → Q (since σ˙ = s (γ˙ ◦ s)), which will be said to be an orientation-preserving reparametrization of σ . Assume that √ γ is parametrized by proper time, i.e. 2K ◦ γ˙ = 1 or, equivalently, s = 2K ◦ σ˙ . Now σ is a base integral curve of E iff σ ¨ = XK ◦ σ˙ + As
(K◦σ) ˙ 2K◦σ˙
=
s s
(K ◦ σ) ˙ (∆ ◦ σ) ˙ . 2K ◦ σ˙
, the above condition amounts to saying σ ¨−
s (∆ ◦ σ) ˙ = XK ◦ σ˙ s
which – in view of the homogeneity properties of a spray – does not differ from γ¨ = XK ◦ γ˙ . and all of their So the motions of (Q, L) are the base integral curves of D orientation-preserving reparametrizations. The oriented orbits of such motions (carrying no distinguished parametrizations) are then the possible world lines of any neutral particle freely falling in the given gravitational field. From the Hamiltonian point of view, the dynamics of (Q, L) will be deduced from the generalized Dirac equation (5.3), which now reads D1 = z ∈ T M1 | ∃ λ ∈ R : z = λ Xφ τM1 (z) . Clearly, not all of the integral curves of D1 are momentum trajectories of the system (consider, e.g., a trivial curve k =const. in M1 ; k is obviously an integral
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curve of D1 , but the corresponding base integral curve γ := πQ ◦ k is not a motion of the system since its tangent lifting γ˙ = T πQ ◦ k˙ = 0 does not even lie on M ). The differential equation of the momentum trajectories will indeed be extracted from D1 by means of (5.4) and (4.14), thus obtaining |T πQ (z)| Xφ τM1 (z) H = z ∈ T M1 | T πQ (z) ∈ M , z = m (remark that satisfies every z belonging to the above H automatically equalities ∗ L T πQ (z) = |T πm g T π (z) = (g◦T π ◦X ) τ (z) = (g◦F φ) τT ∗ Q (z) = Q Q φ T Q Q (z)| τT ∗ Q (z)). The integral curves of H will all be deduced from the integral curves of 1 := Im( 1 Xφ ) by eliminating their distinguished parametrization. D m M1 1 ⊂ H and then each integral curve of D 1 To this end, first remark that D is an integral curve of H (the corresponding base integral curve turns out to be parametrized by proper time). Now let χ := k ◦ s (with s > 0) an orientation-preserving reparametrization 1 . of an integral curve k of D From 1 Im k ⊂ M1 , k˙ = Xφ ◦ k (6.4) m we obtain ˙ ⊂ M , χ˙ = Im χ ⊂ M1 , Im (T πQ ◦ χ)
|T πQ ◦ χ| ˙ Xφ ◦ χ m
(6.5)
s (since χ˙ = s (k˙ ◦ s) and T πQ ◦ χ˙ = m (g−1 ◦ χ)), which proves that χ is an integral curve of H. Conversely, if χ satisfies (6.5), then k := χ ◦ s−1 with s = |T πQ ◦ χ| ˙ is immediately seen to satisfy (6.4). 1 and all So the momentum trajectories of (Q, L) are the integral curves of D of their orientation-preserving reparametrizations.
7 Concluding remarks In conclusion, we have exposed a unitary approach to the dynamics of classical mechanical systems subject to constraints – all embodied in the implicit dynamic equation T – deriving from the degeneratness of their geometrical structures and/or imposed ‘a priori’ as (nonlinear) restrictions on position-velocities with or without constraint reactions. The focal result is to have exhibited – for such systems – a mathematical structure shared by Lagrangian and Hamiltonian dynamics, both being governed by laws extracted, through suitable second-order conditions, from certain generalized Dirac equations in implicit form.
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For regular systems, we have seen that such second-order conditions are hidden by the fact that they turn out to be incorporated into the relevant Dirac equations. For nonregular systems (such as in Relativity), Dirac equations may not incorporate second-order conditions and then fail – on their own – to express the laws of dynamics. The explicit intervention of second-order conditions – well known, in a form or another, on the Lagrangian side of dynamics – had never been pointed out before (as far as we know) on the Hamiltonian side. The consequence of the above situation is that, on the one hand, generalized Dirac equations are the common area where Lagrangian and Hamiltonian problems – such as integrability, symmetries and conserved momentum mappings, reductions and reconstructions, evolution laws of observables and quantization – should firstly be treated, but, on the other hand, all of the possible results should then be adapted to real dynamics by taking second-order conditions into due consideration.
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[25] M.R. Menzio and W.M. Tulczyjew, Infinitesimal symplectic relations and generalized Hamiltonian dynamics, Ann. Inst. Henri Poincar´e, 28 (4), 349–367 (1978). [26] C. Møller, The theory of Relativity, Clarendon Press, Oxford, 1952. [27] W. Sarlet, F. Cantrijn and D.J. Saunders, A geometrical framework for the study of nonholonomic Lagrangian systems, J. Phys. A.: Math. Gen. 28, 3253– 3268 (1995). [28] W.M. Tulczyjew, Geometric Formulations of Physical Theories, Bibliopolis, Napoli, 1989. [29] A.J. Van Der Schaft, Implicit Hamiltonian systems with symmetry, Rep. Math. Phys. 41, 203–221 (1998). [30] A.M. Vershik and L.D. Fadeev, Differential geometry and Lagrangian mechanics with constraints, Soviet Physics-Doklady 17 (1), 34–36 (1972). [31] A.M. Vershik and L.D. Fadeev, Lagrangian mechanics in invariant form, Sel. Math. Sov. 1, 339–350 (1975). F. Barone Dipartimento Interuniversitario di Matematica Universit´ a di Bari Via Orabona, 4 70125 Bari Italy R. Grassini Dipartimento di Matematica e Applicazioni R. Caccioppoli Universit´ a di Napoli Federico II Via Cintia, Monte S. Angelo 80126 Napoli Italy Communicated by Vincent Rivasseau submitted 29/12/00, accepted 08/09/01
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Ann. Henri Poincar´e 3 (2002) 331 – 345 c Birkh¨ auser Verlag, Basel, 2002 1424-0637/02/020331-15 $ 1.50+0.20/0
Annales Henri Poincar´ e
Perturbations of the Wigner–Von Neumann Potential Leaving the Embedded Eigenvalue Fixed J. Cruz-Sampedro∗, I. Herbst† and R. Mart´ınez-Avenda˜ no
Abstract. We investigate the Schr¨ odinger operator H = −d2 /dx2 +(γ/x) sin αx+V , acting in Lp (R), 1 ≤ p < ∞, where γ ∈ R \ {0}, α > 0, and V ∈ L1 (R). For |γ| ≤ 2α/p we show that H does not have positive eigenvalues. For |γ| > 2α/p we show that the set of functions V ∈ L1 (R), such that H has a positive eigenvalue embedded in the essential spectrum σess (H) = [0, ∞), is a smooth unbounded sub-manifold of L1 (R) of codimension one. R´esum´e. On examine l’op´erateur de Schr¨ odinger H = −d2 /dx2 + (γ/x) sin αx + V u γ ∈ R \ {0}, α > 0, et V ∈ L1 (R). Si |γ| ≤ d´ efini dans Lp (R), 1 ≤ p < ∞, o` 2α/p, on montre que H n’a aucune valeur caract´eristique positive. Si |γ| > 2α/p, on montre que l’ensemble des fonctions V ∈ L1 (R), telles que H a une valeur caract´eristique positive immerg´ee dans le spectre essentiel σess (H) = [0, ∞), est a un. une sous-vari´et´ e lisse non-born´ee de L1 (R) de codimension ´egale `
1 Introduction In this paper we consider Schr¨ odinger operators of the form HQ,p = −
d2 + Q, dx2
(1.1)
acting in Lp (R), 1 ≤ p < ∞, where Q = W + V , W (x) = (γ/x) sin αx, α > 0 and γ ∈ R \ {0} are constants, and V is a real-valued function in L1 (R). To give a precise definition of the operator HQ,p we use the Feynman-Kac formula. For f ∈ ∪p≥1 Lp (R) and t ≥ 0 we define t Q(b(s))ds f (b(t)) , UQ (t)f (x) = Ex exp −
(1.2)
0
where Ex denotes the expectation with respect to Brownian motion starting at x with Brownian transition function given by exp −(x − y)2 /4t √ , x, y ∈ R, t ≥ 0. (1.3) pt (x, y) = 4πt ∗ Research † Research
partially supported by CONACYT, 32146-E, Mexico partially supported by NSF grant DMS-96000056.
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We define HQ,p to be the negative of the infinitesimal generator of the C0 - semigroup (UQ,p (t); t ≥ 0), 1 ≤ p < ∞, defined for f ∈ Lp (R) and t ≥ 0 by UQ,p (t)f = UQ (t)f . Various classes of operators which contain the ones defined above have been investigated in [6, 12, 16, 17, 18, 26, 28, 30], and it is well known that the spectrum σ(HQ,p ) is p-independent and that σess (HQ,p ) = [0, ∞) for all p ≥ 1. Schr¨ odinger operators of the form (1.1) were introduced by Wigner and VonNeumann [29] in order to construct an example of a Schr¨ odinger operator, acting in L2 (R3 ), with a spherically symmetric potential which vanishes at infinity and possesses a positive eigenvalue embedded in the continuum. The significance of the Wigner-Von Neumann example lies in the fact that at the time it contradicted physical intuition, which predicted that bound states of positive energy could not occur if the potential tended to zero at infinity. In this paper we study the structure of the set of functions V ∈ L1 (R) for which the operator HQ,p has a positive eigenvalue. Our main result is: Theorem 1.1 Let HQ,p be as in (1.1). If |γ| ≤ 2α/p, then HQ,p does not have positive eigenvalues. If |γ| > 2α/p, then the set of functions V ∈ L1 (R) such that HQ,p has a positive eigenvalue embedded in the essential spectrum σess (HQ,p ) = [0, ∞) is a smooth unbounded sub-manifold of L1 (R) of codimension one. In addition, if V belongs to this sub-manifold then α2 /4 is the unique positive eigenvalue of HQ,p . It is well known that the eigenvalues in the discrete spectrum are, in an appropriate setting, stable under perturbations. On the other hand, it is also known that embedded eigenvalues in the continuum are rather unstable [1, 2, 9, 10, 20]. In [2] Agmon, Herbst, and Skibsted prove that generically, in a Baire category sense, arbitrarily small perturbations of a generalized N -body Hamiltonian remove all non-threshold eigenvalues embedded in the continuum, and conjecture that the set of perturbations that preserve a non-threshold embedded eigenvalue is something like a differentiable manifold. The result presented in this paper shows that the above conjecture is true for the simplest Schr¨ odinger operators which possess an eigenvalue embedded in the continuum. A similar result for p = 2 was announced in [11] without proof. For α > 0 and γ ∈ R \ {0}, let M (α, γ) be the set of functions V ∈ L1 (R) such that, for some k > 0, the differential equation −ψ + γ
sin αr ψ + V ψ = k 2 ψ, r
r ∈ R,
(1.4)
has a nonzero solution that goes to zero as |r| goes to infinity. We say that a function ψ is a solution of this differential equation if it is continuously differentiable, ψ is absolutely continuous, and (1.4) holds almost everywhere. Local existence of solutions to (1.4) is well known. We also prove
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Theorem 1.2 Let M (α, γ) be as defined above. Then M (α, γ) is a smooth unbounded sub-manifold of L1 (R) of codimension one. In addition, if V ∈ M (α, γ) then k = α/2. Using the terminology of [19], M (α, γ) is the set of functions V in L1 (R) such that HQ,2 has a half-bound state of positive energy. To prove the results stated above we determine, following Cassell [7], the exact asymptotic behavior at infinity of the solutions to (1.4) and then identify the set of functions V in L1 (R) that produce positive eigenvalues of HQ,p with the zero set of a smooth function on L1 (R) for which zero is a regular value. Schr¨ odinger operators with eigenvalues in the continuous spectrum have also been investigated in [3, 14, 23], and the asymptotic behavior of the solutions of (1.4) for various classes of potentials has also been studied in [4, 7, 13, 15, 22]. For perturbations of embedded eigenvalues in situations which are relevant to the automorphic Laplacian and N -body Schr¨ odinger operators see [2, 5, 9, 10, 20, 25]. In a different context, results of the type presented here have been obtained in [21]. This paper is organized as follows. In Section 2 we investigate the asymptotic behavior at infinity of solutions to (1.4) and establish the existence of solutions that vanish at infinity. In Section 3 we prove the main results. In the Appendix we establish the connection between the eigenfuctions of HQ,p and the solutions of (1.4) that belong to Lp (R). Acknowledgments. We thank S. Sontz for useful comments.
2 Existence of Solutions that Vanish at Infinity In this section we follow Cassell [7] to determine the asymptotic behavior as r goes to infinity of the solutions to (1.4). We will prove Theorem 2.1 For α > 0, γ ∈ R \ {0}, k > 0, and V ∈ L1 (R) we have: i) If k = α/2, then (1.4) has solutions φ and ψ such that, as r goes to +∞,
with
φ(r) = cos kr + o(1)
and
ψ(r) = sin kr + o(1),
φ (r) = −k sin kr + o(1)
and
ψ (r) = k cos kr + o(1).
ii) If k = α/2, then (1.4) has solutions φ and ψ such that, as r goes to +∞, φ(r) = r−γ/2α (cos kr + o(1))
and
ψ(r) = rγ/2α (sin kr + o(1)),
φ (r) = −kr−γ/2α (sin kr + o(1))
and
ψ (r) = krγ/2α (cos kr + o(1)).
with
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Proof. Setting ξ(r) = ψ(r/k), σ = α/k > 0, and η = γ/k ∈ R \ {0} we find that ξ satisfies sin σr ξ + W ξ = ξ, (2.1) −ξ + η r where W (r) = V (r/k)/k 2 ∈ L1 (R). Using the transformation cos r − sin r ξ x= , sin r cos r ξ we see that (2.1) is equivalent to x = A(r)x,
where A(r) = a(r) and a(r) = −η
(2.2)
sin2 r , − sin r cos r
sin r cos r − cos2 r
sin σr − W (r). r
Next we write A(r) = (η/r)G(r) + R(r), where R(r) is the L1 -matrix given by
sin r cos r R(r) = −W (r) − cos2 r and G(r) ≡ with
g1 (r) g3 (r)
sin2 r , − sin r cos r
(2.3)
g2 (r) , −g1 (r)
(2.4)
1 g1 (r) = − (cos(σ − 2)r − cos(σ + 2)r), 4 1 g2 (r) = − (2 sin σr − sin(σ + 2)r − sin(σ − 2)r), 4
and
1 (2 sin σr + sin(σ + 2)r + sin(σ − 2)r). 4 Now we decompose G as G = G1 + G2 , where G1 = 0 and G2 = G for σ = 2, and 1 −4 0 G1 = 1 0 4 g3 (r) =
and
for σ = 2.
1 G2 (r) = 4
cos 4r 2 sin 2r + sin 4r
−2 sin 2r + sin 4r − cos 4r
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Setting S(r) = I + (η/r)G∗2 , a crude approximation to a solution of S = (η/r)G2 S, where r G∗2 (r) ≡ G2 (u)du, 0
−1
we find that if a is large then S(r) ∞. Hence setting
exists for r ≥ a and sup{S(r)−1 : r > a} <
˜ = S −1 ((η/r)2 (GG∗2 − G∗2 G1 ) + RS + (η/r2 )G∗2 ) R ˜ we have that ˜ ∈ L1 (a, ∞) and defining B = (η/r)G1 + R we have that R SB = AS − S .
(2.5)
Therefore setting x = S(r)y and using (2.5) we find that (2.2) is equivalent to y = B(r)y.
(2.6)
To finish the proof we proceed as follows: ˜ ∈ L1 (a, ∞). Hence, proceeding as in the proof of i) If σ = 2 then B = R Theorem XI.65 of [22] we find that, as r goes to +∞, (2.6) has a fundamental matrix X = I + o(1), where I denotes the 2 × 2 identity matrix. Thus (2.1) has solutions ψ1 , ψ2 such that ψ1 (r) = cos r + o(1), ψ1 (r) = − sin r + o(1),
ψ2 (r) = sin r + o(1), and
ψ2 (r) = cos r + o(1),
from which i) of Theorem 2.1 follows. ii) If σ = 2 and γ > 0, then the change of variables τ = η log r transforms (2.6) into dϕ = (G1 + L) ϕ, (2.7) dτ where L is in L1 (τ0 , ∞) for some τ0 independent of V . It is easily verified that this last system of O.D.E.s satisfies the conditions of a theorem due to Levinson. See Theorem 8.1 in Ch. 3 of [8]. Thus, as τ goes to +∞, (2.7) has solutions ϕ1 and ϕ2 such that 1 0 lim exp(τ /4)ϕ1 (τ ) = , and lim exp(−τ /4)ϕ2 (τ ) = . 0 1 τ →∞ τ →∞ Hence (2.6) has solutions of the form 1 y1 = r−η/4 + o(1) , 0
y2 = rη/4
0 + o(1) , 1
(2.8)
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and therefore (2.1) has solutions ψ1 , ψ2 such that ψ1 (r) = r−η/4 (cos r + o(1)), ψ1 (r) = −r−η/4 (sin r + o(1)),
ψ2 (r) = rη/4 (sin r + o(1)), and
ψ2 (r) = rη/4 (cos r + o(1)).
Changing the signs that need to be changed, we see that the same result is true when γ < 0, and thus ii) of Theorem 2.1 follows. Remark. Clearly an analogous result holds in a neighborhood of −∞.
3 Proof of the main result First we prove Theorem 1.2 and then Theorem 1.1. Proof of Theorem 1.2. Clearly we may assume γ > 0. Let M (α, γ) be as in the statement of Theorem 1.2. First we show that M (α, γ) is nonempty. By Theorem 2.1, for any given V ∈ L1 (R) we may choose nonzero solutions ψ− and ψ+ of (1.4) with k = α/2, such that ψ− (r) goes to zero as r goes to −∞, and ψ+ (r) goes to zero as r goes to +∞. By the same theorem we can also choose a > 0 such that ψ− (−a)ψ+ (a) > 0. Now, if we define ψ(r) = ψ− (r) for r ≤ −a, ψ(r) = ψ+ (r) for r ≥ a, and ψ(r) = ϕ(r) for |r| ≤ a, where ϕ is any C 2 function of constant sign that smoothly joins ψ− and ψ+ on [−a, a], and set V˜ (r) = V (r) for |r| ≥ a and V˜ (r) = (α2 /4) − (γ/r) sin αr + ϕ /ϕ for |r| ≤ a, then V˜ ∈ L1 (R) and ψ is a nonzero continuously differentiable function which goes to zero as |r| goes to infinity and satisfies −ψ + γ
sin αr ψ + V˜ ψ = (α2 /4)ψ, r
a.e in R.
Hence V˜ ∈ M (α, γ). In addition, it follows from the construction of V˜ that M (α, γ) is unbounded in L1 (R). Furthermore, if V belongs to M (α, γ), then in view of Theorem 2.1 we must have k = α/2. To complete the proof of Theorem 1.2 we only need to show that M (α, γ) is a smooth sub-manifold of L1 (R) of codimension one. This is proved in the following lemma. Lemma 3.1 Let M (α, γ) be as in Theorem 1.2. Then there exists a C ∞ function F : L1 (R) −→ R such that zero is a regular value of F and M (α, γ) = F −1 ({0}). Proof. For every V ∈ L1 (R) let ψ+ be the solution of −ψ + γ
sin αr ψ + V ψ = (α2 /4)ψ, r
r ∈ R,
(3.1)
which coincides for large positive r with the function φ given in (ii) of Theorem 2.1. First we will show that ψ+ and ψ+ depend smoothly on V . For r0 ∈ R, let
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Xr0 be the Banach space of continuous functions ϕ from [r0 , ∞) into R2 with the norm ϕr0 ≡ sup ϕ(τ ) exp (τ /4) < ∞. τ ≥r0
with respect to V It is easy to verify that the smoothness of ψ+ and ψ+ follows from the fact that for all r0 ∈ R, the solution ϕ1 of (2.7) that satisfies 1 lim exp(τ /4)ϕ1 = , 0 τ →+∞
is a smooth a function of L ∈ L1 (R) with values in Xr0 . To prove this last define Φ : L1 (R) × Xτ0 → Xτ0 as ∞ Φ(L, ϕ)(τ ) = ϕ(τ ) − ψ1 (τ ) + Ψ(τ )Ψ−1 (s)L(s)ϕ(s)ds, τ
1 , and where, τ0 is as in (2.7), ψ1 (τ ) = exp(−τ /4) 0 exp(−τ /4) 0 Ψ(τ ) = . 0 exp(τ /4) Note that Φ(L, ϕ) = 0 if and only if ϕ = ϕ1 . Next we fix L0 ∈ L1 (R) and let ϕ0 ∈ Xτ0 be so that Φ(L0 , ϕ0 ) = 0. We prove that if τ0 is sufficiently large, then Φ(L, ϕ) = 0 implicitly defines ϕ1 as a smooth function of L, with values in Xτ0 , on a neighborhood of L0 . Since Φ(·, ·) is jointly smooth, by the implicit function theorem it suffices to show that for τ0 sufficiently large the operator d2 Φ(L0 , ϕ0 ) is invertible from Xτ0 onto Xτ0 . Clearly for all h ∈ Xτ0 ,
(d2 Φ(L0 , ϕ0 ))(h) = h + P (h),
where P (h)(τ ) ≡
∞
τ
Ψ(τ )Ψ−1 (s)L0 (s)h(s)ds.
Using the definition of Ψ it is easily verified that ∞ L0 (s)ds, P (h)τ0 ≤ hτ0 τ0
for all h ∈ Xτ0 ,
from which the invertibility of d2 Φ(L0 , ϕ0 ) for large τ0 follows. Thus ϕ1 is smooth in L in a neighborhood O of L0 in the Banach space Xτ0 . Since ϕ1 (τ0 ) is smooth in L, the smoothness of ϕ1 as a function from O to Xr0 follows from the fact that the solutions to the initial value problem dϕ = (G1 + L)ϕ, dτ
ϕ(τ0 ) = ϕ1 (τ0 ),
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are smooth in L and the initial value ϕ1 (τ0 ). This last can be proved in the usual way using the integral equation. Analogous arguments show that the solution ψ− of (3.1) that satisfies ψ− (r) = |r|−γ/2α (cos kr + o(1))
ψ− (r) = −k|r|−γ/2α (sin kr + o(1)),
and
. as r → −∞, is a smooth function of V and so is ψ−
Now we define F : L1 (R) → R as ψ (r, V ) F (V ) = + ψ+ (r, V )
ψ− (r, V ) . ψ− (r, V )
This function is well defined since the Wronskian of any pair of solutions of (3.1) is constant as a function of r. Moreover V ∈ M (α, γ) if and only if F (V ) = 0; or equivalently, if and only if ψ− = λψ+ , where λ = 0 is a function of V , constant as a function of r. Thus M (α, γ) = F −1 ({0}). Since F is a smooth function of V , to finish the proof it remains to show that zero is a regular value of F , that is to say that for every V ∈ M (α, γ) we have dF (V ) = 0. Differentiating F with respect to V we find that for every V and h in L1 (R) we have dψ (r, V )(h) ψ− (r, V ) ψ+ (r, V ) dψ− (r, V )(h) , dF (V )(h) = + + (3.2) (r, V )(h) ψ− (r, V ) ψ+ (r, V ) dψ− (r, V )(h) dψ+ where d indicates differentiation with respect to V and the prime differentiation with respect to r, with V fixed. In order to prove that dF (V ) is not zero we note first that, for any fixed a ∈ R, (r, V ) = ψ+ (a, V ) + ψ+
r
a
Λ(t, V )ψ+ (t, V )dt,
where Λ(r, V ) ≡ (γ/r) sin αr + V − (α2 /4). Using the fact that for any interval [c, d] the map V → ψ+ (·, V ), from L1 (R) to the space C[c, d] is smooth we have r r (r, V )(h) = dψ+ (a, V )(h) + h(t)ψ+ (t, V )dt + Λ(t, V )dψ+ (t, V )(h)dt. dψ+ a
a
(r, V )(h) is absolutely continuous with derivative Thus dψ+ (r, V )(h)) = h(r)ψ+ (r, V ) + Λ(r, V )dψ+ (r, V )(h) (dψ+
a.e.
Now for any fixed b ∈ R and β ≥ b we consider β ψ+ (t, V )(dψ+ (t, V )(h)) dt b
= b
β
(ψ+ (t, V ))2 h(t) + Λ(t, V )ψ+ (t, V )dψ+ (t, V )(h)dt.
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Integrating by parts we also have β ψ+ (t, V )(dψ+ (t, V )(h)) dt b
β = dψ+ (t, V )(h)ψ+ (t, V ) − b
b
β
ψ+ (t, V )dψ+ (t, V )(h)dt.
2
Since ψ+ (t, V ) is not a jointly C -function of t and V , in order to perform another integration by parts we show first that for any t ∈ R and V and h in L1 (R) we have (t, V )(h) = (dψ+ (t, V )(h)) . (3.3) dψ+ t To prove (3.3) just note that ψ+ (t, V ) = ψ+ (c, V ) + ψ+ (τ, V )dτ . Since ψ+ (·, V ) c
is smooth in V as a function in C[c, d] for any d > c, we can differentiate with respect to V under the integral sign and obtain t dψ+ (τ, V )(h)dτ, dψ+ (t, V )(h) = dψ+ (c, V )(h) + c
(τ, V dψ+
)(h) is continuous in τ . Thus (3.3) follows imwhere for fixed V and h, mediately. So another integration by parts yields β (ψ+ (t, V ))2 h(t) + Λ(t, V )ψ+ (t, V )dψ+ (t, V )(h))dt = b β β (t, V )(h) − ψ+ (t, V )dψ+ (t, V )(h) ψ+ (t, V )dψ+ b b β + Λ(t, V )ψ+ (t, V )dψ+ (t, V )(h)dt, b
which gives β β 2 ψ+ (t, V ) h(t)dt = ψ+ (t, V )dψ+ (t, V )(h) − ψ+ (t, V )dψ+ (t, V )(h) . b
b
Taking β to infinity and utilizing the fact that for fixed V and h, the functions , dψ+ , and dψ+ all approach zero at infinity we obtain ψ+ , ψ+ ∞ ψ+ (t, V )2 h(t)dt = ψ+ (b, V )dψ+ (b, V )(h) − ψ+ (b, V )dψ+ (b, V )(h). b
Analogously, b ψ− (t, V )2 h(t)dt = ψ− (b, V )dψ− (b, V )(h) − ψ− (b, V )dψ− (b, V )(h). −∞
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Finally, combining (3.2) with these last two identities and using the fact that for V ∈ M (α, γ) we have ψ− = λψ+ , λ = 0, we find that ∞ 1 b 2 dF (V )(h) = λ ψ+ (t, V ) h(t)dt + ψ− (t, V )2 h(t)dt λ b −∞ ∞ = ψ− (t, V )ψ+ (t, V )h(t)dt, −∞
for all h ∈ L1 (R). Therefore, if V ∈ M (α, γ) then dF (V ) is the linear functional on L1 (R) defined by the function ψ− ψ+ ∈ L∞ (R) \ {0}. Proof of Theorem 1.1. For p ≥ 1, α > 0, and γ ∈ R \ {0}, let Mp (α, γ) be the set of functions V ∈ L1 (R) for which the operator HQ,p has a positive eigenvalue. It follows from Theorem 2.1 and Proposition A.1 that ∅, if |γ| ≤ 2α/p, Mp (α, γ) = M (α, γ), if |γ| > 2α/p.
A
Appendix
Here we establish the connection between the eigenfunctions of the operator HQ,p and the decaying solutions of (1.4). Below we use Duhamel’s formula [27] in the form t (UQ (t − u)ϕ, QU0 (u)f )du, (A.1) (ϕ, UQ (t)f ) = (ϕ, U0 (t)f ) − 0
C0∞ (R)
∞
p
and f ∈ L (R) ∩ L (R), where Q is as in (1.1) and UQ (t) is for ϕ ∈ as introduced in (1.2). Formula (A.1) is readily established by an approximation argument starting with bounded Q. Here ∞ (φ, ψ) = φ(x)ψ(x)dx. −∞
The main result of this section is Proposition A.1 Let p, Q, and HQ,p be as in (1.1). Then f ∈ Lp (R) is an eigenfunction of HQ,p corresponding to the eigenvalue λ ∈ R if and only if f is a differentiable function that vanishes at infinity, such that f is absolutely continuous on every finite interval of R and −f + Qf = λf
a.e.
(A.2)
Proof. Suppose f ∈ Lp (R) is a differentiable function that vanishes at infinity, such that f is absolutely continuous on every finite interval of R and that (A.2) is satisfied. We will show that UQ,p (t)f = exp(−λt)f,
t ≥ 0.
(A.3)
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For any given ϕ ∈ C0∞ we define s ≥ 0.
g(s) = (ϕ, UQ (s)f ), We show first that for any s ≥ 0 we have lim
t→0+
g(s + t) − g(s) = −λg(s). t
Set ψ = UQ (s)ϕ. Using Duhamel’s formula and the fact [6] that (φ, UQ (s)ξ) = (UQ (s)φ, ξ), for φ ∈ Lp (R) and ξ ∈ Lp (R), we find that g(s + t) − g(s) t
= =
UQ (t) − 1 ψ, f t U0 (t) − 1 ψ, f t t 1 − (UQ (t − u)ψ, QU0 (u)f )du. t 0
(A.4)
We show next that as t → 0+ the right side of (A.4) goes to (ψ, f ) − (ψ, Qf ) = −λg(s). In fact, using the kernel pt (x, y) of U0 (t) introduced in (1.3) we have 1 ∞ U0 (t) − 1 f = ψ, pt (x, y)(f (y) − f (x))dy ψ, t t −∞ ∞ 1 = ψ, pt (0, y)(f (x + y) − f (x))dy t −∞ y ∞ 1 = ψ, pt (0, y) (y − u)f (x + u)dudy 0 t −∞ ∞ y (f (x + u) − f (x)) dudy , = ψ, f + pt (0, y) (y − u) t −∞ 0 where in the third equality we have used Taylor’s formula y f (x + u) − f (x) = yf (x) + (y − u)f (x + u)du. 0
√ √ Setting z = y/ t and then u = tw we find that
∞
y (f (x + u) − f (x)) dudy = pt (0, y) (y − u) t −∞ 0 z ∞ p1 (0, z) (z − w)(f (x + τ w) − f (x))dw dz, −∞
where τ ≡
√
t.
0
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Hence ∞ y (f (x + u) − f (x)) ψ, dudy pt (0, y) (y − u) t −∞ ∞ 0 ∞ |z| 1 2 ≤ √ |ψ(x)| exp(−z /4) 2|z||f (x + τ w) − f (x)|dw dz dx 4π −∞ −∞ −|z| ∞ ∞ ∞ 1 ≤ √ |ψ(x)| 2z exp(−z 2 /4)|f (x + τ w) − f (x)|dz dw dx 4π −∞ −∞ |w| ∞ ∞ 2 = √ |ψ(x)| exp(−w2 /4)|f (x + τ w) − f (x)|dw dx π −∞ −∞ Thus, using (A.2), the fact that f ∈ Lp (R) ∩ L∞ (R), and the dominated convergence theorem we see that the right side of the last inequality goes to zero as t → 0+ and therefore U0 (t) − 1 lim ψ, f = (ψ, f ). t t→0+ By the continuity of the function (UQ (u)ψ, Qf ) with respect to u, the second term on the right side of (A.4) approaches −(ψ, Qf ) since 1 t
0
t
(UQ (t − u)ψ, Q(U0 (u) − 1)f ) du
goes to zero as t → 0+ . To see this last we use the fact that UQ (t) maps L∞ (R) into L∞ (R) and that UQ (t)L∞ →L∞ ≤ C, for small t, that Q = W + V , with V ∈ L1 (R) and W ∈ Lp (R), for p > 1, and that (U0 (t) − 1)f converges uniformly to zero as t → 0+ since f vanishes at infinity and hence is uniformly continuous on R. Thus we have proved that the right derivative D+ g of the function g(s) satisfies D+ g(s) = −λg(s), for all s ≥ 0. It follows that D+ (exp(λs)g(s)) = 0 for s ≥ 0 and therefore [24] that g(s) = exp(−λs)g(0), which proves (A.3). Suppose now that f ∈ Lp (R) satisfies (A.3). Then [6, 26] f ∈ L∞ (R), f is continuous, vanishes at infinity, and f exists and belongs to L2loc (R). By Duhamel’s formula, for every ϕ ∈ C0∞ (R) we have (ϕ, UQ (t)f ) = (ϕ, U0 (t)f ) −
0
t
(UQ (t − u)ϕ, QU0 (u)f ) du.
Differentiating this last at t = 0, using (A.3), we obtain −λ(ϕ, f ) = (ϕ , f ) − (ϕ, Qf ) and thus (ϕ , f ) = (ϕ, (λ − Q)f ) for all ϕ ∈ C0∞ (R). Standard approximation arguments show that f is almost everywhere equal to an absolutely continuous function and that (A.2) is satisfied.
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References [1] S. Agmon, On Perturbation of Embedded Eigenvalues, Partial Differential Equations and Mathematical Physics (Copenhagen, 1995; Lund, 1995), 1–14 Progr. Nonlinear Differential Equations Appl., 21, Birkh¨ auser Boston, Boston, MA, 1996. [2] S. Agmon, I. Herbst and E. Skibsted, Perturbations of Embedded Eigenvalues in the Generalized N-body Problem, Comm. Math. Phys. 122, 411–438 (1989). [3] M. Arai and J. Uchiyama, On the von Neumann and Wigner Potentials, J. Diff. Eq. 157 2, 348–372 (1999) . [4] F. Atkinson, The Asymptotic Solutions of Second Order Differential Equations, Ann. Mat. Pura Appl. 37, 347–378 (1954). [5] E. Balslev and A. Venkov, Selberg’s Eigenvalue Conjecture and the Siegel Zeros, Aarhus University preprint, 1998. [6] R. Carmona, Regularity Properties of Schr¨ odinger and Dirichlet Semigroups, J. Funct. Anal. 33, 259–296 (1979). [7] J. S. Cassell, The Asymptotic Integration of some Oscillatory Differential Equations, Quart. J. Math. Oxford (2), 33, 281–296 (1982). [8] E. Coddington and N. Levinson, Theory of Ordinary Differential Equations, McGraw-Hill, 1955. [9] Y. Colin de Verdiere, Pseudo-Laplaciens I, Ann. Inst. Fourier, Grenoble, 32, 3, 275–286 (1982). [10] Y. Colin de Verdiere, Pseudo-Laplaciens II, Ann. Inst. Fourier, Grenoble, 33, 2, 87–113 (1983). [11] J. Cruz and R. Mart´ınez, Perturbations of the Wigner-von Neumann Potential, Memorias del XXVII Congreso Nacional, Soc. Mat. Mex., Aport. Matem. 16, 1995. [12] E. B. Davies, Lp Spectral Independence and L1 Analyticity, J. London Math. Soc. (2) 52, 1, 177–184 (1995). [13] J. Dollard and C. Friedman, Product Integrals and the Schr¨ odinger Equation, J. Math. Phys. 18, 1598–1607 (1977). [14] M. Eastham and H. Kalf, Schr¨ odinger-type Operators with Continuous Spectra, Pitman Advanced Publishing Program, 1982. [15] W. Harris and D. A. Lutz, A Unified Theory of Asymptotic Integration, J. Math. Anal. Appl. 57, 3, 571–586 (1977).
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[16] R. Hempel and J. Voigt, The Spectrum of a Schr¨ odinger Operator in Lp (Rν ) is p-Independent, Comm. Math. Phys. 104, 243–250 (1986). [17] R. Hempel and J. Voigt, On the Lp -Spectrum of Schr¨ odinger Operators, J. Math. Anal. Appl. 121, 138–159 (1987). [18] I. W. Herbst and A. D. Sloan, Perturbations of Translation Invariant Positivity Preserving Semigroups on L2 (RN ), Transactions of the American Mathematical Society, 236, 325–360 (1978). [19] D. B. Hinton, M. Klaus and K. Shaw, Embedded Half Bound States for Potentials of Wigner-von Neumann Type, Proc. London Math. Soc. (3) 62, 3, 607–646 (1991). [20] R. Phillips and P. Sarnak, Automorphic Spectrum and Fermi’s Golden Rule, Journal D’analyse Math´ematique, Vol. 59 (1992). [21] J. P¨ oschel and E. Trubowitz, Inverse Spectral Theory, Academic Press Inc., 1978. [22] M. Reed and B. Simon, Methods of Modern Mathematical Physics, Vol. I-IV, Academic Press Inc., 1978. [23] P. Rejto and M. Taboada, A Limiting Absorption Principle for Schr¨ odinger Operators with Generalized Von Neumann-Wigner Potentials II. The Proof, J. Math. Anal. Appl. 208, 2, 311–336 (1997). [24] L. Royden, Real Analysis, Macmillan Publishing Co., 1968, Second Edition. [25] B. Simon, Resonances in n-body Quantum Systems with Dilation Analytic Potentials and the Foundations of Time-Dependent Perturbation Theory, Ann. of Math. 97, 247–274 (1973). [26] B. Simon, Schr¨ odinger Semigroups, Bulletin (New Series) of the AMS, 7, 3, 447–526, (1982). [27] M. Taylor, Partial Differential Equations I, Applied Math Sciences 115, Springer, 1996. [28] J. Voigt, Absorption Semigroups, their Generators, and Schr¨ odinger Semigroups, J. Funct. Anal. 67, 167–206 (1986). ¨ [29] J. Von Neumann and E. P. Wigner, Uber Merkw¨ urdige Diskrete Eigenwerte Z. Phys., 30, 465–467 (1929). [30] R. Weder, The Unified Approach to Spectral Analysis, Comm. Math. Phys. 60, 291–299 (1978).
Vol. 3, 2002
Perturbations of the Wigner–Von Neumann Potential
J. Cruz-Sampedro Instituto de Ciencias B´ asicas e Ingenier´ıa Pachuca, Hgo., 42090 Mexico email: [email protected] I. Herbst Department of Mathematics University of Virginia Charlottesville, VA 22903 U.S.A. email: [email protected] R. Mart´ınez-Avenda˜ no Department of Mathematics Michigan State University East Lansing, MI 48824-1020 U.S.A. email: [email protected] Communicated by Gian Michele Graf submitted 26/01/01, accepted 18/05/01
To access this journal online: http://www.birkhauser.ch
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Ann. Henri Poincar´e 3 (2002) 347 – 387 c Birkh¨ auser Verlag, Basel, 2002 1424-0637/02/020347-41 $ 1.50+0.20/0
Annales Henri Poincar´ e
Mathematical Theory of Quantum Tunneling Decay at Positive Temperature I.M. Sigal and B. Vasilijevic
1 Introduction The goal of this paper is to initiate mathematical theory of tunneling at positive temperatures. The mathematical framework we develop here rests on the notion of free resonance energy, F (β) (where β is inverse temperature), which we introduce in this paper. This quantity plays the role of the resonance energy at zero temperature. We prove that the probability of escape of the particle from a potential well due to tunneling can be expressed as P (t) = 1 − p(t) where roughly : 2 tΓ Γt Γt e− ≤ p(t) ≤ e− + O (1.1) ∞ modulo O Λ2 + O t , where Λ is an exponentially small (in 1 ) quantity specified in Section 8 (see Theorem 8.1 below). Here is a small parameter related to the Planck constant divided by 2π and Γ = −2Im(F (β)) ,
(1.2)
the ”width” of the free energy. The formula above (but with F (β) understood on the basis of expression (1.5) for Γ given below) is taken for granted in physics literature and is used in condensed matter physics (see e.g. [A,BFGLV,CL,KL,La,LO13]) and cosmology (see e.g. [L1-2],[VS]) in order to analyze the tunneling process. However, it was never justified or analyzed systematically, not to mention rigorously. For instance, the expression for Γ (see (1.5) below) or for F (β) was never connected directly to the underlying Schroedinger operator as it is done in our work. Furthermore, we give a semi-classical bound on Γ: 0 < Γ ≤ C−6 e−
Sβ
(1.3)
for some C > 0. Here Sβ is the action of the instanton of period β (see (2.19) and discussion preceding it). Since the notion of temperature pertains to equilibrium states of systems with infinite number of degrees of freedom while the tunneling is obviously a nonequilibrium process and a quantum particle has only three degrees of freedom we
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have to clarify what we mean by tunneling at a positive temperature. The latter term refers to the process of tunneling of a particle which is either in a contact with a reservoir (i.e. a system of infinite degrees of freedom, e.g. photon or phonon gas) which at time t = 0 is in a state of equilibrium at temperature T , or which is initially thermalized. In the case of initial thermalization the temperature is introduced through the initial condition and has no other effect on the dynamics of the system. This corresponds to the following physical situation: the system is prepared by putting it in contact with a reservoir at temperature T, and at t=0 reservoir is removed and the particle system is left to evolve on its own, or, put differently, the effect of the reservoir on the particle is ignored. This physical situation is of interest in its own right as well as for the reason of giving a good approximation to the process of tunneling with a thermal reservoir included. Indeed, usually the coupling between the particle and the reservoir is, on one hand, sufficiently strong so that the reservoir maintains the particle ”inside the well” in a state of (approximate) equilibrium and, on the other hand, is sufficiently weak so that it yields only a small perturbation to the tunneling process. In other words, usually we have: Ttunneling Trelaxation Tpart ,
(1.4)
where Ttunneling = Γ−1 and Trelaxation = (coupling constant)−1 are the characteristic times of the tunneling and relaxation, respectively, and Tpart is the character istic time of the particle system, say ∆E , where ∆E is a mean gap between energy levels. Thus (for small enough temperatures) the coupling between the particle and the reservoir during the process of tunneling can be neglected in the leading approximation ([GWH]). Now, we address the resonance nature of the process of tunneling. Naively one can think about tunneling at positive temperature as follows. Initially the particle is inside the well and is in each of the ”well states” ψn with the Gibbs n probabilities Z −1 ·e−βE , where En is the energy of the state ψn and β is the inverse temperature and Z = n e−βEn the normalization factor. Since the particle can tunnel from each state ψn , these states must be replaced by the resonance states , where Enres ≈ En and Γtunneling with the resonance eigenvalues Enres − ıΓtunneling n n is the ”width” of the resonance level. The quantity 1 Γtunneling is interpreted as a n probability of decay per unit time due to the tunneling from the level ψn . Thus it is not surprising that the probability of tunneling per unit time at temperature T = β1 is given approximately by: 1
e−βEn Γtunneling n −βE n e
(1.5)
which coincides in the leading order semiclassically with our expression (1.2). In fact, this is the tunneling probability per unit time – expression (1.5) is derived in physics literature.
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In this paper we study the tunneling of initially thermalized particle and derive relations (1.1)-(1.3) in this case. This allows us to use a fairly elementary analysis based on spectral theory of self-adjoint and non-self-adjoint Schr¨odinger operators. Thus we consider a quantum particle in Rd moving under the influence of an external potential V (x), which is a real function on Rd . The dynamics of such a particle is described by the Schr¨odinger operator: H = −2 ∆ + V (x) . d acting in the space L2 Rd . Here ∆ = i=1 ∂x2i is the Laplacian operator on Rd , we use dimensionless units and consider (whose origin is in the Planck’s constant or coupling constant) as a dimensionless small parameter. (The latter is done for simplicity as there are other natural parameters related to the shape of potential barrier which do the same job as ). We also set mass to 12 . We assume that V (x) is such that H is a self-adjoint operator. Moreover we assume that V (x) has a local minimum at xo such that V (xo ) > lim|x|→∞ V (x) (see the figure below).
V (x)
Thus V (x) has a compact well region separated by a finite barrier from the unbounded domain in which the values of the potential are below the bottom of the well. (Technical restrictions on V (x) are given in Section 2.) We are interested in the probability of decay of states which are initially localized in the well. Since we are studying the dynamics at positive temperature, we have to consider mixed states of the particle. These are given by density matrices, ρ, i.e. non-negative, trace-class operators on L2 Rd normalized as T r(ρ) = 1. The evolution of these states is given by the von Neumann equation: ı ρ(t) ˙ = − [H, ρ(t)]
(1.6)
subject to the initial condition ρ(0) = ρo , where ρo is an approximate Gibbs state of the well (see Section 2 for an exact definition).
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The probability that the state ρ(t) which is initially localized in the well region, W , is still localized there at time t is given by: p(t) = T r(χρ(t)) ,
(1.7)
where χ is the characteristic function, χ = χ(W ), of the well W . Then the probability of the particle escaping from the well is given by P (t) = 1−p(t) = T r (χρ(t)), where χ = 1 − χ. p(t) is the quantity we study in this paper and for which we prove relations (1.1)-(1.2). Our paper is a first step on the road of understanding tunneling decay at positive temperatures. To get a glimpse of difficulties ahead, notice that the operator on the right hand side of evolution equation (1.6) is hyperbolic so, to begin with, analysis of this equation is a rather subtle question. In this paper we avoid facing difficult analytical issues by exploiting the commutator structure of the operator mentioned (which is equivalent to separation of variables). The next step would be to consider right hand sides which are perturbations of commutator operators since effective potentials coming from reservoirs are not commutators anymore. Fully including reservoir would come after that. Finally, we give a thumbnail sketch of history of the subject (for more details see [HuSi1]). Rigorous analysis of quantum resonances started with papers [AC,BC] which introduced the method of complex dilations and [S1] which used this method to initiate a rigorous theory of resonances. The dilation framework was extended to general vector-field transformations in [S2,Si1,Hu1,Cy]. The first rigorous treatment of the tunneling process was given in [S3] while the existence and estimates of tunneling resonances related to local minima of the potential (shape resonances) were proven in [CDKS, HeSj, HS1]. The dynamics of quasiclassical (and in particular tunneling) resonances was described in [BZ, MS] (see also earlier papers [Sk,H2,GS,SW]). The modern treatment of tunneling in physics literature began with papers [La,Co,LS] for T = 0 and [CL,A,LO1-2] for T > 0. The physics literature on the subject is immense and we refer the interested reader to a review [BFGLV] for some of the references. The text is organized as follows. In Section 2 we describe the class of Schr¨ odinger operators under consideration and our main results. In Section 3 we quickly review and elaborate on the theory of complex deformations. In Sections 45 we present our theory of resonance free energy for Schr¨ odinger operators. In Section 6 we prove exponential bounds on eigenfunctions of interest. In Section 7 we study the resonances and their spectral projections. In Section 8 we undertake our analysis of the decay probability P (t) and prove relation (1.1). In Section 9 we prove a quasiclassical bound (1.3) on the imaginary part of the resonance free energy. Now we list some standard notation we use in the text below: 1 2 2 , Au := u, Au and x := 1 + |x| χ(Ω) = characteristic f unction of a set Ω ⊂ Rd .
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For a closed hypersurface S in Rd , int(S) and ext(S) denote the interior and exterior domains of Rd separated by S. The symbol C stands for a constant (different in different inequalities) which is independent of all parameters involved, and in particular of , β and t. Acknowledgment. The authors are grateful to V.S. Buslaev for many illuminating discussions.
2 Hamiltonians and results In this section we describe the class of Schr¨odinger operators which we consider in this paper and present our main results. The Schr¨ odinger operators are given by the differential expressions: H = −2 ∆ + V (x) (2.1) extended to functions on L2 Rd . Here we use dimensionless variables, is a small (dimensionless) parameter descending from the Planck constant and the energy and length scales in the problem (for α-decay, is, very roughly, between 1/10 and 1/500 depending on the size of the nucleus) and V (x), the potential, is a real C 2 function such that the operator H (or, more precisely, its closure) is self-adjoint on its natural domain. Now we formulate a set of conditions on the potentials V (x), various subsets of which are required for different results of this paper: (A) (Local trapping) V (x) has a strict local minimum, say at x = 0, V (0) ≥ 0, and lim|x|→∞ V (x) exists and is less than V (0). (B) (Existence of a barrier) Condition (A) implies that there exists λb > λo = V (0) such that ∀E ∈ (λo , λb ):
x ∈ Rd | V (x) = E = Si (E) ∪ So (E) , where Si (E) and So (E) are codimension 1 disjoint hypersurfaces such that Si (E) ⊂ int(So (E)) (inner and outer classical turning surfaces at energy E). For simplicity we assume that Si (E) and So (E) are closed. (C) (Exterior analyticity) There is λ1 ∈ (λo , λb ) such that V (x) is So (λ1 )-exterior analytic in the sense that V is a restriction to Rd of a function, also denoted by V , which is analytic in the truncated cone ΓSo (λ1 ) , where: ΓS = {z ∈ Cd | Re(z) ∈ ext(S) , Im(z) ≤ α dist(Re(z), S) + α1 } ,
(2.2)
for some α, α1 > 0 and a smooth closed hypersurface S in Rd . (D) (Exterior non-trapping) V (x) is exterior non-trapping at energies λ ∈ [λo , λ1 ] in the sense that for all λ ∈ [λo , λ1 ] there exists a So (λ)-exterior (definition is given below) vector field v on Rd and numbers a, b, c > 0, c > a such that: −v · ∇V + a(λ − V ) ≥ b on ext (So (λ))
(2.3)
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and
1 Dv + Dv T ≥ c on ext (So (λ)) (2.4) 2 where Dv is the derivative (Jacobi matrix) of the vector field v. (E) (Technical condition) For a vector field vλ (x) as in (D) let ϕθ,λ (x) := x+θvλ (x) and Vθ,λ (x) := V (ϕθ,λ (x)). By condition (C), Vθλ (x) has an analytic continuation in θ from R into a strip { z ∈ C | |Im(z)| ≤ 4} for some 4 > 0 (see also Section 3). We assume that this continuation satisfies the estimates: Re(Vθ,λ (x)) ≥ −C1
(2.5)
|Im(Vθ,λ (x))| ≤ C2 |Im(θ)| ,
(2.6)
uniformly in θ in the strip above. (F) Either our system is one-dimensional or radially symmetric or the temperature is sufficiently low (an exact bound will be given below). We give the definition of exterior vector field (see conditions (C), (D)). Definition 2.1 (Exterior vector field) Let S be a smooth closed hypersurface in Rd . Then a smooth vector field v(x) is called exterior to a surface S if it has the following properties: • v(x) = 0 on and inside of S, • Dv(x) → 1 as |x| → ∞. Discussion of conditions (D) and (F) We give examples of potentials and vector fields for which Condition (D) is satisfied. Assume in the ext (So (λ)) a potential V (x) behaves as: V (x) = c + r−α (2.7) with c < λo = V (0) and α > 0 and assume that a vector field v is of the form v(x) = g(r)ˆ x where r = |x|. Then in this region we have, for any λ ≥ λo : −v · ∇V + a(λ − V ) = αg(r)r−α + a λ − c − r−α = (αg − a)r−α + a(λ − c) 1 a(λo − c) ≡ b > 0 ≥ 2 provided a ≤ αg or r is sufficiently large in this region. Thus (2.3) holds for these potentials and vector fields with b = 12 a(λo − c) > 0. Now we discuss Condition (F). Its origin is the following property used in this paper: Proposition 2.2 (Eigenvalue gap) Assume conditions (A), (B) and (F) hold. Then for all λ ∈ (λo , λ1 ) there exists a smooth potential Vλ : Rd → R with the following properties:
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a. Vλ (x) = V (x) for x ∈ int (So (λ )), b. Vλ (x) is strictly increasing to ∞ with |x| for x ∈ ext (So (λ )), c. Vλ is exterior analytic with respect to So (λ − δ), with δ 1, d. Eigenvalues of Hλ := −2 ∆ + Vλ (x) in (0, λ) are separated by at least c. Proof. Clearly we can find a potential Vλ satisfying conditions (a)-(c). A difficult part is to satisfy condition (d). For one-dimensional systems the eigenvalue gap property holds generically. Indeed, in that case, semiclassically the eigenvalues En = En () of the reference Hamiltonian Hλ := −2 ∆ + Vλ (x) are given by the Bohr-Sommerfeld quantization condition I(E) = 2πn, where I(E) is the action at energy E, p · dq ,
I(E) =
(2.8)
ϕE
with ϕE a classical trajectory at energy E. Thus for the gap ∆En := En+1 − En we have: 2π + o() . (2.9) ∆En = I (En ) Note that I(E) is the area inside the curve t → (x(t), p(t)) where x(t) and p(t) are the classical coordinate and momentum. The derivative, I (E), is bounded below and above for our range of energies. Hence we have that: C −1 ≤ ∆En ≤ C
(2.10)
for some constant C dependent only on the potential Vλ (x). In the case of radially symmetric potentials, the separation of variables in spherical coordinates reduces the problem to the one-dimensional case. Finally redoing the estimates of [HS,Section 11.3] while explicitly tracing out the constants we conclude that a potential Vλ satisfying (a) and (b) of Proposition 2.2 (i.e. having a single, global, non-degenerate minimum) has the (O()) eigenvalue gap property in the energy range 2
[λo , λo + 3 + ]
for some 4 > 0 ,
where λo = min Vλ (x). It is shown below (see Eqn.(2.20) and the paragraph 2 preceding it with λ1 = 3 + ) that this leads to the following restriction on β: β ≥ β1 , 2
where β1 is the ”period” of the trajectory at energy λ1 = 3 + .
The operator Hλ := −2 ∆ + Vλ (x) (see Proposition 2.2) will be called the reference Hamiltonian. Denote by Eλ k , ψλ k and Pλ k , eigenvalues, eigenfunctions
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and eigenprojections of Hλ , respectively. Without loss of generality we can choose Vλ to satisfy: γ |Vλ (x)| ≤ C x (2.11) for some γ > 0. Reference Hamiltonians Hλ will be used to approximate H in various ways (the most important of which will occur later in this section) and λ will be chosen depending on this approximation. Let S(x, y, τ ) be the action of the instanton - the classical particle in the potential −V (x) (i.e. moving in imaginary time) - going from x to y in time τ : τ 2 |ϕ(s)| ˙ + V (ϕ(s)) ds (2.12) S(x, y, τ ) = 0
where ϕ(s) is the classical path in the potential −V (x), 2ϕ¨ = ∇V (ϕ), starting at s = 0 at x and ending at s = τ at y. This action expressed as a function of the 2 instanton energy, −E = |ϕ| ˙ − V (ϕ), is given by A(x, y, E) := S(x, y, τ ) − τ E|τ :∂τ S=E .
(2.13)
Alternatively, due to the Jacobi theorem, A(x, y, E) is the geodesic distance, i.e. the length of minimal geodesic, between the points x and y in the Agmon Riemannian metric: ds2 = 4(V (x) − E)+ dx2 (2.14) (the Agmon distance). Conversely, S(x, y, τ ) is found by minimizing the function φ(x, y, E) := A(x, y, E) + τ E
(2.15)
with respect to E. In the standard way we define the Agmon distance between two sets, say B and C: A(B, C; E). Denote by A(E) the Agmon distance at the energy E between the turning surfaces Si (E) and So (E): A(E) := A (Si (E), So (E), E) .
(2.16)
For each β > 0 we define the energy Eβ as a minimizer of the function: φ(E, β) := 2A(E) + βE .
(2.17)
This energy satisfies the critical point equation: Eβ :
∂ (2A(E)) = −β . ∂E
(2.18)
Consequently, Eβ is the energy of instanton of the ”period” β (more precisely, β is the time of two trips between the turning surfaces) and Sβ := φ(Eβ , β) = S (Si (Eβ ), So (Eβ ), β) is the action of this instanton.
(2.19)
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Since A(E) is monotonically decreasing and since −2A (E) is the period, τ (E), of the periodic orbit at the energy E and therefore −A (λo ) = ∞, we conclude that for β sufficiently large, the function φ(E) has a minimum Eβ such that Eβ → λo as β → ∞. Furthermore under our assumptions the period τ (E) decreases as E increases, i.e. the function φ(E) is convex and consequently Eβ is a unique minimizer. Let τ1 = τ (λ1 ) = β1 be the period of the trajectory at the energy λ1 . Due to conditions (C)-(D) we will consider only the energies E ≤ λ1 . Hence the periods τ = β we deal with are bounded as τ = β ≥ τ1 = β1
(2.20)
i.e. β ≥ β1 . Next we formulate precisely the problem we consider. Now we can formulate the problem. The retaining probability p(t) (which is actually the probability that the particle is still in the well at time t) is: p(t) = tr(χwλ ρ(t))
(2.21)
with χwλ = χ (int (So (λ + 2δ))), characteristic function of int (So (λ + 2δ)), and where ρ(t) is the density operator at time t, i.e. the solution to the von Neumann equation: ı (2.22) ρ(t) ˙ = − [H, ρ(t)] with the initial condition ρ(0) = ρo , where ρo =
Eλ1 n ≤λ1
e−βEλ1 n Pλ1 n , Zo (β)
(2.23)
where Zo (β) = Eλ n ≤λ1 e−βEλ1 n . Density matrix (2.23) will be called the Gibbs 1 state at inverse temperature β in the well. Now we are ready to formulate our first main result: Theorem 2.3 Assume conditions (A)-(F) on the Hamiltonian H. To simplify the statement we also assume that β is sufficiently large so that technical condition (8.62) of Section 7 is satisfied. Let Γ = Γ(β) := −2Im(F (β)) where F (β) is the resonance free energy introduced in Definition 4.4 (Section 4). Then: 2 tΓ Γt Γt e− ≤ p(t) ≤ e− + O , (2.24) modulo O(−4 Λ2 ) + O tially small (in
1 )
∞ t
and with Γ = Γ (1 + O(Λ)). Here Λ is an exponen-
quantity to be specified in Theorem 8.1.
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Our second main result is: Theorem 2.4 Assume conditions (A)-(F) for the Hamiltonian H. Then: Γ(β) ≤ C−6 e−
Sβ
,
(2.25)
where Sβ is the action of the instanton of ”period β” (see Eqn. (2.19)). Theorems 2.3 and 2.4 are proven in Sections 8 and 9, respectively. Remark. There is a statement analogous to Theorem 2.4 in physics literature while there is a non-rigorous derivation, using Feynman path integrals, of asymptotics of a quantity closely related to Γ(β).
3 Deformation family for H Now, we construct a deformation family for the Hamiltonian described in the previous section. We will use Hunziker type (infinitesimal) deformations. Given λ ∈ [λo , λ1 ] let vλ be a So (λ)-exterior vector field on Rd . Let (θ ∈ R) ϕθ (x) = x + θvλ (x). Define a one-parameter group of unitary operators Uθ on L2 (Rd ) by: 1
and:
(Uθ ψ)(x) = (det(Dϕθ )) 2 ψ(ϕθ (x)) ,
(3.1)
Hθλ = Uθ HUθ−1 .
(3.2)
Due to condition (C) Hθλ has an analytic continuation, in θ, into a neighborhood of 0, Ω, and the resulting family (for which we keep the same notation Hθλ ) is an analytic family of type-A in the sense of Kato (for proof see [HS2,Corollary 18.5]). Then, Aguilar-Combes theorem tells us that ([HS2,Chapter 18]): σess (Hθλ ) = (1 + θ)−2 [0, ∞) + V (∞) and that the discrete spectrum of Hθλ , σd (Hθλ ), represents resonances of H (independent of the vector field vλ (x)). Note that the explicit expression for Hθλ is ([Si], [HS2, Chapter 18]): Hθλ = p · Aθ (x) · p + gθ (x) + Vθλ (x) ,
(3.3)
where we let Vθλ (x) = V (ϕθ (x)) and: Aθ (x) gθ (x)
Jθ (x)
= (Jθ (x))−1 (Jθ (x))−1T 2 2 −1 J (x)∇ ln det Jθ (x) = 4 θ 2 + div Jθ−1T (x)Jθ−1 (x)∇ ln det Jθ (x) 2 = Dϕθ (x) .
(3.4) (3.5)
(3.6)
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In what follows we will need the semigroup e−βHθλ , β ≥ 0. To define such a semigroup we establish the sectoriality property of Hθλ . Recall the definition of that property: Definition 3.1 A closed operator A is called sectorial if its numerical range N (A) = {u, Au|u ∈ D(A), u = 1} is contained in a sector opening to the right, i.e. π N (A) ⊆ z + a | |arg(z)| < − α (3.7) 2 for some 0 < α <
π 2
and a ∈ R.
Now, it is a standard result of the theory of semigroups ([Pa]) that sectorial operators generate differentiable Co -semigroups. Indeed, the Co -semigroup property is showed using Hille-Yosida theorem in conjunction with the standard resolvent estimate for closed operators: R(z, A) ≤ (dist(z, N (A)))−1 ,
(3.8)
for z ∈ N (A). Next we prove the sectoriality property of Hθλ : Proposition 3.2 Let H satisfy conditions (D) and (F). Then for |Im(θ)| sufficiently small, Hθλ is sectorial with the sector: { z ∈ C | |Im(z)| ≤ C |Im(θ)| (Re(z) + C1 )} , where C = 2 max (M, C2 ), M = 4 supx Dv T (x) + Dv(x) (· is the operator norm of a matrix). Proof. Subindex λ is fixed and omitted from the notation in this proof. Since Dϕθ = 1 + θDv we have (α = Im(θ)): DϕTθ Dϕθ = 1 + ıαDv T (1 + ıαDv) = 1 + ıα Dv + Dv T − α2 Dv T Dv and so DϕTθ Dϕθ is invertible for small |α| and its inverse is Aθ (x). 1 1 Next let A = 1 − α2 Dv T Dv, B = Dv T + Dv, K = A− 2 BA− 2 . Then: Aθ (x)
1
−1
=
A− 2 (1 + ıαK)
=
G∗ (1 − ıαK) G ,
1
1
A− 2 = A− 2
1 − ıαK − 1 A 2 1 + α2 K 2
− 1 1 where G = 1 + α2 K 2 2 A− 2 . For |α| = |Im(θ)| sufficiently small: Re (Aθ (x)) = G∗ G ≥ 12 1 and Im (Aθ (x)) = −αG∗ KG ⇒ Im (Aθ (x)) ≤ M α , i.e. −αM 1 ≤ Im (Aθ (x)) ≤ αM 1, where M = 4 supx B(x).
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Since Re (Vθ ) ≥ −C1 , |Im(Vθ )| ≤ C2 |α| and gθ is bounded by 14 p · Re (Aθ ) · p (cf. (3.5)) we have: Re Hθ u
p · Re (Aθ ) · pu + gθ u + Re (Vθ )u 1 2 2 pu − C1 u 4
= ≥
and Hence:
2 2 . |Im Hθ u | ≤ α M pu + C2 u Im Hθ + C1 + 1 2
≤
u
≤
2 2 α max(M, C2 ) pu + u 1 4α max(M, C2 )Re Hθ + C1 + , 2 u
which proves the proposition.
By a standard result (see [HS2,Theorem 18.6]), similar to the second part of Proposition 4.3 below, the isolated eigenvalues of Hθλ are independent of the vector field vλ used and therefore are independent of λ: zn are independent of λ .
(3.9)
4 Resonance partition function and resonance free energy In this section we introduce our key concept – the resonance free energy. To motivate our approach we review first the standard case of a quantum system in a confining (real) potential V (x), i.e. V (x) → ∞ as |x| → ∞ and is bounded below. In this case the Schr¨ odinger operator H = −2 ∆ + V (x) is self−βH adjoint and the heat semigroup e is of the trace-class for β > 0. As it is usual in dealing with systems at positive temperature (again, to justify what follows one brings the particle in contact with a thermal reservoir at the given temperature), we introduce the (quantum mechanical) partition function as: Z(β) = tr e−βH . (4.1) All thermodynamic quantities can be expressed in terms of Z(β). Of particular interest to us is the quantity known as the (Helmholtz) free energy at temperature T = β −1 which is defined as: F (β) = −
1 ln Z(β) . β
(4.2)
It plays the role of the ground state energy for open systems (i.e. it characterizes −βH the stable equilibrium), while the Gibbs state ρβ = eZ(β) plays the role of the ground state. This interpretation is supported by the following theorem:
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Theorem 4.1 (Feynman-Kac) Let H be a self-adjoint operator, bounded below with purely discrete spectrum. Then as β → ∞, F (β) → inf[σ(H)]
(4.3)
i.e. as temperature goes to zero the free energy converges to the ground state energy of the system. In fact something stronger is true: ρβ → Po ,
(4.4)
where Po is the spectral projection of H on the eigenspace associated with Eo . Thus the Gibbs state ρβ converges to the pure ground state Po . Now we consider the situation in which a particle in question is not confined and can (and does) escape to infinity. The latter process is characterized by the presence of continuum in the spectrum of the corresponding Schr¨ odinger operator. Consequently, the heat semigroup e−βH is not of trace-class for any β and the notions of partition function and free energy do not make sense. However, there are situations, described below, in which closely related concepts can be introduced. To prepare for this definition we need the following: Definition 4.2 (Spectral deformation family) Let S be a strip in C along R. A one-parameter group of unitary operators U = {Uθ | θ ∈ R} will be called a spectral deformation family for H, if the following conditions are satisfied: • There is a dense subset, D, of the domain of H, D(H), which is invariant under Uθ for all θ ∈ R, • The family Hθ u = Uθ HUθ−1 u, ∀u ∈ D, has an analytic continuation, from θ real into the strip S, and D is a core for Hθ , θ ∈ S (from now on Hθ , θ ∈ S, will also denote the closure of the operator above as defined first on D), • For all θ ∈ S ∩ C+ : σ(Hθ ) ⊂ C− , i.e. the spectrum of Hθ is in the closed lower half-plane. Now we consider a Schr¨odinger operator H = −2 ∆ + V (x) on L2 (Rd ) for which there exists a spectral deformation family so that we can define the complex deformation Hθ of H. Given an open set Λ ⊂ Rd , such that λo := inf V (x) < λb := inf V (x) , x∈Λ
x∈∂Λ
we associated with it the resonance partition function Z(β) as Z(β) = T r PθΛ e−β Hθ ,
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where PθΛ is a spectral projection associated with Λ which will be defined below. We will also show below that the right hand side of the above expression is independent of θ for Im(θ) > 0 (provided Hθ is sectorial and type-A analytic operator) and, under rather general condition, of the spectral deformation family, Uθ , used. Now we define the spectral projection PθΛ (called the resonance spectral projection for Λ below) associated with Λ ⊂ Rd as 1 R(z, Hθ ) dz , PθΛ := 2πı γθΛ
where γθΛ is a closed simple curve having the following properties for Im(θ) > 0: • γθΛ ⊂ ρ(Hθ ) (the resolvent set of Hθ ), • int (γθΛ ) σess (Hθ ) = ∅, • γθΛ = ∂ {z ∈ C | − 4 < Im(z) < 4 , λo < Re(z) < λb } with 4 Im(θ). Above int (γ) designates the interior of the region enclosed by a closed curve γ. The spectral projection is designed to project onto the resonances of H ”coming” from the region Λ of Rd . It is shown in Section 7 that under Conditions (A)-(F) the operators Hθ , δ b Im(θ), λo ≤ Im(θ) > 0, have no eigenvalues in the strip e− ≤ Im(z) ≤ 10 Re(z) ≤ λb for some δ > 0 and for b given in (2.3) and therefore the resonance projection will not depend on 4 or on a particular shape of the contour γθΛ . Discussion. Another, related way to define a resonance partition function is as follows Z(β) := T r χe−β Hθ χ where χ is the characteristic function of the region Λ and the deformation Hθ is done with a help of a vector field exterior to a surface enclosing Λ. One can show that both definitions give exponentially (in 1 ) close quantities. We derive – in a fairly general framework – some key properties of the resonance partition function, Z(β). To this end we need a representation for the heat semigroup e−β Hθ in terms of the resolvent of Hθ . A sectoriality property is sufficient for such a representation. Proposition 4.3 Let U = {Uθ | θ ∈ R} be a spectral deformation family for H in S and let Hθ be the corresponding deformation of H. Assume that there is an open set Ω ⊆ S such that Hθ , θ ∈ Ω, is sectorial andanalytic type A in the sense of −β Hθ Kato. Then the function Z(β) := T r PθΛ e is independent of θ. Moreover, the resonance partition function is independent of the spectral deformation family used, as long as the above conditions on Hθ are satisfied. −1
Proof. Since Hθ is analytic type A in θ ∈ Ω, i.e. (Hθ − z) is an analytic operator function, and, by the integral representation of the exponent e−βHθ in terms of
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the resolvent (Hθ − z)−1 (see [Pa], Thm 1.7.7; it is here that the of sectoriality −β Hθ −β Hθ for Hθ is used), so is e . This implies analyticity of Z(β) = T r PθΛ e θ ∈ Ω. Furthermore, the same integral formula and the definition of Hθ imply for all t ∈ R: (4.5) P(θ+t)Λ e−βHθ+t = Ut PθΛ e−β Hθ Ut−1 and so Z(β) is independent of Re(θ) (by the cyclicity of the trace). Therefore being analytic it is independent of θ ∈ Ω. On the other hand, the independence of resonances on the deformation family is a direct consequence of ([HS2],Theorem 16.4(3)). Remark. • Due to equation (4.5), the set Ω can always be considered to be a strip. With the above definition of partition function we now define the resonance free energy for H as a corresponding free energy: Definition 4.4 (Resonance free energy) We define the resonance free energy as: F (β) = −
1 ln Z(β) , β
(4.6)
where we take the principal branch of the logarithm.
5 Feynman-Kac theorem for resonances In this section we show that the resonance free energy converges to the resonance eigenvalue corresponding to the ground state as the temperature goes to zero. To do this we extend the Feynman-Kac theorem to the resonance case. Theorem 5.1 (Feynman-Kac for resonances) Let Hθ be sectorial with angle δ > 0. If there is only one (possibly degenerate) eigenvalue zo satisfying:
then as β → ∞ ,
Re(zo ) = inf {Re(σd (Hθ ))} ,
(5.1)
F (β) → zo ,
(5.2)
i.e. resonance free energy converges to the ground state resonance energy of the system at zero temperature (there is no true ground state since system is unstable). Moreover, (in uniform topology): lim ρθβ = Pθo ,
β→∞
(5.3)
where ρθβ = Z(β)−1 PθΛ e−β Hθ and Pθo is the projection onto the eigenspace of Hθ corresponding to the eigenvalue zo . The proof of this theorem is rather standard, it is based on the Riesz formula for eigenprojections and appropriate contour deformation. We omit it here.
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6 Exponential decay of deformed eigenfunctions In this section we prove exponential bounds on eigenfunctions of elliptic operators of the form K = p · a(x) · p + W (x). Such bounds go back to Deift-Hunziker-SimonVock ([DHSV]) and Agmon ([Ag]) (see [HS2] for a textbook exposition). The reason we give details here is that we have to find explicit dependence of constants on the parameter . Recall the notation x =
2
1 + |x| and Aϕ = ϕ, Aϕ.
Theorem 6.1 Assume Re(a) > 0 and let z be an eigenvalue of K with an eigenfunction ψ. Let S be a smooth closed hypersurface such that: Re(W (x)) ≥ Re(z) + ν ,
(6.1)
on ext(S), for some ν > 0. Then the eigenfunction ψ satisfies: 256 γ α f ˜ e ψ ≤ pψ + ψ x χ να2 α
(6.2)
where γ ≥ 0, χ ˜α is the characteristic function of the set: {x ∈ ext(S) | dist(x, S) ≥ α} and f (x) is the geodesic distance from S to x in the Riemannian metric:
dsW,b = (Re(W (x)) − Re(z) − ν)+ b−1 ij dxi dxj
(6.3)
i,j 1
1
with b = Re(a) + (Re(a))− 2 (Im(a))2 (Re(a))− 2 . Proof. First we consider the case of γ = 0. Let χα = η η(t) is a C 2 function η(t)
=
1 , t ∈ [1, ∞)
=
3 0 , t ∈ (−∞, ] 4
1
α dist(x, int (S))
where
and such that |∂ s η| ≤ 32 for s = 1, 2. Thus the cut-off function is supported in the domain { x ∈ ext (S) | dist(x, S) ≥ 34 α}. f f f Let ϕ = χα e ψ and K f := e Ke− . First we want to show: f K − z ϕ ≥ ν ϕ . (6.4) To that effect use: f K − z ϕ ≥ ϕ−1 Re K f − z ϕ and computing K f = (p + ı∇f ) a (p + ı∇f ) + W : Re K f ϕ = p · Re(a) · p − ∇f · Re(a) · ∇f + Re(W ) + Aϕ
(6.5)
(6.6)
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where A = Re { ı (p · a · ∇f + ∇f · a · p)} = − (p · Im(a) · ∇f + ∇f · Im(a) · p) . Now, by Schwartz inequality (a1 = Re(a), a2 = Im(a)): 1 − −1 Aϕ = 2Re a1 2 a2 ∇f ϕ, a1 2 pϕ 1 1 − − ≤ 2 a1 2 a2 ∇f ϕ a1 2 pϕ 1 −1 −1 ≤ 4 p · a1 · pϕ + ∇f a1 2 a22 a1 2 ∇f . 4 ϕ
(6.7)
Equations (6.6) and (6.7) with 4 = 1 imply: Re K f − z ϕ ≥ Re(W ) − ∇f · b · ∇f − Re(z)ϕ −1
−1
where, recall, b = a1 + a1 2 a22 a1 2 . Hence: 2 Re K f − z ϕ ≥ ν ϕ ,
(6.8)
provided: ∇f · b · ∇f ≤ Re(W ) − Re(z) − ν
(6.9)
α
on the support of χ , which is satisfied for f and z given in the theorem (see below). Now (6.4) follows from equations (6.5) and (6.8), provided (6.9) holds. On the other hand: f f f (K − z)ϕ = (K f − z)χα e ψ = e (K − z)χα ψ f = e [K, χα ]ψ = [K, χα ]ψ (6.10) where we have f = 0 on supp(∇χα ). β used that −|β| α Since ∂ χ ≤ 32α and since:
2 [K, χ] = 2 2∂i χ aij ∂j + ∂i aij ∂j χ + aij ∂ij χ i,j
we obtain
f (K − z)ϕ ≤ 128 α−1 pψ + 2 α−2 ψ . f
(6.11)
Equations (6.4), (6.11) and the definition ϕ := χα e ψ imply (6.2). The only thing left to prove is the statement (6.9). First observe that by condition (6.1), the right hand side of (6.9) is nonnegative on ext (S).
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Now, let f (y) := ρ(x, y) ≡ inf γ∈Pxy Lo (γ) where Lo (γ) is the length of the curve γ in the Agmon metric for the potential Re(W ) and energy E = Re(z) + ν. Then the length (in the Riemannian metric defined at the beginning of the theorem) of curve γ is: 1 1 φ(γ(τ )) b− 2 γ (τ ) dτ , L(γ) = 0
1 2
where φ(x) = (Re(W (x)) − E)+ and |·| stands for the Euclidean norm in Rd . Now, 1 1 2 b ∇f = sup h, b 2 ∇f |h|=1
= ≤
1 1 f y + 4b 2 h − f (y) |h|=1 →0 4 1 1 sup lim ρ y + 4b 2 h, y |h|=1 →0 4 sup lim
1 sup lim L (γo ) , →0 4 |h|=1 1 where γo (τ ) = (1 − τ )y + τ y + 4b 2 h . By the definition of L(γo ) we have: ≤
L (γo ) = 4 |h|
0
The last two relations give: 1 2 ≤ sup ∇f lim b |h|=1 →0
1
φ (γo (τ )) dτ .
1
φ (γo (τ )) dτ = φ(y) .
0
1
Since φ(y) = (Re(W (y)) − E)+2 and since E = Re(z) + ν, the last inequality implies (6.9), which completes the proof of the theorem for γ = 0. Now we consider the case γ > 0. We define: f
γ
K f,γ := e x K x
−γ
f
e− ,
f where, recall, x = 1 + |x|2 , and ϕ = χα xγ e ψ. Then, since ∇ x = x−1 x: −2 −2 K f,γ = p + ı∇f + ıγ x x a p + ı∇f + ıγ x x + W = K f,γ=0 + ıγ x
−2
x · a · (p + ı∇f )
+ (p + ı∇f ) · a · ıγ x Thus:
−2
x − 2 γ 2 x
−4
x·a·x .
−2 −4 Re K f,γ = Re K f,γ=0 − 2γ x x · a · ∇f − 2 γ 2 x x · a · x
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which for small enough implies: 1 2 Re K f,γ − z ϕ ≥ ν ϕ , 2 for ϕ the same as above, provided (6.9) holds. From here the proof is parallel to the one of the case γ = 0. Indeed, continuing from the (6.10): f f,γ γ f γ (K − z)ϕ = (K f,γ − z)χα x e ψ = e x (K − z)χα ψ f γ = e x [K, χα ]ψ = M γ [K, χα ]ψ (6.12) where M = supy∈supp(∇χα ) y and we, again, used that f = 0 on supp(∇χα ). We apply Theorem 6.1 to the operators Hλ , the reference Hamiltonian introduced in Section 2, and Hλ θ = Uθ Hλ Uθ−1 , the deformation of the reference Hamiltonian using a So (λ)-exterior vector field vλ (as in Section 3). Theorem 6.2 Let λo < λ < λ1 and δ > 0 and let Eλ n be an eigenvalue of Hλ . Then the corresponding eigenfunction, ψλ n , satisfies: C γ fn (6.13) x χe ψλ n ≤ 6 ecδ ψλ n δ 2 where γ ≥ 0, χ is the characteristic function of the set ext Si (Eλ n + δ ) , and fn (x) = Aλ (Si (Eλ n ), x, Eλ n ) is the (Vλ ) Agmon distance, at energy Eλ n , from Si (Eλ n ) to x. Proof. Denote E = Eλ n , V = Vλ and A = Aλ . We apply Theorem 6.1 with K = Hλ . In this case we have a = 1 and W (x) = V (x), which implies b = 1 and dsW,b = dsV , the Agmon metric for V and E + ν. For an eigenvalue E = Eλ n of Hλ , take S = Si (E + ν) and α = dist (Si (E + ν), Si (E + 2ν)) . Then Theorem 6.1 with S = Si (E + ν) implies the estimate C γ f˜n pψλ n + ψλ n , x χe ψλ n ≤ να α where f˜n (x) = A (Si (E + ν), x, E + ν). We show now that: √ ˜ fn (x) − fn (x) ≤ C ν .
(6.14)
(6.15)
Indeed, by the triangle inequality |A (Si (E + ν), x, E + ν) − A (Si (E), x, E)| ≤
|A (Si (E + ν), x, E + ν) − A (Si (E + ν), x, E)| + A (Si (E + ν), Si (E), E) .
(6.16)
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Next:
Ann. Henri Poincar´e
A (Si (E + ν), Si (E), E) ≤ L(γo )
(6.17)
where γo is a straight interval from x ∈ Si (E + ν) to x ∈ Si (E) and L(γ) = φ(γ) |γ |
(6.18)
1
where φ(y) = (V (y) − E)+2 . Use: L(γo ) ≤ sup φ(y) |x − x|
(6.19)
and, by the Mean Value Theorem, ν = V (x ) − V (x) = ∇V (x) · (x − x) ,
(6.20)
for some x in the interval between x and x . Take x such that (x − x) is parallel to ∇V (x). Then: |x − x| ≤ Cν . (6.21) So, (6.17), (6.19) and (6.21) imply: A (Si (E + ν), Si (E), E) ≤ C(V )ν .
(6.22)
√ |A (Si (E + ν), x, E + ν) − A (Si (E + ν), x, E)| ≤ C ν .
(6.23)
Finally, we have:
The last three inequalities imply (6.15). Now we estimate α. Let x ∈ Si (E + ν) and x ∈ Si (E + 2ν). Then, ν = |V (x ) − V (x)| = |∇V (x)(x − x)| ≤ sup |∇V (y)| |x − x|
(6.24)
y∈Ω
where Ω = int (Si (E + ν)) ∩ ext (Si (E + 2ν)) .
(6.25)
This implies:
ν = c(V )ν . (6.26) supΩ |∇V | Next we estimate pψλ n . Since V (x) ≥ λo = V (0) by our construction, we α≥
have:
Hλ = p2 + V ≥ p2 + λo
and thus
2 2 pψλ n = ψλ n , p2 ψλ n ≤ ψλ n , (Hλ − λo )ψλ n = (E − λo ) ψλ n .
This leads to the estimate: pψλ n ≤ 2
E − λo ψλ n .
(6.27)
Take now ν = 12 δ . Then estimates (6.14), (6.26), (6.15) and (6.27) imply (6.13).
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Let now Hλ θ be the complex deformation of the reference Hamiltonian Hλ with a vector field vλ exterior to the surface { x ∈ Rd | Vλ (x) = λ} where λ = λ −δ. Here δ is the same as in condition (F) of Section 2. In case of K = Hλ θ we have: Theorem 6.3 For any δ > 0, the eigenfunction, ψλ nθ of the operator Hλ θ , corresponding to an eigenvalue Eλ n satisfies: C γ fn (6.28) x χe ψλ nθ ≤ 6 ecδ ψλ nθ δ 2 where γ ≥ 0, χ is the characteristic function of the set ext Si (Eλ n + δ ) and fn (x) is the geodesic distance from Si (Eλ n ) to x in the Agmon metric for the potential Vλ at energy Eλ n . Proof. The operator Hλ θ can be expressed as in (3.4)-(3.6) but with the potential Vθλ = V ◦ ϕθλ replaced by Vλθ . Thus it can be identified with the operator K if we set a(x) := Aθ (x) and W (x) := gθ (x) + Vλθ (x). Now we express the matrix b entering (6.3) in terms of the vector field v = vλ (λ = λ − δ). We calculate: −1 −1 a = Aθ (x) = (1 + θDv)−1 1 + θDv T = 1 + θ Dv + Dv T + θ2 Dv T Dv and W (x) = Vθλ (x) + gθ (x). Now, if θ = ıϕ: −1 a = 1 + ıϕ Dv + Dv T − ϕ2 Dv T Dv and letting A = 1 − ϕ2 Dv T Dv and B = Dv + Dv T : −1 1 1 1 1 A− 2 . a = A− 2 1 + ıϕA− 2 BA− 2 Thus: 1
Re(a) = A− 2
1 2 −1 1 1 A− 2 1 + ϕ2 A− 2 BA− 2
and 1
Im(a) = −ϕA− 2
1
1
A− 2 BA− 2 −1 1 2 A 2 . 1 1 + ϕ2 A− 2 BA− 2
Then a = 1 and gθ = 0 in int (So (λ)) and therefore f (x) is the (V, Eλ n +δ)-Agmon distance from S to x for all x ∈ int (So (λ)). The rest of the proof follows the lines of the proof of Theorem 6.2. Let χwλ be the characteristic function of ext (So (λ )). Define: γ
Λnλ λ := x χwλ Pλ n ,
(6.29)
where γ is the same as in (2.11) and λ ≥ λ > Eλ n . This function is used to estimate the difference Vθλ − Vλ of potentials and certain cut-off functions applied to Pλ n (see Section 7). Let furthermore An := A (Si (Eλ n ), So (Eλ n ), Eλ n ).
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Proposition 6.4 Let λ ≥ λ ≥ Eλ n and δ := λ − Eλ n . Then: Λnλ λ ≤ C−6 e
Cδ
e−
An
.
(6.30)
Proof. Denote E = Eλ n . Since Vλ = V on int (So (λ )) and λ ≥ λ , Theorem 6.2 with δ = 2 implies that: Λnλ λ ≤ C−6 e−
A(Si (E),So (λ ),E )
.
(6.31)
Next we show that: |A (Si (E), So (λ ), E) − A (Si (E), So (E), E)| ≤ Cδ .
(6.32)
Indeed, by the triangle inequality |A (Si (E), So (λ ), E) − A (Si (E), So (E), E)| ≤ A (So (E), So (λ ), E) .
(6.33)
Similarly to (6.22) we obtain A (So (E), So (λ ), E) ≤ C(V )δ ,
(6.34)
so (6.32) follows. Now, (6.31) and (6.32) and the notation An := A (Si (E), So (E), E) imply (6.30). In a similar fashion we prove the estimate for: Λnλ λ θ := xγ χwλ Pλ nθ ,
(6.35)
where Pλ nθ = Uθ Pλ n Uθ−1 . Proposition 6.5 Let λ , λ and δ be as in Proposition 6.4. Then: Λnλ λ θ ≤ C−6 e
Cδ
e−
An
.
(6.36)
7 Stability and resonance estimates Let H = −2 ∆ + V (x), where V (x) satisfies conditions (A)-(F). In this section we deal with the question of existence of resonances and with estimates on the resonance eigenvalues and eigenprojections of H. The latter are eigenvalues and eigenprojections of the operator Hθλ , Im(θ) > 0, which is the spectral deformation of H introduced in Section 3. To study eigenvalues of Hθ = Hθλ in the strip { z ∈ C | Re(z) ∈ [0, λ )} , λ > λ, we compare Hθ with Hλ . Our derivations are similar to those of [Si2] and [HS2]. We begin with a basic resolvent estimate which underpins our analysis.
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Theorem 7.1 Assume conditions (B)-(D). Assume numbers α and ϕ = Im(θ) satisfy the inequalities 3 α bϕ 2 , bϕ ≥ α , (7.1) where b is the same as in (2.3). Then the relations: dist (z, σ(Hλ )) ≥ α and Im(z) > −
1 ϕb , Re(z) > λo , 10
(7.2)
imply: z ∈ ρ (Hθλ ) and R (z, Hθλ ) ≤
C α
(7.3)
with C independent of α, λ, λ and . Proof. The statement follows from Theorem 20.7 of [HS2] or Theorem 10.2 of [S2] applied to the operator Hg = −1 U (H − V (0)) U−1 , where g = −1 and (U f )(x) = 2 f d
√ x .
This theorem allows us to define the eigenprojections associated with clusters of eigenvalues of Hθλ splitting from the eigenvalues of Hλ . By Proposition 2.2, for any eigenvalue Eλ n of Hλ there is a contour Γn around Eλ n which is at the distance at least c from σ(Hλ ). Thus to satisfy the first condition in (7.2) we take α2 = c in (7.1)-(7.3). Let Pλ n be the eigenprojection of Hλ corresponding to the eigenvalue Eλ n . 1 R (z, Hλ ) dz . (7.4) Pλ n = 2πı Γn
On the other hand by Theorem 7.1, Γn ⊂ ρ(Hθλ ). Hence we can also define Pθλn :=
1 2πı
Γn
R (z, Hθλ ) dz
(7.5)
which, in principle, can be zero, but which is not as we show in the next statement. Recall the function Λnλ λ is defined in (6.35). As was mentioned above, Λnλ λ is used to control the potential difference Wθλ := Vθλ − Vλ and certain cut-off functions. Indeed, observe that Wθλ = Wθλ χ (So (λ )), provided λ ≥ λ. Here recall χ(Ω) denotes the characteristic function of a set Ω. Hence by condition (2.11): Wθλ Pλ ,n ≤ CΛnλ λ . (7.6)
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Proposition 7.2 (Closeness of projections) Let λ ≤ λ < λ and χ be supported in ext (So (λ )) and χ = 1 − χ. Assume −2 Λnλ λ 1. Then for all Eλ n < λ1 : (7.7) Pλ n (Pθλn − Pλ n ) Pλ n = O −3 Λ2nλ λ , −2 Pλ n (Pθλn − Pλ n ) = O Λnλ λ = (Pθλn − Pλ n ) Pλ n , (7.8) (7.9) (χPλ n − Pλ n χ) Pλ n = O (Λnλ λ ) χPλ n
=
O (Λnλ λ ) ,
(7.10)
Furthermore, Pλ n can be everywhere replaced by the deformed reference projections Pλ θλn = Uθ Pλ n Uθ−1 with the estimates remaining the same, provided λ − λ ≥ δ, where δ is the same as in condition (F). Proof. We use the notation Hθ = Hθλ and Wθ = Wθλ . Let us first prove the statement (7.8). Using equations (7.4) and (7.5) and the second resolvent identity we obtain: 1 (Pθλn − Pλ n ) Pλ n = 2πı R (z, Hθ )Wθ Pλ n R (z, Hλ ) dz . (7.11) Γn
This equation together with estimates (7.3) (Theorem 7.1) and (7.18) implies (7.8). To prove (7.7) we use equation (7.11) and use the second resolvent equation: R (z, Hθ ) = R (z, Hλ ) + R (z, Hλ )Wθ R (z, Hθ )
(7.12)
to obtain: P
λ n
(Pθλn − P
λ n
)P
λ n
+
=
1 2πı
1 2πı
Γn
Γn
Pλ n R (z, Hλ )Wθ Pλ n R (z, Hλ ) dz
Pλ n R (z, Hλ )Wθ R (z, Hθ )Wθ Pλ n R (z, Hλ ) dz .
Now using that Pλ n Wθ Pλ n ≤ Λ2nλ λ and Pλ n Wθ ≤ Λnλ λ and using that both resolvents are bounded as in Theorem 7.1 we have finally: Pλ n (Pθλn − Pλ n ) Pλ n = O −3 Λ2nλ λ . This proves (7.7). Equation (7.9) follows from (7.10): χP −P χ = P χ−χP . And equation (7.10) follows from the definition (6.29). The second part of the proposition is proven in exactly the same way. Proposition 7.2 implies in particular that a disc around Eλ n of radius α2 = c contains eigenvalues of Hθλ of the total multiplicity not less than the multiplicity of Eλ n .
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Next, as in [Si2,Proposition 10.3] or [HS2,Eqn.(20.55)] one can show that for the Riesz projections defined in (7.4) and (7.5): Pθλn − Pλ n → 0 as → 0 . Consequently, the total multiplicity of eigenvalues of Hθλ inside the disc, D Eλ n , α2 , around Eλ n of radius α2 is equal to the multiplicity of Eλ n . α Let zn be the eigenvalues of Hθλ inside the disc D Eλ n , 2 , i.e. those eigenvalues which split from the eigenvalue Eλ n of Hλ , and let Pθλn be the corresponding rank one projections (we repeat zn according to their multiplicities). Let Hλ θλ be the spectral deformation of the operator Hλ (with the same vector field as Hθλ ) and let Pλ θλn = Uθ Pλ n Uθ−1 , the corresponding eigenprojections. According to [HS2,Section 22.3], the latter can be written as: Pθλn = N −1 P˜θλn ,
(7.13)
where P˜θλn := Pθλn Pλ θλn Pθλn and N := T r (Pλ θλn Pθλn ), for appropriately chosen, mutually orthogonal eigenprojections Pλ θλn of Hλ θλ such that
Pλ θλn = Pλ θλn . (7.14) n 2 ˜ (To check that Pθλn are projections one computes P˜θλn = N Pθλn .) We need the following refinement of Proposition 7.2.
Proposition 7.3 Under the conditions of Proposition 7.2 and with the same notation: (7.15) Pλ θλn (Pθλn − Pλ θλn ) Pλ θλn = O −3 Λ2nλ λ , Pλ θλn (Pθλn − Pλ θλn )
= O −2 Λnλ λ = (Pθλn − Pλ θλn ) Pλ θλn ,
χPλ θλn = O (Λnλ λ )
where the indices n and n eigenvalue Eλ n of Hλ .
(7.16) (7.17)
label the eigenvalues of Hθλ which split from the
Due to relation (7.13) a proof of this proposition is reduced to Proposition 7.2. = Pλ θλn and For instance using the shorthand Λn = Λnλ λ , Pθn = Pθλn and Pθn similarly for n → n , we obtain due to Proposition 7.2 that: N
= =
T r (Pθn ) + T r (Pθn Pθn (Pθn − Pθn ) Pθn ) 2 1 + O Λn
and (Pθn Pθn Pθn − Pθn ) Pθn
=
(Pθn − Pθn ) Pθn Pθn
=
+ Pθn Pθn (Pθn − Pθn ) P −2 θn O Λn
(7.18)
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−2 which combine into (Pθn − Pθn Λn , i.e. the second part of Eqn. ) Pθn = O (7.16). Denote the eigenvalues of Hθλ inside the disc D Eλ n , α2 by zn . Proposition 7.4 Assume that the conditions of Proposition 7.2 hold. Let n be such that Eλ n < λ1 and let zn be the eigenvalues of Hθλ in the disc |z − Eλ n | ≤ α2 . Then: (7.19) |zn − Eλ n | ≤ CΛ2nλ λ , where the constant C is independent of . Proof. This proposition can be deduced from the proof of Theorem 22.6 of [HS2], of estimate (14.14) of [Si2]. Below we prove the consequent estimate on Im(zn ), the only result which we need. In the proof below we denote H = Hθλ , P = Pθλn , P = Pλ θλn , E = En , W = Wθ , P˜ = P˜θλn := Pθλn Pλ θλn Pθλn and Λ = Λnλ λ . First we observe that the definition after (7.13) yields: T r H P˜ = T r (HP P ) = E T r (P P ) + T r (P W P ) = E T r P˜ + T r (P W P ) + T r (P W (P − P )P ) . (7.20) Due to Proposition 7.2 and estimate (7.6) we have T r H P˜ = E T r P˜ + O(Λ2 ) .
(7.21)
On the other hand we have
and therefore
N 2 := T r (P P ) = T r (P ) + T r (P (P − P )P ) ,
(7.22)
N 2 = T r (P ) + O(Λ2 ) = T r (P ) + O(Λ2 ) .
(7.23)
Equations (7.21) and (7.23) together with the notation P˜ = N −1 P (see (7.13)) imply that T r (HP ) = E T r (P ) + O(Λ2 ) , (7.24) and therefore
zn T r (Pn ) = E T r (P ) + O(Λ2 ) ,
(7.25)
n
and thus
Im(zn )T r (Pn ) = O(Λ2 ) .
(7.26)
n
Since Im(zn ) ≤ 0, we have that Im(zn ) = O(Λ2 ) ∀n which completes the task we set out for.
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Theorem 7.5 (Estimates on the resonance eigenvalues) The parameter λ can be chosen so that λ ≥ Eλ n and |zn − Eλ n | ≤ C−6 e−
2An
(7.27)
where, recall, An := A (Si (Eλ n ), So (Eλ n ), Eλ n ). 1
Proof. To begin with we take λ = λ ≥ Eλ n . Pick λ such that 1 λ − λ 3 . Then by Proposition 6.4, −4 Λnλ λ 1 for sufficiently small. Consequently, the conditions of Proposition 7.4 are satisfied and therefore estimate (7.19) holds. Combining this estimate with estimate (6.30) of Proposition 6.4 with λ = λ , we obtain 2An Cδ |zn − Eλ n | ≤ C−6 e e− , (7.28) where δ := λ − Eλ n . Finally, we push λ = λ exponentially close to Eλ n . We can do this since by (7.28), Eλ n changes exponentially (in 1/) little when λ changes on the scale O(1). So given Eλn with λ = Eβ , we first take λ = Eλn and then adjust it exponentially little to have λ ≥ Eλ n and λ − Eλ n ≤ e− for some 4 > 0. This Cδ makes δ = O e− so that the factor e in (7.28) can be absorbed into a constant. Hence (7.27) results.
8 Tunneling probability and resonance free energy Our goal in this section is to prove Theorem 2.3, i.e. to estimate the probability that particle initially localized in the well of the volcano-type potential at temperature T escapes through the barrier. We use the definitions and notation from Sections 24. Let λ1 − δ ≥ λ > Eβ , where λ1 and δ are the same as in conditions (C) and (F) of Section 2, respectively. Note that λ1 ≥ λ + δ. Let zn and En be the eigenvalues of the operators Hθλ and Hλ1 , respectively, and Γn = −2Im(zn ). Recall that the eigenvalues, zn , of Hθλ are independent of λ. Before proceeding with the results of this section let us define the follow 2 (2) ing quantities: Γ = = with pn = En <λ1 pn Γn and Γ En <λ1 pn Γn − Γ −βEk −1 −βEn e . Ek <λ1 e The first main result of this section is the following: Theorem 8.1 Assume conditions (A)-(F) on the Hamiltonian H. Then with the definitions given in the preceding paragraph we have: 2 (2) Γo t t Γ Γt − Γt − − e ≤ p(t) ≤ e + O e , (8.1) 2 ∞ . Here Λ2 = En <λ1 pn Λ2nλ1 λ1 . modulo O(−4 Λ2 ) + O t
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Proof. In the proof below the parameter λ1 is fixed. Henceforth we use the following abbreviations H = Hλ1 , Pn = Pλ1 n and Λn = Λnλ1 λ1 . First we prove the following intermediary proposition: Proposition 8.2 We have for all t ∈ R+ :
Γn t p(t) = pn e−
(8.2)
En <λ1
∞ . modulo O −4 Λ2 + O t ∞ . Proof. In this proof we use the notation A = B to stand for A = B + O t . For the sake of keeping notation relatively simple we assume that the eigenvalues En of H are non-degenerate. Our analysis extends to the degenerate case by using Proposition 7.3 instead of Proposition 7.2. First note that the solution to initial value problem (2.22) is (H is timeıHt ıHt independent) ρ(t) = e− ρo e . This and Eqn. (2.4) imply: p(t) =
pn An
(8.3)
En <λ1
ıHt ıHt An = T r χe− Pn e
(8.4)
χ = χw = characteristic f unction of int (Si (λ1 )) .
(8.5)
where
with Pick a partition of unity { g = g(H) , g = g(H)}, g + g = 1: g ∈ Co∞ (∆) and g(µ) = 1 , µ ∈ ∆1 .
(8.6)
Here ∆ and ∆1 are closed intervals of the size c and satisfying En ∈ ∆1 ⊂⊂ ∆ and ∆ ∩ σ(H ) = {En }.
g
∆1 En ∆
λ
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We omit the subindex n in An , Pn , Λn and Λn : A = An , P = Pn and Λ = Λn . Write A as: A = Agg + Agg + Agg + Agg , (8.7) ıHt ıHt . Aξη = T r χe− ξP ηe
where
(8.8)
Since P g(H ) = 0, we have P g = P (g(H) − g(H )). Using the Helffer-Sjoestrand operator calculus (see [DiSj], [HuSi2]) we write: 1 ˜ g(H) = 2πı − z)−1 , (8.9) dg(z)(H where g˜ is an almost analytic extension of g and dg˜ = ∂ ˜g dx dy) and similarly for g(H ). This and the second resolvent identity give: 1 ˜ (H − z)−1 P W (H − z)−1 . (8.10) dg(z) P g = 2πı Using this formula and (7.6), we find that: P g = O(Λ) = gP . Hence we obtain
(8.11)
ıHt ıHt Agg ≡ T r χe− gP P g e = O(Λ2 ) . !" !"
(8.12)
O(Λ) O(Λ)
Now we estimate the term:
ıHt ıHt . Agg ≡ T r χe− gP ge
Estimate (8.11) gives right away that Agg = O(Λ). To obtain a better estimate Agg = O(Λ2 ) we have to work harder. First we transform by the Stone theorem: ∞ ıλt ıHt . (8.13) Agg = dλg(λ)e− T r χ (R(λ + ı0, H) − R(λ − ı0, H)) P ge −∞
There exists an almost analytic extension g˜(z) of g(λ) into C satisfying: ∞
∂˜ g = O (|Im(z)| ) , supp ∂˜ g ⊂ { z ∈ C | Re(z) ∈ ∆\∆1 }
(8.14) (8.15) −1
(see [DiSj] and [HuSi2]). By Green’s theorem and since (H − z) has no poles in C− (cf. [BZ]): ∞ ıλt ızt . (H − λ + ı0)−1 e− g(λ) dλ = (H − z)−1 e− d˜ g (z) = 0 (8.16) −∞
C−
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where we have used that: ∂˜ g(z)e
− ızt
∞
= O (|Im(z)| ) e
− ızt
∞ ızt =O e− t
(8.17)
for Im(z) < 0. This takes care of the second term on the right hand side of (8.13). Now we analyze the first term on the right hand side of (8.13). We use the complex deformation for the term under the trace. Let vλ be an So (λ)-exterior vector-field satisfying conditions (D) and (F), which was used for the definition of Hθλ , and let Uθ = Uθλ be the corresponding deformation group. By conditions := Uθ P Uα−1 have analytic continuations (C) and (F) the operators Hθλ and Pθα in θ and in θ and α, respectively, as type-A families. Denote by Pθn = Pθλn the eigenprojections corresponding to zn . As above, when it does not cause a confusion we omit the subindex n: Pθn = Pθ , and also set Hθ = Hθλ . Now for θ real we can insert the operators Uθ−1 Uθ = 1 inside the trace and continue the result in θ to obtain: ıHt ıHt T r χR (λ + ı0, H)P ge = T r χR (λ, Hθ )Pθ0 ge
(8.18)
for Im(θ) > 0 and λ ∈ supp(g), where we used that χUθ−1 = χ since vλ ≡ 0 on supp(χ), and where we removed +ı0 from the resolvent since R (λ, Hθ ) is analytic in λ ∈ supp(g). We estimate the integral:
∞
−∞
R (λ, Hθ )e−
ıλt
g(λ) dλ .
Define a domain Ω enclosed by the real axis and a smooth curve Γ = Γ + Γ where (∆ is an open interval containing ∆ and 4 a small number ≥ ): Γ = {(x, −4) | x ∈ ∆ } and Γ is as in the picture below:
∆ Γ
Ω Γ
Γ
(8.19)
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Using Green’s theorem for the domain Ω and using that a. Hθ , Im(θ) > 0, has no spectrum in Ω ∩ supp ∂˜ g and b. g˜ = 0 on Γ c. R (z, Hθ ) ≤ C−1 on Γ and on supp(∂˜ g) by Theorem 20.7 of [HS2] or Theorem 10.2 of [Si2] since the distance from Γ ∪ supp(∂˜ g ) to σ(Hλ ) is greater than c we find that
R (λ, Hθ )e−
ıλt
g(λ) dλ
= 2πı
ızj t
zj ∈Ω∩σ(Hθ )
−
Pθj e−
Γ
R(z, Hθ )e−
+ Ω
ızt
R(z, Hθ )e−
g˜(z) dz
ızt
(8.20)
d˜ g (z) ,
where Pθj is the eigenprojection of Hθ corresponding to zj ∈ σpp (Hθ ), Im(θ) > 0. ∞ − t as The second term on the right is O e and the third term is O t above. We can assume that: Ω ∩ σ(Hθ ) = {zn } . Using (8.13) and (8.18) and that χUθ = χ we obtain: ∞ ıλt ıHt dλ g(λ)e− T r χR (λ, Hθ )Pθ0 ge Agg = −∞ ∞ ıλt ıHt . dλ g(λ)e− T r χR(λ − ı0, H)P ge −
(8.21)
(8.22)
−∞
Thus by (8.16) and (8.20), (8.21) we have:
where
ızn t . Agg = 2πıe− B ,
(8.23)
ıHt . B := T r χPθ Pθ0 ge
(8.24)
Now we estimate B, using Proposition 7.2 and (8.11) (see also (8.35) below): ıHt B = T r χ (Pθ − Pθθ ) Pθθ Pθ0 ge +C ! " !"
O(−2 Λ)
O(Λ)
(8.25)
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ıHt where C := T r χPθ0 ge . Now using that χPθ0 = χP and inserting a cut-off function g 1 = g1 (H) such that g1 g = g and g(λ) = 0 near λ = En , we obtain: ıHt . (8.26) C = T r g 1 χP ge We write C as (in estimates below we use Proposition 7.2 and (8.11) with g and with g 1 ) C
=
ıHt T r g 1 (χP − P χ) P P g e ! " !"
O(Λ)
+
O(Λ)
ıHt T r g 1 P χ P g e . ! " !"
O(Λ)
(8.27)
O(Λ)
Hence C = O Λ2 and therefore B = O −2 Λ2 . This implies by (8.23): . . Agg = O −2 Λ2 and, similarly, Agg = O −2 Λ2 . Equations (8.7), (8.12) and (8.28) imply: . A = Agg + O −2 Λ2 .
(8.28)
(8.29)
Now we investigate the term Agg . We write using the Stone theorem as before (see (8.8)): ∞ ∞ ıµt ıλt Agg = dλ dµ e− e g(λ)g(µ) × (8.30) −∞
−∞
T r (χ (R(λ + ı0, H) − R(λ − ı0, H)) P (R(µ + ı0, H) − R(µ − ı0, H))) . ∞ as above. The terms involving R(λ − ı0, H) and R(µ + ı0, H) contribute O t Also as above we deform the terms R(λ + ı0, H) and R(µ − ı0, H) to obtain: ∞ ∞ ıµt ıλt . Agg = dλ dµ e− e g(λ)g(µ) −∞
−∞
T r (χR (λ, Hθ )Pθα R (µ, Hα )) ,
(8.31)
where, recall, Pθα = Uθ P Uα−1 , Im(θ) > 0 and Im(α) < 0. Using the Green’s formula as above we find ızn t ız n t . Pα ) . Agg = (2πı)2 e− e T r (χPθ Pθα
Here we used that σpp (Hα ) = σpp (Hθ ).
(8.32)
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Now we prove the following: Lemma 8.3
Ao := T r (χPθ Pθα Pα ) = T r (P ) + O −4 Λ2 .
(8.33)
. Let χ = 1 − χ. We write: Proof. In this proof we set Pθ := Pθθ
Ao
= = =
T r (χPθ Pθα Pα ) T r (χPθα ) + T r (χ (Pθ − Pθ ) Pθα ) +T r (χ (Pθ − Pθ ) Pθα (Pα − Pα ))
T r (P ) − T r (χP ) + T r (Pθ (Pθ − Pθ ) Pθ ) − T r (Pθ χ (Pθ − Pθ ) Pθ ) +T r (χ (Pθ − Pθ ) Pθ Pθα Pα (Pα − Pα ))
Now we apply the following three estimates of the second part of Proposition 7.2 and definition of Λ = Λn in (6.29)): Pθ (Pθ − Pθ ) Pθ = O −3 Λ2 (8.34) −1 Pθ (Pθ − Pθ ) = O Λ = (Pθ − Pθ ) Pθ (8.35) χP
=
O (Λ) ,
to the terms above in a straightforward manner to obtain (8.33).
(8.36)
Now equation (8.32), the above lemma and the notation Γn = −2Im(zn ) give:
Γn t . Agg = (2πı)2 e− T r (P ) + O −4 Λ2 ,
(8.37)
which together with (8.29) implies (we restore now the subindex n): Γn t . An = e− T r (Pn ) + O −4 Λ2n
(8.38)
which together with (8.3), the convention that the eigenvalues En are counted together with their multiplicities and the relation En <λ1 pn = 1, imply the statement of the proposition. Now we recall the following definitions:
Γ= Γn p n ,
(8.39)
En <λ1
Γ(2) =
(Γn − Γ)2 pn ,
(8.40)
En <λ1
with pn = Zo (β)−1 e−βEn . Note that Γ is the (truncated) Gibbs average of Γn ‘s.
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Next we use the following easy inequality: Lemma 8.4 Let n pn = 1, pn ≥ 0 and M = supn f (Γn ) < ∞. Then:
pn f (Γn ) ≤ f (Γ) +
n
M pn (Γn − Γ)2 . 2 n
(8.41)
Proof. Performing the Taylor expansion of f (Γn ) around Γ: 1 f (Γn ) = f (Γ) + (Γn − Γ)f (Γ) + (Γn − Γ)2 f (Γ∗ ) 2 1 ≤ f (Γ) + (Γn − Γ)f (Γ) + M (Γn − Γ)2 2 and then averaging by pn ’s we get the desired inequality. ∞ : Estimate (8.2) implies that modulo O −4 Λ2 + O t p(t) = po (t) where po (t) =
En <λ1
pn e−
Γn t
(8.42)
and Λ2 =
pn Λ2n .
(8.43)
En <λ1
Using Lemma 8.4 for the upper bound and Jensen’s inequality for the lower bound we obtain: 1 t2 (2) − Γo t Γt Γt e− ≤ po (t) ≤ e− + Γ e , (8.44) 2 2 where Γo = minj Γj . Equations (8.42) and (8.44) imply (8.1). Using that the maximum of the second term on the right hand side is reached at t∗ = Γ2o we obtain: Γ(2) Γt Γt (8.45) e− ≤ po (t) ≤ e− + 2 2 Γ∗ for 0 ≤ t ≤ T , where Γ∗ = max T −1 , 12 Γo . Now we are ready to connect the Gibbs average of tunneling probabilities, Γ, to the free resonance energy F (β), namely to Γ = −2Im(F (β)) (F (β) is given in Definition 4.4). Theorem 8.5 Assume conditions (A)-(F) are satisfied. Then: 3 2 , Γ = Γ 1 + O Λmax where Λmax = maxEn <λ1 Λnλ1 λ1 with λ1 the same as in condition (C).
(8.46)
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Proof. First we consider the case βΛ2max ≤ 4 for some 4 1. Let Z1 = Re(Z(β)) and Z2 = Im(Z(β)). We will use the formula: Im (ln Z(β)) = arctan
Z1 Z2
.
(8.47)
Expanding arctan we find for F (β) = − β1 ln Z(β): Z2 +O −Im (F (β)) = βZ1 Now we estimate
Z2 Z1 .
Since Z(β) = Z1 =
En <λ1
Z2 Z1
Z2 =
e−βRe(zn ) cos
e−βRe(zn ) sin
En <λ1
.
(8.48)
e−βzn we have:
En <λ1
and
3
βΓn 2
βΓn 2
(8.49)
(8.50)
where we used the definition Γn = −2Im(zn ). By Proposition 6.4 the conditions of Proposition 7.4 are satisfied for sufficiently small. The latter proposition with λ = λ1 implies: Re(zn ) = En + O Λ2n , (8.51) where, recall, En = Eλ1 n and Λn = Λnλ1 λ1 . Using this relation, we find furthermore that
βΓn Z1 = e−βEn 1 + O β 2 Λ4n cos 2 En <λ1
(8.52) e−βEn 1 + O β 2 Γ2n + O β 2 Λ4n . = En <λ1
Since Γn = O Λ2n by Proposition 7.4, we have finally that: Z1
=
e−βEn 1 + O β 2 Λ4n
En <λ1
=
En <λ1
e
−βEn
1 + O β 2 Λ4max .
(8.53)
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Similarly we compute for Z2 : Z2
=
e−βEn 1 + O βΛ2n sin
En <λ1
=
e
−βEn
En <λ1
=
βΓn 2
Ann. Henri Poincar´e
2 2 2 2 βΓn + O β Γn + O β Λ n Γn 2
β −βEn Γn + Γn O βΛ2max . e 2
(8.54)
En <λ1
Dividing the latter expression by the former we find: −1
Z2 β −βEn −βEn Γn + Γn O βΛ2max = e e Z1 2 En <λ1 En <λ1 2 4 × 1 + O β Λmax . Remembering the definition Γ := membering (8.48) we arrive at:
En <λ1
Γn e−βEn
En <λ1
e−βEn
−1
−2Im(F (β)) = Γ (1 + O(4)) ,
(8.55) and re(8.56)
where recall βΛ2max ≤ 4 1 by an assumption. Now we consider the case βΛ2max ≥ 4. Denote λ10 = Re(z1 ) − Re(zo ). We have in this case:
−β(zn −zo ) −βzo Z(β) = e e 1+ Eo <En <λ1
=
e
−βzo
1 + O e−βλ10 ,
(8.57)
and therefore we have for F (β) = − β1 ln Z(β), 1 −βλ10 e F (β) = zo + O . β Since βΛ2max ≥ 4, this gives F (β) = zo + O
Λ2max − Λλ2 10 e max 4
(8.58)
.
Observe now that λ10 = ω + o() for some ω independent of . Next using the definition −1
−βEn −βEn Γ= pn Γn , where pn = e e , En <λ1
En <λ1
(8.59)
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we derive
383
Γ = Γo + O Γ2max e−βEo ,
(8.60)
where Γmax = maxEn <λ1 Γn . Observe that Eo = ω + o() for some ω > 0 independent of (where we used that V (0) ≥ 0). The last two relations imply that: λ10 − Eo − −2Im(F (β)) = Γ + O Γ2max e Λ2max + O 4−1 Λ2max e Λ2max . 3
2 and observing that in this case the error term in (8.61) Now, taking 4 = Λmax is much smaller than in (8.56), we conclude that (8.46) holds.
Corollary 8.6 Assume β is so large that: Γ(2) ≤ CΓ2
and
Λ3max ≤ CΛ2
(8.62)
then (8.1) becomes: e modulo O(Λ2 ) + O
− Γt
∞ t
≤ p(t) ≤ e
− Γt
+O
tΓ
2 (8.63)
.
Proof of Theorem 2.3. The statement of Theorem 2.3 follows from Theorems 8.1 and 8.5, and Corollary 8.6.
9 Semiclassical bound By now we have developed all the necessary machinery for estimating the upper bound on the tunneling probability Γ(β) = −2Im(F (β)). Theorem 9.1 (= Theorem 2.4) Assume conditions (A)-(F) for the Hamiltonian H. Then: Sβ (9.1) Γ(β) ≤ C−6 e− , where Sβ is the action of the instanton of period β (see Eqn. (2.19)). Proof. The statement follows from equation (8.46) and Proposition 9.2 below. Before proceeding to Proposition 9.2 we define a constant C1 by the inequality: N (λ1 ) := card { m | Em ≤ λ1 } ≤ C1 − 2 , d
(9.2)
where En := Eλ1 n . Proposition 9.2 Γ ≤ C1 −6 e−
Sβ
.
(9.3)
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Proof. By the definition:
Γ = Zo (β)−1
e−βEn Γn .
(9.4)
En ≤λ1
Using (7.27), we obtain: Γ ≤ C−6 Zo (β)−1
e−
φ(En ,β)
.
(9.5)
En ≤λ1
where φ(E, β) is defined in (2.17). Since, by the definition, Eβ minimizes φ(E, β) and by (9.2):
φ(En ,β) φ(En ,β) e− ≤ max e− N (λ1 ) En ≤λ1
En ≤λ1
≤
e−
φ(Eβ ,β)
C1 − 2 . d
The last two estimates together with the inequality:
e−βEn ≥ e−βEo , Zo (β) :=
(9.6)
(9.7)
En ≤λ1
and the relation Sβ = φ(Eβ , β) (see (2.19)) imply (9.3).
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I.M. Sigal Department of Mathematics University of Toronto Toronto, ON M5S 3G3 Canada email: [email protected] B. Vasilijevic Department of Mathematics University of Toronto Toronto, ON M5S 3G3 Canada Communicated by Gian Michel Graf submitted 16/06/01, accepted 10/10/01
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Ann. Henri Poincar´e 3 (2002) 389 – 409 c Birkh¨ auser Verlag, Basel, 2002 1424-0637/02/020389-21 $ 1.50+0.20/0
Annales Henri Poincar´ e
Localization for Random Perturbations of Periodic Schr¨ odinger Operators with Regular Floquet Eigenvalues Ivan Veseli´c Abstract. We prove a localization theorem for continuous ergodic Schr¨ odinger operators Hω := H0 +Vω , where the random potential Vω is a nonnegative Anderson-type perturbation of the periodic operator H0 . We consider a lower spectral band edge of σ(H0 ), say E = 0, at a gap which is preserved by the perturbation Vω . Assuming that all Floquet eigenvalues of H0 , which reach the spectral edge 0 as a minimum, have there a positive definite Hessian, we conclude that there exists an interval I containing 0 such that Hω has only pure point spectrum in I for almost all ω.
1 Introduction and results Localization Already in the fifties Anderson [1] concluded by physical reasoning that some random quantum Hamiltonians on a lattice should exhibit localization in certain energy regions. That is to say that the corresponding self-adjoint operator has pure point spectrum in these energy intervals. Since then mathematical physicists developed a machinery to prove rigorously this phenomenon from solid state physics. Most of them used the so-called multi scale analysis (MSA) introduced in a paper by Fr¨ ohlich and Spencer [14] to prove a weaker form of localization at low energies for the discrete analogue of the Schr¨ odinger operator. This quite complicated reasoning was streamlined by von Dreifus and Klein [44]. The underlying lattice structure made the MSA easier to apply to discrete Hamiltonians but soon adaptations for continuous Schr¨ odinger operators followed [29, 23, 6, 24]. We prove in Theorem 1.1 a localization result for energies near internal spectral edges of a periodic Schr¨ odinger operator H0 which is perturbed by an Anderson-type potential Vω . Unlike [2, 21] our results are not restricted to a special disorder regime of the random coupling constants in Vω . Instead we assume that the periodic operator H0 has regular Floquet eigenvalues. This behaviour is commonly assumed among physicists. Recent results by Klopp and Ralston indicate that it is generic [27]. In the remainder of this section we introduce our model, state the main Theorem 1.1 and the technical Proposition 1.2 on which it is based. Section 2 explains how to deduce Theorem 1.1 from Proposition 1.2, in Section 3 we describe the functional calculus with almost analytic functions, Section 4 contains a compari-
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son result between the integrated density of states on finite cubes and on the whole of Rd and the last Section 5 deals with periodic (or more generally quasi-periodic) boundary conditions which are necessary to complete the proof of Proposition 1.2. Two technical proofs are placed in an appendix.
The model On the Hilbert space L2 (Rd ) we consider a self-adjoint operator H := Hω made up of a periodic Schr¨ odinger operator H0 and a random perturbation Vω Hω := H0 + Vω .
(1)
Here H0 := −∆ + V0 is the sum of the negative Laplacian and a Zd -periodic potential V0 ∈ Lploc (Rd ) with p = 2 if d ≤ 3, p > 2 if d = 4 and p ≥ d/2 if d ≥ 5. Such a potential is an infinitesimal perturbation of −∆ so the sum is self-adjoint with domain D(−∆) = W22 (Rd ), the Sobolev space of L2 -functions whose second derivative is also in L2 (cf. [34, 35]). The random perturbation is of Anderson type ωk u(x − k) , (2) Vω (x) := k∈Zd
where (ωk )k∈Zd is a collection of independent identically distributed (i.i.d.) random variables on the probability space (Ω, P), called coupling constants. Their distribution has a bounded density with support [0, ωmax ] for some ωmax > 0. The non-negative single site potential u has to decay exponentially and have an uniform lower bound on some open subset of Rd , more precisely u ≥ δ1 χΛ , δ1 > 0 where Λ := Λs := {x ∈ Rd | x∞ < s/2}, s > 0 and
χΛ1 u(· − l)Lp ≤ δ2 e−δ3 l , δ2 , δ3 > 0 .
(3)
Hω is an ergodic operator and we infer from [19, 4] or [33] that there exists a set σ ⊂ R such that σ = σ(Hω ) for almost all ω ∈ Ω, i.e. the spectrum of Hω is almost surely non-random. In the same sense σac , σsc and σpp are ω-independent subsets of the real line. Under some mild assumptions the periodic background operator H0 has a spectrum with band structure, i.e. σ(H0 ) = n∈N [En− , En+ ] , E1− ≤ E1+ ≤ E2− ≤ . . ., − where for some n we have open spectral gaps, i.e. En+ < En+1 (cf. [9, 39, 35]). We assume that there exist positive numbers a, b and b with [0, a] ⊂ σ(H0 ), [−b, 0[⊂ ρ(H0 ) and [−b , 0[⊂ ρ(Hω ) . Since 0 is in the support of the density of ω0 it follows that 0 ∈ σ(Hω ). In this case we say that 0 is a lower band edge of the periodic operator, which is preserved by the positive random perturbation Vω .
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H0 can be decomposed into a direct integral via an unitary transformation U (cf. [39, 35]) ∗ U H0 U = H0 |θΛ1 dθ . (4)
L
[−π,π]d
Here H0 |θΛ1 is the same formal differential expression as H0 acting on functions f ∈ W22 (Λ1 ) with θ-boundary conditions, i.e. for all j = 1, . . . , d we have a phase shift in the corresponding direction: f (x+ej ) = eiθj f (x) where xj = −1/2. It is an operator with discrete spectrum, which consists of the so-called Floquet eigenvalues E1 (θ) ≤ . . . ≤ En (θ) ≤ . . .
n∈N.
These are Lipschitz-continuous on [−π, π]d . In fact they ”generate” the bands of the spectrum of H0 σ(H0 ) = En (θ) . n∈N θ∈[−π,π]d
There is a finite set of indices N ⊂ N (cf. [39]) such that En (θ) = 0 for some θ ∈ [−π, π]d =⇒ n ∈ N . Since 0 is a lower band edge of σ(Hω ), En (θ) = 0 has to be a minimum of En (·). If for all n ∈ N , En (·) has only quadratic minima at 0 (i.e. the Hessian of En (·) at any minimum with value 0 is positive definite) we say that H0 has regular Floquet eigenvalues at 0.
Results Our result on localization at an lower internal spectral band edge is the following Theorem 1.1 If H0 has regular Floquet eigenvalues at 0 and Hω is constructed as above, then there exists a number E0 > 0 such that almost surely [0, E0 ] ⊂ σpp (Hω ), [0, E0 ] ∩ σc (Hω ) = ∅ . The proof of the theorem is based on the following proposition. Proposition 1.2 Assume that H0 has regular Floquet eigenvalues at 0 and Hω is constructed as above. Then for all q > 0 and α ∈]0, 1[ there exists a l0 := l0 (q, α) ∈ N such that for all l ≥ l0 we have −α P{ω| σ(Hω |per [= ∅ } ≤ l−q . Λl ) ∩ [0, l
Here the index ”per” denotes periodic boundary conditions on the cube Λl .
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The statement of Proposition 1.2 remains true if we replace the periodic boundary −π π d conditions by general θ-boundary conditions with θ ∈ [ 2l+1 , 2l+1 ] , cf. (4) and (27). The proof of the proposition is given in Sections 3 to 5. It uses the existence of Lifshitz-tails of the integrated density of states (IDS) of the ergodic operator Hω if H0 has regular Floquet eigenvalues, which was proved by Klopp in [25], who also noted that his result could be used for a localization proof. Theorem 1.1 is proved using the MSA. Since this technique is well understood by now [6, 21, 41] we only sketch it to show how Proposition 1.2, which is the main technical novelty of this paper, enters. This is done in Section 2, where also a discussion of previous results can be found. Remark 1.3 At any lower band edge one can prove localization under the analogous assumptions. Here E = 0 was chosen only for notational simplicity. If the Anderson-type perturbation Vω is negative our theorem can be used to establish localization on any upper band edge with regular Floquet eigenvalues. If the underlying Zd is replaced by some other Euclidean lattice Γ := {γ ∈ Rd | γ =
d
βj aj , β ∈ Zd } ,
j=1
where {aj }dj=1 is a basis of Rd , the same theorem and proposition are valid by a simple modification of the proofs. In any case we will use the maximum norm when considering lattice points k or γ in Zd or Γ, i.e. |γ| := γ∞ := max{|γj |, j = 1, . . . , d}, where (γ1 , . . . , γd ) ∈ Rd are the components of γ. An inspection of our proofs and the papers [25, 26] and [21, 46] shows that Proposition 1.2 and Theorem 1.1 extend to single site potentials u with sufficiently fast polynomial decay (in Lp -sense), cf. (12). Example 1.4 Finally we give an example of a periodic operator which has only regular Floquet eigenvalues at all band edges. Thus we know that our condition in the above theorem is fulfilled and we can prove localization at any lower band edge. Let V0 satisfy the conditions posed above on the periodic potential and let it be a sum of potentials Vj which are periodic in the jth coordinate direction and constant in all the others; more precisely V0 (x) :=
d
Vj (xj )
j=1
where Vj : R → R is a periodic function and x = (x1 , . . . , xd ) ∈ Rd . Then both H0 and H0 |θΛ1 can be decomposed into a direct sum of one-dimensional operators. For these it is known that all Floquet eigenvalues are regular [9, 25]. As the eigenvalues of the direct sum are just sums of the eigenvalues of the one-dimensional operators it is clear that the former also have to be regular.
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Corollary 1.5 Let the ergodic operator Hω := −∆+V0 +Vω be constructed as above and the periodic potential be decomposable, i.e. V0 (x) :=
d
Vj (xj ) .
j=1
Let E be a lower spectral band edge of the periodic operator H0 := −∆ + V0 at a spectral gap which is not closed by the perturbation Vω . Then there exists an interval I E such that almost surely I ⊂ σpp (Hω ), σc (Hω ) ∩ I = ∅. Acknowledgments. The author would like to thank F. Klopp for his hospitality at Universit´e Paris 13, for stimulating discussions, as well as for many detailed explanations concerning his paper [25], W. Kirsch, under whose guidance this research was undertaken and H. Najar, K. Veseli´c and R. Muno for valuable comments.
2 Multi scale analysis and associated ideas In this section we explain how Theorem 1.1 is deduced from Proposition 1.2 and discuss previous localization results. An intermediary step in the proof of localization is the establishing of the exponential decay of the resolvent sup χx R(()χy L(L2 (Rd )) ≤ const e−c|x−y| for almost all ω , =0
(5)
where R := R(() := (Hω − E − i()−1 is the resolvent of Hω near an energy value E in the energy interval I ⊂ R for which we want to prove localization. The χx and χy are characteristic functions of unit cubes centered at x, respectively at y. This bound can be used to rule out absolutely continuous spectrum [30] and is interpreted as absence of diffusion [14, 29] in the energy region I if (5) holds for all E ∈ I. It turns out that the finite size resolvent RΛ (() := (Hω |Λ − E − i()−1 is easier approachable than R(() on the whole space. Here Hω |Λ is the restriction of Hω to L2 (Λ) with some appropriate boundary conditions (b.c.); the use of Dirichlet or periodic b.c. is most common. However the operator Hω |Λ is not ergodic and for its resolvent an estimate like (5) can be expected to hold only with a probability strictly smaller than one. This is the place where MSA enters. It is an induction argument over increasing length scales lj . They are defined recursively by lj+1 := [ljζ ]3 , where [ljζ ]3 is the greatest multiple of 3 smaller than ljζ . The scaling exponent ζ has to be from the interval ]1, 2[. On each scale one considers the box resolvent Rj (() := RΛlj (() and proves its exponential decay with a probability which tends to 1 as j → ∞. We outline briefly the ingredients of the MSA as it is given in [6, 21] or [4].
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First we explain some notation which is used afterwards. Let δ > 0 be a small constant independent of the length scale lj and φj (x) ∈ C 2 a function which is identically equal to 0 for x with x∞ > lj − δ and identically equal to one for x with x∞ < lj − 2δ. The commutator W (φj ) := [−∆, φj ] := −(∆φj ) − 2(∇φj )∇ is a local operator acting on functions which live on a ring of width δ near the boundary of Λj := Λlj . We say that a pair (ω, Λj ) ∈ Ω × B(Rd ) is m-regular, if sup W (φj )Rj (()χlj /3 L ≤ e−mlj .
(6)
=0
Here · L is the operator norm on L2 (Λj ) and χlj /3 the characteristic function of Λlj /3 := {y| y∞ ≤ lj /6}. Thus the distance of the supports of ∇φj and χl/3 is at least lj /3 − 2δ ≥ lj /4. −1/4 Let q0 > 0 and m0 ≥ const l0 . The starting point of the MSA is the estimate (H1)(l0 , m0 , q0 )
P{ω| (ω, Λ0 ) is m0 -regular} ≥ 1 − l0q0
which serves as the base clause of the induction. The induction step consists in proving (7) (H1)(lj , mj , qj ) =⇒ (H1)(lj+1 , mj+1 , qj+1 ) For the mass of decay mj+1 and the probability exponent qj+1 on the scale lj+1 the following estimates are valid ∀ξ > 0 ∃c1 , c2 , c3 independent of j such that log lj+1 4lj c1 mj+1 ≥ mj 1 − − c2 − lj+1 lj lj+1 2d lj+1 1 −ξ qj+1 2q lj+1 ≤ c3 lj j + lj+1 . lj 2
(8) (9)
For the recursion clause (7) a Wegner estimate [45] is needed: (H2)
P{ω| d(σ(Hω |Λ ), E) ≤ η} ≤ CW η|Λ|2
for all boxes Λ ⊂ Rd and all η > 0, such that [E −η, E +η] is contained in a suitable small energy interval near the spectral band edge (cf. Theorem 3.1 in [21]). Here |Λ| stands for the Lebesgue measure of the cube Λ. The deterministic part of the induction step uses the geometric resolvent formula [6, 17] φΛ (HΛ − z)−1 = (HΛ − z)−1 φΛ + (HΛ − z)−1 W (φΛ )(HΛ − z)−1
(10)
for z ∈ ρ(HΛ ) ∩ ρ(HΛ ) and φΛ ∈ C 2 with support in Λ ⊂ Λ . It gives the estimate χl/3 (· − x)R3l (()χl/3 (· − y)L ≤ (3d e−ml )3|x−y|l
−1
−4
R3l (()L
(11)
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if no two disjoint non-regular boxes Λl ⊂ Λl with center in 3l Zd ∩ Λ3l exist for ω. In our case l := lj is the length scale on which the exponential decay of the resolvent is already known and l := lj+1 the scale on which we want to prove qj+1 (bounded by it. By the estimates (H1), (H2) we have with probability 1 − lj+1 the inequality (9)) exponential decay on the length scale lj+1 with mass mj+1 (bounded as in (8)). We stated above the ingredients of the MSA as they are valid if u is compactly supported. If the single site potential is of long range type as in (3) one has to use the adapted MSA from the papers [21, 46]. Once the estimate (H1) is established on all length scales lj , j ∈ N, one infers an exponential decay estimate for the resolvent on the whole of Rd . Afterwards one uses a spectral averaging technique (cf.[6]) based on ideas of Kotani, Simon, Wolf and Howland to conclude localization [28, 38, 18]. An alternative version of the MSA can be found in the recently published book [41], see also [47, 48]. Recent papers concentrate on proofs for the Wegner estimate and the initial length scale decay of the resolvent. At the same time adaptations of the MSA for various random Schr¨ odinger operators, as well as Hamiltonians governing the motion in classical physics appeared [10, 11, 7, 40]. Recently Najar [32] obtained analog results to [25] and the present paper concerning Lifshitz tails and localization for acoustic operators. We discuss briefly some results for quantum mechanical Hamiltonians. In [24] Klopp proved a Wegner lemma for energies at the infimum of the spectrum which applies to an Anderson perturbation Vω with single site potentials u that are allowed to change sign, cf. also [43, 16]. For Vω a Gaussian random field a Wegner estimate was shown in [12]. Its main feature is that no underlying lattice structure of Vω is needed. This result allows one to conclude localization for the corresponding Schr¨ odinger operator at low energies [13]. Kirsch, Stollmann and Stolz proved in [21] (cf. also [46]) a Wegner estimate with only polynomial decay conditions on the single site potential u and deduced a localization result for Hamiltonians with long range interactions. They require |u(x)| ≤ const (|x| + 1)−m for some m > 4d .
(12)
The resolvent decay estimate (H1) for some initial length scale can be proved with semiclassical techniques. Using the Agmon metric one can achieve rigorously decay bounds with what is called among physicists WKB-method [6, 17]. However this reasoning is only applicable for energies near the bottom of the spectrum. The so-called Combes-Thomas argument [5] allows one to infer the following inequality χx (H − z)−1 χy L ≤ [const d(σ(H), z)]−1 e−const d(σ(H),z) |x−y|
(13)
where H is a self-adjoint Schr¨ odinger operator on L2 (Rd ) and z ∈ ρ(H). It was first applied to multiparticle Hamiltonians [5], but it is also useful in our case, as
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soon as we get a lower bound on d(σ(Hω |Λ ), z). Thus it is sufficient to prove an estimate like P{ω| d(σ(Hω |Λl ), I) < l−α /2 } ≤ l−q (14) for some α ∈]0, 1/4]. Such a bound follows immediately from Proposition 1.2 with I := [0, 12 l−α [, for l > (2b )−1/α . Now Inequality (13) implies the initial scale estimate (H1) with m0 ≥ const l−1/4 for l large and E ∈ I, cf. [21, Lemma 5.5]. The constant depends on the energy and the potential, but not on l and m0 . Two possibilities were used to deduce (14). The first is to assume a special disorder regime, more precisely to demand a sufficiently fast decay of the density g of the distribution of ω near the endpoints 0 and ωmax of supp g: ∃τ > d/2 : ∀ small ( > 0 g(s)ds ≤ (τ , respectively 0
ωmax
ωmax −
g(s)ds ≤ (τ
depending on whether one wants to consider a lower or upper band edge. This approach was used in [2, 21]. Its shortcoming is that it excludes quite a few distributions, e.g. the uniform distribution on [0, ωmax ]. The other way to prove (14), which we pursue, is to use the existence of Lifshitz tails of the integrated density of states at the edges of the spectrum. One defines the IDS usually as follows: N (E) := :=
lim N (Hω |D Λ , E)
(15)
lim |Λ|−1 #{ eigenvalues of Hω |D Λ below E} ,
(16)
Λ Rd Λ Rd
i.e. as the limit of the normalized counting function of eigenvalues of a box Hamil2 D tonian. Here Hω |D Λ is the restriction of Hω to L (Λ) with Dirichlet b.c. As Hω |Λ has compact resolvent and hence discrete spectrum, definition (15) makes sense. N (E) is almost surely ω-independent and the use of Dirichlet b.c. in its definition implies [20] N (E) = sup N (Hω |D (17) Λ , E) . Λ Rd
One says that N (·) exhibits Lifshitz tails at some spectral edge E if lim
E→E
d log | log |N (E) − N (E)|| =− . log |E − E| 2
(18)
At the infimum of the spectrum, i.e. for E = inf σ(Hω ), (17) and (18) imply −d/4 #{eigenvalues of Hω |D ) Λ in [E, E]} ≤ |Λ|N (E) ≤ |Λ| exp(−cE
ˇ sev inequality since N (E) = 0. This estimate was used in [24] together with a Cebiˇ to prove (H1) at the bottom of the spectrum, see also [29].
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If one considers an internal band edge E, Lifshitz asymptotics are not so easy to exploit since (17) cannot be directly used to bound |N (Hω |Λ , E) − N (Hω |Λ , E)| . Therefore a comparison technique between N (·) and N (Hω |Λ , ·) is needed. In the one-dimensional case Mezincescu [31] proved Lifshitz tails at internal band edges as well as a comparison lemma for the IDS (Lemma 2, in Section 4). This proof relies on the delicate analysis of Dirichlet eigenfunctions of Hω |Λ and their roots. The results in [31] make a localization proof in the one-dimensional case possible [42]. We prove in Section 4 an approximation result (Theorem 4.1) for the IDS of the multi-dimensional operator Hω , which enables us to prove Proposition 1.2. In our case however periodic b.c. seem to be more efficient than Dirichlet b.c. since Hω is a perturbation of a periodic operator. In [25] it was proved that the IDS of Hω exhibits Lifshitz asymptotics at a lower band edge E if before the perturbation Vω the Floquet eigenvalues of the periodic background operator H0 at E were regular. Thus our approximation theorem can be applied to conclude localization.
3 The Helffer-Sj¨ ostrand formula: Functional calculus with almost analytic functions In this section we introduce the Helffer-Sj¨ostrand formula (19) which is exploited in Section 4 to prove the IDS approximation result. For an self-adjoint operator on L2 (Rd ) and a complex-valued measurable function f : R → C one can define the operator f (A) with domain D(f (A)) := {ψ ∈ L2 (Rd )| f (A)ψ ∈ L2 (Rd )} via the spectral theorem. The latter is normally proved using Riesz’ representation theorem for C(K)∗ , where K is a compact metric space, and the Cayley-transform if A is unbounded. Helffer and Sj¨ ostrand [15] proved the following representation formula ∂ f˜ 1 (z)(z − A)−1 dz ∧ d˜ z (19) f (A) := 2πi C ∂ z¯ if f is smooth and compactly supported. Here f˜ : C → C denotes an almost analytic extension of f : R → C. Davies [8] uses equation (19) as a starting point to develop systematically a functional calculus equivalent to the standard one. For further details on the material of this section see his book. Definition 3.1 For n ∈ N and f ∈ C0n+1 (R, C) define the almost analytic extension (of order n) f˜ : C → C by n r (iy) s(x, y) , (20) f˜(x, y) := f˜n (x, y) := f (r) (x) r! r=0
398
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where we used the convention z := x√+ iy := (x, y) ∈ C. The cutoff function s is defined with the abbreviation x := x2 + 1 by the formula y s(x, y) := t , t ∈ C0∞ (R) x with t(x) = 0 for |x| > 2, t(x) = 1 for |x| < 1 and t ∞ ≤ 2. With this choice of the almost analytic extension formula (19) holds true. If the support of f is contained in [−R, R], f˜ vanishes outside the set {z ∈ C| x ∈ supp f, |y| < 2R + 2}. A calculation of the derivatives shows 1 ∂ f˜n ∂ f˜n ∂ f˜n (z) = +i (z) (21) ∂ z¯ 2 ∂x ∂y =
n 1 (n+1) 1 (iy)n (iy)r f s(x, y) + (sx (x, y) + isy (x, y)) . (x) f (r) (x) 2 n! 2 r! r=0
By calculating the partial derivatives of s we see |sx + isy | ≤
6 χ{ x<|y|<2 x} , x
(22)
√ which shows that they vanish for |y| ≤ 1 since always x = x2 + 1 ≥ 1. Putting the bounds together we get n ∂ f˜ 1 (n+1) 3 |y|r n (x, y) ≤ |f χ{ x<|y|<2 x} . (23) s| |y|n + |f (r) | ∂ z¯ 2n! x r! r=0 Later on f will be an approximation of the characteristic function χ[0,E] . It is going to have support inside [−E/2, 2E] and be equal to 1 on [0, E]. One can choose f in such a way that f (n) ∞ ≤ CE −n and |f |n :=
n
˜ −n f (n) ∞ ≤ CE
(24)
i=1
for sufficiently small E. The constants C, C˜ are independent of E.
4 IDS approximation theorem In this section we bound the difference of the IDS of the ergodic operator Hω and its periodic approximation Hω,l which will be defined shortly. The estimate is contained in Theorem 4.1 which is the main technical result of this paper. Furthermore, it enables us to show in Theorem 4.6 that the IDS of the periodic
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approximation Hω,l exhibits a kind of Lifshitz tail, if the IDS N of the original operator Hω does so. The periodic approximation Hω,l is defined by Hω,l (x) := H0 (x) +
ωk˜ u(x − k)
(25)
k∈Zd
where k˜ := k (mod(2l + 1)Zd ). For any l ∈ N and ω ∈ Ω it is a (2l+1)Zd -periodic operator. Our assumptions on u and ω ensure that it is an infinitesimally small perturbation of H0 , uniformly in l and ω. Hence it is a lower bounded symmetric operator which is self-adjoint on the domain W22 (Rd ). Its IDS is defined by (cf. [35, 36, 39]) χ{En (θ)<E} dθ . (26) Nω,l (E) := N (Hω,l , E) := (2π)−d n∈N
Bl
Here E ∈ R is an energy value, En (θ) is the n-th eigenvalue of Hω,l |θΛ2l+1 and
−π π θ ∈ Bl := , 2l + 1 2l + 1
d (27)
if Hω is Zd -ergodic. For some other Euclidean lattice it has to be replaced by the basic cell of the corresponding dual lattice Γ∗ := {γ ∗ ∈ (Rd )∗ = Rd | ∀γ ∈ Γ : γ ∗ · γ ∈ 2πZ}. We prove the following approximation result: Theorem 4.1 Let Hω be defined as in Section 1 and Hω,l as above. Denote by N , respectively Nω,l the corresponding IDS’. For a real valued function g ∈ C0n+1 with support in [−1/2, 1/2] we have ≤ const. | supp g| |g|n+1 l−n+2d+1 E g(x)dN (x) − g(x)dN (x) ω,l R
R
for sufficiently large l ∈ N. The proof is split into several lemmata. Remark 4.2 and Lemma 4.3 are taken from Section 5.2 of [25]. We denote with χl the characteristic function of the periodicity cell Λ2l+1 := {x ∈ Rd | x∞ ≤ l + 1/2} of Hω,l and by χl,γ (x) := χl (x − γ) its translation by γ ∈ Zd . Remark 4.2 Note that one can infer from [3, 4],[33] and [25] the following equalities g(x) dN (x) = E (Tr χ0 g(Hω )χ0 ) (28) R
respectively
R
g(x) dNω,l (x) = (2l + 1)−d (Tr χl g(Hω,l )χl ) .
(29)
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Using the decomposition χl =
χ 0, k ,
k∈Zd ,|k|<2l+1
the (2l + 1)Zd -periodicity of Hω,l and the i.i.d. property of (ωk )k∈Zd one gets E g(x) dNω,l (x) = E (Tr χ0 g(Hω,l )χ0 ) . (30) R
Since Hω,l is uniformly lower bounded there exists a λ ≥ 0 such that Id ≤ λ + Hω,l and Id ≤ λ + Hω for all l, ω. From [37] we know that the operator χl (λ + Hω,l )−q (z − Hω,l )−1 is trace class for all q > d/2. Using results from the appendix of [22] we infer χ0,β (z − Hω )−1 (λ + Hω )−q χ0 Tr
≤
C˜1 exp(−|y| |β|/C˜1 ) |y|
(31)
for some C˜1 ≥ 1 independent of ω. This estimate is in fact a sophisticated version of the Combes-Thomas argument which we encountered already in Section 2. A simple resolvent estimate gives χ0 (z − Hω,l )−1 Tγ u χ0,β+γ L(L2 (Rd )) ≤
C˜1 χ0,β+γ uLp , |y|
(32)
where Tγ is the translation by γ ∈ Zd . As the single site potential u decays exponentially, inequality (32) gives a exponential bound in −|γ + β|. If one assumes that u decays polynomially with a sufficiently negative exponent, one still can carry trough the proof of Theorem 4.1. Lemma 4.3 If g ∈ C0n+1 and f˜ is an almost analytic extension of f (x) := (λ + x)q g(x), one has E g(x) dNω,l (x) − g(x) dN (x) R R ˜ ∂ f C1 ≤ |y|−2 (x, y) χ0,γ+β uLp exp(−|y| |β|/C1 ) dx dy ∂ z¯ 2π C d β∈Z γ∈Zd ,|γ|>l
Proof. We use without explicit reference the equations collected in the above Remark 4.2 and the Helffer-Sj¨ ostrand formula (19). Let N q > d/2. If we multiply i g(Hω,l ) = 2π
C
∂ f˜ (z)(z − Hω,l )−1 (λ + Hω,l )−q dz ∧ d¯ z ∂ z¯
(33)
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by the characteristic function χ0 of Λ1 we get a trace-class operator and consequently ∂ f˜ i (z) Tr(χ0 (z − Hω,l )−1 (λ + Hω,l )−q χ0 ) dz ∧ d¯ Tr(χ0 g(Hω,l )χ0 ) = z . (34) 2π C ∂ z¯ The same formula holds with Hω substituted for Hω,l . To bound the trace of χ0 (Hω,l − Hω )χ0 in mean we estimate χ0 (z − Hω,l )−1 (λ + Hω,l )−q χ0 − χ0 (z − Hω )−1 (λ + Hω )−q χ0 Tr ≤ Σ1 + Σ2 by the two summands
Σ1 = χ0 (z − Hω,l )−1 − (z − Hω )−1 (λ + Hω )−q χ0 Tr −1 = (ωγ˜ − ωγ )u(x − γ) (z − Hω )−1 χ0 (z − Hω,l ) γ∈Zd ,|γ|>l
−q × (λ + Hω ) χ0
Tr
and Σ2
= =
χ0 (z − Hω,l )−1 (λ + Hω,l )−q − (λ + Hω )−q χ0 Tr q χ0 (z − Hω,l )−1 (λ + Hω,l )m−q−1 (ωγ˜ − ωγ )u(x − γ) m=1
−m × (λ + Hω ) χ0
γ∈Zd ,|γ|>l
,
Tr
where in the last equality we used an iterated resolvent formula. Since |ωγ˜ − ωγ | ≤ ωmax and by standard bounds for the trace norm · Tr we have χ0 (z − Hω,l )−1 u(x − γ)χ0,β 2 d Σ1 ≤ ωmax L(L (R )) β∈Zd γ∈Zd ,|γ|>l
× χ0,β (z − Hω )−1 (λ + Hω )−q χ0 Tr C1 ≤ χ0,γ+β uLp exp(−|y| |β|/C1 ) |y|2 d d β∈Z γ∈Z ,|γ|>l
As Σ2 can be bounded in the same way, our lemma is proved.
Up to now we followed the proof of Theorem 5.1 of [25] almost literally. From now on we need sharper and more explicit estimates because later we will have to take the limit l → ∞ simultaneously with an approximation g → χ[0,E] . Special care is needed because the parameters E and l are functions of each other.
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Ann. Henri Poincar´e
Lemma 4.4 If we choose the constant C2 sufficiently large and C3 sufficiently small (depending only on d, δ2 , δ3 and C1 ), we have for all y with 0 = |y| ≤ 3: χ0,γ+β uLp exp(−|y| |β|/C1 ) ≤ C2 e−C3 |y|l |y|−2d . (35) β∈Zd
γ∈Zd |γ|>l
The proof of this and the following lemma are given in the appendix. Lemma 4.5 Let f be in C n+1 ([−1/2, 1/2]) and f˜ its almost analytic extension of order n. There exists a l1 := l1 (d, n, C3 ) < ∞ such that we have for all l ≥ l1 : ˜ ∂f (x, y) |y|−2d−2 e−C3 |y|l dx dy ≤ 2C3−n+2d+2 |f |n+1 | supp f | l−n+2d+1 . ¯ C ∂z We have to bound the derivatives of f := (λ+·)q g in terms of the derivatives of g itself. A simple calculation using Leibniz’ formula shows |f |n+1 ≤ C4 |g|n+1 , where C4 depends only on n, q and λ. We collect the estimates of Lemma 4.3, 4.4 and 4.5 and write down the needed inequalities for our difference of integrals with respect to N and Nω,l . E g(x)dN (x) − g(x)dN (x) ω,l ˜ 1 C1 ∂ f ≤ dx dy χ0,β+γ uLp exp(−|y| |β|/C1 ) (x, y) 2π C |y|2 ∂ z¯ γ∈Γ, β∈Γ
|γ|>l
≤
∂ f˜ 1 dx dy δ2 C1 (x, y) C2 |y|−2d−2 exp(−C3 |y|l) ∂ z¯ 2π C
≤
δ2 C1 C2 | supp f | |f |n+1 l−n+2d+1 πC3n−2d−2
≤ C5 | supp f | |g|n+1 l−n+2d+1 if we choose l ≥ l1 and set C5 := the constant on the rightern side.
δ2 C1 C2 C4 . πC3n−2d−2
This proves Theorem 4.1 with C5 as
The IDS approximation result (Theorem 4.1) gives information about Nω,l if properties of N are known. Exploiting this fact, we want to show that Nω,l is ”small” in the energy region where N exhibits a Lifshitz tail. To this end take g ∈ C0n+1 (R, [0, 1]) with g(x) = 1 for all x ∈ [0, E] and support in [−E/2, 2E]. Moreover let g have minimal derivative in the sense of inequality (24). We estimate g dNω,l E [Nω,l (E) − Nω,l (0)] ≤ E g dNω,l − g dN . (36) ≤ g dN + E
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Let E/2 be smaller than the gap width b below the spectral band edge 0. Since supp N = σ(Hω ) a.s. (c.f.[33]) it follows for l ≥ l1 E [Nω,l (E) − Nω,l (0)] ≤ N (2E) − N (0) + C6 E −n l−n+2d+1 ,
(37)
where we used Theorem 4.1 and equation (24). If N has Lifshitz asymptotics at the lower band edge 0, as defined in equation (18), there exists an energy value E1 such that N (E) − N (0) ≤ exp(−E −d/4 ) ∀E ∈ [0, E1 ] . (38) Together with (37) this gives E [Nω,l (E) − Nω,l (0)] ≤ e−(2E)
−d/4
+ C6 E −n l−n+2d+1 ∀E ∈ [0, E1 /2] .
(39)
For α ∈]0, 1[ we set E := 2l−α . This implies E [Nω,l (E) − Nω,l (0)]
≤ exp(−(4l−α )−d/4 ) + C6 (2l−α )−n l−n+2d+1 = exp(−4−d/4 lαd/4 ) + C6 2αn l−n(1−α)+2d+1 ≤ C6 2n l−n(1−α)+2d+1
(40)
if l ≥ l2 := l2 (d, n, α, C6 , b , E1 ). Thus we have proven that the Lifshitz tail of N implies a similar asymptotic behaviour of the IDS of the periodic approximation Hω,l as stated in the following Theorem 4.6 Let N and Nω,l be the IDS of Hω and Hω,l respectively, n ∈ N and α ∈]0, 1[. If N has a Lifshitz tail at the lower band edge 0, there exist a C7 < ∞ such that (41) E [Nω,l (2l−α ) − Nω,l (0)] ≤ C7 l−n(1−α)+2d+1 for sufficiently large l.
5 Sparsity of states near the lower band edge We want to estimate the probability of finding an eigenvalue of Hω,l (θ) in a small energy interval I 0, assuming that N exhibits a Lifshitz tail at 0. Here Hω,l (θ) := Hω,l |θΛ2l+1 = Hω |θΛ2l+1 is the operator Hω,l restricted to L2 (Λ2l+1 ) with θ-boundary conditions. The following lemma allows to bound this probability using the IDS of Hω,l . Lemma 5.1 dθ P({ω| σ(Hω,l (θ)) ∩ [0, E[ = ∅}) ≤ (2π)d E (Nω,l (E) − Nω,l (0)) . θ∈Bl
404
I. Veseli´c
Proof.
θ∈Bl
Ann. Henri Poincar´e
dθ P({ω| σ(Hω,l (θ)) ∩ [0, E[ = ∅})
≤
|Λ2l+1 |
=
|Λ2l+1 | E ( θ∈Bl dθ (N (Hω,l (θ), E) − N (Hω,l (θ), 0))
Fubini’s theorem
=
(2π)d E (Nω,l (E) − Nω,l (0))
equations (15,26)
θ∈Bl
dθ E (N (Hω,l (θ), E) − N (Hω,l (θ), 0))
ˇ Cebyˇ sev inequality
Since the MSA works with specific boundary conditions, e.g. periodic ones, we have to get rid of the average over θ ∈ Bl in the last bound. This is possible using the Lipschitz-continuity in θ of the eigenvalues of Hω,l (θ). Lemma 5.2 For any fixed θ0 ∈ Bl and E < 1 we have P({ω| σ(Hω,l (θ0 )) ∩ [0, E[ = ∅}) ≤
(2π)d E (Nω,l (E + C9 l−1 ) − Nω,l (0)) . |Bl |
(42)
Proof. The eigenvalues of Hω,l (θ) are Lipschitz continuous in θ, so we have : |Ej (Hω,l (θ)) − Ej (Hω,l (θ ))| ≤ Ξj,l |θ − θ | for some Ξj,l > 0. One can choose the Ξj,l independent of j and l only as a function of Ej (Hω,l (θ)). As we consider only eigenvalues in the energy interval [0, E[⊂ [0, 1[ even this dependence can be eliminated. Thus we can find Ξ > 0 such that Ξ ≥ Ξj,l ∀l, j . Now we can estimate : P({ω| σ(Hω,l (θ0 )) ∩ [0, E[ = ∅}) = P({ω| ∃j ∈ N : Ej (Hω,l (θ0 )) ∈ [0, E[ }) dθ P({ω| ∃j ∈ N : Ej (Hω,l (θ0 )) ∈ [0, E[ }) (43) = |B l| θ∈Bl If Ej (Hω,l (θ0 )) ∈ [0, E[ then Ej (Hω,l (θ)) ∈ [0, E + Ξ diam(Bl )[ ∀θ ∈ Bl . Using diam(Bl ) ≤ C8 l−1 we bound (43) by dθ P({ω| ∃j ∈ N : Ej (Hω,l (θ)) ∈ [0, E + C9 l−1 [ }) |B | l θ∈Bl dθ = P({ω| σ(Hω,l (θ)) ∩ [0, E + C9 l−1 [= ∅ }) |B | l θ∈Bl ≤
(2π)d |Bl |−1 E (Nω,l (E + C9 l−1 ) − Nω,l (0))
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We choose now 0 < α < 1 and E := l−α similarly as before. Thus for l ≥ l3 the bound E + C9 l−1 ≤ 2l−α is valid, with l3 depending on α and C9 . As the IDS is monotone increasing in the energy, this implies Nω,l (E + C9 l−1 ) ≤ Nω,l (2l−α ) . If N has Lifshitz tails, we estimate as in Theorem 4.6: E (Nω,l (2l−α ) − Nω,l (0)) ≤ C7 l−n(1−α)+2d+1 for l ≥ l2 . In this way we obtain from Lemma 5.2 P({ω| σ(Hω,l (θ0 )) ∩ [0, l−α [ = ∅}) ≤ C10 l−n(1−α)+3d+1
(44)
since |Bl |−1 ≤ const ld where the constant depends only on the dimension. The probability in (44) can be bounded by l−q for arbitrary q > 0 if −n(1 − α) + 3d + 1 ⇐⇒ n(1 − α)
< −q > q + 3d + 1
(45)
and l ≥ l4 := l4 (d, n, α, q, C10 ) is sufficiently large. It is obvious that for any 0 < α < 1 we can choose n in such a way that the relation (45) is valid. Similarly, for any fixed n > q + 3d + 1 it is possible to choose α sufficiently small, so that (45) holds. Particularly we can choose α from ]0, 1/4[. Recall that if H0 has regular Floquet eigenvalues at the lower spectral band edge 0, the IDS N of Hω := H0 + Vω exhibits Lifshitz asymptotics at 0. Thus we proved Proposition 1.2 with l0 := max4i=1 li .
6 Appendix Proof of Lemma 4.4 By comparing the Euclidean and sup-norm, the sum in (35) can be bounded by a constant times the integral dx dξ e−δ3 κx+ξ2 e−|y|κx2 /C1 , κ := d−1/2 . (46) Rd
ξ2 >l
Substituting x = (|y|δ3 κ/6C1 )(2x+ξ), ξ = (|y|δ3 κ/6C1 )ξ, using the parallelogram identity for · 2 and |y| ≤ 3, C1 ≥ 1 we estimate (46) by 2d
3C1 δ3 κ|y|
2d
dx
δ3 κ dξ e−x 2 −ξ 2 ≤ const |y|−2d exp − |y|l 12C1
Rd ξ 2 >δ3 κ|y|l/6C1
where the constant depends only on d, δ3 and C1 .
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Proof of Lemma 4.5 We use inequality (23) and consider first the term: n 3 |y|r χ{ x<|y|<2 x} . dx dy|y|−2d−2 exp(−C3 |y|l) |f (r) (x)| x r! C r=0
(47)
The properties of x, s and f ensure x ≥ 1, 1 < |y| < 3, thus n 3r (47) ≤ 6 dx dy e−C3 l |f (r) | ≤ 60 | supp f | |f |n e−C3 l . r! supp f [1,3] r=0 Now we turn our attention to the other summand in (23)
dx supp f
dy |y|n−2d−2 e−C3 |y|l
|f (n+1) (x)| 2n! ≤ C3−n+2d+2 |f |n+1 | supp f | l−n+2d+1 . (48)
For sufficiently large l, i.e. l ≥ l1 (d, n, C3 ), we have (47) + (48) ≤ 2C3−n+2d+2 |f |n+1 | supp f | l−n+2d+1 .
References [1] P. Anderson, Absence of diffusion in certain random lattices, Phys. Rev. 109, 1492 (1958). [2] J.-M. Barbaroux, J.-M. Combes and P. D. Hislop, Localization near band edges for random Schr¨ odinger operators, Helv. Phys. Acta 70 (1-2), 16–43 (1997). [3] R. Carmona, Random Schr¨ odinger operators. In P. L. Hennequin, editor, Ecole d’Et´e de Probabilit`es de Saint-Flour, volume 1180 of Lecture Notes in Mathematics, Springer, Berlin, 84. [4] R. Carmona and J. Lacroix, Spectral Theory of Random Schr¨ odinger Operators, Birkh¨ auser, Boston, 1990. [5] J.-M. Combes and L. Thomas, Asymptotic behaviour of eigenfunctions for multiparticle Schr¨ odinger operators, Commun. Math. Phys. 34, 251–270 (1973). [6] J.-M. Combes and P. Hislop, Localization for some continuous, random Hamiltionians in d-dimensions, J. Funct. Anal. 124, 149–180 (1994). [7] J.-M. Combes, P. D. Hislop and A. Tip, Band edge localization and the density of states for acoustic and electromagnetic waves in random media, Ann. Inst. H. Poincar´e Phys. Th´eor. 70(4), 381–428 (1999).
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[8] E. B. Davies, Spectral Theory and Differential Operators, Cambridge University Press, Cambridge, 1995. [9] M. Eastham, The spectral theory of periodic differential operators, Scottish Academic Press, Edinburgh, 1973. [10] A. Figotin and A. Klein, Localization of classical waves. I. Acoustic waves, Comm. Math. Phys. 180(2), 439–482 (1996). [11] A. Figotin and A. Klein, Localization of classical waves. II. Electromagnetic waves, Comm. Math. Phys. 184(2), 411–441 (1997). [12] W. Fischer, T. Hupfer, H. Leschke and P. M¨ uller, Existence of the density of states for multi-dimensional continuum Schr¨ odinger operators with Gaussian random potentials, Comm. Math. Phys., 190(1), 133–141 (1997). [13] W. Fischer, H. Leschke and P. M¨ uller, Spectral localization by Gaussian random potentials in multi-dimensional continuous space, J. Statist. Phys. 101(5-6), 935–985 (2000). [14] J. Fr¨ ohlich and T. Spencer, Absence of diffusion in the Anderson tight binding model for large disorder or low energy, Commun. Math. Phys., 88, 151–184 (1983). [15] J. Helffer and J. Sj¨ ostrand, Equation de Schr¨ odinger avec champ magn´etique et equation de Harper, In H. Holden and A. Jensen, editors, Schr¨ odinger Operators, Lecture Notes in Physics 345, Berlin, 1989. Springer. [16] P. D. Hislop and F. Klopp, The integrated density of states for some random operators with nonsign definite potentials. http://www.ma.utexas.edu/mp arc, preprint no. 01-139, 2001. [17] P. D. Hislop and I. Sigal, Introduction to spectral theory: with Applications to Schr¨ odinger Operators, Springer, New York, 1996. [18] J. S. Howland, Perturbation theory of dense point spectra, J. Func. Anal. 74, 52–80 (1987). [19] W. Kirsch, Random Schr¨ odinger operators, In H. Holden and A. Jensen, editors, Schr¨ odinger Operators, Lecture Notes in Physics 345, Berlin, 1989. Springer. [20] W. Kirsch and F. Martinelli, On the density of states of Schr¨ odinger operators with a random potential, J. Phys. A: Math. Gen. 15, 2139–2156 (1982). [21] W. Kirsch, P. Stollmann and G. Stolz, Localization for random perturbations of periodic Schr¨ odinger operators, Random Oper. Stochastic Equations, 6(3), 241–268 (1998). available at http://www.ma.utexas.edu/mp arc, preprint no. 96-409.
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[22] F. Klopp, An asymptotic expansion for the density of states of a random Schr¨ odinger operator with Bernoulli disorder, Random Oper. and Stoch. Equ. 3, 315–331 (1995). [23] F. Klopp, Localisation pour des op´erateurs de Schr¨odinger al´eatoires dans L2 (Rd ): un mod`ele semi-classique, Ann. Inst. Fourier (Grenoble) 45(1), 265– 316 (1995). [24] F. Klopp, Localization for some continuous random Schr¨ odinger operators, Commun. Math. Phys. 167, 553–569 (1995). [25] F. Klopp, Internal Lifshits tails for random perturbations of periodic Schr¨ odinger operators, Duke Math. J. 98(2), 335–396 (1999). [26] F. Klopp, Internal Lifshitz tails for Schr¨ odinger operators with random potentials, to appear in Journ. Math. Phys. [27] F. Klopp and J. Ralston, Endpoints of the spectrum of periodic operators are generically simple, preprint at site http://www.ma.utexas.edu/mp arc, 2000. [28] S. Kotani and B. Simon, Localization in general one-dimensional random systems II: continuum Schr¨odinger operators, Commun. Math. Phys. 112, 103–119 (1987). [29] F. Martinelli and H. Holden, On absence of diffusion near the bottom of the spectrum for a random Schr¨ odinger operator on L2 (Rν ), Commun. Math. Phys. 93, 197–217 (1984). [30] F. Martinelli and E. Scoppola, Remark on the absence of the absolutely continuous spectrum for d-dimensional Schr¨ odinger operator with random potential for large disorder or low energy, Commun. Math. Phys. 97, 465–471 (1985). [31] G. A. Mezincescu, Internal Lifshitz singularities for one dimensional Schr¨ odinger operators, Commun. Math. Phys. 158, 315–325 (1993). [32] H. Najar, Lifshitz tails for random acoustic operators. preprint available at www-math.math.univ-paris13.fr/prepub/pp2001/pp2001-03.html, 2001. [33] L. A. Pastur and A. L. Figotin, Spectra of Random and Almost-Periodic Operators, Springer Verlag, Berlin, 1992. [34] M. Reed and B. Simon, Methods of Modern Mathematical Physics II, Fourier Analysis, Self-Adjointness, Academic Press, San Diego, 1975. [35] M. Reed and B. Simon, Methods of Modern Mathematical Physics IV, Analysis of Operators. Academic Press, San Diego, 1978. [36] M. A. Shubin, Spectral theory and index of elliptic operators with almostperiodic coefficients, Russ. Math. Surveys 34, 109–157 (1979).
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[37] B. Simon, Schr¨ odinger semigroups, Bull. Am. Math. Soc. 7, 447–526 (1982). [38] B. Simon and T. Wolff, Singular continuous spectrum under rank one perturbations and localization for random Hamiltonians, Comm. Pure Appl. Math. 39, 75–90 (1986). [39] J. Sj¨ ostrand, Microlocal analysis for the periodic magnetic Schr¨ odinger equation and related questions, In Microlocal analysis and applications, volume 1495 of Lecture Notes in Mathematics. Springer Verlag, Berlin, 1991. [40] P. Stollmann, Localization for random perturbations of anisotropic periodic media, Israel J. Math. 107, 125–139 (1998). [41] P. Stollmann, Caught by disorder: A Course on Bound States in Random Media, volume 20 of Progress in Mathematical Physics. Birkh¨auser, July 2001. [42] I. Veseli´c, Lokalisierung bei zuf¨ allig gest¨orten periodischen Schr¨ odingeroperatoren in Dimension Eins, Diplomarbeit, Ruhr-Universit¨at Bochum, 1996. [43] I. Veseli´c, Wegner estimate for some indefinite Anderson-type Schr¨odinger operators with differentiable densities, preprint, http://www.ma.utexas.edu/mp arc/, 2000, to appear in Lett. Math. Phys. [44] H. von Dreifus and A. Klein, A new proof of localization in the Anderson tight binding model, Commun. Math. Phys. 124, 285–299 (1989). [45] F. Wegner, Bounds on the DOS in disordered systems, Z. Phys. B 44, 9–15 (1981). [46] H. Zenk, Anderson localization for a multidimensional model including long range potentials and displacements, Preprint-Reihe des Fachbereichs Mathematik at the Johannes Guteberg-Universit¨ at Mainz, 1999. [47] F. Germinet and A. Klein, Bootstrap multiscale analysis and localization in random media, Comm. Math. Phys. 222 (2), 415–448 (2001). [48] F. Germinet and A. Klein, A characterization of the Anderson metal-insulator transport transition, http://www.ma.utexas.edu/mp arc/, preprint 2001. Ivan Veseli´c Fakult¨ at f¨ ur Mathematik Ruhr-Universit¨ at Bochum D-44780 Bochum Germany email: [email protected] Communicated by Jean Bellissard submitted 12/09/00, accepted 06/12/01
Ann. Henri Poincar´e 3 (2002) 411 – 433 c Birkh¨ auser Verlag, Basel, 2002 1424-0637/02/030411-23 $ 1.50+0.20/0
Annales Henri Poincar´ e
Insertion and Elimination: the Doubly Infinite Lie Algebra of Feynman Graphs A. Connes and D. Kreimer
Abstract. The Lie algebra of Feynman graphs gives rise to two natural representations, acting as derivations on the commutative Hopf algebra of Feynman graphs, by creating or eliminating subgraphs. Insertions and eliminations do not commute, but rather establish a larger Lie algebra of derivations which we here determine.
Introduction The algebraic structure of perturbative QFT [1, 2, 3, 4] gives rise to commutative Hopf algebras H and corresponding Lie-algebras L, with H being the dual of the universal enveloping algebra of L. L can be represented by derivations of H, and two representations are most natural in this respect: elimination or insertion of subgraphs. Perturbation theory is indeed governed by a series over one-particle irreducible graphs. It is then a straightforward question how the basic operations of inserting or eliminating subgraphs act. These are the basic operations which are needed to construct the formal series over graphs which solve the Dyson–Schwinger equations. We give an account of these actions here as a further tool in the mathematician’s toolkit for a comprehensible description of QFT. We introduce these structures by first considering the case of undecorated rooted trees. In that case one is led naturally to the two basic operations of grafting and trimming using the relation between the Hopf algebras Hcm and Hrt [2]. The Hopf algebra Hcm is neither commutative nor cocommutative but admits a finite set of generators with simple relations. The basic relation ([2]) between a commutative subalgebra 1 of Hcm and the Hopf algebra Hrt was obtained using the ”natural growth Hcm operation” on trees. By extending this ”natural growth operation” to the grafting of arbitrary trees we show how to enlarge Hrt to a Hopf algebra Hrtt whose 1 relation to Hrt is the same as the relation of Hcm with Hcm . In particular it is neither commutative nor cocommutative. We show that it is obtained as a ”bicrossed product” construction from a doubly infinite Lie algebra of rooted trees, similar to the Lie algebra of formal vector fields. Since most of the information is then contained in that Lie algebra, which can be concretely described from grafting and trimming operations, we then turn to Feynman graphs, and only discuss the Lie algebra aspect in that case.
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1 Undecorated rooted trees The Hopf algebras Hcm and Hrt Let us first recall the constructions of the basic Hopf algebras involved in [5] and [2], and compare their properties. As an algebra Hcm is noncommutative but finitely generated. It is generated by three elements Y , X, δ1 . To describe the relations between these three generators, one lets δn , n ≥ 1 be defined by induction by [X, δn ] = δn+1 ∀ n ≥ 1 ,
(1)
then the presentation of the relations in Hcm is the following, [Y, X] = X, [Y, δn ] = n δn , [δn , δm ] = 0 ∀ n, m ≥ 1 .
(2)
The coproduct ∆ in Hcm is defined by ∆ Y = Y ⊗ 1 + 1 ⊗ Y , ∆ X = X ⊗ 1 + 1 ⊗ X + δ1 ⊗ Y , ∆ δ1 = δ1 ⊗ 1 + 1 ⊗ δ1 (3) and the equality, ∆(h1 h2 ) = ∆h1 ∆h2
∀ hj ∈ HT .
(4)
The Hopf algebra Hcm is neither commutative nor cocommutative but is 1 obtained in a simple manner from the commutative subalgebra Hcm generated by the δn . Theorem ([5]) Let G2 be the group of formal diffeomorphisms of the real line of the form ψ(x) = x + o(x). For each n, let γn be the functional on G2 defined by γn (ψ −1 ) = (∂xn log ψ (x))x=0 . The equality Θ(δn ) = γn determines a canonical isomorphism Θ of the Hopf alge1 with the Hopf algebra of coordinates on the group G2 . The Hopf algebra bra Hcm Hcm is the bicrossed product associated to the formal decomposition G = G1 G2 associated to the decomposition Lie G = Lie G1 +Lie G2 of formal vector fields in their affine part (Lie G1 ) and nilpotent part (Lie G2 ). The Hopf algebra Hrt of rooted trees is commutative but not finitely generated. Recall that a rooted tree T is, by definition, a finite, connected, simply connected, one dimensional simplicial complex with a base point ∗ ∈ ∆0 (T ) = {set of vertices of T }. This base point is called the root. By the degree of the tree we mean (5) |T | = Card∆0 (T ) = # of vertices of T . By a simple cut of a rooted tree T we mean a subset c ⊂ ∆1 (T ) of the set of edges of T such that for any x ∈ ∆0 (T ) the path (∗, x) only contains at most one element of c .
(6)
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Thus what is excluded is to have two cuts of the same path or branch. Given a cut c the new simplicial complex Tc with ∆0 (Tc ) = ∆0 (T ) and ∆1 (Tc ) = ∆1 (T )\c ,
(7)
is no longer connected, unless c = ∅. We let Rc (T ) be the connected component of ∗ with the same base point and call it the trunk. We endow each other connected component, called a cut branch, with the base point coming from the cut. We obtain in this way a set (with multiplicity) of rooted trees, which we denote by Pc (T ). We let Σ be the set of rooted trees up to isomorphism, and let Hrt be the polynomial commutative algebra generated by the symbols δT , T ∈ Σ . One defines a coproduct on Hrt by ∆ δT = δT ⊗ 1 + 1 ⊗ δT +
c
(8)
δTi ⊗ δRc (T ) ,
(9)
Pc (T )
where the last sum is over all non trivial simple cuts (c = ∅) of T , while the product is over the cut branches. Pc (T )
Equivalently, one can write (9) as ∆ δT = δT ⊗ 1 +
c
δTi ⊗ δRc (T ) ,
(10)
Pc (T )
where the last sum is over all simple cuts. This defines ∆ on generators and it extends uniquely as an algebra homomorphism, ∆ : Hrt → Hrt ⊗ Hrt . (11) The first basic relation between the Hopf algebras Hcm and Hrt is the Hopf algebra homomorphism ([2]) obtained using the ”natural growth” operator N defined as the unique derivation of the commutative algebra Hrt such that, δT (12) N δT = where the trees T are obtained by adding one vertex and one edge to T in all possible ways without changing the base point. It is clear that the sum (12) contains |T | terms. Theorem ([2]) The equality Λ(δn ) = N n (δ∗ ) determines a canonical homomor1 phism Λ of the Hopf algebra Hcm into the Hopf algebra Hrt .
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This theorem suggests, as we did in [2] to enlarge the Hopf algebra Hrt in the 1 same way as Hcm is naturally enlarged to Hcm , by adjoining the elements Y, X implementing both the grading and the natural growth operators. We shall now show that it is indeed possible to do much more by extending the natural growth operator N to the grafting of arbitrary trees.
The derivations NT of Hrt Let us first extend the construction of the natural growth operator N to get operators NT labeled by arbitrary trees. For a given rooted tree T we consider the unique derivation NT of Hrt such that, for any t ∈ Σ, δ(t ∪v T ) (13) NT (δt ) = v
where in the summation, v runs through the vertices v ∈ ∆0 (t) and where the rooted tree t = t ∪v T is obtained as the union of t and T , with the root ∗ of T identified with v. One has ∆1 (t ∪v T ) = ∆1 (t) ∪ ∆1 (T ),
(14)
root(t ∪v T ) = root(t),
(15)
and the number of vertices of (t ∪v T ) is |t | = |t| + |T | − 1 .
(16)
When T = ∗ has one element we see that N∗ (δt ) = |t| δt
(17)
thus we get the derivation Y . When T = is the rooted tree with one edge, we just get the natural growth operation: N = N . Since NT is extended as a derivation one has NT
δti =
n
δt1 . . . N (δtk ) . . . δtn .
(18)
1
Let us now prove Lemma ∆(NT (a)) = NT ⊗ id + id ⊗ NT +
c
Pc (T )
δtj ⊗ NRc (T ) ∆(a) .
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Proof. Both sides of the equation are linear maps from Hrt to Hrt ⊗ Hrt which satisfy the derivation rule, ρ(ab) = ρ(a) ∆(b) + ∆(a) ρ(b). Indeed Hrt is a commutative algebra so that the multiplication by a product of δtj ⊗ 1 does not alter the derivation rule. Thus it is enough to check the lemma for a = f δt , t ∈ Σ. Now, by definition of the coproduct, ∆ NT (δt ) = NT (δt ) ⊗ 1 + 1 ⊗ NT (δt ) +
v0 ,c
δtj ⊗ δRc (t )
where v0 varies in ∆0 (t) and the c varies through simple cuts of t = t ∪v0 T . / Rc (t ) i.e. Let us first consider the partial sum over pairs (v0 , c) with v0 ∈ v0 ∈ ∪ ti . This means that the segment [∗, v0 ] is cut somewhere and hence that c ∩ T = ∅ since otherwise the cut would not be simple. We thus have c ⊂ t so that we can view c as a cut of t. Thus Rc (t ) = Rc (t). Also v0 ∈ ∪ ti and the sum over v0 decomposes as a sum over i and yields for each i the value δti . (19) δtj = NT (δti ) j=i
Thus, since NT is a derivation, the partial sum gives
NT
c (cut of t)
Now this equals NT
c Pc (t)
δti ⊗ δRc (t) .
(20)
Pc (t)
δti ⊗δRc (t) and we can group this sum with NT (δt )⊗
1, using NT (1) = 0 to get (NT ⊗ id) ∆(δt ) ,
(21)
which is the first term in the right hand side of the equation of the lemma. We then consider the partial sum over pairs (v0 , c) with v0 ∈ Rc (t ) and c ∩ T = ∅. Then c is a cut of t as above, while v0 now varies among the vertices of Rc (t). One has ti = ti and Rc (t ) = Rc (t) ∪v0 T . Thus the sum over v0 replaces δRc (t) by NT (δRc (t) ) without touching the δti . We can group this with 1 ⊗ NT (δt ) and get (id ⊗ NT ) ∆(δt ) , (22) which is the second term in the right hand side of the equation of the lemma. We are now left only with the partial sum over pairs (v0 , c) such that c ∩ T = ∅ (in which case v0 ∈ Rc (t )). Let us then fix the nonempty simple cut of T , c = c ∩ ∆1 (T ) ,
(23)
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and show that the corresponding partial sum is equal to δtj ⊗ NRc (T ) (∆ δt ) .
(24)
tj ∈Pc (T )
Since ∆1 (t ) = ∆1 (t) ∪ ∆1 (T ), one has c = c1 ∪ c where c1 now varies among (possibly empty) simple cuts of t. Moreover v0 now varies in Rc1 (t). To each ε ∈ c = c1 ∪ c there is a corresponding fallen branch tε . For ε ∈ c1 it is a fallen branch of t for c1 while for ε ∈ c it is a fallen branch of T for c . Thus the product of fallen branches is δti δtj . (25) ti ∈Pc1 (t)
tj ∈Pc (T )
One has ∆ δt = δt ⊗ 1 +
c1
Pc1 (t)
δti ⊗ δRc1 (t) ,
where c1 varies among (possibly empty) simple cuts of t. Let P =
(26) tj ∈Pc (T )
δtj and
let us look at the terms in (P ⊗ NRc (T ) )(∆ δt ) .
(27)
The term δt ⊗ 1 does not contribute since N (1) = 0. When we apply P ⊗ NRc (T ) to the term δtj ⊗ δRc1 (t) in ∆ δt , we get Pc1 (t)
v0
P
Pc1 (t)
δtj ⊗ δRc1 (t) ∪v0 Rc (t ) ,
(28)
where v0 varies in Rc1 (t). With t = t ∪v0 T , one has Rc1 (t) ∪v0 Rc (T ) = Rc (t ), for c = c1 ∪ c . Thus we get the corresponding term of ∆(NT (δt )), namely P
δtj ⊗ δRc (t ) .
(29)
tj ∈Pc1 (t)
Taking the sum over pairs (v0 , c1 ) such that v0 ∈ Rc1 (t) yields the required equality and completes the proof of the lemma. It is then natural to enlarge the Hopf algebra Hrt by introducing new generators XT , T ∈ Σ such that [XT , δt ] = NT (δt ) (30)
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Insertion and Elimination: the Lie Algebra of Feynman Graphs
and with coproduct rule given by ∆ XT = XT ⊗ 1 + 1 ⊗ XT +
c
δtj ⊗ XRc (T ) .
417
(31)
Pc (T )
This coproduct is superficially similar to (26), but the right hand side now involves both the δ’s and the X’s. In order to complete the presentation of the extended Hopf algebra Hrtt , we need to compute the Lie bracket of the derivations NT . This is straightforward and given by
Lemma [NT1 , NT2 ] =
NT2 ∪v2 T1 −
v2 ∈∆0 (T2 )
NT1 ∪v1 T2 .
(32)
v1 ∈∆0 (T1 )
We are dealing with derivations of Hrt and it is thus enough to consider the action of both sides on δt . One has NT1 (NT2 (δt )) = NT1 (δt∪v0 T2 ) = δ(t∪v0 T2 )∪v1 T1 =
v0 ∈∆0 (t)
v0 ∈∆0 (t) v1 ∈∆0 (T2 )
v0 ∈∆0 (t) v1 ∈∆0 (t∪v0 T2 )
δt∪v0 (T2 ∪v1 T1 ) +
δt∪v0 T1 ∪v1 T2 .
v0 ,v1 ∈∆0 (t) v0 =v1
The last term is symmetric in T1 , T2 and thus does not contribute to the commutator which is thus given by the formula of the lemma. We can thus complete the presentation of the Hopf algebra Hrtt by the rule [XT1 , XT2 ] = XT2 ∪v2 T1 − XT1 ∪v1 T2 , (33) v2 ∈∆0 (T2 )
v1 ∈∆0 (T1 )
and define Hrtt as the enveloping algebra of the Lie algebra which is the linear span of the XT , δt , T, t ∈ Σ, with bracket given by (33), (30) and the commutativity of the δ’s. We define a coproduct on Hrtt by (26) and (31). We thus get Theorem Endowed with the above structure Hrtt is a Hopf algebra. The equalities Λ(δn ) = N n (δ∗ ), Λ(Y ) = X∗ , Λ(X) = (X ) determine a canonical homomorphism Λ of the Hopf algebra Hcm in the Hopf algebra Hrtt . The best way to comprehend the Hopf algebra structure of Hrtt is to consider the natural action of Hrtt as an algebra on the dual of Hrt , obtained by transposition. The compatibility of the algebra structures dictates the Hopf algebra structure, by transposing multiplication to comultiplication. Combining the basic Hopf algebra identity, m(S ⊗ Id)∆ = with equation (31) yields the following explicit formula for the antipode S(XT ), T ∈ Σ, S(XT ) = −XT − S(δtj )XRc (T ) , (34) c
Pc (T )
using the known formula for S(δtj ) in the subalgebra Hrt .
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The reader should note that S 2 = 1 for the antipode S in Hrtt , as this algebra is neither commutative nor cocommutative, comparable to the situation in Hcm . Indeed, we now have a large supply of natural growth operators in generalization of that situation. Let ∆(XT ) = ∆(XT ) − XT ⊗ 1 − 1 ⊗ XT . For the multiple application of that subtracted coproduct, we can still uniquely write n
∆ (XT ) = XT ⊗ · · · ⊗ XT ... ,
.
n+1 s
It is obvious that the Hopf algebra endomorphism S 2 fulfills S 2 (δt ) = δt , while for the generators XT we have Proposition S 2 (XT ) = XT + NT (δT ) + S(δT )NT (δT ).
Proof. In the above notation, S(XT ) = −XT − S(δT )XT , and also S(δT )XT = δT S(XT ). Thus S 2 (XT ) =
S[−XT − S(δT )XT ]
= =
XT + S(δT )XT − S(XT )δT XT + S(δT )XT − δT S(XT ) − [S(XT ), δT ]
= =
XT − [S(XT ), δT ] XT + NT (δT ) + S(δT )NT (δT ),
2
using ∆ .
It is of course desirable to extend to the Hopf algebra Hrtt the description of Hcm as a bicrossed product associated to the decomposition Lie G = Lie G1 +Lie G2 of the Lie algebra of formal vector fields. Our next task will be to describe the Lie algebra L that will play the role of the Lie algebra of formal vector fields. As a preliminary remark, let us relate the Lie algebra structure L1 on the XT given by (33) to an operand P. This insertion operand [6] underlies the preLie structure, whose antisymmetrization is the Lie bracket (33). The operad is obtained by considering as elements of P(n) a pair of rooted tree t and a bijection, σ : {1, . . . , n} → ∆0 (t) .
(35)
We then define t ◦i t as t ∪σ(i) t , for i ∈ {1, . . . , n} and where the new bijection is obtained by shifting the labels of the vertices σ(i+1) . . . σ(n) to i+n , . . . , n+n −1 as well as the labels of the vertices σ (1) . . . σ (n ) to i, i + 1, . . . , i + n − 1.
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2
3
419
2 4
1
3 7
5
3
1
2
=
1
6
5
3
9
4
7
8 6
One has a natural action of Sn the group of permutations of {1, . . . , n} which replaces σ by σ ◦ π −1 , i.e. replaces the labeling σ −1 of the vertices by π ◦ σ −1 . One checks that (36) tπ ◦π(i) tρ = (t ◦i t )α where α is obtained from the permutations π of {1, . . . , n}, ρ of {1, . . . , n } and i ∈ {1, . . . , m} by acting by ρ in {i, i + 1, . . . , i + n − 1} and by π after collapsing the above interval to {i}. One also checks the following two equalities for λ ∈ P(%), µ ∈ P(m), ν ∈ P(n) (λ ◦i µ) ◦j+m−1 ν = (λ ◦j ν) ◦i µ , (λ ◦i µ) ◦i−1+j ν = λ ◦i (µ ◦j ν) ,
1≤i<j≤%,
(37)
1 ≤ i ≤ %, 1 ≤ j ≤ m.
(38)
The first is the independence of two graftings at two distinct vertices, and the second is a kind of associativity of grafting.
The Lie algebra L We shall now describe the Lie algebra L = L1 + L2 playing the role of the Lie algebra of formal vector fields in the case of rooted trees, i.e. bearing the same relation to Hrtt as the Lie algebra of formal vector fields does to Hcm . We already know the Lie subalgebra L1 of the XT ’s. The Lie algebra L2 is the Lie algebra of primitive elements in the dual of Hrt . In order to obtain L we consider the natural actions of both L1 and L2 as derivations of the commutative algebra Hrt . We already saw the action N of L1 . The action of L2 is the canonical action of the Lie algebra of primitive elements of the dual of Hrt on the commutative algebra Hrt . It is given by the following derivations MT of Hrt MT (a) = ZT ⊗ id, ∆(a)
∀a ∈ H,
(39)
∗ given by the linear where, for T ∈ Σ, ZT is the primitive element of the dual Hrt form on Hrt which vanishes on any monomial δ1 δ2 ..... δn except for δT , with
ZT , δT = 1.
(40)
One has ZT (ab) = ZT (a) ε(b) + ε(b) ZT (b) so that by construction MT is a derivation of H.
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The Lie bracket of the ZT ’s is given by the Lie algebra of rooted trees, i.e. [ZT1 , ZT2 ] = (n(T1 , T2 ; T ) − n(T2 , T1 ; T )) ZT , (41) where n(T1 , T2 ; T ) is the number of cuts c of T of cardinality one (|c| = 1) such that Pc (T ) = T1 , Rc (T ) = T2 . For a = δt we get MT (δt ) = δRc (t) if t = T (42) |c|=1 Pc (t)=T
and MT (δT ) = 1 .
(43) 1
Thus MT (δt ) = 0 unless t = T or t admits an edge ε ∈ ∆ (t) such that Pε (t) = T .
ε T=
t=
By construction M is a representation of the Lie algebra L2 in the Lie algebra of derivations D of Hrt which preserve the linear span D = {Σ λT δT + λ1 1}, D (D) ⊂ D.
(44)
Similarly the representation N of L1 is given by derivations fulfilling (44). In order to show that L = L1 + L2 is a Lie algebra, let us now compute the commutator MT1 NT2 − NT2 MT1 .
(45)
Let us first consider the case where T1 and T2 are not comparable, i. e. we assume that T1 = Pc (T2 ) for all cuts c, |c| = 1 of T2 and that T1 = t1 ∪v T2 for any tree t1 and vertex v ∈ ∆0 (t1 ) . Let us show that in that case MT1 and NT2 actually commute. The nonzero terms in MT1 NT2 (t) are given by δPε (t ∪v0 T2 ) for a vertex v0 ∈ ∆0 (t) and an edge ε ∈ ∆1 (t ∪v0 T2 ) such that Pε (t ∪v0 T2 ) = T1 . Now ∆1 (t ∪v0 T2 ) = ∆1 (t) ∪ ∆1 (T2 ), and if ε ∈ ∆1 (T2 ) would yield a nonzero term, then T1 would appear as Pε (T2 ). Thus ε ∈ ∆1 (t). Next if v0 ∈ / Rε (t) then v0 ∈ Pε (t) and Pε (t ∪v0 T2 ) = Pε (t) ∪v0 T2 . But by hypothesis this cannot be T1 so we get 0. The only remaining case is v0 ∈ Rε (t) so that Pε (t ∪v0 T2 ) = Pε (t) while Rε (t ∪v0 T2 ) = Rε (t) ∪v0 T2 , thus we get MT1 NT2 (t) = δRε (t) ∪v0 T2 . (46) v0 ∈∆0 (t),ε∈∆1 (t) v0 ∈Rε (t),Pε (t)=T1
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But we have MT1 (t) =
δRε (t)
421
(47)
ε∈∆1 (t),Pε (t)=T1
and NT2 (MT1 (t)) =
ε∈∆1 (t),v0 ∈Rε (t) Pε (t)=T1
δRε (t) ∪v0 T2 .
(48)
Thus we see that if T1 and T2 are not comparable we get MT1 NT2 = NT2 MT1 .
(49)
In general, given t, T1 , T2 ∈ Σ we define the integers N (t, T2 ; T1 ) and M (T1 , T2 ; t) by N (t, T2 ; T1 ) = NT2 (δt ), ZT1 (50) and M (T1 , T2 ; t) = MT1 (δT2 ), Zt .
(51)
By construction N (t, T2 ; T1 ) is the number of times T1 occurs as t ∪v T2 while M (T1 , T2 ; t) is the number of times T1 occurs as Pc (T2 ) with |c| = 1 and Rc (T2 ) = t. We then get Lemma [MT1 , NT2 ] =
t
N (t, T2 ; T1 ) Mt +
M (T1 , T2 ; t) Nt .
(52)
t
First assume |T1 | ≥ |T2 | so that T1 cannot be a Pc (T2 ), for |c| = 1 and M (T1 , T2 ; t) = 0. Then the same computation of [MT1 , NT2 ](δt ) as above gives the sum of the δRε (t) such that T1 occurs as a Pε (t) ∪v0 T2 . Fixing then t1 = Pε (t) we see that we obtain the sum of the Mt1 with multiplicity given by the number of solutions of t1 ∪v T2 = T1 . (53) Next assume that |T1 | < |T2 | so that T1 can occur as Pc (T2 ), |c| = 1, but cannot occur as t1 ∪v T2 , so that N (t, T2 ; T1 ) = 0. Then in the above computation of [MT1 , NT2 ](δt ) the case v0 ∈ Pε (t) above only gives 0 and the only nonzero contribution comes when ε ∈ ∆1 (T2 ). One then has Rε (t ∪v0 T2 ) = t ∪v0 Rε (T2 ) and Pε (t ∪v0 T2 ) = Pε (T2 ) which must be T1 to yield a non zero result. Thus we obtain the sum of the δRε (t ∪v0 T2 ) where Pε (T2 ) = T1 . This equals the sum of the δt ∪v0 Rε (T2 ) and hence, letting t2 = Rε (T2 ) the sum of the M (T1 , T2 ; t2 ) Nt2 (δt ). We need to take care of (43), i.e. to consider the case where MT1 is applied to some t ∪v0 T2 = T1 which only occurs when |T1 | ≥ |T2 |. For each such term one takes c = ∅ so the above discussion does not apply, but one can check that the additional contribution to both sides of (52) do agree when evaluated on t fulfilling (53) for some v ∈ ∆0 (t).
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We can now define the full Lie algebra L of rooted trees by introducing new generators of the form, Z−t where t is a rooted tree, and extending the Lie bracket (41) based on the above lemma. We associate Z−T with −NT and ZT with MT and work out the Lie brackets so that we get a representation. In particular the elements Z0 , Z−1 now become Z0 = Z−∗ , Z−1 = Z−T .
(54)
We use the − sign, −NT to get that the commutator with Z−∗ does give the grading of the Lie algebra. Indeed if we apply (52) for T2 = ∗ we get [−N∗ , MT ] = |T | MT ,
(55)
[−N∗ , NT ] = (1 − |T |) NT .
(56)
while one has
Theorem L = L1 + L2 is a Lie algebra. The Hopf algebra Hrtt is the bicrossed product associated to the decomposition L = L1 + L2 . As a final remark, note that the Lie subalgebra L2 generated by the ZT is naturally isomorphic to a subalgebra of L1 generated by the Z−T . Indeed one lets ∗T be the new rooted tree given by
T (57)
Then the following map is an inclusion L2 ⊂ L1 , ZT →
1 Z−(∗T ) . ST
(58)
By (41) we see that this is a Lie algebra homomorphism since the grafting at ∗ gives a symmetric result, which drops out of the bracket:
T1
T2 .
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2 Graphs Formal Definitions We only consider graphs without self-loops: no edge starts and ends in the same vertex. We allow for multiple edges though: two vertices might be connected by more than one edge. First, we define n-particle irreducible (n-PI) graphs. Definition A n-particle irreducible graph Γ is a graph such that upon removal of any set of n of its edges it is still connected. Its set of edges is denoted by Γ[1] and its set of vertices is denoted by Γ[0] . Edges and vertices can be of various different type. The type of an edge is often indicated by the way we draw it: (un-)oriented straight lines, curly lines, dashed lines and so on. These types of edges, often called propagators in physicists parlance, are chosen in accordance to Lorentz covariant wave equations: the propagator as the analytic expression assigned to an edge is an inverse wave operator with boundary conditions typically chosen in accordance with causality. The types of vertices are determined by the types of edges to which they are attached: Definition For any vertex v ∈ Γ[0] we call the set fv := {f ∈ Γ[1] | v ∩ f = ∅} its type. Note that fv is a set of edges. Of particular importance are the 1PI graphs. They do not decompose into disjoint graphs upon removal of an edge. Note that any n-PI graphs is also (n − 1)PI, ∀n ≥ 1. A graph which is not 1-PI is called reducible. Also, any connected graph is considered as 0-PI. A further notion needed is the one of external and internal edges. Definition An edge f ∈ Γ[1] is internal, if {vf } := f ∩ Γ[0] is a set of two elements. So, internal edges connect two vertices of the graph Γ. Definition An edge f ∈ Γ[1] is external, if f ∩ Γ[0] is a set of one element. As we exclude self-loops, this means that an external edge has an open end. Thus external edges are associated with a single vertex of the graph. These edges correspond to external particles interacting in the way prescribed by the graph. There are obvious gluing operations combining 1PI graphs into reducible graphs, by identifying two open ends of edges of the same type originating from different 1PI graphs. We will make no use of reducible graphs here but note that the Hopf and Lie algebra structures could be set up in this context as well.
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Γ[1] obviously decomposes into the set of internal edges and the set of external edges of a graph Γ, [1] [1] Γ[1] = Γext ∪ Γint . We now turn to the possibilities of inserting graphs into each other. Our first requirement is to establish bijections between sets of edges so that we can define gluing operations. Definition We call two sets of edges I1 , I2 compatible, I1 ∼ I2 , iff they contain the same number of edges, of the same type. Compatibility is an equivalence relation. We will use it to glue graphs into each other. To compare vertices, we look at the adjacent edges: Definition Two vertices v1 , v2 are of the same type, if fv1 is compatible with fv2 . Quite often, we will shrink a graph to a point. The only useful information still available after that process is about its set of external edges: [1]
Definition We define res(Γ) to be the result of identifying Γ[0] ∪ Γint with a point in Γ. An example is
. [1]
[1]
Note that res(Γ)[1] ≡ res(Γ)ext ∼ Γext . By construction all graphs which have compatible sets of external edges have the same residue. [1] If the set Γext is empty, we call Γ a vacuum graph, if it contains a single element we call the graph a tadpole graph. Vacuum graphs and tadpole graphs will be discarded in most of what follows. If this set contains two elements, we call Γ a self-energy graph, if it contains more than two elements, we call it an interaction graph. Further we restrict ourselves to graphs which have vertices such that the cardinality of their types is ≥ 2. If needed, for example in the presence of external fields, this can be relaxed. A further important notion is the gluing of graphs into each other. It is the opposite of the shrinking of a graph to its residue. While in that process, a graph is reduced to a vertex of a specified type, we can replace any vertex v ∈ Γ[0] of [1] type fv by a graph γ, as long as fv ∼ γext - a vertex will be replaced by a graph which has external edges compatible with its type. To specify such a gluing of γ into Γ we first have to choose an internal vertex [1] v where we wish to glue. If the type of v is incompatible with γext , we define the result to vanish. If the two sets of edges are compatible, we will have in general to choose a bijection between the two sets of edges. Summing over all places and
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bijections defines an operation Γ - γ which sums over all ways of inserting γ into Γ. We impose a normalization such that topologically different graphs are generated with unit multiplicity. The following picture illustrates this process.
. Proposition This gluing operation is pre-Lie. Proof. It suffices to show that for 1PI graphs Γi , i = 1, 2, 3, we have Γ1 - (Γ2 - Γ3 ) − (Γ1 - Γ2 ) - Γ3 = Γ1 - (Γ3 - Γ2 ) − (Γ1 - Γ3 ) - Γ2 . This is elementary using that both sides reduce to the sum over all ways of gluing Γ2 and Γ3 simultaneously into Γ1 at disjoint places. Note that this pre-Lie operation can be extended to the insertion at internal edges (self-energies). Furthermore, external structures [3] can be incorporated easily, using colored types of vertices. Choices of types of lines and vertices are typically dictated by a chosen QFT, where, in particular, one often only considers superficially divergent graphs. External structures reflect their power counting degree of divergence. We let LF G be any such chosen Lie-algebra generated from this pre-Lie product, and HF G be the commutative Hopf algebra which we obtain as the dual of the universal enveloping algebra of LF G .
Derivations on the Hopf algebra [k]
We have the decomposition of HF G by the bidegree HF G = ⊕∞ k=0 HF G , reduced [0] to scalars ∈ HF G by the counit. The linear basis of HF G is denoted by HF G,L . It is spanned by generators δΓ , where Γ is a 1PI graph. Elements of HF G are polynomials in these commutative variables. We write ZΓ for the dual basis of the universal enveloping algebra with pairing K ZΓ , δΓ = δΓ,Γ ,
where on the rhs we have the Kronecker δ K , and extend the pairing by means of the coproduct ZΓ1 ZΓ2 , X = ZΓ1 ⊗ ZΓ2 , ∆(X).
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For X = i ci Γi , we extend by linearity so that δX = i ci δΓi , and similarly for ZX . Quite often, we want to refer to the graph(s) which index an element in HF G or LF G . For that purpose, for each element in HF G and each element in LF G we introduce a map to graphs: ZX = X, δX = X.
Further, we write ∆(X) = i X(i) ⊗ X(i) for the coproduct in the Hopf algebra HF G . The Lie algebra LF G gives rise to two representations acting as derivations on the Hopf algebra HF G : ZΓ+ × δX = δXΓ and
ZΓ− × δX =
i
ZΓ , X(i) X(i) .
Furthermore, any term in the coproduct of a 1PI graph Γ determines gluing data Gi such that Γ = Γ(i) -Gi Γ(i) , ∀i. Here, Gi specifies vertices in Γ(i) and bijections of their types with the elements of Γ(i) such that Γ is recovered from its parts:
. The first line gives a term (i) in the coproduct, decomposing this graph into its only divergent subgraph (assuming we have chosen φ3 in six dimensions, say) and the corresponding cograph, the second line shows the gluing Gi for this term, in this example. We want to understand the commutator [ZΓ+1 , ZΓ−2 ], acting as a derivation on the Hopf algebra element δX . To this end introduce Z[Γ1 ,Γ2 ] × δX = ZΓ2 , X(i) X(i) - G i Γ1 . i
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Here, the gluing operation Gi still acts such that each topologically different graph is generated with unit multiplicity.
. Note that if res(Γ1 ) ∼ res(Γ2 ), Z[Γ1 ,Γ2 ] ×δX vanishes, as the existence of a bijection [1] between edges adjacent to Γ2 in X and Γ1,ext requires the compatibility of the residues of Γ1 , Γ2 . Let X, Y be related as Z[Γ1 ,Γ2 ] × δX = Y , for a 1PI graph X. Then, Y is a sum of say k 1PI graphs. We immediately have thanks to our gluing conventions Proposition Z[Γ2 ,Γ1 ] × δY = k X. Let us now consider [Z[Γ1 ,Γ2 ] , Z[Γ3 ,Γ4 ] ] × δX . We first define Y234 := {Y ∈ HF G,L |ZΓ2 , Z[Γ3 ,Γ4 ] × δY = 1} and Y412 := {Y ∈ HF G,L |ZΓ4 , Z[Γ1 ,Γ2 ] × δY = 1}. Let ∆Γ : HF G → HF G ⊗ HF G be the map X(i) ⊗ [X(i) -Gi Γ] X→
(59)
i
and let us write ∂2 for the map X → ZΓ+2 , X. Then Z[Γ1 ,Γ2 ] × δX = (∂2 ⊗ id) ◦ ∆Γ1 which justifies the shorthand notation 1+ ∂2 X for the above. Then, the desired commutator is [1+ ∂2 3+ ∂4 − 3+ ∂4 1+ ∂2 ]X. Let us consider 1+ ∂2 3+ ∂4 X first. We want to compare it with 1+ 3+ ∂2,4 X. These are the terms generated by shrinking Γ2 , Γ4 at disjoint places, and gluing Γ1 for the residue of Γ2 , and Γ3 for the residue of Γ4 .
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What we now need to know is the commutator 1+ [∂2 , 3+ ]∂4 . There are two cases: i) Γ2 is a proper subgraph of Γ3 , Γ2 ⊂ Γ3 . Then 1+ ∂2 3+ ∂4 X = (1+ ∂2 Γ3 )+ ∂4 X + 1+ 3+ ∂2,4 X. such that X(i) = Y , Y ∈ Y234 , we have a contribuii) Γ2 ⊂ Γ3 . Then, for any X(i) + tion as 3 ∂4 Y = Γ2 , and by the previous proposition, Γ2 = 4+ ∂3 Y . Hence
1+ ∂2 3+ ∂4 X = 1+ ∂4+ ∂3 2 X + 1+ 3+ ∂2,4 X. Consider now 3+ ∂4 1+ ∂2 X. Similarly, we find two cases: i) Γ4 is a proper subgraph of Γ1 , Γ4 ⊂ Γ1 . Then 3+ ∂4 1+ ∂2 X = (3+ ∂4 Γ1 )+ ∂2 X + 3+ 1+ ∂4,2 X. such that X(i) = Y , Y ∈ Y412 , we have a contribuii) Γ4 ⊂ Γ1 . Then, for any X(i) + tion as 1 ∂2 Y = Γ4 , and by the proposition again, Γ4 = 2+ ∂1 Y . Hence
3+ ∂4 1+ ∂2 X = 3+ ∂2+ ∂1 4 X + 3+ 1+ ∂4,2 X. As 1+ 3+ ∂2,4 X = 3+ 1+ ∂4,2 X, we get for the commutator, returning to the full fledged notation, [Z[Γ1 ,Γ2 ] , Z[Γ3 ,Γ4 ] ] =
+Z[Z[Γ
1 ,Γ2 ]
−Z[Z[Γ
3 ,Γ4 ]
− Z[Γ3 ,Z[Γ
×δΓ4 ]
,Γ2 ] + Z[Γ1 ,Z[Γ
×δΓ2 ]
×δΓ3 ,Γ4 ] ×δΓ1
−δΓK2 ,Γ3 Z[Γ1 ,Γ4 ]
2 ,Γ1 ] 4 ,Γ3 ]
+
δΓK1 ,Γ4 Z[Γ2 ,Γ3 ] .
Let us check that this bracket fulfills a Jacobi identity. Equivalently, we can check that Z[Γ1 ,Γ2 ] - Z[Γ3 ,Γ4 ] := Z[Z[Γ
1 ,Γ2 ]
×δΓ3 ,4]
+ Z[Γ1 ,Z[Γ
4 ,Γ3 ]
×δΓ2 ]
defines a right or left pre-Lie product. Indeed, we find, returning to our shorthand
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notation: (1+ ∂2 3+ ∂4 )5+ ∂6 − 1+ ∂2 (3+ ∂4 5+ ∂6 ) = +(1+ (∂2 3))+ ∂4 5+ ∂6 + 1+ ∂4+ ∂3 2 5+ ∂6 =
−1+ ∂2 (3+ ∂4 5)+ ∂6 − 1+ ∂2 3+ ∂6+ ∂5 4 + ((1+ ∂2 3)+ ∂4 5)+ ∂6 + (1+ ∂2 3)+ ∂6+ ∂5 4
a
b1
+
+
+
+ (1 ∂4+ ∂3 2 5) ∂6 + 1 ∂6+ ∂5 (4+ ∂3 2)
b2
c3
− (1+ ∂2 (3+ ∂4 5))+ ∂6 − 1+ ∂6+ ∂3+ ∂ 5 2
4 b3
+
c1
+
+
− (1 ∂2 3) ∂6+ ∂5 4 − 1 ∂(6+ ∂5 4)+ ∂3 2
a
c2
The two ”a” terms cancel, while the terms b1 , b2 , b3 add up to a contribution (1+ 3+ ∂2,4 5)+ ∂6 which is symmetric under exchange of the index pair (1, 2) with (3, 4). This term only contributes when Γ2 appears as a subgraph of Γ5 . The terms c1 , c2 , c3 add up to a contribution 1+ ∂6+ 4+ ∂5,3 2 which only contributes when Γ5 appears as a subgraph of Γ2 , and is symmetric under exchange of the index pair (3, 4) with (5, 6). The bi -terms and the ci terms are mutually exclusive. Furthermore, when the bi terms contribute, we get a right pre-Lie product, while when the ci terms contribute, we get a left pre-Lie product. In all cases, we then fulfill the Jacobi identity. Hence, we have established the following theorem: Theorem For all 1PI graphs Γi , s.t. res(Γ1 ) = res(Γ2 ) and res(Γ3 ) = res(Γ4 ), the bracket [Z[Γ1 ,Γ2 ] , Z[Γ3 ,Γ4 ] ] =
+Z[Z[Γ
1 ,Γ2 ]
−Z[Z[Γ
3 ,Γ4 ]
− Z[Γ3 ,Z[Γ
×δΓ4 ]
,Γ2 ] + Z[Γ1 ,Z[Γ
×δΓ2 ]
×δΓ3 ,Γ4 ] ×δΓ1
−δΓK2 ,Γ3 Z[Γ1 ,Γ4 ]
2 ,Γ1 ] 4 ,Γ3 ]
+
δΓK1 ,Γ4 Z[Γ3 ,Γ2 ] .
defines a Lie algebra of derivations acting on the Hopf algebra HF G via δX Z[Γi ,Γj ] × δX = ZΓ2 , δX(i) Γ , (i) Gi 1 I
where the gluing data Gi are normalized as before. The Kronecker δ K terms just eliminate the overcounting when combining all cases in a single equation.
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We note that Z[Γ,Γ] × δX = kΓ δX , where kΓ is the number of appearances of Γ in X and where we say that a graph Γ appears k times in X if k is the largest integer such that Γk ⊗ id, ∆(δX ) is non-vanishing. Furthermore, we note that I : Z[Γ1 ,Γ2 ] → Z[Γ2 ,Γ1 ] is an anti-involution such that I([Z[Γ1 ,Γ2 ] , Z[Γ3 ,Γ4 ] ]) = −[I(Z[Γ1 ,Γ2 ] ), I(Z[Γ3 ,Γ4 ] )], by inspection. We have [Z[Γ1 ,Γ2 ] , Z[Γ2 ,Γ1 ] ] = Z[Γ1 ,Γ1 ] − Z[Γ2 ,Γ2 ] . Further structural analysis is left to future work. By construction, we have Proposition ZΓ+ ≡ Z[Γ,res(Γ)] , ZΓ− ≡ Z[res(Γ),Γ] . Also, we immediately conclude − Corollary [ZX , ZY− ] = −Z − +
+ [ZX ,ZY ]
.
Finally, we get the desired commutator Corollary [Z[Γ1 ,res(Γ1 )] , Z[res(Γ2 ),Γ2 ] ] =
+Z[Z[Γ
1 ,res(Γ1 )]
−Z[Z[res(Γ
2 ),Γ2]
=
×δres(Γ2 ) ,Γ2 ] ×δΓ1
1 ),Γ1 ]
,res(Γ1 )] + Z[Γ1 ,Z[Γ
2 ,res(Γ2 )]
×δΓ2 ]
×δres(Γ1 ) ]
K −δres(Γ Z + δΓK1 ,Γ2 Z[res(Γ2 ),res(Γ1 )] 1 ),res(Γ2 ) [Γ1 ,Γ2 ] K δres(Γ Z + δΓK1 ,Γ2 Z[res(Γ2 ),res(Γ1 )] 1 ),res(Γ2 ) [Γ1 ,Γ2 ]
−Z[res(Γ2 ),Z[res(Γ
1 ),Γ1 ]
=
− Z[res(Γ2 ),Z[res(Γ
×δΓ2 ]
− Z[Z[res(Γ
2 ),Γ2]
×δΓ1 ,res(Γ1 )]
K δres(Γ Z + δΓK1 ,Γ2 Z[res(Γ2 ),res(Γ1 )] 1 ),res(Γ2 ) [Γ1 ,Γ2 ] −ZZ− − ZZ+ . [res(Γ1 ),Γ1 ] ×δΓ2 [res(Γ2 ),Γ2] ×δΓ1
We can now make contact with derivations in the Hopf algebra of rooted trees. Let us consider the Hopf algebra of iterated one-loop self-energies in massless Yukawa theory in four dimensions. There is a one-to-one correspondence Θ between iterated
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one-loop fermion self-energy graphs and undecorated rooted trees:
. Let Γ2 , Γ3 be arbitrary such fermion self-energy graphs and let Γ4 be the one-loop self-energy graph, and Γ1 be its residue, a two-point vertex with two fermionic external legs. Note that res(Γ4 ) = res(Γ2 ) = res(Γ3 ) = Γ1 . The isomorphism Θ to undecorated rooted trees delivers the previous result on undecorated rooted trees: Indeed, Θ(Z[Γ3 ,Γ4 ] ) = N (Θ(Γ3 )), and Θ(Z[Γ1 ,Γ2 ] ) = M (Θ(Γ2 )). We have, using the previous theorem, Θ([Z[Γ1 ,Γ2 ] , Z[Γ3 ,Γ4 ] ])
= [M (Θ(Γ2 )), N (Θ(Γ3 ))] = Θ Z[Z [res(
+Z[res(
),Γ2 ]×δΓ 3
),Z[
−δΓK2 ,Γ3 Z[res(
,
] ,Γ3 ]×δΓ 2
),
]
]
,
in accordance with the results of the previous section. We used the fact that the residue of a graph contains no subgraph, Z[Γ3 ,
]
× δres(
)
= 0,
and that Z[Γ2 ,res(
)]
×δ
= 0.
The above uses natural growth by identifying the root of a tree with any foot of another tree. We can also work out from our general results the commutator of other derivations, using, for example, natural growth by connecting with an extra edge the root of a tree to all the vertices of another one.
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Conclusions We only considered the Lie algebra aspect for Feynman graphs. A bicrossed structure can be constructed as well, say by enlarging HF G to HF GG using appropriate insertion of subgraphs as a natural growth. The algebraic structures here provided cover all operations which one encounters in the perturbative expansion of a quantum field theory: insertion and elimination of subgraphs. While the construction of local counter terms demands the elimination of subgraphs γ by res(γ) on the expense of multiplication with their counter terms SR (γ) [3], the Dyson–Schwinger quantum equations of motions require that any local interaction, described by a vertex v, can as well be mediated by any graph Γ with res(Γ) = v, and hence the insertion of Γ for v in all possible ways determines naturally the series of Feynman graphs providing a fixed point for those equations.
References [1] D. Kreimer, On the Hopf algebra structure of perturbative quantum field theories, Adv. Theor. Math. Phys. 2, 303 (1998) [arXiv:q-alg/9707029]. [2] A. Connes and D. Kreimer, Hopf algebras, renormalization and noncommutative geometry, Commun. Math. Phys. 199, 203 (1998) [arXiv:hep-th/9808042]. [3] A. Connes and D. Kreimer, Renormalization in quantum field theory and the Riemann-Hilbert problem. I: The Hopf algebra structure of graphs and the main theorem, Commun. Math. Phys. 210, 249 (2000) [arXiv:hepth/9912092]. [4] A. Connes and D. Kreimer, Renormalization in quantum field theory and the Riemann-Hilbert problem. II: The beta-function, diffeomorphisms and the renormalization group, Commun. Math. Phys. 216, 215 (2001) [arXiv:hepth/0003188]. [5] A. Connes and H. Moscovici, Hopf algebras, Cyclic Cohomology and the Transverse Index Theory, Commun. Math. Phys. 198, 199 (1998) [arXiv:math.DG/9806109]. [6] Martin Markl, Steven Shnider and Jim Stasheff, Operads in Algebra, Topology and Physics, AMS, 2002.
Vol. 3, 2002
Insertion and Elimination: the Lie Algebra of Feynman Graphs
Alain Connes IHES and Coll`ege de France Le Bois-Marie 35, route de Chartres F-91440 Bures-sur-Yvette France email: [email protected] Dirk Kreimer Center for Mathematical Physics Boston University 111 Cummington St. Boston, MA 02215 USA email: [email protected] Communicated by Vincent Rivasseau submitted 25/01/02, accepted 12/03/02
To access this journal online: http://www.birkhauser.ch
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Annales Henri Poincar´ e
Large Momentum Bounds from Flow Equations Ch. Kopper and F. Meunier∗
Abstract. We analyze the large momentum behaviour of 4-dimensional massive euclidean ϕ4 theory using the flow equations of Wilson’s renormalization group. The flow equations give access to a simple inductive proof of perturbative renormalizability. By sharpening the induction hypothesis we prove new and, as it seems, close to optimal bounds on the large momentum behaviour of the correlation functions. The bounds are related to what is generally called Weinberg’s theorem.
1 Introduction The high energy or momentum behaviour of correlation functions in quantum field theory is of immediate physical interest. It is reflected in the high energy behaviour of measurable quantities such as interaction cross sections. It is also related to questions of theoretical consistency such as unitarity [1]. Four dimensional field theories of physical relevance to this day have been analyzed rigorously in truncated form only, in particular in perturbation theory. The main reason for this is related to the fact that physical quantities calculated within these theories have to be renormalized, i.e. reparametrized, since when expressed in the original bare parameters of the theory they diverge. In the framework of perturbation theory renormalization can be carried out in full rigour. A particularly attractive tool for performing the renormalization proof is the flow equation of the Wilson renormalization group [2]. The proof is considerably simplified as compared to the traditional Feynman diagram based proofs, and at the same time the technical question of eliminating infinities is traced back to the physical problem of analyzing the renormalization group flow of the theory. The statement of renormalizability of the theory then can be phrased as follows: On fixing the physical structure (i.e. the field and symmetry content) and on fixing a finite number of relevant parameters by physical renormalization conditions the perturbative correlation functions of the theory are finite. From these remarks it is obvious that the analysis of the large momentum behaviour of the correlation functions cannot be performed rigorously before settling the renormalization issue. Historically Weinberg [3] performed his famous analysis of the high energy behaviour of euclidean Feynman amplitudes about ten years before the achievement of rigorous renormalization theory. His conviction that the renormalization procedure would not invalidate his results was confirmed in the 70ies, in particular through the work of Berg`ere, Lam and Zuber [4]. Their result is ∗
Theodor Morschheuser zum 100. Geburtstag gewidmet
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of the following form: For a given Feynman diagram with given euclidean external momenta p1 , . . . , pn the associated Feynman amplitude I(λ) := I(λp1 , . . . , λpn ) for λ large, has the following asymptotic expansion I(λ) =
−∞ s max
ars λr (ln λ)s .
r=rmax s=0
The powers of logarithms are related to the number of renormalization operations performed on the graph, whereas the leading Weinberg power rmax is the maximal scaling dimension of all subgraphs which are irrigated by the flow of large external momenta. As regards the technique of proof, it is based on the Zimmermann forest formula in parametric space together with the Mellin transform. Berg`ere, de Calan and Malbouisson [4] generalized the previous result to the situation where only a subset of momenta is scaled by λ , using the complete Mellin transform, which permits to read off the coefficients of the asymptotic expansion for a given graph by translation of the integration contour of the Mellin transformed integrand. The rigorous analysis of the renormalization problem in the complete Mellin transform representation was performed by de Calan, David and Rivasseau [4]. Our results are related to those of Weinberg and followers. Since the flow equations do not require cutting up perturbative amplitudes into Feynman amplitudes, the result is stated for the full amplitude, and it depends on the geometry of the set of external momenta only. It is written directly in general form such that the bound can also be read off in situations where only subsets of momenta grow large. We restrict our considerations to the simplest item of a renormalizable field theory in four dimensions. The flow equations have been used to prove renormalizability of most theories of physical interest, including theories with massless fields, and also nonabelian gauge theories [6]. The present considerations could then be extended to those theories to prove strict UV bounds. The method of proof is in accord with the standard flow equation inductive proofs. It uses sharpened induction hypotheses incorporating the improvement of UV behaviour when momentum derivatives are applied to the correlation functions. In closing we note that a considerably more ambitious program would be to confirm analogous bounds on Schwinger functions beyond perturbation theory, if they exist, which is supposedly not the case for ϕ44 -theory. A strictly renormalizable model for which existence of the Schwinger functions has been proven is the twodimensional massive Gross-Neveu-model [7]. In the perturbative framework we could prove analogous bounds for that model without much change (performing the necessary changes of power counting dimensions). In the flow equation framework the hard combinatoric problems of the nonperturbative analysis so far could be solved only to the degree of establishing the de Calan-Rivasseau type large order bounds for the expansion [8]. A full construction in momentum space of the GrossNeveu model, if feasible, certainly will need considerable effort. In an inductive proof it will certainly be crucial to dispose of sharp bounds in momentum space
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as presented here noting that unprecise bounds used in inductive proofs tend to proliferate into all combinatoric coefficients of the problem. Finally we emphasize that the flow equations have been used extensively in recent years beyond the field of mathematical physics, in theoretical physics and phenomenology. In different forms and for various truncation schemes they were used to analyze perturbative and nonperturbative problems of high energy physics, statistical and solid state physics and other fields. For a recent review see [9].
2 Renormalization and large momentum bounds from the Flow Equations 2.1
The Flow equation framework
Renormalization theory based on the Wilson flow equation (FE) has been exposed quite often in the literature [5]. So we will introduce it rather shortly. The object studied is the regularized generating functional LΛ,Λ0 of connected (free propagator) amputated Green functions (CAG). The upper indices Λ and Λ0 enter through the regularized propagator 2
2
2 +m2 − p +m 1 −p Λ Λ2 2 0 C (p) = 2 {e − e } p + m2 d4 p or its Fourier transform Cˆ Λ,Λ0 (x) = p C Λ,Λ0 (p) eipx , with p := R4 (2π) 4 . We assume 0 ≤ Λ ≤ Λ0 ≤ ∞ so that the Wilson flow parameter Λ takes the role of an infrared (IR) cutoff1 , whereas Λ0 is the ultraviolet (UV) regularization. The full and their Fourier propagator is recovered for Λ = 0 and Λ0 → ∞ . For the “fields” δ δ 4 −ipx transforms we write ϕ(x) ˆ = p ϕ(p) eipx , δϕ(x) = (2π) . For our ˆ p δϕ(p) e purposes the fields ϕ(x) ˆ may be assumed to live in the Schwartz space S(R4 ). For finite Λ0 and in finite volume the theory can be given rigorous meaning starting from the functional integral Λ,Λ0 Λ,Λ0 (ϕ)+I ˆ ) ˆ ˆ e−LΛ0 ,Λ0 (φˆ + ϕ) = dµΛ,Λ0 (φ) . (1) e−(L
Λ,Λ0
ˆ denotes the (translation invariant) Gaussian measure On the rhs of (1) dµΛ,Λ0 (φ) Λ,Λ0 ˆ with covariance C (x). The functional LΛ0 ,Λ0 (ϕ) ˆ is the bare action including counterterms, viewed as a formal power series in the renormalized coupling g . Its general form for symmetric ϕ44 theory is g Λ0 ,Λ0 L (ϕ) ˆ = d4 x ϕˆ4 (x) 4! (2) 3 1 1 1 + d4 x { a(Λ0 )ϕˆ2 (x) + b(Λ0 ) (∂µ ϕ) ˆ 2 (x) + c(Λ0 )ϕˆ4 (x)} , 2 2 4! µ=0 1 Such a cutoff is of course not necessary in a massive theory. The IR behaviour is only modified for Λ above m.
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the parameters a(Λ0 ), b(Λ0 ), c(Λ0 ) fulfill a(Λ0 ) = O(g) ,
b(Λ0 ), c(Λ0 ) = O(g 2 ) .
(3)
They are directly related to the standard mass, wave function and coupling constant counterterms. On the lhs of (1) there appears the normalization factor Λ,Λ0 which is due to vacuum contributions. It diverges in infinite volume so e−I that we can take the infinite volume limit only when it has been eliminated. We do not make the finite volume explicit here since it plays no role in the sequel. For a more thorough discussion see [5] (in particular the last reference). The FE is obtained from (1) on differentiating w.r.t. Λ . It is a differential equation for the functional LΛ,Λ0 : ∂Λ (LΛ,Λ0 + I Λ,Λ0 ) 1 δ ˆ 0 ) δ LΛ,Λ0 − 1 δ LΛ,Λ0 , (∂Λ C Λ,Λ ˆ 0 ) δ LΛ,Λ0 . = , (∂Λ C Λ,Λ 2 δ ϕˆ δ ϕˆ 2 δ ϕˆ δ ϕˆ
(4)
By , we denote the standard scalar product in L2 (R4 , d4 x) . Changing to momentum space and expanding in a formal powers series w.r.t. g we write (with slight abuse of notation) LΛ,Λ0 (ϕ) =
∞
0 g r LΛ,Λ (ϕ) . r
r=1 0 From LΛ,Λ (ϕ) we then obtain the CAG of order r in momentum space as r 0 0 |ϕ≡0 = δ (4) (p1 + . . .+ pn ) LΛ,Λ (2π)4(n−1) δϕ(p1 ) . . . δϕ(pn ) LΛ,Λ r r,n (p1 , . . . , pn−1 ) , (5)
where we have written δϕ(p) = δ/δϕ(p). Note that by our definitions the free two 0 0 (ϕ) . This means that LΛ,Λ vanishes. point function is not contained in LΛ,Λ r 0,2 This is important for the set-up of the inductive scheme, from which we will prove renormalizability below. The FE (4) rewritten in terms of the CAG (5) takes the following form 1 w Λ,Λ0 0 (∂Λ C Λ,Λ0 (k))∂ w LΛ,Λ ∂Λ ∂ Lr,n (p1 ,...pn−1 ) = r,n+2 (k,−k,p1 ,...pn−1 ) 2 k 1 w1 Λ,Λ0 w3 Λ,Λ0 w2 Λ,Λ0 − ∂ Lr1 ,n1 (p1 ,...,pn1 −1 )(∂ ∂Λ C (p )) ∂ Lr2 ,n2 (pn1 ,...,pn ) , 2 r +r =r, 1
2
ssym
w1 +w2 +w3 =w n1 +n2 =n+2
(6) where p = −p1 − . . . − pn1 −1 = pn1 + . . . + pn .
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Here we have written (6) directly in a form where also momentum derivatives of the CAG (5) are performed, and we used the shorthand notations w
∂ :=
n−1
3
i=1 µ=0
|wi | =
(
∂ wi,µ ) with w = (w1,0 , . . . , wn−1,3 ), ∂pi,µ
wi,µ , |w| =
|wi | , wi,µ ∈ N0 .
µ
The symbol ssym means summation over those permutations of the momenta p1 , . . . , pn , which do not leave invariant the subsets {p1 , . . . , pn1 −1 }
and
{pn1 , . . . , pn }.
Note that the CAG are symmetric in their momentum arguments by definition. The simple inductive proof of the renormalizability of ϕ44 theory [5] gives the following bounds, which serve at the same time as induction hypotheses: A) Boundedness B) Convergence
0 |∂ w LΛ,Λ p)| ≤ κ4−n−|w| P1 (log r,n (
0 p)| ≤ |∂Λ0 ∂ w LΛ,Λ r,n (
κ | p| ) P2 ( ) , m κ
1 6−n−|w| Λ0 | p| κ P3 (log ) P4 ( ) . Λ30 m κ
(7) (8)
Here and in the following we set κ = Λ + m and use the shorthand p = (p1 , . . . , pn−1 )
and
| p| = sup{|p1 |, . . . , |pn |}.
The Pi denote polynomials with nonnegative coefficients, which depend on r, n, |w|, m, but not on p , Λ, Λ0 . The degree of P1 can be shown to be bounded by r + 1 − n/2 for n ≥ 4 and by r − 1 for n = 2 . The statement (8) implies 0 p) to exist to all renormalizability, since it proves the limits limΛ0 →∞, Λ→0 LΛ,Λ r,n ( orders r . But the statement (7) has to be obtained first to prove (8).
2.2
Renormalization together with large momentum bounds
The inductive scheme used to prove (7,8) will also be used to obtain the new bounds. What we need is a sharpened induction hypothesis, and better control of the high energy improvement generated by derivatives acting on the Green functions. We denote by p1 , . . . , pn a set of external momenta with p1 + . . . + pn = 0 ,and we introduce
(n) pk | / J ⊂ {1, . . . , n} − {i, j} . (9) ηi,j (p1 , . . . , pn ) = inf |pi + k∈J (n)
Thus ηi,j is the smallest subsum of external momenta which contains pi and which does not contain pj . Our new bounds are then given by
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Proposition 1 For 0 ≤ Λ ≤ Λ0 , κ = Λ + m , and for n ≥ 4 n
1 | p| κ |w| log(sup( , )) , Pr,n |w | (n) i κ m sup(κ, ηi,j )
(10)
|p| κ 2−|w| |w| 0 for n = 2 : |∂ w LΛ,Λ Pr,2 log(sup( , )) . r,2 (p)| ≤ sup(|p|, κ) κ m
(11)
0 |∂ w LΛ,Λ p)| ≤ κ4−n r,n (
i=1 i=j
|w|
Here Pr,n are (each time they appear possibly new) polynomials with nonnegative coefficients which depend on r, n, |w|, m , but not on p , Λ, Λ0 . They are of degree |r − 1 − n/2| if n = 2, |w| ≥ 3 |w| |r − n/2| if n = 2, |w| ≤ 2 or if n = 4, |w| ≥ 1 deg Pr,n ≤ |r + 1 − n/2| otherwise. Proof. We will use the standard inductive scheme which goes up in r and for given r descends in n , and for given r, n descends in |w| starting from some arbitrary |w|max . The rhs of the FE is then prior the lhs in the inductive order, and the bounds can thus be verified for suitable boundary conditions on integrating the rhs of the FE over Λ , using the bounds of the proposition. To start the induction note that 0 LΛ,Λ r,n ≡ 0 for n > 2r + 2 (as follows from the connectedness). Terms with n + |w| ≥ 5 are integrated down from Λ0 to Λ, since for those terms we have the boundary conditions at Λ = Λ0 following from (2) 0 ,Λ0 ∂ w LΛ (p1 , . . . pn−1 ) = 0 for n + |w| ≥ 5 , r,n
whereas the terms with n + |w| ≤ 4 at the renormalization point - which we choose at zero momentum for simplicity - are integrated upwards from 0 to Λ, since they are fixed at Λ = 0 by (Λ0 -independent) renormalization conditions, which define the relevant parameters of the theory. From symmetry considerations we deduce the absence of nonvanishing renormalization constants apart from those appearing in (3). The Schl¨ omilch or integrated Taylor formula permits us to move away from 0 the renormalization point, treating first L0,Λ r,4 and then the momentum derivatives 0 of L0,Λ r,2 , in descending order. Note that j in (10) is arbitrary, so the bound arrived at will be in fact |∂
w
0 LΛ,Λ p)| r,n (
≤κ
4−n
inf
j,1≤j≤n
n i=1 i=j
1 | p| κ P |w| log(sup( , )) . (n) |wi | r,n κ m sup(κ, ηi,j )
We will choose j = n since the proof is independent of this choice.
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A) n + |w| ≥ 5: A1) n ≥ 4 : Integrating the FE (6) w.r.t. the flow parameter κ from κ to Λ0 + m gives the following bound for the first term on the rhs of the FE Λ0 +m n−1 2 2 1 − p Λ+m 4 4−(n+2)−3 2 dκ κ d pe (n+2) |wi | κ ) i=1 sup(κ , ηi,n | p| |p| κ |w| (Λ = κ − m) × Pr,n+2 log(sup( , , )) κ κ m Λ0 +m n−1 p2 1 p ≤ dκ d4 ( ) e− Λ2 κ3−n−|w| (n+2) η |wi | κ κ i=1 sup(1, i,n ) κ | p| |p| κ |w| ×Pr,n+2 log(sup( , , )) κ κ m n−1 1 | p| κ |w| ≤ κ4−n , )) , log(sup( P r,n+2 |w | (n) i κ m i=1 sup(κ, ηi,n ) which satisfies the required bound. Here we used the important inequality: k k 2 1 1 ≤ c(k) d4 x e−x P(log |x|) sup(1, |x + ai |) sup(1, |ai |) i=1 i=1
(12)
for suitable c(k) > 0 . This inequality will again be used in the subsequent consid2 erations. It is easily established using the rapid fall-off of e−x . The required bound on the second contribution from the rhs of the FE (6) 0 is established when using the induction hypothesis for the terms ∂ w1 LΛ,Λ r1 ,n1 and w2 Λ,Λ0 ∂ Lr2 ,n2 . The only new ingredient needed is a bound for the derivatives of the regularization factor appearing in this second term: |∂ w e−
q2 +m2 Λ2
q2
| ≤ c(|w|) κ−|w| e− Λ2
for suitable c(|w|) > 0 . Note also that by the induction hypothesis |r + 1 − n/2| if n = 4, w = 0 or if n ≥ 6 |w2 | 1| deg Pr|w + deg P ≤ r2 ,n2 1 ,n1 |r − n/2| if n = 4, |w| ≥ 1 in all cases (also if |w| ≥ 1 , and w1 , w2 = 0 ). A2) The case (n = 2,w = 3) which is simpler due to the appearance of one external momentum only is treated analogously. B) n + |w| ≤ 4: For the relevant terms of dimension ≤ 4 the induction hypothesis is easily verified at zero momentum where it agrees with the results from [5] 2 . 2 We note that when performing the integration over κ from m to Λ + m for the terms with |w| n + |w| = 4 there appears a logarithm, which is the origin of the polynomial Pr,n , present also at zero momentum.
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To extend it to general momenta we shall choose a suitable integration path from zero to the momentum configuration considered. B1) For n = 2 we proceed in descending order of |w| starting from |w|max . We use 1 w Λ,Λ0 0 0 (p) = ∂ L (0) + | pµ dλ ∂µ ∂ w LΛ,Λ ∂ w LΛ,Λ r,2 r,2 r,2 (λp)| µ
0
and bound the second term with the aid of the induction hypothesis by |
pµ
0
µ
|p| |p|
1
0
κ inf(1, |p| )
0
1
0 dλ ∂µ ∂ w LΛ,Λ r,2 (λp)| ≤
dλ |p| κ |w|+1 log(sup( , )) ≤ |w|−1 Pr,2 κ m sup(λ|p|, κ)
dλ κ|w|−1
+
1
κ inf(1, |p| )
dλ |p| κ |w|+1 P , )) log(sup( r,2 κ m (λ|p|)|w|−1
|p| κ |w| ≤ |p|2−|w| Pr,2 log(sup( , )) . κ m B2) To prove the proposition for (n = 4, w = 0 ) we will use repeatedly the Lemma For λ ∈ [0, 1] and x, y ∈ Rd , if |x + y| ≥ |x| then |λx + y| ≥ λ|x|. Proof. |λx + y| ≥ |x + y| − |(1 − λ)x| ≥ |x| − (1 − λ)|x| = λ|x| .
In fact, the case n = 4, w = 0 will be treated by distinguishing four different situations as regards the momentum configurations. We use the previously established bounds for the case n = 4, w = 1. These bounds are in terms of the (4) functions ηi,j from (9). Assuming (without loss of generality) |p4 | ≥ |p1 | , |p2 | , |p3 | (4)
we realize that ηi,4 can always be realized by a sum of at most two momenta from the set {p1 , p2 , p3 } . It is then obvious that the subsequent cases ii) and iv) cover all possible situations. The cases i) and iii) correspond to exceptional configurations for which the bound has to be established before proceeding to the general ones. The four cases are i) {p1 , p2 , p3 } = {0 , q , v} (4)
ii) {p1 , p2 , p3 } such that inf i ηi,4 = inf i |pi | iii) {p1 , p2 , p3 } = {p , −p , v} (4)
iv) {p1 , p2 , p3 } such that inf i ηi,4 = inf j =k |pj + pk | .
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i) To prove the proposition in this case we use an integrated Taylor formula: |Lr,4 (0, q, v)| ≤ |Lr,4 (0, 0, 0)| +
µ
0
1
dλ |qµ ∂qµ Lr,4 (0, λq, λv)| + |vµ ∂vµ Lr,4 (0, λq, λv)| .
The second term is bounded using the induction hypothesis: i=2,3
|pi |
1
dλ 0
|p4 | κ 1 , P log sup . r,4 (4) κ m sup(κ, η (λ) ) 1
(13)
i,4
We have written η(λ) for the η-parameter in terms of the scaled variables pλ2 = (4) (4) λq , pλ3 = λv . We directly find η2,4 (λ) = λ|q| , η3,4 (λ) = λ|v| and thus obtain the following bound for (13) κ 1 inf(1, |q| ) |p4 | κ dλ dλ 1 + Pr,4 , |q| log sup + q→v κ κ κ m 0 inf(1, |q| ) λ|q| |q| + κ |v| + κ |q| κ |p4 | κ |v| κ 1 + log + log , ≤ log sup + Pr,4 κ |q| κ κ |v| κ κ m |p κ | 4 0 , ≤ Pr,4 log sup , κ m which ends the proof of case i). (4)
ii) We assume without loss of generality inf i ηi,4 = |p1 | . We use again an integrated Taylor formula along the integration path (pλ1 , pλ2 , pλ3 ) = (λ p1 , p2 , p3 + (4) (4) (1 − λ) p1 ) . By the Lemma we find η1,4 (λ) = |pλ1 | = λ|p1 |, η3,4 (λ) ≥ λ|p1 | . The boundary term for λ = 0 is bounded through i). For the second term we bound 1 dλ p1,µ ∂p1,µ − ∂p3,µ L(pλ1 , pλ2 , pλ3 ) µ 0 1 1 |p4 | κ 1 1 , dλ + Pr,4 log sup ≤ |p1 | (4) (4) κ m 0 sup(κ, η1,4 (λ) ) sup(κ, η3,4 (λ) ) 1 inf(1, |pκ | ) 1 |p4 | κ dλ dλ 1 + , Pr,4 log sup ≤ |p1 | , κ κ m 0 inf(1, |pκ | ) λ|p1 | 1
which gives the required bound similarly as in i). iii) We choose the integration path (pλ1 , pλ2 , pλ3 ) = (λ p, −p, v ). Here we assume without restriction that |v| ≤ |v − (1 − λ)p| , otherwise we interchange the role of
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v and −v . The boundary term leads again back to i). The integral second term is cut into four pieces
1
= 0
inf(1/2, |pκ | ) 1
0
+
1/2
inf(1/2, |pκ | )
+
1
sup(1/2,1− |pκ | ) 1
1/2
+
1 0
dλ of the
1
sup(1/2,1− |pκ | )
.
1
(4)
They are bounded in analogy with ii) using η1,4 (λ ) = λ|p1 | for λ ≤ 1/2 ,
(4) η1,4 (λ )
= (1 − λ)|p1 | for λ ≥ 1/2 , relations easily established with the aid of the Lemma. (4)
iv) We assume without loss of generality inf i ηi,4 = |p1 + p2 | and integrate along (pλ1 , pλ2 , pλ3 ) = (p1 , −p1 + λ(p1 + p2 ), p3 ). The boundary term has been bounded (4) in iii). Using the Lemma again we find inf η2,4 (λ) = λ|p1 +p2 | , and the integration term is then bounded through 1 dλ (p1,µ + p2,µ )∂p2,µ L(pλ1 , pλ2 , pλ3 ) µ 0 inf(1, κ ) λ |p1 +p2 | 1 |p4 | κ dλ dλ 1 ≤ |p1 + p2 | + , Pr,4 log sup κ λ|p1 + p2 | κ m 0
inf(1, |p
κ ) 1 +p2 |
which gives the required bound as before.
Bounds like those of Proposition 1 can also be proven using regularizations different from the one applied here. To analyze properties of Green functions in Minkowski space it is useful to have regulators which stay bounded for large momenta in the whole complex plane. An example is k k 2 2 Λ Λ 1 0 − . C Λ,Λ0 (p) = 2 p + m2 p2 + m2 + Λ20 p 2 + m2 + Λ 2 One realizes that an inequality analogous to (12) in this case requires that 2k > |w|max + 2 . Since |w|max should be at least 3 (to be able to perform the renormalization proof for the two point function), we need k ≥ 3 . Then the proof can be performed as before.
2.3
Weighted trees and large momentum fall-off
In this section we want to show that for n ≥ 6 the n-point functions of symmetric massive ϕ44 fall off for large external momenta. The following definitions are required for a precise formulation of these fall-off properties. A 4-tree of order r is defined to be a connected graph without loops and with a set of r ≥ 1 vertices of coordination number 4. The tree has n external
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lines with n = 2r + 2, which are assumed to be numbered, and it has a set I of internal lines with |I| = r − 1 . We then denote by T 4,n the set of all 4-trees with n external lines. A weighted 4-tree is a 4-tree with a weight µ(I) = 2 attached to each I ∈ I . We now define for 1 ≤ k ≤ n − 4 k-times reduced (weighted) trees obtained from (weighted) 4-trees: A 0-times reduced tree is a 4-tree. A k-times reduced tree T (k) is obtained from a (k − 1)-times reduced tree T (k−1) through the following process: i) by suppressing one external line of T (k−1) , ii) by diminishing by one unit the weight of one among those internal lines of T (k−1) , which are adjacent to the vertex where the external line was suppressed (there are at least 1 and at most 3 lines of this type), iii) by suppressing any internal line I from the tree if it has acquired µ(I) = 0 through this process, and fusing the two adjacent vertices into one, iv) by suppressing the vertex from which the external line has been removed, in case this vertex has acquired coordination number 2 through this removal. If two internal lines have been attached to this vertex, they are fused into a single one and their weights are added. If one internal line had been attached to this vertex, it had necessarily weight 0 and was removed through iii). It is then easy to realize that a k-times reduced tree T (k) with n external lines has the following properties: a) It is a tree. b) Its vertices have coordination numbers 3 or 4. (k) c) The weight µ(I) attached satisfies to each internal line I ∈ I of T i) µ(I) ∈ {1, 2} , ii) I∈I µ(I) = n − 4 . The set of weighted reduced trees with n external lines is denoted by T n,µ . We will use these trees to bound the lhs of the FE in terms of the rhs. To the external lines of a tree T n,µ ∈ T n,µ we associate n external incoming p) for the thus assigned tree. Let momenta p = (p1 , . . . , pn ) and write T n,µ ( then p(I) be the (uniquely fixed, by momentum conservation) momentum flowing through the internal line I ∈ I . For given κ the weight factor of an (assigned p) (shortly T ) is defined as weighted) tree T n,µ ( 1 . g κ (T ) = (sup(κ, p(I))µ(I) I∈I(T )
Our statement on the fall-off of the n-point-functions is then the following Proposition 2 For n ≥ 4 (and with κ = Λ + m ) | p| κ Λ,Λ0 κ |Lr,n ( , p)| ≤ sup g (T ) Pr,n log sup , κ m T ∈T n,µ ( p) where deg Pr,n ≤ r + 1 − n/2 .
(14)
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Remark. We could prove without hardly any change a slightly sharper version of Proposition 2, by restricting the sup in (14) to 2k -times reduced trees with k = r + 1 − n/2 . For k sufficiently large, both sets of trees become equal, however. Proof. We again apply the standard inductive scheme. In starting we note that the Lr,n vanish for n > 2r + 2 and are given by a sum over 4-tree graphs for n = 2r + 2 , which obviously satisfy the bounds of the proposition. We also note that for n = 4 Proposition 2 follows from Proposition 1. Thus we assume n ≥ 6 . i) We bound the first term on the rhs of the FE (6), integrated over κ : | p| |p| κ g κ (T ) Pr,n+2 log sup , , κ κ m T ∈T n+2,µ ( p,p,−p) κ Λ0 +m p2 p 1 4−n−1 ≤ dκ (κ ) d4 ( ) e− κ2 µ(I) |p(I)| κ κ κ ) sup(1, I∈I(Tmax κ ) | p| |p| κ × Pr,n+2 log sup , , κ κ m Λ0 +m | p| κ dκ 1 κ κ , = g (T2 )Pr,n+2 log sup ˆ i )| κ i=1,2 sup(1, |p(I κ m κ κ ) Λ0 +m | p | κ dκ κ κ ≤ g (T2 )Pr,n+2 log sup , (15) κ κ m κ
Λ0 +m
dκ κ3
d4 p e−
p2 +m2 Λ2
sup
with the following explanations: The integral over p/κ was bounded with the aid κ of the inequality (12). By Tmax we denote a tree T ∈ T n+2,µ ( p, p, −p) of maximal weight for given κ . Then we denote by T2κ ( p) or shortly T2κ a twice reduced κ κ , obtained by suppressing the two external lines from Tmax , which tree of Tmax carried the momenta p, −p , and by diminishing the weight of two internal lines I1 , I2 , adjacent to the respective vertices by one unit (it may happen that the two vertices and/or lines are identical). And we set pˆ(Ii ) := p(Ii )|p,−p=0 . Now we note that Λ0 +m κ κ | p| κ dκ g (T ) Pr,n+2 log sup , κ g κ (T κ ) κ m κ (16) | p| κ | p| ˜ , ≤ 1 + | log( )| Pr,n+2 log sup κ κ m and thus obtain finally the required bound for (15) (15) ≤
sup T ∈T n,µ ( p)
| p| κ , g (T ) Pr,n log sup . κ m κ
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ii) To bound the second term on the rhs of (6) we use the inequality 2 −p −2 −3 κ−1 ≤ (sup(κ, |p |)) κ exp Λ2 to obtain straightforwardly the following bound for any given term (with n1 , n2 ≥ 4 )3 in the sum appearing on the rhs of (6): |p 1 | κ κ , g (T ) P log sup 2 1 r1 ,n1 κ m sup(κ, |p |) T1 ∈T n1 ,µ1 ( p1 ) |p 2 | κ , g κ (T2 ) Pr2 ,n2 log sup × sup , κ m T2 ∈T n2 ,µ2 ( p2 ) κ−1
sup
(17)
where we used the notations of (6) and p 1 := (p1 , . . . , pn1 −1 , p ) , p 2 := κ κ (−p , pn1 , . . . , pn ) . We pick two trees T1,max and T2,max , which realize the sup’s κ κ κ in (17) and define the tree T to be given by T1,max ∪ T2,max ∪ , , where , is κ κ the internal line of the new tree T joining T1,max and T2,max . This line carries the momentum −p (cf. (6)). We attach the weight 2 to , . We obviously have T κ ∈ T n,µ . Therefore integrating (17) from κ to Λ0 + m (using again (16)) the result is bounded by |p 1 | κ κ κ g (T ) Pr1 ,n1 log sup , κ m | p| |p 2 | κ , × Pr2 ,n2 log sup (18) log κ + 1 κ m | p| κ , ≤ sup g κ (T ) Pr,n log sup . n,µ κ m T ∈T ( p) The present bounds seem close to optimal. They show for example that the high energy behaviour is not deteriorated if only one single external momentum becomes small, since our trees do not contain vertices of coordination number 2. In particular for n small (6, 8, . . . ) the number of weighted trees to be considered and thus the bound is easily explicited. For n = 6 we find three different trees4 , 3 If
e.g. n1 = 2 for the first term, we use the bound (11) and then 2 |p| κ |p| κ κ−1 , , ≤ κ−1 Pr,2 log sup , 2 sup(|p|, κ) Pr,2 log sup κ m κ m sup(κ, |p|) and retain the contribution of the second term to verify the bound as in (17,18). 4 When taking into account the Remark after Proposition 2, one finds that for n = 6, r = 2 only the last of the 3 weight factors above appears, a fact in accord with (trivial) direct calculation. For r ≥ 3 we again obtain all 3 types of trees.
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up to permutations of the external momenta. Their weight factors g m (T ) are −2 sup(m, |p1 + p2 + p3 |) , −1 , sup(m, |p1 + p2 |) sup(m, |p1 + p2 + p3 |) −1 sup(m, |p1 + p2 |) sup(m, |p3 + p4 |) . From them, from the geometry of the external momenta and from Proposition 2, we read off the bound on the six point function.
References [1] J. Zinn-Justin, Quantum Field Theory and Critical Phenomena, Clarendon Press,Oxford, 3rd ed. 1997. C.Itzykson et J.B.Zuber, Quantum Field Theory, Mc Graw Hill, New York (1980). [2] K.Wilson and J.B. Kogut, The Renormalization Group and the ε-Expansion, Phys. Rep. 12C, 75–199 (1974). J. Polchinski, Renormalization and Effective Lagrangians, Nucl. Phys. B231, 269–295 (1984). [3] S. Weinberg, High Energy behaviour in Quantum Field theory, Phys. Rev. 118, 838–849 (1960). [4] M.C. Berg`ere and Y.M.P. Lam, Asymptotic Expansion of Feynman Amplitudes I, Commun. Math. Phys. 39, 1–32, (1974). M.C. Berg`ere and Y.M.P. Lam, Asymptotic Expansion of Feynman Amplitudes – the divergent case, preprint FU Berlin, FUB-HEP-May 74/9. M.C. Berg`ere and J.B. Zuber, Renormalization of Feynman Amplitudes and Parametric Integral Representation, Commun. Math. Phys. 35, 113–140 (1974). M.C. Berg`ere, C. de Calan and A.P.C. Malbouisson, A Theorem on Asymptotic Expansion of Feynman Amplitudes, Commun. Math. Phys. 62, 137–158 (1978). C. de Calan, F. David and V. Rivasseau, Renormalization in complete Mellin representation of Feynman amplitudes, Commun. Math. Phys. 78, 531–544 (1981) . [5] G. Keller, Ch. Kopper and M. Salmhofer, Perturbative Renormalization and effective Lagrangians in ϕ44 , Helv. Phys. Acta 65, 32–52 (1991). Ch. Kopper, Renormierungstheorie mit Flußgleichungen. Shaker-Verlag, Aachen, 1998. M. Salmhofer, Renormalization, an Introduction, Springer, Berlin, 1999. Ch. Kopper, V.F. M¨ uller and Th. Reisz, Temperature Independent Renormalization of Finite Temperature Field Theory, Ann. Henri Poincar´e 2, 387–402 (2001). [6] G. Keller and Ch. Kopper, Renormalizability Proof for QED Based on Flow Equations, Commun. Math. Phys. 176, 193–226 (1996). Ch. Kopper and V.F. M¨ uller, Renormalization Proof For Spontaneously Broken Yang-Mills Theory with Flow Equations, Commun. Math. Phys. 209, 477–516 (2000).
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[7] K. Gawedzki and A. Kupiainen, Gross-Neveu model through convergent perturbation expansions, Commun. Math. Phys. 102, 1–30 (1985) . J. Feldman, J. Magnen, V. Rivasseau, R. S´en´eor, A renormalizable field theory: the massive Gross-Neveu model, Commun. Math. Phys. 103, 67–103 (1986). M. Disertori, V. Rivasseau, Continuous constructive Fermionic renormalization, Ann. Henri Poincar´e 1, 1–58 (2000). [8] C. de Calan and V. Rivasseau, Local existence of the Borel transform in Euclidean Φ44 , Commun. Math. Phys. 82, 69–100 (1981). G. Keller, Local Borel summability of Euclidean Φ44 : A simple Proof via Differential Flow Equations, Commun. Math. Phys. 161, 311–323 (1994). [9] C. Bagnuls and C. Bervillier, Exact renormalization group equations, an introductory review, Phys. Rep. 348, 91 (2001). Christoph Kopper and Fr´ed´eric Meunier Centre de Physique Th´eorique Ecole Polytechnique F-91128 Palaiseau France email: [email protected] email: [email protected] Communicated by Vincent Rivasseau submitted 06/11/01, accepted 11/01/02
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Ann. Henri Poincar´e 3 (2002) 451 – 482 c Birkh¨ auser Verlag, Basel, 2002 1424-0637/02/030451-32 $ 1.50+0.20/0
Annales Henri Poincar´ e
Multi-Dimensional Semi-Dispersing Billiards: Singularities and the Fundamental Theorem P. B´ alint, N. Chernov, D. Sz´ asz and I. P. T´ oth Abstract. The fundamental theorem (also called the local ergodic theorem) was introduced by Sinai and Chernov in 1987, see [S-Ch(1987)] and an improved version in [K-S-Sz(1990)]. It provides sufficient conditions on a phase point under which some neighborhood of that point belongs to one ergodic component. This theorem has been instrumental in many studies of ergodic properties of hyperbolic dynamical systems with singularities, both in 2-D and in higher dimensions. The existing proofs of this theorem implicitly use the assumption on the boundedness of the curvature of singularity manifolds. However, we found recently ([B-Ch-Sz-T(2000)]) that, in general, this assumption fails in multidimensional billiards. Here the fundamental theorem is established under a weaker assumption on singularities, which we call Lipschitz decomposability. Then we show that whenever the scatterers of the billiard are defined by algebraic equations, the singularities are Lipschitz decomposable. Therefore, the fundamental theorem still applies to physically important models – among others to hard ball systems, Lorentz gases with spherical scatterers, and Bunimovich-Reh´ aˇ cek stadia.
1 Introduction In contrast to smooth dynamical systems, billiards have singularities which make the application of the classical methods substantially more difficult. One reason is that in the neighbourhood of orbits tangent to the obstacles (the so-called tangent singularities) the derivative of the Poincar´e section map diverges. Nevertheless, Sinai’s celebrated 1970 result demonstrated that, at least for d = 2, the hyperbolicity caused by the strictly convex scatterers overcomes the harmful effect of singularities. In fact, he showed that 2D dispersing billiards, i. e. those with strictly convex obstacles, are ergodic and even K-mixing [S(1970)]. Multidimensional geometry is, however, essentially richer so it is not surprising that it had taken 17 years until Chernov and Sinai [S-Ch(1987)] could extend Sinai’s original result to multidimensional dispersing billiards. This remarkable achievement was a corollary of their local ergodicity theorem, often called the fundamental theorem, formulated for semi-dispersing billiards, i. e. those with convex scatterers. Their theorem got slightly generalized with the clarification of some technical details and conditions by Kr´ amli, Sim´ anyi and Sz´ asz [K-S-Sz(1990)] in 1990. The considerations in the proof of the local ergodicity theorem are local. As a matter of fact, by assuming the boundedness (from above) of the curvature of all images of the tangencies, which is a straightforward fact for d = 2, it became
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possible to assume that they are linear objects, at least locally. However, in the recent paper of the present authors, [B-Ch-Sz-T(2000)] it has been discovered that for d ≥ 3, in the neighbourhoods of tangent orbits the images of tangencies (and of other smooth one-codimensional submanifolds of the phase space) develop a pathological behaviour contradicting the boundedness of the curvatures. Therefore for its own interest but also for its various important consequences it became an absolute necessity to correct the original arguments and this is the sole aim of this work. Indeed, instead of the boundedness property of the aforementioned curvatures we formulate a new condition, the so called Lipschitz-decomposability condition. Roughly speaking it requires that the singularities can be decomposed into a finite number of graphs of locally Lipschitz functions with the boundaries of these graphs being not too wild. This assumption, together with the other requirements of the local ergodicity theorem, is already sufficient to save the old proof. The next question is, of course, when this new condition holds. Fortunately, we can verify it under one additional requirement: we assume that the scatterer boundaries are algebraic. Luckily enough, the main examples of multidimensional semi-dispersing billiards are all algebraic. Just think – first of all – of hard ball systems [SSz(1999)], [Sim(2001)], of the Lorentz process with spherical scatterers ([H(1974)], [Sz(2000)]), of general algebraic cylindrical billiards [Sz(1994)], [Sim(2002)], and of the multidimensional stadia designed by Bunimovich and Rehacek [B-R(1998)]. For keeping our exposition possibly short, we rely heavily on that of [K-SSz(1990)]. In section 2, we summarize the necessary notations and prerequisites from the aforementioned work. Section 3 is devoted to the study of singularities. In particular, in subsection 3.1 we briefly recall the pathological behaviour described in [B-Ch-Sz-T(2000)]. Then, in subsection 3.2 we present the aforementioned Lipschitz decomposability property of the singularities. Based upon this assumption, in section 4 we reformulate the local ergodicity theorem and discuss in detail where and how the classical proof of [S-Ch(1987)] and [K-S-Sz(1990)] should be modified. Finally, in section 5 it is shown that the Lipschitz decomposability property holds for algebraic billiards. Though here we use some simple ideas from algebraic geometry and from geometric measure theory, the arguments are still elementary.
2 Prerequisites The methods in this paper, though quite elementary, come from different branches of mathematics. Throughout the arguments we try to keep the exposition selfcontained. More details on the basic notions from algebra or geometric measure theory can be found in the books [B-C-R(1987)], [Sh(1974)], [St(1973)] and [F(1969)], [Fa(1985)], [Fa(1990)]; respectively. We would also like to fix one notation: for any subset in a Riemannian manifold H ⊂ M, H [δ] shall denote its δ-neighborhood: H [δ] := {x ∈ M | ρ(x, H) ≤ δ}.
(2.1)
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Multi-Dimensional Semi-Dispersing Billiards
In this subsection we summarize some basic properties of semi-dispersing billiards. Our aim is to introduce the most important concepts and fix the notation in order to keep the exposition of the paper self-contained. For a more detailed description see the literature, especially [K-S-Sz(1990)]. A billiard is a dynamical system describing the motion of a point particle in a connected, compact domain Q ⊂ T| d . In general, the boundary of the domain in assumed to be piecewise C 3 -smooth, however, later on we impose the further restriction of algebraicity on the billiard (cf. section 5). Inside Q the motion is uniform while the reflection at the boundary ∂Q is elastic. As the absolute value of the velocity is a first integral of motion, the phase space of the billiard flow is fixed as M = Q × S d−1 – in other words, every phase point x is of the form x = (q, v) with q ∈ Q and v ∈ Rd , |v| = 1. The Liouville probability measure µ on M is essentially the product of the Lebesgue measures, i.e. dµ = const. dqdv. The resulting dynamical system (M, {S t , t ∈ R}, µ) is the billiard flow. Let n(q) denote the unit normal vector of a smooth component of the boundary ∂Q at the point q, directed inwards Q. Throughout the paper we restrict our attention on semi-dispersing billiards: we require that for every q ∈ ∂Q the second fundamental form K(q) of the boundary component be non-negative. The boundary ∂Q defines a natural cross-section for the billiard flow. Consider namely ∂M = {(q, v) | q ∈ ∂Q, v, n(q) ≥ 0}. This set actually has a natural bundle structure (cf. [B-Ch-Sz-T(2000)]). In this paper we use the arising Riemannian metric ρ on ∂M . The billiard map is defined as the first return map on ∂M . The invariant measure for the map is denoted by µ1 , and we have dµ1 = const. | v, n(q) | dqdv. Throughout the paper (except for subsection 5.1) we work with this discrete time dynamical system. Its ergodicity implies that of the flow (see [K-S-Sz(1990)]). Singularities. Consider the set of tangential reflections, i.e. R := {(q, v) ∈ ∂M | v, n(q) = 0}. It is easy to see that the map T is not continuous at the set T −1 R. As a consequence, the singularity set for a higher iterate T n is R(n) = ∪ni=1 R−i , where in general Rk = T k R. Generally it was assumed in the literature that the set R(n) is a finite collection of smooth and compact submanifolds of the Poincar´e phase space ∂M . However, for multi-dimensional semi-dispersing billiards these manifolds can be treated as submanifolds of ∂M only in a topological sense (see section 3).
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Remark 2.1 Above the (tangential) singularities have been introduced for the Poincar´e section map T . But in a part of the proof following remark 5.3 they will also be needed for the flow. In fact, the aim of the aforementioned remark is just to hint how this extension of the singularities is understood. For completeness we mention that in case the boundary ∂Q is only piecewise smooth, further singularities – the multiple collisions – arise. At such points neither n(q) and, as a consequence, nor the flow dynamics is uniquely defined, thus we can speak about several “branches” of a trajectory. The singularity set must also be treated with a little more care. For this reason, in all cases we will denote by R+ the set of all singular phase points, which can be points of R or multiple collision points supplied with the possible outgoing velocities. (See [Sim(2001)] and its references for details). In the present paper we consider only tangential singularities. Multiple collisions can be treated in an analogous way, although the main difficulty – the blow-up of the derivative of the dynamics – does not appear here. We introduce some more notation. For any n ∈ N, ∆n stands for the set of doubly singular phase points up to order n, i.e. x ∈ ∂M belongs to ∆n whenever there are indices k1 = k2 , |ki | ≤ n such that both T k1 x and T k2 x are elements of R. We are mainly interested in phase points with regular or with at most once singular trajectories, thus we consider the following sets: ∂M 0 := ∂M \ Rn ∂M ∗ :=
∂M \
n∈Z ∞
∆n
n=1
∂M 1 :=
∂M ∗ \ ∂M 0 .
(2.2)
As to regular and at most once singular phase points of the flow, the sets M 0 , M ∗ and M 1 refer to flow-images of ∂M 0 , ∂M ∗ and ∂M 1 , respectively. Different notions of norms and metrics. In billiard theory several notions of metrics and distances are used. Let us assume that two phase points x = (q, v) and x = (q , v ) and a vector in the tangent plane at x, w = (δq, δv) are fixed. In all calculations presented in the paper we use the Euclidean norm w = |δq|2 + |δv|2 and the generated Euclidean distance ρ(x, x ). The measure on ∂M corresponding to this Riemannian metric (generated by the volume form) is simply the Lebesgue measure const. dq dv. However, in several other statements referred (see e.g. [K-SSz(1990)], especially the Erratum) two other metrics come about. For their definition we fix the notation for two d − 1 dimensional linear subspaces in Rd : T , the one orthogonal to n(q) and J , the one orthogonal to v. Furthermore we introduce the linear operator V : J → T which is simply the projection parallel to v. (On details see [B-Ch-Sz-T(2000)].) This way we may define the invariant norm of a vector: wi = |V −1 δq|2 + |δv|2 and the generated invariant distance ρi (x, x ). The name ’invariant’ comes from
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the fact that the measure corresponding to this Riemannian metric (via the volume form) is the invariant measure dµ1 = const. | v, n(q) | dq dv. Note that w| v, n(q) | ≤ wi ≤ w, thus the two distances are equivalent if we can ensure | v, n(q) | ≥ c for some constant c. This happens throughout the proof of the fundamental theorem (cf. section 4) where we work in a neighborhood of an interior point x ∈ ∂M and thus the two metrics are (locally) equivalent. The third metric-type quantity is the so-called p-metric wp = |V −1 δq|. Even though this is a degenerate metric in general (that is the reason for the name ’p’ – pseudo), it is non-degenerate when restricted to vectors w corresponding to convex fronts (cf. [K-S-Sz(1990)], [B-Ch-Sz-T(2000)]). Its importance is related to the fact that the most convenient way of handling hyperbolicity issues is in terms of the p-metric (see e.g. Lemma 2.2). Related to the above mentioned metrics there are two ways of measuring distance of a phase point x = (q, v) from the set of tangential reflections R. z(x) = ρi (x, R) is simply the distance in terms of ρi . Alternatively we may consider tubular neighborhoods Ur (of radii r) of the flow trajectory starting out of x in the configuration space Q. Then define ztub (x) as the supremum of radii r for which the tube does not intersect the set of singular reflections (see [S-Ch(1987)] and [KS-Sz(1990)], especially the Erratum). It is not difficult to see that z(x) ≤ ztub (x). Hyperbolicity. Besides the presence of singularities the most important feature of semi-dispersing billiard dynamics is that it is – at least locally and non-uniformly – hyperbolic. A highly important consequence of this fact is the abundance of local invariant manifolds. The notion of a local invariant manifold will be used in the traditional sense, i.e. a C 1 -smooth, connected submanifolds γs ⊂ ∂M is a local stable manifold at x ∈ ∂M iff (i) (ii)
x ∈ γs ∃K(γs ), C(γs ) > 0 such that for any y1 , y2 ∈ γs ρ(T n y1 , T n y2 ) ≤ K exp(−Cn)ρ(y1 , y2 ).
(2.3)
Local stable manifolds for the inverse dynamics T −1 will be referred to as local unstable manifolds. The treatment of hyperbolicity is traditionally related to local orthogonal manifolds (or fronts) and sufficient phase points. These objects are defined in the flow phase space the following way. Let x = (q, v) ∈ M \ ∂M and consider a C 2 -smooth codimension 1 sub manifold Σ ⊂ Q \ ∂Q such that q ∈ Σ and v = v(q) is the normal vector to Σ at q. Denote by Σ the normal section of the unit tangent bundle on Q restricted to Σ . Σ is called a local orthogonal manifold or simply a front. A front is said to be (strictly) convex whenever its second fundamental form BΣ (y) ≥ 0 (BΣ (y) > 0) for every y ∈ Σ. Let us consider a nonsingular finite trajectory segment for the flow: S [a,b] x, where a < 0 < b and a, b, 0 are not moments of collision.
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N0 (S [a,b] x), the neutral subspace at time 0 for the segment S [a,b] x is defined as follows: N0 (S [a,b] x) := {
w ∈ Rd : ∃(δ > 0)s.t.∀α ∈ (−δ, δ) v(S a (q(x) + αw, v(x))) = v(S a x)& v(S b (q(x) + αw, v(x))) = v(S b x)}.
Observe that v(x) ∈ N0 (S [a,b] x) is always true, the neutral subspace is at least 1 dimensional. Neutral subspaces at time moments different from 0 are defined by Nt (S [a,b] x) := N0 (S [a−t,b−t] (S t x)), thus they are naturally isomorphic to the one at 0. The non-singular trajectory segment S [a,b] x is sufficient if for some (and in that case for any) t ∈ [a, b] : dim(Nt (S [a,b] x)) = 1. A point x ∈ M 0 is said to be sufficient if its entire trajectory S (−∞,∞) x contains a finite sufficient segment. Singular points are treated by the help of trajectory branches (see [K-S-Sz(1990)]): a point x ∈ M 1 (this precisely means that the entire trajectory contains one singular reflection) is sufficient if both of its trajectory branches are sufficient. All these concepts have their natural counterparts for the billiard map phase space ∂M . For example, a smooth piece Σ ∈ ∂M of the image of a local orthogonal manifold in M is referred to as a front as well. Hyperbolicity is related to the following simple phenomena. Near sufficient phase points hyperplanes in Q orthogonal to the flow evolve into strictly convex fronts. Convex fronts remain convex under time evolution. The importance of this is shown by the Lemma below. Before formulating it we introduce one more n is the derivative of the (nth power of the) dynamics T n restricted notation: Dy,Σ to the front Σ. Lemma 2.2 (Equivalent of Lemma 2.13 from [K-S-Sz(1990)].) For every x ∈ ∂M 0 for which the trajectory is sufficient there exists a neighborhood U (x) and a constant 0 < λ(x) < 1 such that • through almost every point y ∈ U (x) there do pass uniformly transversal local stable and unstable manifolds γ s (y) and γ u (y) of dimension d − 1; • for any y ∈ U (x) and any convex front Σ± passing through ±y: τ −1 p < λ(x), (D±y,Σ ±)
(2.4)
where τ ∈ Z+ is the first return time to U (x). More details about local hyperbolicity and semi-dispersing billiards in general can be found in [K-S-Sz(1990)].
3 Singularities In several papers that appeared, singularities were assumed – either explicitly or implicitly – to consist of smooth 1-codim submanifolds of the phase space. Often,
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even a uniform bound on the curvature was assumed, independent of the order of the singularity. This is true for 2D billiards. However, it is not true in higher dimensions. In this section we present a counter-example in a 3-dim dispersing billiard. Already the curvature of R−2 has no upper bound, i.e. the curvature blows up near a point where the singularity manifold is not even differentiable. After this example we propose another property which, in most applications, can replace the bounded curvature assumption. We conjecture that this property: the Lipschitz decomposability of singularities holds for multi-dimensional semi-dispersing billiards.
3.1
Counter-example for bounded curvature
In this section we recall our example from [B-Ch-Sz-T(2000)] showing that even in a 3D dispersing billiard, already the two-step singularities have no bounded curvature. The proof given in [B-Ch-Sz-T(2000)] was rather implicit. We started with the indirect assumption that the curvature was bounded, and found that Claim. The two-step singularity intersects the one-step singularity tangentially at every point of their intersection, except for a one-codimensional degeneracy, where the intersection is not tangential. This claim obviously contradicts the bounded curvature assumption. We do not repeat here the calculations of [B-Ch-Sz-T(2000)], but rather we present the concrete situation where this pathological behaviour appears. Since this example deals with a very explicitly given billiard configuration, we will not use the complicated notations of the other sections: we will denote R−k simply with Sk (k ≥ 0). plane
n1
sphere2
sphere1 n2
Figure 1: The studied billiard configuration Consider the situation demonstrated on Figure 1. To present the example as transparent as possible the first scatterer, the surface where the trajectories start
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out is a plane – thus it is not strictly convex. Nevertheless this modification has no significance. We are in 3 dimensions, so take a standard 3D Cartesian coordinate system. Let the zeroth ’scatterer’ be the {z = 0} plane. Let the first scatterer be the sphere with center O1 = (0, −1, 1) and radius R = 1. Let the second scatterer be the sphere with center O2 = (1, 0, 2) and radius R = 1. We look at the component of the phase space corresponding to the zeroth scatterer, near the phase point (x0 = 0, y0 = 0, vx0 = 0, vy0 = 0). Of course, vz0 = 1, and the trajectory is the z axis. The counterexample mentioned in the Claim is the intersection of S1 , the inverse image of the first scatterer, and S2 , the second inverse image of the second scatterer, both considered on the zeroth scatterer in (the neighbourhood of) the origin. We are mainly interested in the singularity manifolds close to a doubly tangent orbit. The calculations of [B-Ch-Sz-T(2000)] show that at the origin S1 and S2 can not be tangent. This is essentially the consequence of the circumstance that in the two points of tangencies (with the first and second spheres) the two normals of incidence are perpendicular to each other. In all other situations S1 and S2 are tangent! Consequently at the origin S2 is not even differentiable. Next we recall a much useful paradigm which is a well known object of algebraic singularity theory: the Whitney umbrella. It not only illustrates better the pathological situation in three dimensions (rather than our counterexample in dimension 4) but also suggests to find the way out: to substitute the condition on the boundedness of curvatures with the Lipschitz decomposability property. The Whitney-umbrella. Consider the one-codimensional set in R3 defined by the polynomial equation: {(x, y, z) ∈ R3 | x2 z = y 2 }, the Whitney-umbrella. ‘One half’ of this set (its intersection with the quadrants {xy ≤ 0}) is shown on Figure 2. For simplicity we use the notations: W2 for this ‘half-umbrella’ and W1 for the {z = 0} plane. Clearly • W2 terminates on W1 (in the points of the x-axis), thus W1 ∩ W2 = ∂W2 . • at every point of the x-axis where x = 0 the intersection of W2 and W1 is tangential. • W2 has smooth manifold structure in its interior; nevertheless, near the origin its curvature is unbounded as the normal vector changes rapidly (actually, the unit normal vector does not even have a well-defined limit at the origin). By these properties the geometry of singularities in the counterexample is analogous to Figure 2.1 W1 corresponds to S1 , W2 corresponds to S2 while the 1 To be precise, the situation on Figure 2. has one dimension less – in contrast to W the 2 singularities are 3-dimensional manifolds – but this has little significance to the analogy.
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Figure 2: The Whitney Umbrella origin corresponds to the set of those doubly tangential reflections where the two radii are orthogonal (this set is one-codimensional in S1 ∩ S2 ). Lipschitz decomposability of the Whitney-umbrella. This analogy also shows that bounded curvature is not needed for the neighbourhood of a manifold to be small. Indeed, the ‘half-umbrella’ W2 can be cut further into two pieces (namely, its intersections with the quadrants {x ≥ 0, y ≤ 0} and {x ≤ 0, y ≥ 0}), each of which is the graph of a Lipschitz function, when viewed from the appropriate direction. Indeed, easy calculations show that if we choose the direction (1, 1, 1) or (−1, −1, 1) to be ‘vertical’ (respectively), these√‘quarter-umbrellas’ become graphs of Lipschitz functions with Lipschitz constant 2. So the whole Whitney-umbrella consists of four such graphs plus a one-dimensional tail. This tail (the negative z axis) has no analogue in the singularities of billiards. It’s only there because the umbrella was defined in an algebraic way. However, it will not spoil our measure-theoretic estimates because it has one dimension less than the rest of the set. Generalization I. First let us consider the first-step singularity S1 . By the notations of the previous counterexample (on details see [B-Ch-Sz-T (2000)]) we may characterize the points (x, y, vx , vy ) belonging to S1 easily. These are precisely those for which d(x, y, vx , vy ) = 1, where d(., ., ., .) is the distance of the point O1 = (0, −1, 1) from the line that passes through the point (x, y, 0) and has direction specified by the velocity components vx , vy . As d is a smooth function of its variables there is no curvature blow-up for S1 – and, for first-step singularities in general. Thus S2 is a pre-image of a smooth one-codimensional compact submanifolds, however, the map under which the pre-image is taken has unbounded derivatives and is highly an-isotropic. Curvature blow-up occurs only at those points of S2 (near its intersection with S1 ) where the map behaves irregularly. In correspondence with the above observation we conjecture that curvature blow-up is not a peculiar feature of S2 , it is present in the pre-images of one-
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codimensional smooth submanifolds in general. Consider for example two-step secondary singularities Γ2 – those phase points for which at the second iterate instead of tangentiality the collision term ( n, v ) is a given constant (see [BCh-Sz-T(2000)] for more detail). In the specific example of subsection 3.1 such secondary singular trajectories are precisely those that touch tangentially a sphere of radius R (R < 1) at the second iterate. It is clear that the geometry of Γ2 is completely analogous to S2 . Generalization II. Our calculations in [B-Ch-Sz-T(2000)] do not use any specialty of the explicitly given billiard configuration. Doubly tangential reflections for which the normal vectors of the scatterers at the consecutive collisions are orthogonal can be found in any multi-dimensional semi-dispersing billiard. Near such trajectories a similar calculation can be performed. Generalization III. All in all, the discovered pathology is general. In addition, the higher step singularities Sk ; (k ≥ 3) may show even wilder behaviour near their intersections. Nevertheless, we strongly conjecture that a nice geometric characterization – suggested by the analogy with the Whitney-umbrella in the case of S2 – can be performed. We have mentioned these generalization to present the reader the picture of singularities we have in mind. Nevertheless, for our further discussion we do not need to verify any of these calculations or generalizations since they are completely independent.
3.2
Lipschitz property of singularities
When treating ergodic or stochastic properties of singular systems, we need to understand the properties of singularities in order to know that their neighbourhood is of small measure. By assuming that the singularities are smooth, e. g. they have bounded curvature, in local considerations one can treat them as planes, by choosing an appropriately small scale. This, of course, implies that the intersection of a (smooth) singularity component and a sphere of radius r has a surface-volume of order rm−1 where m = 2d − 2 is the dimension of the phase space. Similarly, the δ-neighbourhood of such a singularity-piece has measure of order rm−1 δ. These properties have been used in several papers without being checked. We now know that the curvature is in general not bounded, so a more careful investigation is essential. To ensure that the regularity properties mentioned hold, we (approximately) propose to assume that the singularities have components which are graphs of Lipschitz functions – instead of assuming they have smooth components. Definition 3.1 A subset H of Rm will be called a Lipschitz graph, if we can choose a Cartesian coordinate system so that H becomes the graph of a Lipschitz function: H = {(x, f (x)) | x ∈ D} with some (measurable) D ⊂ Rm−1 and f : Rm−1 → R Lipschitz-continuous.
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Being a Lipschitz graph ensures that H is rectifiable, and that for its surfacevolume one has µ(H) ≤ Cµ(D) where the constant C depends only on the Lipschitz constant of f . The main property of Lipschitz graphs is shown by the following very basic Lemma 3.2 Let D ⊂ Rm−1 arbitrary, f : Rm−1 → R Lipschitz-continuous with Lipschitz-constant L. Let H = {(x, f (x)) | x ∈ D} ⊂ Rm . Denote by Lm the Lebesgue-measure in Rm , and by Lm−1 the Lebesgue-measure in Rm−1 . Denote by H [δ] the δ-neighbourhood (in Rm ) of H. Then ¯ + o(δ) (3.1) Lm (H [δ] ) ≤ 2δ L2 + 1Lm−1 (D) Proof. Just notice that H [δ] ⊂ {(x, y) | x ∈ D[δ] , |y − f (x)| ≤ δ
L2 + 1},
where D[δ] is the δ-neighbourhood of D in Rm−1 . This implies Lm (H [δ] ) ≤ 2δ L2 + 1Lm−1 (D[δ] ) ¯ as δ → 0. which gives the lemma, since Lm−1 (D[δ] ) → Lm−1 (D)
To precisely formulate the property that we propose instead of smoothness of the singularities, we need the following two definitions: Definition 3.3 (cf. [F(1969)]) A function f : D ⊂ Rm−1 → R will be called locally Lipschitz (with Lipschitz constant L), if for any x ∈ D there exists a neighbourhood is Lipschitz (with Lipschitz U ⊂ Rm−1 of x such that the restricted function f D∩U
constant L). In all our applications D will be open. Notice that in this case, f typically ¯ in a continuous way. cannot even be extended to D Definition 3.4 H ⊂ Rm will be called a (one-codimensional) open locally Lipschitz graph (with Lipschitz constant L), if we can choose an appropriate Cartesian coordinate system so that H becomes the graph of a locally Lipschitz function: H = {(x, f (x) | x ∈ D} with some D ⊂ Rm−1 and f : D → R locally Lipschitz (with constant L). We will be mainly interested in the case when the domain D is an open set in Rm−1 , then – H will be called an open locally Lipschitz graph (even though it is not an open set in Rm ), ¯ \ H. – and we will denote by ∂H the boundary of H as of a surface: ∂H = H
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Now we are able to define the regularity property that should replace the smoothness of singularities. This property, called ‘Lipschitz decomposability’ will be defined for subsets of Rm here. For Lipschitz decomposability of subsets of a Riemannian manifold, see Remark 3.6. Definition 3.5 Consider H ⊂ Rm , and L ∈ R. H will be called ‘Lipschitz decomposable’ (one-codimensional) subset with constant L if it can be decomposed into a finite number of open locally Lipschitz graphs and a small remainder set in the following way: There exist H ∗ and H1 , ..., HK such that: • H⊂
K
¯i H
H ∗,
i=1
• Hi ∩ Hj = ∅ for any i = j, • every Hi is a one-codimensional open locally Lipschitz graph (with constant L), • L
m
(
K
[δ] ∂Hi ) ∪ H
∗
= o(δ).
i=1
The set H ∗ is included in the decomposition for technical reasons: we want to allow for sets H having parts of strictly higher codimension. This occurs generically if H is an algebraic subvariety of Rn – cf. subsection 3.1 on the one dimensional tail of the Whitney-umbrella and section 5. Nevertheless we would like to note that such higher codimensional parts are not present in the singularities of semidispersing billiards. Remark 3.6 Lipschitz decomposability in Riemannian manifolds. Throughout the paper – and in particular in conjecture 3.7 below – subsets of a compact Riemannian manifold M are considered. For H ⊂ M Lipschitz decomposition is understood in terms of coordinate charts. To be more precise, let us fix some convention related to the atlas {Ut , ψt }Tt=1 for M first. It is important that M is compact thus we may consider a finite atlas. We say that the atlas is bi-Lipschitz if all charts ψt : Ut → Rm are bi-Lipschitz maps, i.e. both ψt and (ψt )−1 are Lipschitz with some constant K > 1. All atlases considered are assumed to be bi-Lipschitz with a fixed constant. This ensures that Euclidean distance on Rm is comparable to Riemannian metric on the manifold, and thus our metric estimates indeed apply in the arguments of section 4. Note that bi-Lipschitzness – with Lipschitz constant arbitrarily close to one – can always be obtained by choosing the coordinate patches sufficiently small. As to the problem of Lipschitz decomposition, we will say that H ⊂ M is Lipschitz decomposable whenever a finite bi-Lipschitz atlas can be chosen, such
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that for all charts ψt (H ∩ Ut ) is Lipschitz decomposable as a subset of Rm , in the sense of Definition 3.5.2 The precise property that we expect the singularities of semi-dispersing billiards to have is formulated in the form of a conjecture: Conjecture 3.7 For any semi-dispersing billiard with a finite horizon there exists an L ∈ R such that for any integer N the set |n|≤N Rn of singularities of order up to N is ‘Lipschitz decomposable’ with constant L. It is worth noting that by introducing “transparent walls” (cf. [S-Ch(1987)]) any semi-dispersing billiard can be reduced to one with a finite horizon. The statement of this conjecture will appear word by word among the conditions of the modified version of the fundamental theorem for semi-dispersing billiards stated in section 4.1. The conjecture will be proven for the utmost important special case of semi-dispersing billiards with algebraic scatterers in section 5. To help the reader understand why this ‘Lipschitz decomposability’ property is defined exactly as it is, we present two more lemmas in this section. These are the lemmas through which the decomposability of singularities will be used. Lemma 3.8 Let H ∈ Rm be a one-codimensional locally Lipschitz graph with H = {(x, f (x)) | x ∈ D}, D ⊂ Rm−1 open, f : D → R locally Lipschitz with constant L. Assume furthermore that Lm (∂H)[δ] = o(δ). Let D ⊂ D arbitrary, H = {(x, f (x)) | x ∈ D }. Then
¯ + o(δ). Lm H [δ] ≤ 2δ L2 + 1Lm−1 D Proof. Let x0 ∈ D, X0 = (x0 , f (x0 )) ∈ H. If dist(x0 , ∂D) > δ then
Bδ (X0 ) ⊂ (x, y) | x ∈ D, dist(x, x0 ) ≤ δ, |y − f (x)| ≤ δ L2 + 1 . On the other hand, if d = dist(x0 , ∂D) ≤ δ then there exists an x1 ∈ ∂D with dist(x0 , x1 ) = d. With this x1 , for every 0 ≤ t < 1 xt := tx1 + (1 − t)x0 ∈ D, otherwise dist(x0 , ∂D) < d would hold. The function g : [0, 1) → R, g(t) := f (xt ) is Lipschitz with constant dL, so g(1) := limt 1 g(t) exists and |g(1) − g(0)|√≤ dL. Obviously X1 := (x1 , g(1)) ∈ ∂H and dist(X0 , X1 ) ≤ d L2 + 1. That is, Bδ (X0 ) ⊂ B(√L2 +1+1)δ (X1 ). Putting everything together, we have
√ 2 (H )[δ] ⊂ (x, y) | x ∈ (D )[δ] ∩ D, |y − f (x)| ≤ δ L2 + 1 ∪ (∂H)[( L +1+1)δ] . (3.2) 2 The delicate question how sensitive this notion of Lipschitz-decomposition is to the choice of the atlas needs further investigation.
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This implies √ 2 Lm (H )[δ] ≤ 2δ L2 + 1Lm−1 (D )[δ] + Lm (∂H)[( L +1+1)δ] .
¯ + o(1) This gives the statement of the lemma since Lm−1 (D )[δ] = Lm−1 D and the second term is o(δ) because of our assumption. In the next lemma, π will denote the projection of Rm to Rm−1 parallel to the last axis: π((x, y)) := x when x ∈ Rm−1 and y ∈ R. Lemma 3.9 Let H ⊂ Rm be a one-codimensional locally Lipschitz graph with H = {(x, f (x)) | x ∈ D}, D ⊂ Rm−1 open, f : D → R locally √ Lipschitz with constant L. Let δ > 0 and G ⊂ Rm be such that dist(G, ∂H) > ( L2 + 1 + 1)δ. Then Lm H [δ] ∩ G ≤ 2δ L2 + 1Lm−1 (π(G)) . Proof. Let H √= H. (3.2) holds just like in the previous lemma. Since dist(G, ∂H) > ( L2 + 1 + 1)δ, this means that
H [δ] ∩ G ⊂ (x, y) | x ∈ D ∩ π(G), |y − f (x)| ≤ δ L2 + 1 , which gives the statement of the lemma.
4 The Fundamental Theorem This section is devoted to the fundamental theorem – or local ergodicity theorem – for semi-dispersing billiards. The two-dimensional case had been settled in Sinai’s celebrated work, [S(1970)]. Seventeen years had elapsed until the multidimensional generalization given by Chernov and Sinai appeared, [S-Ch(1987)]. It offered a quite involved, but in essence very transparent formulation of the theorem and a delicate proof. A self-contained exposition of the original ideas with detailed conditions and arguments were provided in [K-S-Sz(1990)] where a slightly more general, the so called “transversal” version of the fundamental theorem was announced — mainly with its application to three-billiards in mind. Several other papers have appeared in the 90s with nice expositions of the theorem, even for classes of dynamical systems more general than the original semi-dispersing billiard setting (eg. Hamiltonian systems with singularities in [L-W(1995)]). However, all of these papers assumed that for all powers of the dynamics the singularity set is a finite collection of one-codimensional smooth and compact submanifolds. Since, as our counterexample shows, this is not the case, it became utmost necessary to replace this assumption. Throughout the section our main reference is [K-S-Sz(1990)]. Actually, our aim is to demonstrate that it is possible to modify the proof presented there to the case when the singularity sets are not smooth but just finitely Lipschitzdecomposable. After formulating the conditions and the statement of the theorem,
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we give a sketch of the proof (that goes along the lines of [K-S-Sz(1990)]) and work out those parts in more detail, where the original argument is to be modified. Our notations introduced mainly in section 2.1 coincide in almost all cases with those of [K-S-Sz(1990)] (we have just altered the original conventions at some places for the sake of simplicity). One more remark: following [K-S-Sz(1990)] the formulation of the theorem presented here (the ‘transversal’ fundamental theorem) is slightly more general than the one usually referred to in the literature.
4.1
Formulation of the theorem
Before its formulation its is important to point out the conditions under which the modified proof of the theorem works. We use the notations introduced in section 2.1. Condition 4.1 (Chernov-Sinai Ansatz, Condition 3.1 from [K-S-Sz(1990)]). For νR+ -almost every point x ∈ R+ we have x ∈ ∂M ∗ and, moreover, the positive semi-trajectory of the point x is sufficient. What follows below is our new condition – Lipschitz decomposability – on singularities. In the original proof smoothness was assumed, even though it was only formulated as a condition for the set of double singularities – see Condition 3.3 from [K-S-Sz(1990)]). Condition 4.2 There exists an L ∈ R such that for every N ∈ N the singularity set Rn is ‘Lipschitz decomposable’ with constant L (cf. Conjecture 3.7).
|n|≤N
Some remarks. • For the set of singular reflections itself the original property remains true: R is a finite collection of smooth compact manifolds of codimension 1. • Condition 4.2 can only be satisfied by semi-dispersing billiards with a finite horizon. However, the infinite horizon case can easily be reduced to the finite horizon case (cf. [S-Ch(1987)], [K-S-Sz(1990)]. • As to the original exposition, one more condition was assumed on the geometry of the scatterers (the regularity of the set of degenerate tangencies Condition 3.2 in [K-S-Sz(1990)]). However, the role of this condition was to guarantee that points belonging to two different smooth components of the singularity set belong to finitely many codimension 2 submanifolds. Now instead of smooth components we have locally Lipschitz graphs and it is enough to require that the δ-neighbourhoods of their boundaries have a volume of o(δ), which is a little less than being two-codimensional (cf. Definition 3.5).
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To formulate the fundamental theorem, we introduce the notion of regular coverings. Note that m = 2d − 2 is the dimension of the (Poincar´e) phase space ∂M . The next definition will not be absolutely precise, for we omit some technical details for the sake of easier understanding. For a precise formulation please see Definition 3.4 in [K-S-Sz(1990)]). Definition 4.3 Let us assume that for a point x ∈ ∂M ∗ and its neighborhood U (x) a smooth foliation U (x) = ∪α∈B d−1 Γα is given. The foliae Γα are d − 1-dimensional manifolds uniformly transversal to all possible local stable manifolds (B d−1 is the standard d − 1-dimensional open ball). The parameterized family of finite coverings G δ = {Gδi | i = 1, ..., I(δ) }
0 < δ < δ0
is a family of regular coverings iff: 1. each Gδi is an open parallelepiped of dimension 2d − 2; 2. the d − 1-dimensional faces of Gδi are all cubes with edge-length δ, moreover, they may belong to two different categories: the s-faces are ’parallel with leaves of the stable foliation’ while the Γ-faces are ’parallel’ with the leaves of the foliation Γ; 3 3. For any point, the maximal number of parallelepipeds covering it is 22d−2 ; 4. if Gδi ∩ Gδj = ∅, then µ1 (Gδi ∩ Gδj ) ≥ c1 δ 2d−2 with c1 independent of δ. Some further convention: Given any Gδi its s-jacket, ∂ s (Gδi ) is the union of those (2d − 3)-dimensional faces of Gδi which contain at least one s-face of it. The Γ-jacket, ∂ Γ (Gδi ) is defined similarly. Clearly, ∂(Gδi ) = ∂ s (Gδi ) ∪ ∂ Γ (Gδi ). We say that a stable manifold γ s (y) intersects Gδi correctly if: ∂(Gδi ∩ γ s (y)) ⊂ ∂ Γ (Gδi ).
Theorem 4.4 (The Fundamental Theorem) We assume that: – – – –
conditions 4.1 and 4.2 are satisfied; a sufficient phase point x ∈ ∂M ∗ in the interior of the phase space is given; a smooth transversal foliation Γ in a neighborhood U0 of x is fixed; a constant 0 < 81 < 1 is chosen.
3 More
precisely if we consider the center of each parallelepiped wiδ ∈ Gδi , the s- and Γ- faces are parallel with the tangent planes Twδ γ s (wiδ ) and Twδ Γ(wiδ ), respectively. i
i
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Then there is a sufficiently small neighborhood U 1 (x) such that for any U (x) ⊂ U 1 (x) and for any family of regular coverings, the covering G δ can be divided into two disjoint subsets, Ggδ and Gbδ (called ‘good’ and ‘bad’), in such a way that: (I) For any Gδi ∈ Ggδ and any s-face E s of it, the set: {y ∈ Gδi | ρ(y, E s ) < 81 δ and γ s (y) intersects correctly} has positive relative µ1 -measure in Gδi . (II) Gδi = o(δ). µ1 Gδi ∈Gbδ
Remark: – With suitable modifications of the proof the theorem applies to all sufficient points x ∈ ∂M ∗ (see [K-S-Sz(1990)]), however, for simplicity here we restrict ourselves to regular phase points.
4.2
Proof of the Fundamental Theorem
Here we would like to give a sketch of the proof following [K-S-Sz(1990)]. For brevity we do not repeat the whole argument. Our aim is to emphasize the main ideas on the one hand and point out those parts where the original proof is to be modified on the other hand. Several arguments apply word by word, as to these, we do not give an exposition, just refer to the original paper. Those steps that need non-trivial modification are emphasized and worked out in detail. Throughout the section we think of the sufficient point x ∈ ∂M 0 and its neighborhood U as being fixed. y usually denotes some point in U . Furthermore, a sufficiently small δ is kept fixed - thus we work with one particular covering G δ . Of course, for every Gδi ∈ G δ we have diam(Gδi ) ≤ mδ where m = 2d − 2 is the dimension of the phase space. As a preparation for the main argument we state two important Lemmas: Lemma 4.5 In correspondence with condition 4.2 let us denote the Lipschitz components of ∪|n|≤N Rn with Ri (i = 1, ..., K), rest with R∗ , and the Lipschitz-constant with L. Consider the set ∆δ,N := {x | ∃i, j ≤ K, , i = j, ρ(x, Ri ) ≤ δ, ρ(x, Rj ) ≤ δ} ∪ {x | ρ(x, R∗ ) ≤ δ}. For all N : µ1 (∆δ,N ) = o(δ). This Lemma plays essentially the role of Lemma 4.6 from [K-S-Sz(1990)]. However, the proof of it is the first point where the original proof of the fundamental theorem had to be modified.
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Proof. Fix an index i and find a coordinate system so that Ri = {(x, fi (x))| x ∈ Di }. where Di ⊂ Rm−1 . πi shall denote the usual projection onto Rm−1 : πi ((x, y)) := x, when x ∈ Rm−1 and y ∈ R. Obviously ∂Di = πi (∂Ri ), so Lm (∂Ri )[δ] ≥
2δLm−1 (∂Di ). So the condition Lm (∂Ri )[δ] = o(δ) implies Lm−1 (∂Di ) = 0. As a consequence for any η > 0 it is possible to find η > 0 such that the (closure of the) open η -neighbourhood of ∂Di inside Di has Lm−1 -measure smaller than η. Let us denote this open neighborhood by Dηi and furthermore ∆iη = (x, fi (x)) | x ∈ Dηi . Now consider the parts of the singularity far away from the borders of the singularities. For different i-s the sets Ri \ ∆iη (i = 1, ..., K) are pairwise disjoint compact sets (as they are continuous images of compact sets). Consequently, for δ small enough the sets (Ri \ ∆iη )[δ] are pairwise disjoint as well. Now for the set mentioned in the Lemma, we can write: ∆δ,N ⊂ ∆η[δ] ∪ (R∗ )[δ] where ∆η =
K i=1
∆iη .
Now apply Lemma 3.8 to get Lm ((∆iη )[δ] ) ≤ 2 This means that
Lm (∆η[δ] ) ≤ 2K
L2 + 1δη + o(δ)
L2 + 1δη + o(δ) [δ]
stand for every η, meaning that Lm (∆η ) = o(δ). Together with Lm ((R∗ )[δ] ) = o(δ) this gives the statement of the lemma. Before formulating the other key Lemma we would like to note that – for Lipschitz-continuous functions are differentiable almost everywhere – in almost every point of a singularity component (i.e. in one particular open locally Lipschitz ˆ it makes sense to talk about their (one-codimensional) tangent planes graph R) ˆ Knowing the behaviour of the tangent plane wherever it exists allows us to TyˆR. ˆ think about the “direction” of the whole R. Lemma 4.6 Given any x ∈ ∂M 0 and any 8 > 0 there is a neighborhood U (x) ⊂ ∂M of x such that for every γ1s , γ2s and any (2d − 3)-dimensional Lipschitz component ˆ of some Rn (n > 0) intersecting U (x) with points y1 , y2 and yˆ, lying on the R ˆ exists: three manifolds, respectively, so that TyˆR (Ty1 γ1s , Ty2 γ2s ) < ˆ < (Ty1 γ1s , TyˆR)
8, 8.
(4.1)
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This Lemma is on the parallelization effect and it is exactly the same as Lemma 4.9 in [K-S-Sz(1990)]– the original argument applies. Nevertheless it might be useful to point out what the second inequality in (4.1) means: there is a (d − 1)dimensional subspace of the tangent space at almost any point of the (2d − 3)ˆ very close to the stable subspace. Note, however, that R ˆ dimensional manifold R may behave extremely widely – i.e. in a non-smooth manner – in the remaining (d − 2) dimensions (in case d ≥ 3). Before the proof we should introduce some more notation. The following two quantities measure the hyperbolicity near the point y ∈ ∂M 0 . Let n −1 −1 κn,0 (y) = inf (D−T p , n y,Σ ) Σ
where the inf is taken over all convex local orthogonal manifolds passing through −T n y. Furthermore denote n κn,δ (y) = inf inf (Dw,Σ )−1 −1 p . Σ w∈Σ
Here the infimum is taken for the set of convex fronts Σ passing through −T n y such that (i) T n is continuous on Σ and (ii) T n Σ ⊂ Bδ (−y). Remark 4.7 (cf. Lemma 5.3 in [K-S-Sz(1990)]and Lemma 2.2 in the present paper) It is not difficult to see that κn,δ (y) is an increasing function of n. Furthermore, for sufficient points y clearly: lim κn,0 (y) = ∞.
n→∞
(Here we do not state in general that κn,0 grows exponentially, linear growth – which is obvious for sufficient points y – is enough.) The following subsets of the neighborhood U
x depend on the constant δ.
Ug Ub
:= {y ∈ U | ∀n ∈ Z+ , ztub (T n y) ≥ (κn,c3 δ (y))−1 c3 δ}; := U \ U g ;
Unb
:= {y ∈ U |ztub (T n y) < (κn,c3 δ (y))−1 c3 δ}
(4.2)
Remark 4.8 Note that for the points y ∈ U g the stable manifold extends to the boundary of Bc3 δ (y), the ball of radius c3 δ around y (cf. Lemma 5.4 from [K-SSz(1990)]). The constant c3 will be chosen in an appropriate way to guarantee that for any y ∈ U g ∩ Gδi the stable leaf γ s (y) intersects Gδi correctly unless it intersects ∂ s Gδi . Furthermore, we introduce the class of permitted functions. Definition 4.9 A function F : R+ → Z+ defined in a neighborhood of the origin is permitted whenever F (δ) ! ∞ as δ " 0. For a fixed permitted function F (δ) we define Uωb = ∪n>F (δ) Unb .
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Most of the statements to come hold for any permitted function F (δ). At one point of the argument we shall fix one particular F (δ). Lemma 4.10 (Tail bound; Lemma 6.1 from [K-S-Sz(1990)]). For any permitted function: µ1 (Uωb ) = o(δ). The measure estimates in the proof of the Tail Bound are related to R, the set of singular reflections. As already mentioned, this set (in contrast to the higher iterates Rn ) is a finite collection of smooth and compact 1-codimensional submanifolds of the phase space. Consequently, there is no need for Lipschitz decomposition here, thus we do not include the proof. Essentially, the original argument from [K-S-Sz(1990)]applies, nevertheless, at the definition of the small set of non-sufficient points a little more care is needed. We would also like to emphasize that the proof of the Tail Bound is the point where the Chernov-Sinai Ansatz (Condition 4.1) is exploited. On more details see [K-S-Sz(1990)]. Remark 4.11 In what follows we will work with distances defined by the Euclidean metric ρ. However, as the interior point x in ∂M is fixed and its neighborhood U (x) is fixed we have | v, n(q) | ≥ c for some positive constant c = c(x) in this neighborhood. Thus the two distances ρ and ρi are equivalent (cf. section 2). Now we can start proving the fundamental theorem by telling explicitly how the collection of parallelepipeds G δ is divided into a good and a bad part. We say Gδi ∈ Gbδ iff (A) either • it intersects more than one Lipschitz component of the singularities of T F (δ) , ˆ ≤ δ, ˆ but ρ(Gδ , ∂ R) • or it intersects only one component R, i • or it intersects the remaining small set R∗ . (B) or it is not of type (A), but it has an s-face E s such that µ1 (Gδi ∩ (E s )[ 1 δ] ∩ Uic ) ≤
83 µ1 (Gδi ), 4
(4.3)
where 83 is a positive constant to be defined later and Uic is the set of points in Gδi with correctly intersecting local stable manifolds. Now we choose one particular permitted function F (δ) : by virtue of Lemma 4.5 there definitely exists a permitted function such that: µ1 (∆(m+1)δ,F (δ) ) = o(δ).
(4.4)
As a consequence the overall measure of bad parallelepipeds of type (A) is o(δ) (such parallelepipeds lie inside the set ∆(m+1)δ,F (δ) ).
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It is time to tell about the choice of our small constants 8i as well. In the formulation of the Fundamental Theorem one particular constant 81 is given. We shall choose three further constants in the following order: 81 → 83 → 84 → 82 . It is utmost important that all of these choices are independent of δ. (they are chosen in the arguments below, 83 in 1., 84 in 2. and 82 in 3.). After all these choices are made we fix the neighborhood U 1 (x) (see the formulation of the Fundamental ˆ of some Rn (n > 0) Theorem) in such a way that for all Lipschitz components R that intersect U 1 (x): (γ1s , γ2s ) < ˆ < (γ1s , R)
82 , 82 .
(4.5)
Such a choice is clearly possible by virtue of Lemma 4.6. Here the second inequalˆ where it makes sense, that is, where R ˆ is ity is understood at every point of R differentiable. One more remark: having fixed the neighborhood U 1 (x) and the foliation Γ uniformly transversal to the stable foliation, it is possible to uniformly compare two different measures for each product-type set inside U 1 (x). More precisely there is a constant c4 > 0 such that given any product-type set, the ratio of its µ1 -measure and its measure that arises as a product of measures in the s− and Γ−directions lies between c−1 4 and c4 . From now on Gδi will always denote a bad parallelepiped of type (B). The proof of the Fundamental Theorem follows from the small arguments to come. 1. Let us first give an estimate from below on the measure of Gδi ∩ (E s )[ 1 δ] where E s is an s-face for the bad parallelepiped Gδi . By the above remark on product measures: d−1 d−1 µ1 (Gδi ∩ (E s )[ 1 δ] ) ≥ c−1 δ ≥ c6 8d−1 µ1 (Gδi ) ≥ 83 µ1 (Gδi ), 4 (81 δ) 1
(4.6)
in case 83 (81 ) is chosen sufficiently small. 2. For estimates from above we fix the constant 84 = 84 (83 ) sufficiently small. The measure of points near the s-jacket (which consists of 2(d − 1) faces of dimension 2d − 3), is: µ1 (Gδi ∩ (∂ s Gδi )[ 4 δ] ) ≤ 2(d − 1)c4 84 δδ 2d−3 ≤
83 µ1 (Gδi ). 4
(4.7)
We need one more estimate of similar type. This is the second point where the original proof has to be modified, and the smoothness/Lipschitzness of singularity components is used. Recall that for a bad parallelepiped of type ˆ of the singularity set for (B) there is at most one Lipschitz component R F (δ) intersecting it. We are interested in estimating the measure of the T 84 δ-neighborhood of this Lipschitz graph inside the parallelepiped. If 84 <
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√ 1 L2 +1+1
(L is the Lipschitz-constant) then, by the construction of type (A) parallelepipeds, Lemma 3.9 can be applied, and gives ˆ [ 4 δ] ) ≤ c5 84 δδ 2d−3 ≤ µ1 (Gδi ∩ (R)
83 µ1 (Gδi ). 4
(4.8)
whenever again 84 (83 ) is small enough. 3. Now we choose 82 (84 ) small enough, so that by (4.5) stable manifolds and singularity components are ‘almost parallel’. Namely, the smallness of 82 should guarantee that for any y ∈ Gδi for which γ s (y) does not intersect correctly we have: ˆ [ 4 δ] ) ∪ (Gδi ∩ (∂ s Gδi )[ 4 δ] ) ∪ Uωb . y ∈ (Gδi ∩ (R)
(4.9)
To see that, given a suitable choice of 82 , the above formula is valid we make two remarks. • First we note that for stable manifolds and singularity components ‘not to approach each other too quickly’, being ‘almost parallel’ is enough ˆ at almost every point of R. • Recalling the definitions from (4.2) and the various notions of distances from section 2 what we see immediately is that the inclusion of (4.9) ˆ [ 4 δ] for the is valid with writing Gδi ∩ (∪n≤F (δ) Unb ) instead of Gδi ∩ R first set. Nevertheless, with a suitable choice of 82 we certainly have ˆ [ 4 δ] ) ⊂ Gδ ∩ (∪n≤F (δ) Unb ) as (i) ztub (x) ≥ z(x) and (ii) the Gδi ∩ (R i Euclidean distance ρ and the distance ρi (in terms of which z(x) is defined) are equivalent, see Remark 4.11. We only need some minor considerations to complete the proof. Observe first that for good parallelepipeds the statement (I) evidently holds. As for (II) we have already shown that bad parallelepipeds of type (A) are of measure o(δ) (recall (4.4)), we shall show the same for those of type (B) as well. Indeed, let us consider a Gδi with an s-face E s for which (4.3) holds. By the arguments 1.-3. above: 83 µ1 (Gδi ∩ Uωb ) ≥ µ1 (Gδi ∩ (E s )[ 1 δ] ∩ Uωb ) ≥ µ1 (Gδi ). 4 Now recall that in a regular covering there are at most 22d−2 parallelepipeds with a non-empty common intersection. Thus: 22d−2 µ1 (Uωb ) ≥
µ1 (Gδi ∩ Uωb ) ≥
8 3 µ1 (Gδi ), 4
denotes the sum over where bad δparallelepipeds of type (B). By the Tail Bound (Lemma 4.10) we have µ1 (Gi ) = o(δ) thus the proof of Theorem 4.4 is complete.
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5 The case of algebraic scatterers The main aim of this section is to show that the singularity submanifolds of algebraic semi-dispersing billiards satisfy the Lipschitz-decomposability property formulated in Conjecture 3.7. Fortunately, the most important examples of semidispersing billiards are algebraic as it has been noted in the introduction. Consequently, the algebraicity condition does not essentially restrict the applicability of the fundamental theorem. For definiteness we will say that the zero-set of a system of polynomial equations is an algebraic variety (we will use these notions over the real ground field). Any (measurable) subset of a k-dimensional algebraic variety will be denoted as a k-dimensional SSAV (for ‘subset of an algebraic variety’). As for the dimension of an algebraic variety, see [Sh(1974)]. We also use the following definition. Definition 5.1 A semi-dispersing billiard is algebraic if it has finitely many scatterers and the boundary of each of these scatterers is a finite union of onecodimensional SSAV-s (as subsets of T| d ⊂ Rd ). Remark 5.2 Assume, in general, that we are given a Riemannian manifold M = Mm and a subset A ⊂ M. We say that A is a k-dimensional weakly algebraic subset of M if it is possible to find an appropriate atlas {Ut , ψt }Tt=1 on M such that, for every t, ψt (Ut ∩ A) (⊂ Rm ) is a k-dimensional SSAV in Rm . BiLipschitzness of the atlas {Ut , ψt }Tt=1 can always be assumed (cf. Remark 3.6) Note that being ‘weakly algebraic’ is really a weak notion due to the high degree of freedom in the choice of the atlas. For example, every smooth curve is 1-dim. weakly algebraic. What follows below in three subsections is a proof of Lipschitz decomposability for the singularities R−n in an algebraic billiard. In subsection 5.1 it is shown that singularities are algebraic as subsets of R2d . This implies that R−n ⊂ ∂M is algebraic in the sense of Remark 5.2 as well.4 The proof is completed in subsections 5.2 and 5.3 where a Lipschitz decomposition is constructed for any (onecodimensional) SSAV of Rm .
5.1
The algebraicity of R−n
Our approach generalizes that of section 3 of [S-Sz(1999)]. Since there a detailed exposition was given, here we are satisfied by referring to the main steps of the complexification of the dynamics. Still we completely explain those parts where our arguments are different. In a nutshell the picture is the following. In [S-Sz(1999)] • the authors were only considering quadratic boundaries since hard ball systems are quadratic billiards; 4 In a small neighbourhood of y ∈ ∂M identify the tangent plane T ∂M with Rm and restrict y the orthogonal projection Π : R2d → Ty ∂M onto ∂M to obtain coordinate charts.
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• and for the quadratic case they elaborated a most detailed algebraic analysis of the situation. Here we do not need such a delicate picture. But on the other hand, we are treating the general algebraic case. The chain of field extensions of [S-Sz(1999)] relied upon the explicit solvability of the arising quadratic equations and applied the related elimination of the square roots. In the general case we rather apply the norm used in Galois theory. We first fix some notation – slightly different from the usual conventions – at this point. According to the definition above, ∂Q = ∪Jj=1 ∂Qj , where both the components ∂Qj and their boundaries are all appropriate dimensional SSAV-s (the decomposition is finer than the one into connected components in Rd ). In other words, for each ∂Qj there is a (non-zero) irreducible polynomial Bj (q) such that ∂Qj ⊂ {q ∈ Rd | Bj (q) = 0}. Note that symbolic collision sequences (5.1) are defined in terms of these algebraic boundary components as well. From this point on it will be suitable to consider orbit segments S [0,T ] x0 , T > 0 of the billiard flow with T sufficiently large. In fact, it will be useful to also drop the condition ||v|| = 1. Consequently, the dimension of our phase space will be 2d (first the phase space will be T| d × Rd and later just R2d ). The symbolic collision sequence of S [0,T ] x0 will be denoted by σ = Σ(S [0,T ] x0 ) = (σ1 , σ2 , . . . , σn )
(n ≥ 0)
(5.1)
Remark 5.3 By definition, (q0 , v0 ) corresponds to the initial, generally noncollision phase point x0 of the flow. Furthermore T k x0 = xk = (qk , vk ) ∈ ∂Qσk for every / ∂M of the flow T x0 ∈ ∂M coincides 1 ≤ k ≤ n (we note that for a phase point x0 ∈ by definition with the first point where the positive semi-orbit of x0 reaches the boundary ∂M ; in [K-S-Sz(1990)] this map was denoted by T + ). By a slight abuse of notation we will keep denoting by R−n (introduced in subsection 2.1) the nth inverse image of R in a 2d-dimensional neighbourhood of x0 . Having fixed σ, we first explore the algebraic relationship between the consecutive xk s. For being able to carry out arithmetic operations on our data, we lift the genuine orbit segment to the covering Euclidean space of the torus. This can be done by a straightforward generalization of the trivial Proposition 3.1 of [S-Sz(1999)]. Proposition 5.4 Let S [0,T ] x0 be an orbit segment of the discretized dynamics. Assume that a certain pre-image (Euclidean lifting) q˜0 ∈ Rd of the position q0 ∈ Td is given. Then there is a uniquely defined Euclidean lifting {˜ qi ∈ Rd |0 ≤ k ≤ n} of the given orbit segment which, when considered in continuous time, is a timecontinuous extension of the original lifting q˜0 . Moreover, for every collision σk
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there exists a uniquely defined integer vector ak ∈ Zd — named the adjustment vector of σk — such that Bσk (˜ qk − ak ) = 0
(1 ≤ k ≤ n).
The orbit segment ω ˜ = {˜ qk |0 ≤ k ≤ n} is called the lifted orbit segment with the system of adjustment vectors A = (a0 , . . . , an ) ∈ Z(n+1)d . In the sequel, <, > denotes Euclidean inner product of d-dimensional real vectors. Our next proposition is also a straightforward extension of Proposition 3.3 of [S-Sz(1999)]. Proposition 5.5 Between the kinetic data corresponding to σk−1 and σk one has the following algebraic relations: — the linear collision equation vk = vk−1 − 2 < vk−1 , nk > nk
(1 ≤ k ≤ n)
(5.2)
where nk is the outer unit normal vector of the scatterer Qσk at the point of impact; — the linear free flight equation q˜k = q˜k−1 + τk vk−1
(1 ≤ k ≤ n)
(5.3)
— where the time slot τk = tk − tk−1 (t0 = 0) in (5.3) is determined by the polynomial equation Bσk (˜ qk−1 + τk vk−1 − ak ) = 0. (5.4) Next we turn to the complexification of the billiard ball map T . Given the pair (Σ, A) = (σ1 , σ2 , . . . , σn ; a0 , a1 , . . . , an ), the equations (5.2), (5.3), (5.4) make it possible to algebraically characterize the kinetic data (˜ qk , vk ) by using the preceding data (˜ qk−1 , vk−1 ). Since — at the moment — we are dealing with genuine, real orbit segments, in this situation the equations have at least one positive, real root τk ; in case of several such roots its selection is unique by the geometry of the problem. Our further arguments, however, also use the algebraic closedness of the arising fields and therefore we complexify the dynamics. From this point on, our approach, though related but nevertheless will already be different from that of [S-Sz(1999)]. Definition 5.6 For n = 0 the field K0 = K(∅; ∅) is the transcendental extension C(B) of the coefficient field C by the algebraically independent formal variables B = {(˜ q0 )j , (v0 )j |1 ≤ j ≤ d} Suppose now that the commutative field Kn−1 = K(Σ ; A ) has already been defined, where Σ = (σ1 , σ2 , . . . , σn−1 ); A = (a0 , a1 , . . . , an−1 ). Consider now the polynomial equation bl τ l + bl−1 τ l−1 + . . . + b0 = 0 (5.5)
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arising from (5.4) with k=n. It defines a new field element τn to be adjoined to the field Kn−1 (of course, b0 , . . . , bl ∈ Kn−1 ). At this point, however, we should be a bit cautious. If the equation 5.5 is irreducible, then all its roots are algebraically equivalent, and τn can denote any of them. If (5.4) is reducible, then we should select a particular irreducible factor of its. Indeed, since we are only interested in the images of R, at each step we choose such an irreducible factor of (5.5) which, when its root τn gets evaluated for real values of x0 , gives us a real root of (5.5) which is actually the real root specified after (5.4). This irreducible factor defines the extension Kn = Kn−1 (τn ). In such a way we are given a chain of extensions K0 , K1 , . . . , Kn where for every k = 1, . . . , n the relation Kk = Kk−1 (τk ) holds. By our construction and by the theorem on the prime element of algebra, Kn can also be expressed as K0 (˜ τn ) for some τ˜n ∈ Kn with minimal polynomial m(α) over K0 . By applying the previous construction we are going to look for an algebraic characterization of R−n . For every x0 ∈ MΣ,A = {x ∈ M | Σ(S [0,T ] x) = Σ, A(S [0,T ] x) = A} one has q˜n ∈ Kn . q˜n can formally be understood as a function q˜n (x0 , τ1 , . . . , τn ) or (by the theorem on the prime element) simply as a function q˜n (x0 , τ˜n ) with values in Kn . We will be considering this function exactly in MΣ,A , that is, where Σ and A are constants. Consider Qσn at the point T n x0 . At this point the submanifolds Bσn (˜ qn − an ) = 0 has a normal vector An which can be expressed by the partial ˜n ∈ R just says that < An, vn >= derivatives of Bσn at q˜n − an . The condition x 0. Here both An and vn are elements of Kn , i. e. formal functions of x0 and τ˜n . Consequently < An, vn >= Φ(˜ τn ) where Φ is a polynomial whose coefficients are rational functions over K0 . Take now the (Galois-) norm (cf. [St(1973)]) of this element i. e. Φ = ΠΦ(˜ τni ) where the product is taken for all roots τ˜ni of the irreducible polynomial m. This norm does not vanish since it is the product of non-zero elements in the normal hull of Kn . Moreover, it is a symmetric polynomial of the elements τ˜ni . As such it can be expressed as a polynomial of the elementary symmetric polynomials of the variables τ˜ni : 1 ≤ i ≤ l. These elementary symmetric polynomials can, however, be easily expressed by the coefficients of m which are elements of K0 . As a consequence, we obtain a non-zero element of K0 . The construction just described generalizes the elimination of the square roots method applied in [S-Sz(1999)]. By our construction it remains also true that this polynomial has real coefficients for our real, dynamical orbit. All in all for every fixed Σ and A the resulting piece of R−n is an algebraic submanifold. From the finiteness of the horizon it is clear that in the case of our real dynamics only a finite number of Σ and A provide a non-empty piece of R−n . In this way we have established Theorem 5.7 R−n is a finite union of one-codimensional SSAV-s in R2d .
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5.2
Multi-Dimensional Semi-Dispersing Billiards
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Dimension and Measure of Algebraic Varieties
The motivation for this section is that we need to estimate the Lebesgue-measure (denoted here by Lm ) of the δ-neighbourhood of an algebraic variety . Actually we only need that Lm (H [δ] ) = o(δ) if H is (at least) two-codimensional, but our results will be more general than that. As we will see, this problem is closely related to the box dimension and the socalled Minkowski-content of H (on box dimension, Minkowski-content and their relation to Hausdorff dimension and measure, see section 3.1 in [Fa(1990)]). To start, let us recall some notions and basic facts related to box dimension. Definition 5.8 Let H be a bounded subset of Rm , 0 ≤ d ∈ R. Then the quantities d
M (H) := lim sup δ→0
Md (H) := lim inf δ→0
Lm (H [δ] ) , δ m−d Lm (H [δ] ) δ m−d
are called the upper and lower d-dimensional Minkowski-content of H. Definition 5.9 Let H be again a bounded subset of Rm , 8 > 0. The set I ⊂ H is called an ε-net in H if H ⊂ I [ε] . We will always be interested in finite ε-nets I, and we will never use that I ⊂ H. Some simple facts: (a) dimH H ≤ dimB H ≤ dimB H d
(b) Hd (H) ≤ Md (H) ≤ M (H) d
(c) If M (H) < ∞, then dimB H ≤ d (d) If dimH H < d then Hd (H) = 0 d
(e) If M (H) < ∞ then Lm (H [δ] ) = O(δ m−d ). (f) If I is an ε-net (finite) in H ∈ Rm then Lm (H [ε] ) ≤ (2ε)m |I|, where |I| is the cardinality of I. Now we turn to the investigation of algebraic varieties. Our proposition will be an easy corollary of the following lemma. For 0 ≤ d ∈ R we will denote the d-dimensional Hausdorff-measure by Hd . ˆ ∩ [0, 1]m , where H ˆ is an algebraic variety. Let k be the Lemma 5.10 Let H = H ˆ Let ε > 0, 0 ≤ d ∈ Z. Let maximum of the degrees of the polynomials defining H.
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c > 1 arbitrary. We claim that if Hd+1 (H) = 0 then, if ε is small enough, there exists a (d · ε)-net I in H with |I| ≤ Nm,d,k,ε :=
d
c
i=0
i
m! (m − i)!
3/2 k m−i
1 . εi
Proof. The proof goes by induction on d, and the induction is based on the following Fact. For every x ∈ [0, 1] let Hx = H ∩ ({x} × [0, 1]m−1 ). Then Hd+1 (H) = 0 implies that for Lebesgue almost every x ∈ [0, 1], Hd (Hx ) = 0. This is an easy consequence of Theorem 5.8 in [Fa(1985)]. The same is true for subsets of H arising by fixing another (than the first) coordinate: for every 1 ≤ l ≤ m if Pxl := [0, 1]l−1 × {x} × [0, 1]m−l and Hxl := Pxl ∩ H then we have Hd (Hxl ) = 0 for L1 -a.e. x. We will take advantage of this by choosing ε arbitrary (later on we will fix ε = √εm ) and fixing K ≤ εc points: 0 = xl,1 < . . . < xl,K = 1, such that xl,j+1 − xl,j ≤ ε and Hd (Hxl l,j ) = 0 for every j. The m · K √ hyperplanes Pxl l ,j : l = 1, . . . , m, j = 1, . . . , K cut H into blocks of diameter ≤ ε m. Notice that if H has a point A in any of these blocks, then √ either it also has one (B) on the surface of the block, so that dist(A, B) ≤ ε m, or the entire component of H containing A is inside the block. 1.) We start the induction with d = 0. The previous construction gives (for any ε ) H0 (Hxl l,j ) = 0, that is, Hxl l,j = ∅ for every l, j. That is, the components are points, and we can certainly find the 0 · ε = 0-net I = H with |I| ≤ k m = Nm,0,k,ε , an upper bound for the number of components coming from Bezout’s theorem. 2.) Suppose we have the statement for some d − 1 ≥ 0. 3.) We prove for d. That is, H ⊂ [0, 1]m , Hd+1 (H) = 0. Apply the previous construction with ε = √εm . The set Hxl l,j is now an algebraic variety in [0, 1]m−1 , the polynomials defining it can be derived from those defining H by fixing a variable. So the√degrees can not grow. We can use the inductive assumption for the mK ≤ mc εm sets Hxl l,j with m → m − 1 and the same k. Thus taking a (d − 1)ε-net on every Hxl l,j according to the inductive assumption, and choosing a point from every component that happens to be entirely inside a block, we get a √ d · ε-net I in H with |I| ≤ mc εm Nm−1,d−1,k,ε + k m = Nm,d,k,ε This lemma leads to the following ˆ ⊂ Rm is an algebraic variety, and k is ˆ ∩[0, 1]m where H Proposition 5.11 If H = H ˆ then s := dimH (H) = the maximum of the degrees of the polynomials defining H, dimB (H) ∈ Z and s
0 < Hs (H) ≤ Ms (H) ≤ M (H) ≤ 2m ss
m! (m − s)!
3/2 k m−s < ∞.
(5.6)
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s
Proof. By (b) we have Hs (H) ≤ Ms (H) ≤ M (H). On the other hand, if we choose d ∈ Z in such a way that Hd+1 (H) = 0 then by the Lemma for any c > 1: d def M (H) = lim supε→0
Lm (H [dε] ) (dε)m−d
(f )
(2dε)m Nm,d,k,ε (dε)m−d
≤ lim supε→0 d m! 3/2 d
Lemma = (5.7)
= lim supε→0 2m dd c k m−i εd−i = i=0 (m−i)! 3/2 m! k m−d < ∞. = cd 2m dd (m−d)! By (c) and (d) this implies dimB (H) ≤ d if s < d + 1 (or even if Hd+1 (H) = 0). This contradicts (a) unless dimB (H) = s ∈ Z (or even if Hs (H) = 0). Now with d = s (5.7) implies the right end of (5.6). Corollary 5.12 If H is a bounded subset of an (at least) two (algebraic) codimensional algebraic variety in Rm , then Lm (H [δ] ) = o(δ). Proof. Knowing from [F(1969)] that the algebraic and Hausdorff dimensions com−2 incide, the proposition actually gives M (H) < ∞ which means (by (e)) that Lm (H [δ] ) = O(δ 2 ).
5.3
Lipschitz decomposability of algebraic varieties
In this subsection our aim is to establish the fact that one-codimensional SSAVs possess the finite Lipschitz decomposability property (in the sense of Definition 3.5). Having already shown the algebraic nature of R−n , this way we find that algebraic billiards satisfy Conjecture 3.7. The main result of the subsection is: Theorem 5.13 Any one-codimensional algebraic variety H is Lipschitz decomposable (in the sense of Definition 3.5) with any constant L > 0. In the following, π shall denote the standard projection of Rm to Rm−1 . That is, π(x, y) = x for any x ∈ Rm−1 , y ∈ R. Proof. We construct the decomposition of H. Fix an arbitrary L > 0. Let I(H) denote the ring of polynomials vanishing on H. Let H ∗ be the set of points in H where the gradient of every polynomial in I(H) vanishes. We know from [BC-R(1987)] that this set at least two (algebraic) codimensional, so Corollary 5.12 ensures that H ∗ is good (for the purpose of Definition 3.5). For the points x ∈ H \H ∗ , there is at least one P ∈ I(H) for which gradP (x) = 0 and the gradients of all polynomials in I(H) are parallel to gradP (x). In the following we will assume H = {x|P (x) = 0} for one such P , only for the sake of more transparent notation. Fix a finite collection of unit vectors v1 , . . . , vN in Rm , such that for any nonzero vector v ∈ Rd , there is a vi for which tan((v, vi )) < L < L. We shall identify those components of H that are Lipschitz graphs as viewed from
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the direction vi . We will omit the index i. The construction clearly depends on the vector v = vi . Having fixed v it is possible to choose an orthogonal coordinate system in Rm such that the mth base vector points in the direction v. For arctan(L ) < φ < arctan(L) and h = cos(φ), consider the following subset of the algebraic variety: H <φ
= {x ∈ H |(gradP (x), v) < φ} = ∂ = x∈H |( P (x))2 > h2 (gradP (x))2 . ∂xm
(5.8)
Note that H <φ ∩ H ∗ = ∅, because the inequality in the definition of H <φ is strict. We claim that for almost every possible φ, ∂H <φ is two-codimensional. Indeed, ∂ <φ =φ 2 2 2 ∂H ⊂H := x ∈ H | ( P (x)) = h (gradP (x)) . ∂xm The intersection of H =φ -s corresponding to different φ-s is H ∗ , which is two-codimensional, so its one-codimensional Hausdorff-measure is zero. However, Proposition 5.11 says that the union of all H =φ -s (which is part of H) has a finite one-codimensional Hausdorff-measure. So apart from a countable number of φ-s, the one-codimensional Hausdorff-measure of H =φ is zero. Since H =φ is algebraic, Proposition 5.11 tells us that almost every H =φ is two-codimensional. We fix H = H <φ with one such arctan(L ) < φ < arctan(L). We will cut H into locally Lipschitz graphs. Let k : Rm−1 → N be the multiplicity of π(H ). Clearly for every x ∈ π(H ) the restriction of P to π −1 (x) is nonzero, so k is bounded by the degree of P , and the Implicit Function Theorem implies that it is lower semicontinuous. So, the set D1 ⊂ Rm−1 where k is maximal, is open. Here we can define the finitely many functions f1,1 , . . . , f1,kmax : D1 → R taking the least, second least, ..., greatest element of π −1 (x) for some x ∈ D1 . the Implicit Function Theorem implies that these functions are locally Lipschitz with constant L and that their graphs are disjoint. Now we claim that the boundary of these graphs is two-codimensional. Indeed, H =φ is two-codimensional and algebraic, so π(H =φ ) is also part of a onecodimensional algebraic variety in Rm−1 . The pre-image (by π) of this variety is one-codimensional in Rm , and the boundary of our graphs is on the intersection of this pre-image with H. This intersection is transversal (ensuring two codimensions) at points of H \ H ∗ , and the rest of the boundary is in H ∗ . Now erase the closure of these graphs from H . So the argument can be repeated with kmax already at least one less. The procedure ends in finitely many steps, and so finitely many open locally Lipschitz graphs are constructed. Their closures cover H by construction, and their boundary is two-codimensional. We carry out this construction for every vi , and get a covering of the entire H \ H ∗ by finitely many locally Lipschitz graphs. To get the sets H1 ,...,HN in
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Definition 3.5 we only need to make these graphs disjoint by subtracting the closure of one from the other.
Acknowledgments The authors express their sincere gratitude to N´andor Sim´ anyi and Andr´ as Sz˝ ucs for illuminating discussions. Special thanks are due to K´ aroly B¨ or¨ oczky, Lajos R´onyai and Endre Szab´ o for supplying some of the ideas that were eventually built into the proofs of the paper. We are also greatly indebted to N´ andor Sim´ anyi for his careful reading of the manuscript and for his most useful remarks. N. Chernov was partially supported by NSF grant DMS-9732728. The financial support of the Hungarian National Foundation for Scientific Research (OTKA), grants T26176 and T32022; and of the Research Group Stochastic of the Hungarian Academy of Sciences, affiliated to the Technical University of Budapest is also acknowledged.
References [B-Ch-Sz-T(2000)] P. B´alint, N. Chernov, D. Sz´ asz and P. T´ oth, Geometry of Multi-dimensional Dispersing Billiards, to appear in Ast´erisque (2000). [B-C-R(1987)] J. Bochnak, M. Coste et M-F. Roy, G´eom´etrie alg´ebrique r´eelle, Springer, 1987. [B-R(1998)] L. A. Bunimovich and J. Reh´aˇcek, How high dimensional stadia look like, Commun. Math. Phys. 197, 277–301 (1998). [Fa(1985)] K. Falconer, The Geometry of Fractal Sets, Cambridge University Press, 1985. [Fa(1990)] K. Falconer, Fractal Geometry, Mathematical Foundations and Applications, John Wiley & Sons, 1990. [F(1969)] H. Federer, Geometric Measure Theory, Springer, 1969. [H(1974)] E. H. Hague, What can we learn from Lorentz models?, Transport Phenomena, Lecture Notes in Physics, Springer 31, 377 (1974). [K-S-Sz(1990)] A. Kr´amli, N. Sim´anyi and D. Sz´ asz, A ”Transversal” Fundamental Theorem for Semi-Dispersing Billiards, Comm. Math. Phys. 129, 535–560 (1990). [L-W(1995)] C. Liverani and M. Wojtkowski, Ergodicity in Hamiltonian Systems, Dynamics Reported 4 (New series), 130–202 (1995). [M(1964)] J. Milnor, On the Betti number of real varieties, Proc. Amer. Math. Soc. 15, 275–280 (1964). [Sh(1974)] I. R. Shafarevich, Basic Algebraic Geometry, Springer, 1974.
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Ann. Henri Poincar´e
[Sim(2001)] N. Sim´ anyi, Proof of the Boltzmann-Sinai Ergodic Hypothesis for Typical Hard Disk Systems, submitted for publication, 2001. arXiv:math.DS/0008241. [Sim(2002)] N. Sim´ anyi, The Complete Hyperbolicity of Cylindric Billiards, Ergodic Theory and Dynamical Systems 22, 281–302 (2002), arXiv:math.DS/9906139. [S-Sz(1999)] N. Sim´anyi and D. Sz´ asz, Hard Ball Systems are Completely Hyperbolic, Annals of Mathematics, 149, 35–96 (1999). [S(1970)] Ya. G. Sinai, Dynamical Systems with Elastic Reflections, Russian Mathematical Surveys, (2) 25, 137–189 (1970). [S-Ch(1987)] Ya. G. Sinai and N. Chernov, Ergodic Properties of Certain Systems of 2–D Discs and 3–D Balls, Russain Mathematical Surveys (3) 42, 181–201 (1987). [Sz(1994)] D. Sz´asz, The K-Property of “Orthogonal” Cylindric Billiards, Commun. Math. Phys. 160, 581–597 (1994). [Sz(2000)] D. Sz´asz (ed.), Hard Ball Systems and the Lorentz Gas, Encyclopedia of Mathematical Sciences 101, Springer (2000). [St(1973)] I. Stewart, Galois Theory, Chapman and Hill, London, 1973. P. B´ alint Alfr´ed R´enyi Institute of the H.A.S. Re´altanoda u. 13-15. H-1053 Budapest, Hungary email: [email protected] N. Chernov Department of Mathematics University of Alabama at Birmingham Birmingham, AL 35294 USA email: [email protected] D. Sz´ asz and I. P. T´ oth Mathematical Institute Technical University of Budapest Egry J´ ozsef u. 1. H-1111 Budapest, Hungary email: [email protected] email: [email protected] Communicated by Eduard Zehnder submitted 07/06/01, accepted 04/02/02
Ann. Henri Poincar´e 3 (2002) 483 – 502 c Birkh¨ auser Verlag, Basel, 2002 1424-0637/02/030483-20 $ 1.50+0.20/0
Annales Henri Poincar´ e
Fluctuations of the Entropy Production in Anharmonic Chains L. Rey-Bellet and L. E. Thomas∗
Abstract. We prove the Gallavotti-Cohen fluctuation theorem for a model of heat conduction through a chain of anharmonic oscillators coupled to two Hamiltonian reservoirs at different temperatures.
1 Introduction The Gallavotti-Cohen fluctuation theorem refers to a symmetry in the fluctuations of the entropy production in nonequilibrium statistical mechanics. It was first discovered in numerical experiments of Evans, Cohen and Morris [8] and then discussed in [9] in the context of thermostated systems. As a mathematical theorem it was proved for Anosov dynamical systems [9, 10]. Soon thereafter the fluctuation theorem was discussed in the context of stochastic dynamical systems first by Kurchan [17] and then, more systematically by Lebowitz and Spohn, and Maes [22, 18]. In particular, Maes discovered a general formulation of the fluctuation theorem in the context of space-time Gibbs measures which covers both Markovian stochastic dynamics and chaotic deterministic dynamics (via a Markov partition). As a mathematical theorem the fluctuation theorem is proven for quite general stochastic models with finite state space, such as lattices gases in a finite box. Relations for the free energy related to the fluctuation theorem have been also discussed in [15, 2]. Among the consequences of the fluctuation theorem is the non-negativity of entropy production although the proof of its positivity is more difficult and is so far proved only in particular examples [7, 20]. We also note that in the related context of open systems, classical and quantum, the production of entropy is discussed at a general level in [27, 13, 24]. Again the non-negativity of entropy production is relatively easy to establish, while the strict positivity has been established only in particular models [7, 14]. In this paper we consider an open system consisting of a finite (but of arbitrary size) chain of anharmonic oscillators coupled at its ends only to reservoirs of free phonons at positive and different temperatures [6, 7, 5, 25, 26]. In particular our model is completely Hamiltonian and its phase space is not compact. ∗ Partially
supported by NSF Grant 980139
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In order to establish the fluctuation theorem, two ingredients are needed: one needs to prove a large deviation theorem for the ergodic average of the entropy production and establish a symmetry of the large deviation functional. The second part is usually relatively straightforward to establish, at a formal level, since it follows from a symmetry of the generator of the dynamics. This formal derivation for models related to ours can be found in [22] and [19]. The first part, proving the existence of the large deviation functional, involves technical difficulties, in particular if the phase space of the model is not compact. In this case large deviation theorems are established provided the system satisfies very strong ergodic properties (such as hypercontractivity) see e.g. [3, 4, 29]. In addition the entropy production is in general an unbounded observable while standard results of large deviations apply only to bounded observables. In this paper we show how to treat these difficulties in the model at hand. The techniques we use are based on the construction of Liapunov functions for certain Feynman-Kac semigroups and Perron-Frobenius-like theorem in Banach spaces. We heavily rely on the strong ergodic properties of our model established in [6, 7, 5] and especially in [26]. The Hamiltonian of the model, as in [6], has the form H = H B + HS + HI .
(1)
The two reservoirs of free phonons are described by wave equations in R with Hamiltonian d
HB
=
H(ϕ, π)
=
H(ϕL , πL ) + H(ϕR , πR ) , 1 dx (|∇ϕ(x)|2 + |π(x)|2 ) , 2
where L and R stand for the “left” and “right” reservoirs, respectively. The Hamiltonian describing the chain of length n is given by HS (p, q) =
n p2 i
i=1
V (q) =
n i=1
U (1) (qi ) +
2
+ V (q1 , · · · , qn ) ,
n−1
U (2) (qi − qi+1 ) ,
i=1
where (pi , qi ) ∈ Rd × Rd are the coordinates and momenta of the ith particle of the chain. The phase space of the chain is R2dn . The interaction between the chain and the reservoirs occurs at the boundaries only and is of dipole-type HI = q1 · dx ∇ϕL (x)ρL (x) + qn · dx ∇ϕR (x)ρR (x) , where ρL and ρR are coupling functions (“charge densities”). Our assumptions on the anharmonic lattice described by HS (p, q) are the following:
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• H1 Growth at infinity: The potentials U (1) (x) and U (2) (x) are C ∞ and grow at infinity like x k1 and x k2 : There exist constants Ai , Bi , and Ci , i = 1, 2 such that lim λ−ki U (i) (λx)
=
Ai x ki ,
∇U (i) (λx)
=
Ai ki x ki −2 x ,
∂ 2 U (i) (x)
≤
(Bi + Ci U (i) (x))
λ→∞ −ki +1
lim λ
λ→∞
1− k2
i
.
Moreover we will assume that k2 ≥ k1 ≥ 2 , so that, for large x the interaction potential U (2) is ”stiffer” than the one-body potential U (1) . • H2 Non-degeneracy: The coupling potential between nearest neighbors U (2) m is non-degenerate: For x ∈ Rd and m = 1, 2, · · ·, let A(m) (x) : Rd → Rd denote the linear maps given by (A(m) (x)v)l1 l2 ···lm =
d l=1
∂ m+1 U (2) (x)vl . · · · ∂x(lm ) ∂x(l)
∂x(l1 )
We assume that for each x ∈ Rd there exists m0 such that Rank(A(1) (x), · · · A(m0 ) (x)) = d . • H3 Rationality of the coupling: Let ρˆi denote the Fourier transform of ρi . We assume that 1 , |ˆ ρi (k)|2 = Qi (k 2 ) where Qi , i ∈ {L, R} are polynomials with real coefficients and no roots on the real axis. We introduce now the temperatures of the reservoirs by choosing initial conditions for the reservoirs. The Hamiltonian of a reservoir is quadratic in Ψ ≡ (φ, π), H = Ψ, Ψ/2, and therefore the Gibbs measure at temperature T , dµT (Ψ) is the Gaussian measure with covariance T · , ·. To construct nonequilibrium steady states we assume that • The initial conditions ΨL = (φL , πL ) and ΨR = (φR , πR ) of the reservoirs are distributed according the gaussian Gibbs measures dµTL and dµTR respectively.
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In order to define the heat flow through the bulk of the crystal we consider the energy of the ith oscillator which we take to be p2 1 (2) U (qi−1 − qi ) + U (2) (qi − qi+1 ) . Hi = i + U (1) (qi ) + (2) 2 2 Differentiating Hi with respect to time, one finds that dHi = Φi−1 − Φi , dt where Φi = is the heat flow from the ith entropy production by
(pi + pi+1 ) ∇U (2) (qi − qi+1 ) (3) 2 to the (i + 1)th particle. We define a corresponding
1 1 σi = − Φi , TR TL where TR and TL are the temperatures of the reservoirs. There are other possible definitions of heat flows and corresponding entropy production that one might want to consider. One might, for example, consider the flows ΦL , ΦR at the boundary of the chains, and define σb = −ΦL /TL − ΦR /TR , or one might take other quantities as local energies. But using conservation laws it is easy to see that all these heat flows have the same average in the steady state. Moreover we will show that all the entropy productions have the same large deviations functionals: the exponential part of their fluctuations are identical. We denote (p(t), q(t)) = (p(t, p0 , q0 , ΨL , ΨR ), q(t, p0 , q0 , ΨL , ΨR )) as the Hamiltonian flow generated by the Hamiltonian (1), and consider the ergodic average 1 t σi t ≡ σi (p(s), q(s)) ds . t 0
The quantity σi (p(s), q(s)) depends on both the initial conditions of the chain and of the reservoirs which, by assumption, are distributed according to thermal equilibrium. By the ergodic theorem proven in [26] there exists a measure dν on R2dn such that σi dν . lim σ i t = t→∞ for all (p0 , q0 ) and dµTL and dµTR almost surely. Moreover σi dν ≡ σν is independent of i and as shown in [7] σν ≥ 0 and σν = 0 if and only if TL = TR . Given a set A ⊂ R, we say that the fluctuations of σi in A satisfy the large deviation principle with large deviation functional I(w) provided inf
w∈Int(A)
1 I(w) ≤ lim inf − log P{σ i t ∈ A} ≤ t→∞ t
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1 lim sup − log P{σ i t ∈ A} ≤ t t→∞
inf
487
I(w) .
w∈Clos(A)
The study of large deviations for σi is based on the moment generating functionals ei (α) given by Rt 1 ei (α) = lim − log dµTL dµTR e−α 0 σi (p(s),q(s)) ds . t→∞ t The main technical result of this paper is Theorem 1.1 Under the assumptions H1-H3, if Tmin Tmin α∈ − , 1+ , Tmax − Tmin Tmax − Tmin e(α) ≡ ei (α) is finite and independent of i and the initial conditions (p0 , q0 ). Moreover e(α) satisfies the relation e(α) = e(1 − α) . As an application of the G¨ artner-Ellis Theorem, see [4], Theorem 2.3.6, we obtain the Gallavotti-Cohen fluctuation theorem. Theorem 1.2 Under the assumptions H1-H3 there is a neighborhood O of the interval [−σν , σν ] such that for A ⊂ O the fluctuations of σi in A satisfy the large deviation principle with a large deviation functional I(w) obeying I(w) − I(−w) = −w , i.e., the odd part of I is linear with slope −1/2. Theorem 1.2 provides information on the ratio of the probabilities of observing the entropy production to be w and −w: roughly speaking we have P{σ i t ∈ (w − ', w + ')} ∼ ewt . P{σ i t ∈ (−w − ', −w + ')} In fact we will prove these theorems for the simplest case |ˆ ρi (k)|2 ∼ (k 2 + γ 2 )−1 , see assumption H3. See [26], Sect. 2 where it is shown how to accommodate higher order polynomials.
2 Fluctuations of the entropy production 2.1
Exponential mixing and compactness
As shown in [6, 26], under condition H3 the dynamics of the complete system can be reduced to a Markov process on the extended phase space consisting of
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the phase space of the chain R2dn and of a finite number of auxiliary variables which we denote as r. As mentioned at the end of section 1, we just consider the simplest case |ˆ ρ(k)|2 ∼ (k 2 + γ 2 )−1 , so that r = (r1 , rn ) ∈ R2d . For higher order polynomials the equations for r given below are replaced by a higher dimensional system of (linear) equations. The resulting equations of motion take the form q˙ p˙
= =
p, −∇q V − ΛT r ,
dr
=
(−γr + Λp) dt + (2γT )1/2 dω .
(4)
Here p = (p1 , · · · , pn ) and q = (q1 , · · · , qn ) denote the momenta and positions of the particle, r = (r1 , rn ) are the auxiliary variables and ω is a standard 2d-dimensional Wiener process. The linear map Λ : Rdn → R2d is given by Λ(p1 , . . . , pn ) = (λp1 , λpn ) and T : R2d → R2d by T (x, y) = (T1 x, Tn y). Here T1 ≡ TL and Tn ≡ TR are the temperatures of the reservoirs attached to the first and nth particles respectively, γ is the constant appearing in ρˆ and λ is a coupling constant equal to ρ L2 . The solution of Eq. (4), x(t) = (p(t), q(t), r(t)) with x ∈ X = R2d(n+1) is a Markov process. We denote T t as the corresponding semigroup T t f (x) = Ex [f (x(t))] , with generator L = γ (∇r T ∇r − r∇r ) + (Λp∇r − rΛ∇p ) + (p∇q − (∇q V (q))∇p ) ,
(5)
and we denote Pt (x, dy) as the transition probability of the Markov process x(t). In [26] we proved that the Markov process x(t) has smooth transition probabilities, in particular it is strong Feller, and that it is (small-time) irreducible: For any t > 0, any x ∈ X and any open set A ⊂ X we have Pt (x, A) > 0. There is a natural energy function associated to Eq.(4), given by G(p, q, r) =
r2 + H(p, q) , 2
which we employ throughout our discussion. In [26] we have constructed a Liapunov function for x(t) from G: Let t > 0 and 0 < θ < max(T1 , Tn )−1 . There exist an E0 and functions κ = κ(E) < 1 and b = b(E) < ∞ defined for E > E0 such that for E > E0 , T t eθG (x) ≤ κ(E)eθG (x) + b(E)1{G≤E} (x) .
(6)
Moreover κ(E) can be made arbitrarily small by choosing E sufficiently large, in fact there exist positive constants c1 = c1 (θ, t) and c2 = c2 (θ, t) such that κ(E) ≤ c1 e−c2 E
2/k2
.
(7)
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By results of [21] it is also shown in [26] that the convergence to the unique stationary state, denoted by µ, occurs exponentially fast: Let H∞,θ denote the Banach space {f ; f ∞,θ ≡ supx |f (x)|e−θG(x) < ∞}. Then there exist constants r > 1 and R < ∞ (8) |T t f (x) − f dµ| ≤ Rr−t f ∞,θ eθG(x) , which means that T t , acting on H∞,θ has a spectral gap. The methods of [21] are probabilistic and rely on a nice probabilistic construction called splitting as well as coupling arguments and renewal theory. Under the condition given here, by taking advantage of the fact that the constant κ in the Liapunov bound (6) can be made arbitrarily small (this is not assumed in [21]), we can prove stronger ergodic properties and also give a direct analytical proof of Eq. (8). Besides the Banach space H∞,θ defined above we also consider the Banach 0 = {f, |f |e−θG ∈ C0 (X)} with norm · ∞,θ ( C0 (X) denotes the set space H∞,θ of continuous functions which vanish at infinity). Furthermore for 1 ≤ p < ∞ we consider the family of Banach spaces Hp,θ = Lp (X, e−pθG(x)dx) and denote · p,θ the corresponding norms. Theorem 2.1 If 0 < θTi < 1, the semigroup T t extends to a strongly continuous 0 quasi-bounded semigroup on Hp,θ , for 1 ≤ p < ∞ and on H∞,θ . For any t > 0, T t 0 is compact on Hp,θ , for 1 < p ≤ ∞ and on H∞,θ . As an immediate consequence of the spectral properties of positive semigroups [11] and the irreducibility of x(t) we have Corollary 2.2 The Markov process x(t) has a unique invariant measure dµ and Eq. (8) holds. Proof. Since T t is a Markovian, compact, and irreducible semigroup the eigenvalue 1 is simple with the constant as the eigenfunction. This shows that the Markov process x(t) has a unique invariant measure. Moreover by the cyclicity properties of the spectrum of a positive semigroup [11], and by the compactness of T t , there are no other eigenvalues of modulus 1. Eq. (8) follows immediately. Proof of Theorem 2.1. In [26], Lemma 3.6, we showed that for some constant C T t eθG ≤ ect eθG provided θTi < 1 (see also Lemma 2.9 below). Therefore for f C ∞ with compact support we have, using Ito’s and Girsanov’s formulas e−θG T t eθG f (x) = Ex eθ(G(x(t))−G(x))f (x) Rt Rt√ 2 = Ex eθ 0 γ(Tr(T )−r ) ds+θ 0 2γT rdω(s) f (x(t)) 2 x(t)) , = Ex eγθTr(T )+γ r˜(θ T −θ)˜r f (˜
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Ann. Henri Poincar´e
where x ˜ is the process with generator ˜ θ = L + 2γθrT ∇r . L ˜ T 1 = γTr(1 − 2θT ). Standard arguments show then A computation shows that L θ that the semigroup associated with the process x ˜ extends to a quasi-bounded and strongly continuous semigroup on Lp (dx), 1 ≤ p < ∞ and on C0 (X). Using the assumption that θTi < 1 and Feynman-Kac formula we see that e−θG T t eθG extends too to a quasi-bounded and strongly continuous semigroup on Lp (dx), 1 ≤ p < ∞ and on C0 (X). This implies immediately that T t extends to a strongly 0 continuous semigroup on Hp,θ , 1 ≤ p < ∞ and H∞,θ . The computation above also t shows that T extends to a quasi-bounded semigroup on H∞,θ . We first prove the compactness of T t for H∞,θ . If f ∈ H∞,θ then |f (x)| ≤ f ∞,θ eθG(x) and by (6) and (7) we obtain |1G≥E T t f (x)|
|T t f (y)| θG(y) {y:G(y)≥E} e
≤
eθG(x)
≤
eθG(x) f ∞,θ
≤
κ(E)eθG(x) f ∞,θ .
sup
T t e(θG(y)) eθG(y) {y:G(y)≥E} sup
(9)
From the bounds (9) and (7) we conclude that the operator 1{G≥E} T t converges uniformly to 0 in H∞,θ as E → ∞. The semigroup T t has a C ∞ kernel since it is generated by a hypoelliptic operator see [26], Proposition 4.1, so, by the ArzelaAscoli theorem 1{G≤E} T t/2 1{G≤E} is compact, for any E. Therefore we obtain T t = lim 1{G≤E} T t/2 1{G≤E} T t/2 , E→∞
where the limit is in the norm sense from (9) above, i.e., T t is the uniform limit of compact operators, hence is compact. 0 The compactness of T t for H∞,θ follows from the same argument. In fact by t 0 Eq.(7), for any t > 0, T H∞,θ ⊂ H∞,θ . To prove the compactness of T t on Hpθ , 1 < p < ∞, we note that |T t f (x)|
= |Ex [f (x(t))]| θ
θ
= |Ex [e q G(x(t)) e− q G(x(t)) f (x(t))]| 1/q 1/p pθ ≤ Ex [eθG(x(t)) ] Ex [e− q G(x(t)) f p (x(t))] . Thus using the bound (7) and the fact that T t is quasi-bounded on H1,θ we obtain p pθ 1G≥E T t f pθ,p ≤ Ex [eθG(x(t)) ] q Ex [e− q G(x(t)) f p (x(t))]e−pθG(x) dx {x:G(x)≥E}
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Fluctuations of the Entropy Production in Anharmonic Chains
≤
sup {x:G(x)≥E} p
Ex [eθG(x(t)) ] eθG(x)
pq
T t (e−
pθ q G
491
f p ) 1,θ
pθ
≤ κ(E) q ect e− p G f p 1,θ p
= κ(E) q ect f pθ,p . As in the case p = ∞, we conclude from the bound (7) that the operator 1G≥E T t converges uniformly to 0 in Hp,θ as E → ∞. Using that the kernel of 1{G≤E} T t 1{G≤E} is bounded, we conclude that T t is compact on Hp,θ for 1 < p < ∞.
2.2
Heat flow and generating functionals
In order to define the heat flows we note that we have d t T H = LT t H = T t (−rΛp) = T t (−λr1 p1 − λrn pn ) . dt Hence we identify Φ0 ≡ −λr1 p1 as the observable describing the heat flow from the left reservoir into the chain and Φn ≡ λrn pn as the heat flow from the chain into the right reservoir. As in the introduction we define the energy Hi of the ith oscillators by Eq.(2), for i ≤ 2 ≤ n − 1, and H1
=
Hn
=
p21 1 + U (1) (q1 ) + U (2) (q1 − q2 ) , 2 2 p2n 1 + U (1) (qn ) + U (2) (qn−1 − qn ) . 2 2
With the heat flows Φi , i = 1, · · · , n, defined as in Eq. (3) we have LHi = Φi−1 − Φi ,
i = 1, · · · , n .
and we define the entropy productions σi , i = 0, · · · , n by 1 1 σi = − Φi i = 0, · · · , n . T1 Tn We now provide several identities involving the generator of the dynamics and the entropy production, which will play a crucial role in our subsequent analysis. Lemma 2.3 Let the function Ri , i = 0, · · · , n be given by
n i 1 r12 1 rn2 + . Hi (p, q) + Hi (p, q) + Ri = T1 2 Tn 2 k=1
Then we have
(10)
k=i+1
σi = γrT −1 r − Tr(γI) + LRi .
(11)
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Proof. This is a straightforward computation.
Remark 2.4 This shows that, up to a derivative, all the entropy productions are equal to the quantity rT −1 r − TrγI which is independent of i and involves only the r-variables. Let LT be the formal adjoint of the operator L given by Eq. (5) LT = γ (∇r T ∇r + ∇r r) − (Λp∇r − rΛ∇p ) − (p∇q − (∇q V (q))∇p ) ,
(12)
and let J be the time reversal operator which changes the sign of the momenta of all particles, Jf (p, q, r) = f (−p, q, r). The following identities can be regarded as operator identities on C ∞ functions. That the left and right side of Eq. (14) actually generate semigroups for some interval of α is a non trivial result which we will discuss in Section 2.3. Lemma 2.5 We have the operator identities eRi JLT Je−Ri = L − σi ,
(13)
e−Ri J(LT − ασi )JeRi = L − (1 − α)σi .
(14)
and also for any constant α
Proof. We write the generator L as L = L0 + L1 with L0 L1
= =
γ (∇r T ∇r − r∇r ) (Λp∇r − rΛ∇p ) + (p∇q − (∇q V (q))∇p ) .
(15) (16)
Since L1 is a first order differential operator we have e−Ri L1 eRi = L1 + (L1 Ri ) = L1 + σi . Using that ∇r Ri = T −1 r we obtain e−Ri L0 eRi
This gives
=
e−Ri γ(∇r − T −1 r)T ∇r eRi
=
γ∇r T (∇r + T −1 r) = LT0 .
e−Ri LeRi = LT0 + L1 + σi = JLT J + σi ,
which is Eq. (13). Since Jσi J = −σi , Eq. (14) follows immediately from Eq. (13). Remark 2.6 In the equilibrium situation, i.e., for T1 = Tn = T , Eq. (14) is eG/T JLT Je−G/T = L ,
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which is simply detailed balance. Eq. (14) can be interpreted in path space in the following manner [18]: Let Π denote the time-reversal in path space on the time interval [0, t]: Π(p(s), q(s), r(s)) = (−p(t − s), q(t − s), r(t − s)) and let dP denote the measure on C([0, t], X) induced by x(t). Then Eq. (13) implies that Rt dP ◦ Π = eRi (x(t))−Ri (x(0))− 0 σi (x(s)) ds . dP
This formula exhibits the fact that the lack of microscopic reversibility is intimately related to the entropy production. We now turn to the study of the large deviations. As shown in [26] the Markov t process x(t) is ergodic. In order to study the large deviations of t−1 0 σi (x(s))ds we consider the moment generating functionals Rt Γix (t, α) = Ex e−α 0 σi (x(s)) ds . Formally the Feynman-Kac formula gives Γix (t, α) = et(L−ασi ) 1(x), but since σi is not bounded, nor even relatively bounded by L, it is not obvious that Γix (t, α) exists for α = 0. Our goal is to prove that Γix (t, α) exists and that the limit 1 e(α) ≡ lim − log Γix (t, α) t→∞ t
(17)
exists and is finite in a neighborhood of the interval [0, 1], and is independent of i and of the initial condition x. The technical difficulty in proving the existence of the limit (17) lies in the fact that the functions σi are unbounded. Standard large deviation theorems for Markov processes (see e.g. [3, 4, 29]) are proven usually under strong ergodic properties for bounded functions and are not directly applicable. Large deviations for unbounded functions are considered in [1] for discrete time countable state space Markov chains under conditions which amount in our case to σ = o(G). In our case this is clearly not satisfied since, in general σ is not bounded by G. But the σi are very special observables, in particular they are intimately linked with the dynamics as shown by the identities Eqs.(13) and (14). The next lemma displays another identity which will be important in our analysis. Lemma 2.7 We have the identity
where
L − ασi = eαRi Lα e−αRi ,
(18)
˜ α − (α − α2 )γrT −1 r − αTr(γI) Lα = L
(19)
˜ α = L + 2αγr∇r . L
(20)
and
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Proof. As in Lemma 2.5 we write the generator L as L = L0 + L1 , see Eqs.(16) and (15). Since L1 is a first order differential operator we have e−αRi L1 eαRi = L1 + α(L1 Ri ) = L1 + ασi .
(21)
Using that ∇r Ri = T −1 r is independent of i we find that
e−αRi L0 eαRi = γ (∇r + αT −1 r)T (∇r + αT −1 r) − r(∇r + αT −1 r) =
L0 + αγ(r∇r + ∇r r) + (α2 − α)γrT −1 r
=
L0 + 2αγr∇r + (α2 − α)γrT −1 r + αTrγI .
Combining Eqs. (21) and (22) gives the desired result.
(22)
Remark 2.8 The identity (18) shows that all operators L−ασi are conjugate to the same operator Lα . This will be the key element to prove that e(α) is independent of i. Furthermore it can be seen from Eqs. (19) and (20) that Lα has the form of L plus a perturbation which is a quadratic form in r and ∇r . Such a perturbation ˜ α has very much the same is indeed nicer than ασi . Also it should be noted that L form as the operator L: they differ only by the coefficient in front of the term r∇r . This fact will allow us to use several results on L obtained in [26].
2.3
Liapunov Function for Feynman-Kac Semigroups
At this point we begin the study of Lα as the generator of a semigroup. Proposition 2.9 If θ and α satisfy the condition −α < θTi < 1 − α ,
(23)
then there exists a constant C = C(α, θ) such that etLα eθG (x) ≤ eCt eθG (x). ˜ α , defined in Eq. (19), for all α ∈ R, is the generator Proof. We note first that L of a Markov process which we denote as x ˜(t). Indeed we have that ˜ α G(x) = Tr(γT ) − (1 + 2α)r2 ≤ C1 + C2 G(x) L Since G grows at infinity, G is a Liapunov function for x ˜(t) and a standard argument [16] shows that the Markov process x ˜(t) is non-explosive. Furthermore we have the bound Lα exp θG(x) = = exp θG(x)γ Tr(θT + αI) + r(θ2 T − (1 − 2α)θ − α(1 − α)T −1 )r ≤ C exp θG(x) , (24) provided α and Ti , i = 1, n satisfy the inequality θ2 Ti − (1 − 2α)θ − α(1 − α)Ti−1 ≤ 0 ,
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or −α < θTi < 1 − α . We denote σR as the exit time from the set {G(x) < R}, i.e., σR = inf{t ≥ 0, G(˜ x(t)) ≥ R}. If the initial condition x satisfies G(x) = E < R, we denote by ˜(t) x ˜R (t) the process which is stopped when it exits {G(x) < R}, i.e., x˜R (t) = x ˜R (t) = x(σR ) for t ≥ σR . Finally we set σR (t) = min{σR , t}. for t < σR and x By Eq. (24), the function W (t, x) = e−Ct eθG(x) satisfies the inequality (∂t + Lα )W (t, x) ≤ 0 and applying Ito’s formula with stopping time to the function W (t, x) we obtain R σR (t) ((α−α2 )γ r˜T −1 r˜−αTr(γI)) ds eθG(˜x(σR (t))) e−CσR (t) − eθG(x) ≤ 0 , Ex e− 0 and thus R σR (t) ((α−α2 )γ r˜T −1 r˜−αTr(γI)) ds eθG(˜x(σR (t))) ≤ eCt eθG(x) . Ex e − 0 x(t)) almost surely Since the Markov process x ˜(t) is non-explosive G(˜ xR (t)) → G(˜ as R → ∞, so by the Fatou lemma we have etLα eθG (x) ≤ eCt eθG (x) . This concludes the proof of Lemma 2.9.
˜ α is the The next three theorems are all consequences of the fact that L generator of a Markov process which is similar to the process generated by L: ˜ α differ only by the coefficient in front of the r∇r term. Therefore Indeed L and L repeating the proofs of [26] we obtain Theorem 2.10 The semigroup etLα has a smooth kernel qα (t, x, y) which belongs to C ∞ ((0, ∞) × X × X). ˜ α satisfies the same H¨ormander-type condition that L proven Proof. The operator L in [26], Proposition 4.1.The result follows then from [12] or [23]. Theorem 2.11 The semigroup etLα is positivity improving for all t > 0. ˜
Proof. The semigroup etLα is shown to be irreducible exactly as etL , see [7, 26] using explicit computation and the Support Theorem of [28]. The statement follows then from the Feynman-Kac formula. As is apparent from the form of Lα we will need estimates on the observable r2 in the sequel. Such estimates were also crucial in [26] for the construction of a Liapunov function.
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Theorem 2.12 Let 0 ≤ α < 1/k2 and let tE = E 1/k2 −1/2 . There exists a set of paths S(x, E, tE ) ⊂ {f ∈ C([0, tE ], X) ; f (0) = x, G(x) = E} , and constants E0 < ∞ and A, B, C > 0 such that for E > E0 P {˜ x ∈ S(x, E, tE )} ≥ 1 − Ae−BE and
0
tE
r˜2 (s) ds ≥ CE 3/k2 −1/2 ,
2α+1/2−1/k2
,
x ˜ ∈ S(x, E, tE ) .
if
(25)
Proof. The proof is exactly as in [26]. One first sets T1 = Tn = 0 in the equations of motion and then, by a scaling argument, Theorem 3.3 of [26], one shows that the deterministic trajectory satisfies the estimate (25). Then one shows, see Proposition 3.7 and Corollary 3.8 of [26], that the overwhelming majority of the random trajectories follows very closely the deterministic ones. We refer the reader to [26] for further details. Remark 2.13 For large energy E, paths satisfying the bound (25) have a very high probability. From Eq. (25) we obtain that, on a time interval of order 1,
t
0
r˜2 (s) ≥ CE 2/k2 ,
for an overwhelming majority of the paths. Theorem 2.14 Let t > 0 be fixed and suppose that α and θ satisfy the condition Eq.(23). There exist a constant E0 and functions κ(E) and b(E) such that for E > E0 etLα eθG (x) ≤ κ(E)eθG(x) + b(E)1{G≤E} (x) . (26) Moreover there exist constants c1 and c2 such that κ(E) ≤ c1 e−c2 E
2/k2
.
Proof. By Proposition 2.9 the function etLα eθG (x) is bounded on any compact set. Therefore to show (26) it suffices to show that Rt −1 Ex e− 0 (α(1−α)γ r˜T r˜−αTr(γI)) ds eθ(G(˜x(t))−G(˜x)) ≤ κ(E) . sup {x : G(x)>E}
Using Ito’s formula we have G(˜ x(t)) − G(x) =
0
t
t γ(Tr(T ) − r˜ ) ds + 2γT r˜dω(s) , 2
0
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and thus we obtain Rt −1 Ex e− 0 (α(1−α)γ r˜T r˜−αTr(γI)) ds eθ(G(x(t))−G(x)) Rt Rt √ −1 = etγTr(θT +αI) Ex e− 0 r˜(α(1−α)γT −γθ(1−2α))r˜ ds e 0 θ 2γT r˜ dω . (27) Using the H¨older’s inequality we find that the expectation on the r.h.s of Eq. (27) can be estimated by 1/q Rt −1 qpθ2 R t √ 2 E e−q 0 r˜(α(1−α)γT −γθ(1−2α))r˜ ds e 2 0 ( 2γT r˜) ds x
p2 θ 2 ×Ex e− 2
Rt √ 0
( 2γT r˜)2 ds p
e
Rt 0
θ(2γT )1/2 r˜ dω)
1/p
1/q Rt −1 2 = Ex e−qγ 0 r˜(α(1−α)T −θ(1−2α)+pθ T )r˜ ds .
(28)
where we have used that the second factor is the expectation of a martingale with expectation 1. If θ and α satisfy the condition (23), then, by choosing p sufficiently close to 1, the quadratic form in the right side of Eq. (28) is negative definite. Using Theorem 2.12 as in Theorem 3.11 of [26] we obtain Rt −1 sup Ex e− 0 (α(1−α)˜rT r˜−αTr(γI)) ds eθ(G(x(t))−G(x)) x∈U C
≤ eγTr(θT +αI) e−CE ≤ c1 e−c2 E
2/k2
2/k2 γTr(α(1−α)T −1 −(1−2α)θ+pθ 2 T )
.
and this concludes the proof of Theorem 2.14. As in Theorem 2.1 we obtain
Theorem 2.15 If α and θ satisfy the condition Eq.(23), then etLα extends to a strongly continuous quasi-bounded semigroup on Hp,θ for 1 ≤ p < ∞ and on 0 0 H∞,θ . Moreover etLα is compact on Hp,θ , 1 < p ≤ ∞ and on H∞,θ . Proof. The proof is a repetition of the proof of Theorem 2.1 and is left to the reader. As a consequence of Theorem 2.15 and of the theory of semigroup of positive operators [11] we obtain Theorem 2.16 If
then
α∈ −
Tmin Tmin , 1+ Tmax − Tmin Tmax − Tmin
1 e(α) = lim − log Γix (t, α) t→∞ t exists, is finite and independent both of i and x.
,
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0 Proof. By Theorem 2.15, etLα generates a strongly continuous semigroup on H∞,θ if −α < θTi < 1 − α . (29)
If α ≤ 0, this implies that |α| < θTmin < θTmax < 1 + |α| and so the set of θ we can choose is non-empty provided α>−
Tmin . Tmax − Tmin
If 0 < α < 1, we can always find θ such that (29) is satisfied. Finally if α > 1 then (29) implies that that Tmin . α<1+ Tmax − Tmin 0 By the definition of Ri , Eq. (10), e−αRi ∈ H∞,θ since −α + θTi < 0. Using now i Lemma 2.7, we see that Γx (t, α) exists and is given by
Γix (t, α) = et(L−ασ) 1(x) = eαRi etLα e−αRi (x) . From Theorem 2.11 the semigroup etLα is an irreducible semigroup of compact 0 operators on the Banach space H∞,θ . From the cyclicity properties of the spectrum of irreducible operators and from the compactness it follows (see [11], Chapter C-III) that there is exactly one eigenvalue e−te(α) with maximal modulus and this eigenvalue is real and simple. The corresponding eigenfunction fα is strictly positive and we denote as Pα the one-dimensional projection on the eigenspace spanned by fα . In particular if g ≥ 0, then Pα g(x) > 0. From compactness it follows that the complementary projection (1 − Pα ) satisfies the bound tLα (30) e (1 − Pα )f (x) ≤ Ce−td(α) f ∞,θ eθG(x) . for some constants C > 0 and d(α) > e(α) and for all t > 0. From Lemma 2.7 and Eq. (30) we obtain, for all x ∈ X, that 1 lim − log Γix (t, α) t 1 1 = lim − log et(L−ασi ) 1(x) = lim − log eαRi etLα e−αRi (x) t→∞ t→∞ t t 1 = lim − αRi (x) + e(α) t→∞ t 1 + lim − log Pα e−αRi (x) + ete(α) etLα (1 − Pα )e−αRi (x) t→∞ t = e(α) .
t→∞
This concludes the proof of Theorem 2.16.
(31)
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Using now the identity (14) we can prove the symmetry of e(α). Theorem 1.1 is then an immediate consequence of the following result. Theorem 2.17 If α∈
−
Tmin Tmin , 1+ Tmax − Tmin Tmax − Tmin
,
(32)
then e(α) = e(1 − α) . Proof. If α is in the interval (32) and −α < θTi < 1 − α then etLα is a strongly 0 continuous compact semigroup on H∞,θ . By Lemma 2.7 et(L−ασi ) = eαRi etLα e−αRi 0 is also a strongly continuous compact semigroup on the Banach space H∞,θ,α = −θG+αRi θG+αRi {f ; |f |e ∈ C0 (x)} with the norm f ∞,θ,α = sup |f |e . The dual semigroup (et(L−ασi ) )∗ is a compact semigroup on the Banach space 0 0 (of measures) (H∞,θ,α )∗ . By Theorem 2.11 (et(L−ασi ) )∗ maps (H∞,θ,α )∗ into meat(L−ασi ) ∗ ) acts as sures with smooth densities and on densities (e
(et(L−ασi ) )∗ (ρ(x)dx) = (et(L
T
−ασi )
ρ(x))dx .
By Lemma 2.5 we have e−Ri et(L−(1−α)σi ) 1(x) = Jet(L
T
−ασi )
−Ri
Je−R (x) .
(33) 0 (H∞,θ,α )∗ .
Since −α < θTi < 1 − α, e is a density of a measure in Since t(L−ασi ) ∗ ) is compact and irreducible with spectral radius e(α) we obtain using (e Eq. (33) e(α)
= = =
T 1 lim − log J(et(L −ασi ) Je−Ri )dx t
1 −Ri t(L−(1−α)σi ) sup lim − log e 1 dx , fe t→∞ t f ≤eθG+αRi
t→∞
e(1 − α) .
In the last equality we have used Theorem 2.16 and the fact that f e−Ri is a finite measure. This concludes the proof of Theorem 2.17. We finally obtain the Gallavotti-Cohen fluctuation theorem Theorem 2.18 There is a neighborhood O of the interval [−σν , σν ] such that for A ⊂ O the fluctuations of σi in A satisfy the large deviation principle with a large deviation functional I(w) obeying I(w) − I(−w) = −w , i.e., the odd part of I is linear with slope −1/2.
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Proof. First we note that e(α) is a real analytic function since it is identified with an eigenvalue of a compact operator. A simple computation gives that d e(α) = σν . dα α=0 The function e(α) is analytic and convex. By the result of [7] it is not identically zero, and so the symmetry the symmetry e(α) = e(1 − α) implies that the set of d the values of dα e(α) is a neighborhood of [−σν , σν ]. The large deviation principle is a direct application of the G¨ artner-Ellis theorem, [4], Theorem 2.3.6. The large deviation functional is given by the Legendre transform of e(α) and so we have I(w)
=
sup {e(α) − αw} = sup {e(1 − α) − αw}
=
sup {e(β) − (1 − β)w} = I(−w) − w .
α
α
β
References [1] S. Balaji and S. P. Meyn, Multiplicative ergodicity and large deviations for an irreducible Markov chain, Stoch. Proc. Appl. 90, 123–144 (2000). [2] G.E. Crooks, Path-ensemble averages in systems driven far from equilibrium, Phys. Rev. E 61, 2361–2366 (2000). [3] J.-D.Deuschel and D.W. Stroock, Large deviations, Pure and Applied Mathematics 137, Boston: Academic Press, 1989. [4] A.Dembo and O. Zeitouni, Large deviations techniques and applications, Applications of Mathematics 38. New-York: Springer-Verlag 1998. [5] J.-P. Eckmann and M. Hairer, Non-equilibrium statistical mechanics of strongly anharmonic chains of oscillators, Commun. Math. Phys. 212, 105–164 (2000). [6] J.-P. Eckmann, C.-A. Pillet and L. Rey-Bellet, Non-equilibrium statistical mechanics of anharmonic chains coupled to two heat baths at different temperatures, Commun. Math. Phys. 201, 657–697 (1999). [7] J.-P. Eckmann, C.-A. Pillet and L. Rey-Bellet, Entropy production in nonlinear, thermally driven Hamiltonian systems, J. Stat. Phys. 95, 305–331 (1999). [8] D.J. Evans, E.G.D. Cohen and G.P. Morriss, Probability of second law violation in shearing steady flows. Phys. Rev. Lett. 71, 2401–2404 (1993).
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[9] G. Gallavotti and E.G.D. Cohen, Dynamical ensembles in stationary state, J. Stat. Phys. 80, 931–970 (1995). [10] G. Gentile, Large deviation rule for Anosov flows, Forum Math. 10 89–118 (1998). [11] G. Greiner, Spectral theory of positive semigroups on Banach lattices, In One-parameter semigroups of positive operators Lecture Notes in Mathematics 1184, Ed. R. Nagel, Berlin: Springer, 1986, pp 292–332. [12] L. H¨ormander, The Analysis of linear partial differential operators, Vol III, Berlin: Springer, 1985. [13] V. Jaksic and C.-A. Pillet, On entropy production in quantum statistical mechanics, Commun. Math. Phys. 217, 285–293 (2001). [14] V. Jaksic and C-A. Pillet, Non-equilibrium steady states of finite quantum systems coupled to thermal reservoirs, Preprint (2001). [15] C. Jarzynski, Hamiltonian derivation of a detailed fluctuation theorem, J. Statist. Phys. 98, 77–102 (2000). [16] R.Z. Has’minskii, Stochastic stability of differential equations, Alphen aan den Rijn—Germantown: Sijthoff and Noordhoff, 1980. [17] J. Kurchan, Fluctuation theorem for stochastic dynamics, J. Phys. A 31, 3719–3729 (1998). [18] C. Maes, The fluctuation theorem as a Gibbs property, J. Stat. Phys. 95, 367–392 (1999). [19] C. Maes, Statistical mechanics of entropy production: Gibbsian hypothesis and local fluctuations, Preprint (2001). [20] C. Maes, F. Redig and M. Verschuere, No current without heat, Preprint (2000) [21] S.P. Meyn and R.L. Tweedie, Markov Chains and Stochastic Stability. Communication and Control Engineering Series, London: Springer-Verlag London, 1993. [22] J.L. Lebowitz and H. Spohn, A Gallavotti-Cohen-type symmetry in the large deviation functional for stochastic dynamics, J. Stat. Phys. 95, 333– 365 (1999). [23] J. Norriss, Simplified Malliavin Calculus, In S´eminaire de probabilit´es XX, Lectures Note in Math. 1204, 0 Berlin: Springer, 1986, pp. 101–130.
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[24] C.-A. Pillet, Entropy production in classical and quantum systems, Markov Proc. Relat. Fields 7, 145–157, (2001). [25] L. Rey-Bellet and L.E. Thomas, Asymptotic behavior of thermal nonequilibrium steady states for a driven chain of anharmonic oscillators, Commun. Math. Phys. 215, 1–24 (2000). [26] L. Rey-Bellet and L.E. Thomas, Exponential convergence to non-equilibrium stationary states in classical statistical mechanics, To appear in Commun. Math. Phys. [27] D. Ruelle, Entropy production in quantum spin systems. Preprint (2000) [28] D.W. Stroock and S.R.S. Varadhan, On the support of diffusion processes with applications to the strong maximum principle. In Proc. 6-th Berkeley Symp. Math. Stat. Prob., Vol III, Berkeley: Univ. California Press, 1972, pp. 361–368. [29] L. Wu, Uniformly integrable operators and large deviations for Markov processes, J. Funct. Anal. 172, 301–376 (2000).
Luc Rey-Bellet and Lawrence E. Thomas Department of Mathematics University of Virginia Kerchof Hall Charlottesville, VA 22903 USA email: [email protected] email: [email protected] Communicated by Jean-Pierre Eckmann submitted 20/10/01, accepted 07/01/02
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Annales Henri Poincar´ e
Modified Wave Operators for the Hartree Equation with Data, Image and Convergence in the Same Space, II K. Nakanishi Abstract. We study modified wave operators for the Hartree equation with a longrange potential |x|−ν , extending the result in [12] to the whole range of the Dollard type 1/2 < ν < 1. We construct the modified wave operators in the whole space of (1 + |x|)−s L2 . We also have the image, strong continuity and strong asymptotic approximation in the same space. The lower bound s > 1 − ν/2 of the weight is sharp from the scaling argument. Those maps are homeomorphic onto open subsets, which implies in particular asymptotic completeness for small data.
1 Introduction In this paper, we continue the study in [12] on asymptotic behavior of solutions for the Hartree equation with a long-range potential |x|−ν : 2iu˙ − ∆u + V (u)u = 0, where
V (u) = KV (x) ∗ |u|2 ,
KV (x) = λ|x|−ν ,
(1.1)
(1.2)
u = u(t, x) : R1+n → C (n ∈ N) is the unknown function and λ ∈ R is a real constant. The main goal in [12] was to obtain results for the modified wave operators in the long range case that are as good as those for the ordinary wave operators in the short range case ν > 1, especially as to the domain, the range and the topology of convergence. In fact, such a result was obtained in [12] in the limiting case ν = 1, which is almost the same as in the short range case except the presence of the modification and the exclusion of the scaling critical case. Before that result, the modified wave operators were defined under much stronger assumptions on the data, while the range and the convergence were given in larger spaces or weaker senses than that for the data. Actually, it was rather recent [3, 4] even that those operators were obtained without any smallness assumption on the data and for ν < 1. However, the argument in [12] strongly depended on the fact that the phase modifier diverges slower than any positive power of t, which might have the readers wonder that the result in [12] for the long range was somewhat special for the borderline case ν = 1 only. For more detail of known results on the modified wave operators, see [3, 4] and the references therein.
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In this paper, we show that the same result as in [12] holds actually in the whole range 1/2 < ν ≤ 1 where the Dollard-type first-order modification suffices. Let U (t) = e−i∆t/2 denote the free propagator and F the Fourier transform. H s denotes the usual inhomogeneous Sobolev space based on L2 . The main result of this paper is the following. We do not consider the case ν = 1, which has been solved in [12]. Theorem 1.1 Let n ≥ 3, 1/2 < ν < 1, 1 − ν/2 < s < 1 and λ ∈ R. Then, for any ψ ∈ FH s , there exists a unique solution u of (1.1) satisfying U (−t)u(t) ∈ C(R; F H s ) and 1 t1−ν F −1 exp V (F ψ) F U (−t)u(t) → ψ (1.3) 2i ν −1 as t → ∞ in F H s . Thus we have the modified wave operator W defined by W : ψ → u(0).
(1.4)
W is a homeomorphism from F H s to an open subset of F H s in the strong topology. We have the same result for the negative time t → −∞. This result is the same as that in [12] except the extension of the range of ν to 1/2 < ν < 1 and the restrictions n ≥ 3 and s < 1. The exception of lower dimensions is related to the Sobolev embedding. Actually, the case n = 2 is required to be excluded only in one place of the estimates, and that restriction may be hopefully avoidable. However, the one dimensional case looks more different. That is because the homogeneous part H˙ s can not dominate any Lebesgue norm when s > n/2, which is always the case when n = 1, while we can choose s < n/2 when n ≥ 2. The restriction s < 1 is much more technical and hardly essential. It is required just because we estimate the H s norm mainly by the spatial difference and we consider only the first-order difference for the sake of simplicity. The basic strategy to construct the modified wave operators is almost the same as in the previous paper [12]; We transform the scattering problem to the initial value problem by the pseudo-conformal inversion, eliminate the diverging oscillation by using the prescribed asymptotic states, and solve the Cauchy problem of thereby modified equation coupled with the evolution equation of the potential term. The essential novelty is in the estimate for the phase terms, where we will see that the divergence in the phases can be cancelled by each other without losing any regularity or decay in time. Since direct calculations of Fourier transform are not helpful for the phase terms, we estimate them in the physical space by using some decomposition arguments in the frequency. The fractional derivative is not so convenient by the same reason, so we will employ the difference operator instead, which is easy to handle in the phase terms and also in the equations. Here we need only the first order of difference, because we can choose s < 1. Other advantages of the difference operator are that we can replace the commutator estimate in
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H s (which played the essential role in [12]) with the trivial chain rule for the difference, and that the difference operator effectively localizes the frequency so that the arguments are indeed free from the particular choice of s and we can obtain some uniform decay estimate for the higher frequency, which will play an important role to obtain the continuity and asymptotic completeness as in [12]. Now we briefly recall our strategy used in the previous paper [12]. We use the well-known transform of the pseudo-conformal inversion: u → u∗ = (it)−n/2 e|x|
2
/(2it)
u(1/t, x/t).
(1.5)
Then the equation (1.1) for u is transformed to the following equation for u∗ : 2iu˙ ∗ − ∆u∗ + |t|ν−2 V (u∗ )u∗ = 0,
(1.6)
and the asymptotic behavior of u as t → ±∞ can be described by that of u∗ as t → ±0, using the relation: U (−1/t)u∗ (1/t) = (2π)−n/2 F U (−t)u(t).
(1.7)
To eliminate the singularity at t = 0 of (1.6), we define the modified field w by u∗ (t) = U (t)eiΦ w(t), where Φ(φ) = V (φ)
|t|ν−2 t . 2(ν − 1)
(1.8)
(1.9)
It is easy to check that eiΦ(φ) φ is the general solution to the ODE given by dropping the non-singular term ∆u∗ from (1.6). It is also easy to see that the second derivative in ∆ will create another singular term if ν ≤ 1/2, so that the above first-order approximation can be valid only when ν > 1/2. The equation for w is the following. 2iw˙ + |t|ν−2 e−iΦ {U (−t)V (u∗ )U (t) − V (φ)}eiΦ w = 0,
(1.10)
where U (−t)V (u∗ )U (t) denotes the operator defined by U (−t)V (u∗ )U (t)ϕ := U (−t)(V (u∗ )U (t)ϕ).
(1.11)
As was mentioned in [12], the advantage of our choice of the modification in (1.8) is that we do not encounter any derivative loss as we do if we choose other modifications such as u∗ (t) = eiΦ w(t). But here we have to square up to the disadvantage that the phase factors remain without exact cancellations, which was disposed with in [12] by relying totally on the fact that the divergence of the phase was only log t. Thus our main problem in this paper is to derive the cancellation estimate at t = 0 of those phase factors, without losing any (extra) regularity or any decay.
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The rest of this paper is organized as follows. First in the next Section 2, we derive the most important estimates with respect to the control of the phase terms. They reduce the necessary estimates on w and V to those for the equation without the phase modifier. In Section 3, we derive several estimates that will be used after that phase elimination. Using those estimates, we can derive some bounds on the difference energy of the modified field in Section 4, and some bounds as well as decay estimates at t = +0 for the potential term in Section 5. Combining those bounds and decay estimates, we can solve the Cauchy problem for the modified equation by the iteration argument in Section 6. These argument can yield some uniform decay of the higher frequency in terms of energy. This uniform estimate effectively reduces any convergence problems in H s to those in L2 . Then we can easily show the continuity properties of the modified wave operator in Section 7, and the openness of it in Section 8. We conclude this introduction by giving some notations used throughout this s denote the inhomogeneous Sobolev space, the homogepaper. H s , H˙ s and B˙ p,q neous Sobolev space and the homogeneous Besov space, respectively (see [1] for the definition). We use the following abbreviation for the norm for the potential term: s+n/2 (1.12) B s := B˙ 2,1 . In general, elements in this space can not be uniquely determined as usual distributions when s > 0. However, we will use this space only for s < 1 and then the elements are uniquely determined up to addition of constants. The readers need not care about this ambiguity, since it will be clear that addition of constants does not matter in each estimate involving this space with s > 0. We also use the following dual space: s−n/2 (1.13) B∗s := B˙ 2,∞ . We will use the following notation to express polynomial bounds. a[b,c] := max(ab , ac ).
(1.14)
For any spatial function u, we denote by ϕI ∗ u the Littlewood-Paley projection on Rn to the frequency of the size |ξ| ∼ I; ϕ= ϕI ∗ ϕ, supp F (ϕI ∗ u) ⊂ {|ξ| ∼ I}. (1.15) I=2j ,j∈Z
δ h ϕ denotes the spatial difference δ h ϕ(x) := ϕ(x + h) − ϕ(x),
(1.16)
with a parameter h ∈ R . For any sequence a and any function F , we denote n
δk a := ak − ak−1 , F (ak+ , ak+ ) :=
F (ak+ ) := F (ak ) + F (ak+1 ), F (ai , aj ), etc.,
i,j=k,k+1
(1.17)
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We will frequently use the above notation for k = 0, 1, even if ak is defined only for k = 0, 1. In particular, we denote δ 1 a = a1 − a0 ,
F (a0+ ) = F (a0 ) + F (a1 ),
δ1 F (a∗ ) := F (a1 ) − F (a0 ), δ1 F (a∗ , b∗ ) := F (a1 , b1 ) − F (a0 , b0 ), etc.
(1.18)
2 Phase estimates In this section, we derive the most important estimates on the phase terms. We will use the following identity: (2.1) S(v, w) := (U (t)v)U (t)w = U (−t)F −1 v(x + tξ)w(x)e−ixξ dx, which is easy to verify by using the explicit formula for U (t): 2 −n/2 e|x−y| /(2it) ϕ(y)dy. U (t)ϕ = ct
(2.2)
The effect of the phase factor is skimmed into S 0 (Φ; v, w) = S 0 (v, w) := S(eiΦ v, eiΦ w) − S(v, w) −1 eiΦ(x+tξ)−iΦ(x) − 1 v(x + tξ)w(x)e−ixξ dx. = U (−t)F
(2.3)
This identity suggests that we may expect the phase factors to cancel each other in the order roughly O(tξ∇ΦL∞ ). However, it is not always possible to estimate the phase term just in L∞ , in particular, when v or w has frequency less than |ξ|. Then the following Lp estimate of the phase factor plays an essential role. We denote Φ (x, tξ) := Φ(x + tξ) − Φ(x),
Ψ(Φ; x, tξ) := eiΦ (x,tξ) − 1.
(2.4)
The parameter α below will be fixed as α = s − ν/2 in the later applications, though we will ignore it for a while, since it has nothing essential to do with the phase estimates. Lemma 2.1 Let n ∈ N, m ≥ 1, 0 ≤ α < 1 and 0 ≤ β ≤ n/2. Assume that (α − 1)m + β < θ < α + n/2,
(2.5)
−α ≤ θ ≤ 1 − α.
(2.6)
Then we have for |ξ| ∼ N , ϕI ∗ Ψ(Φ)Ln/β |tN 2 |θ N −β max (N/I)β−θ+dα (tα ΦB 2α )d . x
1≤d≤m
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(α − 1)m + β + α < θ + σ < α + n/2.
(2.7)
Instead of (2.5), assume that
Then we have for |ξ| ∼ N , ϕI ∗ (Ψ(Φ1 ) − Ψ(Φ0 ))Ln/β |tN 2 |θ N σ−β x
max (N/I)β−θ+dα
1≤d≤m+1
× (tα Φ1 − Φ0 B 2α−σ )
(2.8)
× (t Φ0 B 2α + t Φ1 B 2α ) α
α
d−1
.
Proof. In this proof, we define the norm in Rd for d ∈ N by |(x1 , . . . , xd )| := |x1 | + · · · + |xd |.
(2.9)
Denote Ψi := Ψ(Φi ) and Ψ := Ψ(Φ). By convexity, it suffices to prove the estimate in the case m ∈ N. Fix ξ ∈ Rn as |ξ| ∼ N . We estimate Ψ by taking the m-th order difference. By the assumptions (2.5), α < 1 and 0 ≤ β ≤ n/2, we can find β ∈ (β, n/2) such that (α − 1)m + β < θ < α + β .
(2.10)
and in case we have (2.7), then we can find β ∈ (β, n/2) such that (α − 1)m + β + α < θ + σ < α + β .
(2.11)
It is easy to find functions χk ∈ S(Rn ), k = 1, . . . , n, satisfying ϕ1 =
n
χk ∗ ϕ1
(2.12)
k=1
such that F δ h = eihξ − 1 does not vanish on the support of F χk when h is the k-th unit vector of Rn . Then δ h is invertible when restricted on the Fourier support of ϕ1 so that ϕ1 ∗ u can be effectively dominated by (δ h )m u with finitely many h of size 1. By dilation, we have the same conclusion for any I. See [1, Lemma 6.2.6] for more details. Thus we can estimate ϕI ∗ uLn/β x
sup (δ h )m uLn/β ,
|h|∼1/I
(2.13)
x
for any function u. By the chain rule for the difference operator, we have (δ h )m ΨLn/β
m
d=1 a∈Nd ,|a|=m
[(δ h )a1 Φ ] · · · [(δ h )ad Φ ]Ln/β , x
(2.14)
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(δ h )m δ1 ΨLn/β m [(δ h )a1 Φ0+ ] · · · [(δ h )ad−1 Φ0+ ][(δ h )ad δ1 Φ ]Ln/β +
m+1
(2.15)
x
d=1 a∈Nd ,|a|=m
[(δ h )a1 Φ0+ ] · · · [(δ h )ad−1 Φ0+ ][δ1 Φ ]Ln/β ,
(2.16)
x
d=2 a∈Nd−1 ,|a|=m
where we denote, for simplicity, [ϕ] :=
m
|ϕ(x + kh)|.
(2.17)
k=0
Let µk ∈ [0, 1],
τk ∈ (0, ak ),
bk ∈ [0, n/2],
τk + µk = bk + γk ,
(2.18)
for k = 1, . . . , d, with the exceptional rule that τd = ad = 0 for (2.16). By convexity, we can find such (µk , τk , bk ) when |µ|, |τ | and |b| are given from the region 0 ≤ |µ| ≤ d, 0 < |τ | < m, 0 ≤ |b| ≤ nd/2, |µ| + |τ | = |b| + |γ|.
(2.19)
We set |b| = β ,
|µ| = θ + αd,
|τ | = β + |γ| − θ − αd,
(2.20)
with γ1 = · · · = γd = 2α
(2.21)
for (2.14), and γ1 = · · · = γd−1 = 2α,
γd = 2α − σ
(2.22)
for (2.15) and (2.16). Then the assumptions (2.6), 0 < β < n/2, (2.10), and (2.11) imply that (|µ|, |τ |, |b|) is in the region (2.19) so that we can choose (µk , τk , bk ) for k = 1, . . . , d satisfying (2.18). Using the difference norm of the Besov spaces, we can dominate the summand in (2.14) by d k=1
|h|τk Φ B˙ τk
n/bk ,∞
d
|tN |µk I −τk ΦB˙ τk +µk
n/bk ,∞
k=1
|tN ||µ| I −|τ | ΦdB 2α = |tN |θ+αd I −β
−αd+θ
= |tN 2 |θ (N/I)αd−θ I −β (|t|α ΦB 2α )d .
ΦdB 2α
(2.23)
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We can bound the summand in (2.15) by d−1
|h|τd δ1 Φ B˙ τd
n/bd ,∞
|h|τk Φ0+ B˙ τk
n/bk ,∞
k=1 d−1
|tN |µd I −τd δ1 ΦB˙ τd +µd
n/bd ,∞
|tN |µk I −τk Φ0+ B˙ τk +µk
(2.24)
n/bk ,∞
k=1
|tN ||µ| I −|τ | δ1 ΦB 2α−σ Φ0+ d−1 B 2α = |tN 2 |θ (N/I)αd−θ I −β
+σ
(|t|α δ1 ΦB 2α−σ )(|t|α Φ0+ B 2α )d−1 .
The summand in (2.16) is dominated by δ1 Φ Ln/bd
d−1
|h|τk Φ0+ B˙ τk
n/bk ,∞
k=1 d−1
|tN |µd δ1 ΦB˙ µd
n/bd ,∞
|tN |µk I −τk Φ0+ B˙ τk +µk
(2.25)
n/bk ,∞
k=1
|tN ||µ| I −|τ | δ1 ΦB 2α−σ Φ0+ d−1 B 2α = |tN 2 |θ (N/I)αd−θ I −β
+σ
(|t|α δ1 ΦB 2α−σ )(|t|α Φ0+ B 2α )d−1 .
Thus we obtain ϕI ∗ ΨLn/β I β
−β
x
Iβ
−β
ϕI ∗ ΨLn/β m
x
|tN 2 |θ (N/I)αd−θ I −β (|t|α ΦB 2α )d
d=1 2 θ −β
|tN | N ϕI ∗ (δ1 Ψ)Ln/β I β x
−β
(2.26)
max (N/I)β−θ+dα (|t|α ΦB 2α )d
1≤d≤m
ϕI ∗ (δ1 Ψ)Ln/β x
2 θ
|tN | N
−β+σ
max (N/I)β−θ−σ+dα
1≤d≤m+1
(2.27)
× (|t|α δ1 ΦB 2α−σ )(|t|α Φ0+ B 2α )d−1 Now we proceed to the main estimates on the bilinear operator S 0 . We also need to estimate the difference of the phase term: S δ (Φ0 , Φ1 ; v, w) := S 0 (Φ1 ; v, w) − S 0 (Φ0 ; v, w).
(2.28)
When estimating the evolution of the potential, we need to estimate the following variant which has an additional decay in time: S ψ (v, w) := S 0 (ψv, w) − S 0 (v, ψw).
(2.29)
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We need also to estimate the effect of phase change for this operator: S ψ,δ (Φ0 , Φ1 ; v, w) := S ψ (Φ1 ; v, w) − S ψ (Φ0 ; v, w).
(2.30)
The following are the main estimates in this paper. Lemma 2.2 Let 0 ≤ θ ≤ α < 1/2, β < n/2, γ < n/2, 0 < β + γ < n/2. (i) Assume (α − 1)n/2 + max(β, γ) < θ.
(2.31)
S 0 (Φ; v, w)B˙ β+γ−2θ−n/2 |t|θ d(Φ)vH˙ β wH˙ γ ,
(2.32)
Then we have 2,1
where we denoted d(Φ) := (|t|α ΦB 2α )m + 1,
(2.33)
where m > n/2 is sufficiently large depending on α, β, γ, θ. (ii) Let σ ∈ [0, 1] satisfy (α − 1)n/2 + max(β, γ) + α < θ + σ < α + β + γ, α ≤ θ + σ.
(2.34) (2.35)
Then we have S δ (Φ0 , Φ1 ; v, w)B˙ β+γ−2θ−σ−n/2 2,1
|t|θ (d(Φ0 ) + d(Φ1 ))|t|α Φ1 − Φ0 B 2α−σ vH˙ β wH˙ γ .
(2.36)
(iii) Assume (2.31) and let θ ∈ (0, 1) and σ satisfy 0 < θ − σ < β + γ.
(2.37)
Then we have S ψ (Φ; v, w)B˙ β+γ−2(θ+θ )+σ −n/2 2,1
θ+θ
|t|
d(Φ)ψB˙ n/2+σ vH˙ β wH˙ γ .
(2.38)
2,∞
(iv) Assume (2.34), (2.35), (2.37), and θ + σ + θ − σ < α + β + γ.
(2.39)
Then we have S ψ,δ (Φ0 , Φ1 ; v, w)B˙ β+γ−2(θ+θ )+σ −σ−n/2 2,1
θ+θ
|t| (d(Φ0 ) + d(Φ1 ))|t|α Φ1 − Φ0 B 2α−σ × ψB˙ n/2+σ vH˙ β wH˙ γ . 2,∞
(2.40)
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Since we have to use this lemma with β = γ = s, the assumption β + γ < n/2 completely exclude possible choice of s when n ≤ 2 since s > 1 − ν/2 > 1/2. When n ≥ 3, we can choose s > 1 − ν/2 < 3/4 satisfying 2s < n/2. The upper bounds in (2.34) and (2.37) can be identified with (2.39) by regarding unused parameters as 0. Proof. Denote Ψ0 := Ψ, Ψδ := Ψ(Φ1 ) − Ψ(Φ0 ), (2.41) Ψψ := Ψ0 ψ , Ψψ,δ := Ψδ ψ , where ψ (x, tξ) := ψ(x + tξ) − ψ(x) and Ψ(Φ) is defined in (2.4). In this proof, we sometimes use the superscript (ψ, 0) instead of ψ. By (2.1), we have (2.42) S ∗ (v, w) = U (−t)F −1 Ψ∗ v(x + tξ)w(x)e−ixξ dx, for ∗ = 0, δ, ψ and (ψ, δ). First we use the Littlewood-Paley decomposition for x to localize every function in the frequency. Let I, J, K, M, N > 0 be dyadic parameters ∈ {2j |j ∈ Z}. For brevity, we denote vJ := ϕJ ∗ v, wK := ϕK ∗ w, etc. Then we have S ∗ (v, w) =
U (−t)ϕN ∗ F −1
Ψ∗I (x, tξ)vJ (x + tξ)w K (x)e−ixξ dx,
(2.43)
I,J,K,N :dyadic
for ∗ = 0, δ. If the operator contains ψ, we have to decompose ψ also. Then we have S ψ,∗ (v, w) = I,J,K,N,M:dyadic
U (−t)ϕN ∗ F
−1
(x, tξ)vJ (x + tξ)w K (x)e−ixξ dx, Ψ∗I ψM
(2.44)
for ∗ = 0, δ. In the former case, we denote Ψ∗I,M := Ψ∗I and let M = 0. In the latter case, we denote Ψ∗I,M := Ψ∗I ψM . The estimate for the summand with the appropriate weights in the dyadic parameters will imply the boundedness of S 0 as S 0 (v, w)B˙ β+γ−2θ−n/2 |t|θ d(Φ)vB˙ β wB˙ γ . 2,∞
2,1
2,1
(2.45)
Then the desired estimate will follow from this via the bilinear real interpolation1 . Thus we need only to estimate the L2ξ norm of R∗ := ϕ˜N (ξ) Ψ∗I,M (x, tξ)vJ (x + tξ)w K (x)e−ixξ dx, (2.46) for each I, J, K, M, N , at least when ∗ = 0 or δ. 1 When a bilinear operator is bounded from X × Y to Z i j i+j for (i, j) = (0, 0), (0, 1), (1, 0), then it is also bounded from (X0 , X1 )θ0 ,r0 × (Y0 , Y1 )θ1 ,r1 to (Z0 , Z1 )θ0 +θ1 ,r for θ0 , θ1 , θ ∈ (0, 1) with θ = θ0 + θ1 and r0 , r1 , r ∈ [1, ∞] with 1/r = 1/r0 + 1/r1 . See [1, 3.13.5(b)].
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Also in the trilinear case S ψ,∗ , the original estimate will follow from that for via the bilinear real interpolation as (ψ, v, w) → S ψ,∗ (v, w), B˙ ∗ × B˙ ∗ × B˙ ∗ → B˙ ∗ 2,1
2,1
2,1
2,∞
⇒
(2.47)
∗ ∗ ∗ ∗ B˙ 2,∞ × B˙ 2,2 × B˙ 2,1 → B˙ 2,2
⇒ ∗ ∗ ∗ ∗ B˙ 2,∞ × B˙ 2,2 × B˙ 2,2 → B˙ 2,1 .
Therefore it suffices to estimate the dyadic pieces R∗ in any case. We have to employ different arguments depending on the frequency size of each function. In the following, we always assume that ξ has size N , i.e., |ξ| ∼ N . Let σ = θ = M = 0 when considering R0 or Rδ . Let σ = 0 when considering R0 or Rψ . Case I: J ∼ K N This is the easiest case since the vw part is bounded in L1 . Let b = max(θ − α + σ, 0). If b > 0, then we have δ1 ΨLn/b δ1 Φ Ln/b (tN )θ+α δ1 ΦB˙ θ+α x
x
2 θ
|tN | N
n/b,1
α−θ
(t δ1 ΦB 2α−σ ), α
(2.48)
where we used the conditions 0 ≤ θ + α ≤ 1 and b ≤ n/2. Similarly, we have ΨLn/b |tN 2 |θ N α−θ (tα ΦB 2α ), x
(2.49)
when σ = 0. If b = 0, then we have, ΨL∞ Φ L∞ x x θ/α
2 θ
θ/α |tN |2α ΦB˙ 2α ∞,1
|tN | (|t| ΦB 2α ) α
θ/α
(2.50)
,
where we need 0 ≤ θ ≤ α. Although we have a similar estimate for δ1 Ψ, we avoid to use it since it is sublinear for the difference δ1 Φ. This is the reason why we assume (2.35) when S ∗ bears δ. In both cases, we have ΨLn/b |tN 2 |θ N σ−b d(Φ),
(2.51)
x
when σ = 0. We estimate the ψ part by the Sobolev embedding as
ψ Ln/(θ −σ ) ψ B σ −θ (tN )θ ψB σ x
|tN 2 |θ N −θ ψB σ ,
(2.52)
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where we need 0 ≤ θ − σ ≤ n/2. Thus we obtain
Ψ∗I,M Ln/(b+θ −σ ) |tN 2 |θ+θ N σ−b−θ D∗ ,
(2.53)
x
where we put
D0 := d(Φ),
Dδ := d(Φ0+ )tα δ1 ΦB 2α−σ ,
(2.54)
Dψ := ψB σ D0 , Dψ,δ := ψB σ Dδ ,
older and the and we need 0 ≤ b + θ − σ ≤ n. Using this estimate and the H¨ Sobolev inequalities, we can estimate R∗ as R∗ L2 N n/2 Ψ∗I,M (x, tξ)vJ (x + tξ)w K (x)L∞ L1 |ξ|∼N x N n/2 sup Ψ∗I,M Ln/(b+θ −σ ) vJ (x + tξ)w K (x)Ln/(n−b−θ +σ ) x
|ξ|∼N
2 θ+θ
|tN |
N
n/2
N
n/2−β−γ+σ−σ
N
σ−b−θ
x
∗ −β−γ+b+θ −σ
D J
2 θ+θ
|tN |
(2.55)
vJ H˙ β wK H˙ γ
∗
D vH˙ β wH˙ γ ,
where we used that −β − γ + b + θ − σ ≤ 0 and J ∼ K N . Thus we obtain the desired estimates in this Case I. Case II: I + M J, K, N This is the case where the phase term has the highest frequency. Then we can use the full strength of the above Lemma 2.1 to have spatial decay of Ψ. However, we have to assume β + γ ≤ n/2 here to get spatial decay only from the phase factor. Let β := max(β, 0) and γ := max(γ, 0). By the assumptions β < n/2, γ < n/2 and 0 < β + γ < n/2, we have 0 < β + γ < n/2. By H¨older’s inequality and the Sobolev embedding, we have R∗ L2 N n/2 sup vJ H˙ β wK H˙ γ Ψ∗I,M Ln/(β +γ ) x
|ξ|∼N
N n/2 J −β+β K −γ+γ vH˙ β wH˙ γ Ψ∗I,M L∞
|ξ|∼N
N n/2 (I + M )−β+β
−γ+γ
(2.56)
n/(β +γ )
Lx
vH˙ β wH˙ γ Ψ∗I,M L∞
|ξ|∼N
n/(β +γ )
Lx
.
First we consider the case M ≥ I, which can occur only for Rψ and Rψ,δ . We estimate Ln/(β +γ ) Ψ∗I Ln/b ψM Ln/(β +γ −b) , (2.57) Ψ∗I ψM x
x
x
where we need β + γ ≥ b. The norm for Ψ is estimated by (2.48) or (2.51), and the norm for ψ is estimated by the Sobolev embedding as
ψM Ln/(β +γ −b) |tN |θ ψM B θ
n/(β +γ −b),1
x
2 θ
|tN | N
−θ
M θ −σ −β
−γ +b
ψB σ ,
(2.58)
where we need β + γ − b ≤ n/2. Then we obtain
Ψ∗I,M Ln/(β +γ ) |tN 2 |θ+θ N σ−b−θ M θ −σ −β x
−γ +b
D∗ .
(2.59)
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After plugging this into (2.56), the exponent of M is θ − σ − β − γ + b ≤ 0, so that we may replace M with N . Then we obtain the desired estimate in this case. Next we consider the case I ≥ M . We estimate Ψ∗I ψM Ln/(β +γ ) Ψ∗I Ln/(β +γ ) ψM L∞ . x x
(2.60)
x
Then the norm for ψ is estimated by the Sobolev embedding as
ψM L∞ (tN )θ ψM B θ |tN 2 |θ N −θ M θ −σ ψB σ , x
|tN 2 |θ N −θ I θ −σ ψB σ ,
(2.61)
where we used the condition θ − σ ≥ 0. As for the Ψ part, we use Lemma 2.1 with sufficiently large m to obtain δ1 ΨI Ln/(β +γ ) x
2 θ
|tN | N σ−β
−γ
(N/I)β
+γ −θ−σ+α
d(Φ0+ )tα δ1 ΦB 2α−σ ,
(2.62)
where we need β + γ ≤ n/2, θ + σ < α + n/2 and θ + α ∈ [0, 1]. We have a similar estimate for ΨI . Thus we obtain Ψ∗I,M Ln/(β +γ ) x
2 θ+θ
|tN |
N σ−σ −β
−γ
(N/I)β
+γ +σ −θ−θ −σ+α
D∗ .
(2.63)
When we plug this estimate into (2.56), the exponent of I becomes −α − β − γ + θ + θ − σ + σ ≤ 0, so that we may replace I with N . Then we obtain the desired estimate in this case. Case III: I, J, M N ∼ K This is the case where only one term has the highest frequency of size N and that term is not the phase term. Then the spatial decay provided by the Sobolev embedding is too weak to treat the lower frequency terms, and so we need further decomposition in the Fourier space. Specifically, we decompose the high frequency term w into functions with Fourier support of size I + M + J. More precisely, let K be the set of disjoint cubes of size I + M + J in Rn that are parallel to the axes such that the union of those cubes covers the whole Rn . For each κ ∈ K, wκ denotes the Fourier restriction of w onto κ, and κ ˜ denotes the cube of size 3(I + M + J) consisting 3n cubes in K with κ as its center. Then
Ψ∗I,M (x, tξ)vJ (x + tξ)w κK (x)e−ixξ dx,
(2.64)
is supported on κ ˜ , which can be verified by first freezing tξ as η and putting η = tξ after the Fourier transform. By this support property and the essential
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orthogonality of κ ˜ , we have
2
∗ 2 ∗ κ −ixξ dx
ΨI,M (x, tξ)vJ (x + tξ)w K (x)e R L2 =
κ∈K L2
2
Ψ∗I,M vJ (x + tξ)w κK (x)e−ixξ dx
κ∈K
(2.65)
L2
(I + M + J)n Ψ∗I,M vJ (x + tξ)w κK (x)2L∞
|ξ|∼N
κ∈K
(L1x ) .
First we consider the case I + M ≤ J. Then the remaining argument is almost the same as in Case I. Indeed, by the same estimate as in (2.55), we have Ψ∗I,M vJ (x + tξ)w κK (x)L1x
|tN 2 |θ+θ N σ−b−θ D∗ J β
−β
vJ H˙ β K γ
−γ
κ wK H˙ γ ,
(2.66)
where we choose β , γ ∈ [0, n/2] such that β + γ = b + θ − σ ∈ [0, n]. Plugging this estimate into the above, we obtain
R∗ L2 |tN 2 |θ+θ N σ−b−θ D∗ J n/2+β κ × vH˙ β wK L2 (κ∈K;H˙ γ ) 2 θ+θ
|tN |
∗
D N
−β
Nγ
−γ
(2.67)
σ−b−θ +n/2+β −β+γ −γ
vH˙ β wH˙ γ ,
where we used that J N ∼ K and n/2 + β − β ≥ 0. Since β + γ = b + θ − σ , the above is the desired estimate in this case. Next we consider the case J ≤ I +M . Then the remaining argument is similar to Case II, but not quite the same. Let β := max(0, β) and γ := max(0, γ). Then we have β ≤ β ≤ max(β, γ), since β + γ > 0. Suppose that M ≥ I, which is possible only when S ∗ carries ψ. Then we have J, I M N ∼ K. As in (2.56) and (2.59), we estimate Ψ∗I,M vJ (x + tξ)w κK (x)L1x Ψ∗I,M Ln/(β +γ ) J β x
−β
|tN 2 |θ+θ N σ−b−θ +γ
vH˙ β K γ
−γ
−γ
M θ −σ −β
κ wK H˙ γ
−γ +b
D∗ J β
(2.68)
−β
κ vH˙ β wK H˙ γ .
When we put this into (2.65), the power of M becomes n/2+θ −σ −β −γ +b ≥ 0, so that we may replace M with N . We may replace J with N also. Then we obtain the desired result. Now we proceed to the final remaining case J, M I N ∼ K. By the H¨older and the Sobolev inequalities, we have Ψ∗I,M vJ (x + tξ)w κK (x)L1x Ψ∗I,M L∞
|ξ|∼N
N
−γ
J
β −β
n/β
Lx
κ vJ L2n/(n−2β ) wK L2 x
κ Ψ∗I,M Ln/β vH˙ β wK H˙ γ . x
(2.69)
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We estimate the ψ part in L∞ x , while we use Lemma 2.1 with m = n/2 to estimate the Ψ part as
δ1 ΨI Ln/β |tN 2 |θ N σ−β (N/I)β
−θ−σ+αn/2
x
Dδ ,
(2.70)
where we need β ≤ n/2, (α − 1)n/2 + β + α < θ + σ < α + n/2 and θ + α ∈ [0, 1]. We have a similar estimate for ΨI , but with the weaker lower bound condition (α − 1)n/2 + β < θ. Thus we obtain
Ψ∗I,M Ln/β |tN 2 |θ+θ N σ−σ −β (N/I)β
−θ−θ +σ −σ+αn/2
x
D∗ ,
(2.71)
where we used (2.61). By (2.65), (2.69) and (2.71), we have
R∗ L2 I n/2 |tN 2 |θ+θ N σ−σ
−β −γ
Jβ
−β
(N/I)β
κ × D∗ vH˙ β wK L2 (κ∈K;H˙ γ ) ,
−θ−θ +σ −σ+αn/2
(2.72)
where the power of I is n/2(1 − α) + θ − β + θ − σ + σ ≥ 0 by our assumptions. So we may replace I with N and J with N . Then we obtain the desired estimate in this case. Case III : I, K, M N ∼ J This case is reduced to the previous one by the symmetry of v and w in the operators S ∗ . We have exhausted all the cases where Ψ∗I,M vJ wK e−ixξ dx interacts VN , which can be easily checked as follows. By the Fourier support property, we have N max(I, J, K, M ), and if the maximum is essentially bigger than N , then it must be essentially attained by at least two of I, J, K, M . If max(I, J, K, M ) < 100(I + M ), then we are in case II. Otherwise, the maximum is attained by J or K. If it is much bigger than N , we come into the case I. If it is essentially the same size as N , then we arrive at case III or III , depending whether the maximum is K or J. We will use the above lemma with 1/2 < ν < 1, n ≥ 3, 1 − ν/2 < s < 3/4 and α = s − ν/2. Then we have 1 − ν < α < 1/2,
(2.73)
so that d(Φ) can be bounded. Since we have (α − 1)n/2 < −s,
(2.74)
the lower bound of θ in (2.31) will never bother us, though it could if we would try the lower dimensional n ≤ 2 case.
3 Phase-free estimates In this section, we derive a few basic estimates to treat those terms that do not include the phase function. They are actually variants of those which played the
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central roles in [12]. The main term after the phase elimination can be given by using the following operator T U (A) := U (−t)AU (t) − A,
(3.1)
where U (−t)AU (t) is defined by U (−t)AU (t)ϕ := U (−t)(AU (t)ϕ) for any spacetime real-valued function A. Then the following multilinear operator is a counterpart of S ψ for the phase-free part T U (A). T ψ (A; v, w) := U (−t)AU (t)v, ψwL2 − U (−t)AU (t)ψv, wL2 .
(3.2)
Lemma 3.1 (i) Let β, γ < n/2, β + γ > 0, θ0 , θ1 ∈ [0, 1] and θ = θ0 + θ1 ≤ 1. Then we have vB˙ γ . (3.3) uvB˙ β+γ−n/2 uB˙ β 2,1/θ
2,1/θ0
2,1/θ1
(ii) Let σ ≤ 0 and −n/2 − σ < β < n/2. Then we have AuH˙ β+σ AB σ uH˙ β .
(3.4)
(iii) Let β ∈ R and θ ∈ [0, 1]. Then we have (U (t) − I)uH˙ β+2θ |t|θ uH˙ β .
(3.5)
(iv) Let θ ∈ [0, 1], σ ≤ θ and −n/2 + 2θ − σ < β < n/2. Then we have T U (A)uH˙ β−2θ+σ |t|θ AB σ uH˙ β
(3.6)
n/2+σ In case σ < θ, we may replace the norm B σ for A with B˙ 2,∞ .
(v) Let θ ∈ [0, 1], σ ≤ θ, σ < θ, σ + σ + β + γ = 2θ, and β, γ < n/2. Then we have (3.7) |T ψ (A; v, w)| |t|θ AB σ ψB n/2+σ vH˙ β wH˙ γ . 2,∞
Proof. (i) : By the bilinear real interpolation, it suffices to consider the dyadic pieces of Littlewood-Paley decomposition. By the Sobolev and the H¨ older inequalities, we have ϕK ∗ ((ϕI ∗ u)(ϕJ ∗ v))L2 min(I, J, K)n/2 ϕI ∗ uL2 ϕJ ∗ vL2 min(I, J, K)n/2 I −β J −γ uH˙ β vH˙ γ .
(3.8)
By the Fourier support property, we have I J ∼ K, J K ∼ I or K I ∼ J. In any case, we have min(I, J, K)n/2 I −β J −γ K n/2−β−γ ,
(3.9)
since β, γ < n/2 and β +γ > 0. Thus we obtain the desired estimate for any dyadic pieces, from which the original estimate follows.
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(ii) : This estimate follows from (i) except the borderline case σ = 0, for which we give another proof. By duality, it suffices to consider the case β ≥ 0. In the Fourier space, we have
β ˜
A(ξ − η)˜ u(η)dη
AuH˙ β |ξ| (3.10)
. L2ξ
We split the η integral into those on the region R1 := {|ξ| |η|} and R2 := {|η| |ξ| ∼ |ξ − η|}. In the region R1 , we may replace the ξ weight by |η|β so that we can estimate the above norm by ˜ L1 |ξ|β u ˜L2 AB 0 uH˙ β . A
(3.11)
In the region R2 , we may replace the ξ weight by |ξ − η|β . Then, by the generalized H¨ older and Young inequalities, we can bound the above norm by ˜ Ln/(n−β),∞ ˜ |ξ|β A v L2n/(n+2β),2 AB 0 vH˙ β .
(3.12)
Thus we obtain the desired result. (iii) : This estimate immediately follows from explicit calculation of the Fourier transform. (iv) : Let γ := 2θ − β − σ. Then we have γ < n/2 and 2θ = β + γ + σ. By duality and the Plancherel identity, it suffices to show that 2 2 ˜ − η)˜ (eit(|ξ| −|η| )/2 − 1)A(ξ u(η)˜ v (ξ)dηdξ |T U (A)u, vL2 | ∼ (3.13) θ |t| AB σ uH˙ β vH˙ γ . We split the double integral region into three regions: R1 := {|ξ − η| |ξ| ∼ |η|}, R2 := {|ξ| |ξ − η| ∼ |η|} and R3 := {|η| |ξ| ∼ |ξ − η|}. In the first region R1 , we have |eit(|ξ|
2
−|η|2 )/2
− 1| |t|θ |ξ − η|θ |ξ + η|θ |t|θ |ξ − η|σ |η|β |ξ|γ ,
(3.14)
where we needed the assumption σ ≤ θ, so that the above integral can be estimated ˜ L1 |ξ|β u | · · · | |t|θ |ξ|σ A ˜(ξ)L2 |ξ|γ v˜(ξ)L2 (3.15) R1 θ |t| AB σ uH˙ β vH˙ γ , as desired. In the second region R2 we have |eit(|ξ|
2
−|η|2 )/2
− 1| |t|θ |ξ − η|σ+γ |η|β ,
then we obtain from the generalized Young and H¨ older inequalities, ˜ Ln/(n−γ),∞ |ξ|β u | · · · | |t|θ |ξ|σ+γ A ˜(ξ)L2 ˜ v (ξ)L2n/(n+2γ),2 R2
|t| AB σ uH˙ β vH˙ γ , θ
(3.16)
(3.17)
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if γ ≥ 0. In the case γ < 0, we have (3.14), so the above argument in the region R1 works also in this region. The remaining region R3 is treated in the same way as R2 . Except the borderline case, we may replace the 1-Besov norm with the ∞-Besov by the bilinear real interpolation. (v): In the Fourier space, we have 2 2 2 2 T ψ (A; v, w) = c (eit(|ξ| −|η| )/2 − eit(|ζ| −|ζ−ξ+η| )/2 ) (3.18) ˜ − ξ)w(ζ)dξdηdζ, ˜ − η)˜ ˜ × A(ξ v (η)ψ(ζ and the phase factor can be rewritten as eit(|ξ|
2
−|η|2 )/2
(1 − e−it(ξ−η)(ξ−ζ) ),
(3.19)
so that it can be bounded by |t|θ |ξ − η|θ |ξ − ζ|θ ,
(3.20)
˜ etc. By the above estimate on the phase, we have Let Aˆ := F −1 |A|, ˜ ∗ |w| ˜ ∗ |˜ v |, |ξ|θ |ψ| ˜ L2 |T ψ (A; v, w)| |t|θ |ξ|θ |A| θ θ ˆ θ ˆ v , (|∇| ψ)w |t| (|∇| A)ˆ ˆ L2
ˆw ˆ v
|t|θ (|∇|θ A)ˆ ˆ
β+σ−θ (|∇|θ ψ) ˙ H
˙ γ+σ −θ H
(3.21)
ˆ n/2+σ w ˆ B σ ˆ |t| A v H˙ β ψ ˆ H˙ γ B θ
2,∞
|t| A θ
Bσ
vH˙ β ψB n/2+σ wH˙ γ , 2,∞
as desired, where we used (i) and (ii) in the fourth inequality.
4 Energy estimate In this section, we derive an L2 bound of difference of the modified field w. We consider only positive small time 0 < t 1. Assume that w = wk (k = 0, 1) solves 2iw˙ + tν−2 T (A, φ)w = 0,
(4.1)
with A = Ak and φ = φk , where Ak is a real-valued space-time function and T (A, φ) = e−iΦ {U (−t)AU (t) − V (φ)}eiΦ , Φ(φ) = V (φ)
tν−1 . 2(ν − 1)
(4.2)
We abbreviate Tk := T (Ak , φk ). We decompose the operator T as T = TΦ + TU + TV ,
(4.3)
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T Φ := e−iΦ U (−t)AU (t)eiΦ − U (−t)AU (t), T U := U (−t)AU (t) − A, T
V
(4.4)
:= A − V (φ).
Then we have a general identity T Φ w, vL2 = A, S 0 (Φ; w, v)L2 .
(4.5)
Now the difference of the solutions w1 and w0 satisfies 2iδ1 w˙ + tν−2 {T0 δ1 w + (δ1 T )w1 } = 0,
(4.6)
so that by the energy identity we have ∂t δ1 w2L2 = tν−2 i(δ1 T )w1 , δ1 wL2 .
(4.7)
We can rewrite this by using (4.5) as (δ1 T )w1 , δ1 wL2 = δ1 A, S 0 (Φ0 ; w1 , δ1 w)L2 + A1 , S δ (Φ0 , Φ1 ; w1 , δ1 w)L2
(4.8)
+ T (δ1 A)w1 , δ1 wL2 + (δ1 A − δ1 V (φ∗ ))w1 , δ1 wL2 . U
We apply Lemma 2.2 (i) to the first term with θ = α − κ, β = s , γ = 0 and κ > 0 sufficiently small. The conditions required in the lemma can be satisfied if n ≥ 2, α < 1/2, 0 < s < 1 and κ > 0 is sufficiently small. Then we obtain |δ1 A, S 0 (Φ0 ; w1 , δ1 w)L2 | δ1 AB 2α−s −2κ S 0 (w1 , δ1 w)B s −2α+2κ ∗
tα−κ d(Φ0 )δ1 AB 2α−s −2κ w1 H˙ s δ1 wL2 . (4.9) For the second term, we take θ = α − 2κ, σ = s + κ, β = s and γ = 0 in Lemma 2.2 (ii). Then we obtain |A1 , S δ (Φ0 , Φ1 ; w1 , δ1 w)L2 | A1 B 2α−3κ S δ (Φ0 , Φ1 ; w1 , δ1 w)B∗−2α+3κ A1 B 2α−3κ t
α−2κ
(4.10)
d(Φ0+ )t δ1 ΦB 2α−s −κ w1 H˙ s δ1 wL2 . α
For the T U part, we have |T U (δ1 A)w1 , δ1 wL2 | T U (δ1 A)w1 L2 δ1 wL2 tα−κ δ1 AB 2α−s −2κ w1 H˙ s δ1 wL2 ,
(4.11)
where we used Lemma 3.1 (iv) in the second inequality, so we need s ≥ α − κ, which is satisfied if s ≥ 1/2.
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For the T V part, we use Lemma 3.1 (ii) to have |(δ1 T V )w1 , δ1 wL2 | (δ1 A − δ1 V (φ∗ ))w1 L2 δ1 wL2 δ1 A − δ1 V (φ∗ )B −s w1 H˙ s δ1 wL2 .
(4.12)
Putting these estimates together, we obtain Lemma 4.1 Let 1/2 < ν < 1, n ≥ 3, 1 − ν < α = s − ν/2 < 1/2, and 1/2 ≤ s < 1. Assume that w = wk , k = 0, 1 satisfies (4.1) with φ = φk and real-valued A = Ak . Let κ > 0 be sufficiently small depending on ν, α, s and n. Then we have for 0 < t 1, |∂t w1 − w0 L2 | tν−2 w1 H˙ s A1 − V (φ1 ) − A0 + V (φ0 )B −s + tα−2κ (D(φ0 ) + D(φ1 ))
(4.13)
× (A1 − A0 B 2α−s −4κ + A1 B 2α−3κ tα Φ1 − Φ0 B 2α−s −κ ) ,
where we denote D(φ) := φm H s + 1,
(4.14)
with sufficiently large m depending on ν, α and s . We can bound d(Φ) by D(φ) since V (φ)B 2α φ2H˙ s
(4.15)
by (3.3), if α = s − ν/2. From now on, we fix α = s − ν/2. By the same argument, we can prove the following. Lemma 4.2 Let 1/2 < ν < 1, n ≥ 3, 1 − ν < α = s − ν/2 < 1/2, and 1/2 ≤ s < 1. Then we have for 0 < t 1, T (A, φ)wL2 wH˙ s (tα−2κ D(φ)AB 2α−s −4κ + A − V (φ)B −s ),
(4.16)
where κ > 0 and D(φ) is as in the above lemma. This implies that w˙ ∈ L1t L2x .
5 Potential estimates In this section, we derive a few estimates on the potential term V (u∗ ) = V (U (t) eiΦ w). As in [3, 4, 12], we should derive a decay estimate (or convergence) at t = +0 for the potential by using the equation. Otherwise, if we would regard V (u∗ ) − V (w(0)) simply as a multiplication to dominate it by w(t) − w(0), then
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we could get a closed estimate only for small data and only for ν = 1. As was indicated by the estimates in the previous section, we need to estimate difference of the potentials also. In this section, we fix α = s − ν/2. Let Ak = V (U (t)eiΦk wk ) for k = 0, 1, where Φk = Φ(φk ). First we derive a bound on the potential. Here we do not need the equation. We decompose A = V (U (t)eiΦ w) as V (U (t)eiΦ w) = KV ∗ S 0 (Φ; w, w) + KV ∗ S(w, w).
(5.1)
For the difference, we have δ1 A =KV ∗ δ1 S 0 (Φ0 ; w∗ , w∗ ) + KV ∗ δ1 S(w∗ , w∗ ) + KV ∗ S δ (Φ0 , Φ1 ; w1 , w1 ).
(5.2)
For the S 0 part, we use Lemma 2.2 (i) with θ = 0, β = γ = s and α = s − ν/2. Then we obtain KV ∗ S 0 (w, w)B 2s−ν ∼ S 0 (w, w)B 2s−n d(Φ)w2H˙ s .
(5.3)
Similarly, choosing θ = 0, β = s and γ = 0, we obtain KV ∗ δ1 S 0 (Φ0 ; w∗ , w∗ )B 2α−s d(Φ0 )w0+ H˙ s δ1 wL2 .
(5.4)
The S part can be easily estimated by using (3.3) as KV ∗ S(w, w)B 2α ∼ S(w, w)B 2s−n U (t)w2H˙ s ∼ w2H˙ s ,
(5.5)
and similarly, KV ∗ δ1 S(w∗ , w∗ )B 2α−s w0+ H s δ1 wL2 .
(5.6)
A lower Besov norm is easily estimated by the Sobolev embedding as V (U (t)eiΦ w)B˙ n/2−ν V (U (t)eiΦ w)B˙ n−ν 1,∞ 2,∞
iΦ 2
|U (t)e w| L1 w2L2 .
(5.7)
Similarly we have KV ∗ δ1 S 0 (Φ0 ; w∗ , w∗ ) + KV ∗ δ1 S(w∗ , w∗ )B˙ n/2−ν 2,∞
w0+ L2 δ1 wL2 .
(5.8)
For the term of the phase change, we use Lemma 2.2 (ii) with θ = 0, β = γ = s, α = s − ν/2, σ = s to obtain KV ∗ S δ (Φ0 , Φ1 ; w1 , w1 )B 2α−s d(Φ0+ )tα δ1 ΦB 2α−s w1 2H˙ s
d(Φ0+ )φ0+ H s δ1 φL2 w1 2H˙ s ,
(5.9)
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where we used (3.3) to estimate δ1 Φ as tα δ1 ΦB 2α−s (φ0 + φ1 )δ1 φB˙ s−n/2 2,1
φ0+ H˙ s δ1 φL2 .
(5.10)
Repeating this argument with θ = 0, β = γ = ν/4 + κ < s and σ = s, we obtain a lower Besov bound KV ∗S δ (Φ0 , Φ1 ; w1 , w1 )B −α−ν+2κ d(Φ0+ )φ0+ H s δ1 φL2 w1 2H˙ ν/4+κ .
(5.11)
If κ is sufficiently small, we have −α − ν + 2κ < −ν. Gathering all estimates, we obtain the following. Lemma 5.1 Let 1/2 < ν < 1, n ≥ 3 and 1 − ν/2 < s < 3/4. Let Ak = V (U (t)eiΦk wk ) with Φk = Φ(φk ) for k = 0, 1. Then we have Ak B −ν ∩B 2s−ν D(φk )wk 2H s , A1 − A0 B −ν ∩B s−ν (D(φ0 ) + D(Φ1 ))(w0 H s + w1 H s )(w1 − w0 L2 + φ1 − φ0 L2 ),
(5.12)
where D is as defined in (4.14). Next we proceed to the decay estimate of the potential at t = +0. Now suppose that wk solves 2iw˙ k + tν−2 T (Ak , φk )wk = 0,
(5.13)
and denote Φk := Φ(φk ). We decompose V (U (t)eiΦ w) − V (w(0)) = V Φ (Φ, w) + V U (w) + V w (w), where
(5.14)
V Φ (Φ, w) := KV ∗ S 0 (Φ; w, w), V U (w) := V (U (t)w) − V (w),
(5.15)
V (w) := V (w) − V (w(0)). w
For the difference, we have δ1 A − δ1 V (w∗ (0)) = δ1 V Φ (Φ∗ , w∗ ) + δ1 V U (w∗ ) + δ1 V w (w∗ ),
(5.16)
where the first term is further decomposed as δ1 V Φ (Φ∗ , w∗ ) = KV ∗ S δ (Φ0 , Φ1 ; w1 , w1 ) + KV ∗ δ1 S 0 (Φ0 ; w∗ , w∗ ).
(5.17)
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We use the equation of w only for the V w part, since the other two parts already have decay properties. In fact, we can use Lemma 2.2 (i) with θ = α − κ and β = γ = s, where κ > 0 is sufficiently small, to obtain V Φ (w)B 2κ S 0 (w, w)B˙ 2κ+ν−n/2 tα−κ d(Φ)w2H˙ s . 2,1
(5.18)
For δ1 V Φ , we can use Lemma 2.2 (ii) with θ = α − κ, β = γ = s − κ, σ = s for the first term and (i) with β = s − 2κ, γ = 0 for the second term to obtain δ1 V Φ (Φ∗ , w∗ )B −s tα−κ d(Φ0+ )tα δ1 ΦB 2α−s w1 2H˙ s−κ + tα−κ d(Φ0 )w0+ H˙ s−2κ δ1 wL2 [1,2]
tα−κ (d(Φ0 ) + d(Φ0+ )φ0+ H˙ s )w0+ H s−κ
(5.19)
× (δ1 wL2 + δ1 φL2 ).
The V U part is treated by Lemma 3.1 as V U (w)B 2κ (U (t) − I)w(U (t) + I)wB˙ 2κ+ν−n/2 2,1
(U (t) − I)wH˙ 2κ+ν−s (U (t) + I)wH˙ s
(5.20)
tα−κ w2H˙ s .
We use the duality to estimate δ1 V U , V w and δ1 V w . For any real-valued Schwartz function ψ ∈ S(Rn ), we have V (U (t)w) − V (w), ψL2 = T U (KV ∗ ψ)w, wL2 .
(5.21)
So we can estimate δ1 V U by using Lemma 3.1 as |δ1 V U (w∗ ), ψL2 | T U (KV ∗ ψ)δ1 wH˙ −s w0+ H˙ s + T U (KV ∗ ψ)w0+ L2 δ1 wL2 tα KV ∗ ψB n/2+s−ν w0+ H˙ s δ1 wL2
(5.22)
2,∞
tα ψB∗s w0+ H˙ s δ1 wL2 , which implies by duality that δ1 V U (w∗ )B −s tα w0+ H˙ s δ1 wL2 .
(5.23)
As for V w , we have from the equation of w, ∂t |w|2 = (2iw, ˙ iw)C = −tν−2 U (−t)A U (t)eiΦ w, ieiΦ w . C
(5.24)
Taking the L2 coupling with the test function ψ, we obtain
∂t |w|2 , ψL2 = −tν−2 A U (t)eiΦ w, iU (t)(ψeiΦ w)L2 = −tν−2 A , S 0 (Φ ; w, iψw) + S(w, iψw)L2 tν−2 tν−2 A , iS ψ (Φ ; w, w)L2 − T ψ (A ; w, iw). =− 2 2
(5.25)
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Taking the difference for k = 0, 1, we obtain −2t2−ν ∂t δ1 |w|2 , ψL2 =δ1 A , iS ψ (Φ1 ; w1 , w1 )L2 + A0 , iS ψ,δ (Φ0 , Φ1 ; w1 , w1 )L2 + A0 , iδ1 S ψ (Φ0 ; w∗ , w∗ )L2
(5.26)
+ T ψ (δ1 A ; w1 , iw1 ) + δ1 T ψ (A0 ; w∗ , iw∗ ) =: I1 + · · · + I5 . The S ψ part in (5.25) is estimated as |A , iS ψ (w, w)L2 | A B 2s−ν S ψ (w, w)B∗ν−2s t2α−κ/2 d(Φ )A B 2α ψB˙ n/2−ν−κ w2H˙ s ,
(5.27)
2,∞
where we applied Lemma 2.2 (iii) with θ = α, θ = α − κ/2, σ = −ν − κ and β = γ = s. As for the term involving T ψ , we estimate it by using (3.7) as |T ψ (A ; w, iw)| t2α−κ A B 2α−κ ψB˙ n/2−ν−κ w2H˙ s . 2,∞
(5.28)
For the S ∗ parts in δ1 |w|2 , we apply Lemma 2.2 with θ = θ = α − κ/2, β = s and σ = s − ν. We set γ = s for I1 and I2 , γ = 0 for I3 , and σ = s for I2 . Then we obtain |I1 | t2α−κ d(Φ1 )δ1 A B s−ν−2κ ψB˙ n/2+s−ν w1 2H˙ s , 2,∞
|I2 | t2α−κ d(Φ0+ )δ1 φ L2 φ0+ H˙ s A0 B 2α−2κ ψB˙ n/2+s−ν w1 2H˙ s , 2,∞
|I3 | t
2α−κ
(5.29)
d(Φ0 )A0 B 2α−2κ ψB˙ n/2+s−ν w0+ H˙ s δ1 wL2 . 2,∞
where we used (5.10) for I2 . For the remaining two terms I4 and I5 , we apply (3.7) with θ = 2α − κ. Then we get |I4 | t2α−κ δ1 A B s−ν−2κ ψB˙ n/2+s−ν wk 2H˙ s , 2,∞
|I5 | t2α−κ A0 B 2α−2κ ψB˙ n/2+s−ν w0+ H˙ s δ1 wL2 .
(5.30)
2,∞
By duality, we obtain ∂t V (w)B κ tα−κ−1 d(Φ )A B 2α ∩B 2α−κ w2H˙ s , [0,1]
[0,1]
(5.31) [1,2]
∂t δ1 V (w)B −s tα−κ−1 d(Φ0+ )φ0+ H˙ s A0 B 2α−2κ w0+ H˙ s × (δ1 A B s−ν−2κ + δ1 φ L2 + δ1 wL2 ).
(5.32)
where we used that 1 − ν < α. In conclusion, we have the following decay estimate on the potential.
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Lemma 5.2 Let 1/2 < ν < 1, n ≥ 3 and 1 − ν/2 < s < 3/4. Let Ak = V (U (t)eiΦk wk ) with Φk = Φ(φk ) for k = 0, 1. Assume that wk solves 2iw˙ k + tν−2 T (Ak , φk )wk = 0,
(5.33)
with real-valued Ak . Let κ > 0 be sufficiently small depending on ν and s. Let α = s − ν/2. Then we have Ak − V (wk (0))B κ tα−κ (D(φk ) + D(φk ))
[0,1] × sup Ak B 2α ∩B 2α−κ wk 2H˙ s ,
(5.34)
t
A1 − V (w1 (0)) − A0 + V (w0 (0))B −s tα−κ (D(φ0 ) + D(φ1 ) + D(φ0 ) + D(φ1 ))
[0,1] × sup A0 B 2α−2κ (w0 H s + w1 H s )[1,2]
(5.35)
t
× (A1 − A0 B 2α−s−2κ + φ1 − φ0 L2 + φ1 − φ0 L2 + w1 − w0 L2 )
where D is as defined in (4.14), and supt should be understood as the operator defined by (sup f )(t) := sup f (s). (5.36) t
0<s
6 Modified wave operators In this section, we construct modified wave operators by solving the Cauchy problem for (1.10). The iteration scheme is the same as in [12]. We may concentrate only on very small positive time 0 < t 1, since the continuation for larger time is easy and well known. Let n ≥ 3, 1/2 < ν < 1 and 1 − ν/2 < s < 3/4, α := s − ν/2. We first construct the modified wave operator W for s < 3/4, and then will show that it is also continuous in the topology of H s , 3/4 ≤ s < 1. Let φ ∈ H s . Let κ > 0 be so small depending on s, ν and n that all the above Lemmas can work and moreover we have α − 10κ > 1 − ν. We want to solve the Cauchy problem for w: 2iw˙ + tν−2 e−iΦ {U (−t)V (U (t)eiΦ w)U (t) − V (φ)}eiΦ w = 0,
(6.1)
with w(0) = φ, where Φ = Φ(φ) is defined by (1.9) as before. We solve this by iteration starting with w0 := φ and Ak := V (U (t)eiΦ wk ), 2iw˙ k + tν−2 T (Ak−1 , φ)wk = 0,
(6.2)
wk (0) = φ, where T is as defined in (4.2). In this paper, we do not use the equation for uk := U (t)eiΦ wk .
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First of all, we have D(φ) < ∞ by definition (4.14). In the following, we often regard such a quantity (depending only on φ) as a constant. From Lemma 5.1, we have (6.3) A0 B −ν ∩B 2α φ2H s . We may drop the term V w when estimating the decay of A0 since w0 is independent of t. Then we obtain from the argument of Lemma 5.2, A0 − V (φ)B κ tα−κ φ2H˙ s .
(6.4)
Now we can start an induction to establish uniform bounds on Ak and wk . Let k ∈ N and suppose that we have wj H s ≤ C0 , Aj B −ν ∩B 2α ≤ C1 , Aj (t) − V (φ)B κ ≤ C2 tα−κ ,
(6.5)
for any j < k. First we presume that we can solve the equation for wk such that wk ∈ C(H s ) with wk (0) = φ. Now we apply Lemma 4.1 with w1 := wk (x + h) and w0 := wk . Then we obtain |∂t δ h wk L2 |
tν−2 wk H˙ s δ h (Ak−1 − V (φ))B −s
+ tα−2κ D(φ)(δ h Ak−1 B 2α−s −4κ + Ak−1 B 2α−3κ tα δ h ΦB 2α−s −κ )
tν−2 |h|s +κ wk H˙ s Ak−1 − V (φ)B κ + tα−2κ (Ak−1 B 2α−3κ + Ak−1 B 2α−3κ φ2H s )
|h|s +κ wk H˙ s t4κ−1 Ak−1 B 2α−3κ + tν−2 Ak−1 − V (φ)B κ ,
(6.6)
where we used (5.10) in the second inequality after estimating the difference by the Besov space B 2α . Let s = s and use the assumed bounds (6.5). Then we obtain |∂t δ h wk L2 | |h|s+κ t4κ−1 (C1 + C2 )wk H˙ s .
(6.7)
By (2.13) this implies |∂t wk H˙ s | ≤ C3 t4κ−1 wk H˙ s ,
(6.8)
where C3 depends on C0 , C1 , C2 and κ. On the other hand, by the L2 conservation, we have (6.9) δ h wk L2 ≤ 2φL2 . Then, by the Gronwall inequality, we obtain 4κ
wk H s ≤ M0 eC4 t φH s ,
(6.10)
where M0 is an absolute constant and C4 is a constant dependent on Ci (i < 4) and κ. If we have chosen C0 > 2M0 φH s , then for sufficiently small time (depending on Ci ), we have wk H s < 2M0 φH s < C0 . (6.11)
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Actually, we have a better regularity for the perturbation. Plugging the above bound into (6.8) and using (2.13), we obtain ϕJ ∗ wk H˙ s ϕJ ∗ φH s + J −κ t4κ C0 (C1 + C2 ).
(6.12)
Now we give a rigorous proof of that we can obtain wk solving the equation (6.2) with wk (0) = φ and belonging to C(H s ). If we assume the initial data at positive time t0 > 0 wk (t0 ) = φ, then it is easy to solve the equation for t > t0 by the standard L2 estimate since we are away from the singularity at t = 0. Then we obtain the uniform bound (6.10) in H s for this approximate solution. 2 Moreover, Lemma 4.2 implies a uniform L1+ ˙ k , which yields uniform t Lx bound of w 2 continuity of wk in L . Thus, by letting t0 → +0, we obtain a solution wk of (6.2) with wk (0) = φ in L∞ (H s ) and C(L2 ). Then the frequency-localized bound (6.12) implies that wk is continuous also in H s . Next we turn to the estimate on Ak . By Lemma 5.1 we have Ak B −ν ∩B 2α wk 2H s ,
(6.13)
and from Lemma 5.2 we obtain [0,1]
Ak − V (φ)B κ tα−κ sup Ak−1 B 2α ∩B 2α−κ wk 2H˙ s t
tα−κ (1 + C1 )wk 2H˙ s .
(6.14)
Thus, if we have chosen C0 , C1 and C2 sufficiently large compared with M0 φH s , then we obtain the above bounds (6.5) uniformly for k and for small t > 0 by induction on k. In conclusion, wk H s ,
Ak B −ν ∩B 2α ,
tκ−α Ak − V (φ)B κ
(6.15)
are bounded for small t > 0 and any k ≥ 0. Next we consider the convergence. We will regard those bounds on wk and Ak obtained above just as constants. We apply Lemma 4.1 with s = s , w1 := wk+1 and w0 := wk . Then we obtain |∂t δk+1 wL2 | t4κ−1 δk AB 2α−s−4κ + tν−2 δk AB −s ,
(6.16)
and then we need to estimate δk A. Applying Lemma 5.1 with A1 := Ak and A0 := Ak−1 , we obtain δk AB −ν ∩B 2α−s δk wL2 .
(6.17)
We also use Lemma 5.2 with A1 := Ak and A0 = Ak−1 . Then we obtain 2α−s−2κ ) + δk wL∞ (L2 ) ) δk AB −s tα−κ (δk−1 AL∞ t t (B 2 . tα−κ δk−1+ wL∞ t (L )
(Notice that δk V (w∗ (0)) = 0 in this case.)
(6.18)
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Putting (6.16)–(6.18) together, we obtain 2 , |∂t δk+1 wL2 | t4κ−1 δk−1+ wL∞ t (L )
(6.19)
By integration in time, we obtain δk+1 wL∞ (0,t;L2 ) t4κ δk−1+ wL∞ (0,t;L2 ) ,
(6.20)
which implies that wk converges in L∞ (0, t; L2 ) for sufficiently small t > 0. Since we have additional regularity for the nonlinear term (6.12), we can enhance the L2 convergence to the H s one. In fact, (6.12) implies that for any ε > 0, there exists some large N depending on φ and ε such that for small t > 0 and for any k we have wk ||ξ|≥N H s < ε,
(6.21)
where ξ denotes the Fourier variable. Then, by the Lebesgue dominant convergence theorem, we obtain H s convergence of wk as k → ∞. The above contraction property (6.20) also implies the uniform continuity of wk as t → +0, first in L2 , and then by the same reasoning as above, in H s . Thus we have the strong limit w∞ of wk in C(H s ), and then by Lemma 5.1, Ak converges to A∞ = V (U (t)eiΦ w∞ ) in B −ν ∩ B 2α , which satisfies A∞ − V (φ)B κ tα−κ .
(6.22)
Using these convergence, it is easy to see that this limit function w∞ solves the equation (6.1) with w∞ (0) = φ as desired. The uniqueness of such a solution follows from the estimate for the difference of two solutions by using Lemma 4.1 as above. Then we obtain the well-defined modified wave operator W via the pseudo-conformal inversion. Since the asymptotic behavior described in the theorem uniquely determines the limit of V (u∗ (t)) as t → +0, the injectivity of the modified wave operator W easily follows. Notice that the regularity gain κ in (6.12) also implies that w∞ is bounded in H s+κ/2 if φ belongs to this space. Then we may apply the energy estimate with s = s + κ/2, getting again certain amount of regularity. It is easy to check that the amount of regularity gain κ can be taken uniformly as long as s is away from 1. Therefore, if φ ∈ H s with s < s < 1, then w is bounded also in H s . Thus W maps F H s into F H s , for any s ∈ (1 − ν/2, 1). We also obtain (6.12) for s in this range. Remark 6.1 In principle, we may consider Sobolev norms higher than 1 by taking higher differences. If we want to go beyond H n/2 , then we also need to jack up the regularity of the potential. We do not pursue this problem in this paper.
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7 Continuity of Modified wave operators In this section, we see the continuity of the modified wave operator in F H s . By the local wellposedness in F H s of the original equation (1.1), it is equivalent to the continuity of the map W ∗ : φ → w(1) in H s , where w is the solution of (6.1) with w(0) = φ obtained above. Notice that here the phase factor Φ also changes depending on φ, and it is the reason why the continuity is not trivial from the above iterative construction, where Φ was fixed. However, we have already derived the necessary estimates in Sections 4 and 5, and so we have just to check that they work. We will show L2 continuity of W ∗ in a bounded set of H s . Once it is obtained, we can easily enhance it to the strong continuity in H s via the frequency-localized uniform bound (6.12), which has been extended to s ∈ (1−ν/2, 1) by the argument at the end of the previous section. Let φk ∈ H s for k = 0, 1 and let wk be the solutions to (6.1) with wk (0) = φk and Φ = Φk := Φ(φk ). We assume that φk are bounded in H s , so that we may regard those norms are dominated by a constant. We apply Lemma 4.1 with s = s. Then we obtain |∂t δ1 wL2 | t4κ−1 (δ1 AB 2α−s−4κ + δ1 φL2 ) + tν−2 δ1 A − δ1 V (φ∗ )B −s ,
(7.1)
where we used (5.10) to estimate δ1 Φ. Then we need to estimate δ1 A. By Lemma 5.1, we have (7.2) δ1 AB −ν ∩B 2α−s δ1 wL2 + δ1 φL2 . Next we use Lemma 5.2 with Ak = Ak and φk = φk . Then we obtain δ1 A − δ1 V (φ∗ )B −s tα−κ sup(δ1 AB 2α−s−κ + δ1 wL2 + δ1 φL2 ) t
t
α−κ
sup(δ1 wL2 + δ1 φL2 ),
(7.3)
t
where we used the above obtained bound (7.2). In conclusion, we obtain |∂t δ1 wL2 | t4κ−1 (δ1 wL∞ (L2 ) + δ1 φL2 ),
(7.4)
which, through integration in time, implies that for small t > 0, δ1 wL2 δ1 φL2 .
(7.5)
Thus we obtain L2 continuity of W ∗ in any bounded set of H s . It is easy to enhance this convergence into the strong one in H s by (6.12), which implies the following. For any ε > 0, there exists a small ball Bε of H s around φ0 and N ∈ N large such that for small t > 0 (independent of ε), we have
w1 (t)||ξ|>N s < ε, (7.6) H
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if w1 (0) ∈ Bε . When combined with the L2 continuity obtained above, this implies the strong continuity of W ∗ in H s . Returning by the pseudo-conformal inversion, we obtain F H s continuity of W .
8 Asymptotic completeness In this section, we show the openness of the modified wave operator W , which implies the asymptotic completeness in a small ball of H s around any solution having the asymptotic profile described by the modified wave operator. Here again we invert the problem by the pseudo-conformal transform. By the local wellposedness of the Hartree equation, it suffices to show the following: Let w0 be a solution of (6.1) with w0 (0) = φ0 and Φ = Φ0 := Φ(φ0 ). Then for some t0 > 0 and any ψ sufficiently close to ψ0 := w0 (t0 ) in H s , there exists a solution w of (6.1) with Φ = Φ := Φ(w (0)), satisfying w (t0 ) = ψ . Moreover, when ψ converges to ψ0 , w (0) also converges to w0 (0) in H s . To find the solution w , we again use the iteration method. Let w0 be given as above. Then we define Ak , Φk and wk inductively by Φk := Φ(φk ), Ak := V (U (t)e 2iw˙ k + t
ν−2
φk := wk (0) iΦk
wk ),
T (Ak−1 , φk−1 )wk = 0,
(8.1)
wk (t0 ) = ψ . We remark that for general data ψ at a fixed t0 > 0, this iteration can not possibly work, since we do not have the asymptotic completeness in the whole space in general. It is essential that we can choose t0 as small as we need and also ψ close to w0 (t0 ) (it suffices to be bounded). First we derive uniform bounds for wk and Ak . Assume that we have wj H s ≤ C0 , Aj B −ν ∩B 2α ≤ C1 , Aj − V (φj )B κ ≤ C2 tα−κ ,
(8.2)
for j < k. It is clear by the result obtained above that we have (8.2) for j = 0. So it suffices to show (8.2) for j = k to get the uniform bounds. Applying Lemma 4.1 with w1 := wk (x + h) and w0 = wk , we obtain
|∂t δ h wk L2 | |h|s+κ tν−2 wk H˙ s Ak−1 − V (φk−1 )B κ
+ tα−2κ D(φk−1 )(Ak−1 B 2α−3κ + Ak−1 B 2α−3κ tα Φk−1 B 2α ) [0,m+2]
|h|s+κ wk H˙ s t4κ−1 (C2 + C0
C1 ). (8.3)
Taking the difference norm of H˙ s , we obtain |∂t wk H˙ s | ≤ C3 t4κ−1 wk H˙ s ,
(8.4)
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where C3 is a constant dependent on Ci (i < 3) and κ. By the Gronwall inequality and the L2 conservation, we obtain a uniform bound 4κ
wk (t)H s ≤ eC4 |t
−t4κ 0 |
ψ H s ,
(8.5)
where C4 is determined by Ci (i < 4) and κ. By Lemma 5.1, we have Ak B −ν ∩B 2α D(φk )wk 2H s ,
(8.6)
and by Lemma 5.2, [0,1]
Ak − V (φk )B κ tα−κ D(φk−1+ ) sup Ak−1 B 2α ∩B 2α−κ wk 2H s t
tα−κ D(φk−1+ )(1 + C1 )wk 2L∞ s. t H
(8.7)
s , and C1 sufNow we choose C0 sufficiently large compared with w0 L∞ t H [0,m] 2 C0 , and then C2 sufficiently large compared ficiently large compared with C0 [2,m+2] (1 + C1 ), and finally we choose t0 sufficiently small. Then we can with C0 make those bounds in (8.5), (8.6) and (8.7) smaller than needed to proceed the induction for (8.2). Thus we obtain the uniform bounds (8.2) for any j. Strictly speaking, we have to carry out this procedure first for s < 3/4 and then extend the bound to general s < 1 in the same way as in the construction of W (see the end of Section 6). Next we show the convergence of wk in L2 . We regard the bounds in (8.2) just as constants. By Lemma 4.1 with w1 := wk+1 and w0 := wk , we have
|∂t δk+1 wL2 | tν−2 δk A − δk V (φ∗ )B −s + t4κ−1 (δk AB 2α−s−4κ + δk φL2 ),
(8.8)
where we used (5.10). Meanwhile, Lemma 5.1 with A1 := Ak and A0 := Ak−1 implies that δk AB −ν ∩B 2α−s δk wL2 + δk φL2 , (8.9) and Lemma 5.2 yields that δk A − δk V (φ∗ )B −s tα−κ sup(δk−1 AB 2α−s−2κ + δk−1+ φL2 + δk wL2 ).
(8.10)
t
Plugging these estimates into (8.8), we obtain 2. |∂t δk+1 wL2 | t4κ−1 δk−1+ wL∞ t L
(8.11)
By integration in time, we obtain 4κ 2 |t 2. δk+1 wL∞ − t4κ 0 |δk−1+ wL∞ t L t L
(8.12)
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In the special case k = 1, we have 4κ 2 |t 2, − t4κ δ2 wL∞ 0 |δ1 wL∞ t L t L
(8.13)
since the δk−1 parts in (8.10) disappear thanks to the equation for w0 . Therefore, if we choose t0 sufficiently small, then wk converges to some function w∞ in L2 for 0 < t < 2t0 . By the same argument as for (6.12), we can derive from (8.3) that ϕJ ∗ wk (t)H˙ s ϕJ ∗ ψ H˙ s + J −κ |t4κ − t4κ 0 |C3 ,
(8.14)
which implies that the above L2 convergence is actually strong in H s . Then w := w∞ is the desired solution of (6.1) with w (t0 ) = ψ and Φ = Φ(w (0)). Moreover, the above contraction property (8.12)–(8.13) implies that if ψ converges to ψ0 in L2 (in a bounded set in H s ), then w converges to w0 in L2 for 0 ≤ t ≤ 2t0 . If ψ converges in H s , then the high frequency part of ψ in (8.14) is uniformly bounded (more precisely, compact in A2 for the dyadic parameter J). Hence (8.14) together with the corresponding estimate for w0 (see (6.12)) implies the convergence of w to w0 in H s for 0 ≤ t ≤ 2t0 . This finishes the proof of the openness of W , and so we have completed the proof of Theorem 1.1.
References [1] J. Bergh and J. L¨ ofstr¨ om, Interpolation spaces, Springer, Berlin/Heidelberg/New York, 1976. [2] J. Ginibre and T. Ozawa, Long range scattering for nonlinear Schr¨ odinger and Hartree equations in space dimension n ≥ 2, Comm. Math. Phys. 151, 619–645 (1993). [3] J. Ginibre and G. Velo, Long range scattering and modified wave operators for some Hartree type equations I, Rev. Math. Phys. 12, 361–429 (2000). [4] J. Ginibre and G. Velo, Long range scattering and modified wave operators for some Hartree type equations II, Ann. Henri Poincar´e 1, 753–800 (2000). [5] J. Ginibre and G. Velo, Long range scattering and modified wave operators for some Hartree type equations III. Gevrey spaces and low dimensions, J. Differential Equations 175, 415–501 (2001). [6] N. Hayashi, K. Kato and P. I. Naumkin, On the scattering in Gevrey classes for the subcritical Hartree and Schr¨ odinger equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 27, 483–497 (1998). [7] N. Hayashi and P. I. Naumkin, Scattering theory and large time asymptotics of solutions to the Hartree type equations with a long range potential, Hokkaido Math. J. 30, 137–161 (2001).
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[8] N. Hayashi, P. I. Naumkin and T. Ozawa, Scattering theory for the Hartree equation, SIAM J. Math. Anal. 29, 1256–1267 (1998). [9] N. Hayashi and Y. Tsutsumi, Scattering theory for Hartree type equations, Ann. Inst. H. Poincar´e Phys. Th´eor. 46, 187–213 (1987). [10] H. Nawa and T. Ozawa, Nonlinear scattering with nonlocal interaction, Comm. Math. Phys. 146, 259–275 (1992). [11] K. Nakanishi, Asymptotically free solutions for short range nonlinear Schr¨ odinger equations, SIAM J. Math. Anal. 32, 1265–1271 (2001). [12] K. Nakanishi, Modified wave operators for the Hartree equation with data, image and convergence in the same space, to appear in Comm. Pure Appl. Anal. [13] T. Ozawa, Long range scattering for nonlinear Schr¨ odinger equations in one space dimension, Comm. Math. Phys. 139, 479–493 (1991). Kenji Nakanishi Graduate School of Mathematics Nagoya University Nagoya 464-8602 Japan email: [email protected] Communicated by Vincent Rivasseau submitted 20/08/01, accepted 04/12/01
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Ann. Henri Poincar´e 3 (2002) 537 – 612 c Birkh¨ auser Verlag, Basel, 2002 1424-0637/02/030537-76 $ 1.50+0.20/0
Annales Henri Poincar´ e
Long Range Scattering and Modified Wave Operators for the Wave-Schr¨ odinger System∗ J. Ginibre and G. Velo
Abstract. We study the theory of scattering for the system consisting of a Schr¨ odinger equation and a wave equation with a Yukawa type coupling in space dimension 3. We prove in particular the existence of modified wave operators for that system with no size restriction on the data and we determine the asymptotic behaviour in time of solutions in the range of the wave operators. The method consists in solving the wave equation, substituting the result into the Schr¨ odinger equation, which then becomes both nonlinear and nonlocal in time, and treating the latter by the method previously used for a family of generalized Hartree equations with long range interactions.
1 Introduction This paper is devoted to the theory of scattering and more precisely to the existence of modified wave operators for the Wave-Schr¨odinger (WS) system i∂ u = − 1 ∆u − Au (1.1) t 2 A = |u|2 (1.2) where u and A are respectively a complex valued and a real valued function defined = ∂t2 − ∆ is the d’Alembertian in space time R3+1 , ∆ is the Laplacian in R3 and 3+1 in R . That system is Lagrangian with Lagrangian density 1 1 1 ¯) − |∇u|2 + (∂t A)2 − |∇A|2 + A|u|2 . L = i (¯ u ∂t u − u ∂t u 2 2 2 Formally, the L2 norm of u is conserved, as well as the energy 1 E(u, A) = dx |∇u|2 + (∂t A)2 + |∇A|2 − A|u|2 . 2
(1.3)
(1.4)
The Cauchy problem for the WS system (1.1) (1.2) is known to be globally well posed in the energy space Xe = H 1 ⊕ H˙ 1 ⊕ L2 for (u, A, ∂t A) [1] [2] [4] [15]. A large amount of work has been devoted to the theory of scattering for nonlinear equations and systems centering on the Schr¨ odinger equation, in particular for nonlinear Schr¨ odinger (NLS) equations, Hartree equations, Klein-Gordon ∗ Work
supported in part by NATO Collaborative Linkage Grant 976047
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Schr¨ odinger (KGS) and Maxwell-Schr¨ odinger (MS) systems. As in the case of the linear Schr¨ odinger equation, one must distinguish the short range case from the long range case. In the former case, ordinary wave operators are expected and in a number of cases proved to exist, describing solutions where the Schr¨odinger function behaves asymptotically like a solution of the free Schr¨ odinger equation. In the latter case, ordinary wave operators do not exist and have to be replaced by modified wave operators including a suitable phase in their definition. In that respect, the WS system (1.1) (1.2) in R3+1 belongs to the borderline (Coulomb) long range case, because of the t−1 decay in L∞ norm of solutions of the wave equation. Such is the case also for the Hartree equation with |x|−1 potential. Both are simplified models for the more complicated Maxwell-Schr¨odinger system in R3+1 , which belongs to the same case, as well as the KGS system in R2+1 . Whereas a well developed theory of long range scattering exists for the linear Schr¨ odinger equation (see [3] for a recent treatment and for an extensive bibliography), there exist only few results on nonlinear long range scattering. The existence of modified wave operators in the borderline Coulomb case has been proved for the NLS equation in space dimension n = 1 [19]. That result has been extended to the NLS equation in dimensions n = 2, 3 and to the Hartree equation in dimension n ≥ 2 [5], to the derivative NLS equation in dimension n = 1 [14], to the KGS system in dimension 2 [20] and to the MS system in dimension 3 [22]. All those results are restricted to the case of small data. In a recent series of papers, [6] [7] [8], we proved the existence of modified wave operators for a family of Hartree type equations with general (not only Coulomb) long range interactions and without any size restriction on the data. The method is strongly inspired by a previous series of papers by Hayashi et al [9] [10] [11] [12] [13] on the Hartree equation. In the latter papers it is proved first in the borderline Coulomb case and then in the whole long range case, that the global solutions of the Hartree equation with small initial data exhibit an asymptotic behaviour for large time that is typical of long range scattering and includes in particular the expected relevant phase factor. The present paper is devoted to the extension of the results of [6] [7] [8] to the WS system and in particular to the proof of the existence of modified wave operators for that system without any size restriction on the data. The method consists in eliminating the wave equation by solving it for A in terms of u and substituting the result into the Schr¨ odinger equation, thereby obtaining a new Schr¨ odinger equation which is both nonlinear and nonlocal in time. The latter is then treated as the Hartree equation in [6] [7] [8], namely u is expressed in terms of an amplitude w and a phase ϕ satisfying an auxiliary system similar to that introduced in [11]. Wave operators are constructed first for that auxiliary system, and then used to construct modified wave operators for the original system (1.1). The detailed construction is too complicated to allow for a more precise description at this stage, and will be described in heuristic terms in Section 2 below. In subsequent papers, the results of the present one will be extended to the case of the MS system.
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We now give a brief outline of the contents of this paper. A more detailed description of the technical parts will be given at the end of Section 2. After collecting some notation and preliminary estimates in Section 3, we study the asymptotic dynamics for the auxiliary system in Section 4 and uncover some difficulties due to the different propagation properties of solutions of the wave and Schr¨ odinger equations. As a preparation for the general case, we construct in Section 5 the wave operators associated with the simplified linear system obtained by replacing (1.2) by the free wave equation A = 0. We then solve the local Cauchy problem at infinity for the auxiliary system in Sections 6 and 7, which contain the main technical results of this paper. We finally come back from the auxiliary system to the original one (1.1) (1.2) and construct the modified wave operators for the latter in Section 8, where the final result is stated in Proposition 8.1. We conclude this section with some general notation which will be used freely throughout this paper. We denote by · r the norm in Lr ≡ Lr (R3 ) and we define δ(r) = 3/2−3/r. For any interval I and any Banach space X, we denote by C(I, X) (resp. Cw (I, X)) the space of strongly (resp. weakly) continuous functions from I to X and by L∞ (I, X) (resp. L∞ loc (I, X)) the space of measurable essentially bounded (resp. locally essentially bounded) functions from I to X. For real numbers a and b, we use the notation a ∨ b = Max(a, b), and a ∧ b = Min(a, b). Furthermore, we define [a ∨ b] = a ∨ b if a = b =a+ε
for some ε > 0 if a = b ,
[a ∧ b] = a + b − [a ∨ b] and [a]+ = [a ∨ 0] . For any interval I ⊂ R+ , we denote by I¯ the closure of I in R+ ∪ {∞} and for any interval I = [a, b) we denote by I+ the interval I+ = [a, ∞). In the estimates of solutions of the relevant equations, we shall use the letter C to denote constants, possibly different from an estimate to the next, depending on various parameters, but not on the solutions themselves or on their initial data. We shall use the notation C(a1 , a2 , · · · ) for estimating functions, also possibly different from an estimate to the next, depending in addition on suitable norms a1 , a2 , · · · of the solutions or of their initial data. Additional notation will be given in Section 3.
2 Heuristics In this section, we discuss in heuristic terms the construction of the modifed wave operators for the system (1.1) (1.2), as it will be performed in this paper. We refer to Section 2 of [6] [7] for general background and for a similar discussion adapted to the case of the Hartree equation. The problem that we want to address is that of classifying the possible asymptotic behaviours in time of the solutions of (1.1) (1.2) by relating them to a set of model functions V = {v = v(v+ )} parametrized by some data v+ and with suitably chosen and preferably simple asymptotic behaviour in time. For each v ∈ V, one tries to construct a solution
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(u, A) of (1.1) (1.2) such that (u, A)(t) behaves as v(t) when t → ∞ in a suitable sense. We then define the wave operator as the map Ω : v+ → (u, A) thereby obtained. A similar question can be asked for t → −∞. We restrict our attention to positive time. The more standard definition of the wave operator is to define it as the map v+ → (u, A)(0), but what really matters is the solution (u, A) in the neighborhood of infinity in time, namely in some interval [T, ∞), and continuing such a solution down to t = 0 is a somewhat different question which we shall not touch here. In cases such as (1.1) (1.2) where the system of interest is a perturbation of a simple linear system, hereafter called the free system, a natural candidate for V is the set of solutions of the free system, parametrized by the initial data v+ at time t = 0 for the Cauchy problem for that system. In the case of the system (1.1) (1.2) one is therefore tempted to consider the Cauchy problem i∂ u = − 1 ∆u u(0) = u+ t 2 (2.1) A = 0 A(0) = A+ , ∂t A(0) = A˙ + , to take v+ = (u+ , A+ , A˙ + ) and to take for v(v+ ) the solution (u, A) of (2.1). Cases where such a procedure yields an adequate set V are called short range cases. They require that the perturbation has sufficient decay in time or equivalently in space. This is the case for instance for the linear Schr¨ odinger equation or for the Hartree equation with potential V (x) = |x|−γ for γ > 1. Such is not the case however for the system (1.1) (1.2). This shows up through the fact that the solution A of the wave equation A = 0 decays at best as t−1 (in L∞ norm), which is the borderline case of nonintegrability in time. That situation corresponds to the limiting case γ = 1 (the Coulomb case in space dimension n = 3) for the linear Schr¨ odinger and for the Hartree equation. A similar situation prevails for the KGS system in space dimension 2 and for the MS system in space dimension 3. In the present case, which is the borderline long range case, the set of solutions of the Cauchy problem (2.1) is inadequate, and one of the tasks that will be performed in this paper (see especially Sections 7 and 8) will be to construct a better set V of model asymptotic functions. Constructing the wave operators essentially amounts to solving the Cauchy problem with infinite initial time. The system (1.1) (1.2) in this form is not well suited for that purpose and we shall now perform a number of transformations leading to an auxiliary system for which that problem can be handled. For additional flexibility we shall first of all allow for imposing initial data at two different odinger and wave equations respectively. With initial times t0 and t1 for the Schr¨ the aim of letting t1 and t0 tend to infinity in that order, we shall take t0 ≤ t1 . We shall then eliminate the wave equation by solving it and substituting the result into the Schr¨ odinger equation. We define ω = (−∆)1/2
,
K(t) = ω −1 sin ωt
,
˙ K(t) = cos ωt
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and we replace the wave equation (1.2) by its solution A = A0 + At11 (|u|2 )
(2.2)
where ˙ A0 = K(t) A+ + K(t) A˙ + t t1 2 dt K(t − t ) |u(t )|2 . A1 (|u| ) =
(2.3) (2.4)
t1
Here A0 is a solution of the free wave equation, with initial data (A+ , A˙ + ) at time t = 0. For t1 = ∞, (A+ , A˙ + ) is naturally interpreted as the asymptotic state for A, in keeping with the previous discussion. The Cauchy problem for the system (1.1) (2.2) with initial data u(t0 ) = u0 is no longer a usual PDE Cauchy problem because A1 depends on u nonlocally in time. A convenient way to handle that difficulty is to first replace that problem by a partly linearized form thereof, namely i∂ u = − 1 ∆u − Au , u (t0 ) = u0 t (2.5) 2 A = A + A (|u|2 ) . 0
1
For given u, (2.5) is an ordinary (linear) Cauchy problem for u . Solving that problem for u defines a map Γ : u → u , and solving the original problem then reduces to finding a fixed point of Γ, which in favourable cases can be done for instance by contraction. We shall make use of that linearization method, not for the equation for u, but for the auxiliary system to be defined below. Aside from the nonlocality in time of the nonlinear interaction term, which can be handled by the previous linearization, the system (1.1) (2.2) is rather similar to the Hartree type equations considered in [6] [7] [8], and we next perform the same change of variables, which is well adapted to the study of the asymptotic behaviour in time. The unitary group U (t) = exp(i(t/2)∆)
(2.6)
which solves the free Schr¨odinger equation can be written as U (t) = M (t) D(t) F M (t) where M (t) is the operator of multiplication by the function M (t) = exp ix2 /2t ,
(2.7)
(2.8)
F is the Fourier transform and D(t) is the dilation operator (D(t)f )(x) = (it)−n/2 f (x/t)
(2.9)
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normalized to be unitary in L2 . We shall also need the operator D0 (t) defined by (D0 (t)f ) (x) = f (x/t) .
(2.10)
We now parametrize u in terms of an amplitude w and of a real phase ϕ as u(t) = M (t) D(t) exp[−iϕ(t)]w(t) .
(2.11)
Substituting (2.11) into (1.1) yields an evolution equation for (w, ϕ), namely
i∂t + (2t2 )−1 ∆ − i(2t2 )−1 (2∇ϕ · ∇ + ∆ϕ) + t−1 B + ∂t ϕ − (2t2 )−1 |∇ϕ|2 w = 0 (2.12) where we have expressed A in terms of a new function B by A = t−1 D0 B .
(2.13)
Corresponding to the decomposition (2.2) of A, we decompose B = B0 + B1t1 (w, w)
(2.14)
where A0 = t−1 D0 B0 and At11 = t−1 D0 B1t1 . One computes easily B1t1 (w1 , w2 ) =
1
t1 /t
dν ν −3 ω −1 sin((ν − 1)ω)D0 (ν)(Re w ¯1 w2 )(νt) .
(2.15)
As in the case of the Hartree equation, we have only one evolution equation (2.12) for two functions (w, ϕ). We arbitrarily impose a second equation, namely a Hamilton-Jacobi (or eikonal) equation for the phase ϕ, thereby splitting the equation (2.12) into a system of two equations, the other one of which being a transport type equation for the amplitude w. For that purpose, we split B1t1 into a short range and a long range parts t1 B1t1 = BSt1 + BL .
(2.16)
in the following way. We take 0 < β < 1 and we define F BSt1 (t, ξ) = χ(|ξ| > tβ )F B1t1 (t, ξ) t1 F BL (t, ξ) = χ(|ξ| ≤ tβ )F B1t1 (t, ξ)
(2.17)
where χ(|ξ|
>
parameter β will satisfy various conditions which will appear later, all of which will be compatible with β = 1/2. We then split the equation (2.12) into the following system of two equations ∂t w = i(2t2 )−1 ∆w + t−2 Q(∇ϕ, w) + it−1 (B0 + BSt1 (w, w))w (2.18) t1 (w, w) ∂t ϕ = (2t2 )−1 |∇ϕ|2 − t−1 BL
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where we have defined Q(s, w) = s · ∇w + (1/2)(∇ · s)w
(2.19)
for any vector field s. The first equation of (2.18) is the transport type equation for the amplitude w, while the second one is the Hamilton-Jacobi type equation for the phase ϕ. Since the right-hand sides of (2.18) contain ϕ only through its gradient, we can obtain from (2.18) a closed system for w and s = ∇ϕ by taking the gradient of the second equation, namely ∂t w = i(2t2 )−1 ∆w + t−2 Q(s, w) + it−1 (B0 + BSt1 (w, w))w (2.20) t1 ∂t s = t−2 s · ∇s − t−1 ∇BL (w, w) . Once the system (2.20) is solved for (w, s), one recovers ϕ easily by integrating the second equation of (2.18) over time. We refer to [6] for details. The system (2.20) will be referred to as the auxiliary system and will play an essential role in this paper. For the same reason as was explained for the partly resolved system (1.1) (2.2), we shall use at intermediate stages a partly linearized version of the system (2.20), namely ∂t w = i(2t2 )−1 ∆w + t−2 Q(s, w ) + it−1 (B0 + BSt1 (w, w))w (2.21) t1 ∂t s = t−2 s · ∇s − t−1 ∇BL (w, w) to be considered as a system of equations for (w , s ) for given (w, s). The first question to be considered is whether the auxiliary system (2.20) defines a dynamics for large time, namely whether the Cauchy problem for that system is locally well posed in a neighborhood of infinity in time, more precisely has a unique solution defined up to infinity in time for sufficiently large t1 and sufficiently large initial time t0 , possibly depending on the size of the initial data. This property was satisfied by the corresponding auxiliary system associated with the Hartree equation and considered in [6] [7]. Here however we encounter serious difficulties associated with the difference of propagation properties of solutions of the Schr¨ odinger and wave equations. In fact a typical solution of the free Schr¨ odinger equation behaves asymptotically in time as (U (t)u+ )(x) ∼ (M DF u+ )(x) = exp(ix2 /2t)(it)−3/2 F u+ (x/t) namely spreads by dilation by t in all directions in the support of F u+ , while by the Huyghens principle A0 remains concentrated in a neighborhood of the light cone, more precisely within a distance R of the latter if the initial data (A+ , A˙ + ) are supported in a ball of radius R. When switching to the new variables (w, B), w tends to a limit when t → ∞ whereas B0 concentrates in a neighborhood of the unit sphere, within a distance R/t of the latter in the previous case of compactly supported data. Note however that for t1 = ∞, B1∞ is expected to tend to a limit like w and not to concentrate like B0 , as can be guessed from (2.15).
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We shall treat the auxiliary system (2.20) by energy methods, and in particular look for w in spaces of the type C([T, ∞), H k ) where H k is the usual Sobolev space based on L2 . In order to treat the nonlinear term B1 (w, w), we shall need a minimal regularity, in practice k > 1. However, when taking H k norms of B0 , the previous concentration phenomenon implies
B0 ; H k ∼ O tk−1/2 which has worse and worse asympotic behaviour in time as k increases. This difficulty manifests itself in the following way: (i) If t0 = t1 < ∞, the available estimates for the sytem (2.20) do not prevent finite time blow up after t0 , even if A0 = 0. This encourages us to take t1 > t0 , and actually the situation becomes slightly better in that case. Nevertheless (ii) the available estimates do not prevent finite time blow up after t1 , which is the same fact as (i) with t0 replaced by t1 , and (iii) if A0 = 0 and if t1 is sufficiently large, the available estimates do not prevent blow up before t1 . A definite improvement occurs however if A0 = 0. (iv) If A0 = 0, the available estimates allow for a proof of existence of solutions in [t0 , t1 ] for t0 sufficiently large and arbitrary t1 > t0 , possibly t1 = ∞. In particular for t1 = ∞, the solutions are defined up to infinity in time. Furthermore, for those solutions, w(t) has a limit w+ as t → ∞. The last case brings us in the same situation as that encountered for the Hartree equation in [6] [7] and could be taken as the starting point for the construction of partial modified wave operators (restricted to the case of vanishing (A+ , A˙ + )) by the same method as in [6] [7]. We shall however refrain from performing that construction and turn directly to the case of nonvanishing (A+ , A˙ + ). In that case, the need to use H k norms with k > 1 for w makes the treatment of A0 nontrivial, even if one drops the interaction term A1 . As a preparation for the general case, we shall therefore first construct the wave operators at the same level of regularity for the simplified system i∂t u = −(1/2)∆u − A0 u (2.22) A0 = 0 namely for a linear Schr¨ odinger equation with time dependent potential A0 satisfying the free wave equation. After the appropriate change of variables u = M Dw
,
A0 = t−1 D0 B0
(2.23)
the Schr¨ odinger equation becomes R(w) ≡ ∂t w − i(2t2 )−1 ∆w − it−1 B0 w = 0 .
(2.24)
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The construction of the wave operators for that equation in L2 , either in the form (2.22) or (2.24) can be easily performed by a simple variant of Cook’s method, and the construction of the wave operators at the level of H k becomes a regularity problem for the previous wave operators. Solving that problem for k ≥ 1 (in fact for k > 1/2) requires special assumptions on the asymptotic states (w+ , A+ , A˙ + ), to the effect that the product B0 w+ decays faster in time in the relevant norms than what would naturally follow from factorized estimates. Those assumptions can be ensured for instance by imposing support properties of w+ , to the effect that w+ = 0 on the unit sphere, and suitable decay of (A+ , A˙ + ) at infinity in space. They will be needed again in the treatment of the general problem. The construction of the modified wave operators in the general case follows the same pattern as for the Hartree equation. The aim is to construct solutions of the auxiliary system (2.20) with suitably prescribed asymptotic behaviour at infinity, and in particular with w(t) tending to a limit w+ as t → ∞. That asymptotic behaviour will be imposed in the form of a suitably chosen pair (W, φ) and therefore (W, S) with S = ∇φ, with W (t) tending to w+ as t → ∞. For fixed (W, S), we make a change of variables in the system (2.18) from (w, ϕ) to (q, ψ) defined by (q, ψ) = (w, ϕ) − (W, φ) (2.25) or equivalently a change of variables in the system (2.20) from (w, s) to (q, σ) defined by (q, σ) = (w, s) − (W, S) , (2.26) and instead of looking for a solution (w, s) of the system (2.20) with (w, s) behaving asymptotically as (W, S), we look for a solution (q, σ) of the transformed system with (q, σ) (and also ψ) tending to zero as t → ∞. Actually for technical reasons, we need to modify the auxiliary system slightly, in the following way. When expanding w = W + q in B1t1 (w, w), we shall replace that quantity by B1t1 ,∞ (w, w) ≡ B1∞ (W, W ) + 2B1t1 (W, q) + B1t1 (q, q) .
(2.27)
We furthermore define the remainders R1 (W, S) = ∂t W − i(2t2 )−1 ∆W − t−2 Q(S, W ) − it−1 (B0 + BS∞ (W, W ))W ∞ R2 (W, S) = ∂t S − t−2 S · ∇S + t−1 ∇BL (W, W ) .
(2.28) (2.29)
Performing the change of variables (2.26) and including the previous technical modification in the system (2.20) yields the modified auxiliary system for the new variables (q, σ). ∂t q = i(2t2 )−1 ∆q + t−2 (Q(s, q) + Q(σ, W )) + it−1 B0 q +it−1 BSt1 ,∞ (w, w)q + it−1 2BSt1 (W, q) + BSt1 (q, q) W − R1 (W, S) (2.30) t1 t1 (W, q) + BL (q, q) − R2 (W, S). ∂t σ = t−2 (s · ∇σ + σ · ∇S) − t−1 ∇ 2BL
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Note that changing B t1 (w, w) to B t1 ,∞ (w, w) changes A by a solution of the free wave equation, so that we are still solving the original system (1.1) (1.2), with however a slightly different A0 as compared with (2.2). For the same reason as for the partly resolved system (1.1) (2.2) and for the auxiliary system (2.20), we shall use at intermediate stages a partly linearized version of the system (2.30), namely ∂t q = i(2t2 )−1 ∆q + t−2 (Q(s, q ) + Q(σ, W )) + it−1 B0 q (2.31) +it−1 BSt1 ,∞ (w, w)q + it−1 2BSt1 (W, q) + BSt1 (q, q) W − R1 (W, S) t1 t1 ∂t σ = t−2 (s · ∇σ + σ · ∇S) − t−1 ∇ 2BL (W, q) + BL (q, q) − R2 (W, S). The construction of solutions (q, σ) tending to zero at infinity for the system (2.30) with t1 = ∞ proceeds in several steps. We assume first that (W, S) and B0 satisfy suitable boundedness properties and that the remainders R1 (W, S) and R2 (W, S) satisfy suitable decay in time. We solve the linearized system (2.31) for (q , σ ) for given (q, σ), both with finite and infinite time t1 and initial time t0 . We then solve (2.30) by proving that the map Γ : (q, σ) → (q , σ ) is a contraction in suitable norms. We also prove that the solution of (2.30) with t0 = t1 < ∞ converges to the solution with t0 = t1 = ∞ when t0 → ∞, a property which is natural in the framework of scattering theory. There remains the task of constructing (W, S) with W (t) tending to w+ as t → ∞, and satisfying the required boundedness and decay properties. This is done by solving the auxiliary system (2.20) with t1 = ∞ approximately by iteration. We restrict our attention to the second iteration, which is sufficient to cover the range 1 < k < 2. The pair (W, S) or equivalently (W, φ) thereby obtained depends only on the asymptotic state w+ . Solving the auxiliary system (2.30) with that (W, S) and with t0 = t1 = ∞ yields a solution (w, s) of the system (2.20) and therefore a solution (w, ϕ) of the system (2.18) with prescribed asymptotic behaviour characterized by (W, S) or (W, φ). That solution depends on (w+ , A+ , A˙ + ). Plugging that solution with w+ = F u+ into (2.11) and substituting u thereby obtained into (2.2) (2.4) with t1 = ∞ yields a solution (u, A) of the system (1.1) (1.2) with prescribed asymptotic behaviour in time explicitly expressed in terms of the asymptotic state (u+ , A+ , A˙ + ). More precisely, that asymptotic behaviour is obtained or rather defined by replacing (w, ϕ) by (W, φ) and |u|2 = |Dw|2 by |DW |2 in (2.11) and (2.2) (2.4) with t1 = ∞, so that 2 actually (u, A) behaves asymptotically as (M D exp(−iφ)W , A0 + A∞ 1 (|DW | )), which plays the role of modified free solution for the system (1.1) (1.2). As a by product of that construction, we can define the map Ω : (u+ , A+ , A˙ + ) → (u, A), which is the required modified wave operator for the system (1.1) (1.2). The main result of this paper, namely the construction of solutions of the system (1.1) (1.2) defined for large time and with prescribed asymptotic behaviour as described above, is stated in full mathematical detail in Proposition 8.1 below. Since however that detail is rather cumbersome, we give here a heuristic description thereof, which can serve as a reader’s guide for that proposition. One starts
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with asymptotic states (u+ , A+ , A˙ + ) which are sufficiently regular in the sense that (i) w+ ≡ F u+ ∈ H k+ for sufficiently large k+ , (ii) the solution A0 of the free wave equation generated by (A+ , A˙ + ) according to (2.3) satisfies the optimal time decay associated with that equation in suitable norms (see (8.9)≡(3.15)) and satisfies an additional joint time decay with w+ , needed to damp light cone interferences (see (8.10)). One then constructs model asymptotic functions (W, S) for the auxiliary system (2.20), depending only on w+ , by solving a truncated version of that system approximately by iteration to second order (see (7.3) (7.5) (7.7)). The main technical result is that one can construct a unique solution (w, s) of the auxiliary system (2.20), defined for large time, and asymptotic to (W, S) in suitable norms (see (8.11) (8.12) (8.13)). One then defines the phases ϕ and φ corresponding to s and S according to s = ∇ϕ and S = ∇φ and one reconstructs (u, A) from (w, ϕ) by (2.11) and (2.2) (2.4) with t1 = ∞. Then (u, A) is a solution of the system (1.1) (1.2), defined for large time, and (u, A) is asymptotic to the modified free solution 2 (M D exp(−iφ)W, A0 + A∞ 1 (|DW | )) in suitable norms (see (8.15)–(8.23)). The auxiliary system (2.18) satisfies a gauge invariance property similar to that of the corresponding system for the Hartree equation used in [6] [7], and the construction of the intermediate wave operator for that system can be made in a gauge covariant way. For brevity we shall refrain from discussing that question in this paper. We now describe the contents of the technical parts of this paper, namely Sections 3–8. In Section 3, we introduce some notation, define the relevant function spaces and collect a number of preliminary estimates. In Section 4, we study the Cauchy problem for large time for the auxiliary system (2.20). We solve the Cauchy problem with finite initial time for the linearized system (2.21) (Proposition 4.1), we prove a number of uniqueness results for the system (2.20) (Proposition 4.2), we prove the existence of a limit w(t) of w+ for suitably bounded solutions of the system (2.20) (Proposition 4.3), we discuss in more quantitative terms the possible occurrence of blow up mentioned above, and we finally solve the Cauchy problem for the system (2.20) with t1 = ∞ and large t0 in the special case A0 = 0 (Proposition 4.4). In Section 5, as a preparation for the construction of the wave operators for the system (2.20) with A0 = 0, we study the existence of wave operators for the linear problem (2.22) in the form (2.24). In particular we prove the existence of L2 -wave operators by a variant of Cook’s method (Proposition 5.2), we prove the H k regularity of those wave operators under suitable decay assumptions of R(W ) for the model function W (Proposition 5.3) and we finally reduce those decay properties to conditions on the asymptotic state (w+ , A+ , A˙ + ). In Section 6 and 7, we study the Cauchy problem at infinity in the general case A0 = 0 for the auxiliary system (2.20) in the difference form (2.30). Under suitable boundedness assumptions on (W, S) and decay assumptions on R1 (W, S) andf R2 (W, S) we prove the existence of solutions for t0 and t1 finite and infinite, first for the linearized system (2.31) (Propositions 6.1 and 6.2) and then for the nonlinear system (2.30) (Proposition 6.3). We then choose appropriate (W, S), prove that they satisfy the
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required assumptions (Lemmas 7.1 and 7.2) and finally state the result on the Cauchy problem at infinity for the system (2.30) in H k for 1 < k < 2 (Proposition 7.1). Finally in Section 8, we construct the wave operators for the system (1.1) (1.2) from the results previously obtained for the system (2.30) and we derive the asymptotic estimates for the solutions (u, A) in their range that follow from the previous estimates (Proposition 8.1).
3 Notation and preliminary estimates In this section we introduce some additional notation and we collect a number of estimates which will be used throughout this paper. We shall use the Sobolev spaces Hrk defined for 1 ≤ r ≤ ∞ by
Hrk = u : u; Hrk ≡ < ω >k u r < ∞ where < · >= (1 + | · |2 )1/2 . The subscript r will be omitted if r = 2. We shall look for solutions of the auxiliary system (2.20) in spaces of the type C(I, X k, ) where I is an interval and X k, = H k ⊕ ω −1 H namely
X k, = (w, s) : w ∈ H k , ∇s ∈ H 2
(3.1) 6
where it is understood that ∇s ∈ L includes the fact that s ∈ L , and we shall use the notation
w; H k = |w|k . (3.2) We shall use extensively the following Sobolev inequalities, stated here in Rn , but to be used only for n = 3. Lemma 3.1 Let 1 < q, r < ∞, 1 < p ≤ ∞ and 0 ≤ j < k. If p = ∞, assume that k − j > n/r. Let σ satisfy j/k ≤ σ ≤ 1 and n/p − j = (1 − σ)n/q + σ(n/r − k) . Then the following inequality holds
ω j u p ≤ C u 1−σ
ω k u σr . q
(3.3)
The proof follows from the Hardy-Littlewood-Sobolev (HLS) inequality ([16], p. 117) (from the Young inequality if p = ∞), from Paley-Littlewood theory and interpolation. We shall also use extensively the following Leibnitz and commutator estimates.
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Lemma 3.2 Let 1 < r, r1 , r3 < ∞ and 1/r = 1/r1 + 1/r2 = 1/r3 + 1/r4 . Then the following estimates hold
ω m (uv) r ≤ C ( ω m u r1 v r2 + ω m v r3 u r4 )
(3.4)
for m ≥ 0, and
[ω m , u]v r ≤ C ω m u r1 v r2 + ω m−1 v r3 ∇u r4
(3.5)
for m ≥ 1, where [ , ] denotes the commutator. The proof of those estimates is given in [17] [18] with ω replaced by < ω > and follows therefrom by a scaling argument. We shall also need the following consequence of Lemma 3.2. Lemma 3.3 Let m ≥ 0 and 1 < r < ∞. Then the following estimate holds
ω m (eϕ − 1) r ≤ ω m ϕ r exp (C ϕ ∞ ) .
(3.6)
Proof. For any integer n ≥ 2, we estimate m n−1 an ≡ ω m ϕn r ≤ C ω m ϕ r ϕ n−1
r ϕ ∞ ∞ + ω ϕ = C a1 bn−1 + an−1 b (3.7) by (3.4), where b = ϕ ∞ and we can assume C ≥ 1 without loss of generality. It follows easily from (3.7) that an ≤ n(Cb)n−1 a1 for all n ≥ 1, from which (3.6) follows by expanding the exponential.
t1 defined by (2.15) (2.17). It We next give some estimates of B1t1 , BSt1 and BL follows immediately from (2.17) that
ω m BSt1 2 ≤ tβ(m−p) ω p BSt1 2 ≤ tβ(m−p) ω p B1t1 2
(3.8)
for m ≤ p and similarly t1 t1
ω m BL
2 ≤ tβ(m−p) ω p BL
2 ≤ tβ(m−p) ω p B1t1 2
(3.9)
for m ≥ p. On the other hand it follows from (2.15) that t1 ( ω m (w1 w ¯2 ) 2 )
ω m+1 B1t1 (w1 , w2 ) 2 ≤ Im
(3.10)
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t1 where Im is defined by
t1 /t t1 −m−3/2 Im (f ) (t) = dν ν f (νt) 1
or equivalently
t1 m+1/2 Im (f ) (t) = t
t1
dt t
−m−3/2
t
(3.11)
f (t )
for t > 0, t1 > 0. Most of the subsequent estimates of B1t1 will follow from (3.8) (3.9) (3.10) and from an estimate of ω m (w1 w ¯2 ) 2 . The latter follows from the HLS inequality if −3/2 < m < 0 and from (3.4) if m ≥ 0. For future reference, we quote the following special case, which will occur repeatedly k t1
ω w 22 (3.12)
ω 2k−1/2 B1t1 (w, w) 2 ≤ C I2k−3/2 and which holds for 0 < k < 3/2. The required estimate
ω 2k−3/2 |w|2 2 ≤ C ω k w 22
(3.13)
follows from the HLS inequality if 2k < 3/2 and from (3.4) if 2k ≥ 3/2, as mentioned above, and from Sobolev inequalities. t1 of B1t1 . We next give a special estimate of the long range part BL Lemma 3.4 Let m > −3/2. Then t1 t1
ω m+1 BL (w1 , w2 ) 2 ≤ C tβ(m+3/2) I−3/2 ( w1 2 w2 2 ) .
(3.14)
Proof. Let f = D0 (ν)Re w1 w ¯2 . From (2.15) (2.17), we estimate t1 /t m+1 t1 −3 β m BL (w1 , w2 ) 2 ≤ dν ν
χ(|ξ| ≤ t )|ξ| F f (ξ) 2
ω 1 t1 /t ≤ dν ν −3 χ(|ξ| ≤ tβ )|ξ|m 2 F f ∞ 1 t1 /t ≤C dν tβ(m+3/2) (w1 w ¯2 )(νt) 1 1
which implies (3.14).
We finally collect some estimates of the solution of the free wave equation A0 = 0 with initial data (A+ , A˙ + ) at time zero, given by (2.3). Lemma 3.5 Let m ≥ 0. Let ω m A+ ∈ L2 , ω m−1 A˙ + ∈ L2 , ∇2 ω m A+ ∈ L1 and ∇ω m A˙ + ∈ L1 . Then the following estimate holds
ω m A0 r ≤ b0 t−1+2/r for all t > 0.
for 2 ≤ r ≤ ∞
(3.15)
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Proof. It suffices to prove (3.15) for m = 0, for r = 2 and r = ∞. For r = 2, it follows from (2.3) that
A0 2 ≤ A+ 2 + ω −1 A˙ + 2
(3.16)
for all t ∈ R. For r = ∞, the result follows from the divergence theorem applied to the standard representation of solutions of the free wave equation in terms of spherical means [21]. The time decay expressed by (3.15) is known to be optimal, and we shall always consider solutions A0 of the free wave equation satisfying those estimates for suitable m. In the applications, we shall use the estimates (3.15) in the equivalent form expressed in terms of B0 defined by (2.13), namely
ω m B0 r ≤ b0 tm−1/r
for 2 ≤ r ≤ ∞ .
(3.17)
4 Cauchy problem and preliminary asymptotics for the auxiliary system In this section, we study the Cauchy problem for the auxiliary system (2.20) and we derive some preliminary asymptotic properties of its solutions. This section illustrates both the method of solution with the help of the linearized version (2.21) of that system and the difficulties arising from the different propagation properties of the Schr¨ odinger and wave equations. In particular we are able to prove the existence of solutions up to infinity in time only if A0 = 0. This section could be the starting point for the construction of partial wave operators with vanishing asymptotic states for the field A, a construction which would be very similar to that of the wave operators for the Hartree equation performed in [6] [7], but which we shall refrain from performing here. The general case of non-vanishing asymptotic states for A will be treated by a similar but more complicated method in Section 6 below. The basic tool of this section consists of a priori estimates for suitably regular solutions of the linearized system (2.21). Those estimates can be proved by a regularisation and limiting procedure and hold in the integrated form at the available level of regularity. For brevity, we shall state them in differential form and we shall restrict the proof to the formal computation. We first estimate a single solution of the linearized system (2.21) at the level of regularity where we shall eventually solve the auxiliary system (2.20). Lemma 4.1 Let 1 < k ≤ :, : > 3/2 and β > 0. Let I ⊂ [1, ∞) be an interval and ¯ Let B0 satisfy the estimates (3.17) for 0 ≤ m ≤ k. Let (w, s), (w , s ) ∈ let t1 ∈ I. k, C(I, X ) with w ∈ L∞ (I, H k ) and let (w , s ) be a solution of the system (2.21) in I. Then the following estimates hold for all t ∈ I:
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w 2 = const.
1/k 1−1/k 1−δ/k δ/k C b0 w 2 |w |k + tk−1−δ/3 w 2 |w |k
t1 (|w|2k ) |w |k (4.1) + C t−2 |∇s| + t−1−β1 Im 1
t1 t1 |∂t |∇s | | ≤ C t−2 |∇s| |∇s | + C t−1+β2 Im ( w 2 |w|k ) + Im (|w|2k ) (4.2) 1 1 −k |∂t |w |k | ≤
where 0 < δ ≤ [k ∧ 3/2], β1 = β[1 ∧ 2(k − 1)] = β(1 − 2[3/2 − k]+ ) , m1 = [k ∧ (2k − 3/2)] = k − [3/2 − k]+ ,
(4.3) (4.4)
β2 = β(: + 1 − k + [3/2 − k]+ ) .
(4.5)
Proof. We omit the superscript t1 in all the proof. We first estimate w . It is clear from (2.21) that w 2 = const. We next estimate ∂t ω k w 2 ≤ t−1 [ω k , B0 ]w 2 +t−2 [ω k , s] · ∇w 2 + (∇ · s)ω k w 2 + ω k ((∇ · s)w ) 2 + t−1 [ω k , BS (w, w)]w 2 . (4.6) The contribution of B0 is estimated by Lemma 3.2 and (3.17) as
[ω k , B0 ]w 2 ≤ C ∇B0 ∞ ω k−1 w 2 + ω k B0 3/δ w r
(4.7) ≤ C b0 t ω k−1 w 2 +tk−δ/3 w r with 0 < δ = δ(r) < k ∧ 3/2. This yields the first term in the RHS of (4.1) by Sobolev inequalities and interpolation. We next estimate by Lemma 3.2
[ω k , s] · ∇w 2 + (∇ · s)ω k w 2 + ω k ((∇ · s)w ) 2 ≤ C ∇s ∞ ω k w 2 + ω k s 3/δ ∇w r + ω k (∇ · s) 3/δ w r where 0 < δ = δ(r) ≤ [(k − 1) ∧ 3/2] and 0 < δ = δ(r ) ≤ [k ∧ 3/2]. Choosing δ = [(k − 1) ∧ 1/2] and δ = [k ∧ 3/2] and using Sobolev inequalities, we continue the previous estimate by · · · ≤ C ∇s ∞ ω k w 2 + ω [k∨3/2] ∇s 2 ω [k∧3/2] w 2
+χ(k > 3/2) ω k ∇s 2 w ∞ ≤ C |∇s| |w |k . (4.8) We next estimate the contribution of BS to (4.6). By Lemma 3.2 and Sobolev inequalities, we estimate
[ω k , BS (w, w)]w 2
≤ C ∇BS (w, w) 3 ω k−1 w 6 + ω k BS (w, w) 3/δ w r ≤ C ω 3/2 BS (w, w) 2 ω k w 2 + ω k+3/2−δ BS (w, w) 2 w r (4.9) where 0 < δ = δ(r) ≤ 3/2. We choose δ = [k ∧ 3/2] and continue (4.9) as follows:
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If k < 3/2, so that δ = k, ··· ≤ ≤ ≤ ≤
C ω 3/2 BS (w, w) 2 ω k w 2 C t−2β(k−1) ω 2k−1/2 B1 (w, w) 2 ω k w 2 C t−2β(k−1) I2k−3/2 ω k w 22 ω k w 2 C t−β1 Im1 |w|2k |w |k
(4.10)
by Sobolev inequalities, by (3.8) (3.12) and by the definitions (4.3) (4.4). If k = 3/2, so that δ = 3/2 − ε, ··· ≤ ≤ ≤ ≤
C ω 3/2+ε BS (w, w) 2 ω 3/2−ε w 2 C t−β(1−2ε) ω 5/2−ε B1 (w, w) 2 ω 3/2−ε w 2
C t−β(1−2ε) I3/2−ε ω (3−ε)/2 w 22 ω 3/2−ε w 2 C t−β1 Im1 |w|2k |w |k
(4.11)
by Sobolev inequalities, by (3.8), by (3.12) with k = (3 − ε)/2 and by (4.3) (4.4). If k > 3/2, so that δ = 3/2 and r = ∞, · · · ≤ C ω k+1 B1 (w, w) 2 t−β(k−1/2) ω k w 2 +t−β w ∞ ≤ C t−β Ik ω k w 2 w ∞ ω k w 2 + w ∞ (4.12) ≤ C t−β1 Im1 |w|2k |w |k by (3.8) (3.10), Lemma 3.2, Sobolev inequalities and (4.3) (4.4). Substituting (4.7) (4.8) (4.10) (4.11) (4.12) into (4.6) yields (4.1). We now turn to the estimate of s , namely to the proof of (4.2). For 0 ≤ m ≤ :, we estimate
∂t ω m+1 s 2 ≤ t−2 [ω m+1 , s] · ∇s 2 + (∇ · s)ω m+1 s 2 +t−1 ω m+2 BL (w, w) 2 . The first bracket in the RHS of (4.13) is estimated by Lemma 3.2 as {·} ≤ C ∇s ∞ ω m+1 s 2 + ω m+1 s 2 ∇s ∞ ≤ C|∇s| |∇s |
(4.13)
(4.14)
by Sobolev inequalities. The contribution of BL for m = : is estimated by (3.9) (3.10) (3.12) and Lemma 3.2 as
ω +2 BL (w, w) 2 ≤ C tβ(+5/2−2k) I2k−3/2 ω k w 22 for k < 3/2 , k β(+1−k) C t for k > 3/2 , Ik ω w 2 w ∞
for k = 3/2 , C tβ(−1/2+ε) I3/2−ε ω (3−ε)/2 w 22 ≤ C tβ2 Im1 |w|2k (4.15)
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in all cases. The contribution of BL for m = 0 is estimated similarly as
∇2 BL (w, w) 2 ≤ C tβ(5/2−k) Ik−3/2 ω k w 2 w 2 for k < 3/2 , C tβ I0 ( w ∞ w 2 )
C tβ(1+ε) I−ε ω 3/2−ε w 2 w 2 β2
≤C t in all cases, with
for k > 3/2 , for k = 3/2 ,
Im1 −k ( w 2 |w|k )
β2 = β (1 + [3/2 − k]+ ) ≤ β2
since : ≥ k. Collecting (4.14) (4.15) (4.16) yields (4.2).
(4.16) (4.17)
We next estimate the difference of two solutions of the linearized system (2.21) corresponding to two different choices of (w, s). We estimate that difference at a lower level of regularity than the solutions themselves. Lemma 4.2 Let 1 < k ≤ :, : > 3/2 and β > 0. Let I ⊂ [1, ∞) be an interval ¯ Let B0 be sufficiently regular, for instance B0 ∈ C(I, H k ). Let and let t1 ∈ I. 3 (wi , si ), (wi , si ) ∈ C(I, X k, ) with wi ∈ L∞ (I, H k ), i = 1, 2, and let (wi , si ) be solutions of the system (2.21) associated with (wi , si ). Define (w± , s± ) = (1/2)(w1 ± w2 , s1 ± s2 ) and (w± , s± ) = 1/2(w1 ± w2 , s1 ± s2 ). Then the following estimates hold for all t ∈ I: ∂t w 2 ≤ C t−2 |∇s− |0 |w |k + Ct−1−β1 I t1 − + m1 −k (|w+ |k w− 2 ) |w+ |k (4.18) ∂t |∇s |0 ≤ C t−2 |∇s+ | |∇s |0 + |∇s− |0 |∇s | − − + +
t1 (|w+ |k w− 2 ) C t−1+β2 Im 1 −k
(4.19)
where β1 , m1 and β2 are defined by (4.3) (4.4) (4.5) and where [3/2 − k]+ ≤ :0 ≤ : − k .
(4.20)
Proof. We again omit the superscript t1 in the proof. Taking the difference of the system (2.21) for (wi , si ), we obtain the following system for (w− , s− ): ∂ w = i(2t2 )−1 ∆w− + t−2 (Q(s+ , w− ) + Q(s− , w+ )) + it−1 B0 w− t −
+it−1 (BS (w+ , w+ ) + BS (w− , w− )) w− + 2BS (w+ , w− )w+ (4.21) −2 −1 s+ · ∇s− + s− · ∇s+ − 2t ∇BL (w+ , w− ) . ∂t s− = t We first estimate w− . From (4.21) we obtain ∂t w−
2 ≤ t−2 Q(s− , w+ ) 2 +2t−1 BS (w+ , w− )w+
2
(4.22)
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where only those terms appear that do not preserve the L2 -norm. We estimate the first norm in the RHS by H¨ older and Sobolev inequalities as follows: If k < 3/2,
Q(s− , w+ ) 2 ≤ C s− 3/(k−1) + ∇ · s− 3/k ω k w+
2 ≤
C ω 3/2−k ∇s− ω k w 2 .
If k = 3/2,
Q(s− , w+ ) 2 ≤ C ω ε ∇s− 2 ω 3/2−ε w+
2 .
If k > 3/2,
Q(s− , w+ ) 2 ≤ C ∇s− 2 ∇w+
3 + w+
∞ ,
and in all cases
) 2 ≤ C|∇s− |0 |w+ |k
Q(s− , w+
(4.23)
provided :0 ≥ [3/2 − k]+ . We estimate the second norm in the RHS of (4.22) by (3.8) (3.10), by Lemma 3.2 and by the H¨ older and Sobolev inequalities as follows: If k < 3/2,
BS (w+ , w− )w+
2 ≤ C ω 3/2−k BS (w+ , w− ) 2 ω k w+
2 ≤ C t−2β(k−1) ω k−1/2 B1 (w+ , w− ) 2 ω k w+
2 −2β(k−1) k k ≤C t Ik−3/2 ω w+ 2 w− 2 ω w+ 2 .
If k = 3/2,
2 ≤ C t−β(1−2ε) I−ε ω 3/2−ε w+ 2 w− 2 ω 3/2−ε w+
2 .
BS (w+ , w− )w+ If k > 3/2,
2 ≤ t−β ∇BS (w+ , w− ) 2 w+
∞
BS (w+ , w− )w+ ≤ C t−β I0 ( w+ ∞ w− 2 ) w+
∞
and in all cases
BS (w+ , w− )w+
2 ≤ C t−β1 Im1 −k (|w+ |k w− 2 ) |w+ |k .
(4.24)
with β1 and m1 defined by (4.3) (4.4). Substituting (4.23) (4.24) into (4.22) yields (4.18). We now turn to the estimate of s− , namely to the proof of (4.19). From (4.21) we estimate for m ≥ 0 ∂t ω m+1 s− 2 ≤ t−2 [ω m+1 , s+ ] · ∇s− 2 + (∇ · s+ )ω m+1 s− 2 + ω m+1 (s− · ∇s+ ) 2 + 2t−1 ω m+2 BL (w+ , w− ) 2 . (4.25)
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If m = 0 (a case which has to be self-estimating if :0 = 0, which is allowed if k > 3/2), we estimate the bracket in the RHS of (4.25) directly as {(m = 0)} ≤ ∇s+ ∞ ∇s− 2 + ∇s− 2 ∇s+ ∞ + ∇2 s+ 3 (4.26) ≤ C |∇s+ | ∇s− 2 + ∇s− 2 |∇s+ | since : > 3/2. If m > 0, we estimate that bracket by Lemma 3.2 and Sobolev inequalities as {·} ≤ C ∇s+ ∞ ω m+1 s− 2 + ω m+1 s+ 3/δ ∇s− r + ω m+1 s− 2 ∇s+ ∞ + s− r ω m+2 s+ 3/δ (4.27) where 0 < δ = δ(r) ≤ 3/2, 0 < δ = δ(r ) ≤ 3/2. The first and third term in the RHS of (4.27) are readily controlled by the corresponding terms in (4.19) for 0 < m ≤ :0 and : > 3/2. The remaining two terms are similarly controlled through Sobolev inequalities provided 0 < δ ≤ [:0 ∧ 3/2] , m + 3/2 − δ ≤ : . 1 ≤ δ ≤ [(:0 + 1) ∧ 3/2] , m + 5/2 − δ ≤ : . Those conditions are easily seen to be compatible in δ and δ for all m, 0 < m ≤ :0 , provided : ≥ [(:0 + 1) ∨ 3/2], which follows from : > 3/2 and : ≥ :0 + k. We finally estimate the contribution of BL (w+ , w− ) by
ω m+2 BL (w+ , w− ) 2 ≤ C tβm ∇2 BL (w+ , w− ) 2 by (3.9) and we estimate the last norm in exactly the same way as in (4.16), thereby obtaining
ω m+2 BL (w+ , w− ) 2 ≤ C tβ2 +βm Im1 −k (|w+ |k w− 2 ) .
(4.28)
Collecting (4.25) (4.26), (4.27) and the discussion that follows, and (4.28) and noting that β2 + βm ≤ β2 for m ≤ :0 ≤ : − k, we obtain (4.19). With the estimates of Lemma 4.1 and 4.2 available, it is an easy matter to solve the Cauchy problem globally in time for the linearized system (2.21). Proposition 4.1 Let 1 < k ≤ :, : > 3/2 and β > 0. Let I ⊂ [1, ∞) be an interval ¯ Let B0 satisfy the estimates (3.17) for 0 ≤ m ≤ k. Let (w, s) ∈ and let t1 ∈ I. C(I, X k, ) with w ∈ L∞ (I, H k ). Let t0 ∈ I and let (w0 , s0 ) ∈ X k, . Then the system (2.21) has a unique solution (w , s ) ∈ C(I, X k, ) with (w , s )(t0 ) = (w0 , s0 ). That solution satisfies the estimates (4.1) (4.2) for all t ∈ I. Two such solutions (wi , si ) associated with (wi , si ), i = 1, 2, satisfy the estimates (4.18) (4.19) for all t ∈ I. Proof. The proof proceeds in the same way as that of Proposition 4.1 of [6], through a parabolic regularization and a limiting procedure, with the simplification that
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the system (2.21) is linear. We define U1 (t) = U (1/t), w (t) = U1 (t)w (t). We first consider the case t ≥ t0 . The system (2.21) with a parabolic regularization added is rewritten in terms of the variables (w , s ) as ∂w = η∆w + t−2 U1 Q(s, U1∗ w ) + it−1 U1 (B0 + BS (w, w))U1∗ w t ≡ η∆w + F (w ) (4.29) −2 −1 ∂t s = η∆s + t s · ∇s − t ∇BL (w, w) ≡ η∆s + G(s ) where the parametric dependence of F , G on (w, s) has been omitted. The Cauchy problem for the system (4.29) can be recast in the integral form t w w 0 F (w ) (t − t ) dt V (t − t ) (4.30) (t) = V + (t ) η 0 η s s0 G(s ) t0 where Vη (t) = exp(ηt∆). The operator Vη (t) is a contraction in X k, and satisfies the bound
∇Vη (t); L(X k, ) ≤ C(ηt)−1/2 . From those facts and from estimates on F , G similar to and mostly contained in those of Lemma 4.1, it follows by a contraction argument that the system (4.30) has a unique solution (w η , sη ) ∈ C([t0 , t0 + T ], X k,) for some T > 0 depending only on |w0 |k , |s0 |˙ and η. That solution satisfies the estimates (4.1) and (4.2) and can therefore be extended to I+ = I ∩ {t : t ≥ t0 } by a standard globalisation argument using Gronwall’s inequality. We next take the limit η → 0. Let η1 , η2 > 0 and let (wi , si ) = (wη i , sηi ), i = 1, 2 be the corresponding solutions. Let (w− , s− ) = (1/2)(w1 − w2 , s1 − s2 ). By estimates similar to, but simpler than those of Lemma 4.2, since in particular (w− , s− ) = 0, we obtain ∂t w−
22 ≤ |η1 − η2 | ∇w1 22 + ∇w2 22 ∂t ∇s− 22 ≤ |η1 − η2 | ∇2 s1 22 + ∇2 s2 22 + Ct−2 ∇s+ ∞ ∇s− 22 . Those estimates imply that (wη , sη ) converges in X 0,0 uniformly in time in the compact subintervals of I+ , to a solution of the original system. It follows then by a standard compactness argument using the estimates (4.1) (4.2) that the limit belongs to C(I+ , X k, ). This completes the proof for t ≥ t0 . The case t ≤ t0 is treated similarly. We now turn to the Cauchy problem for the auxiliary system (2.20). Because of the difficulties described in Section 2, the problem of existence of solutions is scattered with pitfalls, as the discussion below will show. On the other hand, the uniqueness problem of suitably bounded solutions is a rather easy matter and we consider that problem first. The proof relies entirely on Lemma 4.2 and therefore does not require any a priori estimate on B0 . The snag of course is that it is difficult to prove the existence of solutions with the required boundedness properties.
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Proposition 4.2 Let 1 < k ≤ :, : > 3/2 and β > 0. Let I ⊂ [1, ∞) be an interval ¯ Let B0 be sufficiently regular, for instance B0 ∈ C(I, H k ). and let t1 ∈ I. 3
(1) Let t0 = t1 < ∞ and let (w0 , s0 ) ∈ X k, . Then the system (2.20) has at most one solution (w, s) ∈ C(I, X k, ) with w ∈ L∞ (I, H k ) and (w, s)(t0 ) = (w0 , s0 ). Let now β2 < 1, where β2 is defined by (4.5), let :0 satisfy (4.20). Let (wi , si ), i = 1, 2 be two solutions of the system (2.20) in I such that (wi , tη−1 si ) ∈ (C ∩ L∞ )(I, X k, ) for some η > 0 and let
wi , L∞ (H k ) ≤ a
,
tη−1 ∇si ; L∞ (H ) ≤ b .
(4.31)
(2) Let t0 ∈ I, t0 ≤ t1 , t0 < ∞ and assume that (w1 , s1 )(t0 ) = (w2 , s2 )(t0 ). Then there exists c = c(a, b) such that if
−(1−β2 ) 1 (1 − (t0 /t1 )α ) ≤ c(a, b) ∨ t−β (4.32) t0 0 where α = [k ∧ 3/2] − 1, then (w1 , s1 ) = (w2 , s2 ). In particular there exists T0 = T0 (a, b) such that if t0 ≥ T0 , then (w1 , s1 ) = (w2 , s2 ). (3) Let t1 = ∞. Assume that w1 − w2 2 tβ2 and |∇(s1 − s2 )|0 tend to zero when t → ∞. Then (w1 , s1 ) = (w2 , s2 ). Proof. If (wi , si ), i = 1, 2 are two solutions of the system (2.20) in C(I, X k, ), then they satisfy the estimates (4.18) (4.19) with (wi , si ) = (wi , si ), which we denote (4.18=) (4.19=) and refrain from rewriting for brevity. The proof consists in exploiting those estimates to prove that (w1 , s1 ) = (w2 , s2 ). We define y = w− 2 and z = |∇s− |0 . Part (1). With t0 = t1 < ∞, the estimates (4.18=) (4.19=) take the general form |∂t y| ≤ f1 (t)z + g1 (t) |∂t z| ≤ f2 (t)z + g2 (t)
t
t0 t t0
dt h1 (t ) y(t )
(4.33)
dt h2 (t ) y(t )
(4.34)
for suitable continuous nonnegative functions f1 , g1 , h1 , f2 , g2 , h2 (actually h1 = h2 , but that is irrelevant). Furthermore y(t0 ) = z(t0 ) = 0. We shall reduce the system (4.33) (4.34) to a standard form where Gronwall’s inequality is applicable. We restrict our attention to the case t ≥ t0 for definiteness. The case t ≤ t0 can be treated similarly. Defining z by t dt f2 (t ) z(t) , z(t) = E(t) z(t) = exp t0
we reduce the system (4.33) (4.34) for (y, z) to a similar system for (y, z), where f2 , g2 and f1 are replaced by 0, E −1 g2 and Ef1 . We can therefore assume that
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f2 = 0. Then z(t) ≤
t
t0
dt g2 (t )
t
t0
dt h2 (t ) y(t ) ≤ G2 (t)
where
G2 (t) =
t
t0
t t0
dt h2 (t ) y(t )
dt g2 (t )
so that ≤
∂t y
f1 (t)G2 (t)
t
t0
dt h2 (t )y(t ) + g1 (t)
≤
(f1 G2 + g1 )
t t0
t
t0
dt h1 (t ) y(t )
dt (h1 ∨ h2 )(t ) y(t )
which is of the same form as (4.33) with f1 = 0. Integrating the latter yields y≤
t t0
dt g1 (t )
t t0
dt h1 (t ) y(t ) ≤ G1 (t)
where
G1 (t) =
t t0
t
t0
dt h1 (t ) y(t )
dt g1 (t ) ,
which together with y(t0 ) = 0 implies y(t) = 0 for all t by an easy variant of Gronwall’s inequality. Substituting that result into (4.34) (with f2 = 0) yields z = 0 and therefore (w1 , s1 ) = (w2 , s2 ). We now turn to the proof of Parts (2) and (3). Introducing the assumption and notation (4.31), changing the variable from ν to t = νt in the definition of t1 Im , and omitting an absolute overall constant, we can rewrite (4.18=) (4.19=) in the form t1 |∂t y| ≤ t−2 az + t−1−β1 +α a2 dt t−1−α y(t ) (4.35) t t1 dt t−1−α y(t ) (4.36) |∂t z| ≤ t−1−η bz + t−1+β2 +α a t
where α = [k ∧ 3/2] − 1 > 0, and the goal is to prove that (4.35) (4.36) with suitable initial conditions imply y = z = 0. Part (2). Let Y = y; L∞ ([t0 , t1 ]) . Then tα
t
t1
dt t−1−α y(t ) ≤ ≤
Y α−1 (1 − (t/t1 )α ) Y α−1 (1 − t0 /t1 )α ) ≡ Y¯ .
(4.37)
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Substituting (4.37) into (4.36) and integrating with z(t0 ) = 0 yields z ≤ exp b η −1 t−η a Y¯ β2−1 tβ2 . 0
(4.38)
Substituting (4.37) (4.38) into (4.35), integrating with y(t0 ) = 0 and taking the Supremum over t in [t0 , t1 ] yields −1 −β1 −1 −(1−β2 ) a2 Y¯ (1 − β ) t + β t Y ≤ exp b η −1 t−η 2 0 1 0 0 which implies Y = 0 and therefore y = z = 0 provided −(1−β2 ) 1 a2 exp b η −1 t−η + β1−1 t−β α−1 (1 − (t0 /t1 )α ) < 1 , (1 − β2 )−1 t0 0 0 (4.39) a condition which follows from (4.32) for suitable c(a, b). Part (3). We now take t1 = ∞. The term bz in (4.36) can be exponentiated as in the proof of Part (2). Since in addition the statement does not involve conditions on a and b, we can and shall assume without loss of generality that b = 0 and a = 1. Let (4.40) ε(t) =Sup tβ2 y(t ) . t ≥t
Then ε(t) is nonincreasing in t and tends to zero as t → ∞. Furthermore for any t0 ∈ I ∞ −(α+β2 ) dt t−1−α y(t ) ≤ ε(t0 )(α + β2 )−1 t0 . (4.41) t0
Let now t0 ∈ I (t0 will eventually tend to ∞), y0 = y(t0 ) and z0 = z(t0 ). We estimate y and z for t ≤ t0 by integrating (4.35) (4.36) (with t1 = ∞, a = 1 and b = 0) between t and t0 . Integrating (4.36) yields −2
z(t) ≤ z0 + (α + β2 ) ≤ ··· +
t0
ε(t0 ) +
t0
dt t t
dt t−1−α y(t )
t
t
−1+β2 +α
t0
dt t−1−α y(t )
t
dt t−1+β2 +α
t
≤ z0 + (α + β2 )−2 ε(t0 ) + (α + β2 )−1 Y (t)
(4.42)
where we have used (4.41) and where
t0
Y (t) =
dt t−1+β2 y(t ) .
(4.43)
t
Substituting (4.42) into (4.35), integrating and using the fact that Y (t) is decreasing in t, we obtain y(t) ≤ y0 + t−1 z0 + (α + β2 )−2 ε(t0 ) + (α + β2 )−1 t−1 Y (t) + y1 (t) (4.44)
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where
y1 (t) =
t0
dt t−1−β1 +α
∞
561
dt t−1−α y(t ) .
t
t
Substituting (4.44) into (4.43) yields y0 β2−1 tβ0 2 + z0 + (α + β2 )−2 ε(t0 ) (1 − β2 )−1 t−(1−β2 ) t0 (α + β2 )−1 dt t−2+β2 Y (t ) + Y1 (t) (4.45)
≤
Y (t)
+
t
where Y1 (t) =
t0
t
dt t−1+β2 y1 (t )
−1−β1 +α
t
−1+β2
∞
dt t dt t dt t−1−α y(t ) t t t t0 ∞ dt t−1−β1 +α+β2 dt t−1−α y(t ) ≤ β2−1 t t t0 t0 dt t−1−β1 dt t−1+β2 y(t ) ≤ β2−1 t−β1 (α + β2 )−2 ε(t0 ) + β2−1 t t t0 dt t−1−β1 Y (t ) . (4.46) ≤ β2−1 t−β1 (α + β2 )−2 ε(t0 ) + β2−1
=
t0
t
Substituting (4.46) into (4.45) yields the following inequality for Y (t): Y (t) ≤ f (t) +
t0
dt g(t ) Y (t )
(4.47)
t
where f (t) = β2−1 ε(t0 ) + z0 (1 − β2 )−1 t−(1−β2 ) + (α + β2 )−2 ε(t0 )
(1 − β2 )−1 t−(1−β2 ) + β2−1 t−β1 g(t) = (α + β2 )−1 t−2+β2 + β2−1 t−1−β1 . Note that f and g are decreasing in t and that g is integrable at infinity. Let ∞ G(t) = g(t ) dt , t t0 ¯ Y (t) = dt g(t ) Y (t ) . t
Then (4.47) can be rewritten as ∂t Y¯ = −gY ≥ −gf − g Y¯
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which is readily integrated with Y¯ (t0 ) = 0 to yield t0 t Y¯ (t) ≤ dt g(t ) f (t ) exp dt g(t ) ≤ f (t) G(t) exp(G(t)) t
t
and therefore by (4.47) again Y (t) ≤ f (t) {1 + G(t) exp(G(t))} . Now G is independent of t0 while f tends to zero when t0 → ∞ for fixed t under the assumptions made. Letting t0 → ∞ then shows that t0 dt t−1+β2 y(t ) −→ 0 when t0 → ∞ Y (t) = t
which implies that y = 0, from which it follows easily that z = 0, and therefore (w1 , s1 ) = (w2 , s2 ). Remark 4.1 The necessity of some condition of the type (4.32) in Part (2) is easily understood on the simpler example t1 dt y(t ) (4.48) |∂t y| ≤ a2 t
with t0 = 0 and y(0) = 0. Defining Y = y; L∞ ([0, t1 ]) we obtain |∂t y| ≤ a2 (t1 − t)Y and therefore by integration Y ≤ a2 Y Sup
0≤t≤t1
which implies Y = 0 if at1 < nishing solution y = sin at.
0
t
dt (t1 − t ) = (1/2)a2 t21 Y
√ 2. However if at1 = π/2, (4.48) admits the nonva-
We next prove another property which follows easily from estimates similar to those of Lemma 4.2, namely the fact that for suitably bounded solutions (w, s) of the auxiliary system, w(t) tends to a limit w+ when t → ∞. Proposition 4.3 Let k > 1, :0 ≤ [3/2 − k]+ and β > 0. Let T ≥ 1, t1 = ∞ and I = [T, ∞). Let B0 satisfy the estimate (3.17) for m = 0. Let (w, s) ∈ C(I, X k,0 ) with (w, tη−1 s) ∈ L∞ (I, X k,0 ) for some η > 0 and let (w, s) satisfy the first equation of the system (2.20). Then there exists w+ ∈ H k such that w(t) tends to w+ weakly in H k and strongly in H k for 0 ≤ k < k when t → ∞. Furthermore the following estimate holds for all t ∈ I:
w(t) − w+ 2 ≤ C t−α1
(4.49)
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where w(t) = U (1/t)w, and α1 = η ∧ [1/2 ∧ k/3] ∧ (β1 + βk) with β1 defined by (4.3). Proof. Let t0 ∈ I, w 0 = w(t 0 ) and a = w; L∞ (I, H k ) ,
b = tη−1 ∇s; L∞ (I, H 0 ) .
The first equation of the system (2.20) can be rewritten as ∂t (w(t) −w 0 ) = t−2 U (1/t) Q(s, w) + it−1 U (1/t) (B0 + BS (w, w)) w where we have omitted the superscript ∞ in BS , and therefore ∂t w(t) −w 0 2 ≤ t−2 Q(s, w) 2 +t−1 B0 w 2 +t−1 BS (w, w)w 2 . (4.50) By exactly the same estimate as in (4.23), we obtain
Q(s, w) 2 ≤ C|∇s|0 |w|k ≤ C ab t1−η .
(4.51)
B0 w 2 ≤ B0 3/δ w r ≤ C ab0 t−[1/2∧k/3]
(4.52)
We next estimate
with δ = δ(r) = [k ∧ 3/2], by (3.17) and Sobolev inequalities, and
BS (w, w)w 2
≤
BS (w, w) 3/δ w r
≤
C ω [3/2−k]+ BS (w, w) 2 |w|k
(4.53)
with the same δ. The last norm of BS in (4.53) is estimated exactly as in the proof of Lemma 4.1 (see (4.10) (4.11) (4.12)) as ∞
ω [3/2−k]+ BS (w, w) 2 ≤ C t−β1 −βk Im (|w|2k ) ≤ C a2 t−β1 −βk . 1
(4.54)
Substituting (4.51) (4.52) (4.53) (4.54) into (4.50) and integrating between t0 and t yields
w(t) − w(t 0 ) 2 ≤ C (t ∧ t0 )−η ab + (t ∧ t0 )−[1/2∧k/3] ab0 + (t ∧ t0 )−β1 −βk a3 from which it follows that w(t) and therefore also w(t) has a strong limit w+ in L2 when t → ∞ and that (4.49) holds. Since in addition w(t) is bounded in H k , it follows by a standard compactness argument that w+ ∈ H k and that w(t) tends to w+ in the other topologies stated in the Proposition.
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We now turn to the problem of existence of solutions of the auxiliary system (2.20), with the aim of proving that that system defines an asymptotic dynamics for large times and preferably up to infinity in time. Here however, we encounter the difficulties described in Section 2 and arising from the different propagation properties of the Schr¨ odinger and wave equations. First of all for t1 = t0 , even if B0 = 0, the estimates of Lemma 4.1 are insufficient to prevent blow up of the solutions in a finite time after t0 , independently of the size of t0 and of the initial data for (w, s) at t0 . In fact, if in the estimates (4.1=) (4.2=) we set b0 = 0, omit the second inequality and take s = 0 in the first one, we obtain the following stronger estimate for y = |w |k = |w|k 1 ∂t y ≤ C t−1−β1 y dν ν −1−m y(νt)p (4.55) t0 /t
where m = m1 + 1/2 > β1 for β ≤ 1, and p = 2. We shall prove that (4.55) does not prevent finite time blow up by showing that equality in (4.55) implies such a blow up. Taking y p instead of y as the unknown function and rescaling t and y, we can take p = 1, t0 = 1 and C = 1 without loss of generality. We are therefore led to consider the equation 1 ∂t y = t−1−β1 y dν ν −1−m y(νt) (4.56) 1/t
or equivalently ∂t y = t−1−β1 +m y
t
dt t−1−m y(t ) .
(4.57)
1
Warning 4.1 Let 0 < β1 < m. Then the solution of the equation (4.57) with initial data y(1) = y0 > 0 blows up in a finite time. The proof will be given in Appendix A. The previous result encourages us to take t1 > t0 and actually the situation improves in that case and in particular we shall prove the existence of solutions defined in [t0 , t1 ] if B0 = 0 in Proposition 4.4 below. Of course for t1 < ∞, by the previous argument, we shall be unable to exclude finite time blow up after t1 . On the other hand, if B0 = 0, we cannot exclude finite time blow up between t0 and t1 if t1 is sufficiently large. Actually, we shall show that equality in a stronger version of (4.1) implies such a blow up. We again drop the inequality (4.2) and take s = 0 in (4.1). Omitting in addition the second term with b0 , we are left with t1 /t 1−1/k −1−β1 −1−m 2 +t y ν y(νt) . (4.58) ∂t y = C y 1
Since the solution of (4.58) is increasing in time for t ≥ t0 , blow up for (4.58) is implied by blow up for the equation ∂t y = C y 1−1/k + t−1−β1 y 3 m−1 (1 − (t/t1 )m ) . (4.59)
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Now if blow up occurs for t ≤ T ∗ for the equation
∂t y = C y 1−1/k + t−1−β1 y 3 m−1 (1 − 2−m )
565
(4.60)
then a fortiori blow up will occur for the equation (4.59) if t1 ≥ 2T ∗. It is therefore sufficient to prove blow up for (4.60), which after rescaling can be rewritten as ∂t y = k y 1−1/k + t−1−β1 y 3 .
(4.61)
Warning 4.2 Let 2k > β1 . Then the solution of the equation (4.61) with initial data y(t0 ) > 0 at time t0 ≥ 1 blows up in a finite time. The proof will be given in Appendix A. The condition 2k > β1 is always satisfied in the present situation. We now prove the main result of this section, namely the existence of solutions of the auxiliary system (2.20) defined up to t1 , possibly with t1 = ∞, for B0 = 0 and for initial data given at sufficiently large t0 < t1 . Proposition 4.4 Let B0 = 0. Let 1 < k ≤ :, : > 3/2 and 0 < β < 1. Let β2 < 1, s0 | . where β2 is defined by (4.5). Let (w0 , s0 ) ∈ X k, and let y0 = |w0 |k and z0 = |∇ Then there exists T0 < ∞ depending on (y0 , z0 ) such that for all t0 ≥ T0 , there exists T < t0 , depending on (y0 , z0 ) and on t0 , such that for all t1 , t0 ≤ t1 ≤ ∞, the auxiliary system (2.20) with initial data (w, s)(t0 ) = (w0 , tβ0 2 s0 ) has a unique solution (w, s) in the interval I = [T, t1 ) such that (w, t−β2 s) ∈ (C ∩ L∞ )(I, X k, ). One can take 1/(1−β2 ) 2/β (4.62) ∨ y0 1 , T0 = C z0 + y02 T = tβ0 2 T01−β2 ,
(4.63)
and the solution (w, s) is estimated for all t ∈ I by |w(t)|k ≤ 2y0 , |∇s(t)| ≤ 2 z0 + C y02 (t0 ∨ t)β2 .
(4.64) (4.65)
Proof. The proof consists in exploiting the estimates of Lemmas 4.1 and 4.2 in order to show that the map Γ : (w, s) → (w , s ), where (w , s ) is defined from (w, s) by Proposition 4.1, is a contraction of a suitable subset of C(I, X k, ) for a suitably time rescaled norm of L∞ (I, X 0,0 ). We first consider the interval I = [t0 , t1 ) and we define the set
R = (w, s) ∈ C(I, X k, ) : w; L∞ (I, H k ) ≤ Y, t−β2 ∇s; L∞ (I, H ) ≤ Z , for Y > 0, Z > 0. Let (w, s) ∈ R and (w , s ) = Γ(w, s). Let y = |w(t)|k , y = |w (t)|k , z = |∇s(t)| and z = |∇s (t)| . From Lemma 4.1, namely (4.1) (4.2) with
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b0 = 0 and with an overall constant omitted, we obtain ∂t y ≤ t−2+β2 Zy + t−1−β1 Y 2 y ∂t z ≤ t−2+β2 Zz + t−1+β2 Y 2 .
(4.66)
Integrating from t0 to t with (y , z )(t0 ) = (y, z)(t0 ) = (y0 , z0 ) where z0 = z0 tβ0 2 , we estimate (4.67) Y = y ; L∞ (I) , Z = t−β2 z ; L∞ (I)
by
−1 −β1 2 2 Y ≤ y0 exp (1 − β2 )−1 t−1+β Z + β t Y 1 0 0 Z ≤ z + β −1 Y 2 exp (1 − β )−1 t−1+β2 Z . 0 2 2 0
We now impose
2 ≥ 2(:n2)−1 Z (1 − β2 ) t1−β 0
β1 tβ0 1 ≥ 2(:n2)−1 Y 2 and choose Y = 2y0
,
Z=
√ 2 z0 + 4β2−1 y02
(4.68)
(4.69)
(4.70)
thereby ensuring that Y ≤ Y , Z ≤ Z, so that the set R is mapped into itself by Γ. The conditions (4.69) can be rewritten as √ 2 ≥ 2 2(:n2)−1 z0 + 4β2−1 y02 (1 − β2 ) t1−β 0 (4.71) β1 tβ0 1 ≥ 8(:n2)−1 y02 and hold for all t0 ≥ T0 for T0 satisfying (4.62) with suitable C. We next show that the map Γ is a contraction on R. We use the notation of Lemma 4.2 and in addition y− = w− (t) 2 , z− = |∇s− (t)|0 , (4.72) Y− = y− ; L∞ (I) , Z− = t−β2 z− ; L∞ (I)
and a similar notation for primed quantities. From Lemma 4.2, in particular (4.18) (4.19), and again with an overall constant omitted, we obtain ∂t y− ≤ t−2 Y z− + t−1−β1 Y 2 Y− (4.73) ∂t z− ≤ t−2+β2 Z(z− + z− ) + t−1+β2 Y Y− and by integration with (y− , z− )(t0 ) = 0, 2 1 Y− ≤ (1 − β2 )−1 t−1+β Y Z− + β1−1 t−β Y 2 Y− 0 0
Z ≤ exp (1 − β2 )−1 t−1+β2 Z (1 − β2 )−1 t−1+β2 ZZ− + β −1 Y Y− . − 2 0 0
(4.74)
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The second inequality in (4.74) reduces to Z− ≤
√ 2 2 (1 − β2 )−1 t−1+β ZZ− + β2−1 Y Y− 0
(4.75)
under the first condition in (4.69) imposed previously. We now ensure that the map Γ is a contraction for the norms defined by (4.72) in the form Y− ≤ (c−1 Z− + Y− )/4 (4.76) Z− ≤ (Z− + c Y− )/4 which implies Z− + c Y− ≤ (Z− + c Y− )/2
(4.77)
by taking c = 8β2−1 Y and imposing the conditions
2 ≥ 8Z (1 − β2 )t1−β 0
,
β1 tβ0 1 ≥ 4Y 2
2 (1 − β2 ) t1−β ≥ 4c Y = 32β2−1 Y 2 0
which follow again from (4.62) for all t0 ≥ T0 . We have proved that Γ maps R into itself and is a contraction for the norms (4.72). By a standard compactness argument, R is easily shown to be closed for the latter norms. Therefore Γ has a unique fixed point in R, which completes the proof for t ≥ t0 . We now turn to the case t ≤ t0 , namely we consider the interval I = [T, t0 ] for some T < t0 . The proof proceeds in exactly the same way, with however slightly different norms. In addition, one has to take into account the following fact: the t1 various integrals Im that occur in (4.1) (4.2) and (4.18) (4.19) involve w and w1 , w2 up to time t1 . In the subinterval [t0 , t1 ], one takes w = w1 = w2 = the solution constructed at the previous step (in particular w− = 0 for t ≥ t0 , so that actually no contribution from the interval [t0 , t1 ] occurs in (4.18) (4.19)). In (4.1) (4.2) the contribution of the interval [t0 , t1 ] is taken into account by using the fact that all t1 the integrals over time in the relevant Im are convergent at infinity and that we shall eventually use the same ansatz |w(t)|k ≤ Y = 2y0 both for t ≤ t0 and t ≥ t0 . With this in mind, we complete the proof by simply giving the computational details. We consider the set
R< = (w, s) ∈ C(I, X k, ); w; L∞ (I, H k ) ≤ Y, ∇s; L∞ (I, H ) ≤ Z . For (w, s) ∈ R< , (w , s ) = Γ(w, s) and y, y , z and z defined as previously, we estimate by Lemma 4.1, again with an overall constant omitted |∂t y | ≤ t−2 Zy + t−1−β1 Y 2 y (4.78) |∂t z | ≤ t−2 Zz + t−1+β2 Y 2
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and by integration from t to t0 with (y , z )(t0 ) = (y, z)(t0 ) = (y0 , z0 ), we obtain y (t) ≤ Y , z (t) ≤ Z with Y ≤ y0 exp t−1 Z + β1−1 t−β1 Y 2
(4.79) Z ≤ z0 + β −1 Y 2 tβ2 exp(t−1 Z) . 2 0 We now impose
t ≥ 2(:n 2)−1 Z β1 tβ1 ≥ 2(:n 2)−1 Y 2
and choose Y = 2y0
,
Z=
√ 2 z0 + 4β2−1 y02 tβ0 2
(4.80)
(4.81)
thereby ensuring that Y ≤ Y , Z ≤ Z so that the set R< is mapped into itself by Γ. The conditions (4.80) can be rewritten as β1 tβ1 ≥ 8(:n 2)−1 y02 (4.82) √ t ≥ 2 2(:n 2)−1 z0 + 4β2−1 y02 tβ0 2 and hold for all t ≥ T with T defined by (4.63) and T0 satisfying (4.62) for suitable C. We next prove that Γ is a contraction on R< for the norm in L∞ (I, X 0,0 ). With the notation of Lemma 4.2 and in addition y− = w− (t) 2 , z− = |∇s− (t)|0 (4.83) Y− = y− ; L∞ (I) , Z− = z− ; L∞ (I)
and a similar notation for primed quantities, we obtain from (4.18) (4.19) | ≤ t−2 Y z− + t−1−β1 Y 2 Y− |∂t y− |∂t z− | ≤ t−2 Z(z− + z− ) + t−1+β2 Y Y− .
By integration between t and t0 , we deduce therefrom Y− ≤ t−1 Y Z− + β1−1 t−β1 Y 2 Y−
Z ≤ t−1 ZZ− + β −1 tβ2 Y Y− exp(t−1 Z) − 2 0
(4.84)
(4.85)
thereby ensuring the contraction in the form (4.76) which implies (4.77) by taking c = 8β2−1 Y tβ0 2 and imposing t ≥ 8Z
,
β1 tβ1 ≥ 4Y 2
,
t ≥ 4c Y = 32β2−1 tβ0 2 Y 2
which hold for all t ≥ T with the choice (4.81) under the conditions (4.62) (4.63). With the previous estimates available, the proof proceeds as in the case t ≥ t0 .
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5 Asymptotics and wave operators for the linear system In this section we study the asymptotic properties of solutions of the linear equation (2.22) in the form (2.24) at the level of regularity of H k with k ≥ 1 for w. In particular we solve the Cauchy problem at infinity, thereby constructing the wave operators in H k . For the linear equation (2.24), the wave operators in L2 can be easily constructed by a variant of Cook’s method and the construction of the wave operators in H k reduces to a regularity problem for the L2 wave operators thereby obtained. As a preliminary to that study, we shall first solve the Cauchy problem for the equation (2.24) with finite initial time. We emphasize the fact that in this section we do not strive after any kind of optimality in the treatment of the linear equation, since we are mainly interested in a form of that treatment that can be incorporated in that of the fully interacting system. Proposition 5.1 Let I = [1, ∞), let k ≥ 1 and let B0 satisfy the estimates (3.17) for 0 ≤ m ≤ k. Let t0 ∈ I and w0 ∈ H k . Then the equation (2.24) has a unique solution w ∈ C(I, L2 ) with w(t0 ) = w0 . Furthermore w ∈ C(I, H k ) ∩ L∞ (I, L2 ) and w satisfies the conservation law
w(t) 2 = const. and the estimate
¯ |w(t)|k ≤ 1 + Ck |t − t0 |(t ∨ t0 )k−1 |w0 |k
(5.1)
where k¯ = k for integer k and k¯ = k + ε with ε > 0 for noninteger k. Proof. It follows easily from standard arguments and from Lemma 3.2 that w exists and satisfies the properties stated except possibly the estimate (5.1), and we concentrate on the proof of the latter, assuming without loss of generality that |w0 |k = 1. We first prove (5.1) by induction for integer k ≥ 1. Let 0 ≤ j ≤ k−1 and yj = ω j w 2 . From (2.24) and from the Leibnitz formula and Sobolev inequalities, we obtain
∂ α B0 ∞ w 2 |∂t yj+1 | ≤ C t−1 ∇B0 ∞ ω j w 2 + |α|=j+1
and therefore by (3.17) |∂t yj+1 | ≤ C b0 yj + tj .
(5.2)
Substituting the induction assumption for yj and integrating (5.2) between t0 and t, we obtain yj+1 ≤ 1 + C b0 1 + (Cj + 1)(t ∨ t0 )j |t − t0 | ≤
1 + Cj+1 |t − t0 |(t ∨ t0 )j
with Cj+1 = Cb0 (Cj + 2). This completes the proof for integer k.
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Let now k = k0 + θ with integer k0 ≥ 1 and 0 < θ < 1. We estimate ∂t ω k w 2 ≤ t−1 [ω k , B0 ]w 2
≤ Ct−1 ∇B0 ∞ ω k−1 w 2 + ω k B0 3/δ w r
(5.3)
by Lemma 3.2, with 0 < δ = δ(r) ≤ 1,
· · · ≤ C b0 ω k−1 w 2 +tk−1−δ/3 ω δ w 2 by (3.17) and Sobolev inequalities. We next interpolate and obtain
· · · ≤ C b0 ω k0 −1 w 1−θ
ω k0 w θ2 +tk−1−δ/3 w 1−δ
∇w δ2 . 2 2 We finally substitute the estimate (5.1) for the integer values k0 − 1, k0 , and 1 and integrate between t0 and t, thereby obtaining
ω k w 2 ≤ 1 + C b0 |t − t0 |(t ∨ t0 )k−1+2δ/3 which yields (5.1) with ε = 2δ/3.
k
The fact that a direct H estimate of the solution does not prevent its H k norm to increase as a power of t is a warning of the fact that the construction of the wave operators at that level of regularity is not trivial. The same fact appeared already in Section 4 above in Warning 4.2 and compelled us to assume B0 = 0 in Proposition 4.4. We next construct the L2 -wave operators for (2.24). Proposition 5.2 Let I = [1, ∞) and let B0 satisfy the estimates (3.17) for m = 0. (1) Let W ∈ C(I, L2 ) with U (1/t)W ∈ C 1 (I, L2 ), satisfying
R(W ) 2 ≤ c0 t−1−λ0
(5.4)
for some λ0 > 0 and for all t ∈ I. Then there exists a unique solution w ∈ C(I, L2 ) of the equation (2.24), such that w(t) − W (t) tends to zero strongly in L2 when t → ∞. Furthermore, for all t ∈ I, −λ0
w(t) − W (t) 2 ≤ c0 λ−1 . 0 t
(5.5)
The solution w is the norm limit in L∞ (I, L2 ) as t0 → ∞ of the solution wt0 of the equation (2.24) with initial condition wt0 (t0 ) = W (t0 ) obtained in Proposition 5.1, and the following estimate holds for all t ∈ I: −λ0
wt0 (t) − w(t) 2 ≤ c0 λ−1 . 0 t0
(5.6)
(2) Let in addition W ∈ L∞ (I, H k ) for some k, 0 < k < 3/2. Then there exists w+ ∈ H k such that W (t) tends to w+ strongly in L2 and weakly in H k when t → ∞, and the following estimate holds for all t ∈ I:
(5.7)
W (t) − w+ 2 ≤ C t−λ0 + t−k/3 .
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Conversely let w+ ∈ H k for some k, 0 < k < 3/2, and let W1 = U ∗ (1/t)w+ . Then W1 satisfies the assumptions of Part (1) with λ0 = k/3. Let W , w+ and W1 be related as above. Then the solutions of the equation (2.24) constructed in Part (1) from W and W1 coincide. (3) Let w+ ∈ L2 . Then the equation (2.24) has a unique solution w ∈ C(I, L2 ) such that w(t) tends to w+ strongly in L2 when t → ∞. Proof. Part (1). Following the sketch of Section 2, we look for w in the form w = W + q, so that q satisfies the equation ∂t q = i(2t2 )−1 ∆q + i t−1 B0 q − R(W )
(5.8)
and therefore the a priori estimate −1−λ0 |∂t q 2 | ≤ R(W ) 2 ≤ c0 λ−1 . 0 t
(5.9)
Define wt0 as in Part (1) and let wt0 = W + qt0 so that qt0 (t0 ) = 0. Integrating (5.9) between t0 and t yields −λ0 0 − t−λ |
qt0 (t) 2 ≤ c0 λ−1 0 |t 0
(5.10)
and therefore, by L2 norm conservation for the difference of two solutions −λ0 0 − t−λ |
qt0 (t) − qt1 (t) 2 = qt0 (t1 ) 2 ≤ c0 λ−1 0 |t1 0
(5.11)
for any t0 and t1 , 1 ≤ t0 , t1 < ∞. This proves convergence of qt0 and therefore of wt0 in norm in L∞ (I, L2 ). Let w be the limit of wt0 . Taking the limit t0 → ∞ in (5.10) yields (5.5), while taking the limit t1 → ∞ in (5.11) yields (5.6). Clearly w satisfies the equation (2.24). Part (2). W satisfies the equation ∂t U (1/t) W = i t−1 U (1/t) B0 W + U (1/t) R(W ) .
(5.12)
From (3.17) and Sobolev inequalities, we obtain
B0 W 2 ≤ B0 3/k W r ≤ C ab0 t−k/3
(5.13)
where a = W ; L∞ (I, H k ) and k = δ(r). Integrating (5.12) between t1 and t2 and using (5.4) and (5.13), we obtain −k/3 −k/3 −λ0 0
U (1/t1 )W (t1 ) − U (1/t2 )W (t2 ) 2 ≤ C |t1 − t2 | + |t−λ − t | (5.14) 1 2 for any t1 and t2 , 1 ≤ t1 , t2 < ∞. Therefore U (1/t)W (t) and therefore also W (t) has a strong limit w+ in L2 , and
(5.15)
U (1/t) W (t) − w+ 2 ≤ C t−k/3 + t−λ0 ,
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from which (5.7) follows. Furthermore by a standard compactness argument, w+ ∈ H k with |w+ |k ≤ a and W (t) tends to w+ weakly in H k . Let now w+ ∈ H k and W1 = U ∗ (1/t)w+ . Then
so that
R(W1 ) = −it−1 B0 U ∗ (1/t) w+
(5.16)
R(W1 ) 2 ≤ C b0 |w+ |k t−k/3
(5.17)
by (5.13). The last statement follows from the fact that W and W1 have the same limit w+ in L2 and from L2 norm conservation for the equation (2.24). Part (3) follows from Parts (1) and (2) by a standard density argument. We next prove that the solutions with asymptotic properties in L2 obtained in Proposition 5.2 exhibit similar asymptotic properties in H k under suitable additional assumptions. Proposition 5.3 Let I = [1, ∞), let k ≥ 1 and let B0 satisfy the estimates (3.17) for 0 ≤ m ≤ k. Let λ > 0 and λ0 > λ + k and let U (1/t)W ∈ C 1 (I, H k ) satisfy the estimates (5.4) and (5.18)
ω k R(W ) 2 ≤ c1 t−1−λ for all t ∈ I. (1) Let w be the solution of the equation (2.24) obtained in Proposition 5.2 part (1). Then w ∈ C(I, H k ) and w satisfies the estimates (5.5) and
ω k (w(t) − W (t)) 2 ≤ C t−λ
(5.19)
for all t ∈ I. (2) Let wt0 be the solution of the equation (2.24) defined in Proposition 5.2 part (1). When t0 → ∞, wt0 converges to w strongly in L∞ ([1, T¯ ], H k ) for 0 ≤ k < k and in the weak ∗ sense in L∞ ([1, T¯ ], H k ) for any T¯ < ∞. Proof. Part (1) will be proved together with the limiting properties stated in Part (2). We know from Proposition 5.1 that wt0 ∈ C(I, H k ). The main point of the proof consists in estimating qt0 = wt0 − W in H k uniformly in t0 for t ≤ t0 . We know already from (5.10) that
qt0 (t) 2 ≤ Y0 t−λ0
(5.20)
k for t ≤ t0 , with Y0 = c0 λ−1 0 . We next estimate y ≡ ω qt0 2 . From (5.8) we obtain ∂t ω k qt0 2 ≤ t−1 [ω k , B0 ]qt0 2 + ω k R(W ) 2 (5.21)
so that by Lemma 3.2, in the same way as in (4.7), ∂t ω k qt0 2 ≤ Ct−1 ∇B0 ∞ ω k−1 qt0 2 + ω k B0 3/δ qt0 r + ω k R(W ) 2
(5.22)
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with 0 < δ = δ(r) < k ∧ 3/2. Using the estimates (3.17), Sobolev inequalities and interpolation with the help of (5.20) and the assumption (5.18), we obtain 1−δ/k δ/k k−1−δ/3−λ0 (1−δ/k) |∂t y| ≤ C b0 Y0 y t 1/k +Y0 y 1−1/k t−λ0 /k + c1 t−1−λ . (5.23) We now define Y ≡ tλ y; L∞ ([1, t0 ]) , substitute that definition into the RHS of (5.23), integrate the latter between t and t0 , and obtain
1−δ/k 1/k Y δ/k (λ + ν1 )−1 t−λ−ν1 + Y0 Y 1−1/k (λ + ν2 )−1 t−λ−ν2 y ≤ Cb0 Y0 +c1 λ−1 t−λ (5.24) where ν1 = (λ0 − λ)(1 − δ/k) − k + δ/3
(5.25)
ν2 = (λ0 − λ)/k − 1
(5.26)
provided λ+ν1 > 0 and λ+ν2 > 0. We impose in addition ν1 ≥ 0, ν2 ≥ 0, multiply (5.24) by tλ , take the Supremum over t and obtain
1−δ/k 1/k Y δ/k + Y0 Y 1−1/k + λ−1 c1 (5.27) Y ≤ λ−1 C b0 Y0 which is uniform in t0 . The condition ν1 ≥ 0 reduces to λ0 ≥ λ + k + 2δk/3(k − δ)
(5.28)
and can be satisfied for λ0 > λ + k by taking δ sufficiently small. It implies ν2 > 0. Changing the notation to x = Y Y0−1 , b = λ−1 Cb0 and c = λ−1 c1 Y0−1 , we rewrite (5.27) as
x ≤ b xδ/k + x1−1/k + c .
(5.29)
Assuming δ ≤ k − 1 without loss of generality and using xθ ≤ εx + ε−θ/(1−θ) for x > 0, ε > 0 and 0 < θ < 1, we obtain from (5.29) x ≤ 2b εx + ε1−k + c and for ε = (4b)−1 or equivalently
x ≤ (4b)k + 2c k Y ≤ 4Cλ−1 b0 Y0 + 2λ−1 c1 ,
(5.30)
which completes the proof of the estimate of qt0 in H k uniformly in t0 for t ≤ t0 .
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Let now T¯ < ∞ and J = [1, T¯ ]. We know from (5.30) that wt0 is estimated in L (J, H k ) uniformly in t0 for t0 ≥ T¯ and that wt0 converges to w in norm in L∞ (J, L2 ) by Proposition 5.2 part (1). It follows therefrom by a standard compactness argument that w ∈ (L∞ ∩ Cw )(J, H k ), that w also satisfies the estimate (5.30) and that wt0 converges to w in the topologies described in Part (2). Strong continuity of w in H k follows from Proposition 5.1. ∞
In order to complete the construction of the wave operators in H k , we now have to construct model functions W satisfying the assumptions (5.4) and (5.18). In view of Proposition 5.2 part (2), we restrict our attention to W of the form W (t) = U ∗ (1/t)w+ for some fixed w+ ∈ H
k+
(5.31)
and we take k+ > 3/2. With that choice, we obtain
R(W ) = R(U ∗ (1/t)W ) = −i t−1 B0 U ∗ (1/t)w+ .
(5.32)
However, with no further assumptions on w+ , we are restricted to λ0 ≤ 1/2 and consequently to k < 1/2. In fact, from (3.17) we obtain
R(W ) 2 ≤ t−1 B0 2 U ∗ (1/t)w+ ∞ ≤ Cb0 t−3/2 |w+ |k+ .
(5.33)
Furthermore, from Lemma 3.2 and (3.17) we obtain for k < 1/2
ω k R(W ) 2
≤ C t−1 ω k B0 2 U ∗ (1/t)w+ ∞ + B0 r ω k U ∗ (1/t)w+ 3/δ ≤ Cb0 t−3/2+k |w+ |k+
(5.34)
with 0 < δ = δ(r) ≤ 3k < 3/2. Together with an extension of Proposition 5.3 to k ≤ 1/2, which we have not performed, the estimates (5.33) (5.34) would allow us to complete the construction of the wave operators for 0 < k < 1/2, with λ0 = 1/2 and 0 < λ < 1/2 − k. In order to cover higher values of k, and in particular for k > 1, as will be needed for the nonlinear system (1.1) (1.2), we shall need additional conditions on w+ and B0 . We first exhibit a set of local sufficient conditions in the form of joint decay estimates for w+ , and B0 , where the nonlocal operator U ∗ (1/t) no longer appears. Lemma 5.1 Let λ0 > 0 and let m ¯ be a nonnegative integer. Let B0 satisfy the ¯ and let estimates (3.17) for 0 ≤ m ≤ m. ¯ Let w+ ∈ H k+ with k+ ≥ 2λ0 ∨ m a+ = |w+ |k+ . Assume that B0 and w+ satisfy the estimates
(∂ α1 B0 ) (∂ α2 w+ ) 2 ≤ b1 t−λ0 +|α1 |+|α2 |/2
(5.35)
for all multi-indices α1 , α2 with 0 ≤ |α1 | ≤ m ¯ and 0 ≤ |α2 | < 2λ0 , and for all t ≥ 1. Then the following estimates hold for all m, 0 ≤ m ≤ m, ¯ and for all t ≥ 1
ω m R(U ∗ (1/t)w+ ) 2
=
t−1 ω m (B0 U ∗ (1/t)w+ ) 2
≤
C (b1 + b0 a+ ) t−1−λ0 +m .
(5.36)
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Proof. By interpolation, it is sufficient to prove (5.36) for integer m. Let α be a multi-index with |α| = m. We estimate
(∂ α1 B0 ) U ∗ (1/t) ∂ α3 w+ 2 . (5.37)
∂ α (B0 U ∗ (1/t)w+ ) 2 ≤ C α1 +α3 =α
If |α3 | < 2λ0 , we expand U ∗ (1/t) through the relation ix −1 j −1 p+θ e − (j!) (ix) ≤ 2(p!) |x| j≤p with p + θ = λ0 − |α3 |/2 and 0 < θ ≤ 1, so that
(∂ α1 B0 ) U ∗ (1/t)∂ α3 w+ 2 ≤ C
t−j (∂ α1 B0 ) ∆j ∂ α3 w+ 2
j<λ0 −|α3 |/2 α1
−(λ0 −|α3 |/2)
+C ∂ B0 ∞ t
ω 2λ0 w+ 2 ≤ C(b1 + b0 a+ )t−λ0 +|α1 |+|α3 |/2 = C(b1 + b0 a+ )t−λ0 +m−|α3 |/2
(5.38)
by (5.35) and (3.17), which proves (5.36) in this case. If |α3 | ≥ 2λ0 , the last norm in (5.37) is estimated by the use of (3.17) as
(∂ α1 B0 ) U ∗ (1/t)∂ α3 w+ 2 ≤ C ∂ α1 B0 ∞ ∂ α3 w+ 2 ≤ C b0 a+ t|α1 | ≤ Cb0 a+ tm−2λ0 since |α1 | = m − |α3 | ≤ m − 2λ0 , which completes the proof of (5.36).
(5.39)
We shall apply Lemma 5.1 with m ¯ = {k}, the smallest integer ≥ k. Then (5.36) with m = 0 and m = k reduces to (5.4) and (5.18) with λ = λ0 − k respectively. For λ0 > k ≥ 1, one can take k+ = 2λ0 . We now give sufficient conditions that ensure the assumption (5.35). We first remark that (5.35) is trivially satisfied under suitable support properties of w+ and of the initial data (A+ , A˙ + ) of the scalar field A0 at time t = 0 (see (2.3)). In fact, assume that (5.40) Supp (A+ , A˙ + ) ⊂ {x : |x| ≤ R} . Then, by the Huyghens principle Supp A0 ⊂ {(x, t) : ||x| − t| ≤ R}
(5.41)
Supp B0 ⊂ {(x, t) : ||x| − 1| ≤ R/t} .
(5.42)
Supp w+ ⊂ {x : ||x| − 1| ≥ η}
(5.43)
so that If on the other hand for some η, 0 < η < 1, then (∂ α1 B0 )∂ α2 w+ = 0 for t ≥ R/η for any multi-indices α1 and α2 , which ensures (5.35) in a trivial way.
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We shall now give more general conditions that ensure (5.35), keeping the support condition (5.43) on w+ , and relaxing the support condition (5.40) on (A+ , A˙ + ) to space decay conditions. Lemma 5.2 Let λ0 > 0, k+ ≥ 2λ0 and let k+ > 3/2. Let w+ ∈ H k+ . Let α1 be a multi-index. (1) Let B0 satisfy (3.17) for m = |α1 | and in addition
χ0 ∂ α1 B0 2 ≤ b2 t−λ0 +|α1 |
(5.44)
where χ0 is the characteristic function of the support of w+ . Then (5.35) holds for any multi-index α2 , 0 ≤ |α2 | ≤ 2λ0 . The constant in (5.35) can be taken as b1 = C(b0 ∨ b2 )a+ . (2) Let w+ satisfy the support property (5.43) and let (A+ , A˙ + ) satisfy the following conditions for all R ≥ R0 for some R0 > 0:
χ(|x| ≥ R)∂ α1 A+ 2
χ(|x| ≥ R)A˙ + 6/5
χ(|x| ≥ R)∂ α1 A˙ + 2
≤ ≤
C R−λ0 +1/2 , C R−λ0 +1/2 if α1 = 0 ,
(5.45) (5.46)
≤
C R−λ0 +1/2
(5.47)
if α1 = 0 ,
where α1 is a multi-index satisfying α1 ≤ α1 , |α1 | = |α1 |−1, and where χ(|x| ≥ R) is the characteristic function of {x : |x| ≥ R}. Then (5.44) holds. Proof. Part (1). We estimate by the H¨ older inequality and interpolation between (3.17) and (5.44)
(∂ α1 B0 ) (∂ α2 w+ ) 2 ≤ χ0 (∂ α1 B0 ) r ∂ α2 w+ 3/δ ≤ (b0 ∨ b2 )t−λ0 +|α1 |+2λ0 δ/3 ∂ α2 w+ 3/δ = (b0 ∨ b2 )t−λ0 +|α1 |+|α2 |/2 ∂ α2 w+ 3/δ
(5.48)
where δ = δ(r) = 3|α2 |/4λ0 , so that 0 ≤ δ ≤ 3/2. The last norm in (5.48) is estimated by |w+ |k+ through Sobolev inequalities since |α2 | + 3/2 − δ ≡ 3/2 + |α2 |(1 − 3/4λ0 ) ranges from 3/2 to 2λ0 when |α2 | ranges from 0 to 2λ0 . Part (2). Using the support properties of w+ and returning to the variable A0 , we see that (5.44) is implied by
χ(||x| − t| ≥ ηt)∂ α1 A0 (t) 2 ≤ b2 t−λ0 +1/2 .
(5.49)
Let now R > 0, let χ1 ∈ C ∞ (R3 ), 0 ≤ χ1 ≤ 1, χ1 (x) = 0 for |x| ≤ 1, χ1 (x) = 1 R be the solution of the wave equation for |x| ≥ 2 and let χR (x) = χ(x/R). Let A AR = 0 with initial data
R , ∂t A R (0) = χR ∂ α1 A+ , χR ∂ α1 A˙ + A at t = 0 .
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R (t) = ∂ α1 A0 (t) for ||x| − t| ≥ 2R so that By the Huyghens principle, A R (t) 2 ≤ A R (t) 2 . (5.50)
χ(||x|−t| ≥ 2R)∂ α1 A0 (t) 2 = χ(||x|−t| ≥ 2R)A It follows now from (3.16) that R (t) 2 ≤ χR ∂ α1 A+ 2 + ω −1 χR ∂ α1 A˙ + 2 .
A
(5.51)
If α1 = 0, we estimate the last norm in (5.51) by
ω −1 χR A˙ + 2 ≤ C χR A˙ + 6/5 .
(5.52)
If α1 = 0, we rewrite ∂ α1 = ∂j ∂ α1 and estimate
ω −1 χR ∂ α1 A˙ + 2
≤ ω −1 ∂j χR ∂ α1 A˙ + 2 + ω −1 (∂j χR )∂ α1 A˙ + 2 ≤ χR ∂ α1 A˙ + 2 +C ∂j χ1 3 χ(|x| ≥ R)∂ α1 A˙ + 2
(5.53) by the Sobolev and H¨ older inequalities. Collecting (5.50)–(5.53) and using the assumption (5.45)–(5.47), we obtain
χ(||x| − t| ≥ 2R)∂ α1 A0 (t) 2 ≤ C R−λ0 +1/2
from which (5.49) follows by taking 2R = ηt. k
Collecting the previous results essentially yields the wave operators in H for the equation (2.24) in the form of Proposition 5.4 below. In that proposition, we have kept the assumptions on B0 in the implicit form of the estimates (3.17) and (5.35). If so desired, those assumptions can be replaced by sufficient conditions on (w+ , A+ , A˙ + ) by the use of Lemmas 3.5 and 5.2. Proposition 5.4 Let k ≥ 1, k+ > 2k, let λ0 and λ satisfy λ > 0, k + λ < λ0 ≤ k+ /2. Let w+ ∈ H k , let B0 satisfy the estimates (3.17) for 0 ≤ m ≤ k, and let (w+ , B0 ) satisfy the estimates (5.35) for all multi-indices α1 , α2 with 0 ≤ |α1 | ≤ k¯ and 0 ≤ |α2 | < 2λ0 , where k¯ is the smallest integer ≥ k. Then the equation (2.24) has a unique solution w ∈ C([1, ∞), L2 ) such that
w(t) − w+ 2 → 0
when t → ∞ .
Furthermore w ∈ C([1, ∞), H k ) and w satisfies the estimates
w(t) − U ∗ (1/t)w+ 2 ≤ C t−λ0
ω k (w(t) − U ∗ (1/t)w+ ) 2 ≤ C t−λ
(5.54) (5.55)
for all t ≥ 1. Proof. The results follow from Propositions 5.2 and 5.3 and from Lemma 5.1. The existence of the wave operators for u in the usual sense at the corresponding level of regularity is an easy consequence of Proposition 5.4. We refrain from giving a formal statement at this stage. The same question will be considered in Section 8 in the case of the interacting system (1.1) (1.2).
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6 Cauchy problem at infinity for the auxiliary system In this section we begin the construction of the wave operators for the auxiliary system (2.20) by solving the Cauchy problem at infinity for that system in the difference form (2.30), for large or infinite initial time, and for a given choice of (W, S) satisfying a number of a priori estimates. The construction of (W, S) satisfying those estimates is deferred to the next section. In the same spirit as in Section 4, we solve the system (2.30) in two steps. We first solve the linearized version of that system (2.31), thereby defining a map Γ : (q, σ) → (q , σ ). We then show that this map is a contraction in suitable norms on a suitable set. The basic tool of this section again consists of a priori estimates for suitably regular solutions of the linearized system (2.31). In order to handle efficiently a non-vanishing B0 , those estimates have to be much more elaborate than those of Section 4. We first estimate a single solution of the linearized system (2.31) at the level of regularity where we shall eventually solve the auxiliary system (2.30). Lemma 6.1 Let 1 < k ≤ :, : > 3/2 and β > 0. Let I ⊂ [1, ∞) be an interval and ¯ Let B0 satisfy the estimates (3.17) for 0 ≤ l ≤ k. Let (U (1/t))W, S) ∈ let t1 ∈ I. k+1,+1 C(I+ , X ) ∩ C 1 (I+ , X k, ) with W ∈ L∞ (I+ , H k+1 ) and let a = W ; L∞ (I+ , H k+1 ) .
(6.1)
Let (q, σ), (q , σ ) ∈ C(I, X k, ) with q ∈ L∞ (I, H k ) ∩ L2 (I, L2 ) if t1 = ∞, and let (q , σ ) be a solution of the system (2.31) in I. Then the following estimates hold for all t ∈ I: |∂t q 2 | ≤ C t−2 a ∇σ 2 +t−1−β a2 I0 ( q 2 ) (6.2) +t−1 a I−1 ( q 2 q 3 ) + R1 (W, S) 2 ,
∂t ω k q 2 ≤ C b0 ω k−1 q 2 +tk−1−δ/3 q r +t−2 a ω k ∇σ 2 + σ ∞ + ∇σ 3 t−2 ∇s ∞ ω k q 2 + ω [k∨3/2] ∇s 2 ω [k∧3/2] q 2 +χ(k > 3/2) ω k ∇s 2 q ∞ +t−1 a2 Ik−1 ω k−1 q 2 + q 2 + I0 ( q 2 ) + ω k−1 q 2 + q 2 +t−1 a Ik−1 ω k q 2 q 3 + I0 ( ∇q 2 q 3 )
+I1/2 ω 1/2 q 2 ω k q 2 + Ik−1/2 ω k−1/2 q 2 + q 2 ∇q 2 +t−1 I1/2 ∇q 22 ω k q 2 +Ik−1/2 ω k q 2 ∇q 2 ∇q 2 + ω k R1 (W, S) 2
(6.3)
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where s = S + σ and 0 < δ = δ(r) ≤ [k ∧ 3/2], |∂t ω m ∇σ 2 | ≤ C t−2 ∇s ∞ ω m ∇σ 2 + ω m ∇s 2 ∇σ ∞
ω m ∇σ 2 ∇S ∞ + σ ∞ ω m ∇2 S 2 +C t−1+β(m+1) a I0 ( q 2 ) + t−1+β(m+5/2) I−3/2 q 22 + ω m ∇R2 (W, S) 2
(6.4)
for 0 ≤ m ≤ :,
|∂t ∇σ 2 | ≤ Ct−2 ∇s ∞ ∇σ 2 + ∇σ 2 ∇S ∞ + ω 3/2 ∇S 2 +C t−1+β a I0 ( q 2 ) + t−1+5β/2 I−3/2 q 22 + ∇R2 (W, S) 2
(6.4)0
where the time parameter is t1 in all the estimating functions Im , and the superscript t1 is omitted for brevity. Remark 6.1. All the norms of (q, σ) and (q , σ ) that appear in (6.2)–(6.4) are controlled by the norms in X k, through Sobolev inequalities. Furthermore all the integrals Im are convergent if t1 = ∞. This follows from boundedness of q in H k in all cases where m > −1/2, namely in all cases but two. The exceptions are ∞ 2 I−3/2 q 2 = dν q(νt) 22 1
in (6.4) and I−1 ( q 2 q 3 ) ≤ C
∞
1
3/2
dν ν −1/2 q 2
1/2
∇q 2
≤ C ∇q; L∞ ([t, ∞), L2 ) 1/2 q; L2 ((t, ∞), L2 ) 3/2 in (6.2), both of which are controlled under the additional assumption that q ∈ L2 (I, L2 ). Finally it is easy to see by estimates similar to, but simpler than, those of Lemma 4.1 that all the norms of the remainders R1 and R2 that occur in (6.2)– (6.4) are finite under the assumptions made on (W, S). Proof of Lemma 6.1 In all the proof, the time superscript in BS , BL and in the various Im is omitted, except in dubious cases. That time superscript is in general t1 , except in BS (W, W ) where it is ∞. Proof of (6.2). From (2.31), we estimate |∂t q 2 | ≤ t−2 Q(σ, W ) 2 +t−1 (BS (q, q) + 2BS (W, q)) W 2 + R1 (W, S) 2 . (6.5)
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We next estimate by (3.8) (3.10) and Sobolev inequalities
Q(σ, W ) 2 ≤ C ∇σ 2 ( ∇W 3 + W ∞ ) ≤ Ca ∇σ 2 , −β
(6.6)
2 −β
BS (W, q)W 2 ≤ Ct W ∞ I0 ( W ∞ q 2 ) ≤ Ca t I0 ( q 2 ) , (6.7)
BS (q, q)W 2 ≤ C W ∞ I−1 ( q 2 q 3 ) ≤ CaI−1 ( q 2 q 3 ) . (6.8) Substituting (6.6) (6.7) (6.8) into (6.5) yields (6.2). Proof of (6.3). From (2.31), we estimate ∂t ω k q 2 ≤ t−1 [ω k , B0 ]q 2 +t−2 [ω k , s] · ∇q 2
+ (∇ · s)ω k q 2 + ω k ((∇ · s)q ) 2 + ω k Q(σ, W ) 2 t−1 [ω k , BS (w, w)]q 2 + ω k (BS (q, q) + 2BS (q, W )) W 2 + ω k R1 (W, S) 2
(6.9)
and we estimate the various terms in the RHS successively. The contribution of B0 is estimated by Lemma 3.2 and by (3.17) exactly as in Section 4 (see (4.7)) and yields
(6.10)
[ω k , B0 ]q 2 ≤ C b0 t ω k−1 q 2 +tk−δ/3 q r . The contribution of Q(s, q ) is estimated by (4.8) as
[ω k , s] · ∇q 2 + (∇ · s)ω k q 2 + ω k ((∇ · s)q ) 2 ≤ C ∇s ∞ ω k q 2 + ω [k∨3/2] ∇s 2 ω [k∧3/2] q 2 +χ(k > 3/2) ω k ∇s 2 q ∞ .
(6.11)
The contribution of Q(σ, W ) is estimated by Lemma 3.2 and Sobolev inequalities as
ω k Q(σ, W ) 2 ≤ C σ ∞ ω k ∇W 2 + ω k σ 6 ∇W 3 + ω k ∇σ 2 W ∞ + ∇σ 3 ω k W 6 (6.12) ≤ C a ω k ∇σ 2 + σ ∞ + ∇σ 3 . The contribution of BS with w = W + q yields a number of terms which we order by increasing number of q or q occurring therein. We first expand BSt1 ,∞ (w, w) = BS∞ (W, W ) + 2BSt1 (W, q) + BSt1 (q, q) . By Lemma 3.2 and Sobolev inequalities, we estimate
[ω k , BS (W, W )]q 2 ≤ C ∇BS (W, W ) ∞ ω k−1 q 2 + ω k+3/2−δ B1 (W, W ) 2 q r
(6.13)
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where we take 0 < δ = δ(r) = (k − 1) ∧ 1/2 so that
q r ≤ C ω k−1 q 2 + q 2 . Furthermore
∇BS (W, W ) ∞ ≤ C ∇2 B1 (W, W ) 1−θ
ω k+3/2 B1 (W, W ) θ2 2 with θ = 1/(2k − 1), and by (3.10) and Lemma 3.2
ω m+1 B1 (W, W ) 2 ≤ C Im ( ω m W 2 W ∞ ) ≤ C a2 which we use with m = 1, k + 1/2 and k + 1/2 − δ. Substituting those estimates into (6.13) yields (6.14)
[ω k , BS (W, W )]q 2 ≤ C a2 ω k−1 q 2 + q 2 . In a similar way, we estimate
[ω k , BS (W, q)]q 2 ≤ C
ω 3/2 B1 (W, q) 2 ω k q 2 + ω k+1/2 B1 (W, q) 2 ∇q 2 .
(6.15)
By Lemma 3.2 again and by (3.10)
ω m+1 B1 (W, q) 2 ≤ Im ( ω m W q 2 ) ,
ω m W q 2 ≤ C W ∞ ω m q 2 + ω m+3/2−δ W 2 q r with 0 < δ = δ(r) = m ∧ 1/2, so that for m ≤ k − 1/2
ω m+1 B1 (W, q) 2 ≤ C a Im ( ω m q 2 + q 2 ) ,
(6.16)
where q 2 can be omitted for m ≤ 1/2. Substituting (6.16) with m = 1/2 and m = k − 1/2 into (6.15) yields
[ω k , BS (W, q)]q 2 ≤ C a I1/2 ω 1/2 q 2 ω k q 2
(6.17) +Ik−1/2 ω k−1/2 q 2 + q 2 ∇q 2 . By Lemma 3.2 and Sobolev inequalities again, we next estimate
[ω k , BS (q, q)]q 2 ≤ C ω 3/2 B1 (q, q) 2 ω k q 2 + ω k+1/2 B1 (q, q) 2 ∇q 2 followed by (see also (3.10))
ω 3/2 B1 (q, q) 2 ≤ CI1/2 ∇q 22
ω k+1/2 B1 (q, q) 2 ≤ CIk−1/2 ω k q 2 ∇q 2
(6.18)
582
so that
J. Ginibre and G. Velo
Ann. Henri Poincar´e
[ω k , BS (q, q)]q 2 ≤ C I1/2 ∇q 22 ω k q 2 +Ik−1/2 ω k q 2 ∇q 2 ∇q 2 .
(6.19)
We now turn to the second contribution of BS to (6.9). By Lemma 3.2 and Sobolev inequalities again
ω k (BS (W, q)W ) 2 ≤ C ω k B1 (W, q) 2 W ∞ (6.20) + ∇B1 (W, q) 2 ω k+1/2 W 2 and by (6.16) with m = k − 1 and m = 0
ω k (BS (W, q)W ) 2 ≤ C a2 Ik−1 ω k−1 q 2 + q 2 + I0 ( q 2 ) . (6.21) Similarly
ω k (BS (q, q)W ) 2 ≤ C
ω k B1 (q, q) 2 W ∞ + ∇B1 (q, q) 2 ω k+1/2 W 2
(6.22)
followed by (see the proof of (6.19))
ω k B1 (q, q) 2 ≤ C Ik−1 ω k q 2 q 3
∇B1 (q, q) 2 ≤ C I0 ( ∇q 2 q 3 ) yields
ω k (BS (q, q)W ) 2 ≤ C a Ik−1 ω k q 2 q 3 + I0 ( ∇q 2 q 3 ) . (6.23) Substituting (6.10) (6.11) (6.12) (6.14) (6.17) (6.19) (6.21) and (6.23) into (6.9) and reordering the contributions of BS by increasing powers of (q, q ) yields (6.3). Proof of (6.4). From (2.31), we estimate ∂t ω m+1 σ 2 ≤ t−2 [ω m+1 , s] · ∇σ 2 + (∇ · s)ω m+1 σ 2 + ω m+1 (σ · ∇S) 2 + t−1 ω m+2 (BL (q, q) + 2BL (W, q)) 2 + ω m+1 R2 (W, S) 2 .
(6.24)
We next estimate by Lemma 3.2 again
[ω m+1 , s]·∇σ 2 ≤ C ∇s ∞ ω m+1 σ 2 + ω m+1 s 2 ∇σ ∞ (6.25)
ω m+1 (σ · ∇S) 2 ≤ C ω m+1 σ 2 ∇S ∞ + σ ∞ ω m+1 ∇S 2 (6.26)
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while by (3.9) and Lemma 3.4
ω m+2 BL (W, q) 2
tβ(m+1) I0 ( W ∞ q 2 ) tβ(m+1) a I0 ( q 2 ) ,
ω m+2 BL (q, q) 2 ≤ C tβ(m+5/2) I−3/2 q 22 . ≤ ≤
(6.27) (6.28)
Substituting (6.25) (6.26) (6.27) (6.28) into (6.24) yields (6.4). For m = 0, the term ω m ∇s 2 ∇σ ∞ can be omitted, and the term σ · ∇S can be estimated in a sllightly different way, thereby leading to (6.4)0 . We next estimate the difference of two solutions of the linearized system (2.31) corresponding to two different choices of (q, σ), but to the same choice of (W, S). As in Section 4, we estimate that difference at a lower level of regularity than the solutions themselves. Lemma 6.2 Let 1 < k ≤ :, : > 3/2 and β > 0. Let I ⊂ [1, ∞) be an interval and ¯ Let B0 be sufficiently regular, for instance B0 ∈ C(I, H k ). Let (W, S) let t1 ∈ I. 3 satisfy the assumptions of Lemma 6.1. Let (qi , σi ), (qi , σi ) ∈ C(I, X k, ) with qi ∈ L∞ (I, H k )∩L2 (I, L2 ), i = 1, 2, if t1 = ∞, and let (qi , σi ) be solutions of the system (2.31) associated with (qi , σi ) and (W, S). Define (q± , σ± ) = (1/2)(q1 ± q2 , σ1 ± σ2 ) and (q± , σ± ) = (1/2)(q1 ± q2 , σ1 ± σ2 ). Then the following estimates hold for all t ∈ I. ∂t q−
2 ≤ C t−2 a ∇σ− 2 +t−2 ω [3/2−k]+ ∇σ− 2 ω [k∧3/2] q+
2 +χ(k > 3/2) ∇σ− 2 q+
∞ + t−1−β a2 I0 ( q− 2 ) +t−1 a q+
3 I0 ( q− 2 ) + I−1 ( q+ 3 q− 2 ) +t−1 q+
6 I−1/2 ( q+ 6 q− 2 ) , (6.29) ∂t ω m ∇σ−
2 ≤ C t−2 ∇s+ ∞ ω m ∇σ−
2 + ∇s+ ∞ ω m ∇σ− 2 + ω m−m +3/2 ∇s+ 2 ω m ∇σ−
2 + ω m+5/2−δ ∇s+ 2 σ− r +C t−1+β(m+1) a I0 ( q− 2 ) + t−1+β(m+5/2) I−3/2 ( q+ 2 q− 2 ) , ∂t ∇σ 2 ≤ C t−2 ∇s+ ∞ ∇σ 2 − −
+ ∇σ− 2 ∇s+ ∞ + ω 3/2 ∇s+ 2
+C t−1+β a I0 ( q− 2 ) + t−1+5β/2 I−3/2 ( q+ 2 q− 2 ) where s+ = S + σ+ ,
s+
=S+
(6.30)
(6.30)0
σ+ ,
0 ≤ m ≤ :0 , m = m ∧ 1/2 , δ = δ(r) = [(m + 1) ∧ 3/2] , [3/2 − k]+ ≤ :0 ≤ : − 1 , and the superscript t1 is again omitted in the estimating functions Im .
(6.31)
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Remark 6.2 Under the assumptions made, all the norms in the first part of the RHS of (6.30) are controlled by |∇s+ | , |∇s+ | , |∇σ− |0 and |∇σ− |0 . Proof. Taking the difference of the system (2.31) for (qi , σi ), or equivalently and more simply rewriting (4.21) with (w− , s− ) = (q− , σ− ), (w− , s− ) = (q− , σ− ), and t1 ,∞ t1 accounting for the replacement of BS (w, w) by BS (w, w), we obtain ∂t q− = i(2t2 )−1 ∆q− + t−2 Q(s+ , q− ) + Q(σ− , w+ ) + it−1 B0 q− (6.32) +it−1 BSt1 ,∞ (w+ , w+ ) + BSt1 (q− , q− ) q− + 2BSt1 (w+ , q− )w+ t1 ∂t σ− = t−2 s+ · ∇σ− + σ− · ∇s+ − 2t−1 ∇BL (w+ , q− ) . . From (6.32) we obtain We first estimate q− ∂t q−
2 ≤ t−2 Q(σ− , w+ ) 2 +2t−1 BS (w+ , q− )w+
2 .
(6.33)
We expand (6.33) by using (w+ , s+ ) = (W, S) + (q+ , σ+ ) , (w+ , s+ ) = (W, S) + (q+ , σ+ )
and we estimate the various terms successively. From (6.6) we obtain
Q(σ− , W ) 2 ≤ C a ∇σ− 2 . By the same estimates as in the proof of (4.23), we next obtain
Q(σ− , q+ ) 2 ≤ C ω [3/2−k]+ ∇σ− 2 ω [k∧3/2] q+
2 +χ(k > 3/2) ∇σ− 2 q+
∞ .
(6.34)
(6.35)
We next estimate by (6.7) (6.8)
BS (W, q− )W 2 ≤ C a2 t−β I0 ( q− 2 ) ,
(6.36)
BS (q+ , q− )W 2 ≤ C a I−1 ( q+ 3 q− 2 ) .
(6.37)
The remaining terms are new. Using (3.10) and Sobolev inequalities, we obtain successively
2 ≤ C a q+
3 I0 ( q− 2 ) , (6.38)
BS (W, q− )q+
BS (q+ , q− )q+
2
≤ C BS (q+ , q− ) 3 q+
6 ≤ C I−1/2 ( q+ 6 q− 2 ) q+
6 .
Substituting (6.34)–(6.39) into (6.33) yields (6.29).
(6.39)
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We next estimate σ− . From (6.32) we obtain
∂t ω m+1 σ 2 ≤ t−2 [ω m+1 , s+ ] · ∇σ 2 + (∇ · s+ )ω m+1 σ 2 − − − (6.40) m+1 −1 m+2 + ω (σ− · ∇s+ ) 2 + 2t ω (BL (W, q− ) + BL (q+ , q− )) 2 and we estimate the various terms successively. From Lemma 3.2 and Sobolev inequalities, we obtain
2 ≤ C ∇s+ ∞ ω m+1 σ−
2
[ω m+1 , s+ ] · ∇σ− m+1 (6.41) + ω s+ 3/m ∇σ− r
≤ C ∇s+ ∞ ω m ∇σ−
2 + ω m−m +3/2 ∇s+ 2 ω m ∇σ−
2 with m = δ(r ) = m ∧ 1/2, and
ω m+1 (σ− ∇s+ ) 2 ≤ C ∇s+ ∞ ω m ∇σ− 2 + ω m+1 ∇s+ 3/δ σ− r
(6.42) ≤ C ∇s+ ∞ ω m ∇σ− 2 + ω m+5/2−δ ∇s+ 2 σ− r with δ = δ(r) = [(m + 1) ∧ 3/2]. The contribution of BL to (6.40) is estimated exactly as in the proof of (6.4) (see (6.27) and (6.28)) by
ω m+2 (BL (W, q− ) + BL (q+ , q− )) 2 ≤ C a tβ(m+1) I0 ( q− 2 )
+tβ(m+5/2) I−3/2 ( q+ 2 q− 2 ) . (6.43) Substituting (6.41) (6.42) (6.43) into (6.40) yields (6.30) and (6.300 ), where one term from (6.41) can be omitted. We now begin the study of the Cauchy problem for the auxiliary system in the difference form (2.30) and for that purpose we first study that problem for the linearized version of that system. For finite initial time t0 , that problem is solved by a minor modification of Proposition 4.1. The following proposition is simply a compilation of that result and of Lemmas 6.1 and 6.2. Proposition 6.1 Let 1 < k ≤ :, : > 3/2 and β > 0. Let I ⊂ [1, ∞) be an interval and ¯ let B0 satisfy the estimates (3.17) for 0 ≤ m ≤ k. Let (U (1/t)W, S) ∈ let t1 ∈ I. k+1,+1 C(I+ , X ) ∩ C 1 (I+ , X k, ) with W ∈ L∞ (I+ , H k+1 ) and define a by (6.1). Let (q, σ) ∈ C(I, X k, ) with q ∈ L∞ (I, H k ) ∩ L2 (I, L2 ) if t1 = ∞. Let t0 ∈ I and let (q0 , σ0 ) ∈ X k, . Then the system (2.31) has a unique solution (q , σ ) ∈ C(I, X k, ) with (q , σ )(t0 ) = (q0 , σ0 ). That solution satisfies the estimates (6.2) (6.3) (6.4) of Lemma 6.1 for all t ∈ I. Two such solutions (qi , σi ) associated with (qi , σi ), i = 1, 2 and with the same (W, S) satisfy the estimates (6.29) (6.30) of Lemma 6.2 for all t ∈ I.
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We shall be eventually interested in solving the Cauchy problem for the auxiliary system (2.30) with infinite initial time t0 . As a preliminary, we need to solve the same problem for the linearized system (2.31). This is done in the following proposition, which of course requires much stronger assumptions on the asymptotic behaviour in time of (W, S) and (q, σ). With the study of the nonlinear system in view, we already make the assumptions that will be needed for that purpose, although they could be slightly weakened for the linear problem. Since we want to take t0 = ∞, we also take t1 = ∞. Proposition 6.2 Let 1 < k ≤ :, : > 3/2. Let β, λ0 and λ satisfy 0<β<1
,
λ>0
,
λ0 > λ + k
,
λ0 > β(: + 1) .
(6.44)
Let t1 = ∞, let 1 ≤ T < ∞, and I = [T, ∞). Let B0 satisfy the estimates (3.17) for 0 ≤ m ≤ k. Let (W, S) satisfy the assumptions of Proposition 6.1 with |W |k+1 ≤ a ,
ω m ∇S 2 ≤ b t1−η+β(m−3/2)
(6.45) (6.46)
for some η > 0 and for 0 ≤ m ≤ : + 1,
R1 (W, S) 2 ≤ c0 t−1−λ0 , ω k R1 (W, S) 2 ≤ c1 t−1−λ , (6.47) (6.48)
ω m ∇R2 (W, S) 2 ≤ c2 t−1−λ0 +β(m+1) for 0 ≤ m ≤ : , for all t ∈ I. Let (q, σ) ∈ C(I, X k, ) satisfy
q 2 ≤ Y0 t−λ0 , ω k q 2 ≤ Y t−λ ,
ω m ∇σ 2 ≤ Z t−λ0 +β(m+1) for 0 ≤ m ≤ : ,
(6.49) (6.50)
for all t ∈ I. Then the system (2.31) has a (unique) solution (q , σ ) ∈ C(I, X k, ) satisfying
q 2 ≤ Y0 t−λ0 , ω k q 2 ≤ Y t−λ ,
ω m ∇σ 2 ≤ Z t−λ0 +β(m+1) for 0 ≤ m ≤ : ,
(6.51) (6.52)
for some Y0 , Y , Z depending on k, :, β, λ0 , λ, a, b, c0 , c1 , c2 , Y0 , Y , Z and T , for all t ∈ I. That solution satisfies the estimates (6.2) (6.3) (6.4) of Lemma 6.1 for all t ∈ I. Two such solutions (qi , σi ) associated with (qi , σi ), i = 1, 2, satisfy the estimates (6.29) (6.30) of Lemma 6.2 for all t ∈ I. The solution (q , σ ) is actually unique in C(I, X k, ) under the condition that (q , σ ) tends to zero in X 0,0 norm when t → ∞. Proof. The proof consists in showing that the solution (qt 0 , σt 0 ) of the linearized system (2.31) with t1 = ∞ and with initial data (qt 0 , σt 0 )(t0 ) = 0 for finite t0 , obtained from Proposition 6.1, satisfies the estimates (6.51) (6.52) uniformly in t0 for t ≤ t0 (namely with Y0 , Y and Z independent of t0 ), and that when t0 → ∞,
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that solution converges on the compact subintervals of I uniformly in suitable norms. We first derive the estimates (6.51) (6.52) for that solution, omitting the subscript t0 for brevity in that part of the proof. Let y0 = q 2
,
y = ω k q 2
,
zm = ω m ∇σ 2 .
(6.53)
We first estimate y0 . Substituting (6.47) (6.49) (6.50) into (6.2), and omitting an overall constant, we obtain |∂t y0 | ≤ a Z t−2−λ0 +β + a2 Y0 t−1−β−λ0 1/2 −1−2λ0 +(λ0 −λ)/2k +a Y03 Y¯ t + c0 t−1−λ0
(6.54)
where Y¯ = Y ∨ Y0 . Integrating (6.54) from t0 to t with y0 (t0 ) = 0, using the fact that λ0 > 1 and λ0 − (λ0 − λ)/2k > k − 1/2 + λ, and defining Y0 = tλ0 y0 ; L∞ ([T, t0 ]) ,
(6.55)
we obtain 1/2 −(k−1/2+λ) T . Y0 ≤ c0 + a Z T −(1−β) + a2 Y0 T −β + a Y03 Y¯
(6.56)
That estimate is manifestly uniform in t0 . We next estimate y , wasting part of the time decay in order to alleviate the computation. In particular when estimating s = S + σ, we use the fact that the time decay of σ is better than that of S by at least a power 1−η. Furthermore in the contributions coming from BS , we eliminate Y0 and λ0 by using Y0 ≤ Y¯ = Y ∨ Y0 and λ0 > λ + k. In particular we estimate
ω m q 2
≤ ≤
1−m/k
m/k
q 2
ω k q 2 Y¯ t−λ0 (1−m/k)−λm/k ≤ Y¯ t−λ+m−k
(6.57)
for 0 ≤ m ≤ k, and similarly m/k −λ0 1−m/k
ω m q 2 ≤ y Y0 t −λ 1−m/k m−k m/k Y0 t ≤t y ≤ tm−k y + Y0 t−λ .
(6.58)
Substituting (6.47) (6.49) (6.50) into (6.3), using (6.57) (6.58) and omitting an overall constant, we obtain 1/k 1−1/k 1−δ/k δ/k k−1−δ/3 |∂t y | ≤ b0 y 0 y + y0 y t +a Z t−2−λ−k+β(k+1) + t−1 bt−η + Zt−1 y + Y0 t−λ +a2 t−2 y + Y0 t−λ + Y¯ t−λ + a t−k−1/2−λ y + Y0 t−λ + Y¯ t−λ Y¯ +t1−2k−2λ Y¯ 2 y + Y0 t−λ + c1 t−1−λ . (6.59)
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The terms linear in y in the RHS of (6.59) can be eliminated by changing variables from y to y exp(−E(t)), where E(t) = bη −1 t−η + Zt−1 + a2 t−1 + (k − 1/2 + λ)−1 aY¯ t−(k−1/2+λ) +(2(k − 1 + λ))−1 Y¯ 2 t−2(k−1+λ)
(6.60)
so that it is sufficient to estimate y from (6.59) with those terms omitted and to multiply the end result by exp(E(t)). With those terms omitted, and with the help of the estimate of y0 , (6.59) can be rewritten as 1−δ/k δ/k k−1−δ/3−λ0 (1−δ/k) 1/k |∂t y | ≤ b0 Y 0 y t + Y 0 y 1−1/k t−λ0 /k +t−1−λ C1 (t)
(6.61)
where C1 (t) = a Z t−(1−β)(k+1) + b t−η + Zt−1 Y0
+ a2 t−1 + aY¯ t−(k−1/2+λ) Y0 + Y¯ + Y¯ 2 Y0 t−2(k−1+λ) + c1 . (6.62) In particular C1 (t) is decreasing in t. The inequality (6.61) is essentially identical with (5.24), up to notational change and replacement of c1 by C1 (t). Proceeding as in Section 5, defining Y = tλ y ; L∞ ([T, t0 ])
and reintroducing the factor exp(E(t)), we obtain (see (5.30)) k Y ≤ exp(E(T )) 4λ−1 b0 Y0 + 2λ−1 C1 (T ) ,
(6.63)
(6.64)
an estimate which is again manifestly uniform in t0 . This completes the proof of (6.51). We next estimate zm for 0 ≤ m ≤ :. By interpolation, it suffices to estimate z0 and z . We define
and
Zm = tλ0 −β(m+1) zm ; L∞ ([T, t0 ])
(6.65)
Z = Sup Zm = Z0 ∨ Z .
(6.66)
0≤m≤
We first estimate z0 . Substituting (6.48) (6.49) (6.50) into (6.4)0 and omitting an overall constant, we obtain |∂t z0 | ≤ b t−1−η + Zt−2 z0 + bZt−1−η−λ0 +β +a Y0 t−1−λ0 +β + Y02 t−1−2λ0 +5β/2 + c2 t−1−λ0 +β .
(6.67)
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Integrating (6.67) from t to t0 , we obtain −1 Z0 ≤ exp b η −1 T −η + Z T −1 (η + λ0 − β) b Z t−η +(λ0 − β)−1 a Y0 + Y02 T −β + c2
(6.68)
where we have used again the fact that λ0 > 5β/2. We next estimate z . Using the inequality
∇σ ∞
3/2
1−3/2
≤ C ω ∇σ 2
∇σ 2 ≤ C t3β/2 t−β ω ∇σ 2 + ∇σ 2 ,
(6.69)
substituting (6.48) (6.49) (6.50) into (6.4) with m = :, and omitting again an overall constant, we obtain
|∂t z | ≤ b t−1−η + Zt−2 z + Z0 t−λ0 +β(+1) +b Z t−1−η−λ0 +β(+1) + a Y0 t−1−λ0 +β(+1) +Y02 t−1−2λ0 +β(+5/2) + c2 t−1−λ0 +β(+1) .
(6.70)
Integrating (6.70) as before, we obtain Z ≤ exp b η −1 T −η + Z T −1 b(Z + Z0 )η −1 T −η + ZZ0 T −1 +ν −1 a Y0 + Y02 T −β + c2 (6.71) where ν = λ0 − β(: + 1) > 0, which together with (6.68) completes the proof of (6.52). We have proved that the solution (qt 0 , σt 0 ) of the system (2.31), vanishing at t0 , satisfies the estimates (6.51) (6.52) for t ∈ [T, t0 ], with Y0 , Y , Z satisfying (6.56) (6.64) (6.66) (6.68) (6.71), which are uniform in t0 . We now prove that (qt 0 , σt 0 ) tends to a limit when t0 → ∞. For that purpose, we first let (qi , σi ), i = 1, 2, be two solutions of the system (2.31) corresponding to the same (q, σ) and defined in an interval [T, t0 ) for some t0 > T . Let (q− , σ− ) = (1/2)(q1 −q2 , σ1 −σ2 ). 2 L norm conservation for q− implies (t) 2 = q− (t0 ) 2
q−
for all t ∈ [T, t0 ] .
(6.72)
Furthermore, the simple case q− = 0, σ− = 0 of (6.30)0 implies
2 ≤ C t−1 b t−η + Z t−1 ∇σ−
2 ∂t ∇σ−
(6.73)
and therefore (t) 2 ≤ exp C η −1 b t−η + Z t−1 ∇σ− (t0 ) 2 .
∇σ−
(6.74)
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Let now T < t1 < t2 < ∞ and let (qi , σi ) = (qt , σt ), i = 1, 2. Then i
i
q− (t) 2 = q− (t1 ) 2 = (1/2) q2 (t1 ) 2 ≤ Y0 t1
−λ0
for all t < t1
(6.75)
by (6.72) with t0 = t1 and (6.51) for q = q2 and t = t1 Similarly, by (6.52) with σ = σ2 and t = t1 , −λ0 +β
(t1 ) = (1/2) ∇σ2 (t1 ) ≤ Z0 t 1
∇σ−
so that by (6.74) with t0 = t1 , −λ +β (t) ≤ exp C η −1 b t−η + Z t−1 Z0 t 1 0
∇σ−
for all t < t1 .
(6.76)
From (6.75) (6.76), it follows that (qt 0 , σt 0 ) converges to a limit (q , σ ) ∈ C(I, X 0,0 ) uniformly on the compact subintervals of I. From the uniform estimates (6.51) (6.52) and from Lemma 6.1, it then follows by a standard compactness argument that (q , σ ) ∈ C(I, X k, ) and that (q , σ ) also satisfies the estimates (6.51) (6.52). Clearly (q , σ ) satisfies the system (2.31). This completes the existence part of the proof. The uniqueness statement follows immediately from (6.72) (6.74) by letting t0 → ∞. We now turn to the main result of this section, namely the fact that for T sufficiently large (depending on (W, S)), the auxiliary system in difference form (2.30) has a solution (q, σ) defined for all t ≥ T and decaying at infinity in a suitable sense. In the same spirit as for Proposition 4.4, this will be done by showing that the map Γ : (q, σ) → (q , σ ) defined by Proposition 6.2 is a contraction in suitable circumstances. According to our intuition of scattering, another natural route towards the same result would be to construct first the solution (qt0 , σt0 ) of the auxiliary system (2.30) vanishing at t0 and to take the limit of that solution as t0 → ∞. That route can also be followed, but it is slightly more complicated than the previous one. One of the complications comes from the fact that the system (2.30) depends on t1 . In view of Warning 4.2, for finite t0 , we expect difficulties if we take t1 > t0 . This prompts us to take t1 = t0 . The comparison of two solutions (qt0 , σt0 ) corresponding to different values of t0 is then complicated by the fact that they do not solve exactly the same system, so that Lemma 6.2 is not directly applicable and additional terms occur in the comparison. On the other hand, for B0 = 0, the construction of the solution (qt0 , σt0 ) of (2.30) is expected to meet difficulties for t ≥ t0 because of Warning 4.1. We shall therefore undertake it for t ≤ t0 only, which is sufficient anyway to take the limit t0 → ∞. That construction proceeds again by a contraction starting from the solutions obtained for the linearized system. The corresponding proof for t0 < ∞ is not significantly simpler than for t0 = ∞, which is another reason why the second method is more complicated than the first one, since in addition to that construction, a limiting procedure is needed.
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We now state the main result and formalize the previous heuristic discussion in the following proposition. Proposition 6.3 Let 1 < k ≤ : and : > 3/2. Let β, λ0 and λ satisfy (6.44) and in addition 1 + λ > β(5/2 − k) . Let B0 satisfy the estimates (3.17) for 0 ≤ m ≤ k. Let (W, S) satisfy the assumptions of Proposition 6.2 in [1, ∞). Then there exists T , 1 ≤ T < ∞, and positive constants Y0 , Y and Z, depending on k, :, β, λ0 , λ, a, b, c0 , c1 and c2 , such that the following holds. (1) For all t0 , T ≤ t0 < ∞, the system (2.30) with t1 = t0 has a unique solution (q, σ) ∈ C(I, X k, ) with I = [T, t0 ] and (q, σ)(t0 ) = 0. That solution satisfies the estimates (6.49) (6.50) for all t ∈ I. (2) The system (2.30) with t1 = ∞ has a unique solution (q, σ) ∈ C(I, X k, ), where I = [T, ∞) satisfying the estimates (6.49) (6.50) for all t ∈ I. (3) Let (qt0 , σt0 ) be the solution defined in Part (1) for t0 < ∞ and let (q, σ) be the solution defined in Part (2) for t0 = ∞. When t0 → ∞, (qt0 , σt0 ) converges to (q, σ) strongly in L∞ (J, X k , ) for 0 ≤ k < k, 0 ≤ : < :, and in the weak-∗ sense in L∞ (J, X k, ) for any interval J = [T, T¯ ] with T¯ < ∞. Proof. Parts (1) and (2). We prove Parts (1) and (2) together, because the proof is exactly the same for both. It consists in showing that the map Γ : (q, σ) → (q , σ ) defined by solving the linearized system (2.31) is a contraction on a suitable subset of C(I, X k, ) in the lower norms used in Lemma 6.2. For t0 < ∞, the map Γ is defined by Proposition 6.1, restricted to those (q, σ) satisfying (q, σ)(t0 ) = 0, with the initial data (q , σ )(t0 ) = 0. For t0 = ∞, the map Γ is defined by Proposition 6.2. The relevant estimates on Γ are those derived in the proof of Proposition 6.2 in the case t1 = ∞. The same estimates also apply to the case t0 = t1 < ∞, which is relevant for Part (1) of this proposition. They are independent of t0 . We define the set R = (q, σ) ∈ C(I, X k, ) : (q, σ)(t0 ) = 0 if t0 < ∞ ,
tλ0 q; L∞ (I, L2 ) ≤ Y0 , tλ ω k q; L∞ (I, L2 ) ≤ Y , Sup tλ0 −β(m+1) ω m ∇σ; L∞ (I, L2 ) ≤ Z .
0≤m≤
(6.77)
We first show that R is stable under Γ for suitable Y0 , Y , Z and for sufficiently large T . Let (q, σ) ∈ R and (q , σ ) = Γ(q, σ). Then (q , σ ) satisfies the estimates are defined by (6.53) (6.55) (6.63) (6.56) (6.64) (6.68) (6.71) where Y0 , Y , Zm (6.65) and/or their extension to t0 < ∞. It is therefore sufficient to ensure that the RHS of (6.56) (6.64) (6.68) (6.71) are not larger than Y0 , Y , Z, and Z respectively.
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For that purpose, it is sufficient to choose Y0 = 2c0 , Z = 2eν −1 (c2 + 4ac0 ) Y = e (4λ−1 b0 )k 2c0 + 4λ−1 (c1 + c0 )
(6.78)
and to take T sufficiently large in the sense that 1/2 −(k−1/2+λ) −(1−β) + a2 Y0 T −β + a Y03 Y¯ T ≤ c0 , aZ T η β η T ≥ 8b , T ≥ 4eZ , a T ≥ Y0 , E(T ) ≤ 1 , C1 (T ) ≤ 2(c1 + c0 ) .
(6.79)
The conditions (6.79) are lower bounds on T expressed in terms of the parameters listed in the Proposition, after substitution of (6.78). We next show that Γ is a contraction on R in the norms considered in Lemma 6.2. Let (qi , σi ) ∈ R and (qi , σi ) = Γ(qi , σi ), i = 1, 2, and define (q± , σ± ) and , σ± ) as in Lemma 6.2. We define in addition (q± y− = q− 2 λ0
z−m = ω m ∇σ− 2
, ∞
Y− = t y− ; L (I) ,
Z− = Sup t 0≤m≤0
(6.80) λ0 −β(m+1)
∞
z−m ; L (I)
(6.81)
and we make similar definitions for the primed quantities. We take :0 = [3/2 − k]+ and estimate y− and z−m by (6.29) (6.30), taking advantage of the fact that m = m in (6.30) for that choice of :0 . Using the fact that Γ maps R into itself and omitting again overall constants, we obtain ∂t y− ≤ a Z− t−2−λ0 +β + Y¯ Z− t−2−λ0 +β(0 +1)−λ + a2 Y− t−1−β−λ0 +a Y¯ Y− t−1−2λ0 +(λ0 −λ)/2k + Y¯ 2 Y− t−1−3λ0 +2(λ0 −λ)/k (6.82) where Y¯ = Y ∨ Y0 ,
−1−η ∂t z−m ≤ b t + Z t−2 z−m + Z− t−λ0 +β(m+1) +a Y− t−1−λ0 +β(m+1) + Y0 Y− t−1−2λ0 +β(m+5/2)
(6.83)
for 0 ≤ m ≤ :0 . Integrating (6.82) (6.83) from t to t0 with (y− , z−m )(t0 ) = 0 and using again the fact that λ0 > λ + k > 1 and λ0 > β(: + 1) > β((:0 + 2) ∨ 5/2), we obtain
Y− ≤ a Z− T −(1−β) + Y¯ Z− T −1−λ+β(0 +1) +a2 Y− T −β + a Y¯ Y− T −(k−1/2+λ) + Y¯ 2 Y− T −2(k−1+λ) ,
(6.84)
−1 −η b η T + Z T −1 Z− Z− ≤ exp b η −1 T −η + Z T −1 . +β −1 a Y− + T −β Y0 Y−
(6.85)
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We now ensure that the map Γ is a contraction for the norms defined by (6.80) (6.81) in the form Y− ≤ c−1 Z− + Y− /4 (6.86) Z− ≤ (Z− + c Y− ) /4 which imply
+ c Y− ≤ (Z− + c Y− ) /2 Z−
(6.87)
−1
by taking c = 8β a and T sufficiently large, depending on the parameters listed in the proposition, in part explicitly and in part through Y0 , Y and Z defined by (6.78). (It is only at this point that we need the condition 1 + λ > β(5/2 − k), in order to ensure that the power of T in the second term in the RHS of (6.84) is negative). We have proved that for sufficiently large T , the map Γ maps R defined by (6.77) into itself and is a contraction for the norms (6.81). By a standard compactness argument, R is closed for the latter norms, and therefore Γ has a unique fixed point in R, which completes the existence part of the proof of Parts (1) and (2). The uniqueness statement of Part (1) is a special case of Proposition 4.2 part (1), while the uniqueness statement of Part (2) follows from Proposition 4.2 part (3) and from the fact that λ0 > 1 > β2 . Part (3). Let T < t1 < t2 < ∞ and let (qi , σi ), i = 1, 2, be the solutions of the ssytem (2.30) obtained in part (1) and corresponding to t0 = t1 = ti respectively. Those solutions satisfy the estimates (6.49) (6.50) for t ≤ ti . Define as before (q± , σ± ) = (1/2)(q1 ± q2 , σ1 ± σ2 ). We shall estimate (q− , σ− ) for t ≤ t1 in the norms considered in Lemma 6.2. In order to alleviate the notation, we omit the prime on t1 , t2 in the rest of the proof. By (6.49) (6.50), we estimate 0
q− (t1 ) 2 = (1/2) q2 (t1 ) 2 ≤ (1/2)Y0 t−λ 1 (6.88) −λ +β(m+1)
ω m ∇σ− (t1 ) 2 = (1/2) ω m ∇σ2 (t1 ) 2 ≤ (1/2)Zt1 0 for 0 ≤ m ≤ :. On the other hand (q− , σ− ) satisfies a system closely related to (6.32) where however (q± , σ± ) = (q± , σ± ) and where additional terms appear because of the different values t1 and t2 occuring in BS and BL . More precisely ∂t q− = i(2t2 )−1 ∆q− + t−2 Q(s+ , q− ) + Q(σ− , w+ ) + it−1 B0 q− +it−1 BSt1 ,∞ (w+ , w+ ) + BSt1 (q− , q− ) q− + 2BSt1 (w+ , q− )w+ −i(2t)−1 BSt2 − BSt1 (q2 , q2 + 2W ) (q2 + W ) (6.89) t1 ∂t σ− = t−2 (s+ · ∇σ− + σ− · ∇s+ ) − 2t−1 ∇BL (w+ , q− ) t2 t1 −1 +t ∇ BL − BL (q2 , q2 + 2W ) .
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We first estimate the additional terms in (6.89) as compared with (6.32). From (6.36)–(6.39) we obtain
BSt2 − BSt1 (q2 , q2 + 2W ) (q2 + W ) 2 ≤ C at−β + q2 3 a I0 ( q2 2 ) + a I−1 ( q2 2 q2 3 ) (6.90) + q2 6 I−1/2 ( q2 2 q2 6 ) where the various Im ’s are taken in the interval [t1 , t2 ]. From (6.49) and Sobolev inequalities, we obtain t2 −1/2−λ0 dt t−3/2−λ0 ≤ Y0 t1/2 t1 (6.91) I0 ( q2 2 ) ≤ Y0 t1/2 t1
I−1 ( q2 2 q2 3 ) ≤ ≤
C Y0 Y¯ t−1/2
t2
dt t−1/2−2λ0 +(λ0 −λ)/2k
t1 −1/2 1/2−2λ0 +(λ0 −λ)/2k t t1
C Y0 Y¯
(6.92)
q2 6 I−1/2 ( q2 2 q2 6 ) ≤ CY0 Y¯ 2 t−λ0 +(λ0 −λ)/k t2 −2λ +(λ −λ)/k × dt t−1−2λ0 +(λ0 −λ)/k ≤ CY0 Y¯ 2 t−λ0 +(λ0 −λ)/k t1 0 0 (6.93) t1
and therefore for t ≤ t1 −β−1/2
BSt2 − BSt1 (q2 , q2 + 2W ) (q2 + W ) 2 ≤ C t−λ0 +1/2 a2 Y0 t1 −λ +(λ −λ)/2k−1/2 −2λ +2(λ0 −λ)/k−1/2 . (6.94) +a Y0 Y¯ t1 0 0 + Y0 Y¯ 2 t1 0 Similarly using (6.43) and I−3/2 q2 22 ≤ Y02 t−1
t2
t1
0 dt t−2λ0 ≤ Y02 t−1 t1−2λ 1
we estimate for t ≤ t1
t2 t1 0 0 tβ(m+1) t−λ (q2 , q2 + 2W ) 2 ≤ C a Y0 + Y02 t3β/2−1 t1−λ
ω m+2 BL − BL . 1 1 (6.95) We define y− and z−m by (6.80), we take again :0 = [3/2 − k]+ , we choose λ0 satisfying (6.96) 1 ∨ (λ0 − 1/2) ∨ β (:0 + 1) < λ0 < λ0 , we define (see (6.81))
Y− = tλ0 y− ; L∞ ([T, t1 ]) , Z− = Sup tλ0 −β(m+1) z−m ; L∞ ([T, t1 ]) (6.97) 0≤m≤0
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and we estimate those quantities in the same way as in the proof of Parts (1) and (2). From (6.89), we obtain differential inequalities for y− , z−m , very similar to (6.82) (6.83) with y− = y− , z−m = z−m with however additional terms estimated by (6.94) (6.95). We integrate those inequalities from t to t1 , with initial condition at t1 estimated by (6.88). We then substitute the result in (6.97) and omitting an overall constant, we obtain finally (see (6.84) (6.85)) Y− ≤ aZ− T −(1−β) + Y¯ Z− T −1−λ+β(0 +1) + a2 Y− T −β +aY¯ Y− T −(k−1/2+λ) + Y¯ 2 Y− T −2(k−1+λ) ¯ −(k−1/2+λ) + Y0 Y¯ 2 t−2(k−1+λ) t−(λ0 −λ0 )(6.98) + Y0 + a2 Y0 t−β 1 + aY0 Y t1 1 1 −1 −η Z− ≤ exp b η −1 T −η + Z T −1 b η T + Z T −1 Z−
−(λ −λ ) t1 0 0 . (6.99) +β −1 a Y− + T −β Y0 Y− + Z + a Y0 + Y02 t−β 1 Proceeding as above, we deduce therefrom that for T sufficiently large and for a suitable constant c
−(λ −λ ) . (6.100) Y− + cZ− ≤ O t1 0 0 From (6.100) it follows that (qt0 , σt0 ) tends to a limit uniformly in compact subintervals of [T, ∞) in the norms (6.80). By a standard compactness argument, that limit belongs to C([T, ∞), X k, ) and satisfies (6.49) (6.50). One sees easily that the limit satisfies the system (2.30) with t1 = ∞, and therefore coincides with the solution obtained in Part (2). Actually, as mentioned before, Part (3) provides an alternative (more complicated ) proof of Part (2).
7 Choice of (W, S) and remainder estimates In this section, we construct approximate solutions (W, S) of the system (2.20) satisfying the assumptions needed for Propositions 6.2 and 6.3 and in particular the remainder estimates (6.47) (6.48), thereby allowing for the applicability of Proposition 6.3, namely for the construction of solutions of the system (2.30). More general (W, S) also suitable for the same purpose, could also be constructed by exploiting the gauge invariance of the system (2.20). We rewrite the remainders as R1 (W, S) = U ∗ (1/t)∂t (U (1/t)W ) − t−2 Q(S, W ) − it−1 (B0 + BS (W, W ))W (7.1) R2 (W, S) = ∂t S − t−2 S · ∇S + t−1 ∇BL (W, W ) .
(2.29) ≡ (7.2)
We recall that t1 = ∞ in R1 , R2 , and we omit t1 from the notation. We construct (W, S) by solving the system (2.20) approximately by iteration. The n-th iteration
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should be sufficient to cover the case λ0 < n. Here we need λ0 > k > 1, and we must therefore use at least the second iteration, which will allow for λ0 < 2. For simplicity, we shall not go any further here. Accordingly we take W = w0 + w1
where
,
S = s0 + s1
∂t U (1/t)w0 = 0
w0 (∞) = w+ ,
∂t s0 = −t−1 ∇BL (w0 , w0 )
s0 (1) = 0 ,
∗ w0 = U (1/t)w+ , t s0 (t) = − dt t−1 ∇BL (w0 (t ), w0 (t ))
so that
(7.3)
(7.4)
(7.5)
1
and
∂t (U (1/t)w1 ) = t−2 U (1/t) Q(s0 , w0 )
w1 (∞) = 0 ,
∂t s1 = t−2 s0 · ∇s0 − 2t−1 ∇BL (w0 , w1 )
s1 (∞) = 0 ,
so that ∞ ∗ dt t−2 U (1/t ) Q(s0 (t ), w0 (t )) w1 (t) = −U (1/t) t ∞ ∞ −2 s (t) = − dt t s (t ) · ∇s (t ) + 2 dt t−1 ∇BL (w0 (t ), w1 (t )) . 1 0 0 t
(7.6)
(7.7)
t
The remainders then become −2 R Q(s (W, S) = −t , w ) + Q(s , w ) + Q(s , w ) 1 0 1 1 0 1 1 −it−1 (B0 + BS (W, W ))W , R2 (W, S) = −t−2 s0 · ∇s1 + s1 · ∇s0 + s1 · ∇s1 + t−1 ∇BL (w1 , w1 ) .
(7.8)
Note that the term with B0 + BS (W, W ) in (7.1) is regarded as short range and not included in the definition of (W, S). We now turn to the derivation of the estimates (6.45)–(6.48). The regularity properties of (W, S) used in Section 6 follow from similar but simpler estimates. We first estimate all the terms not containing B0 . Lemma 7.1 Let 0 < β < 1, k+ ≥ 3, w+ ∈ H k+ and a+ = |w+ |k+ . Then the following estimates hold: |w0 |k+ ≤ a+ (7.9) C a2+ :n t for 0 ≤ m ≤ k+
ω m s0 2 ≤ (7.10) C a2+ tβ(m−k+ ) for m > k+ ,
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|w1 |k+ −1 ≤ C a3+ t−1 (1 + :n t) ,
4 −1 2 for 0 ≤ m ≤ k+ − 1 C a+ t (1 + :n t) m
ω s1 2 ≤ C a4+ t−1+β(m+1−k+ ) (1 + :n t) for k+ − 1 < m < k+ − 1 + β −1 , −3 3 for 0 ≤ m ≤ k+ − 2 C(a+ ) t (1 + :n t) m
ω R2 (W, S) 2 ≤ C(a+ ) t−3+β(m+2−k+ ) (1 + :n t)2 for k+ − 2 < m < k+ − 2 + β −1 ,
597
(7.11)
(7.12)
(7.13)
ω m (Q(S, w1 ) + Q(s1 , w0 )) 2 ≤ C(a+ ) t−1 (1 + :n t)2
ω m (BS (W, W )W ) 2 ≤ C(a+ ) t−β(k+ −m+1)
for 0 ≤ m ≤ k+ − 2 (7.14) for 0 ≤ m ≤ k+ − 1 . (7.15)
Proof. (7.9) is trivial. (7.10). By (3.10), Lemma 3.2 and (3.9) we estimate
ω m s0 2 ≤
t
1
dt t−1 ω m+1 BL (w0 (t ), w0 (t )) 2
t C dt t−1 Im ( ω m w0 (t ) 2 w0 (t ) ∞ ) ≤ Ca2+ :n t 1 for 0 ≤ m ≤ k+ ≤ t C dt t−1+β(m−k+ ) Ik+ ω k+ w0 (t ) 2 w0 (t ) ∞ ≤ Ca2+ tβ(m−k+ ) 1 for m > k+ . (7.11). By Lemma 3.2 and (7.10), we estimate
Q(s0 , w0 ) 2 ≤ C ∇s0 2 ω 3/2 w0 2 + w0 ∞ ≤ C a3+ :n t
ω k+ −1 Q(s0 , w0 ) 2 ≤ C ω k+ s0 2 ω 3/2 w0 2 + w0 ∞
+ ω 3/2 s0 2 + s0 ∞ ω k+ w0 2 ≤ C a3+ :n t from which (7.11) follows by integration. (7.12). By Lemma 3.2 and (7.10) we estimate
ω m (s0 · ∇s0 ) 2 ≤ C ω m+1 s0 2 ω 3/2 s0 2 + s0 ∞ ≤
Ca4+ (:n t)2
for m ≤ k+ − 1
Ca4+ tβ(m+1−k+ ) :n t
for m > k+ − 1 .
(7.16)
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On the other hand
ω m+1 BL (w0 , w1 ) 2 m m 4 −1 CIm ( ω w0 2 w1 ∞ + ω w1 2 w0 ∞ ) ≤ Ca+ t (1 + :n t) for m ≤ k+ − 1 ≤ k −1 β(m+1−k ) + Ct Ik+ −1 ω + w0 2 w1 ∞ + ω k+ −1 w1 2 w0 ∞ for m > k+ − 1 . ≤ Ca4+ tβ(m+1−k+ )−1 (1 + :n t) (7.17) (7.12) now follows from (7.16) and (7.17) by integration provided β(m+1−k+ ) < 1. (7.13). By Lemma 3.2 again, and by (7.10) (7.12) we estimate
ω m (s0 · ∇s1 + s1 · ∇s0 + s1 · ∇s1 ) 2 ≤
C ω m+1 s0 2 ω 3/2 s1 2 + s1 ∞ + ω m+1 s1 2 ω 3/2 s0 2 + s0 ∞
+ ω 3/2 s1 2 + s1 ∞ for m ≤ k+ − 2 C(a+ ) t−1 (1 + :n t)3 ≤ C(a+ ) t−1+β(m+2−k+ ) (1 + :n t)2 for k+ − 2 < m < k+ − 2 + β −1 . (7.18) On the other hand
ω m+1 BL (w1 , w1 ) 2 m 6 −2 2 for m ≤ k+ − 1 CIm ( ω w1 2 w1 ∞ ) ≤ Ca+ t (1 + :n t) ≤ Ctβ(m+1−k+ ) Ik+ −1 ω k+ −1 w1 2 w1 ∞ ≤ Ca6+ tβ(m+1−k+ )−2 (1 + :nt)2 for m > k+ − 1. (7.19) (7.13) now follows from (7.8) (7.18) (7.19). (7.14). By Lemma 3.2 again, and by (7.10) (7.11) (7.12) we estimate
ω m (Q(s0 , w1 ) + Q(s1 , w0 ) + Q(s1 , w1 )) 2
≤ C ω m+1 s0 2 ω 3/2 w1 2 + w1 ∞ + ω m+1 w1 2 ω 3/2 s0 2
+ s0 ∞ + ω 3/2 s1 2 + s1 ∞ + ω m+1 s1 2 ω 3/2 w0 2 + w0 ∞
+ ω 3/2 w1 2 + w1 ∞ + ω m+1 w0 2 ω 3/2 s1 2 + s1 ∞ ≤ Ca5+ t−1 (1 + :n t)2 (1 + a2 t−1 (1 + :n t)) from which (7.14) follows. (7.15). As previously, we estimate for 0 ≤ m ≤ k+
ω m BS (W, W ) 2 ≤ ω m BS (w0 , w0 ) 2 +2 ω m BS (w0 , w1 ) 2 + ω m BS (w1 , w1 ) 2
(7.20)
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≤ Ctβ(m−k+ ) t−β Ik+ ω k+ w0 2 w0 ∞ + Ik+ −1 ω k+ −1 w0 2 w1 ∞ + ω k+ −1 w1 2 ( w0 ∞ + w1 ∞ ) (7.21) ≤ Ctβ(m−k+ ) t−β a2+ + t−1 a4+ (1 + :n t) + t−2 a6+ (1 + :n t)2 by (7.11). Therefore
BS (W, W )W 2 ≤ BS (W, W ) 2 W ∞ ≤ C(a+ ) t−β(k+ +1)
ω k+ −1 (BS (W, W )W ) 2 ≤ C ω k+ −1 BS (W, W 2 W ∞ ≤ C(a+ ) t−2β + BS (W, W ) ∞ ω k+ −1 W 2
(7.22)
which yields (7.15) by interpolation. We next estimate the terms in R1 containing B0 .
Lemma 7.2 Let 0 < β < 1. Let 1/2 < λ0 < 2 and k+ ≥ 2λ0 ∨ 3. Let B0 satisfy the estimates (3.17) for 0 ≤ m ≤ 2, let w+ ∈ H k+ and assume that B0 and w+ satisfy the estimate (5.35) for all multi-indices α1 , α2 with 0 ≤ |α1 | ≤ 2 and 0 ≤ |α2 | < 2λ0 . Then the following estimate holds for all m, 0 ≤ m ≤ 2, and all t ≥ 1. (7.23)
ω m (B0 W ) 2 ≤ C t−λ0 +m . ¯ = 2. Proof. The contribution of w0 to (7.23) is estimated by Lemma 5.1 with m In order to estimate the contribution of w1 , we decompose w1 = w1 + w1 where ∞ w1 = − dt t−2 Q(s0 (t ), w+ ) t ∞ w1 (t) = (1 − U ∗ (1/t)) dt t−2 U (1/t )Q(s0 (t ), w0 (t )) t ∞ dt t−2 (1 − U (1/t ))Q(s0 (t ), w0 (t )) + Q(s0 (t ), (1 − U (1/t ))w+ ) . + t
We first consider B0 (t)w1 (t) = −
∞
t
dt t−2 s0 (t ) · B0 (t)∇w+ + (1/2)(∇ · s0 )(t )B0 (t)w+
We estimate
B0 (t)w1 (t)
2 ≤
t
∞
dt t−2 s0 (t ) ∞ B0 (t)∇w+ 2
+ (∇ · s0 )(t ) ∞ B0 (t)w+ 2
≤C t
−λ0 +1/2
≤ C t−λ0 −1/2 (1 + :n t) by (5.35) and (7.10).
∞
dt t−2 :n t
t
(7.24)
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Similarly, we estimate
∆(B0 (t)w1 (t))
2 ≤ C
t
∞
dt t−2 ∆s0 3 B0 ∇w+ 6
+ ∇s0 ∞ ∇(B0 ∇w+ ) 2 + s0 ∞ ∆(B0 ∇w+ ) 2 + ∆∇ · s0 2 B0 w+ ∞ (7.25) + ∇2 s0 6 ∇(B0 w+ ) 3 + ∇ · s0 ∞ ∆(B0 w+ ) 2 where s0 = s0 (t ) and B0 = B0 (t), and therefore by (7.10)
∆(B0 w1 ) 2 ≤ C a2+ t−1 (1 + :n t) ( B0 ∞ + ∇B0 3 ) a+ + (∆B0 )∇w+ 2 + (∆B0 )w+ 2
≤ C a2+ a+ b0 t−1/3 + b1 t−λ0 +3/2 (1 + :n t)
(7.26)
by (3.17) and (5.35). We next estimate the contribution of w1 . By the same estimates as for w1 (see the proof of (7.11)) we obtain |w1 |k+ −1 ≤ C a3+ t−1 (1 + :n t) |w1 |k+ −3 ≤ C a3+ t−2 (1 + :n t) where we have used the fact that the factors (1 − U (∗) (1/t)) can be replaced by t−1 ∆ for the purpose of the second estimate, and therefore
ω m w1 2 ≤ C a3+ t−2+m/2 (1 + :n t) for 0 ≤ m ≤ k+ − 1
(7.27)
by interpolation. By Lemma 3.2 and (3.17) we then obtain
ω m (B0 w1 ) 2 ≤ C ( ω m B0 ∞ w1 2 + B0 ∞ ω m w1 2 ) ≤ C b0 a3+ t−2+m (1 + :n t) .
(7.28)
Collecting (7.24) (7.26) (7.28) and the estimates of B0 w0 coming from Lemma 5.1 yields (7.23) for 0 ≤ m ≤ 2. We can now collect Proposition 6.3 and Lemmas 7.1 and 7.2 to obtain the main technical result on the Cauchy problem for the auxiliary system in the difference form (2.30). We again keep the assumptions on B0 in the implicit form of the estimates (3.17) and (5.35), which can however be replaced by sufficient conditions on (w+ , A+ , A˙ + ) by the use of Lemmas 3.5 and 5.2. Proposition 7.1 Let 1 < k ≤ : and : > 3/2. Let β, λ0 and λ satisfy 0 < β < 2/3
,
λ>0
,
λ + k < λ0 < 2
,
λ0 > β(: + 1) .
(7.29)
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Let k+ satisfy k+ ≥ k + 2
k+ ≥ 2λ0
,
β(k+ + 1) ≥ λ0
,
,
β(: + 3 − k+ ) < 1 . (7.30)
Let w+ ∈ H k+ , let B0 satisfy the estimates (3.17) for 0 ≤ m ≤ k and let (w+ , B0 ) satisfy the estimates (5.35) for all multi-indices α1 , α2 with 0 ≤ |α1 | ≤ 2 and 0 ≤ |α1 | < 2λ0 . Let (W, S) be defined by (7.3) (7.5) (7.7). Then (1) (W, S) satisfy the estimates (6.45) (6.46) (6.47) (6.48), with 0 < η < 1−3β/2 in (6.46). (2) All the statements of Proposition 6.3 hold. Proof. It follows from Lemmas 7.1 and 7.2 that all the assumptions of Proposition 6.3, and in particular the estimates (6.45)–(6.48), are satisfied.
8 Wave operators and asymptotics for (u, A) In this section we complete the construction of the wave operators for the system (1.1) (1.2) and we derive asymptotic properties of solutions in their range. The construction relies in an essential way on Proposition 7.1. So far we have worked with the system (2.20) for (w, s) and the first task is to reconstruct the phase ϕ. Corresponding to S = s0 + s1 , we define φ = ϕ0 + ϕ1 where t ∞ ϕ0 = − dt t−1 BL (w0 (t ), w0 (t )) (8.1) ϕ1 = −
1
∞
t
dt (2t2 )−1 |s0 (t )|2 + 2
t
∞
∞ dt t−1 BL (w0 (t ), w1 (t ))
(8.2)
so that s0 = ∇ϕ0 and s1 = ∇ϕ1 . Let now (q, σ) be the solution of the system (2.30) constructed in Proposition 6.3 part (2) and let (w, s) = (W, S) + (q, σ). We define ∞ dt (2t2 )−1 (σ · (σ + 2S) + s1 · (s1 + 2s0 )) (t ) ψ=−
t
∞
+ t
∞ ∞ ∞ dt t−1 (BL (q, q) + 2BL (W, q) + BL (w1 , w1 )) (t )
(8.3)
which is taylored to ensure that ∇ψ = σ, given the fact that s0 , s1 and σ are gradients. The integral converges in H˙ 1 , as follows from (6.49) (6.50) and from the estimate (see the proof of (6.4)) ∂t σ 2 ≤ t−2 ∇σ 2 ( s ∞ + ∇S 3 ) + t−1 a I0 ( q 2 ) +t−1+3β/2 I−3/2 q 22 + R2 (W, S) 2
≤ C t−2−λ0 +β (1 + :n t) + t−1−λ0 + t−1−2λ0 +3β/2 + t−3 (1 + :n t)3 ≤ C t−1−λ0 .
(8.4)
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Furthermore, this implies that
∇ψ 2 = σ 2 ≤ C t−λ0 .
(8.5)
Finally we define ϕ = φ + ψ so that ∇ϕ = s, and (w, ϕ) solves the system (2.18). For more details on the reconstruction of ϕ from s, we refer to Section 7 of [6]. We can now define the wave operators for the system (1.1) (1.2) as follows. We start from the asymptotic state (u+ , A+ , A˙ + ) for (u, A). We define w+ = F u+ , we define (W, S) by (7.3) (7.5) (7.7) and B0 by (2.3) (2.13), namely ˙ A+ + K(t) A˙ + = t−1 D0 B0 . A0 = K(t) We next solve the system (2.30) with t1 = ∞ and with initial time t0 = ∞ for (q, σ) by Proposition 6.3, part (2), we define (w, s) = (W, S) + (q, σ) and we reconstruct ϕ from s as explained above, namely ϕ = ϕ0 + ϕ1 + ψ with ϕ0 , ϕ1 and ψ defined by (8.1) (8.2) (8.3). We finally substitute (w, ϕ) thereby obtained into (2.11) (2.2), thereby obtaining a solution (u, A) of the system (1.1) (1.2). The wave operator is defined as the map Ω : (u+ , A+ , A˙ + ) → (u, A). In order to state the regularity properties of u that follow in a natural way from the previous construction, we introduce appropriate function spaces. In addition to the operators M = M (t) and D = D(t) defined by (2.8) (2.9), we introduce the operator J = J(t) = x + it ∇ , (8.6) the generator of Galilei transformations. The operators M , D, J satisfy the commutation relation iM D ∇=J M D . (8.7) For any interval I ⊂ [1, ∞) and any k ≥ 0, we define the space X k (I) = u : D∗ M ∗ u ∈ C(I, H k ) = u :< J(t) >k u ∈ C(I, L2 )
(8.8)
where < λ >= (1+λ2 )1/2 for any real number or self-adjoint operator λ and where the second equality follows from (8.7). We now collect the information obtained for the solutions of the system (1.1) (1.2) and state the main result of this paper as follows. Proposition 8.1 Let 1 < k ≤ :, : > 3/2. Let β, λ0 and λ satisfy 0 < β < 2/3
,
λ>0
,
λ + k < λ0 < 2
,
λ0 > β(: + 1) .
(7.29)
Let k+ satisfy k+ ≥ k + 2
,
k+ ≥ 2λ0
,
β(k+ + 1) ≥ λ0
,
β(: + 3 − k+ ) < 1 . (7.30)
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Let u+ ∈ F H k+ , let w+ = F u+ and a+ = |w+ |k+ . Let (A+ , A˙ + ) ∈ H k ⊕ H k−1 . Let A0 defined by (2.3) satisfy the estimates
ω m A0 (t) r ≤ b0 t2/r−1
(3.15) ≡ (8.9)
for 0 ≤ m ≤ k, 2 ≤ r ≤ ∞ and all t ≥ 1, and the estimates
(∂ α1 A0 ) ((∂ α2 w+ ) (x/t)) 2 ≤ b1 t−λ0 +(1+|α2 |)/2
(8.10)
for all multi-indices α1 , α2 with 0 ≤ |α1 | ≤ 2 and 0 ≤ |α2 | < 2λ0 . Let (W, S) be defined by (7.3) (7.5) (7.7). Then (1) There exists T , 1 ≤ T < ∞ such that the auxiliary system (2.20) with t1 = ∞ has a unique solution (w, s) ∈ C([T, ∞), X k, ) satisfying
w(t) − W (t) 2 ≤ C t−λ0
(8.11)
ω k (w(t) − W (t)) 2 ≤ C t−λ
(8.12)
m
ω (s(t) − S(t)) 2 ≤ C t
−λ0 +βm
for 0 ≤ m ≤ : + 1 .
(8.13)
(2) Let φ = ϕ0 + ϕ1 be defined by (8.1) (8.2), let ϕ = φ + ψ with ψ defined by (8.3) and (q, σ) = (w, s) − (W, S). Let (2.11) ≡ (8.14)
u = M D exp(−iϕ)w
and define A by (2.2) (2.3) (2.4) with t1 = ∞. Then u ∈ X k ([T, ∞)), (A, ∂t A) ∈ C([T, ∞), H k ⊕ H k−1 ), (u, A) solves the system (1.1) (1.2) and u behaves asymptotically in time as M D exp(−iφ)W in the sense that it satisfies the following estimates:
u(t) − M (t) D(t) exp(−iφ(t))W (t) 2 ≤ C(a+ , b0 , b1 )t−λ0
(8.15)
|J(t)|k (exp(iφ(t, x/t))u(t) − M (t) D(t) W (t)) 2 ≤ C(a+ , b0 , b1 )t−λ
(8.16)
u(t) − M (t) D(t) exp(−iφ(t))W (t) r ≤ C(a+ , b0 , b1 )t
−λ0 +(λ0 −λ)δ(r)/k
(8.17)
for 0 ≤ δ(r) = (3/2 − 3/r) ≤ [k ∧ 3/2]. Define in addition 2 A2 = A − A0 − A∞ 1 (|DW | ) .
(8.18)
2 Then A behaves asymptotically in time as A0 + A∞ 1 (|DW | ) in the sense that A2 satisfies the following estimates:
A2 (t) 2 ≤ C(a+ , b0 , b1 ) t−λ0 +1/2 .
(8.19)
Furthermore, for 3/2 < k(< 2):
∇A2 (t) 2 ≤ C(a+ , b0 , b1 ) t−λ0 −1/2
(8.20)
ω k ∇A2 (t) 2 ≤ C(a+ , b0 , b1 ) t−λ−k−1/2 ,
(8.21)
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while for (1 <)k < 3/2:
∇A2 (t) 2 ≤ C(a+ , b0 , b1 ) t−λ0 −1/2 + t−2λ0 −1/2+(λ0 −λ)3/2k
(8.22)
ω 2k−1/2 A2 (t) 2 ≤ C(a+ , b0 , b1 )t−λ−2k+1 t−λ + tk−3/2 .
(8.23)
A similar result holds for k = 3/2 with a tε loss in the decay. Proof. The proof follows from Propositions 6.3 part (2) and from Proposition 7.1, supplemented with the reconstruction of ϕ described above in this section, except for the estimates (8.15)–(8.17) on u and (8.19)–(8.23) on A. In particular the estimate (8.10) is nothing but the estimate (5.35) expressed in terms of A0 instead of B0 while the estimates (8.11) (8.12) (8.13) are essentially (6.49) (6.50) supplemented with (8.4) (8.5). We next prove the estimates (8.15) - (8.17) on u. From (8.14) with ϕ = φ + ψ and from (8.7), it follows that
|J|m (exp(i D0 φ)u − M DW ) 2 = ω m w e−iψ − W 2 (8.24) For m = 0, we estimate
w exp(−iψ) − W 2
≤ w (exp(−iψ) − 1) 2 + w − W 2 ≤ w 3 ψ 6 + q 2 ≤ C t−λ0
by (8.5), a Sobolev inequality and (8.11). This proves (8.15). For m = k, we estimate by Lemma 3.2
ω k (exp(−iψ)w − W ) 2 ≤ C ω k (exp(−iψ) − 1) 3 w 6 + exp(−iψ) − 1 ∞ ω k w 2 + ω k (w − W ) 2 ≤ C ω k−1/2 σ 2 exp (C ψ ∞ ) ∇w 2 + ( σ 2 ∇σ 2 )
+ ω k q 2 ≤ C t−λ0 +β(k−1/2) + t−λ0 +β/2 + t−λ ≤ C t−λ
1/2
ω k w 2
by Lemma 3.3, by the Sobolev inequality
ψ ∞ ≤ C ( σ 2 ∇σ 2 )1/2 and by (8.12) (8.13). This proves (8.16). The estimate (8.17) follows immediately from (8.15) (8.16) and from the inequality
f r = t−δ(r) D∗ M ∗ f r ≤ C t−δ(r) ω δ(r) D∗ M ∗ f 2 = C t−δ(r) |J(t)|δ(r) f 2 .
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We finally prove the estimates (8.19)–(8.23) on A. It follows from the definitions (2.2) (2.3) (2.4) (8.18) and from (2.13) (2.14) that A2 = t−1 D0 B1∞ (q, q + 2W ) .
(8.25)
It is therefore sufficient to estimate B1∞ (q, q + 2W ). We omit the superscript ∞ for brevity. We first estimate by (3.10)
B1 (q, q + 2W ) 2 ≤ C I−1 ω −1 (q(q + 2W )) 2 ≤ C I−1 ( q 2 q + 2W 3 ) ≤ C t−λ0
(8.26) k
by Sobolev inequalities and by (8.11), since q + 2W is bounded in H and a fortiori in L3 . This proves (8.19). For k > 3/2, we estimate similarly
∇B1 (q, q + 2W ) 2 ≤ I0 ( q 2 q + 2W ∞ ) ≤ C t−λ0
(8.27)
by (3.10) and (8.11), since q + 2W is bounded in L∞ in that case. Furthermore, by (3.10), Lemma 3.2 and Sobolev inequalities
ω k+1 B1 (q, q + 2W ) 2 ≤ C Ik ω k q 2 ( q ∞ + W ∞ )
+ ∇q 2 ω k+1/2 W 2 ≤ C t−λ (8.28) by (8.11) (8.12). The last two inequalities imply (8.20) and (8.21) respectively. For k < 3/2, we must estimate B1 (q, q) and B1 (q, W ) separately because q is no longer controlled in L∞ . We estimate as before
∇B1 (q, W ) 2 ≤ I0 ( q 2 W ∞ ) ≤ C t−λ0 by (8.11), while
∇B1 (q, q) 2 ≤ I0 q 24 ≤ C t−2λ0 +(λ0 −λ)3/2k
by (8.11) (8.12), which together imply (8.22). We next estimate by (3.12) and (8.12)
ω 2k−1/2 B1 (q, q) 2 ≤ C I2k−3/2 ω k q 22 ≤ C t−2λ
(8.29)
while by (3.10) and Lemma 3.2
ω 2k−1/2 B1 (q, W ) 2 ≤ C I2k−3/2 ω 2k−3/2 q 2 W ∞
+ q r ω 2k−3/2 W 3/δ
≤ C I2k−3/2 ω 2k−3/2 q 2 W ∞ + ω 3/2 W 2
by Sobolev inequalities, with 1/2 < δ = δ(r) = 2k − 3/2 < 3/2, · · · ≤ C t−λ−(λ0 −λ)(3/2k−1) ≤ C t−λ−3/2+k
(8.30)
by interpolation between (8.11) and (8.12). Now (8.23) follows from (8.29) and (8.30).
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We conclude this section with some remarks on variations which can be made or attempted in the formulation of Proposition 8.1. Remark 8.1 We have stated the assumptions on (A+ , A˙ + ) in an implicit way in the form of conditions on the solution A0 of the free wave equation that they generate. Sufficient conditions for (8.9) and (8.10) to hold directly expressed in terms of (A+ , A˙ + ) and possibly w+ can be found in Lemma 3.5 and Lemma 5.2, but those conditions are far from being optimal (especially those of Lemma 5.2). Remark 8.2 The available regularity for A is stronger than stated, as follows from 2 the assumption (8.9) on A0 , from the simple estimate on A∞ 1 (|DW | ) 2 −m+1/2
ω m A∞ 1 (|DW | ) 2 ≤ C(a+ ) t
for 0 ≤ m ≤ k+ , and from the remainder estimates (8.19)–(8.23). Remark 8.3 The asymptotic behaviour in time of the scalar field A differs in an important way from that of a solution of the free wave equation. In fact A behaves asymptotically in time as 2 A ∼ A0 + A∞ 1 (|DW | ) .
Replacing W by w+ as a first approximation in the last term, one obtains 2 −1 D0 B1∞ (w+ , w+ ) A∞ 1 (|Dw+ | ) = t
with B1∞ (w+ , w+ ) constant in time. This yields a contribution to A which spreads by dilation by t and decays as t−1 in L∞ norm. That contribution can in no obvious sense be regarded as small as compared with A0 . Remark 8.4 One might be tempted to look for simpler asymptotic forms for u and for A by replacing for instance W by w+ in (8.15) (8.16) (8.18) and/or by omitting a few factors U (∗) (1/t) in (7.5) (7.7). This however would introduce errors at least O(t−1 ) and spoil the t−λ0 decay in (8.11) (8.15) (8.19) (8.20) (8.22). Acknowledgements. We are grateful to Professor Yves Meyer for enlightening conversations.
Appendix A In this appendix, we prove Warnings 4.1 and 4.2. Proof of Warning 4.1 One sees easily that (4.57) with y(1) = y0 > 0 has a unique maximal increasing solution y ∈ C 1 ([1, T ∗ ), R+ ) for some T ∗ > 1. We shall argue by contradiction by showing that if T ∗ is sufficiently large, then y(t) is infinite for some t < T ∗ . By integration, (4.57) with y(1) = y0 is converted into the integral equation 1 y(t) = y0 exp (m − β1 )−1 t−β1 . (A.1) dν y(νt) ν −1−m − ν −1−β1 1/t
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We shall prove by induction that (A.1) implies a sequence of lower bounds y(t) ≥ an tαn with αn rapidly growing and an not too small. We start with a0 = y0 , α0 = 0. Substituting that lower bound into (A.1) yields y ≥ y0 exp{y0 h(t)} where −1
h(t) = (m − β1 ) −1
= (m − β1 )
t
−β1
1 1/t
dν ν −1−m − ν −1−β1
−1 m−β1 − β1−1 + m−1 β1−1 t−β1 m t
so that y ≥ a1 tα1 provided
:n a1 /y0 ≤ (m − β1 )−1 −α1 :n τ + y0 (m−1 τ − β1−1 )
(A.2)
where τ = tm−β1 ≥ 1. The minimum of the RHS is attained for τ = mα1 /y0 , which we take > 1, and we can then take (A.3) a1 = y0 exp (m − β1 )−1 β1−1 y0 (e y0 /m α1 )α1 /(m−β1 ) . Here α1 is an arbitrary fixed parameter, which we take large. In particular we impose α1 > (m−1 y0 ∨ 2β1 ). At the following steps of the iteration, it will be sufficient to replace (A.1) by the lower bound obtained by letting m decrease to β1 , namely 1 dν y(νt) ν −1−β1 |:n ν| (A.4) y(t) ≥ y0 exp t−β1 1/t
or equivalently y(t) ≥ y0 exp
t
dt y(t ) t−1−β1 :n(t/t ) .
(A.5)
1
We now describe the determination of (an+1 , αn+1 ) = (a , α ) from (an , αn ) = (a, α). Substituting the induction assumption into (A.5), we obtain for α > β1 (a condition that will be ensured below) t y ≥ y0 exp a dt t−1−β1 +α :n(t/t ) 1 = y0 exp a(α − β1 )−2 (τ − 1 − :n τ )
where τ = tα−β1 > 1. This implies y ≥ a tα provided :n a /y0 ≤ a(α − β1 )−2 (τ − 1 − :n τ ) − α (α − β1 )−1 :n τ = θ ( a(τ − 1) − ( a + 1):n τ )
(A.6)
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a = a/α (α − β1 ) .
(A.7)
The minimum over τ of the last member of (A.6) is attained for aτ = a + 1, and it suffices to impose :n(a /y0 ) ≤ θ(1 − ( a + 1):n( a + 1)/ a) ≡ θ(:n a − f ( a)) which allows us to take a = y0 aθ provided f ( a) ≡ ( a + 1):n( a + 1) − a :n a−1≤0 a condition which is easily seen to hold for a ≤ 1/2. Finally we can take θ α = θ(α − β1 ) , a = y0 a/θ(α − β1 )2 provided
a ≤ θ(α − β1 )2 /2 .
(A.8) (A.9)
So far θ is a free parameter. For definiteness we choose θ = 2, so that after coming back to the original notation, (A.8) (A.9) become αn+1 = 2(αn − β1 ) , an+1 =
y0 a2n /4(αn
(A.10) 4
− β1 ) , 2
an ≤ (αn − β1 ) .
(A.11) (A.12)
(A.10) is readily solved by αn = 2β1 + 2n−1 (α1 − 2β1 ) . (A.12) is harmless and holds for all n if it holds for n = 1 and if y0 ≤ 4(α1 − β1 )2 , which can be arranged by taking α1 sufficiently large. (A.11) can be rewritten as 4 2 an y 0 an+1 y0 2(αn − β1 ) y02 a2n = 2 ≥ (A.13) 64(αn+1 − β1 )4 64 (αn − β1 )8 αn+1 − β1 64(αn − β1 )4 by (A.10). Let now t ≥ 1 and define un = an tαn −2β1 y0 /64(αn − β1 )4 . It follows from (A.10) (A.13) that un+1 ≥ u2n and in particular un ≥ 1 for all n if u1 ≥ 1, namely if t is sufficiently large in the sense that tα1 −2β1 ≥ (a1 y0 )−1 64(α1 − β1 )4 .
(A.14)
For such t, the condition un ≥ 1 can be rewritten as y(t) ≥ an tαn ≥ t2β1 y0−1 64(αn − β1 )4 ≥ 4t2β1 y0−1 24n (α1 − 2β1 )4 .
(A.15)
Since the last member of (A.15) tends to infinity with n, such a t cannot be smaller than T ∗ , which proves finite time blow up.
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Remark A1. Since the RHS of (4.56) and (4.57) is decreasing in β1 and increasing in m, blow up in finite time for (β1 , m) implies blow up in finite time for (β1 , m) with β1 ≤ β1 and for (β1 , m ) with m ≥ m, while the opposite situation prevails as regards the existence of global solutions. Actually it is easy to see that (4.56) or (4.57) admits global solutions for small data if β1 > 0 and m ≤ β1 . When coming back to the original equation (4.55), the condition of small data becomes a condition of large t0 . Proof of Warning 4.2. We want to prove finite time blow up for (4.61) with y(t0 ) = y0 > 0. Omitting the second term in the RHS and integrating the remaining inequality, we obtain
k 1/k y ≥ y0 + t − t0 ≥ (t − t0 )k . (A.16) We next keep (A.16), omit the first term in the RHS of (4.61) and change t to t + t0 . It is then sufficient to prove blow up for y ≥ tk (A.17) ∂t y ≥ (t + t0 )−1−β1 y 3 . For that purpose, we show by induction that y satisfies y(t) ≥ yn (t) ≥ an tαn (t + t0 )−(1+β1 )γn ,
(A.18)
starting with a0 = 1, α0 = k and γ0 = 0 given by (A.17). We obtain t yn+1 = dt (t0 + t )−1−β1 yn3 (t ) 0 t 3 dt (t0 + t )−(1+β1 )(3γn +1) t3αn ≥ an 0
≥ a3n (t0 + t)−(1+β1 )(3γn +1) t3αn +1 (3αn + 1)−1 thereby ensuring (A.18) at the level n + 1 if we choose αn+1 = 3αn + 1
,
γn+1 = 3γn + 1 ,
an+1 = a3n /(3αn + 1) .
(A.19) (A.20)
(A.19) is readily solved by αn = 3n (k + 1/2) − 1/2 so that or equivalently
,
γn = (3n − 1)/2
(A.21)
an+1 ≥ a3n (k + 1/2)−1 3−(n+1)
(A.22)
bn+1 ≥ b3n
(A.23)
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where bn = an 3−n/2−3/4 (k + 1/2)−1/2 .
(A.24)
un = bn tαn +1/2 (t0 + t)−(1+β1 )(γn +1/2) .
(A.25)
Let now t > 0 and
It follows from (A.19) (A.23) that un+1 ≥ u3n and in particular that un ≥ 1 for all n if u0 ≥ 1. The condition u0 ≥ 1 reduces to t2k+1 (t0 + t)−(1+β1 ) ≥ 33/2 (k + 1/2)
(A.26)
and holds for t sufficiently large if 2k > β1 . For such a t, by (A.18) y ≥ an tαn (t0 + t)−(1+β1 )γn ≥ (k + 1/2)1/2 3n/2+3/4 t−1/2 (t0 + t)(1+β1 )/2 . (A.27) Since the last member of (A.27) tends to infinity with n, such a t cannot be smaller than the maximal time T ∗ of existence of the solution y of (4.61), which proves finite time blow up.
References [1] A. Bachelot, Probl`eme de Cauchy pour des syst`emes hyperboliques semilin´eaires, Ann. IHP (Anal. non lin.), 1, 453–478 (1984). [2] J. B. Baillon and J. M. Chadam, The Cauchy problem for the coupled Schr¨ odinger-Klein-Gordon equations, in “Contemporary Developments in Continuum Mechanics and Partial Differential Equations”, G. M. de La Penha and L. A. Medeiros eds, North-Holland, Amsterdam, 1978. [3] J. Derezinski and C. G´erard, Scattering theory of classical and quantum N particle systems, Springer, Berlin, 1997. [4] I. Fukuda and M. Tsutsumi, On coupled Klein-Gordon-Schr¨ odinger equations II, J. Math. Anal. Appl., 66, 358–378 (1978). [5] J. Ginibre and T. Ozawa, Long range scattering for nonlinear Schr¨ odinger and Hartree equations in space dimension n ≥ 2, Commun. Math. Phys., 151, 619–645 (1993). [6] J. Ginibre and G. Velo, Long range scattering and modified wave operators for some Hartree type equations I, Rev. Math. Phys., 12, 361–429 (2000). [7] J. Ginibre and G. Velo, Long range scattering and modified wave operators for some Hartree type equations II, Ann. H.P., 1, 753–800 (2000).
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[8] J. Ginibre and G. Velo, Long range scattering and modified wave operators for some Hartree type equations III, Gevrey spaces and low dimensions, J. Diff. Eq., 175, 415–501 (2001). [9] N. Hayashi, E. I. Kaikina and P. I. Naumkin, On the scattering theory for the cubic nonlinear Schr¨ odinger and Hartree type equations in one space dimension, Hokka¨ıdo Math. J., 27, 651–667 (1998). [10] N. Hayashi and P. I. Naumkin, Asymptotics for large time of solutions to the nonlinear Schr¨ odinger and Hartree equations, Am. J. Math., 120, 369–389 (1998). [11] N. Hayashi and P. I. Naumkin, Scattering theory and large time asymptotics of solutions to Hartree type equations with a long range potential, preprint, 1997. [12] N. Hayashi and P. I. Naumkin, Remarks on scattering theory and large time asymptotics of solutions to Hartree type equations with a long range potential, SUT J. of Math., 34, 13–24 (1998). [13] N. Hayashi, P. I. Naumkin and T. Ozawa, Scattering theory for the Hartree equation, SIAM J. Math. Anal., 29, 1256–1267 (1998). [14] N. Hayashi and T. Ozawa, Modified wave operators for the derivative nonlinear Schr¨ odinger equation, Math. Ann., 298, 557–576 (1994). [15] N. Hayashi and W. von Wahl, On the global strong solutions of coupled KleinGordon-Schr¨ odinger equations, J. Math. Soc. Japan, 39, 489–497 (1987). [16] L. H¨ormander, The Analysis of Linear Partial Differential Operators, Vol I, Springer, Berlin, 1983. [17] T. Kato and G. Ponce, Commutator estimates and the Euler and NavierStokes equations, Comm. Pure Appl. Math., 41, 891–907 (1988). [18] C. Kenig, G. Ponce and L. Vega, The initial value problem for a class of nonlinear dispersive equations, in Functional-Analytic Methods for Partial Differential Equations, Lect. Notes Math., 1450, 141–156 (1990). [19] T. Ozawa, Long range scattering for nonlinear Schr¨ odinger equations in one space dimension, Commun. Math. Phys., 139, 479–493 (1991). [20] T. Ozawa and Y. Tsutsumi, Asymptotic behaviour of solutions for the coupled Klein-Gordon-Schr¨ odinger equations, in Spectral and Scattering Theory and Applications, Adv. Stud. in Pure Math., Jap. Math. Soc., 23, 295–305 (1994). [21] W. Strauss, Nonlinear Wave Equations, CMBS Lecture notes 73, Am. Math. Soc., Providence, 1989.
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[22] Y. Tsutsumi, Global existence and asymptotic behaviour of solutions for the Maxwell-Schr¨ odinger system in three space dimensions, Commun. Math. Phys., 151, 543–576 (1993). Jean Ginibre Laboratoire de Physique Th´eorique∗ Universit´e de Paris XI Bˆatiment 210 F-91405 Orsay Cedex France email: [email protected] Giorgio Velo Dipartimento di Fisica Universit` a di Bologna and INFN Sezione di Bologna Italy email: [email protected] Communicated by Rafael D. Benguria submitted 13/07/01, accepted 27/03/02
To access this journal online: http://www.birkhauser.ch
∗ Unit´ e
Mixte de Recherche (CNRS) UMR 8627
Ann. Henri Poincar´e 4 (2002) 739 – 756 c Birkh¨ auser Verlag, Basel, 2002 1424-0637/02/040739-18
Annales Henri Poincar´ e
Resonance Free Domains for Non Globally Analytic Potentials A. Martinez∗
Abstract. We study the resonances of the semiclassical Schr¨ odinger operator P = −h2 ∆ + V near a non-trapping energy level λ0 in the case when the potential V is not necessarily analytic on all of Rn but only outside some compact set. Then we prove that for some δ > 0 and for any C > 0, P admits no resonance in the domain 1 Ω = [λ0 −δ, λ0 +δ]−i[0, Ch log (h−1 )] if V is C ∞ , and Ω = [λ0 −δ, λ0 +δ]−i[0, δh1− s ] if V is Gevrey with index s. Here δ > 0 does not depend on h and the results are uniform with respect to h > 0 small enough.
1 Introduction The location of quantum resonances has been a very rich source of researches during these last years, giving rise to a wide literature, including estimates on the number of resonances in various domains of the complex plane and relationship between resonances and the underlying classical geometry: see e.g. [GeSj, HeSj, SjZw] and references therein. Concerning more specifically the resonances of the semiclassical Schr¨ odinger operator, one of the first results about their location is probably the one which asserts the absence of resonance in some fixed complex neighborhood of any non trapping energy level: see e.g. [BCD, GeSj]. However such a result has been proved under the assumption that the potential is analytic everywhere on Rn , while the theory of resonances (and the notion of non trapping level as well) exists for potentials which may be analytic outside a compact set only. As far as we know, the only similar results concerning such potentials are those very recent of [Ro1] in the Gevrey case, and [TaZw] for compactly supported potentials (in the latter case a simpler proof can be found in [Bu] but leading to a less optimal result). Moreover, a semiclassical theory of resonances also exists for potentials which are sum of a globally analytic function and an exponentially decaying smooth function: see [Na1]. In this case the absence of resonance holds in a neighborhood of size O(h) (h = semiclassical parameter) of any non trapping level (see [Na1] Theorem 5.1). Here we plan to provide a rather simple proof which generalizes the result of [TaZw] to long range smooth potentials. Incidentally, our method also gives an alternative (and simpler) proof of the result of [Ro1] (which has been shown to be ∗ Investigation
supported by University of Bologna. Funds for selected research topics.
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optimal in [Ro2]). The idea is to use an almost analytic extension of the potential (as defined in [MeSj]) and to prove for it an equivalent of the exponential microlocal weighted estimates introduced in [BeMa]. Because of the non-analyticity, some extra terms will appear involving the ∂ of the potential, but they are rather well controlled if we stay sufficiently close to the real. These estimates together with a construction of a ‘global’ escape function permit to obtain the results. In the next section we give the precise statement of our main results. Then in Section 3 we introduce the a priori estimate that we use in Sections 4 and 5 to prove the main results. Finally, Section 6 contains the proof of the a priori estimate.
2 Notations and Results We consider the n-dimensional semiclassical Schr¨ odinger operator P = P (h) = −h2 ∆ + V (x) where h > 0 is small enough and the potential V satisfies the following hypothesis: (H1) V ∈ C ∞ (Rn ) is real valued and there exists some compact set K ⊂ Rn such that V is analytic on K C := Rn \K and can be extended as a holomorphic function on D := {x ∈ Cn ; |Imx| < δ0 Rex , Rex ∈ K C } for some constant δ0 > 0. Moreover, V (x) → 0 as |Re x| → ∞ , x ∈ D. Then the continuous spectrum of P is R+ , and one can define the resonances of P near the positive real axis as the complex eigenvalues of the so-called distorded operator Pθ (θ > 0 small enough) defined in the following way (see e.g. [Hu]): Let F : Rn → Rn be a smooth vector-field such that F (x) = 0 on the compact K given in (H1) and F (x) = x for |x| large enough, and for ν ∈ R small enough consider the unitary operator Uν on L2 (Rn ) defined by: Uν ϕ(x) = det(1 + νdF (x))−1/2 ϕ(x + νF (x)). Then we see by (H1) that the coefficients of the differential operator Pν := Uν P Uν−1 depend analytically on ν near 0, and can therefore be extended in a unique way to small complex values of ν. In particular for ν = iθ (θ > 0 small enough) we get a differential operator Pθ := Piθ and by the Weyl perturbation theorem we see that the essential spectrum of Pθ is given by: σess (Pθ ) = e−2iθ R+ .
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In particular the spectrum σ(Pθ ) of Pθ is discrete in the sector Sθ := {λ ∈ C ; Reλ > 0 , −2θ < Arg λ ≤ 0} and standard arguments (see e.g. [AgCo, HeMa]) show that σ(Pθ ) ∩ Sθ does not depend on the particular choice of the vector-field F and for any 0 < θ < θ small enough one has (for any h > 0): σ(Pθ ) ∩ Sθ = σ(Pθ ) ∩ Sθ . Then the resonances of P in Sθ are by definition the eigenvalues of Pθ there, or equivalently the eigenvalues of Pθ there, for any θ > θ small enough. Now we fix some energy level λ0 > 0 and we assume that it is non-trapping for the underlying classical dynamic, that is: (H2) For all (x, ξ) ∈ R2n such that ξ 2 + V (x) = λ0 , one has | exp tHp (x, ξ)| → ∞ as |t| → ∞. Here Hp := 2ξ∂ξ − ∂x V (x)∂x denotes the Hamilton field of p := ξ 2 + V (x). Then we have: Theorem 2.1 Under assumptions (H1) and (H2), there exists δ > 0 such that for any C > 0, P (h) admits no resonance in the complex domain Ω(h) := [λ0 − δ, λ0 + δ] − i[0, Chlog (h−1 )] for all h > 0 small enough. Now let us assume more regularity on V in a neighborhood of K, namely that it is a Gevrey function of index s ≥ 1 in the sense that: (H3) There exists C1 > 0 such that for all α ∈ Nn the following estimate: 1+|α|
|∂ α V (x)| ≤ C1
(α!)s
holds uniformly in a neighborhood of K. In this case we recover in a simpler way the result of [Ro1], namely: Theorem 2.2 Under assumptions (H1), (H2) and (H3), there exists δ > 0 such that P (h) admits no resonance in the complex domain Ωs (h) := [λ0 − δ, λ0 + δ] − 1 i[0, δh1− s ] for all h > 0 small enough. Observe that when s = 1 one recovers a well-known result for analytic potentials.
3 Microlocal Exponential Weighted Estimates for Operators with Non Analytic Symbol The purpose of this section is to generalize a result of [BeMa] to the case of operators with non analytic symbol. In the next sections these estimates, together with the construction of a global escape function, will permit to define a h-dependent norm on L2 (Rn ) for which Pθ (θ = θ(h) convenient) will satisfy a global ellipticitytype energy inequality.
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For the sake of simplicity, and since we do not need more in this paper, here we consider differential operators only, with smooth bounded coefficients. More precisely we consider the operator A defined by the oscillatory integral: x+y 1 i(x−y)ξ/h , ξ; h u(y)dydξ Au(x) = OpW (a)u(x) := a e h (2πh)n 2 (u ∈ S (Rn )) where a is a polynomial of degree d with respect to ξ, whose coefficients are smooth functions of x on Rn , uniformly bounded together with all their derivatives. Let G ∈ C0∞ (R2n ) be a compactly supported smooth real-valued function on 2n R . We set H(x, ξ) = G(x, ξ) − ξ · ∂ξ G(x, ξ) and, for t ∈ R small enough: ΛtG = {(x + it∂ξ G(x, ξ), ξ − it∂x G(x, ξ)) ; (x, ξ) ∈ R2n } which is a I-Lagrangian submanifold of C2n (that is Lagrangian with respect to Im(dζ ∧ dz)) and R-symplectic if t is small enough (also observe that tdH(x, ξ) = −Imζdz |ΛtG ). In particular one can define L2 (ΛtG ) e.g. by using the measure dRe (z, ζ) which is equivalent to (Re (dζ ∧ dz))∧n on ΛtG . We also introduce the F.B.I. transform T defined by: 2 T u(x, ξ) = c ei(x−y)ξ/h−(x−y) /2h u(y)dy n
2
3n
where c = 2− 2 (πh)− 4 . Then T maps S (Rn ) into S (R2n ) ∩ e−ξ /2h H(Cnx−iξ ) where H(Cnx−iξ ) denotes the space of entire function with respect to x − iξ on 2
Cn , and it is also an isometry from L2 (Rn ) to L2 (R2n ) ∩ e−ξ /2h H(Cnx−iξ ) (see e.g.[Ma1]). In particular, T u is an entire function of (x, ξ), and since G is compactly supported, it is not difficult to see that L2 (ΛtG )∩RanT = L2 (R2n )∩Ran T = T (L2 (Rn )) (the norms are equivalent with h-dependent constants that are exponentially large in general). In order to state the main result of this section, we also need to recall the notion of almost analytic extension introduced by A. Melin and J. Sj¨ ostrand in [MeSj] and subsequently widely used in the literature. Let f ∈ Cb∞ (Rn ), the space of smooth functions on Rn which are uniformly bounded together with all their derivatives. For (x, y) ∈ R2n , one considers the formal series: F (x, y) =
(iy)α ∂ α f (x). α! n
α∈N
Then, considering y as a small vector-parameter, this series can be resumed up to O(|y|∞ ) in a standard way (see e.g. [Ho, Ma1]), and gives rise to a smooth function
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f(x, y) on R2n of the type: f(x, y) =
(iy)α ∂ α f (x) 1 − χ(ε|α| /|y|) α! n
(3.1)
α∈N
where χ ∈ C0∞ (R) is a fixed cutoff function that is equal to 1 near 0, and (εk )k∈N is a decreasing sequence of positive numbers converging to 0 sufficiently rapidly (depending only on the L∞ -norms of ∂ α f , α ∈ Nn ). Then, f satisfies: f(x, 0) = f (x) and
∂ ∂ +i ∂x ∂y
f(x, y) = O(|y|∞ )
(3.2)
(3.3)
as |y| → 0, uniformly with respect to x ∈ Rn . Because of the latter property, f is called an almost analytic extension of f . Moreover, in the case when f is globally 1 with C > 0 a large enough Gevrey with index s > 1 one can take εk = Ck s−1 constant, and then (3.3) is improved into: 1/(s−1) ∂ ∂ (3.4) +i f(x, y) = O e−δ/|y| ∂x ∂y for some positive constant δ (depending on C, which in turn depends on the L∞ norms of ∂ α f , α ∈ Nn ). The previous discussion permits in particular to consider an almost analytic extension of the coefficients of a, giving rise to a function a= a(x, ξ) polynomial with respect to ξ with coefficients which are smooth functions of x on a complex strip around Rn of the type Γε := {x ∈ Cn ; |Im x| < ε} (ε > 0 arbitrary), and satisfying: a |R2n = a ∂ a = O(|Im x|∞ )ξ d ∂x uniformly on Γε × Rn . In particular, since we have taken a function G compactly supported, a is well defined on ΛtG . Then, the main technical tool for proving our theorems is the following estimate: Proposition 3.1 Let m ∈ C ∞ (ΛtG ) such that ∂ γ m = O(ξ ) for some 1 ≥ 0 and for all γ ∈ N2n . Then there exists an operator R(t, h) : C0∞ (R2n ) → C ∞ (R2n ) that can be extended to a continuous operator : ξ −(d++σ) L2 (R2n ) → ξ −σ L2 (R2n )
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for any σ ∈ R, and such that for all u, v ∈ C0∞ (Rn ) one has (denoting H(α) := H(Reα) when α ∈ ΛtG ), me−tH/h T Au, e−tH/h T v L2 (ΛtG ) = (m a + R(t, h))e−tH/h T u, e−tH/h T v L2 (ΛtG ) with
σ
ξ R(t, h)ξ −(d++σ)
L(L2 (ΛtG ))
= O h + h−3n/2 |t|∞ eJ0 |t|/h
for any σ ∈ R and uniformly with respect to t and h both small enough. Here J0 := 2 Sup(|G| + |∇G|2 ). R2n
Moreover, if the coefficients of A are globally Gevrey with index s > 1, the estimate on R(t, h) can be improved into:
1/(s−1)
σ = O h + h−3n/2 e−ε/t eJ0 |t|/h
ξ R(t, h)ξ −(d++σ) 2 L(L (ΛtG ))
for some positive constant ε. Proof. Essentially, the proof proceeds as in [BeMa] Section 3, with the difference that the various changes of contour give rise, by Stokes formula, to extra terms involving the ∂ of the integrands. However, since it is rather technical, we postpone the details of the proof to Section 7. As a corollary, we also obtain the following result: Corollary 3.2 For any u ∈ C0∞ (Rn ) one has: e−tH/h T Au2L2(ΛtG )
=
ae−tH/h T u2L2(ΛtG ) +O h + h−3n |t|∞ e2J0 |t|/h ξ d e−tH/h T u2L2(ΛtG )
uniformly with respect to t and h both small enough. Moreover, if the coefficients of A are globally Gevrey with index s > 1, the previous estimate can be improved into: e−tH/h T Au2L2 (ΛtG ) = ae−tH/h T u2L2(ΛtG ) 1/(s−1) +O h + h−3n e−ε/t e2J0 |t|/h ξ d e−tH/h T u2L2 (ΛtG ) for some positive constant ε.
4 Proof of Theorem 2.1 We start by constructing a kind of ‘global escape function’ in the same spirit as in [GeMa]. First we fix δ > 0 and R > 1 such that for |x| ≥ R and (x, ξ) ∈ p−1 ([λ0 − δ, λ0 + δ]) one has: Hp [x · ξ] = 2ξ 2 − x∇x V (x) ≥ λ0 .
(4.1)
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We can also assume that R is sufficiently large so that K ⊂ {|x| < R − 1}. Then one can construct as in [GeMa] a real-valued function f ∈ Cb∞ (R2n ) such that Hp f ≥ 0 everywhere, Hp f = 1 for |x| ≤ R and ξ 2 + V (x) close to λ0 , and Hp f = 0 for |x| ≥ R + 1. We also specify our choice of the vector-field F by taking it as: F (x) = χ1 (x)x where χ1 ∈ C ∞ (Rn ), 0 ≤ χ1 ≤ 1, χ1 (x) = 1 for |x| ≥ R. If we denote by pθ the Weyl-symbol of the distorded Hamiltonian Pθ defined in Section 2, we see that it is a second order polynomial with respect to ξ with coefficients in Cb∞ (Rnx ) and thus we can consider an almost analytic extension pθ of pθ as defined in Section 3. Then we set G(x, ξ) := χ2 (x)χ3 (ξ)f (x, ξ) (4.2) where χ2 , χ3 ∈ C0∞ (Rn ; [0, 1]), χ2 (x) = 1 for |x| ≤ R + 1, χ3 (ξ) = 1 for ξ 2 ≤ λ0 + Sup |V | + δ. In particular we have χ2 (x)Hp f (x, ξ) = Hp f (x, ξ). Moreover χ2 can be chosen arbitrarily flat and we denote: µ := Sup |f Hp χ2 |
(4.3)
which will be fixed small enough later. The following lemma will allow us to obtain an ellipticity estimate of pθ on ΛtG for a suitable choice of t, θ: Lemma 4.1 For any (x, ξ) ∈ R2n one has: pθ (x+it∂ξ G, ξ−it∂x G) = p(x, ξ)−iHp (θF (x)ξ+tG)+O (t2 + θ2 + h)ξ 2 (4.4) uniformly with respect to h, t and θ (all of them small enough). Moreover, for (x, ξ) ∈ p−1 ([λ0 − δ, λ0 + δ]) one has: Hp (θF (x)ξ + tG) ≥ θλ0 χ1 (x) + (t − C2 θ)Hp f (x, ξ) − µt where C2 :=
Sup p−1 ([λ0 −δ,λ0 +δ])
(4.5)
|(x · ξ)Hp χ1 | (< ∞).
Proof. By construction we have: pθ (x, ξ) = p0θ (x, ξ) + O(hξ 2 ) with
p0θ (x, ξ) = p x + iθF (x), (1 + iθdF (x))−1 ξ .
Considering also an almost analytic extension p of p (together with an almost analytic extension of F ), we obtain by a Taylor expansion: pθ (x + it∂ξ G, ξ − it∂x G)
= p(x + iθF (x) + it∂ξ G, ξ − iθdF (x)ξ − it∂x G) +O (tθ + θ2 + h)ξ 2
and formula (4.4) follows easily from the definition of p and by observing that (θF (x) + t∂ξ G, θdF (x)ξ + t∂x G) = (∂ξ G1 , ∂x G1 ) with G1 := θF (x)ξ + tG.
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Moreover, on p−1 ([λ0 − δ, λ0 + δ]) we have: Hp (θF (x)ξ + tG) = θχ1 Hp (x · ξ) + tχ2 Hp f + θx · ξHp χ1 + tf Hp χ2 and since Hp χ1 is supported in {χ2 Hp f = 1}, the estimate (4.5) follows by using (4.1)-(4.3). Now we take t = 2C2 θ and we choose χ2 in such a way that µ≤
1 Min{C2 , λ0 }. 4C2
In this case, since χ1 + Hp f ≥ 1 everywhere, the estimate (4.5) gives: Hp (θF (x)ξ + tG) ≥
θ Min{C2 , λ0 } 2
on p−1 ([λ0 − δ, λ0 + δ]). As a consequence, (4.4) allows to deduce the existence of a positive constant C3 such that if θ and h are small enough and if ρ ∈ [λ0 − δ , λ0 + δ ] − i[0, δ θ] with δ > 0 small enough, then, θ + C3 h for all C3 ξ 2 (x, ξ) such that |Re ( pθ (x + it∂ξ G, ξ − it∂x G) − ρ) | ≤ . C3 Im ( pθ (x + it∂ξ G, ξ − it∂x G) − ρ) ≤ −
(4.6)
2
pθ − ρ) | ≤ ξ (Note that |ξ| remains bounded on Σ := ΛtG ∩ {|Re ( C3 } if C3 > 1). Then, we apply Proposition 3.1 with A = Pθ − ρ, t = 2C2 θ and θ = Chlog (h−1 ) where C >> 1 is arbitrary. Since for any N >> 1 one has h−3n |t|2N eJ0 |t|/h = O(hN −CJ0 −3n ) = O(h), Proposition 3.1 gives in this case, = Im( pθ − ρ)e−tH/h T u, e−tH/h T u Im e−tH/h T (Pθ − ρ)u, e−tH/h T u ΛtG
ΛtG
+ O(h)ξ 2 e−tH/h T uΛtG e−tH/h T uΛtG and thus, we obtain from (4.6) (since also h << θ),
Im e−tH/h T (Pθ − ρ)u, e−tH/h T u
≥
ΛtG
θ −tH/h e T u2Σ C3
−C4 ξ 2 e−tH/h T u2ΣC
(4.7)
for some positive constant C4 and for any u ∈ C0∞ . On the other hand, by Corollary 3.2 we also have, e−tH/h T (Pθ − ρ)u2ΛtG ≥
1 ξ 2 e−tH/h T u2ΣC − C5 he−tH/h T u2Σ C5
(4.8)
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for some other positive constant C5 . We deduce from (4.7)-(4.8) the existence of a constant C6 > 0 such that, C6 e−tH/h T u2Σ ≤ e−tH/h T (Pθ − ρ)uΛtG · e−tH/h T uΛtG −1 hlog (h ) (4.9) +ξ 2 e−tH/h T u2ΣC ξ 2 e−tH/h T u2ΣC ≤ C6 e−tH/h T (Pθ − ρ)u2ΛtG (4.10) +he−tH/h T u2Σ and thus, inserting (4.10) into (4.9) and taking h small enough, C7 e−tH/h T u2Σ ≤ e−tH/h T (Pθ − ρ)uΛtG · e−tH/h T uΛtG hlog (h−1 ) (4.11) +e−tH/h T (Pθ − ρ)u2ΛtG which, inserted again into (4.10) and summed up, gives, ξ 2 e−tH/h T u2ΛtG
≤
C8 e−tH/h T (Pθ − ρ)uΛtG · e−tH/h T uΛtG hlog (h−1 ) +e−tH/hT (Pθ − ρ)u2ΛtG , (4.12)
where C7 , C8 are positive constants. We finally obtain from (4.12), ξ 2 e−tH/h T uΛtG ≤
C9 e−tH/h T (Pθ − ρ)uΛtG hlog (h−1 )
(4.13)
for any u ∈ C0∞ (Rn ), with some other constant C9 > 0. By density this last estimate extends for all u ∈ H 2 (Rn ) and proves that the equation (Pθ − ρ)u = 0 does not admit a non trivial solution in L2 (Rn ). The proof of Theorem 2.1 is complete.
5 Proof of Theorem 2.2 We assume s > 1 and we refer to [BeMa] for a similar (but simpler) proof in the case s = 1. The arguments of the previous section are unchanged until (4.6), then 1 the difference consists in the choice of θ only: here we take θ = δh1− s and we 1 observe that if δ is chosen small enough, then for t = 2C2 θ = 2C2 δh1− s we have: 1/(s−1) 1/s (5.1) eJ0 |t|/h = O e−ε /(2h ) h−3n e−ε/t for some ε > 0 related to ε, δ and C2 . Moreover, the cut-off function χ1 used in the definition of F (x) can be taken Gevrey of index s, and this permits to apply
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the Gevrey version of Proposition 3.1 and Corollary 3.2 with A = Pθ − ρ. The same procedure as in the previous section then leads to ξ 2 e−tH/h T uL2(ΛtG ) ≤
C10 e−tH/h T (Pθ 1−1/s h
− ρ)uL2 (ΛtG )
(C10 > 0 constant) and we conclude again that the equation (Pθ − ρ)u = 0 admits 1 no non-trivial solution in L2 (Rn ) when θ = δh1− s and ρ ∈ [λ0 −δ , λ0 +δ ]−i[0, δ θ] (δ, δ > 0 small enough). This completes the proof of Theorem 2.2.
6 Proof of Proposition 3.1 and Corollary 3.2 Denote
I := me−tH/h T Au, e−tH/h T v L2 (ΛtG ) .
Using the definitions of A and T we obtain: c2 y + x Φ/h I= , η u(x )v(y )d(x , η, y, y , Reα) e m(α)a (2πh)n 2
(6.1)
where the integral runs over {(α, y, y , x , η) ∈ ΛtG × R4n }, and where (denoting α = (αx , αξ ) ∈ Cn × Cn for α in ΛtG ): 1 1 Φ = −2tH(Reα)+i(αx −y)αξ −i(αx −y )αξ − (αx −y)2 − (αx −y )2 +i(y −x)η. 2 2 (6.2) Then we observe that by construction of a we have for all Y = (y, η) ∈ R2n and α = (αx , αξ ) ∈ ΛtG (and denoting Xs := sY + (1 − s)α, 0 ≤ s ≤ 1): 1 ∂ a ∂ a (Xs ) − Imα (Xs ) ds (Y − Re α) a(Y ) − a(α) = ∂Reα ∂Im α 0 1 ∂ a ∂ a (Xs ) + 2iImα (Xs ) ds = (Y − α) ∂Re α ∂α 0 = (α − Y ) · b(α, Y ) + r(α, Y ) (6.3) where b and r are C ∞ on Γε × Rn × R2n (polynomial with respect to αξ and η) and satisfy the following estimates for all γ ∈ N4n : ∂γ b = ∂γ r =
O((αξ , η) d ) O(µ(Im αx )(αξ , η) d )
uniformly on Γε × Rn × R2n . Here we have used the following notation: µ(t)
= |t|∞ in the C ∞ case;
µ(t)
= e−ε/|t|
1/(s−1)
(some ε > 0) in the Gevrey case,
and we shall continue to use it all along the proof.
(6.4)
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Inserting (6.3) into (6.1) we obtain: I = m ae−tH/h T u, e−tH/h T v L2 (ΛtG ) + R1 + R2 with y + x , αξ − η · b1 m(α)u(x )v(y )d(x , η, y, y , Reα) eΦ/h αx − 2 (6.5) c2 y + x Φ/h , η m(α)u(x )v(y )d(x , η, y, y , Reα) (6.6) R2 = e r α, (2πh)n 2 where we have denoted b1 = b α, y+x , η . Now the arguments of [BeMa] Section 2 3 show the existence of a smooth vector-field of the type: y + x ∂ ∂ ∂ L := L α, , η, , , 2 ∂Reα ∂y ∂η R1 =
c2 (2πh)n
with coefficients O((αξ , η) d ) together with all their derivatives (actually polynomial with respect to η), such that: y + x , αξ − η · b1 = hL(eΦ/h ). αx − 2 Inserting this inside (6.5), making integrations by parts and denoting: y + x , η = t L[m] f α, 2
(6.7)
(where t L denotes the transposed of L), we obtain R1 = he−tH/h Tf u, e−tH/h T v L2 (ΛtG )
with Tf u(α) := c
ei(αx −y)αξ /h−(αx −y)
2
/2h
OpW h (f (α, ·))u(y)dy.
With the same notation we also have: R2 = e−tH/h Tmr u, e−tH/h T v L2 (ΛtG ) . Then we introduce the formal adjoint of T as: 2 ∗ T w(x) = c e−i(x−y)ξ/h−(x−y) /2h w(x, ξ)dxdξ
(6.8)
where the (oscillatory) integral runs over R2n in the general case when w ∈ S (R2n ). In the case when w = T u with u ∈ C0∞ (Rn ), one can make a change
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of contour of integration in (6.8) so that the new contour becomes ΛtG , and using that u = T ∗ T u we see that R1 and R2 can be rewritten as: R1 = h αξ −σ Kf αξ d++σ e−tH/h T u, e−tH/hT v 2 L (ΛtG ) −σ d++σ −tH/h −tH/h R2 = αξ Kmr αξ e T u, e Tv 2 (6.9) L (ΛtG )
where we have used the following notation: Kg = αξ σ e−tH/h Tg T ∗ etH/h αξ −(d++σ) for g = f, mr. Now, in view of (6.9), it is clear that Proposition 3.1 is a consequence of the following lemma: Lemma 6.1 The operators Kf and Kmr are bounded on L2 (ΛtG ), and their norms satisfy: Kf L(L2 (ΛtG )) = O 1 + h−3n/2 µ(t)eJ0 |t|/h Kmr L(L2 (ΛtG )) = O h−3n/2 µ(t)eJ0 |t|/h + h∞ . Proof. The distribution kernel of Kg is given by the oscillatory integral: c2 αξ σ y + x Φ1 /h , η d(x , y, η) g α, e Kg (α, β) = (2πh)n βξ d++σ 2
(6.10)
with Φ1
=
1 1 t[H(Reβ) − H(Re α)] + Φ2 − (αx − y)2 − (βx − x )2 2 2
where Φ2 := iαx αξ − iβx βξ + i(η − αξ )y + i(βξ − η)x (α, β ∈ ΛtG ). At first we perform repeated integrations by parts in (6.10) by using the operator: −1 × L1 := 1 + |η − αξ |2 + |η − βξ |2 + |y − x |2 × (1 + (η − αξ ) · hDy + (βξ − η) · hDx + (y − x ) · hDη ) which satisfies : L1 (eΦ2 /h ) = eΦ2 /h . This permits to obtain for any N ≥ 1: c2 αξ σ Kg (α, β) = (6.11) eΦ1 /h gN (α, β, x , y, η)d(x , y, η) (2πh)n βξ d++σ with
y + x ,η gN (α, β, x , y, η) = ( L2 ) g α, 2
L2 = e−[(αx −y)
t
2
N
+(βx −x )2 ]/2h
L1 e[(αx −y)
2
+(βx −x )2 ]/2h
(6.12)
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In particular, gN satisfies to the estimates:
γ
(αx − y, βx − x ) N y + x
gN = O
∂ g α, 2 , η . (αξ − η, βξ − η, y − x ) N
751
(6.13)
|γ|≤N
We denote by gN an almost analytic extension of gN constructed from an almost analytic extension g of g via (6.12) (so that it satisfies (6.13) too), and, for C > 0 large enough (to be specified later on) we consider the following family of contours in C3n : (η − αξ )/η − αξ y y Cs : R3n x → x + iCs · (βξ − η)/βξ − η η η (y − x )/y − x for s = (s1 , ..., s3n ) ∈ [0, |t|]3n . Then, along Cs we have: |y − x |2 |η − αξ |2 |η − βξ |2 + + Re Φ1 ≤ |t|Ft (α, β) − (Min sj ) j y − x η − αξ η − βξ 1 − |Reαx − y|2 + |Reβx − x |2 4 with Ft (α, β) = |G(Re β) − G(Re α)| + 4|t| |∇G(Re α)|2 + 4|∇G(Reβ)|2 . (Here we have used estimates such as: |(x − Reβx )Im βξ | ≤ 18 |x − Reβx |2 + 2t2 |∇G(Re β)|2 .) Therefore, since ∂ gN = (t L2 )N ∂ g, we obtain from (6.11) by Stokes’ formula: |η−α |2 |η−β |2 (y−x )2 − C|t| + η−αξ + η−βξ h y−x ξ ξ Kg (α, β) = O(h−5n/2 e|t|Ft (α,β)/h ) e (6.14) C|t|
×e−[|αx −y|
2
+|βx −x |2 ]/4h
+O(h−5n/2 e|t|Ft (α,β)/h ) Sup Ωt
|(αξ , η) −(d+) ∂ γ ∂ g |
|γ|≤N
−[|αx −y|2 +|βx −x |2 ]/4h
e (αξ , η) d++σ L(d(x , y, η)) d++σ (α − η, β − η, y − x ) N ξ ξ Ωt βξ stands for C(|t|,...,|t|), Ωt := Cs and L(d(x , y, η)) denotes the ×
where C|t|
αξ σ gN d(x , y, η) βξ d++σ
s∈[0,|t|]3n 3n
Lebesgue measure in C . Let C > 0 be such that Supp G ⊂ {|(x, ξ)| ≤ C }. Then, as in [BeMa], we split ΛtG × ΛtG in 4 regions:
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1st region: |Reα| ≥ C and |Reβ| ≥ C In this region Ft (α, β) = 0 and thus, if we take N large enough (6.14) gives there (using also (6.13) and the fact that (αξ −η, βξ −η) −N αξ M βξ −M = O(1) if N >> M ): 2 |t| |αξ −βξ | 2 − −ν|αx −βx | /h Kg (α, β) = O(h−n )e h αξ −βξ Sup |(αξ , η) −(d+) ∂ γ g| 2
+O(h−3n/2 )
−ν|αx −βx | /h
Ωt
e Sup αξ − βξ n+1 Ωt
|γ|≤N
|(αξ , η) −(d+) ∂ γ ∂ g |
|γ|≤N
for some ν > 0 and any fixed N ≥ 1 large enough. In particular, if we use (6.4) and the properties of the almost analytic extensions: 2 −ν|αx −βx |2 /h |t| |αξ −βξ | 2 e − −ν|α −β | /h x x Kf (α, β) = O h−n e h αξ −βξ + h−3n/2 µ(t) αξ − βξ n+1 2 e−ν|αx −βx | /h Kmr (α, β) = O h−3n/2 µ(t) (6.15) αξ − βξ n+1 2nd region: |Reα| ≥ 2C and |Re β| ≤ C In this region we have |α − β| ≥ Therefore, denoting
C 1 1 |Re α| ≥ + |Re α|. 2 2 4
(6.16)
J0 := 2 Sup(|G| + |∇G|2 ) R2n
we see on (6.14) that if we have chosen C and N sufficiently large we have in this region: |(αξ , η) −(d+) ∂ γ g| Kg (α, β) = O(e−|tRe α|/h ) Sup C|t|
|γ|≤N
+O(h−3n/2 eJ0 |t|/h )Re α −(2n+1) Sup Ωt
|(αξ , η) −(d+) ∂ γ ∂g| .
|γ|≤N
In particular: Kf (α, β) Kmr (α, β)
= O(e−|tRe α|/h + h−3n/2 µ(t)eJ0 |t|/h Reα −(2n+1) ) = O(h−3n/2 µ(t)eJ0 |t|/h Re α −(2n+1) )
(6.17)
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3rd region: |Reα| ≤ C and |Re β| ≥ 2C We obtain in the same way: Kf (α, β)
=
O(e−|tRe β|/h + h−3n/2 µ(t)eJ0 |t|/h Reβ −(2n+1) )
Kmr (α, β)
=
O(h−3n/2 µ(t)eJ0 |t|/h Re β −(2n+1) )
(6.18)
4th region: |Reα| ≤ 2C and |Reβ| ≤ 2C In this last region we try to apply the stationary phase theorem to (6.10). The unique stationary point (given by ∂y Φ1 = ∂x Φ1 = ∂η Φ1 = 0) is: x y = x = αx +β − 2i (αξ − βξ ) 2 αξ +βξ i η = 2 + 2 (αx − βx ) In order to be able to apply the stationary phase theorem with complex-valued phase function (see [MeSj] p.145), we need that the critical point remains close enough to the real (it already remains bounded because so are α and β in this region). That is, we need that |α − β| is small enough. But if it is not the case (that is if |α − β| ≥ δ for some δ > 0), then (6.14) gives as for the 2nd and 3rd regions (possibly by increasing the value of the constant C again):
Kf (α, β)
=
O(e−C
|t|/h
+ h−3n/2 µ(t)eJ0 |t|/h )
Kmr (α, β)
=
O(h−3n/2 µ(t)eJ0 |t|/h )
(6.19)
for some arbitrary large constant C > 0. Otherwise, if |α − β| ≤ δ with δ small enough, then Theorem 2.3 of [MeSj] can be applied and gives for any N ≥ 1 (recalling that c2 ∼ h−3n/2 ): Kg (α, β) = h−nbN (α, β, h)eϕ(α,β)/h + O(hN )
(6.20)
where ϕ(α, β) is the critical value of Φ1 , that is: 1 1 i ϕ(α, β) = t(H(β) − H(α)) + (αx − βx )(αξ + βξ ) − (αx − βx )2 − (αξ − βξ )2 2 4 4 and bN is a bounded symbol which can be estimated by the derivatives of g at the critical point, up to order 2(N + n) (note that such an accuracy is not really needed for Proposition 3.1, but in any case we think that the result of Lemma opKg is interesting in itself). Moreover, using the definition of H we can see that 1 1 Reϕ(α, β) = − Re (αx − βx )2 − Re(αξ − βξ )2 + tψ(α, β) 4 4 with ψ(α, β)
=
=
1 G(β) − G(α) − Re(βx − αx )(∂x G(α) + ∂x G(β)) 2 1 − Re (βξ − αξ )(∂ξ G(α) + ∂ξ G(β)) 2 2 O(|α − β| ).
(6.21)
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As a consequence, if we take t small enough we deduce from (6.4), (6.20) and (6.21): 2 Kf (α, β) = O h−n e−ν|α−β| /h + hN 2 Kmr (α, β) = O h−n (µ(t) + µ(|α − β|)(e−ν|α−β| /h + hN (6.22) with ν > 0 constant and N ≥ 1 arbitrary. (Here we have also used that µ(t + s) = O(µ(t/2) + µ(s/2)) for s, t ≥ 0.) Now, gathering the estimates (6.15), (6.17), (6.18), (6.19) and (6.22), we obtain uniform bounds for both |Kg (α, β)|dRe α and |Kg (α, β)|dRe β ΛtG
ΛtG
(g = f, mr) which, by the Schur lemma, give the following estimates on the operators Kf Kmr : Kf L(L2 (ΛtG )) = O 1 + h−3n/2 µ(t)eJ0 |t|/h Kmr L(L2 (ΛtG )) = O h−3n/2 µ(t)eJ0 |t|/h + h∞ . this completes the proof of Lemma 6.1 and of Proposition 3.1.
To prove Corollary 3.2, we first apply Proposition 3.1 with m = 1 and v = Au. We obtain: e−tH/h T Au2L2 (ΛtG ) = ae−tH/h T u, e−tH/hT Au + R3 with
R3 = O h + h−3n/2 µ(t)eJ0 |t|/h ξ d e−tH/h T u · e−tH/h T Au.
a. This Then we apply again Proposition 3.1 but this time with v = u and m = gives: ae−tH/h T u2L2(ΛtG ) + R4 (6.23) e−tH/h T Au2L2(ΛtG ) = with R4
=
(6.24) O h + h−3n/2 µ(t)eJ0 |t|/h × × ξ d e−tH/h T u2 + ξ d e−tH/h T u · e−tH/h T Au .
In particular, we deduce from these two last inequalities (and the fact that ae−tH/h T u = O(ξ d e−tH/h T u)): e−tH/h T Au = O 1 + h−3n/2 µ(t)eJ0 |t|/h ξ d e−tH/h T u
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Resonance Free Domains for Non Globally Analytic Potentials
which, inserted into (6.24) gives R4 = O h + h−3n µ(t)e2J0 |t|/h ξ d e−tH/h T u2 . Then Corollary 3.2 follows from (6.23) and (6.25).
755
(6.25)
Acknowledgements. The impulse for this paper has been given to us by Johannes Sj¨ ostrand in suggesting us to first use the analytic distorsion before applying the microlocal weighted estimates. For this reason we are very indepted to him. Moreover, we thank B. Helffer for having read carefully the paper and having suggested to us several improvements in the presentation. Finally, we thank Jean-Fran¸cois Bony and J. Sj¨ ostrand for having pointed out to us a mistake in a previous proof of the main result.
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A. Martinez, Estimates on Complex Interactions in Phase Space, Math. Nachr. 167, 203–254 (1994).
[MeSj]
A. Melin and J. Sj¨ostrand, Fourier Integral Operators with Complex Valued Phase Functions, Springer Lecture Notes in Math., 459, 120–223 (1974).
[Na1]
S. Nakamura, Shape Resonances for Distorsion Analytic Schr¨ odinger Operators, Comm. Part. Diff. Eq. 14, 1385–1419 (1989).
[Na2]
S. Nakamura, On Martinez’ Method of Phase Space Tunneling, Rev. Math. Phys. 7 No. 3, 431–441 (1995).
[Ro1]
M. Rouleux, Absence of Resonances for Semiclassical Schr¨ odinger Operators with Gevrey Coefficients, Hokkaido Mathematical Journal 30, 475– 517 (2001).
[Ro2]
M. Rouleux, Resonances for a semiclassical Schr¨ odinger operator near a non-trapping energy level, Publ. of the RIMS, Kyoto Univ. 34, 487–523 (1998).
[Sj]
J. Sj¨ ostrand, Singularit´es analytiques microlocales, Ast´erisque No 95 (1982).
[SjZw]
J. Sj¨ ostrand and M. Zworski, Asymptotic distribution of resonances for convex obstacles, Acta Math. 183 No.2, 191–253 (1999).
[TaZw]
S.H. Tang and M. Zworski, Resonance Expansions of Scattered Waves, Comm. Pure App. Math. 53 (10), 1305–1334 (2000).
Andr´e Martinez Universit` a di Bologna Dipartimento di Matematica Piazza di porta San Donato, 5 40127 Bologna Italy email: [email protected] Communicated by Bernard Helffer submitted 05/02/02, accepted 06/05/02
Ann. Henri Poincar´e 3 (2002) 613 – 634 c Birkh¨ auser Verlag, Basel, 2002 1424-0637/02/040613-22
Annales Henri Poincar´ e
Locality in Free String Field Theory – II J. Dimock
Abstract. We study the covariant free bosonic string field theory and explore its locality (causality) properties. We find string fields which are strictly local and covariant, but act on an unconstrained Hilbert space with an indefinite inner product. From these we construct observable fields which act on the physical Hilbert space with a definite inner product. These are shown to be approximately local.
1 Introduction 1.1
Overview
We continue to explore the locality properties of free bosonic string field theory. The guiding question is whether one can find string fields which satisfy the field equation, are strictly local (causal), and are Lorentz covariant. Success in this quest would mean one could define an algebra of local observables of the type that one usually considers in quantum field theory [11]. We do not necessarily expect to succeed in this quest, but do expect that the ways in which we fall short will be of interest. In an earlier work [16], [15], [5], the problem was considered in the light cone gauge and string fields were found which were local with respect to the center of mass coordinate. However these fields were not Lorentz covariant. In the present paper we work with a Lorentz covariant formalism right from the start. It is the socalled “old covariant quantization” in which one quantizes first and then imposes the constraint. Before imposing constraints, we are able to construct string field operators which are Lorentz covariant and local in the sense that the commutator of two fields vanishes when the center of mass coordinates are spacelike separated. However the field operators act on a Hilbert space with an indefinite inner product. Once the constraints are imposed one obtains a definite inner product. On this space we also define covariant field operators called observable fields. For these observable fields we establish an approximate locality property. Our results seem to be consistent with the treatment of Hata and Oda [12] who work in a BRST formalism. A general account of string field theory can be found in Thorn [20] Another goal of this work is to solidify the mathematical foundations of covariant string theory. For earlier work in this direction see Grundling and Hurst [10].
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Lorentz invariant measures
We start by developing some facts about Lorentz invariant measures on the mass shells. (See also [18]). For any real number r let Vr = {p ∈ Rd − {0} : p2 + r = 0} .
(1)
Here p2 = p · p = µν ηµν pµ pν = −(p0 )2 + | p |2 is the Lorentz inner product. The Lorentz group is all nonsingular linear transformations preserving p2 and it acts on Vr . First we define a Lorentz invariant volume element on Vr . Let τ = dp0 ∧ 1 dp ... ∧ dpd−1 be the volume element in Rd . With α(p) = −p2 define σ to be the unique d − 1 form on Vr such that dα ∧ σ = τ .
(2)
If Λ is a proper Lorentz transformation we have d(α◦Λ)∧Λ∗ σ = Λ∗ τ . But α◦Λ = α by definition and Λ∗ τ = τ since det Λ = 1. Thus dα ∧ Λ∗ σ = τ and hence Λ∗ σ = σ. Now Vr f σ is defined for continuous functions f with compact support on Vr . The map f → Vr f σ is positive and hence there is a positive measure µr on Vr such that f dµr = f σ. (3) Vr
The Lorentz invariance of σ implies the invariance of µr . Now we comment on some specific representations of this measure, first for r ≥ 0. In this case the hyperboloid has two sheets which are Vr± = {p ∈ Vr : ±p0 > 0} . Lemma 1 For r ≥ 0, let f have compact support on Vr± and let ωr (p) = Then d p . f dµr = f (±ωr ( p ), p ) 2ωr ( p) Vr±
(4) | p |2 + r. (5)
Proof. For Vr± we can take global coordinates p = (p1 , ..., pd−1 ). In these coordinates we have σ=
1 1 dp1 ∧ ...dpd−1 = 0 dp1 ∧ ...dpd−1 . ∂α/∂p0 2p
(6)
This is the form on Vr± . Pulling it back to Rd−1 with the inverse coordinate p ) = (±ωr ( p ), p ) we have function φ± ( φ∗± (σ) =
±1 dp1 ∧ ... ∧ dpd−1 . 2ω( p)
(7)
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Now φ+ is orientation preserving so Vr+ f σ = (f ◦ φ+ )φ∗+ (σ). On the other hand φ− is orientation reversing so Vr− f σ = − (f ◦ φ− )φ∗− (σ). In either case we get the stated result. Another representation uses light-cone coordinates and now we allow all r ∈ R. Light-cone coordinates pˆ = (p) are defined on Rd by pˆ = (p− , p˜, p+ ) where p± = 2−1/2 (p0 ± pd−1 ) and p˜ = (p1 , ..., pd−2 ). For any function f let fˆ = f ◦ −1 be the expression in light-cone coordinates. Also define Vr± = {p ∈ Vr : ±p+ > 0}
(8)
either in the original coordinates or in light-cone coordinates depending on the context. For r ≥ 0 these are again the two components of Vr , but for r < 0 they are just open sets in Vr . Lemma 2 For any r, let f have compact support on Vr± . Then 2 |˜ p| + r d˜ pdp+ + . f dµr = fˆ , p ˜ , p + 2p 2|p+ | Vr± Proof. The integral f σ can be evaluated as (−1 )∗ τ = τ we have dˆ α∧σ ˆ = τ . On Vr± we can − 1 d−2 τ = dp ∧ dp ... ∧ dp ∧ dp+ and since α ˆ (p) expressed in these coordinates σ ˆ=
(9)
fˆ σ ˆ where σ ˆ = (−1 )∗ σ. Since take coordinates (p+ , p˜) . Since = 2p+ p− − |˜ p|2 . We find that
1 1 dp1 ∧ ... ∧ dpd−2 ∧ dp+ = + dp1 ∧ ... ∧ dpd−2 ∧ dp+ . ∂α ˆ /∂p− 2p
(10)
p|2 + r)/2p+ , p˜, p+ ) defined on The inverse coordinate function is φ(p+ , p˜) = ((|˜ + d−1 the half spaces ±p > 0 in R . The pull back φ∗ (ˆ σ ) to these half spaces has the same form. The function φ is orientation preserving for Vr+ and orientation reversing for Vr− . Thus the integral is evaluated as ± (fˆ ◦ φ)φ∗ (ˆ σ ) respectively, and in either case we get the stated result. For r ≤ 0 the sets Vr± do not cover all of Vr . However suppose we define p± j = ± + −1/2 0 j 2 (p ± p ) with j = 1, ..., d − 1 and define more sets Vr,j = {p ∈ Vr : ±pj > 0} ± ˜ = {p ∈ Vr : ±p− and Vr,j j > 0}. On each of these sets we can prove a result similar to (9). These sets do cover Vr and by taking a partition of unity subordinate to this covering we can express any integral as a sum of integrals of the type (9). Lemma 3 Let f be a function on on Rd − {0} which is continuous and has compact support. Let fr be the restriction to Vr . Then f= fr dµr dr . (11) Vr
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± − d ˜± Proof. Let Ur,j = {p ∈ Rd : ±p+ j > 0} and Ur,j = {p ∈ R : ±pj > 0}. These cover R − {0} and by introducing a subordinate partition of unity it suffices to prove the theorem assuming that f has compact support in one of these sets, for example + Ur+ defined by p+ = p+ d−1 > 0. Then fr has compact support in Vr for all r. We − + − 2 make the change of variables p ↔ r with r = 2p p − |˜ p| and then we have 2 d˜ pdp+ dr |˜ p| + r + fˆ(p− , p˜, p+ )dp− d˜ pdp+ = fˆ , p ˜ , p + 2p 2|p+ | (12) = fr dµr dr . Vr
2 The single string 2.1
Pre-constrained
We now undertake the covariant quantization of the single string. The construction is mostly standard [9]. However, one novelty is that the center of mass momentum is treated as a genuine quantum observable with a distribution of values. Most treatments take a fixed center of mass momentum. For the open string in Rd the coordinates of the quantum string should be operators X µ (τ, σ) defined for (τ, σ) ∈ R × [0, π] and satisfying the wave equation 2 µ ∂ X ∂2X µ − =0 (13) ∂τ 2 ∂σ 2 with Neumann boundary conditions on [0, π]. The operators X µ and the string momentum P µ = ∂X µ /∂τ are supposed to satisfy the equal τ commutation relations [[X µ (σ, τ ), P ν (σ , τ )] = iπδ(σ − σ )η µν . (14) Corresponding to reparametrization invariance we impose the constraints
∂X ∂X ± ∂σ ∂τ
2 =0.
(15)
That is we ask for states which are annihilated by these operators. Finally we π ask that the center of mass xµ = π −1 0 X µ (τ, σ)dσ be parametrized in a forward moving direction. This means we require that π the constant center of mass momentum pµ = dxµ /dτ , also written pµ = π −1 0 P µ (τ, σ)dσ, should satisfy p0 =
dx0 >0. dτ
(16)
Classically one can find solutions of the wave equation by expanding in eigenfunctions of the Laplacian with Neuman boundary conditions, that is in a cosine
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series. One finds that X µ (σ, τ ) = xµ + pµ τ + i
αµn e−inτ
n=0
617
cos nσ . n
(17)
The quantum operator will be given by the same expression. It formally satisfies the commutation relations (14) if we ask that xµ , pµ , αµn be operators satisfying the commutation relations [xµ , pν ] =iη µν [αµm , ανn ] =mδm+n η µν .
(18)
Here is a construction of these operators. Consider the Hilbert space L2 (Rd ). On this let xµ be the multiplication operator, let pν = −i∂/∂xν and let space ν µν p = ν η pν (the spacetime representation), or else let pν be the multiplication operators and let xµ = i∂/∂pµ, etc. (the momentum representation). In either case these satisfy (18). A convenient dense domain for these operators is S(Rd ), the Schwartz space of smooth rapidly decreasing functions. This is invariant under the Fourier transform and connects the two representations. Next let 2 = 2 (N, Cd ) be the space of square summable maps f : N → d C . This has the usual inner product (f, g) and also an indefinite inner product < f, g >. They are (f, g) =
< f, g >=
∞ n=1 µ ∞
fµn gµn (19) ηµν fµn gνn .
n=1 µν
They are related by < f, g >= (f, Jg) where (Jg)µ = ηµµ gµ . (Thus J = η , 2 but without the geometric interpretation). ∞ Let2 Fj ( ) be the j-fold symmetric 2 2 tensor product of and let F ( ) = j=0 Fj ( ) be the bosonic Fock space over 2 . Any unitary operator U on 2 induces a unitary ⊗j U on Fj (2 ) and hence an operator Γ(U ) on F (2 ). We define an indefinite inner product on F (2 ) by < f, g >= (f, J g) where J = Γ(J). For any operator O on F (2 ), let O∗ be the adjoint with the definite inner product, and let O† be the adjoint with the indefinite inner product. Hence (f, Og) = (O∗ f, g) and < f, Og >=< O† f, g >. They are related by O∗ = J O† J . Next define annihilation operators a(f ), b(f ) on the n-fold symmetric tensor product by a(f )(f1 ⊗ ... ⊗ fj ) = j < f, f1 > f2 ⊗ ... ⊗ fj (20) b(f )(f1 ⊗ ... ⊗ fj ) = j(f, f1 ) f2 ⊗ ... ⊗ fj . By restriction these define operators on the symmetric subspace and hence on F (2 ). We have a(f ) = b(Jf ). The adjoints satisfy a† (f ) = b∗ (f ). We have [b(f ), b∗ (g)] = (f, g) and [a(f ), a† (g)] =< f, g >.
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In 2 there is a standard basis eµm defined by (eµn )νm = δνµ δm,n . We define for n>0 √ αµn = n a(eµn ) (21) √ αµ−n = n a† (eµn ) . These satisfy (18) since < eµn , eνm >= η µν δn,m . Let D0 be the dense subspace of F (2 ) generated by applying a finite number of operators αµ−n to the no excitation state Ω0 = (1, 0, 0...). Now consider the Hilbert space L2 (Rd ) ⊗ F(2 ) ≈ L2 (Rd , F (2 )) .
(22)
Besides the usual inner product this space has an indefinite inner product inherited from F (2 ) and defined by < ψ, χ >= < ψ(p), χ(p) > dp . (23) The operators xµ , pµ in the momentum representation, and αµn all act on this space. Now we can define the coordinate operator X µ (σ, τ ) by (17). It is well defined provided we interpret it as a distribution in σ and to restrict a nice domain like S(Rd ) ⊗ D0 . It does satisfy (13),(14). On the same domain (xµ )† = xµ , (pµ )† = pµ , (αµn )† = αµ−n . and hence (X µ (σ, τ ))† = X µ (σ, τ ). We digress to discuss representations Lorentz group. First on 2 there of the ν is a representation defined by (Λf )µn = ν Λµ fνn which preserves the indefinite inner product. This induces an operator Γ(Λ) on F (2 ) which also preserves the indefinite inner product. It is not bounded but is at least defined on vectors with a finite number of entries. Finally for a ∈ Rd and a proper Lorentz transformation Λ we define U (a, Λ) on L2 (Rd , F (2 )) by (U (a, Λ)ψ)(p) = e−ip·a Γ(Λ)ψ(Λ−1 p) .
(24)
This is well-defined if ψ takes values in the domain of Γ(Λ). The operators U (a, Λ) give a representation of the inhomogeneous Lorentz group which preserves the indefinite inner product since Lebesgue measure is Lorentz invariant. We note that U (a, Λ)−1 a(f )U (a, Λ) = a(Λ−1 f ). Since Λ−1 eµn = ν (Λ−1 )νµ eνn = ν Λµν eνn this implies U (a, Λ)−1 αµn U (a, Λ) = Λµν ανn . (25) We also have x → µ
ν
µ ν ν Λ νx
+ a , and p → µ
µ
U (a, Λ)−1 X µ (σ, τ ) U (a, Λ) =
ν
ν
Λµν pν and thus
Λµν X ν (σ, τ ) + aµ .
(26)
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Now we turn to the constraint operators (15). Passing to the Fourier components one finds the operators ∞ 1 2 p + α−n · αn 2 n=1 1 = αm · p + αm−n · αn 2
L0 = Lm
(27) m = 0 .
n=m,0
These are well defined on S(Rd ) ⊗ D0 and satisfy L†m = L−m . Instead of asking for states ψ satisfying Ln ψ = 0 for all n we make the standard modification and ask for states satisfying (L0 − 1)ψ =0 Lm ψ =0
m>0.
(28)
As usual when quantizing a parametrized theory, the dynamics are contained in the constraints. Suppose we define the operator M 2 (not really a square) on a dense domain in F (2 ) by (29) M 2 = 2(N − 1) where N is the excitation operator which can be written in any of the following forms N=
∞
α−n · αn
n=1
=
n ηµν a† (eµn )a(eνn )
(30)
n,µν
=
n b∗ (eµn )b(eµn ) .
n,µ
The constraint (L0 − 1)ψ = 0 can be written (p2 + M 2 )ψ = 0. In the spacetime representation we have (31) (− + M 2 )ψ = 0 . It the Klein - Gordon equation for an F (2 ) valued function, and gives the evolution in time. The operator M 2 is identified as a mass operator. The next result shows that M 2 is self-adjoint and has spectrum −2, 0, 2, 4, 6... with finite multiplicity. Lemma 4 N is self-adjoint and has spectrum 0, 1, 2, ... with finite multiplicity. Proof. Let {Nµ,n } be a finite sequence of positive integers indexed by µ = 0, 1, ..., d − 1 and n = 1, 2, ..., with at most finitely many Nµ,n = 0. For each such sequence we define a vector (b∗ (eµ ))Nµ,n n ψ({Nµ,n }) = (32) Ω0 . Nµ,n ! µ,n
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This is a orthonormal basis for F (2 ) and they are eigenfunctions of N since
nNµ,n ψ({Nµ,n }) . (33) N ψ({Nµ,n }) = µ,n
This gives the self adjointness and the spectrum. The finite multiplicity follows since for any positive integer n∗ there are only a finite number of sequences with ∗ µ,n nNµ,n = n .
2.2
Reconfigured
The constraint (31) cannot be satisfied in L2 (Rd , F (2 )). (If we project onto an eigenvalue r of M 2 it says (p2 +r)ψ = 0. This is not possible since p2 has continuous spectrum). To impose the constraint we will have to modify the Hilbert space. To begin we write this space as a direct integral over the various mass shells. For the moment our purpose is just to motivate Definition 1 below, so we pass over various technicalities such as the exact definition of the direct integral in this case. (See however Nielsen [17] ). Let · denote the norm in the Fock space F (2 ) defined by the definite inner product. For ψ ∈ L2 (Rd , F (2 )) the norm squared can be written as ∞ ψ(p)2 dp = ψ(p)2 dµr (p) dr . (34) Rd
−∞
Vr
2
This follows from (11) extended to L functions. Then we can make the identification ⊕ 2 d 2 L2 (Vr , F (2 ), dµr ) dr (35) L (R , F ( ), dp) = where ψ ∈ L2 (Rd , F (2 ), dp) is identified with the map r → ψr ( the restriction of ψ to Vr ). The indefinite inner product on F (2 ) induces the same on L2 (Vr , F (2 ), dµr ) and we have, again by (11), ∞ < ψ, χ >= < ψr , χr > dr . (36) −∞
Thus the decomposition can be regarded as a decomposition of indefinite inner product spaces. The operators M 2 , Lm act on the space L2 (Vr , F (2 ), dµr ) and we have the decompositions ⊕ 1 (−r + M 2 ) dr L0 − 1 = 2 (37) ⊕ Lm = Lm dr . This means for example that (Lm ψ)r = Lm ψr . Since the Lorentz group acts on Vr the operators U (a, Λ) act on L2 (Vr , F (2 ), dµr ), and they preserve the indefinite
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inner product since the measure is Lorentz invariant. Denoting this representation by Ur (a, Λ) we have ⊕ U (a, Λ) = Ur (a, Λ) dr . (38) To impose the constraint we first make a minimal reconfiguration of the Hilbert space so it can accept the constraints. Since M 2 has spectrum in −2, 0, 2, 4, ... the constraint (L0 − 1)ψ = 0 only has a chance for r in this set. Accordingly we pick out these values from the direct integral and form a direct sum. At this stage we also impose the forward moving condition p0 > 0 as much as possible. For r ≥ 0 we pick out the positive energy part Vr+ of the mass shell. This is not possible for r < 0 without losing the representation of the Lorentz group, and so we leave it alone. We these ideas in mind we make the following definitions after (35), (36), (37), (38). Definition 1 1. The Hilbert space for the single string is Hr H= (39)
r=−2,0,2,... 2
Hr =L (+)
where Vr
(Vr(+) , F (2 ), dµr ) (+)
= Vr+ for r ≥ 0 and Vr
= Vr for r < 0 .
2. For ψ, χ in H with components ψr , χr an indefinite inner product is defined by < ψ, χ >= < ψr , χr > . (40) r
3. The constraint operators are defined by 1 (−r + M 2 ) L0 − 1 = 2 r
Lm =
Lm .
(41)
r
4. A representation of the inhomogeneous Lorentz group is defined by U (a, Λ) = Ur (a, Λ) .
(42)
r
Let us be more precise about the domains of N, M 2 , Lm . We will define them as closed operators on Hr and then the above equations define them as closed operators on H. As we have noted N (or M 2 ) is self-adjoint on F (2 ) and we define N on Hr by (N ψ)(p) = N ψ(p) with domain
N ψ(p)2 dµr (p) < ∞ . (43) D(N ) = ψ ∈ Hr : ψ(p) ∈ D(N ) a.e. p ,
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Similarly for each p let Lm (p) be the closure of the operator (27) defined on D0 ⊂ F(2 ). (It is closable since the adjoint L∗m (p) = J L−m J is densely defined). Then define Lm on Hr by (Lm ψ)(p) = Lm (p)ψ(p) with domain
D(Lm ) = ψ ∈ Hr : ψ(p) ∈ D(Lm (p)) a.e. p , Lm (p)ψ(p)2 dµr (p) < ∞ . (44) With these domains we have: Lemma 5 N and M 2 are self adjoint, and Lm is closed. Proof. Start with the second. Let ψj ∈ D(Lm ) satisfy ψj → ψ and Lm ψj → χ. Then there exist a subsequence such that for almost every p we have ψjk (p) → ψ(p) and Lm (p)ψjk (p) → χ(p). (This is a corollary of the proof that vector-valued Lp spaces are complete [13]). Since Lm (p) is closed it follows that ψ(p) ∈ D(Lm (p)) and Lm (p)ψ(p) = χ(p) for almost every p. Hence ψ ∈ D(Lm ) and Lm ψ = χ. Thus Lm is closed. The same argument shows that N is closed. It is also symmetric and since it has a dense set of analytic vectors, for example C0∞ ⊗ D0 , it is self-adjoint.
2.3
Constrained
Now let H be the subspace of H satisfying the constraints (28). We have H = Hr (45)
r=−2,0,2,...
Hr ={ψ ∈ Hr : (−r + M 2 )ψ = 0, Lm ψ = 0 for m > 0} . Note that a function ψ ∈ H−2 is in H−2 iff both N ψ = 0 and Lm ψ = 0 which is true iff ψ takes values in F0 (2 ) ≈ C. Thus H−2 = L2 (V−2 , F0 (2 ), dµ−2 ). These are the tachyons. Next we consider the isotropic or spurious elements in H which are defined by H = H ∩ (H )⊥ . Here the orthogonal subspace is defined by the indefinite inner product. Vectors in ψ ∈ H satisfy < ψ, ψ >= 0. The subspace has the form H = Hr r (46) Hr =Hr ∩ (Hr )⊥ .
Now let Hphys = H /H . We identify Hphys =
Hrphys
r
(47)
Hrphys =Hr /Hr . The indefinite inner product on H lifts to Hphys and is the direct sum of the the inner products on Hrphys lifted from Hr .
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Now we establish the famous no-ghost theorem. Our method is to reduce the result to a statement pointwise in p and then quote the literature. Lemma 6 For d = 26, < ., . > is positive definite on Hrphys and Hphys . (+)
Proof. It suffices to prove the result on Hrphys . Now Hr = L2 (Vr be regarded as a constant fiber direct integral ⊕ F (2 )dµr (p) . Hr = (+)
, F (2 ), dµr ) can
(48)
Vr
Indeed the former can be taken as the definition of the latter, so ⊕this just amounts (−r+M 2 )dµr (p) to a change in notation. We have decompositions (−r+M 2 ) = ⊕ and Lm = Lm (p)dµr (p). The constrained space can be characterized as ⊕ Hr = H (p)dµr (p) (+) (49) Vr 2 2 2 H (p) ={ψ ∈ F( ) : (p + M )ψ = 0, Lm (p)ψ = 0 for m > 0} . (+)
This means ψ ∈ Hr iff ψ(p) ∈ H (p) for almost every p ∈ Vr . Since eigenvalues of M 2 have finite multiplicity, H (p) is finite dimensional. Note also H (p) ⊂ D0 . ⊕ Next we have (Hr )⊥ = (H (p))⊥ dµr (p) and it follows that ⊕ Hr = H (p)dµr (p) (+) (50) Vr H (p) =H (p) ∩ H (p)⊥ . Now for ψ, χ ∈ Hr we have ψ − χ ∈ Hr if and only if ψ(p), χ(p) ∈ H (p) satisfy ψ(p) − χ(p) ∈ H (p) for almost every p. Thus equivalence classes can be defined pointwise which we write as ⊕ Hphys (p)dµr (p) Hrphys = (+) (51) Vr phys H (p) = H (p)/H (p) . Now it suffices to prove that the inner product is positive definite on Hphys (p). A proof of this can be found in Frenkel, Garland, and Zuckerman [6]. They also compute the dimension of this space and show it depends only on p2 . For the original proofs see Brower [2] and Goddard and Thorn [8]. Lemma 7 U (a, Λ) determines a unitary representation of the inhomogeneous Lorentz group on Hphys . Proof. Ur (a, Λ) is defined on all of Hr . It preserves Hr since we have [Lm , Ur (a, Λ)] = 0 by (25). If χ ∈ (Hr )⊥ then for ψ ∈ Hr we have < ψ, Ur (a, Λ)χ >= < Ur (a, Λ)−1 ψ, χ >= 0 and so Ur (a, Λ)χ ∈ (Hr )⊥ . Thus Ur (a, Λ) preserves Hr and so it lifts to Hrphys . Since it is still inner product preserving it is unitary.
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Physical States (+)
We want to exhibit some non-trivial smooth elements of Hr ⊂ L2 (Vr , F (2 ), dµr ). These then determine well-behaved vectors in Hrphys and Hphys . We start with the fact that the spaces H (p) are non-trivial (finite dimensional ) vector spaces. Lemma 8 Γ(Λ) is a bijection from H (p) to H (Λp) . Proof. First note that Γ(Λ) is defined on H (p) since H (p) ⊂ D0 . Then we use (25) to conclude that Γ(Λ)−1 M 2 Γ(Λ) = M 2 and that Γ(Λ)−1 Lm (p) Γ(Λ) = Lm (Λ−1 p). This gives the result. (+)
(+)
and a smooth Lemma 9 For any q ∈ Vr there is a neighborhood U of q in Vr family Λ(p, q) of Lorentz transformations defined for p ∈ U such that Λ(p, q) q = p. Proof. The proof uses the following well-known fact (see for example [21]). Let G be a Lie group with closed subgroup H. Let π : G → G/H be the projection onto the coset space. Then there are local smooth sections. That is for any point in G/H there is a neighborhood U and a smooth map s : U → G such that π ◦ s = id. In our case let L be the proper Lorentz transformations (L = a component of SO(d − 1, 1)). Fix q and let H be the subgroup which leaves q invariant. (e.g. (+) if r > 0 then H = SO(d − 1)). Since L acts transitively on Vr we have that the (+) map Λ → Λq from L to Vr lifts to a diffeomorphism ΛH → Λq from the coset (+) space L/H to Vr . This identifies the two spaces. Now from the general result there is a neighborhood U of H in L/H and a map s : U → L such that s(ΛH)H = ΛH for all ΛH ⊂ U . Equivalently we can (+) regard U as a neighborhood of q in Vr and have a map s : U → L satisfying s(p)q = p for all p ∈ U . Defining Λ(p, q) = s(p) we have the result. Lemma 10 Let q, U , and Λ(p, q) be as above. 1. Γ(Λ(p, q)) is a bijection from H (q) to H (p). 2. There exist ψ ∈ C ∞ (U, F (2 )) such that ψ(p) ∈ H (p) for all p ∈ U . (+)
3. There exist ψ ∈ C0∞ (Vr ψ ∈ Hr .
, F (2 )) such that ψ(p) ∈ H (p) for all p ∈ Vr , i.e.
Proof. The first follows from lemma 8. For the second let ψ0 ∈ C0∞ (U, H (q)) and take ψ(p) = Γ(Λ(p, q))ψ0 (p). Multiplying by χ ∈ C0∞ (U ) gives a functions satisfying the third condition. By adding functions for different neighborhoods U we get a rich class of functions.
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Remarks. 1. There is another way to look at this result. Let ξ the set of all pairs {(p, ψ)} where p ∈ Vr and ψ ∈ H (p), and let ξU be the subset of pairs with p ∈ U . We have smooth mappings (p, ψ) → (p, Γ(Λ(p, q)−1 )ψ) from ξU to U × H (q) and hence from ξU to U × Cs where s is the dimension of H (q). Covering (+) Vr by such charts gives ξ the structure of a smooth vector bundle. Such bundles have smooth sections which is our result. 2. If r > 0 then the √ neighborhood U can be taken to be all of Vr+ . In this case we can take q = ( r, 0, 0, ..) and let Λ(p, q) be the standard boost to p. ( See Weinberg [22], equation (2.5.24) for the explicit formula).
3 String field theory We develop the string field theory by taking the dynamical equation satisfied by the single string wave equation, specializing to real solutions, treating these solutions as a classical Hamiltonian system, quantizing this system, and then finally imposing the constraints. This is “second quantization” , and the exact status of this process is always a little ambiguous . . . which quantization was the genuine quantization? Whatever attitude one takes one ends up at the same place. In any case the quantization process is just meant to be suggestive of a true quantum theory. Our formulation of the problem uses techniques which have been useful in the study of quantum field theory in curved spacetime [14], [3], [4].
3.1
String field equation
We start by defining real elements of F (2 ). These are vectors satisfying Cψ = ψ where C is some conjugation on F (2 ). A conjugation is an anti-linear isometry satisfying C 2 = 1. We also want our conjugation to satisfy [C, M 2 ] = 0 and [C, J ] = 0. Then also < ψ, χ > =< Cψ, Cχ >. For example one could take C0 = Γ(c0 ) where c0 is the usual conjugation c0 ψ = ψ¯ on 2 . In the following we just suppose that some C satisfying the above conditions has been chosen. Now we study the Klein-Gordon equation: (− + M 2 )U = 0
(52)
for functions U : Rd → F (2 ). Given real F0 , G0 ∈ C0∞ (Rd−1 , F (2 )) there is a unique smooth real solution U such that U = F0 and ∂U/∂x0 = G0 on some surface x0 = t0 , called a Cauchy surface. The solution has compact support on any other Cauchy surface x0 = t. Such solutions will be called regular. Associated with this equation there is a real bilinear form. For any functions U (x) = U (x0 , x ) and V (x) = V (x0 , x ) it is defined by ∂U ∂V , V > dx . (53) σt (U, V ) = < U, 0 > − < ∂x ∂x0 x0 =t
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Green’s identity states that for t > s σt (U, V ) − σs (U, V ) = (< U, (− + M 2 )V > − < (− + M 2 )U, V >)dx .
(54)
s<x0
Thus if U, V are regular solutions then σt (U, V ) is independent of t and is just denoted σ(U, V ). This form is skew symmetric and non-degenerate on the space of regular solutions, i.e. it is symplectic. This symplectic form will be the basis on quantization. But first we develop some facts about fundamental solutions for our KG equation. The following results are standard for the scalar KG equation. Advanced and retarded fundamental solutions E ± are operators on functions F ∈ C0∞ (Rd , F (2 )) defined by eip·x 1 F˜ (p)dp . (55) (E ± F )(x) = 2 2 d/2 (2π) Γ± ×Rd−1 p + M The p0 contour Γ± is the real line shifted slightly above/below the real axis. The exact choice does not matter because F˜ is entire and rapidly decreasing in real directions. Also for x ∈ Rd let us define J ± (x) = {y : (x − y)2 < 0, ±(x0 − y 0 ) > 0} to be the past or future of x. For A ⊂ Rd define J ± (A) = ∪x∈A J ± (x). Lemma 11 For F ∈ C0∞ (Rd , F (2 )) we have (−+M 2 )E ± F = F and supp(E ± F ) ⊂ J ± (suppF ). Proof. Let Pr be the operator on F (2 ) which is the projection onto the eigenspace M 2 = r. On the range of Pr we have E ± F = Er± F where (Er± F )(x) =
1 (2π)d/2
Γ± ×Rd−1
eip·x ˜ F (p)dp . p2 + r
(56)
These are the advanced/retarded fundamental solutions for the Klein-Gordon equation with mass r and they satisfy (− + r)Er± F = F and supp(Er± F ) ⊂ J ± (suppF ). Now E ± are fundamental solutions since Pr (− + M 2 )E ± F = (− + r) ± Er Pr F = Pr F . We have supp(E ± F ) ⊂ ∪r supp(Pr E ± F ) . But Pr E ± F = Er± Pr F and supp(Er± Pr F ) ⊂ J ± (supp(Pr F )) ⊂ J ± (suppF ) and hence the result. The propagator function is defined by E = E + − E − . Then U = EF is a regular solution. In fact we have: Lemma 12 1. U is a regular solution of (− + M 2 )U = 0 iff it can be written U = EF with F ∈ C0∞ (Rd , F (2 )) 2. F ∈ C0∞ (Rd , F (2 )) satisfies EF = 0 iff F = (− + M 2 )H for some H ∈ C0∞ (Rd , F (2 )) .
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Proof. Let U be a regular solution and let θ = θ(x0 ) be smooth and satisfy θ = 1 for x0 sufficiently positive and θ = 0 for x0 sufficiently negative. Define F = (− + M 2 )(θU ) = −(− + M 2 )((1 − θ)U ). Then F has compact support. Next note that θU = E + F since the difference satisfies the KG equation and vanishes in the distant past. Similarly −((1 − θ)U ) = E − F . Taking the difference of the last two equations gives U = EF . This proves the first claim. For the second suppose F = (− + M 2 )H. Then EF = (− + M 2 ) EH = 0. On the other hand if EF = 0, then H = E ± F has compact support and (− + M 2 )H = F . The next identity establishes a connection between any solution and its values on any Cauchy surface (all in the sense of distributions). We define < U, F > = < U (x), F (x) > dx . (57) Lemma 13 For F ∈ C0∞ (Rd , F (2 )) and any regular solution U : σ(U, EF ) = < U, F >
(58)
or equivalently For F, G ∈ C0∞ (Rd , F (2 )) σ(EF, EG) = < EF, G > = − < F, EG > . Proof. By Green’s identity we have for t < 0 + + σ0 (U, E F ) − σt (U, E F ) =
< U (x), F (x) > dx .
(59)
(60)
t<x0 <0
Letting t → −∞ we get an expression for σ0 (U, E + F ). Similarly we get an expression for σ0 (U, E − F ) . They are σ0 (U, E ± f ) = ± < U (x), F (x) > dx . (61) ∓x0 >0
Take the difference to obtain the result. We next want to make a connection with the single string Hilbert space H = ⊕r Hr . Given F ∈ C0∞ (Rd , F (2 )) we define ΠF ∈ H by specifying that (ΠF )r ∈ Hr is obtained by taking the Fourier transform, projecting onto the (+) (+) subspace M 2 = r with Pr , and then restricting to Vr . More precisely for p ∈ Vr we define √ (62) (ΠF )r (p) = 2π Pr F˜ (p) . We will need to exclude tachyons, so we restrict to functions F which take values in F+ (2 ) ≡ (F0 (2 ))⊥ ≡ Fj (2 ) . (63) j≥1
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We have M 2 ≥ 0 on this subspace. Hence for such F , (ΠF )−2 = 0 and hence ΠF ∈ H+ the no-tachyon subspace of H: Hr . (64) H+ = r≥0
Lemma 14 For (real) F, G ∈ C0∞ (Rd , F+ (2 )) σ(EF, EG) =< EF, G >= 2Im < ΠF, ΠG > .
(65)
Proof. Only the second identity needs proof. We compute with Er = Er+ − Er− < EF, G >= < Er Pr F, Pr G > r≥0
=
(66) 2 Im < (ΠF )r , (ΠG)r >= 2 Im < ΠF, ΠG > .
r≥0
The second step follows since for r ≥ 0 and any F, G 1 ˜ < Er F, G >= − dp < F˜ (¯ p), G(p) > 2 p +r Γ+ −Γ− d p ˜ r ( − c.c.} p ), p ), G(ω p ), p ) > = − 2πi{ < F˜ (ωr ( 2ωr ( p) √ √ ˜ +> . = 2 Im < 2π F˜ |Vr+ , 2πG|V r
(67)
In the second step we evaluated the contour integral by taking residues at p0 = ˜ ˜ ˜ ±ω( p ). We also used < F˜ (p), G(p) > =< C F˜ (p), C G(p) > =< F˜ (−p), G(−p) > for p real , a consequence of the reality of F, G.
3.2
String field operator
Now we quantize solutions of the string field equation. We take as our phase space the space of all regular solutions Φ of (− + M 2 )Φ = 0 with symplectic form σ(Φ, Φ ) defined previously. For each solution U there is a function Φ → σ(Φ, U ) on the phase space. We quantize these functions by replacing them by operators on a complex Hilbert space, also denoted σ(Φ, U ), which are required to satisfy [σ(Φ, U ), σ(Φ, V )] = iσ(U, V ) .
(68)
This looks more familiar if we identify solutions with their data on some Cauchy surface. Then the operators are σ(Φ0 , Π0 ; F0 , G0 ) = Φ0 (G0 ) − Π0 (F0 ) and the commutator would be written [σ(Φ0 , Π0 ; F0 , G0 ), σ(Φ0 , Π0 ; F0 , G0 )] = iσ(F0 , G0 ; F0 , G0 ) .
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A special case is the standard [Φ0 (G0 ), Π0 (F0 )] = i < G0 , F0 >. The full spacetime field operator is obtained from the operators σ(Φ, U ) just as in the classical case. Following (58) we define the field operator as a distribution by Φ(F ) = σ(Φ, EF ) .
(69)
Then Φ(F ) satisfies the field equation and has a local commutator as the next result shows. Lemma 15 let σ(Φ, U ) be a family of operators indexed by regular solutions U of the KG equation and satisfying (68). Then the operators Φ(F ) = σ(Φ, EF ) defined for F ∈ C0∞ (Rd , F (2 )) satisfy Φ((− + M 2 )F ) =0 [Φ(F ), Φ(G)] = − i < F, EG > .
(70)
Furthermore every operator valued distribution Φ(F ) satisfying (70) arises in this way. Proof. The field equation follows from E(− + M 2 )F = 0 and the commutator follows from the identity (59). For the converse given Φ(F ) we define σ(Φ, U ) = Φ(F ) for any F such that U = EF . To see that this is well defined we have to show that if EF1 = EF2 then Φ(F1 ) = Φ(F2 ), or if EF = 0 then Φ(F ) = 0. But we have seen that EF = 0 implies F = (− + M 2 )H and hence the result follows. The operators σ(Φ, U ) have the commutator (68) again by the identity (59). Remarks. 1. Since σ(U, V ) is a symplectic form, representations of (68) do exist on general principles. Thus string field theories exist. Furthermore the spacetime field Φ(F ) defined by (69) is strictly local because if supp(F ) and supp(G) are spacelike separated, then supp(EF ) and supp(G) do not overlap and hence [Φ(F ), Φ(G)] = −i < F, EG >= 0. All this holds without suppressing the negative mass part of the equation! 2. However this is not the end of the story. We actually want the particular representation in which time translation is unitarily implemented with positive energy. (One can think of this as the forward moving condition again). Choosing a particular representation requires a complex structure or a “oneparticle structure” on phase space. These are equivalent to expressing the symplectic form σ(EF, EG) as the imaginary part of an inner product on some complex Hilbert space. But if we suppress the tachyon then this has already been accomplished in (65) where it is written as 2 Im < ΠF, ΠG > . Furthermore it is this choice which is associated with positive energy as we shall see.
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These considerations lead to the following definition. Tachyons are completely suppressed. We bypass σ(Φ, U ) and go directly to operators Φ(F ) satisfying (70). Also we enlarge the class of test functions from compact support to the Schwartz space of smooth rapidly decreasing functions. The Hilbert space is the Fock space over the no-tachyon single string Hilbert space H+ : K = F (H+ ) .
(71)
This has the indefinite inner product < Ψ, Ξ >= (Ψ, Γ(J )Ξ). States in K with finitely many entries are denoted Kf . Definition 2 (The String Field). For F ∈ S(Rd , F+ (2 )) we have ΠF ∈ H+ and we define on Kf Φ(F ) = a† (ΠF ) + a(ΠF ) . (72) Theorem 1 1. The string field satisfies Φ(F )† = Φ(F ), and Φ((− + M 2 )F ) = 0, and has the commutator [Φ(F ), Φ(G)] = −i < F, EG >. 2. There is positive energy representation U(a, Λ) of the inhomogeneous Lorentz group on K such that U(a, Λ)Φ(F )U(a, Λ)−1 = Φ(Fa,Λ )
(73)
where Fa,Λ (x) = Γ(Λ)F (Λ−1 (x − a)). Proof. The field equation is satisfied since Π(− + M 2)F = 0. The commutator is evaluated as [[Φ(F ), Φ(G)] = 2i Im < ΠF, ΠG >= −i < F, EG >
(74)
since the identity (65) holds for S as well as C0∞ . The representation is defined by U(a, Λ) = Γ(U (a, Λ)). We compute U(a, Λ)Φ(F )U(a, Λ)−1 = a† (U (a, Λ)ΠF ) + a(U (a, Λ)ΠF ) = a† (ΠFa,Λ ) + a(ΠFa,Λ ) = Φ(Fa,Λ ) .
(75)
As noted we have the following corollary: Corollary 1 ( Locality). If F, G have spacelike separated supports [Φ(F ), Φ(G)] = 0. Now we impose the constraint, and just as for the single string this will ˆ m be the Fourier transform of Lm , give us a positive definite inner product. Let L ˆ that is Lm is given by (27) but with pµ = −i∂/∂xµ . We would like to select states which are annihilated by Lm Φ for m > 0. However, just as for the GuptaBeuler quantization of the electromagnetic field [19] we must compromise and only impose the condition on the negative frequency part of the field defined by Φ− (F ) = a(ΠF ). This is defined and anti-linear for complex test functions. We ˆ m Φ− )(F ) ≡ Φ− (L ˆ −m F ). This is fulfilled by taking look for states annihilated by (L
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the subspace
K = F (H+ ),
631
H+ = H ∩ H+ .
ˆ −m F )Ψ = 0 since if ψ ∈ All Ψ ∈ K satisfy the required a(ΠL
H+
(76) then
ˆ −m F, ψ >=< L−m ΠF, ψ >=< ΠF, Lm ψ >= 0 < ΠL
(77)
Thus K is our constrained space, something we might have guessed directly. Next let K = K ∩ (K )⊥ be the isotropic vectors in K and define Kphys = K /K .
(78)
This space inherits an indefinite inner product from K . Lemma 16 For d = 26 the inner product on Kphys is positive definite and we have the identification of Hilbert spaces: phys Kphys = F (H+ ),
phys H+ = H+ /H+ .
(79)
Proof. H+ = H ∩H+ is a closed subspace of H+ and so we can write H+ = H+ ⊕M where M is the orthogonal complement with respect to the definite inner product. phys and gives an identification of M with H+ The projection onto M has kernel H+ which preserves the indefinite inner product. Thus the inner product is positive . Now we have the definite on M, and of course zero if either entry is in H+ identification of Hilbert spaces [7] K = F (H+ ⊕ M) = F (H+ ) ⊗ F(M) .
(80)
Under this identification Γ(J ) = Γ(J ) ⊗ Γ(J ) and so the induced inner product satisfies < Ψ1 ⊗ Ψ2 , Ψ1 ⊗ Ψ2 >=< Ψ1 , Ψ1 >< Ψ2 , Ψ2 >. Splitting F (H+ ) = ) ⊕ F+ (H+ ) and using F0 (H+ ) ≈ C we have F0 (H+ K = F (M) ⊕ (F+ (H+ ) ⊗ F(M))
(81)
supplied with the natural indefinite inner product. Every component of F+ (H+ )⊗ F (M) has at least one factor in H+ and so we can identify K = F+ (H+ ) ⊗ F(M) .
(82)
phys Kphys = F (M) = F (H+ ).
(83)
Thus These identifications preserve the indefinite inner product. Since the inner product phys is positive definite on H+ it is positive definite on Kphys . For certain test functions the string field operator Φ(F ) on K determines an operator on Kphys . We define Definition 3 F ∈ S(Rd , F+ (2 )) is a constrained test function if ΠF ∈ H+
To get real constrained test functions it is useful to pick a particular conjugation on F (2 ). It is C1 = Γ(c1 ) where c1 on 2 is defined by (c1 f )0n = f¯0n and (c1 f )kn = −f¯kn for k = 1, ..., d − 1. For the next result real means C1 ψ = ψ.
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Lemma 17 Non-trivial (real) constrained test functions exist Proof. Take r ≥ 0 and choose ψ0 ∈ C0∞ (Vr+ , F (2 )) so that ψ0 (p) ∈ H (p) for all p ∈ Vr+ . Then ψ defines an element of Hr and hence an element of H+ .We have seen that such functions exist in lemma 10. Next define ψ(ωr ( p ), p ) = ψ0 (ωr ( p ), p ) + C1 ψ0 (ωr ( p ), − p) .
(84)
This satisfies C1 ψ(ωr ( p ), p ) = ψ(ωr ( p ), − p ) and is still an element of Hr . This is so since C1 α0n C1 = α0n and C1 αkn C1 = −αkn and hence C1 Lm (ωr ( p ), p )C1 = Lm (ωr ( p ), − p ). We will find F so that ΠF = ψ. First write ψ(ωr ( p ), p ) = h( p) for a function h ∈ C0∞ (Rd−1 , F (2 )). (Or Rd−1 − {0} if r = 0). Let χ ∈ C0∞ (R) be real and satisfy χ(0) = 1. We define F by specifying that the Fourier transform be F˜ (p0 , p ) = (2π)−(1/2) χ(−(p0 )2 + | p |2 + r)h( p) .
(85)
Then F˜ is smooth and has compact support and so F ∈ S(Rd , F (2 )). Since h( p ) ∈ RanPr we have (ΠF )r = ψ and (ΠF )r = 0 for r = r as required. Since elements of RanPr have no zero component in Fock space, this is true of h( p ) and hence F (x). Thus F takes values in F+ . Finally we have C1 F˜ (p) = F˜ (−p) and hence C1 F (x) = F (x) so F is real. This completes the proof. Recall that Kf is the subspace of K with a finite number of entries. Similarly define Kf and Kf and Kfphys = Kf /Kf . One can identify Kfphys with a dense subspace of Kphys . Theorem 2 (Observable fields for d = 26) 1. Let F ∈ S(Rd , F+ (2 )) be a constrained test function. Then Φ(F ) on Kf lifts to an operator Φ(F ) on Kfphys called an observable field 2. These satisfy [Φ(F ), Φ(G)] = −i < F, EG >. 3. The representation U(a, Λ) on K lifts to a unitary representation U(a, Λ) on Kphys and U(a, Λ)Φ(F )U(a, Λ)−1 = Φ(Fa,Λ ) . Proof. Since ΠF ∈ H+ we have that Φ(F ) preserves Kf . It also preserves Kf since if Ψ ∈ Kf and Ξ ∈ Kf then < Ξ, Φ(F )Ψ >= %Φ(F )Ξ, Ψ& = 0. Since Kf is dense in K we have < Ξ, Φ(F )Ψ >= 0 for all Ξ ∈ K and hence Φ(F )Ψ ∈ Kf . Hence Φ(F ) acts on Kfphys . The commutator follows from the commutator on Kf For the covariance first note that F is constrained if and only if Fa,Λ is constrained. This follows from the identity ΠFa,Λ = U (a, Λ)ΠF and the fact that U (a, Λ) preserves H+ . The operator U(a, Λ) preserves Kf since U (a, Λ) preserves H+ . We argue as before that it also preserves Kf and so it lifts. The unitarity follows since it is inner product preserving, and the identity lifts from the identity on Kf .
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Remark. According to this theorem the observable fields have a local commutator. But can the fields themselves be localized? That is, are there constrained test functions F ∈ C0∞ (Rd , F+ (2 ))? Or is there some other way to get strictly localized operators? These are open questions. Without strictly localized fields we cannot get a vanishing result like Corollary 1. The best we can do is the following approximate result. If F, G are constrained then it estimates the commutator for observable fields on Kphys . Otherwise it refers to fields on K. Corollary 2 let F, G ∈ S(Rd , F+ (2 )), and let a be in the spacelike region |a0 | < (1 − >)|a |. Then as |a| → ∞, we have for any n > 0 [Φ(Fa ), Φ(G)] = −i < Fa , EG >= O(|a|−n ) . Proof. Since EG is bounded we have F (x − a)dx | < Fa , EG > | ≤O(1) supp(EG) ≤O(1) (1 + |x − a|)−n−d−1 dx supp(EG)
(86)
(87)
≤O(1)d(a, supp(EG))−n ≤O(|a|−n ) . In the last step we use the fact that supp(EG) is contained in a set of the form {x ∈ Rd : |x | ≤ x0 + C}. We omit the details.
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[9] M. Green, J. Schwartz and E. Witten, Superstring Theory, Cambridge University Press (1987). [10] H. Grundling and C.A. Hurst, The operator quantization of the open bosonic string field algebra, Commun. Math. Phys., 473–525 (1993). [11] R. Haag, Local Quantum Physics, Springer-Verlag (1992). [12] H. Hata and H. Oda, Causality in covariant string field theory, Phys. Lett. B 394, 307–314 (1997). [13] E. Hille and R.S. Phillips, Functional analysis and semi-groups, American Mathematical Society, Providence, (1957). [14] C. Isham, Quantum field theory in curved spacetimes: a general mathematical framework, in Differential Geometrical Methods in Mathematical Physics II, K. Bleuler, H. Petry, A. Reetz, eds, Springer-Verlag (1978). [15] D. Lowe, Causal properties of free string field theory, Phys. Lett. B326, 223– 230 (1994). [16] E. Martinec, The light cone in string theory, Class. Quant. Grav. 10, L187– L192 (1993). [17] O. Nielsen, Direct integral theory, Marcel Dekker, New York, (1980). [18] M. Reed and B. Simon, Methods of Modern Mathematical Physics II, Academic Press, New York, (1975). [19] F. Strocchi and A. Wightman, Proof of the charge superselection rule in local relativistic quantum field theory, J. of Math. Phys. 15, 2198–2224 (1974). [20] C. Thorn, String field theory, Phys. Rep. 175, 1–101 (1989). [21] F. Warner, Foundations of Differentiable Manifolds and Lie Groups, SpringerVerlag (1983). [22] S. Weinberg, The Quantum Theory of Fields I, Cambridge University Press (1995). J. Dimock1 Dept. of Mathematics SUNY at Buffalo Buffalo, NY 14226 U.S.A. email: [email protected]ffalo.edu Communicated by Vincent Rivasseau submitted 06/03/01, accepted 18/04/02
1 Research
supported by NSF Grant PHY0070905
Ann. Henri Poincar´e 3 (2002) 635 – 657 c Birkh¨ auser Verlag, Basel, 2002 1424-0637/02/040635-23
Annales Henri Poincar´ e
The State Space of Perturbative Quantum Field Theory in Curved Spacetimes S. Hollands and W. Ruan Abstract. The space of continuous states of perturbative interacting quantum field theories in globally hyperbolic curved spacetimes is determined. Following Brunetti and Fredenhagen, we first define an abstract algebra of observables which contains the Wick-polynomials of the free field as well as their time-ordered products, and hence, by the well-known rules of perturbative quantum field theory, also the observables (up to finite order) of interest for the interacting quantum field theory. We then determine the space of continuous states on this algebra. Our result is that this space consists precisely of those states whose truncated n-point functions of the free field are smooth for all n = 2, and whose two-point function has the singularity structure of a Hadamard fundamental form. A crucial role in our analysis is played by the positivity property of states. On the technical side, our proof involves functional analytic methods, in particular the methods of microlocal analysis.
I Introduction The perturbative construction of self-interacting quantum field theories in Minkowski spacetime was put on a completely rigorous mathematical footing in the works by Bogliubov, Parasiuk, Hepp, Zimmermann and other people [1] in the late sixties and early seventies. The issue of generalizing these constructions to curved spacetimes was first analyzed by Bunch and collaborators [2, 3]. These authors showed, within the context of Euclidean quantum field theory on Riemannian curved spaces, that if a theory is “perturbatively renormalizable” in flat space, then it remains so in curved space. However, while the perturbative definition of a quantum field theory on flat Euclidean space gives rise, via a “Wick rotation”, to the definition of a corresponding theory on Minkowski space, no such connection holds for curved Lorentzian spacetimes, which, apart from a few special classes of spacetimes such as static ones, do not possess a corresponding real Riemannian section. This means that Euclidean methods cannot directly be used for the definition of interacting quantum field theories in most Lorentzian spacetimes. Significant progress in perturbative construction of interacting quantum field theories on an arbitrary globally hyperbolic Lorentzian spacetime has recently been made by [4, 5, 6], using the mathematical tools of “microlocal analysis” [7]. In [4], the authors demonstrated that the formally infinite Wick-polynomials of a free field can be given a well-defined sense as operator-valued distributions via a normal ordering prescription. In [5], they then constructed time-ordered products of these Wick-polynomials. As in Minkowski spacetime, some “renormalization
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ambiguities” necessarily arise in the definition of the time-ordered products in curved spacetime, and moreover, unlike in Minkowski space, renormalization ambiguities also arise in the definition of the Wick-polynomials in curved spacetime. If one demands that these quantities be locally constructed from the metric in a generally covariant way, have a certain scaling behavior under a rescalings of the metric and have a suitable dependence under variations of the metric, then it can be shown [6], that these renormalization ambiguities are reduced to a finite number of free parameters.1 Moreover, a detailed analysis of the nature of these renormalization ambiguities [6] leads to the conclusion that interacting quantum field theories in globally hyperbolic spacetimes have the same classification into ones that are perturbatively renormalizable and ones that are not as in Minkowski space. The (smeared) Wick-polynomials and their (smeared) time-ordered products may be regarded as members of some abstract *-algebra, W.2 The Wickpolynomials and time-ordered products in W which satisfy the above additional locality, covariance and scaling requirements can be used to define, via the usual perturbation expansions familiar from Minkowski space, the quantities of interest in the interacting theory. The infinite sums occurring in these perturbation expansions do not by themselves define elements of the algebra W. However, if these sums are truncated at some arbitrary finte order, then the so obtained truncated expressions are elements of the algebra W. This algebra therefore contains all observables of interest in the interacting theory up to an arbitrary finite order in perturbation theory. In this work we investigate the space of quantum states on W, that is, the space of states for the perturbatively defined interacting quantum field theory. Here, by a state we mean a linear functional ψ : W → C which is normalized so that ψ(I) = 1, where “I” denotes the identity element in W, and which is positive in the sense that ψ(A∗ A) ≥ 0 for any element A in W. The above algebraic notion of state is related to the usual Hilbert space notion of state, but it is more general: Given a representation of W on some Hilbert space, one can consider a vector or density matrix state3 in this Hilbert space as defining a corresponding algebraic state on W. However, it is well-known that not all algebraic states—and not even all physically interesting ones—can be obtained in this manner from some specific Hilbert space representation4. 1 It turns out [6] that the normal ordered Wick-polynomials and their time-ordered products defined in [4, 5] necessarily fail to be locally constructed out of the metric in a covariant manner. A construction of Wick-polynomials that are locally defined in terms of the metric in a covariant manner and have the above additional properties was given in [6]. Local covariant time-ordered products were constructed in [8]. 2 While the construction of this algebra involves the choice of a quasi-free Hadamard of the corresponding free field theory, it turns out [6] that different choices for this state give rise to isomorphic algebras. Therefore, as an abstract *-algebra, W is independent of that choice. 3 Actually, we must restrict ourselves here to the vector or density matrix states contained in some common, dense invariant domain. 4 For example, in Minkowski space, the standard thermal state at some finite temperature of
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It was conjectured in [6] that the space of states on W consists precisely of those positive, normalized linear functionals ψ whose truncated n-point functions of the free field are smooth for n = 2 and whose two-point function of the free field is of Hadamard form. The main result of our paper (Thm. III.1) is that this conjecture is correct with regard to the states ψ on W that are continuous with respect to some natural topology on the algebra W. In order to clarify the status of this continuity requirement, we note that, if no restriction at all was placed on the state ψ, then its n-point functions of the free field would be merely linear functionals on the space of testfunctions, but would not even have to be distributions, which is always assumed even in such general frameworks as the Wightman-axioms. On the other hand, the continuity requirement that we shall impose on the states under consideration will automatically imply that these n-point functions are at least distributions. In addition, since any element of W can be obtained as a limit of elements in the subalgebra A spanned by finite products of (smeared) free fields, a continuous state on W is uniquely determined by its restriction to the subalgebra A, that is, by its n-point functions of the free field. This is in complete agreement with the philosophy behind the so-called “point-splitting” prescription for renormalizing Wick-products such as the stress energy operator—which is an element of W, but not of A—wherein one defines the action of a state on a Wick-product as the limit of the expectation values of suitable “point-split” quantities, which are elements of A. Clearly, such a prescription implicitly involves a continuity assumption about the action of states on the algebra W, which, as one can show, is a special case of the general continuity assumption considered in this paper. An equivalent way to express our result is to say that only those states on the algebra A of free fields have a continuous extension to the algebra W of observables in perturbation theory whose truncated n-point functions are smooth for n = 2 and whose two-point function is of Hadamard form. It has long been known from the theory of renormalizing the stress energy operator that there exist many states for the free field whose action cannot be extended in a reasonable (that is, continuous) way from free fields to the stress energy operator. Our result puts this observation into a much more general perspective. We note that our result does not hold in general for functionals on W which are continuous but which are not positive: One can construct continuous functionals on W whose truncated n-point functions are not smooth. For simplicity and definiteness, we will here consider only the case of a Hermitian scalar field. However, the generalization of our results to other types of fields should be possible. The organization of this paper is as follows: We first review the definition of the basic algebra of free fields, A. After that, we recall the definition of the truncated n-point functions of a state on A and of Hadamard states, thereby the free field gives rise to a state on W. But this state cannot be regarded in any reasonable sense as arising from a density matrix state in the vacuum representation.
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giving a brief summary of some notions from microlocal analysis that shall be needed later on. We then briefly review the construction of the algebra W and recall how the topology on W is defined. After that, we present our main result, Thm. III.1. The proof of that result makes up the rest of the paper. Some parts of this proof are moved to an appendix. Acknowledgements. We would like to thank R. M. Wald for helpful discussions during the early stages of this research. We are indebted to him in particular for suggesting to us that the positivity property of states should play a crucial role in the proof of our main result. We are also grateful to K. Fredenhagen for suggsting to us a substantial improvement of our original proof of Lemma IV.1. S. Hollands was supported by NFS grant PHY00-90138 to the University of Chicago.
II Preliminaries II.1 Definition of the minimal algebra A of observables for the free KleinGordon field A free classical Hermitian Klein-Gordon field on a curved spacetime is a real valued solution of the equation (✷ − ξR − m2 )ϕ = 0, (1) where ✷ = |g|−1/2 ∂µ |g|1/2 g µν ∂ν is the wave-operator in curved space, R is the curvature scalar, and where m, ξ are real parameters. It is known that this equation possesses unique advanced and retarded fundamental solutions on any (timeoriented) globally hyperbolic spacetime. These are determined by the equations (✷ − ξR − m2 )∆adv = (✷ − ξR − m2 )∆ret = δ,
(2)
and by the requirement that the support of ∆adv respectively ∆ret consists of pairs of points (x1 , x2 ) such that x2 is in the causal past respectively future of x1 . The theory of a quantized free Klein-Gordon field on globally hyperbolic spacetimes [9, 10] can be described in different ways. For our purposes, it is essential to use an algebraic approach. In this approach one starts with an abstract *-algebra, A, of quantum observables for the free field theory. Several choices for A are possible.5 We here take A to be the *-algebra generated by the identity I and the smeared field operators ϕ(f ), sometimes formally written as ϕ(x)f (x) |g|1/2 d4 x, with the following relations: • Linearity: f → ϕ(f ) ∈ A is complex linear. • Klein-Gordon: The field operators satisfy the Klein-Gordon equation in the sense that ϕ((✷ − ξR − m2 )f ) = 0. 5 We note that the choice for A used in this paper is not the same as in [9, 10], where the authors work instead with the algebra generated by exponentiated smeared field operators. Such a choice has some advantages, but would not be convenient for our purposes.
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• Hermiticity: The quantum field is Hermitian, ϕ(f )∗ = ϕ(f¯). • Commutation Relations: [ϕ(f1 ), ϕ(f2 )] = i∆(f1 , f2 ) · I,
(3)
where ∆ = ∆adv − ∆ret . We will subsequently consider more general observables than those contained in the algebra A. We therefore refer to this algebra as the “minimal algebra”.
II.2 States on A Given a state ψ on A, one defines its “n-point functions”, ψn , as the n times multilinear functionals on the space of testfunctions given by ψn (f1 , f2 , . . . , fn ) = ψ(ϕ(f1 )ϕ(f2 ) . . . ϕ(fn )).
(4)
Every state on A is uniquely determined by the collection of its n-point functions. One also defines the “truncated n-point functions”, ψnT , of a state ψ. For the first few n, these are given by ψ1T (f ) = ψ1 (f ), ψ2T (f1 , f2 ) = ψ2 (f1 , f2 ) − ψ1 (f1 )ψ1 (f2 ), ψ3T (f1 , f2 , f3 ) = ψ3 (f1 , f2 , f3 ) − ψ1 (f1 )ψ2 (f2 , f3 )− ψ1 (f2 )ψ2 (f1 , f3 ) − ψ1 (f3 )ψ2 (f1 , f2 ) + 2ψ1 (f1 )ψ1 (f2 )ψ1 (f3 ). (5) Their definition for general n is as follows. Denote by In the set of partitions P of the set {1, . . . , n} into pairwise disjoint, ordered subsets r1 , . . . , rj . If r is a set in P , then we denote its elements by r(1), . . . , r(|r|), where |r| is the number of elements in the set r. Note that by definition r(i) < r(j) if i < j. With this notation, the truncated n-point functions are implicitly defined by ψnT (fr(1) , fr(2) , . . . , fr(|r|)). (6) ψn (f1 , f2 , . . . , fn ) = P ∈In r∈P
Note that the sum always contains the term ψnT (f1 , . . . , fn ) corresponding to the trivial partition consisting only of the set {1, . . . , n}. Therefore, once the truncated n-point functions have been defined for 1, . . . , n−1, one can solve the above relation for ψnT in terms of ψn and the lower order truncated n-point functions. A state on A is called “quasi-free” if its truncated n-point functions are all zero except for n = 2. A standard example for a quasi-free state is the vacuum state in Minkowski-space, and, more generally, all states constructed from some set of “positive frequency solutions” to the Klein-Gordon equation. It is a consequence of the definition (6) that the odd n-point functions of a quasi-free state vanish, and
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that the even n-point functions can be expressed solely in terms of the two-point function. These expressions can be summarized in the formula6 1
ψ(eiϕ(f ) ) = e− 2 ψ2 (f,f ) ,
(7)
valid for any quasi-free state ψ. A state ψ on A is said to be of “Hadamard form” if its two-point function has no spacelike singularities, and if it locally can be written in the form ψ2 (x1 , x2 ) = U (x1 , x2 )σ −1 + V (x1 , x2 ) ln σ + W (x1 , x2 ).
(8)
Here, σ is the signed, squared geodesic distance between the points x1 and x2 , U and V are certain smooth functions defined in terms of the metric and the coupling parameters that can be uniquely determined by imposing the wave Klein-Gordon equation on (8). W is a smooth function that is not uniquely determined, but depends on the state in question. The -prescription for the singular terms σ −1 and ln σ is the same as for the usual vacuum two-point function in Minkowski space. Strictly speaking, the quantities U, V and W are well defined only for real analytic spacetimes, so the above definition of Hadamard states needs to be modified in spacetimes which are only smooth. For a discussion of this issue and a precise formulation of the statement that “there are no spacelike singularities”, we refer the reader to [10]. It is an immediate consequence of the definition that if ψ and ω are two Hadamard states, then the difference between the corresponding two-point functions, ψ2 − ω2 , is smooth. There exists an alternative, equivalent characterization of Hadamard states in terms of the so-called “wave front set” of its associated two-point function [11], which plays an important role in this work. In order to state what this characterization is, we first recall the concept of the wave front set of a distribution. Let u be a smooth function on Rn with compact support. Then it is known that the Fourier transform 7 of u is rapidly decaying, that is, for any N , there is a constant CN such that | u(k)| ≤ CN (1 + |k|)−N for all k ∈ Rn . (9) Let now u be a compactly supported distribution. Then it is known that the Fourier transform u is polynomially bounded in k. However, it is no longer true in general that u is rapidly decaying in all directions. The directions in k-space, for which there exists no conic neighborhood (that is, a neighborhood which is invariant under multiplication by positive scalars) such that (9) holds, are called “singular directions of u” and are denoted by Σ(u). Note that Σ(u) is by definition a conic set, that is, a set which is invariant under multiplication by positive scalars. 6 Actually, expressions like eiϕ(f ) are not elements in our algebra A, since this algebra contains by definition only finite sums of products of smeared free fields. What is meant by (7) (and other similar formulas in the sequel) is the set of equalities obtained by expanding both sides of the equation in a formal power series and by equating the corresponding terms in these series. R 1 7 Our convention for the Fourier transform is u b(k) = u(x)e+ikx dn x. n/2 (2π)
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Let now u be an arbitrary distribution on an open set X ⊂ Rn , not necessarily of compact support. Then one can define the singular directions of u near some point x by localizing u with a smooth function of compact support for which χ(x) = 0, by defining Σχ (u) = Σ(χu). If one now shrinks the support of χ to the point x, then one obtains the singular directions of u at the point x, defined as Σx (u) = Σχ (u) . (10) χ(x)=0
The wave front set, WF(u), of u is just the union of all nonzero singular directions of u, (11) WF(u) = {(x, k) ∈ X × (Rn \{0}) | k ∈ Σx (u)}. Note that it follows directly from the definition of the wave front set of a distribution u is given by a smooth density if and only if WF(u) is empty. It can be demonstrated that the wave front set transforms covariantly under a change of coordinates, that is, if φ is a smooth one–to–one map on X, then (φ(x), k) ∈ WF(u) is t equivalent to (x, [Dφ(x)]t k) ∈ WF(φ∗ u), where Dφ = ∂φ ∂x , where means the trans∗ pose of a matrix and where φ u denotes the pull-back of a distribution, defined by analogy with the pull-back of a smooth density. This makes it possible to define in an invariant way the wave front set of a distribution on a smooth manifold X. The above transformation law then shows that WF(u) is intrinsically a conic subset of the cotangent bundle T ∗ X minus its zero section. It is common to define, for every closed conic set Γ ⊂ T ∗ X \ {0}, the subspace DΓ (X) = {u ∈ D (X) | WF(u) ⊂ Γ} of the space D (X) of all distributions on X. Having introduced the wave front set of a distribution, we can now state the promised alternative characterization of Hadamard states: Namely, a state ψ is Hadamard if the wave front set of its two-point function has the following form: WF(ψ2 ) = {(x1 , k1 ; x2 , −k2 ) ∈ T ∗ (M × M )\{0} | (x1 , k1 ) ∼ (x2 , k2 ), k1 ∈ (V+ )x1 }. (12) The notation (x1 , k1 ) ∼ (x2 , k2 ) means that x1 and x2 can be joined by a nullgeodesic and that the covectors k1 and k2 are cotangent and coparallel to that null-geodesic. (V+ )x denotes the closed forward lightcone in the cotangent space at the point x, defined as the set of all future directed timelike or null covectors in the cotangent space at x. The closed backward lightcone, (V− )x , is defined similarly. For later purposes, we also set V± = ∪x∈M (V± )x . For the algebras considered in this paper, there holds the so-called GNStheorem, which says that, given an algebraic state ψ on the algebra, there is a *-representation πψ of the algebra on a Hilbert space Hψ containing vector |Ωψ , which is determined, up to equivalence, by the relation ψ(A) = Ωψ |πψ (A)|Ωψ , required to hold for all algebraic elements A. This representation is commonly called the “GNS-representation” of the state ψ. For the case of the algebras considered in this paper, the GNS-representations corresponding to different states are in general inequivalent.
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II.3 Definition of the extended algebra W In the previous subsections, we have introduced a minimal algebra, A, of observables for a free Klein-Gordon field, and we have introduced the notions of quasi-free states and of Hadamard states on this algebra. The algebra A contains the observables corresponding to the smeared n-point functions of the free field, A = ϕ(f1 )ϕ(f2 ) . . . ϕ(fn ) (and finite linear combinations thereof). If one wants to define a nonlinear quantum field theory perturbatively off the free field theory, one must consider additional observables such as (smeared) Wick-polynomials of the free field and (smeared) time-ordered products of these fields. However, none of these observables are contained in A. In order to include these additional observables, we consider, besides the minimal algebra A, an enlarged algebra of observables, W, that contains A and that also contains, among others, elements corresponding to (smeared) Wick-polynomials of free fields and time-ordered products of these fields. The construction of the algebra W was first given by [5] and was later formalized in [12] for the case of Minkowski spacetime. The straightforward generalization of [12] to curved spacetimes can be found in [6]. The construction of W initially depends on the choice of some quasi-free Hadamard state ω. One can show however [6] that different choices for ω give rise to isomorphic algebras W, so in this sense W does not depend on the specific choice for ω. The observables in the interacting field theory (defined perturbatively off the free field theory) are given in terms of the well-known formal power series expansions in the coupling constant, whose coefficients are elements of the algebra W. The infinite sums occurring in these expressions are, of course, not by themselves elements of W. These series are believed not to converge, and are at best expected to approximate the “true, nonperturbative quantities” well only up to some finite order, after which they diverge. For this reason, one is only interested, even in principle, in the calculation of the interacting observables up to some finite order in perturbation theory anyway. The latter observables are elements of our algebra W, and we therefore take the view that W should be regarded as the algebra of observables which are of interest in perturbative quantum field theory. For the convenience of the reader, we now recall the basic steps in the definition of W. Let ω be a quasi-free Hadamard state on the minimal A, which we shall keep fixed for the rest of this work. The minimal algebra A contains the normal ordered smeared n-point functions of the free field, defined as8 : ϕ⊗n (⊗i fi ) :ω ≡ : ϕ(f1 )ϕ(f2 ) . . . ϕ(fn ) :ω
∂n = n G( ti fi ) i ∂t1 ∂t2 . . . ∂tn i
, (13) t1 =t2 =···=0
8 Actually, the fields ϕ(f ) appearing in the expression below should be understood as the representers of these algebraic elements in the GNS-representation of the state ω.
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where
1
G(f ) = e 2 ω2 (f,f ) eiϕ(f ) .
643
(14)
Explicitly, : ϕ(f ) :ω
=
ϕ(f ),
: ϕ(f1 )ϕ(f2 ) :ω : ϕ(f1 )ϕ(f2 )ϕ(f3 ) :ω
= =
ϕ(f1 )ϕ(f2 ) − ω2 (f1 , f2 ) · I, ϕ(f1 )ϕ(f2 )ϕ(f3 ) − ω2 (f1 , f2 )ϕ(f3 ) − ω2 (f1 , f3 )ϕ(f2 ) − ω2 (f2 , f3 )ϕ(f1 )
(15)
for the first few values of n. If t is a smooth testfunction on M n , we also define the elements A = : ϕ⊗n (t) :ω = : ϕ(x1 )ϕ(x2 ) . . . ϕ(xn ) :ω t(x1 , x2 , . . . , xn ) |g(xi )|1/2 d4 xi Mn
(16)
i
in the minimal algebra A. In order to obtain an algebra which is large enough to contain the observables of interest in perturbative quantum field theory, one would like to smear the fields : ϕ⊗n :ω not only with smooth testfunctions t, but also in addition with certain compactly supported testdistributions. Now, smearing the operator-valued distributions : ϕ⊗n :ω with a distribution involves taking the pointwise product of two distributions. As it is well-known, the pointwise product of two distributions is in general meaningless. While it is therefore impossible to smear the : ϕ⊗n :ω with an arbitrary compactly supported distribution t, it turns out to be possible to smear it with distributions t contained in a subclass En of the class of all compactly supported distributions (here the Hadamard property of ω enters). This subclass is most conveniently described in terms of the wave front set, En = {symmetric, compactly supported distributions t on M n with WF(t) ⊂ T ∗ M n \(V+n ∪ V−n )}. (17) Definition II.1 W is the *-algebra generated by the elements A of the form (16) with t ∈ En . By construction, the extended algebra W contains the minimal algebra A, but it also contains additional elements such as for example normal ordered Wickpowers of the field at the same spacetime point. These are defined as follows: Let (18) t(x1 , x2 , . . . , xn ) = f (x1 )δ(x1 , x2 , . . . , xn ), where f is a compactly supported testfunction and where δ is the covariant deltafunction on the product manifold M n . Then one can show that t ∈ En . The
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algebraic element : ϕ⊗n (t) :ω is just the smeared n-th normal ordered Wick power of the free field at the same spacetime point, : ϕn (f ) :ω = : ϕ⊗n (t) :ω ,
(19)
as previously defined in [4]. More generally, it can be shown [5], that W also contains time-ordered products of normal ordered Wick-products. Using Wick’s theorem, one can show that the product of two elements in W of the form (16) can again be written as a finite sum of elements of this form. This shows in particular that any element in W arises as a finite sum of elements of the form (16) with tn ∈ En plus a multiple of the identity operator. For later purposes, we would like to have a suitable notion of the continuity of states on the algebras W and A. In order to define such a notion, we must first equip W (and therefore also A) with a topology. In other words, we must explain what we mean by the statement that a sequence {Aκ } of elements in W converges to an element A. Such a topology has been defined in [6], we here briefly indicate the main idea. One first defines a notion of convergence of a sequence {tκ } in the spaces of distributions En defined above in (17). Namely, such a sequence is said to converge to a distribution t if (a) the support of tκ is contained in some compact set K for all κ, (b) tκ → t weakly in the sense of distributions, (c) there is a closed conic set Γ ⊂ T ∗ M n \(V+n ∪ V−n ) such that WF(tκ ) ⊂ Γ for all κ, (d) for any properly supported pseudo differential operator P with µsupp(P ) ∩ Γ = ∅, we have that P tκ → P t in the sense of compactly supported smooth functions. Remark. It is common to say that a sequence of distribtuions {tκ } satisfying (b) through (d) “converges to t in the sense of DΓ ”. For an explanation of the notion of a pseudo differential operator and the related technical terms appearing in item (d), we refer the reader to [7]. It can be shown that t is again an element in En , so these spaces are complete with respect to the above topology. Having defined a notion of sequential convergence within the spaces En , we now define a notion of sequential convergence in the algebra W as follows. Let {Aκ } be a sequence of generators in W, defined by distributions tκ ∈ En as in (16). Then we say that the sequence {Aκ } converges to an element A of the form (16), if tκ → t in En . The so defined notion of covergence for the generators of W generalizes to arbitrary sequences in W, because every element of this algebra can be written as a finite linear combination of the generators.
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A state ψ on W is said to be continuous, if ψ(Aκ ) → ψ(A) whenever the sequence {Aκ } converges to A. We note that, since the space of smooth testfunctions on M n is dense in the space En , the algebra A is dense in W in the above topology. Therefore we have the important result that continuous states on W are completely determined by their restrictions to A. If we consider sequences {tκ } satifying (a) through (d) with Γ = ∅, then we get that the n-point functions of a continuous state must be continuous in the Laurent-Schwartz topology on the space of smooth testfunctions. Thus, we find in particular that the n-point functions of a continuous state are distributions.
III The state space of W The aim of this section is to characterize the space of continuous states on the algebra W. We first note that, since W ⊃ A, every continuous state ψ on W gives rise, by restriction, to a state on the minimal algebra A whose n-point functions are distributions. However, the opposite is not true, namely it is not true that every such state on A can be extended to a continuous state on W. This may be seen for example by considering the smeared normal ordered Wick-power : ϕ2 (f ) :ω , which is an element of W, but which is not an element of A. Now if ψ is a state on A with distributional n-point functions, then its action—provided it can be defined—on this Wick-power must be given by the limit ψ(: ϕ2 (f ) :ω ) = lim (ψ2 − ω2 )(x1 , x2 )f (x1 )δκ (x1 , x2 )|g(x1 )|1/2 |g(x2 )|1/2 d4 x1 d4 x2 , (20) κ→∞
M2
where {δκ } is a suitable sequence of smooth functions tending to the deltadistribution in the product manifold M × M . (Note that this prescription is just a reformulation of the usual “point-splitting” method, as explained for example in [9].) However, this limit will only exist and be independent of the particular choice of sequence {δκ } if the distribution ψ2 − ω2 is at least continuous at the diagonal {(x, x) | x ∈ M } in the product manifold M × M . There are many states on A which do not have this property and which therefore do not extend to W. The precise characterization of the space of continuous states on W is as follows: Theorem III.1 (i) Let ψ be a continuous state on W. Then the two-point function of the free field must be of Hadamard form and the truncated n-point functions of the free field must be smooth for n = 2. (ii) Conversely, if ψ is a state on A whose two-point function is of Hadamard form and whose truncated n-point functions are smooth for all n = 2, then ψ extends to a (necessarily unique) continuous state on W.
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Remarks. 1. The results of [5, 6] imply that any quasi-free Hadamard state on A can be extended to a continuous state on W. Clearly, this is a special case of item (ii) of the above theorem, since quasi-free states by definition have vanishing truncated n-point functions for n = 2. 2. B. S. Kay has shown (unpublished manuscript) that the N -particle states with smooth mode functions in the GNS-representation of any quasi-free Hadamard state are Hadamard and have smooth truncated n-point functions for n = 2. By (ii) of our theorem, these states therefore extend to continuous states on W. 3. On Robertson-Walker spacetimes, there exists the notion of “adiabatic vacuum states” on A, introduced by Parker and defined in a mathematically rigorous way by [13]. The two-point function of such a state differs from that of a Hadamard state typically by a term which is a number of times differentiable, but which is not smooth [14]. By our theorem, adiabatic states therefore do not possess an extension to continuous states on W. The same remark applies to the class of states recently introduced by Junker and Schroe [15]. 4. In [4], the authors introduce a “microlocal spectrum condition”, which generalizes to curved spacetimes the usual spectrum condition imposed on the n-point function of a field theory in the context of the Wightman-axioms. The content of this condition is to require that wave front set of the n-point functions of an admissible state should have a specific form. The microlocal spectrum condition is known to hold for the n-point functions of the free field in any quasi-free Hadamard state. It is an easy consequence of our result that the microlocal spectrum condition holds in fact for the n-point functions of any continuous state on W. 5. We shall from now on only deal with continuous states. Therefore, for simplicity, whenever we speak of “states”, we shall mean “continuous states”. Proof. We begin with the proof of (i). Let thus ψ be a continuous state on W. We need to show that the two-point function, ψ2 , is of Hadamard form, and that the truncated n-point functions, ψnT , are all smooth, except for n = 2. Let Ψn (f1 , f2 , . . . , fn ) ≡
ψ (: ϕ(f1 )ϕ(f2 ) . . . ϕ(fn ) :ω ) .
(21)
Then we have Lemma III.1 Let ψ be a state on A. Then the following statements are equivalent: (i) ψ2 is Hadamard and ψnT are smooth for n = 2. (ii) The distributions Ψn are smooth for all n.
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Proof. The proof of the lemma is based on the following combinatorical formula, which we shall prove in the appendix: n ∂ (ψnT − ωnT )(f1 , f2 , . . . , fn ) = n ln ψ G( ti f i ) , (22) i ∂t1 ∂t2 . . . ∂tn i t1 =t2 =···=0
where G(f ) is defined in (14). If one carries out the differentiations in formula (22) and uses that the functional f → ψ(G(f )) is the generating functional for the hierarchy of
distributions {Ψ1 , Ψ2 , . . . , Ψn , . . . }, as well as the standard relation ln(1 + x) = k≥1 (−1)k+1 xk /k, then one obtains the formula (ψnT −ωnT )(f1 , f2 , . . . , fn ) =
(−1)|P |−1 (|P |−1)!
P ∈In
Ψ|r| (fr(1) , fr(2) , . . . , fr(|r|)).
r∈P
(23) Thus, if Ψn is smooth for all n, then so is ψnT − ωnT . For n = 1 this means that ψ1 is smooth. For n = 2 this means that ψ2 − ω2 − ψ1 ⊗ ψ1 is smooth, and hence that ψ2 is Hadamard. For n ≥ 3 this shows that ψnT is smooth, since ωnT = 0 for all n ≥ 3. We have thus shown the implication (i) =⇒ (ii) of the Lemma. The implication (ii) =⇒ (i) can be shown similarly by solving (23) for Ψn in terms of ψkT − ωkT with k ≤ n. It thus remains to be shown that Ψn is smooth for all n. We begin by showing that the wave front set of Ψn is not arbitrary. Lemma III.2 Let ψ be a continuous state on W. Then necessarily WF (Ψn ) ⊂ V+n ∪ V−n
for all n.
(24)
Proof. Given in the Appendix. In order to show that the wave front set of the distributions Ψn is in fact empty, we proceed by an induction in n. For n = 0 there is nothing to prove, since Ψ0 = 1, which is clearly smooth. Let us therefore assume that Ψk is smooth for all k ≤ n − 1. We need to prove that also Ψn is smooth. For this, it is necessary to gain some information about the Fourier transform, χ n Ψn (l1 , . . . , ln ), in directions such that either all li are in the future lightcone at some points xi or all li are in the past light cone of some points xi , where χn is a smooth bump function whose support is localized around the point (x1 , . . . , xn ) in the product manifold M n . We prepare the ground with the following three lemmas. Lemma III.3 Let ψ be a state on A. Then there holds 2 |ψn (f1 , . . . , fn )| ≤ ψ2n f1 , . . . , fn , f¯n , . . . , f¯1 , for all n and all testfunctions.
(25)
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By the the Cauchy-Schwartz inequality |ψ(A)| ≤ ψ(AA∗ ) 2
(26)
for all A ∈ A. The statement of the lemma then follows by setting A = ϕ (f1 ) · · · ϕ (fn ). For the next lemma, we introduce the following notation. We denote by P a partition of the set {1, . . . , n} into disjoint ordered pairs {(i1 , j1 ), . . . (i|P | , j|P | )}, meaning that i < j for all (i, j) ∈ P . The number of pairs in the partition P is denoted by |P |. The set of all such partitions for a given n is denoted by Pn . If 1 ≤ k ≤ n, then we write k ∈ P if the partition P contains a pair (i, j) such that either k = i or k = j. Lemma III.4 Let ψ be a state on A. Then there holds (−1)|P | Ψn−2|P | (⊗k∈P ω2 (fi , fj ) ψn (f1 , f2 , . . . , fn ) = / fk ) P ∈Pn
(27)
(i,j)∈P
for all testfunctions f1 , f2 , . . . , fn . Proof. Recall that G(f ), defined in (14), is the generating functional for the Wick products : ϕ(x1 )ϕ(x2 ) . . . ϕ(xn ) :ω . Therefore ψ(G(f )) is the generating functional for the distributions Ψn . By repeatedly using the identity i
eiϕ(f1 ) eiϕ(f2 ) = eiϕ(f1 +f2 ) e− 2 ∆(f1 ,f2 ) , it is straightforward to calculate that
ψ eit1 ϕ(f1 ) . . . eitn ϕ(fn ) 1 = exp ti tj ω2 (fi , fj ) + t2i ω2 (fi , fi ) ψ G ti f i . 2 i<j i i
(28)
(29)
Applying (−i)n ∂ n /∂t1 . . . ∂tn to both sides of this equation and setting t1 , . . . , tn to zero then yields the formula claimed in the lemma. Lemma III.5 Let ψ be a continuous state on W, and let n ≥ 1. Then WF(ψ2n ) does not contain any elements of the form (x1 , k1 , . . . , xn , kn , xn , −kn , . . . , x1 , −k1 )
with ki ∈ (V− )xi for all i.
(30)
Proof. As a preparation, let us start by introducing some notation. Let (i1 , . . . , ir ) be tuple of natural numbers with 1 ≤ i1 < i2 < . . . ir ≤ 2n. For each such a tuple, we define a map φ(i1 ,i2 ,...,ir ) : M 2n → M r by φ(i1 ,i2 ,...,ir ) (x1 , x2 , . . . , x2n ) ≡ (xi1 , xi2 , . . . , xir ).
(31)
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With this notation, Eq. (27) can be rewritten as ψ2n = (−1)|P | φ∗(k1 ,...,k2n−2|P | ) Ψn−2|P | · φ∗(i,j) ω2 , P ∈P2n
(32)
(i,j)∈P
where {k1 , . . . , k2n−2|P | } is the set of numbers in {1, . . . , 2n} which are not contained in the partition P , and where the pull-back of a distribution is defined by analogy with the pull-back of a smooth density. Note that the distributions φ∗(i,j) ω2 etc. are by definition distributions on M 2n , so the products in formula (32) denote the pointwise product of distributions on M 2n . Using now formulas [7, I, Thms. 8.2.10 and 8.2.4] for the wave front of products and pull-backs of distributions, we get the estimate WF(ψ2n ) ⊂
P ∈P2n
WF φ∗(k1 ,...,k2n−2|P | ) Ψn−2|P | ·
φ∗(i,j) ω2
(i,j)∈P
φ∗(k ,...,k φ∗(i,j) WF (ω2 ) ∪ {0} WF Ψn−2|P | ∪ {0} + ⊂ 1 2n−2|P | ) P ∈P2n
⊂
P ∈P2n
(i,j)∈P
2n−2|P | 2n−2|P | φ∗(k ,...,k (V+ ∪ V− )+ 1 2n−2|P | )
φ∗(i,j) (V+ × V− ) ,
(33)
(i,j)∈P
where we have used that WF(ω2 ) ⊂ V+ × V− since ω is Hadamard, and that WF(Ψk ) ⊂ V+k ∪ V−k , by Lem. III.2. Hence, in order to prove the lemma, it is sufficient to demonstrate that if a vector (x, l) ∈ T ∗ M 2n of the form (x1 , l1 , . . . , x2n , l2n ) ≡ (x1 , k1 , . . . , xn , kn , xn , −kn , . . . , x1 , −k1 ) with ki ∈ (V− )xi for all i
(34)
is in the set 2n−2|P |
φ∗(k1 ,...,k2n−2|P | ) (V+
2n−2|P |
∪ V−
)+
φ∗(i,j) (V+ × V− )
(35)
(i,j)∈P
for some partition P , then ki = 0 for all i. So let (x, l) be in the set (35) for some P . This implies that (a) li ∈ V+ , lj ∈ V− for all (i, j) ∈ P . / P or li ∈ V− for all i ∈ / P. (b) Either li ∈ V+ for all i ∈ Now property (a), together with the specific form of (x, l) given by (34), implies that li = lj = 0 whenever (i, j) ∈ P . Combining this with property (b), we see
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that the li must either be all in V− or all in V+ . Using again the specific form of (x, l), we conclude that this is only possible when all li = 0, implying that all ki = 0. We have now gathered enough information to show that Ψn is smooth for all n. By the induction hypothesis, we know that Ψk is smooth for all k ≤ n − 1. We want to use this to obtain an estimate for the Fourier transform of χn Ψn , where χn is a smooth function with compact support that will be specified momentarily. From Lemmas III.4 and III.3 we get the inequality ψ2n (f1 , . . . , fn , f¯n , . . . , f¯1 ) ≥ ¯ (−1)|P |+|P | Ψn−2|P | (⊗k∈P / fk )Ψn−2|P | (⊗k ∈P / fk ) P,P ∈Pn
ω2 (f¯j , f¯i )
(i,j)∈P
ω2 (fi , fj ). (36)
(i ,j )∈P
We now specialize the above inequality to testfunctions fj of the form fj (x) =
1 ηj (x)eilj x . (2π)2
(37)
Here, ηj are real-valued smooth bump functions whose support is contained in some chart, lj are vectors in R4 and the expression lj x denotes the scalar product in R4 between lj and the coordinate components of x in the above chart. With this choice for fj , the above inequality can be rewritten as 2 |χ n Ψn (l1 , . . . , ln )| ≤ χ2n ψ2n (l1 , . . . , ln , −ln , . . . , −l1 )− (−1)|P |+|P | χn−2|P | Ψn−2|P | −lk1 , . . . , −lkn−2|P | P,P ∈Pn ,P,P =∅
| Ψn−2|P | lk1 , . . . , lkn−2|P × χn−2|P | × χ 2 ω2 (−lj , −li ) (i,j)∈P
χ 2 ω2 (li , lj ), (38)
(i ,j )∈P
where {k1 , . . . , k2n−2|P | } is the set of numbers in {1, . . . , n} which are not con tained in the partition P and where {k1 , . . . , k2n−2|P | } is the set of numbers in {1, . . . , n} which are not contained in the partition P . The smooth functions χk denote suitable tensor products of k factors of the functions ηi . For example, in the expression χ 2 ω2 (li , lj ), the function χ2 should be taken to be χ2 = ηi ⊗ηj ; in the expression χ 2n ψ2n (l1 , . . . , ln , −ln , . . . , −l1 ), the function χ2n should be taken to be χ2n = η1 ⊗ . . . ηn ⊗ ηn ⊗ . . . η1 , etc. We would now like to argue that the right side of inequality (38) is rapidly decaying in directions for which all li are in some conic neighborhood of (V− )xi ,
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with at least one li = 0, provided the support of the functions ηi is localized sharply enough around points xi . For this, we first look at the terms in the sum on the right side of inequality (38). Each term in this sum contains at least one factor of either the form χ 2 ω2 (li , lj ) or χ 2 ω2 (−lj , −li ), where (i, j) ∈ P or (i , j ) ∈ P . Provided that not all the covectors li , lj , li , lj occurring in these factors are zero, these factors give us rapid decay of the corresponding term in the sum. This is because, by the Hadamard property of ω2 , −N |χ 2 ω2 (k1 , k2 )| ≤ CN (1 + |k1 | + |k2 |)
(39)
for all N and all directions (k1 , k2 ) in some conic neighborhood of (V+ )x1 ×(V+ )x2 ∪ (V− )x1 × (V− )x2 , provided χ2 is localized sufficiently sharply around (x1 , x2 ). If all the li , lj , li , lj occurring in the factors χ 2 ω2 (li , lj ) or χ 2 ω2 (−lj , −li ) are zero, then at least one of the covectors lk with k ∈ / P and at least one of the covectors lk with k ∈ / P must be nonzero, since otherwise all the li would be zero. Let us first assume that P is not the empty set. Then we have χn−2|P | Ψn−2|P | −lk1 , −lk2 , . . . , −lkn−2|P | ≤ CN
1+
−N |lk |
(40)
k∈P /
for all N and suitable constants CN , since the distributions Ψn−2|P | are smooth by the inductive assumption. If P = ∅, then P is not the empty set, and we get | Ψn−2|P | . In summary, we have an estimate of the above form for the term χn−2|P shown that each term in the sum on the right side of (38) contains at least one factor which is rapidly decaying. Therefore the whole sum is rapidly decaying in directions such that either all li ∈ (V+ )xi or all li ∈ (V− )xi and not all li = 0, provided the functions ηi are localized sufficiently sharply around the points xi . By Lemma III.5, the first term on the right side of (38) is rapidly decaying in directions for which all li are in a conic neighborhood of (V− )xi , provided the functions ηi are localized sharply enough around xi . Hence, we have altogether found that −N |li | (41) 1+ |χ n Ψn (l1 , l2 , . . . , ln )| ≤ CN i
in directions for which all li are in a neighborhood of (V− )xi , provided the functions ηi are localized sharply enough around xi . Now, since (: ϕ⊗n (t) :ω )∗ = : ϕ⊗n (t¯) :ω ,
(42)
for all testfunctions t, the distributions Ψn are real, in the sense that Ψn (t) = Ψn (t¯) for all testfunctions t. Hence, |χ n Ψn (l1 , . . . , ln )| = |χn Ψn (−l1 , . . . , −ln )|, and the inequality (41) must therefore also hold if all the li in that inequality are replaced by −li , that is, (41) must also hold in directions for which all li are in some neighborhood of the cones (V+ )xi . Therefore, since we already know that
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WF(Ψn ) ⊂ V+n ∪ V−n , we get from this that the distribution χn Ψn has in fact no singular directions at all, provided the supports of ηi are sufficiently sharply localized around xi . But this implies that WF(Ψn ) = ∅, as we wanted to show.
We next prove (ii). Let us thus assume ψ is a state on A for which ψ2 is Hadamard and for which ψnT are smooth for all n = 2. By Lemma III.1, this implies that the distributions Ψn are smooth. We are thus allowed to define an action of ψ on elements of W by the formula ψ(: ϕ⊗n (t) :ω ) ≡ Ψn (t),
(43)
for all t ∈ En and all n. It is easily checked that this formula defines a linear, normalized and continuous functional on W which extends the action of ψ on A. Moreover, this functional is also positive, since it is continuous and positive on A, which is a dense subspace of W.
IV Appendix IV.1 Proof of formula (22) Let {h1 , h2 , . . . , hn , . . . } denote some hierarchy of symmetric distributions and let in hn (f, f, . . . , f ) (44) H(f ) = 1 + n! n≥1
be the corresponding generating functional. Then the “linked cluster theorem” (see e.g. [16, pp 125]) states that the corresponding truncated distributions are given by ∂n T ln H( ti fi ) . (45) hn (f1 , f2 , . . . , fn ) = n i ∂t1 ∂t2 . . . ∂tn i
t1 =t2 =···=0
We would like to apply this result to the hierarchies {ψ1 , ψ2 , . . . , ψn , . . . } and {ω1 , ω2 , . . . , ωn , . . . } of the n-point functions of the states ψ and ω. However, these are not symmetric and therefore the linked cluster theorem is not directly applicable. Instead, we first apply the linked cluster theorem to the hierarchies of symmetrized n-point functions, {ψ1S , ψ2S , . . . , ψnS , . . . } and {ω1S , ω2S , . . . , ωnS , . . . }, where the superscript “S” stands for symmetrization. This gives us [(ψnS )T − (ωnS )T ](f1 , . . . , fn )
∂n iϕ( i ti fi ) iϕ( i ti fi ) ln ψ e − ln ω e = n i ∂t1 . . . ∂tn t1 =t2 =···=0 n ∂ = n ln ψ G( ti f i ) , (46) i ∂t1 . . . ∂tn i
P
P
t1 =t2 =···=0
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where we have used the definition of G(f ), Eq. (14), as well as the relation 1 ω(eiϕ(f ) ) = e− 2 ω2 (f,f ) , which holds because ω is quasi-free. The desired relation (22) then follows if we can show that (ψnS )T − (ωnS )T = ψnT − ωnT
(47)
for all n. The demonstration of (47) makes up the rest of this subsection. Relation (47) can be checked immediately for n = 1 and n = 2. For n ≥ 3 it reduces to (48) (ψnS )T = ψnT since ωnT = (ωnS )T = 0 for n ≥ 3. In order to see (48), we first note that the truncated n-point functions of any state on A are symmetric for n ≥ 3, ψnT = (ψnT )S , as one can show by a straightforward inductive argument using the commutation relation (3) for the free field. Eq. (48) thus follows from the fact that for any hierarchy {h1 , h2 , . . . , hn , . . . } (not necessarily symmetric) there holds (hTn )S = (hSn )T
(49)
for all n. To see this, we argue as follows. Let P = {r1 , . . . , rk }, and let Q (rj ) denote the set of all permutations of rj ≡ (rj (1) , . . . , rj (nj )) where nj = |rj |. Let Q (P ) = {σ1 · · · σk | σj ∈ Q (rj ) for j = 1, . . . , k} . Let I (n1 , . . . , nk ) denote the subset of partitions which has k members {r1 , . . . , rk } such that for each i = 1, . . . , k, |rj | = nj for all j. For any fixed P ∈ I (n1 , . . . , nk ), we then get S 1 h|r| (x1 , . . . , xn ) = hnj xσ(1) , . . . , xσ(n) n! r∈P
P ∈I(n1 ,...,nk ) σ∈Q(P )
=
n1 ! · · · nk ! n!
rj ∈P
hSnj xrj (1) , . . . , xrj (nj ) .
P ∈I(n1 ,...,nk ) rj ∈P
(Note that the left hand side is independent of P ∈ I (n1 , . . . , nk ).) From this we conclude that S S h|r| (x1 , . . . , xn ) = h|r| (x1 , . . . , xn ) P ∈In r∈P
{n1 ,...,nk } P ∈I(n1 ,...,nk )
=
r∈P
hSnj xrj (1) , . . . , xrj (nj )
{n1 ,...,nk } P ∈I(n1 ,...,nk ) rj ∈P
=
P ∈In r∈P
hSnj xrj (1) , . . . , xrj (nj )
(50)
where the sum {n1 ,...,nk } is over all possible set of positive integers such that n1 + . . . + nk = n.
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We now use this formula to prove Eq. (49). Suppose that this equation is true for 1, 2, . . . , n. We now show that it must also be true for n + 1. By the induction hypothesis, for any P ∈ In+1 which is not P0 = {(1, 2, . . . , n + 1)} and any r ∈ P , we have that (hS|r| )T xr(1) , . . . , xr(|r|) = (hT|r| )S xr(1) , . . . , xr(|r|) . (51) Hence, (hSn+1 )T (x1 , . . . , xn )
= hSn+1 (x1 , . . . , xn+1 ) −
P ∈In+1 ,P =P0 r∈P
=
hSn+1
(x1 , . . . , xn+1 ) −
(hS|r| )T xr(1) , . . . , xr(|r|) (hT|r| )S xr(1) , . . . , xr(|r|)
P ∈In+1 ,P =P0 r∈P
= hSn+1 (x1 , . . . , xn+1 ) −
S hT|r| (x1 , . . . , xn+1 )
P ∈In+1 ,P =P0 r∈P
=
(hTn+1 )S
(x1 , . . . , xn+1 ) ,
where in the first line we have used the definition of the truncated n-point functions, in the second line we have used the induction hypothesis, and where in the third line we have applied formula (50), applied to the hierarchy {hT1 , hT2 , . . . , hTn , . . . }. This completes the induction.
IV.2 Proof of Lemma III.2 Let Γ be a closed conic subset of T ∗ M n \(V+n ∪ V−+ ) and let {tκ } be a sequence of smooth functions on M n , whose support is contained in some compact subset of M n for all κ, and which converges to some t in the sense of DΓ . Then, the sequences Aκ = : ϕ⊗n (tκ ) :ω by definition converges in W. Therefore, since the state ψ is assumed to be continuous on W, ψ(Aκ ) = Ψn (tκ ) is a convergent sequence for κ → ∞. We need to show that this implies that WF(Ψn ) ⊂ V+n ∪ V−n . This immediately follows from the following general result. Lemma IV.1 Let u ∈ D (Rn ) and let Γ be a closed conic set in Rn × (Rn \{0}). Assume that u has the following property. For every sequence of smooth functions {fκ } such that fκ → f in DΓ (Rn ) and such that supp(fκ ) ⊂ K, with K a compact subset of Rn , we have that {u(fκ )} is a convergent sequence. Then {0} ∈ / WF(u)+Γ. Proof. Let A be a properly supported pseudo differential operator and with µsupp(A) ⊂ Γ. Then, if {fκ } is any sequence of distributions on Rn converging weakly to some f ∈ D (Rn ) in the sense of distributions, it follows that Afκ → Af in the sense of DΓ (Rn ) and that supp(Afκ ) ⊂ K , where K is some compact subset
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of Rn . Therefore, the sequence u(Afκ ) = At u(fκ ) is convergent, by the assumption of the lemma, where At is the formal adjoint of the pseudo differential operator A. We claim that it follows from this that At u is in fact smooth. Assuming for the moment that this has been shown to be true, we get from the characterization [7] of the wave front set that char(B) ⊂ char(At ) = −char(A) WF(u) = Bu∈C ∞
µsupp(A)⊂Γ
µsupp(A)⊂Γ
= T ∗ Rn \(−Γ), because the set of properly supported pseudo differential operators A with µ supp(A) ⊂ Γ contains elements whose characteristic, char(A), is contained in an arbitrarily small conic neighborhood of T ∗ Rn \Γ. Since the set T ∗ Rn \(−Γ) contains no element (x, k) such that k + k = 0 for some (x, k ) ∈ Γ, this then proves the lemma. It thus remains to be shown that the compactly supported distribution v = At u is smooth. This would immediately follow if the Fourier transform, v(k) was rapidly decaying. Let us assume on the contrary that v (k) is not rapidly decaying. Then there is an integer N > 0 and a sequence {ξκ } in Rn such that |ξκ | → ∞ and v (ξκ )| ≥ κ for κ = 1, 2, . . . (52) (1 + |ξκ |)N | Define fκ by fκ (x) =
1
N
n/2
(2π)
(1 + |ξκ |) ρ(x)eiξκ x ,
κ = 1, 2, . . . ,
where ρ ∈ C0∞ (Rn ) is a positive function, identically 1 in supp (v). Then, for any φ ∈ D(Rn ), since ρφ is rapidly decaying, N ) |fκ (φ)| = (1 + |ξκ |) ρφ(ξ κ → 0 as κ → ∞. This implies that fκ → 0 weakly in D (Rn ), and therefore v(fκ ) → 0. Thus, N
(1 + |ξκ |) | v (ξκ )| = |v(fκ )| → 0, which is in contradiction with (52).
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References [1] See for example the lectures given at the 1975 Erice summer school, collected in: “Renormalization Theory,” G. Velo and A. S. Wightman, eds., NATO ASI Series C 23, Reidel, Dodrecht, 1976. [2] T. S. Bunch, BPHZ renormalization of λΦ4 field theory in curved space-times, Ann. of Phys. 131, 118 (1981). [3] T. S. Bunch, P. Panangaden and L. Parker, On renormalization of λΦ4 in curved space-time I, J. Phys. A: Math. Gen. 13, 901–918 (1980), On renormalization of λΦ4 in curved space-time II, J. Phys. A: Math. Gen. 13, 919–932 (1980). [4] R. Brunetti, K. Fredenhagen and M. K¨ ohler, The microlocal spectrum condition and Wick polynomials on curved spacetimes, Commun. Math. Phys. 180, 633–652 (1996). [5] R. Brunetti and K. Fredenhagen, Microlocal Analysis and Interacting Quantum Field Theories: Renormalization on physical backgrounds, Commun. Math. Phys. 208, 623–661 (2000). [6] S. Hollands and R. M. Wald, Local Wick polynomials and time ordered products of quantum fields in curved spacetime, Commun. Math. Phys. 223, 289– 308 (2001) [gr-qc/0103074]. [7] L. H¨ ormander, The Analysis of Linear Partial Differential Operators I–IV, Springer-Verlag, Berlin 1985. [8] S. Hollands and R. M. Wald, Existence of local covariant time ordered products of quantum fields in curved spacetime, [gr-qc/0111108]. [9] R. M. Wald, Quantum Field Theory in Curved Spacetime and Black Hole Thermodynamics,” The University of Chicago Press, Chicago 1994. [10] B. S. Kay and R. M. Wald, Theorems on the uniqueness and thermal properties of stationary, nonsingular, quasifree states on spacetimes with a bifurcate Killing horizon, Phys. Rep. 207, 49 (1991). [11] M. J. Radzikowski, Micro-Local Approach to the Hadamard condition in QFT on Curved Space-Time, Commun. Math. Phys. 179, 529–553 (1996). [12] M. D¨ utsch and K. Fredenhagen, Algebraic quantum field theory, perturbation theory, and the loop expansion, [hep-th/0001129]; Perturbative algebraic field theory, and deformation quantization, [hep-th/0101079]. [13] C. L¨ uders and J. Roberts, Local quasiequivalence and adiabatic vacuum states, Commun. Math. Phys. 134, 29 (1990).
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[14] S. Hollands, The Hadamard condition for Dirac fields and adiabatic states on Robertson-Walker spacetimes, Commun. Math. Phys. 216, 635 (2001) [grqc/9906076]. [15] W. Junker and E. Schroe, Adiabatic vacuum states on general spacetime manifolds: Definition, construction and physical properties, [math-ph/0109010]. [16] Handbuch der Physik, Band XII, “Thermodynamik der Gase,” SpringerVerlag 1958 Stefan Hollands Department of Physics Enrico Fermi Institute University of Chicago 5640 Ellis Ave. Chicago, IL 60367 U.S.A. email: [email protected] Weihua Ruan Department of Mathematics Computer Science and Statistics Purdue University Calumet Hammond, IN 46323 U.S.A. email: [email protected] Communicated by Klaus Fredenhagen submitted 24/08/01, accepted 15/04/02
To access this journal online: http://www.birkhauser.ch
Ann. Henri Poincar´e 3 (2002) 659 – 671 c Birkh¨ auser Verlag, Basel, 2002 1424-0637/02/040659-13
Annales Henri Poincar´ e
Algebraic Structure of n-Body Systems E. Sorace
Abstract. A general method to easily build global and relative operators for any number n of elementary systems if they are defined for 2 is presented. It is based on properties of the morphisms valued in the tensor products of algebras of the kinematics and it allows also the generalization to any n of relations demonstrated for two. The coalgebra structures play a peculiar role in the explicit constructions. Three examples are presented concerning the Galilei, Poincar´e and deformed Galilei algebras.
1 Introduction It has been recently found that the renormalization procedure in Quantum Field Theory is intrinsically determined by an Hopf algebra whose essential constituent has been soon recognized to be the set of the parameters of the classical group of Virasoro with their law of composition [1]. The presence and the utility of basic algebraic concepts even in such an elementary problem as the search of the so called “relative variables” in classical and quantum mechanics will be illustrated here. In this note indeed we introduce a method, whose use is based on the coalgebraic structure of canonical commutation relations, which allows for the explicit construction of the collective, i.e. global and relative, canonical operators for any number n of elementary systems in a given kinematics once one has been able to operate this transformation for n = 2. It is also shown that there are classes of relations between single system operators and the collective ones that once they hold for two of them, then they are straightforward extended to any n. This is done by using the same algorithm which generates the transformation of the operators. The construction is a priori possible in any situation in which operators are constructed in terms of single algebra generators and the separation of the “global operators” is necessary. The presentation is self explanatory, very euristic and constructive. We tacitly suppose the existence of any object necessary for the results. The examples are thus essential, not only to illustrate the physical utility of the method but also to show that it is rather flexible and not mathematically void. In section 1 we give the general definitions and results. In sections 2, 3 we present examples from usual Galilei and Poincar´e kinematics while useful applications are devised for a quantum algebra too in 4. We use always 1d algebras, thus avoiding the rotations whose consideration is not essential in exemplifying the method. In 5 there are some concluding remarks.
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2 Collective operators Let us consider an algebra A, whose elements can be represented as hermitian operators on an Hilbert space, endowed with algebra morphism Γ : A → A ⊗ A, Γa = Σ(a1 )i ⊗ (a2 )i ∀aA, Γ(ab) = ΓaΓb,
(a1 )i , (a2 )i A,
∀a, bA,
(2.1)
With such a morphism Γ one can combine the two operators of the single systems in collective ones, we call global. The set of operators acting on the same space must be completed to conserve the number of the operators of the original set of the single systems, (such sets include too the Poisson algebra of functions on a symplectic manifold). This involves the introduction of the aforementioned relative operators which complement the global ones in the set of the collective operators. We thus introduce, by assuming it exists, another homomorphic map δ on A: δ : A → A ⊗ A,
δ(ab) = δaδb, ∀a, bA,
ΓA ⊕ δA = A ⊗ 1 ⊕ 1 ⊗ A
Moreover we impose the commutation property . Γaδb − δbΓa = [Γa, δb] = 0 ∀a, bA
(2.2)
(2.3)
so that Γ and δ implement exactly a transformation we may call canonical. If the generators of A satisfy canonical relations we may call canonical the operators recovered by the transformation. As a direct consequence of the more general result below it is possible to produce all the collective canonical operators for n-body by using only Γ and δ and the right tensor multiplication ⊗1. Indeed let us given n morphisms, Γj : A → A ⊗ A, (j = 0, . . . , n − 1), not necessarily different, and a morphism δ satisfying (2.2) and (2.3), with Γ = Γj , ∀j(0, n − 1). Let us now consider the following expressions: (n−2)
δa ⊗ 1⊗ Γ
(n−1)
, (n−3)
δa ⊗ 1⊗
, (n−4)
Γ(n−1) Γ(n−2) δa ⊗ 1⊗
,
.................................... Γ(n−1) . . . . . . . . . . . . Γ(1) δa, Γ(n−1) . . . . . . . . . . . . Γ(1) Γ(0) a
(2.4)
where the following notations have been introduced: (j) (j) Γ(0) = Γσ(0) , Γ(j) = Γσ(j) ⊗ id⊗ with x⊗ ≡ x ⊗ . . . x ⊗ x, (j factors x), and σ is any permutation of i, (i = 0, . . . , n − 1).
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The following proposition then holds: “Each one of the previous expressions realizes the algebra A in the n-fold tensor (n) product A⊗ . Moreover the n realizations are all mutually commuting.” The demonstration is by recurrence. So let us suppose the proposition is true for n and apply to the left of each one of the n previous expressions 2.4 the map (n) (n+1) so that the new expressions satisfy Γ(n) . This map is a morphism A⊗ → A⊗ the same algebraic relations as the previous ones. Let us now complete the set with (n−1) the new expression δa⊗1⊗ : to end the proof it must be shown that it commutes with the n new expressions. By construction the last ones are n-multilinear sums of elements of the form (Γσ(n) a(1) ) ⊗ a(2) . . . ⊗ a(n) , a(k) A, k(1, n). The commutators to evaluate are sums of elements: (n−1) = Γσ(n) a(1) , δa ⊗ a(2) . . . ⊗ a(n) = 0. (Γσ(n) a(1) ) ⊗ a(2) . . . ⊗ a(n) , δa ⊗ 1⊗ The proposition being true for n = 2 is thus demonstrated for any n. If one deals, as it is very probable, with only one morphism Γ = Γj , ∀j(0, n− 1) then the (2.4) give a straightforward procedure to generate the n-body expressions from the two-body ones. By the way in this case the proposition holds even if we change everywhere in (2.4) Γ with δ. Let us show now a general implication of the use of algorithmic definitions in generating relations for n-body once they hold for 2. So suppose that for 2 peculiar elements a, b A (a may be equal or not to b) it exists a function R on A ⊗ A × A ⊗ A × A ⊗ A with values in A ⊗ A, which can be extended on the direct product of growing tensor powers of A, which makes explicit a relation between global and the relative operators built on a and b in the form, e.g.: (2.5) Γ(0) a = R(a1 , a2 , δb) where we have introduced the notation: zi = 1 ⊗ 1 ⊗ 1 · · · ⊗ z ⊗ 1 ⊗ . . . · · · ⊗ 1, with z acting on the i-th space. Let us now apply to both sides of (2.5) one time the right multiplication ⊗1 and another time the operation from the left Γ(1) . We thus get by exploiting (j) identities like f (a ⊗ 1) = f (a) ⊗ 1, with a, f (a)A⊗ , j integer : Γ(0) a ⊗ 1 = R(a1 , a2 , δb ⊗ 1),
Γ(1) Γ(0) a = R(Γ(0) a ⊗ 1, a3 , Γ(1) δb)
where now a1 , a2 must be read as a ⊗ 1 ⊗ 1, 1 ⊗ a ⊗ 1 and R is valued in A ⊗ A ⊗ A. The explicit relation between the 3-body collective operators and the 3 single body ones is therefore: Γ(1) Γ(0) a = R(R(a1 , a2 , δb ⊗ 1), a3 , Γ(1) δb)
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It is now straightforward to find, by iterating n − 2 times the two previous operations, that for the n-body the following implicit relation between all the single and all the collective operators derived from a and b (the tensor product domain and codomain of R is extended at each step) holds: Γ(n−2) . . . Γ(1) Γ(0) a = R(Γ(n−3) . . . Γ(1) Γ(0) a ⊗ 1, an , Γ(n−2) . . . Γ(1) δb),
(2.6)
and the general solution in the computable form of a recursive function is: Γ(n−2) ...Γ(1) Γ(0) a (n−2) (n−3) ,a3 ,Γ(1) δb ⊗ 1⊗ ,... ,an ,Γ(n−2) ...Γ(1) δb = R R ... R a1 ,a2 ,δb ⊗ 1⊗ (2.7) which eventually recovers an explicit expression by taking into account the concrete form of the 2-body initial relation (2.5). Let us remark now that a could be any expressions of the generators so that there can be interesting cases in which R is simply a primitive recursive function. Moreover the demonstration deals only with the elements a, b and those expressions derived from them by using Γ, δ so that (2.7) could hold even if Γ, δ don’t fulfil their defining properties on all A. As is well known a morphism ∆A → A ⊗ A is called coproduct when the coassociativity holds: (∆ ⊗ id)∆a = (id ⊗ ∆)∆a, ∀aA .
(2.8)
In this case A is a coalgebra, all the Lie and quantum algebras stay in this category. A property to notice in this context is that owing to the coassociativity (2.8) the (n) action of ∆ can be univocally iterated to any A⊗ There are also algebras where the coassociativity is fulfilled only modulo some equivalence: the quasi-coalgebras, and the quasi-coassociative morphism is the quasi-coproduct. When we deal with n representations (rps from now on) of A we can, by means of the map ∆ and id recover a set of global operators on the product space satisfying exactly the original algebra of the single components La , independently of the order of ∆ and id. In any Lie algebra the coproduct simply reads in algebraic terms: ∆La = La ⊗ 1 + 1 ⊗ La . When there is a basis of an algebra A in which ∆ gets this form it is called a primitive coproduct. If ∆ is invariant after the interchange of the two base spaces in the tensor product it is called cocommutative, any element built in terms of the generators of a Lie algebra clearly shares this property. The “barycenter formulas” of the classical kinematics are tied to the canonical coassociative coproduct. Thus the starting point in the research of the collective operators must be, if it exists, the coproduct. But one cannot find in general a δ satisfying (2.3) with Γ = ∆; clear examples are given by semisimple Lie algebras. A near solution to this problem could existe.g. for nonsemisimple Lie algebras with non null first
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class of cohomology, where an arbitrary scalar variation can be given to the action of the global morphism on some generators and a quasi-coassociative Γ results. Sometimes we can thus satisfy (2.3), at the price however of the non univocity of some global operators. Actually to proceed with Γ = ∆ seems physically reliable only when one deals with global operators of no direct physical meaning. Anyway it must be remarked again that the collective set is completely defined once the morphisms Γj , δ, whichever they are, have been done. This will be illustrated from the three examples we present in the next sections which share different degrees of complexity.
3 The Galilei Algebra Despite its ubiquitous presence in the contemporary Physics, as symmetry of the non relativistic Q.M., the literature on the Galilei group is not huge, and even in general presentations [2] the space devoted to collective coordinates is not large. Moreover in last times physical results and researches mainly concerned the classical and quantum statistical mechanics and the field theory implications of the Galilean invariance [3, 4, 5, 6]. Therefore it maybe that an extensive treatment of the collective position operators in 2-body Galilei kinematics must be searched yet in [7]. Thus it will be instructive to apply firstly our method to the 1d Galilei group. The mass is chosen to be a Lie generator, this implies the use of non projective representations with the advantages that the Galilean symmetry is seen from the physicist viewpoint, see e.g. [8], and that this is the form necessary to obtain the deformed version [9]. We start thus with the 3 generators Lie algebra gh(1) : [B, P ] = iM,
[M, B] = [M, P ] = 0;
(3.9)
It is the algebra of the purely spatial 1d extended Galilean transformations where B is the boost, P the momentum and the central generator M is the mass. If one defines, by exploiting the localization with respect to the center, the position generator X = B/M one gets: [X, P ] = i1,
[M, X] = [M, P ] = 0;
(3.10)
where 1 is the identity element of the enveloping algebra U (gh(1)). The first commutator of (3.10) define a couple of Heisenberg canonical operators. But the Lie coalgebraic structure in the Heisenberg commutator is not . compatible with X primitive if P is primitive owing to ∆1 = 1 ⊗ 1. Indeed once the momenta have been summed the corresponding positions must be linearly combined with arbitrary coefficients whose sum is 1. This is recovered by exploiting the algebraic status of M. In fact in the Lie algebra (3.9) the coproduct amounts simply to: ∆P = P1 + P2 ,
∆B = B1 + B2 ,
∆M = M1 + M2 .
(3.11)
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Consequently one has for X: ∆X = ∆B/∆M = (M1 X1 + X2 M2 )/(M1 + M2 ) .
(3.12)
It is therefore very sensible to think in this case to the Heisenberg canonical set as a coalgebra with three generators. A good well known map δ is given by δP = (P ⊗ M − M ⊗ P )/(∆M ) =
P1 M2 − M1 P2 , M1 + M2
δX = X ⊗ 1 − 1 ⊗ X = X1 − X2 , M1 M2 δM = M ⊗ M/(∆M ) = , M1 + M2
(3.13)
The expressions (2.4), with Γj = ∆, are then the usual canonical Jacobi coordinates. Anyway the algebra (3.9) is a sub-algebra of the full Galilei Lie algebra g one gets by adding a fourth Lie generator E, the energy, whose non zero commutator is: [B, E] = iP . The center of g is generated besides M even by the quadratic Casimir C = 2M E − P 2 . It is now obvious that the coproduct of the energy cannot commute with all the relative operators. But the generator E cannot be written as a commutator and we can put ΓE = ∆E. Thus the set of collective operators can be completed by introducing a global energy ΓE and a relative energy δE. They can be found by imposing that 2ΓM ΓE − ΓP 2 and 2δM δE − δP 2 are Casimir. The result is: ΓE =
M1 E1 + M2 E2 + P1 P2 M1 + M2
(3.14)
and
δE = (M ⊗ E + E ⊗ M ) − P ⊗ P )/(∆M ) (3.15) M1 E2 + M2 E1 − P1 P2 = = E1 + E2 − ΓE . M1 + M2 The definition of ΓE is anyway coassociative modulo global Galilei invariant operators. The application of (2.4) gives the expressions for any n. Let us notice that (3.15) is in a form where the general recursive formula (2.7) is trivially explicited so that we have for any n: ΣEj
= Γ(n−2) . . . Γ(0) E + Γ(n−3) . . . Γ(1) δE ⊗ 1 + (n−3)
· · · + Γ(1) δE ⊗ 1⊗
+ δE ⊗ 1⊗
(n−2)
(3.16)
Let us observe also that by choosing a = b = P 2 /(2M ) and then recovering from the 2-body that R(x, y, z) = x + y − z the formula (2.7) gives immediately that the
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sum of the n single kinetic energies transforms in the identical formal expression in terms of the collective Jacobi set. It must be remarked that the map δ (3.13), doesn’t satisfy coassociativity nor it is a coaction, (i.e.: (∆ ⊗ id)δa is not equal to (id ⊗ δ)δa, ∀aA). Indeed it is just the initial support for the action of the globalizing and injecting operations. It can be shown by direct calculation that there is no coassociative δ producing all the previous properties in the Galilei algebra. Of course one can introduce functions of the masses as factors in the definition of δ, as, e.g., in the analysis of the 1d integrable many-body Schr¨ odinger equation by McGuire [10]. A δ actually quasi-coassociative can be obtained in this way, with a lack of completeness however as relative and global masses happen to be the same. It is worth noticing that the analogous coproduct and the same role of the mass hold in the three-dimensional situation, where the Heisenberg set can be derived again by a sub-algebra of the extended Galilei. In this case the expressions of the 2-body collective operators can be much more composite, following the dynamical problems one has to face. But, as shown before, once the collective expressions have been found for 2 the algorithm to give expressions for n is straightforward.
4 The Poincar´e Algebra The proposals about the localization and the canonical operators of the position in special relativity are not univocal, see e.g. [11] and references therein. We adopt here the one, firstly studied in [7], based on the Weyl algebra, analyzed and exploited in [12] where the hamiltonian dynamics of 1 and 2 scalar or spinning relativistic particles was written. Coulomb and Schwartzschwild type 2-body interactions were covariantly introduced in the mass square and the dynamics of two scalar particles completely solved, with results in very good agreement with field calculations (see [13] also). Some very encouraging quantistic estimates were also done for 2 and 3 interacting scalar particles [14]. Moreover operators with identical expressions, although there the Weyl algebra is included in the conformal one, have been independently rediscovered and proposed as the quantum observables of relativistic spinning particles in many recent papers see [15] and references therein. We exploit the cohomological based possibility of adding a 2-body Weyl invariant operator to the global dilatator defined by the primitive coproduct. Thus the coassociativity holds only modulo Weyl invariant operators and the global operators involving the dilatators are strictly dependent on the order of the Γ and id. We discuss now the (1, 1)d situation. The analogous in the Poincar´e kinematics of the (2.2) is given by the E(1, 1) Lie algebra : [B, P ] = iE,
[B, E] = iP,
[E, P ] = 0.
However, to get a time operator, the starting point of our procedure must be the Weyl algebra, obtained by adding as fourth Lie generator the dilatator D: [D, P ] = −iP,
[D, E] = −iE,
[D, B] = 0 .
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We construct then two commuting Heisenberg pairs by defining the two “Lorentz (1,1)-vectors”: (P, E) and (X, T ), X = M −2 (DP + BE),
T = M −2 (DE + BP )
(4.17)
where M 2 = E 2 −P 2 is the Casimir of E(1, 1) and one has [X, P ] = i, [T, E] = −i, all the other commutators being zero. Those operators are the building blocks of the 1-body. It must be remarked however that the dynamics of such systems must be generated by Hamiltonians conserving the Poincar´e invariant mass and that the maximal invariance can be the Poincar´e symmetry, not the Weyl one, because in any situation the physical time T at least must change with any evolution parameter: all that is done in quite natural manner in this framework. The projection on the irreducible rps of E(1, 1) is indeed the equivalent of the classical reduction procedure on the fixed mass sub-variety. Let us now discuss the 2-body collective scheme. It reads ΓE = E1 + E2 ,
ΓP = P1 + P2 ,
ΓB = B1 + B2
and ΓD = D1 + D2 + c
where c is an arbitrary element in the center of the global Weyl in the tensor product, allowed because D never appears on the right member of the commutations relations (this happens in the (3, 1)d case also). We have thus: ΓM = ((ΓE)2 − (ΓP )2 )1/2 and the “quasi-coproduct” of X, T is given by ΓX = (ΓM )−2 (((µ1 )2 X1 + (µ2 )2 X2 ) − t(P1 E2 − P2 E1 ) + (2i + c)(P1 + P2 )) ΓT = (ΓM )−2 (((µ1 )2 T1 + (µ2 )2 T2 ) + r(P1 E2 − P2 E1 ) + (2i + c)(E1 + E2 )) where (µA )2 = (EA )2 − (PA )2 + (E1 E2 − P1 P2 ) so that (µ1 )2 + (µ2 )2 ) = (ΓM )2 , and it is r = X1 − X2 ,
t = T1 − T2 .
q = (P1 − P2 )/2,
u = (E1 − E2 )/2 .
Let us choose c = −2i − (ut − qr): it is straightforward to show that (Γ ⊗ id)ΓD − (id ⊗ Γ)ΓD is again an operator invariant under the global 3-body Weyl algebra. A good set of relative operators is then obtained by adding the definitions δX = r˜ = (ΓE r − ΓP t)/(ΓM ), δT = r¯ = (ΓE t − ΓP r)/(ΓM ),
δP = q˜ = (ΓE q − ΓP u)/(ΓM ) δE = q¯ = (ΓE u − ΓP q)/(ΓM )
Together with ΓX, ΓT , and ΓP, ΓE they give a complete set of canonical and “covariant”(invariant in this 1d case) operators as a direct calculation can confirm. The relevant property of this set is the existence of a relation: (ΓM )2 = (((M1 )2 + (˜ q )2 )1/2 + ((M2 )2 + (˜ q )2 )1/2 )2
(4.18)
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recovered by eliminating δE from the collective expressions of (M1 )2 and (M2 )2 and solving in (ΓM )2 . By projecting on definite values (M1 )2 = m21 , (M2 )2 = m22 one recovers for the relativistic 2-body a rigorous hamiltonian formulation in terms of one global time, while the relative time δT = r¯ is ignorable and can be chosen a posteriori to reconstruct the dynamics in the higher dimension. At this point one can introduce interactions depending on |˜ r |. Clearly the physical description is given at this level, the galilean limit too must be checked there. It is now possible to extend straightforward (4.18) to any number of massive Poincar´e representations because it is given explicitly in the form of relation (2.5). The absence of angular momenta in those (1, 1)d models avoid any problem of formal covariance (as opposed to the commutativity of the components of the position, see [12]). It is thus possible to construct recursively, by adopting the formulas (2.4) for the n-body and the corresponding expressions (2.7) with nested square roots, a genuine relativistic hamiltonian system of n interacting particles, with n given masses and one global physical time.
5 The Quantum framework The definitions (2.4), (2.6) depend on a canonical map and thus they can be in principle applied to any coalgebra. The crucial problem is to find a good map δ for the 2-body system. It is thus interesting to analyze from this view point the operators of the quantum version of the Galilei algebra [9], where (2.3) cannot be completely realized. This deformed algebra has found physical applications directly as kinematical symmetry of many-body quantum dynamics on lattice [9]. Moreover its unitary irreducible rps have been studied by inducing on the non commutative space of parameters and they appear in agreement with those of Heisenberg on the lattice, but the recovering of unitary irreducible rps in the usual way in the common space of the product of two is rather problematic, notwithstanding the algebra has a real form although rather unconventional [17]. Thus let us introduce the coalgebra gha (1) having the same 3 generators and algebraic relations as gh(1) and non trivial coproduct of B and M: ∆P = P ⊗ 1 + 1 ⊗ P ∆B = B ⊗ exp(iaP ) + exp(−iaP ) ⊗ B ∆M = M ⊗ exp(iaP ) + exp(−iaP ) ⊗ M
(5.19)
where the length a is the deformation parameter and having, defined again X = B/M , one gets ∆X =
∆B M1 X1 + M2 X2 exp(−ia(P1 + P2 )) = . ∆M M1 + M2 exp(−ia(P1 + P2 ))
(5.20)
We take obviously Γ = ∆ but let us observe that ∆M is a Casimir of the algebra ∆A but it is not a central element of A ⊗ A. Its expression implies that it is
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impossible to get δ such that both δX and δP commute with ∆X, ∆P and even with ∆M . The map δ we define is the following: δX = X1 − X2 ΓM i δP = log a (M1 + M2 ) M1 M2 δM = (M1 + M2 )
(5.21)
and we have two couple of commuting canonical operators, although not a direct product of the two triples. Indeed there is a deformed commutator: [δX, ΓM ] = aΓM . By looking at the structure of the expressions (2.4) one sees that in this case they produce n distinct realizations of the algebra which however are not commuting between them. It must be remarked again that ∆M, ∆X are given a priori and δX has the form necessary to commute with total momentum while the remaining expressions have correct relations. Thus the previous choice must be accepted and one has to pay the price of a deformation of canonicity, starting from n = 3, in the collective formulation. A quasi-associative energy E completes the Galilean deformed algebra. The resulting nonstandard [B, E] = (i/a) sin(aP ) determines a Casimir C = M E − (1/a2 )(1 − cos(aP )), from which we define the deformed kinetic energy: T = (1/(M a2 ))(1 − cos(aP )) . It is then straightforward to obtain for the 2-body operators: T1 + T2 = (1/(∆M a2 ))(1 − cos(a∆P )) + (1/(δM a2 ))(1 − cos(a δP )) = ∆T + δT . (5.22) We are again in a situation where an explicit elementary expression of the (2.6) exists and the previous anomalies cannot affect the result given by (2.7). Indeed we are using only the abelian coalgebra generated by P and M , with their coproducts. Therefore we can be sure of the existence of the set of trigonometric identities which state in the deformed case the same theorem about the kinetic energies as in the classical one: ΣTj = ∆(n−2) . . . ∆(0) T + ∆(n−3) . . . ∆(1) δT ⊗ 1+ · · · + ∆(1) δT ⊗ 1⊗
(n−3)
(n−2)
+ δT ⊗ 1⊗
.
(5.23)
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This is the kinetic part of a lattice Hamiltonian. If one searches for values of observables such that the kinetic energy is given only by the barycenter term the result is that all the relative momenta must be zero, i.e.: ∆(0) M = M1 + M2 ∆(j) . . . ∆(1) ∆(0) M = ∆(j−1) . . . ∆(1) ∆(0) M + Mj+2 , j ∈ (1, n − 2).
(5.24)
It has been demonstrated that when all the masses are equal the system (5.24) gives exactly the Bethe conditions for the momenta of n-magnons bound states of the XXX model and the right spectrum of the energy [9]. It is possible to introduce in the same way as in the classical case the global energy and the relative one δE: δE =
m1 T 2 + m2 T 1 m1 E2 + m2 E1 + δT − m1 + m2 m1 + m2
whose non deformed limit is (3.15). The global energy is E1 + E2 − δE, which like ∆T - doesn’t commute with δX. A sum rule formally identical to (3.16) can be written however.
6 Concluding remarks An intuitive method of constructing collective classical canonical coordinates or quantum mechanical operators for n-body on the ground of their expressions for n = 2 has been precisely formulated and demonstrated by means of algebra morphisms, constructed on the basis of the coalgebra of the systems. Examples from Galilei, Poincar´e and deformed Galilei are discussed. An interesting result is the ability of writing immediately for n relations calculated for 2. A further point worth to be studied is the way to apply the algorithm in field theory and the possible connection to the integrability suggested by section 4. Preliminary analysis of those problems are in fieri. Concerning the coproduct it must be stressed that its possible substitution by the morphism Γ is essential in allowing a rigorous and physically good description of the many-body relativistic systems in our approach to the Poincar´e systems. From this view point the inclusion of the Weyl in the larger conformal algebra as in [15] may generate problems, because in that case there is no space to substitute the coproduct of D with a morphism having c = 0. This remark leads us again to enhance a very general point sometimes ignored in the practice, owing to the long monopoly of the Lie primitive structures; i.e. that a complete knowledge of an algebra can be obtained only by the knowledge of the coalgebra too. All that is very important in those attempts to grasp quantum gravity by means of noncommutative geometries, implied e.g. by the introduction of deformed relativistic kinematics, strongly supported in last years by the preliminary astrophysical measures concerning gamma-ray bursts and the possible violation
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of the GZK threshold in cosmic rays, see [17, 18] and references therein. A deep analysis of the collective operators connected to the proposed deformations of the Poincar´e kinematics could then be very useful in formulating their phenomenological implications. Indeed one exotic relation of dispersion is in itself not enough, but if it is accompanied by the emergence of 2-body spectra deduced from noncocommutative coalgebra it will be read as a clear signature of a noncommutative space-time. Acknowledgements. I thanks M. Tarlini for very helpful discussion and valuable criticisms.
References [1] D. Kreimer, Adv. Th. Math. Phys. 2.2, 303 (1998); A. Connes, D. Kreimer, Comm. Math. Phys. 199, 203 (1998). [2] A.W. Wightman, Rev. Mod. Phys. 34., 845 (1962); J.M. Levy Leblond, Galilei Group and Galilean Invariance in “Group Theory and Its Applications”, edited by E.M. Loebl, (Academic Press, New York and London, 1971), Vol. II, pp 221–299. [3] H. Narnhofer, W. Thirring, Int. Journ. Mod. Phys. A 6., 2937 (1991). [4] Y. Takahashi, Fortschr. Phys. 36, 63 and 81 (1988). S.R. Corley, O.W. Greenberg, Jour. Math. Phys. 38, 571 (1995). [5] A. Harinandrath and R. Kundu, Int. Journ. Mod. Phys. 13, 4591 (1998). [6] L.O. Raifeartaigh and V.V. Sreedhar, “The maximal invariance group of fluid dynamics and explosion-implosion duality” hep-th/0007199v2. [7] D.J. Almond, Ann. I. Henry Poincar´e A 19, 105 (1973). [8] S. Weinberg, “Quantum Field Theory”, (Cambridge U.P., New York, 1995), Vol.I, ch.2. [9] F. Bonechi, E. Celeghini, R. Giachetti, E. Sorace and M. Tarlini, Phys. Rev. B 46, 5727 (1992). [10] J.B. Mc Guire, J. Math. Phys. 5.4, 622 (1964). [11] M. Czachor, Phys. Rev. A 55, 72 (1997); R. Omn´es, J. Math. Phys. 38, 708 (1997). [12] E. Sorace and M. Tarlini, Nuovo Cimento B 71, 98 (1982); and L. Nuovo Cimento 35, 1 (1982).
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[13] E. Sorace, L. Nuovo Cimento 28, 175 (1980); R. Giachetti, E. Sorace, Nuovo Cimento B 63, 666 (1981). [14] E. Sorace, M. Tarlini, Nuovo Cimento B 82, 29 (1984). [15] M.-T. Jaekel and S. Reynaud, Annalen Phys. 9, 589 (2000). [16] F. Bonechi, N. Ciccoli, R. Giachetti, E. Sorace and M. Tarlini, Lett. Math. Phys. 49, 17 (1999). [17] G. Amelino-Camelia, J. Ellis, N.E. Mavromatos, D.V. Nanopoulos and S. Sarkar, Nature 393, 763 (1998); G. Amelino-Camelia, “Testable scenario for Relativity with minimum length”, hep-th/0012238. [18] T. Kifune, Astrophys. J. Lett. L21-L24, 518 (1999); R. Aloisio, A.F. Grillo, “Cosmic rays and the structure of space time” astro-ph/0008941. Emanuele Sorace Universit` a di Firenze INFN Firenze Dipartimento di Fisica via G. Sansone 1 I–50019 Sesto Fiorentino (FI) Italy email: sorace@fi.infn.it Communicated by Klaus Fredenhagen submitted 11/06/01, accepted 15/04/02
To access this journal online: http://www.birkhauser.ch
Ann. Henri Poincar´e 3 (2002) 673 – 691 c Birkh¨ auser Verlag, Basel, 2002 1424-0637/02/040673-19
Annales Henri Poincar´ e
Uniform Estimates of the Resolvent of the Laplace-Beltrami Operator on Infinite Volume Riemannian Manifolds. II F. Cardoso∗ and G. Vodev
Abstract. We prove uniform weighted high frequency estimates for the resolvent of the Laplace-Beltrami operator on connected infinite volume Riemannian manifolds under some natural assumptions on the metric on the ends of the manifold. This extends previous results by Burq [3] and Vodev [8].
1 Introduction and statement of results The purpose of this paper is to extend the results in [8] to more general Riemannian manifolds (which may have cusps). Let (M, g) be an n-dimensional unbounded, connected Riemannian manifold with a Riemannian metric g of class C ∞ (M ) and a compact C ∞ -smooth boundary ∂M (which may be empty), of the form M = X0 ∪ X1 ∪ X2 , where X0 is a compact, connected Riemannian manifold with a metric g|X0 of class C ∞ (X 0 ) with a compact boundary ∂X0 = ∂M ∪ ∂X1 ∪ ∂X2 , ∂M ∩ ∂X1 = ∅, ∂M ∩ ∂X2 = ∅, ∂X1 ∩ ∂X2 = ∅, Xk = [rk , +∞) × Sk , rk 1, with metric g|Xk := dr2 + σk (r), k = 1, 2. Here (Sk , σk (r)), k = 1, 2, are n − 1 dimensional compact Riemannian manifolds without boundary equipped with families of Riemannian metrics σk (r) depending smoothly on r which can be written in any local coordinates θ ∈ Sk in the form k k gij (r, θ)dθi dθj , gij ∈ C ∞ (Xk ). σk (r) = i,j
Denote Xk,r = [r, +∞) × Sk . Clearly, ∂Xk,r can be identified with the Riemannian manifold (Sk , σk (r)) with the Laplace-Beltrami operator ∆∂Xk,r written as follows ∂θi (pk gkij ∂θj ), ∆∂Xk,r = −p−1 k i,j k k 1/2 where (gkij ) is the inverse matrix to (gij ) and pk = (det(gij )) = (det(gkij ))−1/2 . Let ∆g denote the Laplace-Beltrami operator on (M, g). We have 2 ∆Xk := ∆g |Xk = −p−1 k ∂r (pk ∂r ) + ∆∂Xk,r = −∂r − ∗ Partially
supported by CNPq (Brazil)
pk ∂r + ∆∂Xk,r . pk
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Throughout this paper given a function p(r, θ), p , p and etc. will denote the first, the second and etc. derivative with respect to r. It is easy to check the identity 1/2
−1/2
pk ∆Xk pk
= −∂r2 + Λk,r + qk (r, θ),
where Λk,r = −
(1.1)
∂θi (gkij ∂θj ),
i,j
and qk is an effective potential given by 2 ∂pk ∂pk ij ∂pk −2 qk (r, θ) = (2pk ) + (2pk )−2 gk + 2−1 pk ∆Xk (p−1 k ). ∂r ∂θ ∂θ i j i,j We make the following assumptions: ∂q1 ∂q2 (r, θ) ≤ Cr−1−δ0 , − (r, θ) ≤ Cr−1 , (1.2) ∂r ∂r with constants C, δ0 > 0. Denote by hk the principal symbol of ∆∂Xk,r , that is, ij hk (r, θ, ξ) = gk (r, θ)ξi ξj , (θ, ξ) ∈ T ∗ Sk . |qk (r, θ)| ≤ C,
i,j
Clearly, −∂hk /∂r can be interpreted as being the second fundamental form of the surface ∂Xk,r . We suppose that (−1)k
C ∂hk (r, θ, ξ) ≥ hk (r, θ, ξ), ∂r r
∀(θ, ξ) ∈ T ∗ Sk ,
(1.3)
with a constant C > 0. In particular, this means that ∂X1,r (resp. ∂X2,r ) is strictly convex (resp. strictly concave) viewed from X1,r (resp. X2,r ). This implies that the commutators (−1)k [∂r , Λk,r ], k = 1, 2, are strictly positive. Denote by G the selfadjoint realization of ∆g on the Hilbert space H = L2 (M, dVolg ) with Dirichlet or Neumann boundary conditions on ∂M . Given s1 , s2 ∈ R, choose a real-valued positive function χs1 ,s2 ∈ C ∞ (M ), χs1 ,s2 = 1 on M \ (X1,r1 +1 ∪ X2,r2 +1 ), χs1 ,s2 = r−sk on Xk,rk +2 . Also, given a > r1 choose a real-valued positive function ηa ∈ C ∞ (M ), ηa = 0 on M \ X1,a , ηa = 1 on X1,a+1 . Our main result is the following Theorem 1.1 Under the assumptions (1.2) and (1.3), for every s1 > 1/2, s2 > 1, there exist positive constants C0 , C > 0, a > r1 so that for z ∈ R, z ≥ C0 , the limit Rs+1 ,s2 (z) := lim+ χs1 ,s2 (G − z + iε)−1 χs1 ,s2 : H → H ε→0
exists and satisfies the bounds Rs+1 ,s2 (z)L(H) ≤ eCz
1/2
,
ηa Rs+1 ,s2 (z)ηa L(H) ≤ Cz −1/2 .
(1.4) (1.5)
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Suppose that there exist metrics σ k (r) depending smoothly on r ∈ (−∞, +∞) such that σ k (r) = σk (r) for r ≥ rk and the resolvents (defined for Im z < 0, Re z > 0) 2 RXk0 (z) := (∆Xk0 − z)−1 : L2comp(Xk0 , dVolgX 0 ) → Hloc (Xk0 , dVolgX 0 ), k
k
k (r), ∆Xk0 denoting the where Xk0 = (−∞, +∞) × Sk with metric gXk0 = dr2 + σ selfadjoint realization of the Laplace-Beltrami operator on Xk0 on the Hilbert space 1/2 L2 (Xk0 , dVolgX 0 ), extend analytically to Im z ≤ e−γ1 |z| , Re z ≥ C1 , γ1 , C1 > 0, k
and satisfy in this region the bounds (with α = 0, 1): ∂zα χRXk0 (z)χL(L2 (Xk0 ,dVolg
X0 k
))
≤ C2 eγ2 |z|
1/2
,
∀χ ∈ C0∞ (Xk0 ),
(1.6)
with some constants C2 , γ2 > 0. As a consequence of Theorem 1.1 we get the following Corollary 1.2 Under the assumptions (1.2), (1.3) and (1.6), the resolvent (defined for Im z < 0, Re z > 0) 2 RM (z) := (G − z)−1 : L2comp (M, dVolg ) → Hloc (M, dVolg ), 1/2
extends analytically to Im z ≤ e−γ|z| , Re z ≥ C0 , and satisfies in this region the bound 1/2 (1.7) χRM (z)χL(H) ≤ Ceγ|z| , ∀χ ∈ C ∞ (M ) of compact support, with some constants C0 , C, γ > 0. Remark. It is easy to see that the above results hold for more general connected Riemannian manifolds of the form M = X0 ∪ X11 ∪ · · · ∪ X1J ∪ X21 ∪ · · · ∪ X2I ,
I ≥ 0, J ≥ 1,
with X1j like X1 , X2i like X2 , and X0 being a compact Riemannian manifold with boundary ∂X0 = ∂M ∪ ∂X11 ∪ · · · ∪ ∂X1J ∪ ∂X21 ∪ · · · ∪ ∂X2I , ∂M ∩ ∂X1j = ∅, ∂M ∩ ∂X2i = ∅, ∂X1j ∩ ∂X2i = ∅, ∂X1j1 ∩ ∂X1j2 = ∅, j1 = j2 , ∂X2i1 ∩ ∂X2i2 = ∅, i1 = i2 . This corollary can be derived from the bounds (1.4) and (1.6) in precisely the same way as in the proof of Theorem 1.2 of [8] and this is why we omit the proof. Another consequence of the above theorem is that we get uniform high frequency resolvent estimates for long-range perturbations of the Euclidean metric. Let O ⊂ Rn , n ≥ 2, be a bounded domain with a C ∞ -smooth boundary Γ and
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a connected complement Ω = Rn \ O. Let g be a Riemannian metric in Ω of the form n gij (x)dxi dxj , gij (x) ∈ C ∞ (Ω). g= i,j=1
We make the following assumption: |∂xα (gij (x) − δij )| ≤ Cα x−δ0 −|α| ,
(1.8)
for every multi-index α, with constants Cα , δ0 > 0, where x := (1 + |x|2 )1/2 and δij denotes the Kronecker symbol. Denote by ∆g the corresponding LaplaceBeltrami operator, i.e. ∆g = −f
−1/2
n
∂xi (f 1/2 g ij ∂xj ),
i,j=1
where (g ij ) is the inverse matrix to (gij ) and f = det(gij ). Denote by G the selfadjoint realization of ∆g on the Hilbert space H = L2 (Ω; dVolg ), dVolg := f 1/2 dx, with Dirichlet or Neumann boundary conditions on Γ. It is not hard to see (e.g. see the appendix of [3] for the proof of an analytic version) that under the assumption (1.8), there exists a global smooth change of variables, (r, θ) = (r(x), θ(x)), for |x| 1, where r ∈ [r0 , +∞), r0 1, θ ∈ S = {y ∈ Rn : |y| = 1}, which transforms the metric g in the form hij (r, θ)dθi dθj , (1.9) dr2 + r2 i,j
where hij ∈ C ∞ satisfy the inequalities |∂rα ∂θβ (hij (r, θ) − h0ij (θ))| ≤ Cα,β r−δ0 −α (1.10) for all multi-indexes α and β. Here i,j h0ij (θ)dθi dθj is the metric on S induced by the Euclidean one. The coordinates (r, θ) are just the normal geodesics coordinates which are well defined outside a sufficiently large compact since the metric g is close to the Euclidean one. In other words, the Riemannian manifold (Ω, g) is isometric to a connected Riemannian manifold (M, g) of the form M = Y0 ∪ Y , where Y0 is a compact connected Riemannian manifold with boundary ∂Y0 = ∂M ∪ ∂Y , ∂M ∩ ∂Y = ∅, and Y = [r0 , +∞) × S, r0 1, with metric given by (1.9) and satisfying (1.10). Therefore, Y is a particular case of the manifold X1 above, and we get the following consequence of Theorem 1.1. Corollary 1.3 Under the assumption (1.8), for every s > 1/2 there exist constants C0 , C > 0 and a 1 so that for z ∈ R, z ≥ C0 , the limit Rs+ (z) := lim+ x−s (G − z + iε)−1 x−s : H → H ε→0
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exists and satisfies the bounds Rs+ (z)L(H) ≤ eCz
1/2
,
χa Rs+ (z)χa L(H) ≤ Cz −1/2 ,
(1.11) (1.12)
where χa denotes the characteristic function of |x| ≥ a. Remark. It is easy to see from the proof that it suffices to have (1.10) for α+|β| ≤ 3. When gij = δij outside some compact the bound (1.11) follows from the results of Burq [2], where he proved a similar bound for the cutoff resolvent. This 2+ 0 was improved in [7] for metrics satisfying gij − δij = O(e−|x| ), /0 > 0. Burq [3] has recently extended his result to long-range metric perturbations assuming that gij admit an analytic extension from {x ∈ Rn : |x| ≥ ρ0 }, ρ0 1, to {z ∈ Cn : |Re z| ≥ ρ0 , |Im z| ≤ γ0 |Re z|}, γ0 > 0. In particular, this implies that if (1.8) holds with α = 0, it holds for any α. He used the complex scaling method to show that there are no resonances in an exponentially small neighbourhood of the real axis. In particular, it follows from [3] that one has an analogue of (1.11) for the cutoff resolvent, which combined with the result of Bruneau-Petkov [1] imply the bound (1.11) itself in that case. Burq [3] has also proved an analogue of (1.12) with χa replaced by the characteristic function of a < |x| < b with b > a 1. Note that the class of manifolds, (M, g), we study includes hyperbolic ones with negative curvature, κ, satisfying C −1 ≤ −κ ≤ C on M for some constant C > 0. In fact, the methods we develop in the present paper apply to infinite volume Riemannian manifolds with infinity consisting of a finite number of two type of ends - elliptic ends (like X1 above) whose number is ≥ 1 and cusps (like X2 above) whose number is ≥ 0. An elliptic end satisfying (1.2) and (1.3) with k = 1 is of infinite volume. The condition (1.2) on the effective potential together with (1.3) guarantee that the (Dirichlet) self-adjoint realization of ∆X1 on L2 (X1 , dVolg ) has no discrete spectrum (except for possibly a finite number of eigenvalues). Moreover, if we consider the generalized geodesic flow in X1 , as (1.3) implies that ∂X1 is strictly convex, every geodesic coming from the infinity of X1 is allowed to hit the boundary either transversally or at a diffractive point, so it escapes back to infinity. This suggests that the operator ∆X1 should have properties typical for the so called nontrapping operators. This in turn suggests that the resolvent of the global operator ∆g cut off on the both sides by a cutoff function supported in X1 should satisfy the same high frequency estimates as does the resolvent of ∆X1 . We show that this is exactly what happens - see the bound (1.5) which without cutoffs is known to hold for nontrapping perturbations. The key point of our proof is the estimate (2.22) proved in Section 2. It seems that the assumptions (1.2) and (1.3) with k = 1 are the weakest ones under which (2.22) holds true. The situation on a cusp X2 is exactly opposite and this is why in (1.5) we cannot take the function η with support on X2 . In fact, the conditions (1.2) and (1.3) with k = 2 do not imply that the volume of X2 must be finite, but we
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will keep the notion cusp in this case as well. Of course, there are finite volume hyperbolic cusps, X2 , (with negative curvature) satisfying (1.2) and (1.3) with k = 2. An interesting example of two dimensional hyperbolic manifolds our results apply to is Xk = [ak , +∞)r × (R \ 3k Z)t , ak , 3k > 0, k = 1, 2, with metrics g|X1 = dr2 + cosh2 rdt2 , g|X2 = dr2 + e−2r dt2 . Note that for such manifolds the bound (1.4) as well as Corollary 1.2 have been already proved in [8], but the bound (1.5) seems to be new. We expect that Theorem 1.1 (or at least (1.4)) holds for more general infinite volume hyperbolic manifolds with a more complex structure at infinity, as for example manifolds with non-maximal cusps. The bound (1.4) is proved in [8] for manifolds which have a similar structure at infinity as the manifold M above, but under the restriction that the metric on the ends Xk , k = 1, 2, is of the form dr2 + pk (r)−2 σk , where σk does not depend on r, and pk (r) are smooth positive functions satisfying conditions analogous to (1.2) and (1.3) above. The fact that we have a separation of variables was used in an essential way in the methods developed in [8]. In the situation we treat in the present paper we do not have such a separation of variables, which requires a different approach. It is based on an idea of Burq [3] which consists of using Carleman estimates outside a sufficiently large compact with a real-valued phase function, ϕ(r), with ϕ (r) > 0, depending on the spectral parameter (in our case λ 1) such that ϕ = O(λ−1 r−1 ) outside another compact (in which region the estimates are no longer of Carleman type). We apply this on the elliptic (infinite volume) end X1 - see Proposition 2.3 which is essentially due to Burq (see Propositions 6.2 and 7.2 of [3]), but here we give a different proof in a little bit more general situation. Moreover, our construction of the phase function ϕ is simpler than that one in [3]. Then the problem is to paste together this estimate with estimates on the compact part of the manifold essentially due to Lebeau-Robbiano [4], [5] (see Proposition 4.1 and also Theorem A.2 of [7]), with weighted estimates at the infinity of X1 (see Proposition 2.4) as well as with weighted Carleman estimates on X2 (see Proposition 3.1). This is carried out in Section 4. Acknowledgements. A part of this work was carried out while the second author was visiting Universidade Federal de Pernambuco, Recife, Brazil, in March-April 2001, and he would like to thank this institution for the hospitality and the nice working conditions. The first author was also partially supported by the agreement Brazil-France in Mathematics - Proc. 69.0014/01-5.
2 Uniform a priori estimates on X1 We begin this section by constructing a real-valued phase function, ϕ, with properties described in Lemma 2.1 below. A similar phase function was first constructed by Burq [3]. Here we simplify this construction (as well as some of his arguments) adapting it to our approach. Let λ 1 be a big parameter, let 0 < δ 1 be independent of λ, and let γ0 > 1 be independent of λ and δ. In what follows, C will denote a positive
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constant independent of λ, while C will denote a positive constant independent of λ and δ. Define the continuous function ϕ 1 (r) so that ϕ 1 (r) = (Ar−δ − 1)1/2 1/δ for r1 ≤ r < a1 = A , ϕ 1 (r) = 0 for r ≥ a1 , where A = (r1 + 2)δ (γ0+ 1)2 /4 + 1. Choose a real-valued function φ ∈ C0∞ ((−1, 1)) such that φ ≥ 0, φ = 1 and φ2 ≤ C0 φ with some constant C0 > 0, and set φ (r) = /−1 φ(r//), 0 < / 1. Let ζ ∈ C0∞ (R) be a real-valued function, ζ ≥ 0, equal to 1 in a small neighbourhood of a1 and to zero outside another small neighbourhood of a1 . Then the function 1 + φ : (ζ ϕ 1 ) ϕ1 = (1 − ζ)ϕ belongs to C ∞ ([r1 , ∞)) and vanishes for r ≥ a1 + 1. Moreover, since ϕ1 ϕ1 → ϕ 1 ϕ 1 = −2−1 δAr−1−δ if r < a1 and to zero if r > a1 as / → 0, taking / > 0 small enough we can arrange −ϕ1 (r)ϕ1 (r) ≤ C δr−1 ,
∀r ≥ r1 .
(2.1)
Also, the choice of φ guarantees the bound ϕ1 (r)2 ≤ Cϕ1 (r),
∀r ≥ r1 .
(2.2)
Define a real-valued function ϕ ∈ C ∞ ([r1 , +∞)) such that ϕ(r1 ) = −1 and ϕ (r) = ϕ1 (r) + λ−1/2 r−1 ϕ2 (r)(1 + λ1/2 ϕ3 (r))−1 , where ϕj ∈ C ∞ ([r1 , +∞)), j = 2, 3, are real-valued functions independent of λ, 0 ≤ ϕj (r) ≤ 1, ϕj (r) ≥ 0, ∀r, chosen so that ϕ2 = 0 for r ≤ a1 , ϕ2 = 1 for r ≥ a1 , r1 + 2 < a1 < a1 ∈ supp (1 − ζ), ϕ3 = 0 for r ≤ a2 , ϕ3 = 1 for r ≥ a2 , a1 + 1 < a2 < a2 . We also require that rϕ3 (r) ≤
1 , 4
∀r.
(2.3)
Moreover, near a1 we choose ϕ2 in the form ϕ2 (r) = exp((a1 − r)−1 ) if r > a1 , which guarantees the inequality ϕ2 (r)2 ≤ Cϕ2 (r),
∀r ≥ r1 .
(2.4)
It is easy also to see that we have the inequalities −1 |ϕj (r)| + |ϕj (r)| + |ϕ ϕ2 (r), j = 1, 3, j (r)| ≤ Cr
|ϕ2 (r)| + |ϕ2 (r)| + |ϕ 2 (r)| ≤ Cϕ1 (r).
(2.5)
Note that the choice of the constant A guarantees that ϕ(r1 + 2) ≥ γ0 . Lemma 2.1 The following inequalities hold for λ ≥ λ0 (δ) 1 and ∀r ≥ r1 : Cλ−1 r−1 ≤ ϕ (r) ≤ Cr−1 ,
−ϕ (r)ϕ (r) ≤ C δr
1/2 −1
|ϕ (r)| ≤ Cλ
r
ϕ (r),
2
−1
(2.6)
,
(2.7) 1/2 −1
ϕ (r) ≤ Cλ
r
ϕ (r),
(2.8)
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|ϕ (r)| ≤ Cλr−1 ϕ (r), (4)
|ϕ
|ϕ (r)| ≤ Cλ1/2 r−1 , 3/2 −1
(r)| ≤ Cλ 2
Ann. Henri Poincar´e
r
ϕ (r),
−1
2λϕ (r) + ϕ (r) ≥ C r
ϕ (r).
(2.9) (2.10) (2.11)
Proof. We have Cλ−1 r−1 ≤ λ−1 r−1 (rϕ1 (r) + ϕ2 (r)) ≤ ϕ (r) ≤ r−1 (rϕ1 (r) + λ−1/2 ϕ2 (r)) ≤ Cr−1 , which proves (2.6). To prove (2.7) observe that ϕ (r) = ϕ1 (r) − λ−1/2 r−2 ϕ2 (r)(1 + λ1/2 ϕ3 (r))−1 +λ−1/2 r−1 ϕ2 (r)(1 + λ1/2 ϕ3 (r))−1 − r−1 ϕ2 (r)ϕ3 (r)(1 + λ1/2 ϕ3 (r))−2 , and hence, in view of (2.1), −ϕ ϕ = −ϕ1 ϕ1 + λ−1/2 r−2 ϕ1 ϕ2 (1 + λ1/2 ϕ3 )−1 −λ−1/2 r−1 (ϕ1 ϕ2 + ϕ1 ϕ2 )(1 + λ1/2 ϕ3 )−1 +λ−1 r−2 ϕ22 (1 + λ1/2 ϕ3 )−2 − λ−1 r−2 ϕ2 ϕ2 (1 + λ1/2 ϕ3 )−2 +λ−1/2 r−2 ϕ22 ϕ3 (1 + λ1/2 ϕ3 )−2 ≤ C δr−1 + Cλ−1/2 r−1 ≤ 2C δr−1 . Moreover, in view of (2.5) we have |ϕ | ≤ Cr−2 ϕ2 (1 + λ1/2 ϕ3 )−1 ≤ Cλ1/2 r−1 ϕ . On the other hand, −2 1/2 (ϕ2 + ϕ2 ϕ3 )−1 , ϕ2 ≤ 4ϕ2 1 + Cr 2 )(1 + λ
and hence (2.8) follows in view of (2.2) and (2.4). Furthermore, we have ϕ = ϕ1 + 2λ−1/2 r−3 ϕ2 (1 + λ1/2 ϕ3 )−1 − 2λ−1/2 r−2 ϕ2 (1 + λ1/2 ϕ3 )−1 +2r−2 ϕ3 (1 + λ1/2 ϕ3 )−2 + λ−1/2 r−1 ϕ2 (1 + λ1/2 ϕ3 )−1 1/2 −r−1 ϕ3 (1 + λ1/2 ϕ3 )−2 + 2λ1/2 r−2 ϕ2 ϕ3 )−3 , 3 (1 + λ
and hence |ϕ | ≤ Cλ1/2 r−1 . On the other hand, in view of (2.5) we have |ϕ | ≤ Cr−2 ϕ2 + λ−1/2 ϕ1 + Cλ1/2 r−2 ϕ2 (1 + λ1/2 ϕ3 )−1 ≤ Cλr−1 ϕ , which proves (2.9). In the same way, −2 3/2 −1 ϕ2 (1+λ1/2 ϕ3 )−1 +Cλ−1/2 (|ϕ2 |+|ϕ2 |+|ϕ r ϕ. |ϕ(4) | ≤ |ϕ 1 |+Cλr 2 |) ≤ Cλ
To prove (2.11) observe that 2λϕ2 + ϕ ≥ 2λϕ21 + 2r−2 ϕ22 (1 + λ1/2 ϕ3 )−2 + ϕ1 − λ−1/2 r−2 ϕ2 (1 + λ1/2 ϕ3 )−1 − r−1 ϕ2 ϕ3 (1 + λ1/2 ϕ3 )−2 .
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For r ≥ a1 + 1 we have ϕ1 = ϕ1 = 0 and hence, in view of (2.3), 2λϕ2 + ϕ ≥ (1 − 2rϕ3 − λ−1/2 )r−2 (1 + λ1/2 ϕ3 )−2 ≥ C r−1 ϕ . For r < a1 + 1 we have ϕ3 = 0 and hence 2λϕ2 + ϕ ≥ 2λϕ21 + ϕ1 + r−2 ϕ22 . Since ϕ1 (a1 + 1) = 0, there exists a0 < a1 + 1 such that |ϕ1 | ≤ (2r)−2 ϕ22 for a0 ≤ r ≤ a1 + 1. Hence, for a0 ≤ r ≤ a1 + 1, 2λϕ2 + ϕ ≥ λϕ2 ≥ Cλ1/2 r−1 ϕ .
(2.12)
For r1 ≤ r ≤ a0 , we have |ϕ1 | ≤ Cϕ21 , which again implies (2.12).
Throughout this section · and ·, · will denote the norm and the scalar product on L2 (S1 ), while the Sobolev space H 1 (X1 , dVolg ) will be equipped with the semiclassical norm given by u2H 1 (X1 ,dVolg ) =
u2L2 (X1 ,dVolg )
+
Dr u2L2 (X1 ,dVolg )
∞
+ r1
p1 g1ij Dθi u(r, ·), Dθj u(r, ·)dr,
i,j
where Dr = (iλ)−1 ∂r , Dθj = (iλ)−1 ∂θj . Denote by L2 (X1 ) and H 1 (X1 ) the spaces equipped with the norms ∞ u2L2(X1 ) = u(r, ·)2 dr, r1 ∞ ij u(r, ·)2 + Dr u(r, ·)2 + u2H 1 (X1 ) = g1 Dθi u(r, ·), Dθj u(r, ·) dr. r1
i,j
It is easy to see that 1/2
uL2 (X1 ,dVolg ) = p1 uL2 (X1 ) ,
1/2
uH 1 (X1 ,dVolg ) p1 uH 1 (X1 ) .
Finally, given an a ≥ r1 and functions u(r, θ), v(r, θ), we denote uL2(∂X1,a ) := u(a, ·),
u, vL2 (∂X1,a ) := u(a, ·), v(a, ·),
u2H 1 (∂X1,a ) := u(a, ·)2 +
g1ij Dθi u(a, ·), Dθj u(a, ·).
i,j
It is clear from the definition of the function ϕ above that there exists an a ≥ r1 such that ϕ (r) = λ−1 r−1 for r ≥ a. The main result in this section is the following
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Theorem 2.2 Let u ∈ H 2 (X1 , dVolg ), u = 0, ∂r u = 0 on ∂X1 , be such that rs (∆X1 − λ2 + iε)u ∈ L2 (X1 , dVolg ) for λ > 0, 0 < ε ≤ 1 and 0 < s − 1/2 1. Then, for a suitable choice of the parameter δ > 0, there exist constants C1 , C2 , λ0 > 0 (independent of λ and ε) so that for λ ≥ λ0 we have eλ(ϕ(r)−ϕ(a)) u2H 1 (X1 \X1,a ,dVolg ) + r−s u2H 1 (X1,a ,dVolg ) ≤ C1 λ−2 eλ(ϕ(r)−ϕ(a))(∆X1 − λ2 + iε)u2L2 (X1 \X1,a ,dVolg ) +C1 λ−2 rs (∆X1 − λ2 + iε)u2L2(X1,a ,dVolg ) − C2 λ−1 Im∂r u, uL2 (∂X1,a ) . (2.13) Proof. Denote 1/2
−1/2
P = p1 (λ−2 ∆X1 − 1 + iε)p1
= Dr2 + Lr − 1 + V + iε,
where 0 < ε = O(λ−2 ), Lr = λ−2 Λ1,r , V = λ−2 q1 , and Pϕ = eλϕ P e−λϕ = P − ϕ (r)2 + λ−1 ϕ (r) + 2iϕ (r)Dr . We will first prove the following Proposition 2.3 Let u ∈ H 2 (X1 \ X1,a ), u = 0, ∂r u = 0 on ∂X1 ∪ ∂X1,a . Then, there exist constants C, λ0 > 0 (independent of λ and ε) so that for λ ≥ λ0 we have (2.14) (ϕ /r)1/2 uH 1 (X1 \X1,a ) ≤ Cλ1/2 Pϕ uL2 (X1 \X1,a ) . Proof. Let ψ(r) ∈ C ∞ ([r1 , a]) be a real-valued function. Integrating by parts one can easily get the identity Re ψPϕ u, uL2 (X1 \X1,a ) = ψDr u, Dr uL2 (X1 \X1,a ) + ψLr u, uL2 (X1 \X1,a ) −(ψ + ψϕ2 − λ−2 q1 + λ−1 ϕ ψ + 2−1 λ−2 ψ )u, uL2 (X1 \X1,a ) . Set
(2.15)
F (r) = −(Lr − 1 + W )u(r, ·), u(r, ·) + Dr u(r, ·)2 ,
where W = λ−2 q1 − ϕ2 + λ−1 ϕ . We have F (r) = = −2Re Lr u(r, ·), u (r, ·) − 2Re Dr2 u(r, ·), u (r, ·) + 2Re (1 − W )u(r, ·), u (r, ·) − [∂r , Lr ]u(r, ·), u(r, ·) − W u(r, ·), u(r, ·) = −2Re Pϕ u(r, ·), u (r, ·) + 4λϕ Dr u(r, ·)2 − 2εIm u(r, ·), u (r, ·) − [∂r , Lr ]u(r, ·), u(r, ·) − W u(r, ·), u(r, ·). Multiplying this identity by ϕ and integrating with respect to r lead to a a a ϕ F dr = −2Re ϕ Pϕ u, u dr + 4λ ϕ Dr u2 dr (2.16) r1 r1 r1 a a a ϕ u, u dr − ϕ [∂r , Lr ]u, udr − ϕ W u, udr. − 2εIm r1
r1
r1
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On the other hand, we have
a
a
ϕ F dr
ϕ F dr = − r1
a
r1
a
ϕ Lr u, udr −
= Re r1
r1 a
+
a
ϕ Dr u, Dr udr −
ϕ (1 − W )u, udr
r1 a
ϕ Pϕ u, udr − 2
= Re
r1 a
683
ϕ Dr u, Dr udr
r1
(λ−1 ϕ2 + λ−1 ϕ ϕ + 2−1 λ−2 ϕ(4) )u, udr,
(2.17)
r1
where we have used (2.15) with ψ = ϕ . Combining (2.16) and (2.17) we get the identity a a 2 (2λϕ + ϕ )Dr u, Dr udr − ϕ [∂r , Lr ]u, udr 2
r1 a
r1 a
+ r1
r1 a
ϕ Pϕ u, u dr + Re
= 2Re
a
ϕ Pϕ u, udr + 2εIm r1
ϕ u, u dr
r1
(−2ϕ2 ϕ + λ−1 ϕ2 + 2λ−1 ϕ ϕ + 2−1 λ−2 ϕ(4) + λ−2 ϕ q1 )u, udr. (2.18)
It is easy to see that (1.3) implies −[∂r , Lr ] ≥
C Lr , r
C > 0,
(2.19)
and hence in view of (1.2) and Lemma 2.1 we conclude from (2.18) a a 2 (ϕ /r)1/2 Dr u2 dr + (ϕ /r)1/2 L1/2 r u dr r1
r1
a
≤ O(λ)
Pϕ u2 dr + Cδ
r1
a
(ϕ /r)1/2 u2 dr,
(2.20)
r1
for λ ≥ λ0 (a, δ) 1, where C > 0 does not depend on λ, δ and a. On the other hand, by (2.15) used with ψ = r−1 ϕ we have a −1 r ϕ (1 + ϕ2 + λ−1 ϕ − λ−1 r−1 ϕ + λ−2 r−2 − λ−2 q1 ) r1
−λ−2 r−2 ϕ + 2−1 λ−2 r−1 ϕ u, u dr a a a = r−1 ϕ Dr u, Dr udr + r−1 ϕ Lr u, udr − Re Pϕ u, r−1 ϕ udr, r1
r1
r1
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and hence, in view of Lemma 2.1 and (1.2), we get 1 a (ϕ /r)1/2 u2 dr 4 r1 a a 1/2 2 1/2 1/2 2 (ϕ /r) Dr u dr + (ϕ /r) Lr u dr + ≤ r1
r1
(2.21) a
Pϕ u2 dr.
r1
Now (2.14) follows from (2.20) and (2.21), provided δ > 0 is taken small enough. Proposition 2.4 Let u ∈ H 2 (X1,a ) be such that rs P u ∈ L2 (X1,a ) for 1/2 < s ≤ (1 + δ0 )/2. Then, ∀0 < γ 1 there exist constants C1 , C2 , λ0 > 0 (which may depend on γ but are independent of λ and ε) so that for λ ≥ λ0 we have r−s u2H 1 (X1,a+1 ) ≤ C1 λ2 rs P u2L2 (X1,a ) − C2 λ−1 Im ∂r u, uL2 (∂X1,a ) + γu2H 1 (X1,a \X1,a+1 ) . (2.22) Proof. Choose a real-valued function φ ∈ C ∞ (R), 0 ≤ φ ≤ 1, such that φ(r) = 0 for r ≤ a + 1/2, φ(r) = 1 for r ≥ a + 2/3 and φ (r) ≥ 0, ∀r. Integrating by parts we get r−2s (Lr − 1 + V )φu, φuL2 (X1,a ) + r−s Dr (φu)2L2 (X1,a ) = Re r−2s P (φu), φuL2 (X1,a ) + 2sλ−2 Re r−2s−1 (φu) , φuL2 (X1,a ) , and hence
−2s r (Lr − 1 + V )φu, φuL2 (X1,a ) + r−s Dr (φu)2L2 (X1,a )
≤ O(λ)P (φu)2L2 (X1,a ) + O(λ−1 ) r−s φu2L2 (X1,a ) + r−s Dr (φu)2L2 (X1,a ) . (2.23) We also have εu2L2(X1,a ) = Im P u, uL2 (X1,a ) − λ−2 Im u , uL2 (∂X1,a ) ≤ γ −1 λrs P u2L2 (X1,a ) + γλ−1 r−s u2L2 (X1,a ) − λ−2 Im u , uL2 (∂X1,a ) , ∀γ > 0, and Dr (φu)2L2 (X1,a ) ≤ 2φu2L2 (X1,a ) + P (φu)2L2 (X1,a ) ≤ 2u2L2(X1,a ) + P u2L2(X1,a ) + O(λ−2 )φ1 u2H 1 (X1,a ) , where φ1 ∈ C0∞ ([a, a + 1]), φ1 = 1 on [a + 1/3, a + 3/4]. Hence, ελ φu2L2 (X1,a ) + Dr (φu)2L2 (X1,a )
(2.24)
≤ Oγ (λ2 )rs P u2L2(X1,a ) + γr−s u2H 1 (X1,a ) − 3λ−1 Im u , uL2 (∂X1,a ) ,
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∀γ > 0. Set E(r) = −(Lr − 1 + V )φu(r, ·), φu(r, ·) + Dr (φu)(r, ·)2 . We have E (r) = −[∂r , Lr ]φu(r, ·), φu(r, ·) − V φu(r, ·), φu(r, ·) −2εIm φu(r, ·), (φu) (r, ·) − 2λIm P (φu)(r, ·), Dr (φu)(r, ·) = −[∂r , Lr ]φu(r, ·), φu(r, ·) − V φu(r, ·), φu(r, ·) −2εIm φu(r, ·), (φu) (r, ·) − 2λIm φP u(r, ·), Dr (φu)(r, ·) −2λIm [P, φ]u(r, ·), φDr u(r, ·) − 2λIm [P, φ]u(r, ·), [Dr , φ]u(r, ·). Since
[P, φ] = [Dr2 , φ] = −λ−2 φ − 2iλ−1 φ Dr ,
we obtain in view of (2.19),
C Lr (φu)(r, ·), φu(r, ·) − ελ φu(r, ·)2 + Dr (φu)(r, ·)2 r
−O(γ)r−2s φu(r, ·)2 + Dr (φu)(r, ·)2
−O(λ−1 ) φ1 u(r, ·)2 + φ1 Dr u(r, ·)2
E (r) ≥
+4φφ Dr u(r, ·)2 − Oγ (λ2 )r2s P u(r, ·)2 . Since φφ ≥ 0, we deduce
C Lr (φu)(r, ·), φu(r, ·) − ελ φu(r, ·)2 + Dr (φu)(r, ·)2 r
−O(γ)r−2s u(r, ·)2 + Dr u(r, ·)2 − Oγ (λ2 )r2s P u(r, ·)2 . (2.25)
E (r) ≥
Integrating (2.25) from t ≥ a to +∞ and using that Lr ≥ 0 and (2.24), we get E(t) ≤ O(γ)r−s u2H 1 (X1,a ) + Oγ (λ2 )rs P u2L2 (X1,a ) − 3λ−1 Im u , uL2 (∂X1,a ) , (2.26) ∀γ > 0. Multiplying (2.26) by t−2s and integrating from a to +∞ yield (with a constant C > 0): ∞ r−2s E(r)dr ≤ O(γ)r−s u2H 1 (X1,a ) a
2
+Oγ (λ )rs P u2L2(X1,a ) − Cλ−1 Im u , uL2 (∂X1,a ) ,
(2.27)
∀γ > 0. On the other hand, multiplying (2.25) by r1−2s , integrating from a to +∞, using (2.23), (2.24) and the identity ∞ ∞ 1−2s r E (r)dr = (2s − 1) r−2s E(r)dr, a
a
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we obtain (with a new constant C > 0): 2 −s u2H 1 (X1,a ) r−s L1/2 r (φu)L2 (X1,a ) ≤ O(γ)r
+Oγ (λ2 )rs P u2L2(X1,a ) − Cλ−1 Im u , uL2 (∂X1,a ) ,
(2.28)
∀γ > 0. Combining (2.23), (2.27) and (2.28), we get (with possibly a new constant C > 0): r−s φu2H 1 (X1,a ) ≤ O(γ)r−s u2H 1 (X1,a ) +Oγ (λ2 )rs P u2L2(X1,a ) − Cλ−1 Im u , uL2 (∂X1,a ) , ∀0 < γ 1, which clearly implies (2.22).
(2.29)
Let u ∈ H 2 (X1 ), u = 0, ∂r u = 0 on ∂X1 , be such that rs P u ∈ L2 (X1 ). Choose a function χ ∈ C ∞ (X1 ) such that χ = 1 on X1 \ X1,a+2 , χ = 0 on X1,a+3 . Applying Proposition 2.3 to the function eλϕ χu (with a replaced by a + 3), we get eλϕ u2H 1 (X1 \X1,a+2 ) ≤ O(λ2 )eλϕ P u2L2 (X1 \X1,a+3 ) + O(1)eλϕ u2H 1 (X1,a+2 \X1,a+3 ) .
(2.30)
Since 1 ≤ eλ(ϕ(r)−ϕ(a)) ≤ Const for a ≤ r ≤ a + 3, we deduce eλ(ϕ(r)−ϕ(a))u2H 1 (X1 \X1,a ) + u2H 1 (X1,a \X1,a+2 ) ≤ O(λ2 )eλ(ϕ(r)−ϕ(a))P u2L2 (X1 \X1,a ) +O(λ2 )P u2L2 (X1,a \X1,a+3 ) + O(1)u2H 1 (X1,a+2 \X1,a+3 ) .
(2.31)
It is easy to see that (2.13) follows from combining (2.22) and (2.31).
3 Uniform a priori estimates on X2 The purpose of this section is to prove the following Proposition 3.1 Let u ∈ H 2 (X2 , dVolg ), u = 0, ∂r u = 0 on ∂X2 . Then ∀δ > 0, 0 < ε ≤ 1, we have r−1−δ eλr
−2δ
uH 1 (X2 ,dVolg ) ≤ Cλ−3/2 eλr
−2δ
(∆X2 − λ2 + iε)uL2(X2 ,dVolg ) , (3.1)
for λ ≥ λ0 with constants C, λ0 > 0 independent of λ, ε and u but depending on δ. Proof. Define the spaces L2 (X2 ) and H 1 (X2 ) analogously to L2 (X1 ) and H 1 (X1 ) introduced in the previous section. Denote ϕ(r) = r−2δ , w = eλϕ u, and 1/2
−1/2
P := p2 (λ−2 ∆X2 − 1 + iε)p2
= Dr2 + Lr − 1 + V + iε,
Pϕ = eλϕ P e−λϕ = P − ϕ (r)2 + λ−1 ϕ (r) + 2iϕ (r)Dr ,
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where 0 < ε = O(λ−2 ), Lr = λ−2 Λ2,r , V = λ−2 q2 . Note that (1.3) implies [∂r , Lr ] ≥
C Lr , r
C > 0.
(3.2)
Clearly, (3.1) is equivalent to the estimate r−1−δ wH 1 (X2 ) ≤ O(λ1/2 )Pϕ wL2 (X2 ) .
(3.3)
Denote by Pϕ∗ the adjoint operator of Pϕ with respect to the scalar product in L2 (X2 ), and set Re Pϕ =
∗ Pϕ +Pϕ , 2
Im Pϕ =
Re Pϕ = Dr2 + Lr − 1 − ϕ (r)2 + V,
∗ Pϕ −Pϕ . 2i
We have
Im Pϕ = ϕ (r)Dr + Dr ϕ (r) + ε.
In view of (1.2) and (3.2), and taking into account that ϕ (r) = −2δr−2δ−1 , ϕ (r) = 2δ(2δ+1)r−2δ−2 , ϕ (r) = −2δ(2δ+1)(2δ+2)r−2δ−3 , it is easy to see that we have, in view of (3.2) and (1.2), λPϕ w2L2 (X2 ) = λ(Re Pϕ )w2L2 (X2 ) + λ(Im Pϕ )w2L2 (X2 ) + iλ[Re Pϕ , Im Pϕ ]w, wL2 (X2 ) ≥ iλ[Re Pϕ , Im Pϕ ]w, wL2 (X2 ) ≥ 2ϕ Dr w, Dr wL2 (X2 ) + 4−ϕ [∂r , Lr ]w, wL2 (X2 ) + 4ϕ2 ϕ w, wL2 (X2 ) − 2ϕ V w, wL2 (X2 )
− O(λ−1 ) r−1−δ Dr wL2 (X2 ) + r−1−δ wL2 (X2 ) 2 −1 ≥ Cr−1−δ Dr w2L2 (X2 ) + Cr−1−δ L1/2 )r−1−δ w2H 1 (X2 ) . r wL2 (X2 ) − O(λ
(3.4) On the other hand, integrating by parts leads to the identity Rer−2−2δ Pϕ w, wL2 (X2 ) = r−1−δ Dr w2L2 (X2 ) +r−2−2δ (Lr −1+V −ϕ2 −4δ(δ+1)λ−1 r−2−2δ −(δ+1)(2δ+3)λ−2 r−2 )w, wL2 (X2 ) , and hence 1 −1−δ 2 r wL2 (X2 ) 2 −2−2δ 2 ≤ r−1−δ Dr w2L2 (X2 ) + r−1−δ L1/2 Pϕ w, wL2 (X2 ) . r wL2 (X2 ) + r Since
−2−2δ 1 r Pϕ w, wL2 (X2 ) ≤ r−1−δ w2L2 (X2 ) + Pϕ w2L2 (X2 ) , 4 we conclude 1 −1−δ 2 2 2 r wL2 (X2 ) ≤ r−1−δ Dr w2L2 (X2 ) + r−1−δ L1/2 r wL2 (X2 ) + Pϕ wL2 (X2 ) . 4 (3.5) Now (3.3) follows from (3.4) and (3.5).
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4 Proof of Theorem 1.1 Let (M0 , g0 ) be a compact, connected Riemannian manifold with a C ∞ -smooth boundary ∂M0 and a metric g0 of class C ∞ (M 0 ). Denote by ∆M0 the (positive) Laplace-Beltrami operator on (M0 , g0 ) and let U ⊂ M0 , U = ∅, be an arbitrary Γ = ∅, Γ = ∅, open domain such that ∂U ∩ ∂M0 = ∅. Suppose that ∂M0 = Γ ∪ Γ, Γ ∩ Γ = ∅, and given 0 < ε0 1 denote M0,ε0 = M0 \ {x ∈ M0 : dist(x, ∂M0 ) ≤ 0,ε0 = M0 \ {x ∈ M0 : dist(x, Γ) ≤ ε0 }. Let U ⊂ M0,2ε0 . The following ε0 }, M proposition is proved in [8] using the interpolation inequalities of Lebeau-Robbiano [4], [5] (see Theorem 3.2 of [8]) and this is why we omit the proof. Proposition 4.1 Let u ∈ H 2 (M0 ) satisfy either Dirichlet or Neumann boundary conditions on Γ. Then, ∀β > 0 ∃Cβ , γβ > 0 (independent of u and λ below but depending on U ) so that we have uH 1 (M f0,ε
0)
≤ Cβ eγβ |λ| (∆M0 − λ2 )uL2 (M0 )
+Cβ eγβ |λ| uH 1 (U) + e−β|λ| uH 1 (M0 \M f0,ε ) , 0
λ ∈ C, |λ| 1.
(4.1)
2 Let u ∈ D(G) be such that χ−1 s1 ,s2 u ∈ L (M, dVolg ), where s1 and s2 are as in Theorem 1.1. Let χ2 ∈ C ∞ (M ), χ2 = 0 on M \ X2,r2 +1 , χ2 = 1 on X2,r2 +2 . Applying Proposition 3.1 (with δ = s2 − 1) to χ2 u yields
r−s2 u2H 1 (X2,r
2 +2 ,dVolg )
≤ ec0 λ (∆g − λ2 + iε)u2L2 (X2 ,dVolg )
+ec0 λ u2H 1 (X2,r
2 +1 \X2,r2 +2 ,dVolg )
.
(4.2)
Let χ1 ∈ C ∞ (M ), χ1 = 1 on M \ X1,r1 +2 , χ1 = 0 on X1,r1 +3 . By Proposition 4.1 applied to the function χ1 u we get χ1 u2H 1 (M\X2,r
2 +2 ,dVolg )
≤ Cβ eγβ λ (∆g − λ2 + iε)χ1 u2L2 (M\X2,r
+e−βλ u2H 1 (X2,r
2 +2 \X2,r2 +3 ,dVolg )
2 +3 ,dVolg )
,
(4.3)
∀β > 0 with Cβ , γβ > 0 independent of λ, ε and u. Hence, u2H 1 (M\(X1,r
1 +2 ∪X2,r2 +2 ),dVolg )
≤ Cβ eγβ λ (∆g − λ2 + iε)u2L2 (M\(X1,r +Cβ λ2 eγβ λ u2H 1 (X1,r ∀β > 0.
1 +2 \X1,r1 +3 ,dVolg )
1 +3 ∪X2,r2 +3 ),dVolg )
+ e−βλ u2H 1 (X2,r
2 +2 \X2,r2 +3 ,dVolg )
, (4.4)
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Combining (4.2) and (4.4), for λ 1, we obtain r−s2 u2H 1 (X2,r
2 +2 ,dVolg )
+ u2H 1 (M\(X1,r
1 +2 ∪X2,r2 +2 ),dVolg )
≤ ec0 λ (∆g − λ2 + iε)u2L2 (X2 ,dVolg ) +e2γ1 λ (∆g − λ2 + iε)u2L2 (M\(X1,r +e2γ1 λ u2H 1 (X1,r
1 +3 ∪X2,r2 +3 ),dVolg )
1 +2 \X1,r1 +3 ,dVolg )
,
(4.5)
with a constant γ1 > 0 independent of λ, ε and u. Let r1 < b1 < b2 < r1 + 1 be such that ϕ(b1 ) < ϕ(b2 ) < 0 and choose χ 1 ∈ C ∞ (M ), χ 1 = 0 on M \ X1,b1 , χ 1 = 1 on X1,b2 . By Theorem 2.2 applied to χ 1 u (with γ0 = γ1 + 1, s = s1 ), we get eλϕ u2H 1 (X1,b \X1,a ,dVolg ) + e2λϕ(a) r−s1 u2H 1 (X1,a ,dVolg ) 2
−2
≤ O(λ
)eλϕ (∆g − λ2 + iε)u2L2 (X1 \X1,a ,dVolg )
+O(λ−2 )e2λϕ(a) rs1 (∆g − λ2 + iε)u2L2 (X1,a ,dVolg ) −Cλ−1 e2λϕ(a) Im ∂r u, uL2 (∂X1,a ) + e−cλ u2H 1 (X1,b
1
\X1,b2 ,dVolg ) ,
(4.6)
with some c, C > 0. Since ϕ(r) ≥ γ1 + 1 for r ≥ r1 + 2, by combining (4.5) and (4.6) one can absorb the last terms in the right-hand sides and conclude r−s2 u2H 1 (X2,r
2 +2 ,dVolg )
+eλϕ u2H 1 (X1,b
2
+ u2H 1 (M\(X1,r
\X1,a ,dVolg )
1 +2 ∪X2,r2 +2 ),dVolg )
+ e2λϕ(a) r−s1 u2H 1 (X1,a ,dVolg )
≤ ec0 λ (∆g − λ2 + iε)u2L2 (X2 ,dVolg ) +e2γ1 λ (∆g − λ2 + iε)u2L2 (M\(X1,r
1 +3 ∪X2,r2 +3 ),dVolg )
+O(λ−2 )eλϕ (∆g − λ2 + iε)u2L2(X1 \X1,a ,dVolg ) +O(λ−2 )e2λϕ(a) rs1 (∆g − λ2 + iε)u2L2 (X1,a ,dVolg ) −Cλ−1 e2λϕ(a) Im ∂r u, uL2 (∂X1,a ) .
(4.7)
On the other hand, by Green’s formula we have − Im ∂r u, uL2 (∂X1,a ) = −Im (∆g − λ2 + iε)u, uL2 (M\X1,a ,dVolg ) − εu2L2 (M\X1,a ,dVolg ) 2 2 ≤ e−βλ ρs2 u2L2 (M\X1,a ,dVolg ) + eβλ ρ−1 s2 (∆g − λ + iε)uL2 (M\X1,a ,dVolg ) , (4.8)
∀β > 0, where ρs ∈ C ∞ (M ), ρs = r−s on X2,r2 +1 , ρs = 1 on M \ X2 .
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Combining (4.7) and (4.8) leads to the estimate e−c1 λ ρs2 u2H 1 (M\X1,a ,dVolg ) + r−s1 u2H 1 (X1,a ,dVolg ) 2 2 ≤ ec2 λ ρ−1 s2 (∆g − λ + iε)uL2 (M\X1,a ,dVolg )
+O(λ−2 )rs1 (∆g − λ2 + iε)u2L2(X1,a ,dVolg ) ,
(4.9)
with some constants c1 , c2 > 0. Hence, 2 χs1 ,s2 uL2 (M,dVolg ) ≤ Ceγλ χ−1 s1 ,s2 (∆g − λ + iε)uL2 (M,dVolg )
(4.10)
for λ ≥ λ0 , with some constants C, λ0 , γ > 0 independent of λ, ε and u, which implies the existence of the limit in Theorem 1.1 as well as the bound (1.4) (with z = λ2 ). Let now (∆g − λ2 + iε)u = 0 in M \ X1,a . Then (4.9) yields r−s1 uL2 (X1,a ,dVolg ) ≤ O(λ−1 )rs1 (∆g − λ2 + iε)uL2(X1,a ,dVolg ) ,
(4.11)
which clearly implies (1.5).
References [1] V. Bruneau and V. Petkov, Semiclassical resolvent estimates for trapping perturbations, Commun. Math. Phys. 213, 413–432 (2000). [2] N. Burq, D´ecroissance de l’´energie locale de l’´equation des ondes pour le probl`eme ext´erieur et absence de r´esonance au voisinage du r´eel, Acta Math. 180, 1–29 (1998). [3] N. Burq, Lower bounds for shape resonances widths of long-range Schr¨ odinger operators, American J. Math., to appear. [4] G. Lebeau and L. Robbiano, Contrˆ ole exact de l’´equation de la chaleur, Commun. Partial Diff. Equations 20, 335–356 (1995). [5] G. Lebeau and L. Robbiano, Stabilization de l’´equation des ondes par le bord, Duke Math. J. 86, 465–490 (1997). [6] G. Vodev, On the exponential bound of the cutoff resolvent, Serdica Math. J. 26, 49–58 (2000). [7] G. Vodev, Exponential bounds of the resolvent for a class of noncompactly supported perturbations of the Laplacian, Math. Res. Lett. 7, 287–298 (2000). [8] G. Vodev, Uniform estimates of the resolvent of the Laplace-Beltrami operator on infinite volume Riemannian manifolds with cusps, Commun. Partial Diff. Equations, 27, 1437–1465 (2002).
Vol. 3, 2002
Uniform Estimates of the Resolvent of the Laplace-Beltrami Operator II
Fernando Cardoso1 Universidade Federal de Pernambuco Departamento de Matem`atica CEP. 50540-740 Recife-Pe Brazil email: [email protected] Georgi Vodev Universit´e de Nantes D´epartement de Math´ematiques UMR 6629 du CNRS 2, rue de la Houssini`ere, BP 92208, 44072 Nantes Cedex 03 France email: [email protected] Communicated by Bernard Helffer submitted 13/11/01, accepted 14/05/02
To access this journal online: http://www.birkhauser.ch
1 Partially
supported by CNPq (Brazil)
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Ann. Henri Poincar´e 3 (2002) 693 – 710 c Birkh¨ auser Verlag, Basel, 2002 1424-0637/02/040693-18
Annales Henri Poincar´ e
R´esonances dans des Petits Domaines ´ pr`es d’une Energie Critique J.-F. Bony R´esum´e. On travaille dans le cadre de l’analyse semi-classique. Consid´erons p(x, hDx ), une perturbation de −h2 ∆ qui est analytique ` a l’infini. On suppose que dans la surface d’´energie E0 > 0, les points critiques du symbole p(x, ξ) forment une sous-vari´et´ e C et que p est non d´eg´ en´ er´ e dans l’espace normal ` a C. En utilisant les r´esultats de [6] et [18], on obtient une majoration du nombre de r´ esonances dans des disques de rayon δ centr´es en E proche de E0 , o` u δ satisfait Ch < δ < 1/C pour une constante C > 0. En g´ en´ eralisant la formule de trace de Sj¨ ostrand qui exprime la trace d’une diff´erence d’op´erateurs en fonction des r´ esonances, on trouve une minoration du nombre de r´esonances proches de E0 .
1 R´esultat Lax et Phillips [15] ont d´efini, de fa¸con rigoureuse, les r´esonances pour des perturbations a` support compact du laplacien. Ils ont montr´e que la matrice de diffusion a un prolongement m´eromorphe. Les r´esonances sont les pˆoles de ce prolongement. Dans cet article, les r´esonances seront d´efinies a` l’aide des dilatations analytiques (voir [1], [24] et [26]). Il existe beaucoup de r´esultats concernant les majorations du nombre de r´esonances. Dans le cadre classique, on peut, par exemple, citer Melrose [16], Sj¨ ostrand [26], Vodev [33], Zworski [34]. Sous diverses hypoth`eses, ils ont progressivement prouv´e la borne a` haute fr´equence Card {λ r´esonance; |λ| ≤ r} = O(rn ). Dans le cadre semi-classique, on cherche `a estimer le nombre de r´esonances de P pr`es d’un niveau d’´energie E0 > 0 quand h tend vers 0. Typiquement P est de la forme P = −h2 ∆ + V (x). Sous des hypoth`eses tr`es faibles, il y a au plus O(h−n ) r´esonances dans un compact fixe autour de E0 (on pourra lire [23]). Dans [22], Sj¨ ostrand a trouv´e, sous certaines hypoth`eses g´eom´etriques, des majorations plus fines dans des bandes de largeur δ autour de l’axe r´eel avec Ch < δ < 1/C pour une constante C > 0. Si E0 est une valeur non-critique d’un op´erateur autoadjoint agissant sur une vari´et´e compacte, il y a au plus O(δh−n ) valeurs propres dans un disque de rayon δ > Ch autour de E0 . Petkov et Zworski [18] ont prouv´e une majoration semblable dans le cadre des r´esonances (Ils ont aussi d´emontr´e dans le cas classique que le nombre de r´esonances dans r + Ω, o` u Ω est un petit rectangle, est major´e
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par Crn−1 (voir [17])). D’autre part, il existe des r´esultats sur l’asymptotique du nombre de valeurs propres pour un op´erateur sur une vari´et´e compacte pr`es d’une surface d’´energie critique (voir [6] et [13]). A l’aide de ces r´esultats, on d´emontrera une majoration du nombre de r´esonances pr`es d’une valeur critique. Il est plus ardu de minorer le nombre de r´esonances. On utilise une formule de trace locale qui g´en´eralise la formule de Poisson : f (λj ), tr(f (P ) − f (P0 )) ≈ o` u f est une fonction holomorphe dans une certaine partie Ω de C, o` u les λj sont les r´esonances de P dans Ω et o` u P0 est un op´erateur qui approche P `a l’infini et qui n’a pas de r´esonances dans Ω. Ce genre de formule de Poisson a ´et´e prouv´ee par Bardos, Guillot et Ralston [2], puis par de nombreux auteurs [18] et [25]. En calculant la trace de la diff´erence (f (P ) − f (P0 )) par une autre m´ethode et en utilisant un argument taub´erien, on prouve une minoration du nombre de r´esonances (voir [27] et [28]). Sous certaines hypoth`eses il est possible de connaˆıtre tr`es pr´ecis´ement les r´esonances proches d’un niveau d’´energie critique. Si P est analytique et si, en dehors d’un point critique non d´eg´en´er´e, le flot du champ hamiltonien de P n’a ostrand [11], pas de trajectoire capt´ee dans la surface d’´energie E0 , Helffer et Sj¨ pour un puits dans une isle, et Sj¨ostrand [21], dans le cas g´en´eral, ont montr´e qu’il y a un nombre fixe de r´esonances dans un disque de taille Ch autour de E0 (voir aussi l’article de Ka¨ıdi et Kerdelhu´e [14]). Dans le cas de la dimension 1, Fujii´e et Ramond [9] ont, entre autres, montr´e qu’il y a de l’ordre de δ ln(1/δ)h−1 r´esonances dans un domaine de taille δ autour de E0 si P = −h2 ∆ + V (x) o` u V (x) est un potentiel a` deux bosses de hauteur E0 . On se place dans le cadre de l’analyse semi-classique et h ∈]0, h0 [ d´esigne le petit param`etre. On utilise la notation z = 1 + |z|2 . Soit m(u; h) une fonction positive sur R2n ×]0, h0 [. On dit que m est une fonction d’ordre si il existe une constante C > 0 telle que m(u; h) ≤ C u − v C m(v; h) pour tous u, v ∈ R2n . On dit qu’une fonction d(x, ξ; h) ∈ C ∞ (R2n ) est un symbole de classe Sa,b (m) si, pour tout α, β ∈ Nn , il existe Cα,β > 0 tel que α β (1.1) ∂x ∂ξ d(x, ξ; h) ≤ Cα,β m(x, ξ; h) x −a|α| ξ −b|β| . On peut alors d´efinir la h-quantification de Weyl de d, not´ee D = Op(d). Pour f ∈ C0∞ (Rn ), on a x + y 1 i(x−y,ξ)/h , ξ; h f (y) dydξ. d e (Op(d)f ) (x) = (2πh)n 2 d(x, ξ; h) est appel´e le symbole de Weyl de l’op´erateur h-pseudo-diff´erentiel D. Si d(x, ξ; h) a un d´eveloppement en puissances de h de la forme d∼ dj (x, ξ)hj , j≥0
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avec dj (x, ξ) ∈ Sa,b (m), on dit que d est un symbole classique et on note d ∈ cl Sa,b (m). d0 (x, ξ) est alors le symbole principal. cl ( ξ 2 )) un op´erateur h-pseudo-diff´erentiel qui co¨ıncide en Soit P ∈ Op(S0,0 dehors d’un compact avec un op´erateur h-diff´erentiel d’ordre deux, aα (x; h)(hDx )α , |α|≤2
o` u aα (x; h) ne d´epend pas de h pour |α| = 2. On admet que P est formellement autoadjoint, c’est a` dire que : ∞ n (P u)vdx = u(P v)dx, ∀u, v ∈ C0 (R ) Rn
Rn
ou que p(x, ξ; h) est r´eel. On suppose que l’op´erateur P est elliptique au sens qu’il existe une constante C > 0 telle que, pour tout x, ξ ∈ Rn , p0 (x, ξ) ≥ |ξ|2 /C − C. Pour d´efinir les r´esonances, on suppose qu’il existe θ > 0 et r > 0 tels que les coefficients aα (z; h) ont un prolongement holomorphe dans l’ensemble Γ = {z ∈ Cn ; |Im z| ≤ θ Re z et |Re z| ≥ r} de classe S0cl (1) (cela signifie que (1.1) est v´erifi´e pour x ∈ Γ et que les d´eveloppements asymptotiques en puissances de h s’´etendent `a Γ de fa¸con holomorphe). En utilisant la formule de Cauchy, on peut cl ( ξ 2 ) pour |x| assez grand. On montrer que p(x, ξ; h) est un symbole de classe S1,1 2 suppose ensuite que l’op´erateur P “tend” vers −h ∆ quand z tend vers +∞ dans le domaine complexe Γ. Cela signifie que aα (z; h)ξ α −→ ξ 2 , |α|≤2
lorsque |ξ| ≤ 1 et |z| → +∞, z ∈ Γ, uniform´ement par rapport `a h. On fait des hypoth`eses tr`es proches de celles de [6]. Soit E0 un r´eel strictement positif. On suppose que l’ensemble des points critiques de p0 (x, ξ) dans la surface d’´energie {p0 (x, ξ) = E0 } est une sous-vari´et´e C de T∗ Rn et que le Hessien de p0 est non d´eg´en´er´e sur le sous-espace normal `a C. En cons´equence, E0 est une valeur critique isol´ee de p0 (x, ξ). Comme P tend vers −h2 ∆ `a l’infini, C est compos´ee d’un nombre fini de composantes connexes compactes C = C1 ∪C2 · · ·∪CN . On note (rj , sj ) la signature du Hessien de p0 sur le sous-espace normal a` Cj . N´ecessairement la codimension de Cj , rj + sj , est plus grande que 1. Remarque. Soit P est un op´erateur h-diff´erentiel sur L2 (Rn ). L’ellipticit´e implique l’existence d’une constante C > 0 telle que aα,0 (x)ξ α ≥ |ξ|2 /C. |α|=2
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On “compl`ete le carr´e” dans p0 (x, ξ) afin d’´eliminer les termes d’ordre 1. Un changement de variables permet alors d’´ecrire p0 (x, ξ) = t ξA(x)ξ + V (x), o` u A(x) est une matrice strictement positive. Les points critiques de C sont donc les points (x, 0) avec V (x) = E0 et V (x) = 0. D’apr`es les hypoth`eses faites sur C, l’ensemble des points critiques de V (x) v´erifiant V (x) = E0 est une sous-vari´et´e de Rn et le Hessien de V (x) est non d´eg´en´er´e sur son sous-espace normal. On en d´eduit, dans ce cas, que rj + sj ≥ n + 1. Les r´esonances sont d´efinies grˆ ace `a la m´ethode des dilatations analytiques (voir [1] et [24]). On prend la convention que les r´esonances ont une partie imaginaire n´egative ou nulle. On note Res P l’ensemble des r´esonances de P compt´ees avec leur multiplicit´e (voir [26]). Ωδ,E est le domaine complexe d´efini grˆ ace au dessin : Im z
✻
E ✲
✛
✻
δ Ωδ,E
✲
Re z
δ ❄
Dans ce travail, on montre la majoration du nombre de r´esonances suivante. Th´eor`eme 1 On se place sous les hypoth`eses pr´ec´edentes. Il existe une constante C > 0 telle que pour h assez petit, E dans un petit voisinage de E0 et Ch < δ < 1/C, on a δh−n Card Res P ∩ Ωδ,E = O(1) . (1.2) |E − E0 | + δ De plus, si pour tout 1 ≤ j ≤ N , rj + sj ≥ 2, Card Res P ∩ Ωδ,E = O(1)δ ln δ + |E − E0 | h−n .
(1.3)
De plus, si pour tout 1 ≤ j ≤ N , max{rj , sj } ≥ 2, Card Res P ∩ Ωδ,E = O(1)δh−n .
(1.4)
Remarque. Le r´esultat de Fujii´e et Ramond [9] montre que (1.3) est optimal en dimension 1. D’autre part, en dimension quelconque, comme il n’est pas exclu que le champ hamiltonien de p0 ait des trajectoires ferm´ees dans la surface d’´energie E0 , (1.4) est optimal. En effet, soient E0 > 0 et P = −h2 ∆ + V (x) avec V (x) ∈ C0∞ (Rn ). On suppose que V (x) = |x − C|2 + C dans un puits et que, dans {V (x) =
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E0 }, le potentiel a un seul point critique, xc , qui est non d´eg´en´er´e. Typiquement, le potentiel est de la forme : V (x) xc
E0
puits x
Le puits engendre de l’ordre de δh−n quasimodes proches de E. Dans ces conditions, un r´esultat de Stefanov [29] (voir aussi [30], [31] et [32]) affirme que Card uC >0 (Res(P ) ∩ Ωδ,E ) > δh−n /C, pour E au voisinage de E0 et Ch < δ < 1/C o` est une constante. Pour obtenir une minoration du nombre de r´esonances, on rajoute une hypoth`ese. On note d le minimum de rj + sj pour 1 ≤ j ≤ N . On suppose que rj et sj ne d´ependent pas de j si j v´erifie rj + sj = d. On a alors Th´eor`eme 2 Il existe une constante C > 0 telle que pour h assez petit et Ch < δ < 1/C, on a d Card Res P ∩ Ωδ ln(1/δ),E0 ≥ δ 2 h−n /C. Pour d´emontrer le th´eor`eme 1, on applique les r´esultats de [3] ou [18] et il suffit de majorer, a` l’aide de techniques d´evelopp´ees dans [6], la norme trace d’un u P # est un op´erateur de r´ef´erence agissant op´erateur de la forme f ((P # −E)/δ), o` sur une vari´et´e compacte. Pour minorer le nombre de r´esonances, on utilise une formule de trace locale, valable dans des petits domaines, qui relie les r´esonances de P `a la trace d’une diff´erence d’op´erateurs. En effectuant une construction B.K.W., on approche e−itP/h par un op´erateur int´egral de Fourier (O.I.F.). On minore alors la trace pr´ec´edente en appliquant la m´ethode de la phase stationnaire. Remarque. Ce travail fait partie de la th`ese de doctorat de l’auteur pr´epar´ee sous la direction de Sj¨ ostrand.
2 Majoration du nombre de r´esonances Comme P tend vers −h2 ∆ `a l’infini, C est compact. Soit donc R0 1 tel que C ⊂ B(0, R0 )×Rn . Les points critiques de p0 sont alors dans une “boˆıte noire”. Soit
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P # un op´erateur de r´ef´erence sur T = (R/R1 Z)n avec R1 R0 qui co¨ıncide avec P pr`es de B(0, R0 ) et dont le symbole n’a pas de point critique hors de B(0, R0 )× Rn avec E0 comme valeur critique (voir [24]). Pour appliquer les r´esultats de [3] ou [18], il suffit de majorer le nombre de valeurs propres de P # dans des domaines de tailles δ autour de E proche de E0 . On suit la d´emonstration de Brummelhuis, Paul et Uribe [6] mais comme on d´esire juste des majorations du nombre de valeurs propres et non un d´eveloppement asymptotique, la situation est plus simple. On choisit f ∈ C0∞ (R; [0, 1]) v´erifiant f (t) = 1 sur [−1, 1] et on estime la norme trace de f ((P # − E)/δ) a` l’aide de techniques similaires a` celles de [6]. On fixe g ∈ S(R; R) (S(R; R) ´etant l’espace de Schwartz des fonctions ind´efiniment d´erivables a` d´ecroissance rapide de R dans R) une fonction positive, strictement positive au voisinage du support de f , et dont la transform´ee de Fourier est `a support compact inclu dans [−T, T ]. Soit k ∈ C0∞ (R) tel que f = g × k. On a # P −E f K= δ # tr P −E # g = O(1) − E )/ε) f ((P 0 δ tr
# P −E # = O(1) tr g f ((P − E0 )/ε) δ # = O(1) tr g (t)ei(th/δ)(P −E)/h f ((P # − E0 )/ε) dt ,
(2.1)
#
o` u ε δ est une petite constante fixe. On approche eisP /h f ((P # − E0 )/ε) par un O.I.F. D’apr`es [12] ou le chapitre IV de [19], il existe localement 1 Us ϕ(x) = (2πh)n
ei(φ(s,x,θ)−y.θ)/ha(s, x, y, θ; h)ϕ(y) dθ dy,
(2.2)
d´efini pour |s| ≤ T v´erifiant les assertions suivantes. La fonction a(s, x, y, θ; h) est un symbole de classe S0,0 (1) a` support compact dans les variables x, y, θ et la phase φ(s, x, θ) satisfait l’´equation eikonale
∂s φ − p# 0 (x, ∂x φ) = 0 φ(0, x, θ) = x.θ
(2.3)
# o` u p# 0 est le symbole principal semi-classique de P . Enfin,
# Us − eisP /h f ((P # − E0 )/ε) = O(h∞ ), tr
uniform´ement pour |s| ≤ T .
(2.4)
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Si h/δ est assez petit, |th/δ| ≤ T pour t dans le support de g (t) et (2.1) devient K = O(1) e−itE/δ g(t) tr Uth/δ dt + O(h∞ ) = O(1)h−n ei(φ(th/δ,x,θ)−x.θ)/he−itE/δ g (t)a(th/δ, x, x, θ; h) dt dx dθ + O(h∞ ). (2.5) / C un point de la projection du support de g a sur R2n Soit (x0 , θ0 ) ∈ x,θ . L’´equation eikonale implique
1 (x, θ) + th/δ (1 − s)φ (sth/δ, x, θ) ds . φ(th/δ, x, θ) − x.θ = th/δ p# t,t 0 0
Comme (x0 , θ0 ) est un point non-critique de p# 0 , il existe un changement de variables (t, x, θ) −→ (t, q = (φ(th/δ, x, θ; δ) − x.θ) δ/th, u), au voisinage de (x0 , θ0 ) a` condition que h δ. Dans ces coordonn´ees (2.5) devient localement −n (2.6) eit(q−E)/δ b(t, q, u; δ, h) dt dq du + O(h∞ ), K = O(1)h o` u b(t, q, u; δ, h) ∈ S0,0 (1) est `a support compact en t, q et u. Une phase stationnaire en t et (q − E) montre que K = O(1)δh−n . Maintenant on suppose que (x0 , θ0 ) ∈ Cj . [6] montre qu’il existe un changement de variables (s, (x, θ)) −→ (s, (u, v, w)), o` u u, v et w varient dans des voisinages de 0 de Rrj , Rsj et R2n−rj −sj respectivement. (x0 , θ0 ) est envoy´e sur (0, 0, 0) et Cj correspond a` u = v = 0. Dans ces variables, la phase s’´ecrit φ(th/δ, x, θ) − x.θ = t(E0 + u21 + · · · + u2rj − v12 − · · · − vs2j )h/δ. Pr`es de (x0 , θ0 ), (2.5) devient it(E −E+u21 +···+u2r −v12 −···−vs2 )/δ −n j j K = O(1)h b(t, u, v, w; δ, h) e 0
= O(1)h−n
b (E0 − E + u2 + · · · + u2 1 rj
dt du dv dw + O(h∞ ) − v12 − · · · − vs2j )/δ, u, v, w; δ, h du dv dw + O(h∞ ), (2.7)
o` u b(t, q, u; δ, h) ∈ S0,0 (1) est `a support compact et b est sa transform´ee de Fourier par rapport a` t.
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On suppose que rj ≥ 1 et sj = 0. En utilisant les coordonn´ees polaires, (2.7) s’´ecrit localement −n K = O(1)h U rj −1 b (E0 − E + U 2 )/δ, U, V ; δ, h dU dV + O(h∞ ) = O(1)h−n U rj −1 c (E0 − E)/δ + U 2 /δ, U ; δ, h dU + O(h∞ ), o` u c v´erifie |c(x, y; δ, h)| ≤ CM x −M , pour tout M > 0, uniform´ement par rapport a y, δ et h. On fait le changement de variables W = U 2 /δ, d’o` ` u rj rj √ K = O(1)δ 2 h−n W 2 −1 c (E0 − E)/δ + W, δW ; δ, h dW + O(h∞ ) rj 2
rj
h−n (E0 − E)/δ 2 −1 + O(h∞ ) rj −1 = O(1)δh−n |E0 − E| + δ 2 + O(h∞ ). = O(1)δ
(2.8)
En particulier, si rj = 1 et sj = 0, on a localement δh−n . K = O(1) |E − E0 | + δ
(2.9)
Dans le cas rj > 1 et sj = 0, (2.8) implique K = O(1)δh−n .
(2.10)
On admet maintenant que rj et sj sont strictement positifs. Comme pr´ec´edemment, K s’´ecrit U rj −1 V sj −1 c (E0 − E + U 2 − V 2 )/δ, U, V ; δ, h dU dV + O(h∞ ), O(1)h−n avec c(t, x, y; δ, h) ∈ S(R3 ) a` support compact en x et y. Un changement de variables transforme K en une somme de termes de la forme = O(1)h−n K c (E0 − E + xy)/δ, x, y; δ, h dx dy, xα y β o` u α + β = rj + sj − 2, α, β ≥ 0 et c v´erifie les mˆemes propri´et´es que c. On suppose que α ≥ β et on note c˘(t, x, y; δ, h) = y β c(t, x, y; δ, h). Alors = O(1)h−n xα dx c˘ (E0 − E + xy)/δ, x, y; δ, h dy K −n α = O(1)h 1|x|≤C1 x dx c˘ (E0 − E + xy)/δ, x, y; δ, h dy |x|≥δ −n α + O(1)h x dx 1|y|≤C1 c˘ (E0 − E + xy)/δ, x, y; δ, h dy |x|≤δ
Vol. 3, 2002
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= O(1)δh−n =
1|x|≤C1 xα−1 dx
|x|≥δ
701
c˘ (E0 − E)/δ + u, x, δu/x; δ, h du
+ O(1)δh−n O(1)δh−n O(1)δ| ln(δ)|h−n
si α > 0 . si α = 0
(2.11)
Dans la formule pr´ec´edente, C1 > 0 est une constante qui d´epend du support de c˘(t, x, y; δ, h). On d´eduit de (2.11) que K = O(1)δh−n ,
(2.12)
si rj ≥ 1, sj ≥ 1 et rj + sj > 2. √ quand α = β = 0. Pour |E − E0 | ≤ δ, (2.11) implique On majore K = O(1)δ| ln(δ)|h−n = O(1)δ ln δ + |E − E0 | h−n . K (2.13) √ Soient |E − E0 | ≥ δ et C2 > 0, on a ∞ = O(1)h−n K dx c (E0 − E + xy)/δ, x, y; δ, h dy |E0 −E|/C2
+ O(1)h−n + O(1)h
dx −|E0 −E|/C2
−n
Comme pour (2.11), ∞ O(1)h−n
|E0 −E|/C2
c (E0 − E + xy)/δ, x, y; δ, h dy
−|E0 −E|/C2
dx −∞
dx
|E0 −E|/C2
c (E0 − E + xy)/δ, x, y; δ, h dy.
(2.14)
c (E0 − E + xy)/δ, x, y; δ, h dy
C1 δ dx c (E0 − E)/δ + u, x, δu/x; δ, h du = O(1)h−n |E0 −E|/C2 x = O(1)δ ln |E0 − E|h−n . (2.15) Le troisi`eme terme de (2.14) se traite de la mˆeme fa¸con. Soient |x| ≤ |E0 − E|/C2 et y dans le support de c. On a alors |xy| ≤ |E0 − E| C1 /C2 et donc |(E0 − E + xy)/δ| ≥
1 |E0 − E| ≥ √ , 2δ 2 δ
a condition que C2 soit choisi assez grand. En utilisant le fait que ` c(z, x, y) ∈ S(R) est `a support compact en x et y, on montre que le deuxi`eme terme de (2.14) est un O(1)δ ∞ h−n . En combinant ceci avec (2.14) et (2.15), on trouve = O(1)δ ln δ + |E − E0 | h−n . On en d´eduit que, si rj = sj = 1, K K = O(1)δ ln δ + |E − E0 | h−n . (2.16)
702
J.-F. Bony
Ann. Henri Poincar´e
3 Minoration du nombre de r´esonances 3.1
Formule de trace locale
Dans cette partie on consid`ere deux op´erateurs Pj (j = 0, 1) qui v´erifient les hypoth`eses de la premi`ere partie. Pour Kj (j = 0, 1) une quantit´e quelconque, on note 1 Kj 0 = K1 − K0 . Si les ajα (x; h) d´esignent les coefficients de Pj , on suppose qu’il existe n > n tel que j 1 aα (z; h) 0 = O(1) z −ne , pour tout z ∈ Γ. Soient W ⊂⊂ Ω deux ouverts relativement compacts de R + i] − ∞, C], avec C > 0 et Ω simplement connexe. Typiquement Ω et W sont de la forme suivante : Im z ✻ ✲ Re z W Ω
On d´efinit J = Ω ∩ R et I = W ∩ R et on suppose que I et J sont des intervalles. On fixe χ ∈ C0∞ (J), ind´ependant de h, tel que χ = 1 au voisinage I. On suppose que E varie dans un compact de ]0, +∞[ et on pose Ωδ = E + δΩ, Wδ = E + δW et χδ (x) = χ((x − E)/δ). Comme dans [4], on suit la d´emonstration de la formule de trace locale de [23], [24] et on d´emontre la formule de trace : Th´eor`eme 3 (Sj¨ ostrand) Il existe une constante C > 0 telle que pour h assez petit et Ch < δ < 1/C, la formule suivante est vraie. Soit f (z; h) une fonction holomorphe dans Ωδ v´erifiant |f (z; h)| ≤ 1 pour z ∈ Ωδ \Wδ . On a alors
1 tr (χδ f )(Pj ; h) 0 =
λ∈ResPj ∩Wδ
1 f (λ; h) + O(1)δ −1 h−n . 0
Le reste O(1)δ −1 h−n est uniforme par rapport a ` f (z; h) et E.
(3.1)
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Estimation d’une trace
En suivant la construction de [24] (voir aussi [4]), on trouve un op´erateur hdiff´erentiel sur L2 (Rn ), P = aα (x; h)(hDx )α , |α|≤2
qui a les propri´et´es suivantes. P est proche de P a` l’infini, car aα (z; h) = O(1) z −n−1 aα (z; h) −
(3.2)
pour tout z ∈ Γ. P n’a pas de r´esonance dans un voisinage de E0 . D’autre part p0 , le symbole principal semi-classique de P , n’a pas de point critique dans les surfaces d’´energie { p0 (x, ξ) = E} o` u E est proche de E0 . On prend T > 0 assez petit et g ∈ C0∞ ([T /2, 2T ]) avec g = 1 pr`es de T . Pour D > 0 fix´e, on pose 2 fg (E) = e−it(E−E0 )/δ g(t)e−(t−T ) D ln(1/δ)/2 dt, (3.3) −1/2 −iT (E−E0 )/δ −(E−E0 )2 /(2Dδ2 ln(1/δ)) f (E) = D ln(1/δ)/2π e e , formellement f (E) = f1 (E). On se donne une troncature χ ∈ C0∞ (]0, +∞[) v´erifiant χ = 1 au voisinage de E0 . On pose P1 = P , P0 = P et on note p0,0 , p1,0 leur symboles principaux. On utilisera la notation suivante k(t) = tE0 + i(t − T )2 Dδ ln(1/δ)/2.
(3.4)
Lemme 4 Il existe une constante C > 0 telle que pour h assez petit, Ch < δ < 1/C et T < 1/C, on a tr fg (Pj )χ(Pj ) 1 ≥ δ d2 h−n ln(1/δ)−1/2 /C. (3.5) 0 Preuve. On peut ´ecrire 1 tr fg (Pj )χ(Pj ) 0 1 =tr eik(t)/δ g(t) e−itPj /δ χ(Pj ) 0 dt 1 =tr eik(t)/δ g(t)e−itP0 /δ χ(Pj ) 0 dt 1 + tr eik(t)/δ g(t) e−itPj /δ 0 χ(P1 )(1 − ϕ(x)) dt − tr eik(t)/δ g(t)e−itP0 /δ χ(P1 )ϕ(x) dt + tr eik(t)/δ g(t)e−itP1 /δ χ(P1 )ϕ(x) dt , o` u ϕ ∈ C0∞ (Rn ) est ´egal `a 1 dans une grande boule.
(3.6) (3.7) (3.8) (3.9)
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J.-F. Bony
Ann. Henri Poincar´e
1 On commence par majorer (3.6). On d´eduit de (3.2), que χ(Pj ) 0 est un cl op´erateur h-pseudo-diff´erentiel de classe S0,0 ( x −n−1 ξ −M ) pour tout M . On 1 −isP0 /h approche e χ(Pj ) 0 par un O.I.F. not´e Us qui v´erifie des formules analogues a (2.2)–(2.4) : ` 1 Us ϕ(x) = (2πh)n
ei(φ0 (s,x,θ)−y.θ)/ha(s, x, y, θ; h) dθ dy,
o` u s est petit, a(s, x, y, θ; h) ∈ S0,0 ( x −n−1 ) est `a support compact en θ et en (x − y). La phase φ0 (s, x, θ) v´erifie
∂s φ + p0,0 (x, ∂x φ0 ) = 0 φ0 (0, x, θ) = x.θ
Enfin, pour s assez petit, 1 Us − e−isP0 /h χ(Pj ) 0 = O(h∞ ). tr
Donc (3.6) devient tr
e
ik(t)/δ
g(t)e
1 χ(Pj ) 0 dt =
−itP0 /δ
e−(t−T )
2
D ln(1/δ)/2
1 (2πh)n
ei(φ0 (th/δ,x,θ)−x.θ+tE0h/δ)/h
g(t)a(th/δ, x, x, θ; h) dt dx dθ + O(h∞ ).
(3.10)
La phase de (3.10) s’´ecrit φ0 (th/δ, x, θ) − x.θ + tE0 h/δ =
1 (1 − s)φ0 t,t (sth/δ, x, θ) ds . th/δ E0 − p0,0 (x, θ) + th/δ 0
Comme E0 est une valeur non-critique de p0,0 et comme 0 n’est pas dans le support de g(t), une phase non-stationnaire implique que (3.6) est un O(δ ∞ h−n ) pour Ch < δ < 1/C avec C assez grand. (3.8) se traite de la mˆeme fa¸con. On calcule maintenant (3.7). La diff´erence des groupes d’´evolution s’´ecrit sous forme int´egrale
1 e−itPj /δ 0
t = iδ
0
1
e−istP1 /δ (P1 − P0 )e−i(1−s)tP0 /δ ds,
cl ( x −n−1 ξ 2 ) d’apr`es (3.2). o` u P1 − P0 ∈ S0,0
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Comme pr´ec´edemment, on appoche les groupes d’´evolution par des O.I.F. et on trouve 1 tr eik(t)/δ g(t) e−itPj /δ 0 χ(P1 )(1 − ϕ(x)) dt 1 1 ei(φ1 (sth/δ,x,θ)−y.θ+φ0((1−s)th/δ,y,ξ)−x.ξ+tE0h/δ)/h = δ(2πh)2n 0 e−(t−T )
2
D ln(1/δ)/2
g(t)a(s, t, x, y, θ, ξ; δ, h)(1 − ϕ(x)) ds dt dx dy dθ dξ + O(h∞ ), (3.11)
o` u la phase φj correspond a` l’op´erateur Pj et a(s, t, x, y, θ, ξ; δ, h) ∈ S0,0 ( y −n−1 ) est `a support compact en θ, ξ et (x − y). Comme pour (3.10), on ´ecrit la phase de (3.11) sous la forme φ1 (sth/δ, x, θ) − y.θ + φ0((1 − s)th/δ, y, ξ) − x.ξ + tE0h/δ /h = (x − y).(θ − ξ)/h
1 (1 − z)φ1 t,t (zsth/δ, x, θ) dz + t E0 − sp1,0 (x, θ) − (1 − s)p0,0 (y, ξ) + sth/δ 0
+ (1 − s)th/δ (1 − z)φ0 t,t (z(1 − s)th/δ, y, ξ) dz /δ = (x − y).(θ − ξ)/h 0 + t E0 − sp1,0 (x, θ) − (1 − s)p0,0 (y, ξ) + h/δf (s, t, x, y, θ, ξ; δ, h) /δ,
1
o` u la fonction f (s, t, x, y, θ, ξ; δ, h) et toutes ses d´eriv´ees sont uniform´ement born´ees sur le support de l’int´egrande de (3.11) car on contrˆ ole les d´eriv´ees de φj (voir [4] ou [19]). On pose u = y − x et v = ξ − θ et on travaille dans les variables (x, θ, u, v). L’int´egrande de (3.11) est `a support compact en θ, u, v et la phase de (3.11) s’´ecrit φ1 (sth/δ, x, θ) − y.θ + φ0 ((1 − s)th/δ, y, ξ) − x.ξ + tE0 h/δ /h = u.v/h + t E0 − sp1,0 (x, θ) − (1 − s)p0,0 (x + u, θ + v)
+ h/δf (s, t, x, x + u, θ, θ + v; δ, h) /δ.
On majore d’abord (3.11) quand |u| < 1/C3 et |v| < 1/C3 o` u C3 > 0 est une grande constante. Le fait que Pj tend vers −h2 ∆ et que ϕ = 1 dans une grande boule, permet d’effectuer des int´egrations par partie dans les variables t, θ et de gagner autant de puissances de δ que souhait´ees pour Ch < δ < 1/C, avec C assez grand. On fait alors une phase stationnaire dans les variables u, v en prenant comme phase u.v/h et en consid´erant tout le reste comme un symbole. On gagne alors un facteur hn quitte a` perdre un nombre fini de puissances de δ. Finalement, la contribution de cette r´egion dans (3.11) est un O(δ ∞ h−n ). Lorsque |u| ≥ 1/C3 (resp. |v| ≥ 1/C3 ), on fait des int´egrations par partie en v (resp. en u) et on montre que la contribution de cette r´egion dans (3.11) est un O(h∞ ) pour Ch < δ < 1/C, avec C assez grand. En d´efinitive, (3.7) est un O(δ ∞ h−n ). Des arguments l´eg`erement diff´erents ont ´et´e d´evelopp´es dans l’appendice de [5].
706
J.-F. Bony
Ann. Henri Poincar´e
Pour ´etudier (3.9), on remplace e−isP1 /h χ(P1 )ϕ(x) par un O.I.F. encore not´e Us qui satisfait le mˆeme type de formules que pr´ec´edemment et en plus a(0, x, y, θ; h) = 1 dans un grand compact. Donc 1 1 eik(t)/δ ei(φ1 (th/δ,x,θ)−x.θ)/h tr fg (Pj )χ(Pj ) 0 = (2πh)n g(t)a(th/δ, x, x, θ; h) dt dx dθ + O(h∞ ). (3.12) On d´ecoupe l’int´egrale (3.12) avec une partition de l’unit´e en (x, θ) et on traite localement chaque terme. On ´etudie (3.12) pr`es de (x0 , θ0 ) ∈ / C. Une d´emonstration analogue `a (2.6) montre que, localement, 1 1 eit(E0 −q)/δ tr fg (Pj )χ(Pj ) 0 = (2πh)n e−(t−T )
2
D ln(1/δ)/2
g(t)b(t, q, u; δ, h) dt dq du.
Puisque 0 n’est pas dans le support de g(t), une phase non-stationnaire implique que, pr`es de (x0 , θ0 ), (3.12) est un O(δ ∞ h−n ). On se place au voisinage de (x0 , θ0 ) ∈ Cj . Comme pour (2.7), on peut localement ´ecrire 1 1 it(−u21 −···−u2rj +v12 +···+vs2j )/δ e tr fg (Pj )χ(Pj ) 0 = (2πh)n e−(t−T )
2
C ln(1/δ)/2
g(t)b(t, u, v, w; δ, h) dt du dv dw,
o` u b(t, u, v, w; δ, h) est une fonction a` support compact, b(0, u, v, w; δ, h) > 1/C4 (avec C4 > 0) pr`es de (x0 , θ0 ) et b(0, u, v, w; δ, h) ≥ 0 sinon. En faisant une phase stationnaire dans les variables (u, v) et une int´egration en w, on trouve localement rj +sj 1 tr fg (Pj )χ(Pj ) 0 = δ 2 h−n ei(sj −rj )π/4 2 e−(t−T ) D ln(1/δ)/2 g(t) c(t; δ, h) + O(δ) dt,
o` u c(t; δ, h) ∈ S0,0 (1) et il existe une autre constante C4 > 0 telle que c(t; δ, h) > 1/C4 sur le support de g(t), si T est assez petit. D’o` u, localement, rj +sj 1 tr fg (Pj )χ(Pj ) 0 = δ 2 h−n ln(1/δ)−1/2 ei(sj −rj )π/4 M (δ, h) + O(δ) , avec 1/C4 < M (δ, h) < C4 . En conclusion, on a N rj +sj 1 δ 2 h−n ln(1/δ)−1/2 ei(sj −rj )π/4 Mj (δ, h) tr fg (Pj )χ(Pj ) 0 = j=1
+ O(δ) + O(δ ∞ h−n ).
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Les termes dominants de cette somme correspondent aux j qui minimisent rj + sj . Pour de tels j, la quantit´e rj − sj est constante d’apr`es l’hypoth`ese suppl´ementaire du th´eor`eme 2. Leur contribution a` la somme s’ajoutent et il vient tr fg (Pj )χ(Pj ) 1 ≥ δ d2 h−n ln(1/δ)−1/2 /C4 , (3.13) 0 o` u C4 > 0 est une autre constante.
3.3
Minoration du nombre de r´esonances
Pour appliquer la formule de trace locale, on d´efinit W =] − α, α[ + i] − β, γ[ Ω =] − α − γ, α + γ[ + i] − β − γ, γ[, o` u α, β et γ seront fix´es ult´erieurement. On fixe ϕ ∈ C0∞ (Ω ∩ R) tel que ϕ = 1 pr`es de W ∩ R. On pose Wδ = E0 + δ ln(1/δ)W , Ωδ = E0 + δ ln(1/δ)Ω, Gδ = R + δ ln(1/δ)Ω et ϕδ (x) = ϕ (x − E0 )/(δ ln(1/δ) . Le lemme 5.1 et la proposition 5.2 de [4] prouvent le lemme suivant. Lemme 5 Soit M > 0. On peut trouver des constantes CM , α, β, γ, D > 0 telles que |fg (E) − f (E)| ≤ CM δ M |fg (E)| ≤ CM δ
si E ∈ Gδ
M
si E ∈ Ωδ \Wδ −1/2
|f (E)| ≤ CM ln(1/δ)
si E ∈ Ωδ
pour δ assez petit. On applique alors le th´eor`eme 3 et on trouve 1 tr fg (Pj )ϕδ (Pj ) 0 =
1 fg (λ) + O(1)δ M−1 h−n
λ∈ResPj ∩Wδ
=
0
fg (λ) + O(1)δ M−1 h−n ,
(3.14)
λ∈ResP1 ∩Wδ
car P0 n’a pas de r´esonance proche de E0 . En utilisant une extension presque analytique de ϕδ (x) − χ(x) et le lemme 5, on d´emontre (voir [4]) que 1 1 tr fg (Pj )ϕδ (Pj ) 0 = tr fg (Pj )χ(Pj ) 0 + O(1)δ M−2 h−n et (3.14) devient 1 tr fg (Pj )χ(Pj ) 0 =
λ∈ResP1 ∩Wδ
fg (λ) + O(1)δ M−2 h−n .
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J.-F. Bony
Ann. Henri Poincar´e
Comme il y a au plus O(h−n ) r´esonances dans Wδ , on a fg (λ) − f (λ) ≤ # Res P1 ∩ Wδ sup fg (E) − f (E) E∈Wδ
λ∈ResP1 ∩Wδ
= O(1)δ M h−n . Donc
1 tr fg (Pj )χ(Pj ) 0 =
Le lemme 4 implique alors
f (λ) + O(1)δ M−2 h−n .
λ∈ResP1 ∩Wδ
|f (λ)| ≥ δ 2 h−n ln(1/δ)−1/2 /C. d
λ∈ResP1 ∩Wδ
Et finalement
d Card Res P1 ∩ Wδ ≥ δ 2 h−n /C.
(3.15)
R´ef´erences [1] J. Aguilar and J.M. Combes, A class of analytic perturbations for one-body Schr¨ odinger Hamiltonians,Comm. Math. Phys. 22, 269–279 (1971). [2] C. Bardos, J.C. Guillot et J. Ralston, La relation de Poisson pour l’´equation des ondes dans un ouvert non born´e, Comm. P.D.E. 7, 905–958 (1982). [3] J.F. Bony, R´esonances dans des domaines de taille h, Inter. Math. Res. Not. 16, 817–847 (2001). [4] J.F. Bony, Minoration du nombre de r´esonances engendr´ees par une trajectoire ferm´ee, Preprint, `a paraˆıtre dans Comm. P.D.E. [5] V. Bruneau and V. Petkov, Meromorphic continuation of the spectral shift function, Preprint. [6] R. Brummelhuis, T. Paul and A. Uribe, Spectral Estimates around a critical level, Duke Math. J. 78(3), 477–530 (1995). [7] M. Dimassi and J. Sj¨ ostrand, Spectral asymptotic in the semi-classical limit, London Math. Soc., Lecture Notes 268, Cambridge Univ. Press (1999). [8] S. Dozias, Op´erateurs h-pseudodiff´erentiels `a flot p´eriodique, Th`ese Paris 13 (1994). [9] S. Fujii´e et T. Ramond, Matrice de scattering et r´esonances associ´ees `a une orbite h´et´erocline, Ann. Inst. Henri Poincar´e 69(1), 31–82 (1998). [10] B. Helffer et A. Martinez, Comparaison entre les diverses notions de r´esonances, Helv. Phys. Acta 60, 992–1003 (1987). [11] B. Helffer et J. Sj¨ ostrand, R´esonances en limite semi-classique, Bull. Soc. Math. France 114 (1986).
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[12] L. H¨ormander, The analysis of linear partial differential operators, Springer Verlag (1985). [13] V. Ivrii, Microlocal analysis and precise spectral asymptotics, Springer Verlag (1998). [14] N. Ka¨ıdi et P. Kerdelhu´e, Forme normale de Birkhoff et r´esonances, Asympt. Anal. 23(1), 1–23 (2000). [15] P.D. Lax and R.S. Phillips, Scattering theory, Academic Press (1967). [16] R.B. Melrose, Polynomial bounds on the number of scattering poles, J. Funct. Anal. 53, 287–303 (1983). [17] V. Petkov and V. Zworski, Breit-Wigner approximation and the distribution of resonances, Comm. Math. Phys. 204, 329–351 (1999). [18] V. Petkov and M. Zworski, Semi-classical estimates of the scattering determinant, Ann. Inst. Henri Poincar´e 2, 675–711 (2001). [19] D. Robert, Autour de l’approximation semi-classique, Birkh¨auser (1987). [20] D. Robert, Semi-classical asymptotics for the spectral shift function, Differential operators and spectral theory, Amer. Math. Soc., Providence, RI, Amer. Math. Soc. Transl. Ser. 2 189, 187–203 (1999). [21] J. Sj¨ ostrand, Semiclassical resonances generated by a non-degenerate critical point, Springer Verlag L.N.M. 1256, 402–430 (1986). [22] J. Sj¨ ostrand, Geometric bounds on the density of resonances for semi-classical problems, Duke Math. J. 48(2), 1–57 (1990). [23] J. Sj¨ ostrand, A trace formula and review of some estimates for resonances, in microlocal analysis and spectral theory, Microlocal Analysis and Spectral Theory, NATO ASI series C, 490, 377–437 (1997). [24] J. Sj¨ ostrand, Resonances for bottles and trace formulae, Math. Nachr. 221, 95–149 (2001). [25] J. Sj¨ ostrand, Asymptotic distribution of eigenfrequencies for damped wave equations, Publ. Res. Inst. Math. Sci. 36(5), 573–611 (2000). [26] J. Sj¨ ostrand and M. Zworski, Complex scaling and the distribution of scattering poles, J. Amer. Math. Soc. 4(4), 729–769 (1991). [27] J. Sj¨ ostrand and M. Zworski, Lower bounds on the number of scattering poles, Comm. P.D.E. 18, 847–857 (1993). [28] J. Sj¨ ostrand and M. Zworski, Lower bounds on the number of scattering poles II, J. Func. Anal. 123(2), 336–367 (1994). [29] P. Stefanov, Quasimodes and resonances : sharp lower bounds, Duke Math. J. 99, 75–92 (1999). [30] P. Stefanov and G. Vodev, Distribution of resonances for the Neumann problem in linear elasticity outside a strictly convex body, Duke Math. J. 78, 677–714 (1995).
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[31] P. Stefanov and G. Vodev, Neumann resonances in linear elasticity for an arbitrary body, Comm. Math. Phys. 176, 645–659 (1996). [32] S.U. Tang and M. Zworski, From quasimodes to resonances, Math. Res. Lett. 5(3), 261–272 (1998). [33] G. Vodev, Sharp bounds on the number of scattering poles for perturbations of the Laplacian, Comm. Math. Phys. 146, 205–216 (1992). [34] M. Zworski, Sharp polynomial bounds on the number of scattering poles, Duke Math. J. 59, 311–323 (1989). Jean-Fran¸cois Bony Universit´e de Paris Sud D´epartement de Math´ematiques UMR 8628 Bˆatiment 425 F-91405 Orsay cedex France email: [email protected] Communicated by Bernard Helffer submitted 05/11/01, accepted 18/01/02
To access this journal online: http://www.birkhauser.ch
Ann. Henri Poincar´e 3 (2002) 711 – 737 c Birkh¨ auser Verlag, Basel, 2002 1424-0637/02/040711-27
Annales Henri Poincar´ e
Weak Disorder Localization and Lifshitz Tails: Continuous Hamiltonians F. Klopp∗
Abstract. This paper is devoted to the study of band edge localization for continuous random Schr¨ odinger operators with weak random perturbations. We prove that, in the weak disorder regime, λ small, the spectrum in a neighborhood of size C · λ of a non-degenerate simple band edge is exponentially and dynamically localized. Upper bounds on the localization length in these energy regions are also obtained. Our results rely on the analysis of Lifshitz tails when the disorder is small; the single site potential need not be of fixed sign. R´esum´e. Ce travail est consacr´e ` a l’´ etude des ´ etats localis´es en bord de spectre pour des op´erateurs de Schr¨ odinger al´eatoires ` a faible d´esordre, λ petit. Nous prouvons que, dans un voisinage de taille C · λ d’un bord simple et non-d´eg´ en´ er´ e du spectre, les ´ etats sont localis´es exponentiellement et dynamiquement. Nous obtenons aussi une majoration de la longueur de localisation. Notre analyse repose sur une ´etude des asymptotiques de Lifshitz quand le d´esordre est petit; le potentiel de simple site n’a pas n´ecessairement un signe constant.
0 Introduction Consider the following continuous Anderson model ωγ Vγ , Hω,λ = −∆ + λVω = −∆ + λ γ∈Zd
where 1. Vγ (·) = V (·−γ) where V : Rd → R is a bounded, measurable and compactly supported function; 2. the average of V does not vanish i.e. V (x)dx = 0; V :=
(0.1)
Rd
3. the random variables (ωγ )γ∈Zd are i.i.d., bounded, non-trivial and have a bounded density; 4. λ is a positive coupling constant. It is well known that the spectrum of Hω,λ is almost surely non random ([30]). Denote it by Σλ = σ(Hω,λ ). Let E− (λ) = inf Σλ . Consider the periodic Schr¨odinger ∗ The author gratefully acknowledges support of the FNS 2000 “Programme Jeunes Chercheurs”. It is a pleasure to thank F. Germinet for useful remarks.
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operator H λ = −∆ + λ
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Vγ
γ∈Zd
and let E(λ) be the infimum of the spectrum of the periodic Schr¨ odinger operator H λ . Define ω = E(ω0 ). As the random variables (ωγ )γ are not trivial, assumption (0.1) ensures that, for some C > 0 and λ sufficiently small (see section 5.1), one has E(λω) − E− (λ) ≥ λ/C. (0.2) Note that E(λω) is the infimum of the spectrum of E(Hω,λ ) = H λω . We prove Theorem 0.1 Fix η ∈ (0, d/(4d+4)). There exists λη > 0 such that, for λ ∈ (0, λη ), with probability 1, one has 1. exponential localization in Iη,λ := [E− (λ), E(λω) − λ1+η ] i.e. • σ(Hω,λ ) ∩ Iη,λ = σpp (Hω,λ ) ∩ Iη,λ ; • σac (Hω,λ ) ∩ Iη,λ = σsc (Hω,λ ) ∩ Iη,λ = ∅; • there exists a > 0 such that, an eigenfunction ψ corresponding to an eigenvalue E ∈ Iη,λ satisfies log |ψ(x)| ≥ a |E − E(λω)|. (0.3) lim inf − |x| |x|→+∞ 2. strong Hilbert-Schmidt dynamical localization in Iη,λ i.e. if Πη (Hω,λ ) denotes the spectral projector of Hω,λ on the interval Iη,λ and X denotes the position operator, then, for B ⊂ Rd bounded, one has 2 q/2 ∀q ≥ 0, E sup |X| f (Hω,λ )Πη (Hω,λ )1B < +∞. (0.4) f ∞ ≤1
2
Here, σpp,ac,sc denote respectively the pure point, absolutely continuous and singular continuous part of the spectrum. The function 1B is the characteristic function of B; · 2 denotes the Hilbert-Schmidt norm. In (0.4), the supremum is taken over bounded, Borel measurable functions on R. Theorem 0.1 is a consequence of the behavior of the integrated density of states near the infimum of the spectrum. More precisely, let ΛL be the cube of center 0 and side length L in Rd , and let Hω,λ|ΛL be the Hamiltonian Hω,λ restricted to the cube ΛL with Dirichlet boundary conditions. Define the integrated density of states of Hω,λ by Nλ (E) = lim
L→+∞
#{eigenvalues of Hω,λ|ΛL ≤ E} . |ΛL |
(0.5)
The limit in (0.5) exists ω-a.e.; it is non-random and non-decreasing ([4, 30]). We prove
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Theorem 0.2 Fix η ∈ (0, d/(4d + 4)). Then, there exists ε > 0 and λη > 0 such that, for λ ∈ (0, λη ), one has −ε
Nλ (E(λω) − λ1+η ) ≤ e−λ .
(0.6)
So, the density of states in the interval Iη,λ is small. This interval is in the fluctuational region of the spectrum (see [28]). That in fluctuation regions, the spectrum is scarce and thus the states are localized is a basic mechanism for localization and has been known for a long time (see e.g. [27, 28]). It also is the only mechanism that has been understood mathematically in dimensions larger than 1; this mechanism is also basic to the understanding of large disorder localization. The asymptotics (0.6) is directly related to the celebrated “Lifshitz tail” behavior (see e.g. [20] for a recent review). The main difference with Lifshitz tails is that Lifshitz tails are asymptotics in the limit when one approaches the band edge, and (0.6) holds on an interval. This is a consequence of the weak disorder limit. Technically speaking the passage from Theorem 0.2 to Theorem 0.1 makes use of multiscale analysis and Wegner estimates (see [34, 9, 12]). These are well known tools in the proof of localization for random Schr¨ odinger operators. In section 3, we explain how we use these tools. As the main result of this paper, we show that the behavior (0.6), found at the bottom of the spectrum, also takes place at simple non-degenerate spectral edges. Hence, at such spectral edges, one also has both exponential and dynamical localization on an interval of size C · λ, and one obtains an estimate on the exponential decay rate of eigenfunctions. This is the content of Theorems 1.1 and 1.3. To complete this section, let us note that results similar to Theorems 0.1 and 0.2 have been obtained for discrete random operators in [21].
1 The results Let W be a bounded Zd -periodic potential and consider the periodic Schr¨ odinger operator H = −∆ + W acting on L2 (Rd ). It is self-adjoint on H 2 (Rd ); let Σ0 be its spectrum. Consider the continuous Anderson model i.e. the random Schr¨ odinger operator defined by Hω,λ = H + λVω = H + λ ωγ Vγ , (1.1) γ∈Zd
where Vγ (·) = V (· − γ), V : Rd → R is a potential and random variables (ωγ )γ∈Zd are independent identically distributed. We assume that. H0.1 V : Rd → R is bounded and decaying faster than |x|−d− for some > 0; H0.2 the random variables (ωγ )γ∈Zd are i.i.d., bounded and non-trivial. Notice that we do not assume that V keeps a fixed sign.
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Assumption (H0) ensures that the potential Vω stays uniformly bounded. Hence, Hω,λ is self-adjoint on H 2 (Rd ). By assumption (H0.2), Hω,λ is ergodic; hence, its spectrum is almost surely independent of ω. We denote it by Σλ . Consider (E+ (0), E− (0)) a gap in the spectrum of H (if E− (0) is the infimum of Σ0 , we set E+ (0) = −∞). For λ sufficiently small, let E− (λ) (resp. E+ (λ)) be the infimum (resp. supremum) of Σλ in [E0 , +∞) (resp. (−∞, E0 ]) where E0 = (E− (0) + E+ (0))/2, see Fig. 1. So, for some δ > 0 and λ sufficiently small, one has E+ (λ) < E− (λ) and Σλ ∩ [E+ (0) − δ, E− (0) + δ] ⊂ [E+ (0) − δ, E+ (λ)] ∪ [E− (λ), E− (0) + δ]. We study the spectrum of Hω,λ near E− (λ) for λ sufficiently small. An analogous study can be done near E+ (λ). To describe our main assumption, we need to E+ (0)
Σλ
E+ (λ)
E− (0)
Σ0
E− (λ)
Figure 1: The band edges for Σ and Σp recall some facts from the Floquet theory of periodic Schr¨ odinger operators.
1.1
Floquet theory of periodic Schr¨ odinger operators
The Floquet spectrum of H is the spectrum of the differential operator H acting on L2loc (Rd ) with quasi-periodic boundary conditions. For θ ∈ T∗ = Rd /Zd , consider the following eigenvalue problem on L2loc (Rd ) Hϕ = Eϕ (1.2) ϕ(x + γ) = ei2πγθ ϕ(x), ∀x ∈ Rd , ∀γ ∈ Zd . As H is elliptic, one knows that the eigenvalues of (1.2) are discrete; when repeated according to multiplicity, we denote them by E0 (θ) ≤ E1 (θ) ≤ · · · ≤ En (θ) ≤ . . . . They are called the Floquet eigenvalues of H. These functions are Lipschitz continuous in the variable θ; when of multiplicity 1, the Floquet eigenvalues are even analytic in θ. Moreover, Weyl’s law tells us that En (θ) → +∞ as n → +∞ (uniformly in θ). In regard of (1.2), the Floquet eigenvalues are Zd -periodic functions of θ. The spectrum of H is given by Σ0 = ∪n≥0 En (T∗ ). So the spectrum
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of H is the union of closed intervals called the bands of the spectrum; the connected components of R \ Σ0 are called the gaps of the spectrum of H. One says that an energy E ∈ Σ0 is simple if there exists exactly one index n ≥ 0 such that, for some θ ∈ T∗ , one has En (θ) = E. It is proved in [22] that, generically, the band edges are simple. One says that E, a simple band edge, is non-degenerate if the Floquet eigenvalue En (·) reaching this band edge has only non-degenerate quadratic extrema at that edge, i.e. if θ is such that En (θ) = E, then θ is a nondegenerate quadratic extremum of En . One can define N0 (E), the integrated density of states of H in the same way as for Hω i.e. by means of (0.5). One proves that 1 N0 (E) = 1{Ej (θ)≤E} dθ. (2π)d T∗ j≥1
The set of points of increase of N0 coincides with the spectrum. The nondegeneracy of band edges can also be characterized in terms of the density of states. Namely, E0 , a simple band edge of H is non-degenerate if and only if |N0 (E) − N0 (E0 )| ≤ C|E − E0 |d/2 as E → E0 , E ∈ Σ0 (see [23]). Details on the material presented here may be found in [31, 26, 33].
1.2
The main assumptions
Let us now state our main assumptions. On the underlying periodic Schr¨ odinger operator, we assume H1 E− (0) is a simple non-degenerate band edge. This condition is known to hold at the bottom of the spectrum of H in any dimension, see [16, 2]. It also holds at any band edge in dimension 1, see [35, 8]. As assumption (H1) is stable under small perturbations, it holds for sufficiently small perturbations of periodic operators with separate variables. In the semiclassical regime, there are also results for the first spectral band [29, 32]. For a more general discussion on the validity of assumption (H1), we refer to [5]. Define the periodic operator Hλ = H + λ
Vγ = H + λV .
(1.3)
γ∈Zd
Let E(λ) be the spectral band edge of H λ closest to E− (0). For λ sufficiently small, it is well defined (see section 5.1). We assume that H2 there exists C > 0 such that, for λ sufficiently small |E(λ) − E(0)| = |E(λ) − E− (0)| ≥ C|λ|.
(1.4)
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It is proved in section 5.1 that, for a given background periodic operator H, assumption (H2) is satisfied for a generic single-site perturbation V . Moreover, (1.4) implies (0.2). Note that, due to the unique continuation principle ([13]), assumption (H2) is satisfied for any non vanishing single site potential having a definite sign.
1.3
The results
We prove Theorem 1.1 Assume (H0), (H1) and (H2). Fix η ∈ (0, d/(4d + 4)). Then, there exists ε > 0 and λη > 0 such that, for λ ∈ (0, λη ), one has −ε
Nλ (E(λω) − λ1+η ) − Nλ (E− (λ)) ≤ e−λ . This result has strong consequences on the spectral behavior of Hω . Indeed, let us assume that H3 the common probability distribution of the random variables (ωγ )γ∈Zd is absolutely continuous with respect to Lebesgue measure and its density is locally absolutely continuous. Under this additional assumption, in [12], it is proved that Theorem 1.2 ([12]) Assume (H0), (H1), (H2) and (H3). Fix 0 < τ < 1. Then, there exists c > 0 and λ0 > 0 such that, for 0 < λ < λ0 , the integrated density of states of Hω,λ , Nλ , is H¨ older continuous of order τ in the interval [E− (λ), E(λω)+ cλ] i.e. there exists Cλ > 0 such that, for (E, E ) ∈ [E− (λ), E(λω) + cλ]2 , one has |Nλ (E) − Nλ (E )| ≤ Cλ |E − E |τ . Theorem 1.2 is a consequence of the Wegner’s estimate proved in [12]; a weaker form of such an estimate is stated in Proposition 3.1. Using Theorems 1.1 and 1.2 (or better said, the Wegner estimate of Proposition 3.1) together with the multi-scale analysis technique (see e.g. [34, 9]), we prove Theorem 1.3 Assume (H0), (H1), (H2) and (H3). Fix η ∈ (0, d/(4d + 4)). There exists λη > 0 such that, for λ ∈ (0, λη ), with probability 1, one has 1. exponential localization in Iη,λ := [E− (λ), E(λω)− λ1+η ] (see the description given in Theorem 0.1); moreover, the eigenfunctions associated to eigenvalues in this interval satisfy (0.3). 2. strong Hilbert-Schmidt dynamical localization in Iη,λ (see the description given in Theorem 0.1).
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Though localization at band edges is a now well known feature of random Schr¨ odinger operators with sign-definite single potentials ([19, 1, 17, 36, 34]), to our knowledge, these are the first results on localization for sign-indefinite potentials at internal band edges. Localization for such potentials had been established at the bottom of the spectrum assuming some Lifshitz behavior or quickly decaying tails for the distributions of the random variables (ωγ )γ∈Zd (see e.g. [19, 3]). To our knowledge, for single site potentials that do not have a definite sign, Lifshitz tails have not been proved to hold, even at the bottom of the spectrum. In the present paper, we prove a kind of Lifshitz behavior at internal non-degenerate band edges without assuming that the single site potential has a definite sign. This assumption has always played a crucial role in the analysis of Lifshitz tails. In the present work, it is replaced with assumption (H2) and the assumption that λ, the disorder, is small. Theorems 0.2 and 0.1 follow immediately from Theorems 1.1 and 1.3. Indeed, for H = −∆ and E− (0) = 0, it is well known that assumption (H1) is fulfilled at the bottom of the spectrum, see e.g. [31, 2]. In this case, one proves E(λω) = λV +o(λ) (see section 5.1); hence, assumption (H2) is fulfilled if and only if V = 0 i.e. if and only condition (0.1) is fulfilled. Analogues of Theorems 1.1 and 1.3 in the case of discrete random Schr¨ odinger operators have been proved in [21]. Finally, let us say that the methods used in the present paper should also apply to other random models, e.g. to Schr¨ odinger operators with weak random magnetic potentials ([10]).
1.4
The outline of the paper
Section 2 is devoted to the proof of Theorem 1.1. This proof is made in four steps. In section 2.1, we introduce the periodic approximations that enable us to estimate Nλ . Section 2.2 is devoted to the recollection of some facts from the Floquet theory of periodic operators. In section 2.3, we show that estimating Nλ comes up to estimating the probability that, restricted to a sufficiently but not too large cube, the operator Hω,λ has an eigenvalue in Iη,λ . This probability is then estimated in section 2.4 by showing that it reduces to large deviation estimates for sums of independent random variables. In section 3, we prove Theorem 1.3 using Theorem 1.1 and Proposition 3.1 that is proved in section 4. At last, in the appendix, section 5, we gathered some useful results.
2 The proof of Theorem 1.1 The scheme of this proof follows the one of the proof of Theorem 1.1 in [21]. We first state an approximation theorem giving precise finite volume approximations to Nλ ; such a statement was derived in [23, 25]. Then, we study the finite volume approximations.
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Periodic approximations
Let n ∈ N \ {0} and define the following periodic Schr¨ odinger operator n =H +λ ωγ V (x − γ − β) = H + λVωn Hω,λ γ∈Zd 2n+1
β∈(2n+1)Zd
where Zd2n+1 := Zd /(2n + 1)Zd . n For ω fixed and n ∈ N∗ , Hω,λ is a (2n + 1)Zd -periodic self-adjoint Schr¨ odinger n be its integrated density of states; it satisfies operator. Let Nω,λ 1 n (E) = dθ. (2.1) Nω,λ (2π)d {θ∈T∗2n+1 ; Ek (n,ω,λ;θ)≤E} k∈N
n and the torus T∗n is where (Ek (n, ω, λ; θ))k≥0 are the Floquet eigenvalues of Hω,λ defined by T∗n = Rd / n1 Zd . We recall Theorem 1.2 from [24] namely
Theorem 2.1 ([24]) Pick α > 0 and I ⊂ R, a compact interval. There exists ν0 > 0 and ρ > 0 such that, for λ ∈ [0, 1], E ∈ I, ν ∈ (0, ν0 ) and n ≥ ν −ρ , one has n n (E + ν/2)) − E(Nω,λ (E − ν/2)) − e−ν E(Nω,λ
−α
≤ Nλ (E + ν) − Nλ (E − ν) ≤ n n (E + 2ν)) − E(Nω,λ (E − 2ν)) + e−ν E(Nω,λ
−α
. (2.2)
In [24], we did not consider the case when the random operator depends on a parameter λ. One easily checks that the proof of Theorem 1.2 of [24] holds locally uniformly in the parameter λ.
2.2
More on Floquet Theory
It is convenient to introduce some notations. Fix θ ∈ Rd . The set of functions ϕ that are locally square integrable (resp. locally in H 2 ) on Rd and that satisfy ϕ(x+γ) = eiγθ ϕ(x), ∀γ ∈ Zd is denoted by L2θ (resp. Hθ2 ); both spaces are endowed with their natural scalar product (see [31]). The operator H acting on the Hθ2 is denoted by H(θ). Pick n ≥ 1. The operator H is Zd -periodic; hence, it is also (2n + 1)Zd -periodic. 2 To stress this point of view, we sometimes call it H n . Let L2n,θ (resp. Hn,θ ) be the 2 set of functions ϕ that are locally square integrable (resp. locally in H ) on Rd and that satisfy ϕ(x + γ) = ei2πγθ ϕ(x), ∀γ ∈ (2n + 1)Zd ; both spaces are endowed with 2 their natural scalar product. The operator H acting on the space Hn,θ is denoted n by H (θ). Notice that, in this case, we can restrict ourselves to θ ∈ T∗2n+1 . The Floquet eigenvalues and eigenvectors of H n (θ) are easily computed in terms of the Floquet eigenvalues and eigenvectors of H(θ).
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Indeed, let (Ek (θ), ϕk (·, θ))k≥0 be the Floquet eigenvalue and eigenvectors for H(θ) i.e. the solutions to the eigenvalue problem (1.2). One checks that, for γ ∈ Zd2n+1 , Hϕk,γ (·, θ) = Ek,γ (θ)ϕk,γ (·, θ) where Ek,γ (θ) = Ek (θ + γ/(2n + 1)), 1 ϕk,γ (·, θ) = ϕk (·, θ + γ/(2n + 1)). (2n + 1)d/2 For γ = (γ1 , . . . , γd ) ∈ Zd and l ≥ 0, define Cγ,l = {x = (x1 , . . . , xd ); ∀1 ≤ i ≤ d, −l −1/2 ≤ xi −(2l +1)γi < l +1/2} (2.3) The vectors (ϕk,γ (·, θ))k,γ form an orthonormal family in L2 (C0,n ) as ϕk,γ (x, θ)ϕk ,γ (x, θ)dx = ϕk,γ (x, θ)ϕk ,γ (x, θ)dk C0,n
=
β∈Zd /(2n+1)Zd
β+C0,0
1 e2πβ(γ−γ )/(2n+1) d (2n + 1) β∈Zd 2n+1 ϕk (x, θ + γ/(2n + 1))ϕk (x, θ + γ /(2n + 1))dx C0,0
= δkk δγγ . (2.4) The family (ϕk,γ (·, θ))k,γ is complete as the family (ϕk (·, θ))k≥0 is. Indeed, for u ∈ L2 (Rd ), one has u= u ˆn (θ)ϕn (·, θ)dθ = u ˆγ,n (θ)ϕγ,n (·, θ)dθ k≥0
T∗
k≥0 γ∈Zd 2n+1
T∗ 2n+1
where u ˆγ,n (θ) = uˆn (θ + γ/(2n + 1)). So, the pairs (Ek,γ (θ), ϕk,γ (·, θ))k≥0,γ∈Zd2n+1 form a family of (2n + 1)Zd -periodic Floquet eigenvalues and eigenvectors of H. We also make use of spectral projectors. We define Π<E (resp. Πn<E , Πn<E (θ), etc) to be the spectral projector associated to the interval (−∞, E) and to the operator H (resp. H n , H n (θ), etc)
2.3
The asymptotics of the density of states
n We now estimate the density of states of Hω,λ for large n. The size of the cube, n, −ρ is chosen of order λ for some large ρ.
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Pick η < η < d/(4d + 4). The density of states Nλ being constant in gaps of Σλ , as Vω is uniformly bounded, for λ sufficiently small, one has ∀E ∈ [E0 , E− (λ)],
Nλ (E− (λ)) = Nλ (E).
By Theorem 2.1, to estimate Nλ (E(λω) − λ1+η ) − Nλ (E− (λ)) to exponential pre n n cision, we only need to estimate E[Nω,λ (E(λω) − λ1+η ) − Nω,λ (E0 )] for n ≥ λ−ρ (ρ sufficiently large). A well known characterization of the spectrum of random Schr¨ odinger operators ([15]) tells us that, with probability one, the spectrum of n Hω,λ is contained in Σλ , the almost sure spectrum of Hω,λ . Hence, with probability n n (E0 ) = Nω,λ (E− (λ)). We write one, Nω,λ
n n E[Nω,λ (E(λω) − λ1+η ) − Nω,λ (E− (λ))] n =E #{eigenvalues of Hω,λ (θ) in [E− (λ), E(λω) − λ1+η ]} . T∗ 2n+1
As Vω is uniformly bounded in supremum norm, for all θ, the number of eigenvalues n (θ) less than E is bounded by the number of eigenvalues of H n (θ) less than of Hω,λ E + C (for some fixed constant C). The computation done in section 2.2 show that this number is bounded by Cnd (for some constant C depending only on E). So, as T∗2n+1 has volume (2n + 1)−d , for some constant C > 0, one has
n n E[Nω,λ (E(λω) − λ1+η ) − Nω,λ (E− (λ))] ≤ CP(Ω(n, λ, η ))
(2.5)
where
∃θ ∈ Rd , n 1+η Ω(n, λ, η ) = ω; ∃E ∈ [E− (λ), E(λω) − λ ], s.t. Hω,λ (θ)ϕ = Eϕ . 2 , ϕ = 1, ∃ϕ ∈ Hn,θ
Floquet theory gives us another characterization of Ω(n, λ, η ), namely, n Ω(n, λ, η ) = ω; σ(Hω,λ ) ∩ [E− (λ), E(λω) − λ1+η ] = ∅ .
(2.6)
By (2.5) and Theorem 2.1, Theorem 1.1 is an immediate consequence of Proposition 2.1 Fix η ∈ (0, d/(4d + 4)) and ρ > d. Then, there exists λη,ρ > 0 and ε > 0 such that, for λ ∈ (0, λη,ρ ), one has −ε
P[Ω(n, λ, η)] ≤ e−λ . where
2n + 1 = [λ−1/2+2η ]o · [λ−η ]o · [λ−ρ ]o and η < η < d/(4d + 4). Here, [·]o denotes the largest odd integer smaller than · .
(2.7)
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The proof of Proposition 2.1
We first reformulate the definition of Ω(n, λ, η). It is convenient to work at extremal points of the spectrum e.g. at the bottom of the spectrum. We first transform our problem so as to be in that case. Therefore, we project out the part of the spectrum below E0 = (E− (0) + E+ (0))/2. We decompose L2 = L2 (Rd ) into L2 = Π≤E0 L2 + Π>E0 L2 and write n Π≤E0 (Hω,λ − E)Π≤E0 λΠ≤E0 Vωn Π>E0 n . (2.8) Hω,λ − E = n λΠ>E0 Vωn Π≤E0 Π>E0 (Hω,λ − E)Π>E0 n − E)Π≤E0 is invertible and its inverse For E ∈ Iη,λ , the operator Π≤E0 (Hω,λ bounded by a constant independent of λ and ω. This follows from the fact that Vω is uniformly bounded and that Π≤E0 H n Π≤E0 ≤ E+ (0)Π≤E0 . n Hence, E ∈ Iη,λ is in the spectrum of Hω,λ if and only if it is in the spectrum of
−1 n n Π>E0 − λ2 Π>E0 Vωn Π≤E0 Π≤E0 (Hω,λ − E)Π≤E0 Π≤E0 Vωn Π>E0 . Π>E0 Hω,λ (2.9) As the second term in (2.9) is bounded by Cλ2 , we see that E ∈ Iη,λ is in the n n spectrum of Hω,λ implies that the operator Π>E0 Hω,λ Π>E0 has spectrum below 1+η 2 the energy E(λω) − λ + Cλ . Thus, we have proved Ω(n, λ, η) ⊂ ω; ∃ϕ ∈ Π>E0 H 2 , ϕ = 1, (2.10) n ϕ, ϕ ≤ E(λω) − λ1+η + Cλ2 . s.t. Hω,λ Let us estimate the probability of this last event. As Vωn is uniformly bounded, perturbation theory tells us that, for some C > 0 and λ sufficiently small, |E− (λ) − E− (0)| + |E(λω) − E− (0)| ≤ C|λ|. Hence, for ϕ as in (2.10), one has H − E− (0)ϕ, ϕ ≤ |E − E− (0)| + λVωn ∞ ≤ Cλ.
(2.11)
Rewrite the condition in (2.10) as n
(H λω − E(λω))ϕ, ϕ ≤ (−λ1+η + Cλ2 )ϕ2 + λ(E(Vωn ) − Vωn )ϕ, ϕ.
(2.12)
Note that E(Vω ) = E(Vωn ) = ωV . By the definition of E(λω), for λ sufficiently n n small, H λω , hence, Π>E0 H λω Π>E0 has no spectrum in the interval [E0 , E(λω)) i.e. n σ(Π>E0 H λω Π>E0 ) ∩ [E0 , E(λω)) = ∅. (2.13) Now, equation (2.12) implies that, for some ϕ ∈ Π>E0 H 2 , (E(Vωn ) − Vωn )ϕ, ϕ ≥ λη /2ϕ2 .
(2.14)
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Indeed, if this is not the case, (2.12) implies that, for some ϕ ∈ Π>E0 H 2 , n
(H λω − E(λω))ϕ, ϕ ≤ (−λ1+η /2 + Cλ2 )ϕ2 . n
which means that, for λ sufficiently small, Π>E0 H λω Π>E0 has spectrum in the interval [E0 , E(λω) − λ1+η /4); this contradicts (2.13). So, taking (2.10) into account, we have proved that Ω(n, λ, η) ⊂ Ω (n, λ, η) where Ω (n, λ, η)= ω; ∃ϕ ∈ Π>E0 H 2 , ϕ = 1, s.t. equations (2.11) and (2.14) hold . Let us now estimate the probability of Ω (n, λ, η). Write ϕ ∈ Π>E0 H 2 using the Floquet decomposition introduced in section 2.2 χp (θ)ϕp (·, θ)dθ. (2.15) ϕ= p≥p0
T∗
The index p0 is the largest index p ≥ 0 so that, for all θ ∈ T∗ , one has Ep−1 (θ) ≤ E0 (we set E−1 (θ) = −∞, ∀θ). By assumption (H.2), E− (0) is a simple non-degenerate Floquet eigenvalue of H i.e., hence, there exists C > 0 such that • for p = p0 , ∀θ ∈ T∗ ,
|Ep (θ) − E− (0)| ≥ 1/C;
(2.16)
• there exists Z = {θj ; 1 ≤ j ≤ nz } such that Ep0 (θj ) = E− (0) and for θ ∈ Rd , one has (2.17) |Ep0 (θ) − E− (0)| ≥ 1/C inf |θ − θj |2 . 1≤j≤nz
We refer to section 5.1 for more details on these properties. Equation (2.11) implies that |Ep (θ) − E− (0)|2 |χp (θ)|2 dθ ≤ Cλ2 . p≥p0
(2.18)
T∗
Fix 2l + 1 = [λ−1/2+2η ]o · [λ−η ]o and 2k + 1 = [λ−ρ ]o where η < η < d/(4d + 4) is fixed as in Proposition 2.1. Note that 2n + 1 = (2l + 1)(2k + 1). Equations (2.18), (2.16) and (2.17) imply that |χp (θ)|2 dθ + |χp0 (θ)|2 dθ ≤ Cλ2 l2 ≤ Cλ2η . p>p0
T∗
1≤j≤nz
|θ−θj |>1/l
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Hence, we write ϕ=
ϕj + ϕe where ϕj =
1≤j≤nz
|θ−θj |≤1/l
χp0 (θ)ϕp0 (·, θ)dθ and ϕe ≤ Cλη . (2.19)
We note that
ϕj 2 = ϕ2 − Cλ2η = 1 − Cλ2η .
(2.20)
1≤j≤nz
Pluging (2.19) into (2.14), for λ sufficiently small, we get (E(Vωn ) − Vωn )ϕj , ϕj ≥ λη /4. 1≤j,j ≤nz
By (2.20), this implies that, for some 1 ≤ j, j ≤ nz , one has (E(Vωn ) − Vωn )ϕj , ϕj ≥ λη /(2nz )2 .
(2.21)
Let us study the functions (ϕj )1≤j≤nz in more detail. We prove Lemma 2.1 Fix 1 ≤ j ≤ nz . For 1 ≤ l ≤ l, there exists ϕ˜j ∈ L2 (Rd ) such that 1. ϕ˜j is constant on each cube Cγ,l = {x = (x1 , . . . , xd ); ∀1 ≤ i ≤ d, −l − 1/2 ≤ xi − (2l + 1)γi < l + 1/2} where γ = (γ1 , . . . , γd ) ∈ Zd ; 2. there exists C > 0 (depending only on ϕp0 ) ϕj (·) − ϕ˜j (·) · ψp0 (·, θj )L2 ≤ Cl /l
(2.22)
where ψp0 (·, θ) is the periodic component of ϕp0 (·, θ) i.e. ϕp0 (·, θ) = eixθ ψp0 (·, θ). We postpone the proof of Lemma 2.1 to complete the proof of Proposition 2.1. We set 2l + 1 = [λ−1/2+2η ]o and 2k + 1 = [λ−ρ ]o · [λ−η ]o where η < η < d/(4d + 4) is fixed as in Proposition 2.1. Note that 2n + 1 = (2l + 1)(2k + 1). Using point (1) of Lemma 2.1, we define (2l + 1)−d/2 aj (β)1(2l +1)β+C0,l . Ψj (x) = ψp0 (x, θj )ϕ˜j (x) = ψp0 (x, θj ) β∈Zd
As ψp0 (·, θj )ϕ˜j (·) ∈ L2 (Rd ), using the periodicity of ψp0 (·, θj ), we compute ψp0 (·, θj )ϕ˜j 2L2 (Rd ) = (2l + 1)−d |aj (β)|2 |ψp0 (x, θj )|2 dx β∈Zd
=
β∈Zd
|aj (β)|2
C0,0
(2l +1)β+C0,l
|ψp0 (x, θj )|2 dx.
724
F. Klopp
Hence, as
|ϕp0 (x, θj )|2 dx =
C0,0
C0,0
Ann. Henri Poincar´e
|ψp0 (x, θj )|2 dx is positive, by (2.22), we get
|aj (β)|2 ≤ Cϕj 2L2 (Rd ) < +∞.
(2.23)
β∈Zd
By (2.22) and the definitions of l and l , the lower bound (2.21) implies that
(E(Vωn ) − Vωn )Ψj , Ψj ≥ λη /(2nz )2 − Cλη .
(2.24)
Let us compute (E(Vωn ) − Vωn )Ψj , Ψj . One has (E(Vωn ) − Vωn )Ψj , Ψj = (ω − ωγ )(2l + 1)−d aj (β)aj (β)
β∈Zd γ∈Zd 2n+1
(2l +1)β+C
V n (x − γ)ψp0 (x, θj )ψp0 (x, θj )dx 0,l
β ∈Zd
β ∈Zd 2k +1
γ∈Zd 2n+1
=
(ω − ωγ )(2l + 1)−d
aj (β + (2k + 1)β )aj (β + (2k + 1)β ) V n (x − γ + (2l + 1)β )ψp0 (x, θj )ψp0 (x, θj )dx. C0,l
as V n is (2n + 1)Zd -periodic and (2l + 1)(2k + 1) = (2n + 1). Then, setting cjj (β ) = aj (β + (2k + 1)β )aj (β + (2k + 1)β ) β ∈Zd
we obtain (E(Vωn ) − Vωn )Ψj , Ψj = (2l + 1)−d
cjj (β )
β ∈Zd 2k +1
γ∈Zd 2n+1
(ω − ωγ )
V n (x − γ + (2l + 1)β )ψp0 (x, θj )ψp0 (x, θj )dx
C0,l
= (2l + 1)−d
γ∈Zd 2l +1
(ω − ωγ+(2l +1)γ )
γ ∈Zd 2k +1
cjj (β )
β ∈Zd 2k +1
V (x − γ + (2l + 1)(β − γ ))ψp0 (x, θj )ψp0 (x, θj )dx n
C0,l
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= (2l + 1)−d
Xjj (γ)
725
(2.25)
γ∈Zd 2l +1
where
Xjj (γ) =
(ω − ωγ+(2l +1)γ )Ajj (γ, γ ),
γ ∈Zd 2k +1
Ajj (γ, γ ) =
cjj (β )
β ∈Zd 2k +1
V n (x − γ + (2l + 1)(β − γ ))ψp0 (x, θj )ψp0 (x, θj )dx.
C0,l
The random variables (Xjj (γ))γ∈Zd
2l +1
are independent and centered (i.e. E(Xjj (γ)) = 0). They are bounded; indeed, one computes |Xjj (γ)| ≤ C
|Ajj (γ, γ )|
γ ∈Zd 2k +1
≤C
β ∈Zd 2k +1
γ ∈Zd 2k +1
|cjj (β )|
|V n (x − γ + (2l + 1)(β − γ ))||ψp0 (x, θj )||ψp0 (x, θj )|dx
C0,l
≤C
β ∈Zd 2k +1
|cjj (β )|
γ ∈Zd 2k +1
|V n (x − γ + (2l + 1)γ )||ψp0 (x, θj )||ψp0 (x, θj )|dx.
C0,l
Using the fact that V n is (2n+1)Zd -periodic, the relation (2n+1) = (2l +1)(2k +1) and estimate (2.23), we obtain |Xjj (γ)| ≤C |aj (β)| · |aj (β)| β∈Zd
≤ Cϕj ϕj
γ ∈Zd
C0,0
≤ Cϕj ϕj < +∞.
γ ∈Zd 2n+1
|V n (x − γ )||ψp0 (x, θj )||ψp0 (x, θj )|dx
C0,0
|V (x − γ )||ψp0 (x, θj )||ψp0 (x, θj )|dx
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By (2.24) and (2.25), to estimate the probability of Ω(n, λ, η), it suffices to estimate the probability that Xjj (γ). λη /(4nz )2 ≤ λη /(2nz )2 − Cλη ≤ (2l + 1)−d γ∈Zd 2l +1
This is a large deviation estimate; it is bounded above in the usual way using exponential Markov inequalities (see e.g. [7, 6]) and yields d 2η −d/2+2dη +2η P λη /(4nz )2 ≤ (2l + 1)−d Xjj (γ) ≤ e−c(l ) λ ≤ e−cλ γ∈Zd 2l +1
recalling the definition of l . As η < η < d/(4d + 4), one has −d/2 + 2dη + 2η < 0. Hence, for some ε > 0 and λ sufficiently small, one has −ε P λη /(4nz )2 ≤ (2l + 1)−d Xjj (γ) ≤ e−λ . γ∈Zd 2l +1
On the other hand, the probability of Ω(n, λ, η) is bounded by the sum over j and j of the probability estimate above. This yields (2.7) and completes the proof of Proposition 2.1. Proof of Lemma 2.1. Recall that ϕj =
T∗
χ ˜p0 (θ)ϕp0 (·, θ)dθ.
where χ ˜p0 = 1|θ−θj |≤1/l · χp0 (see (2.19)). Using the periodic components, we write ϕj = χ ˜p0 (θ)ei··θ dθ ψp0 (·, θj ) + χ ˜p0 (θ)ei··θ (ψp0 (·, θ) − ψp0 (·, θj ))dθ. T∗
T∗
As discussed in section 5.1, under our assumptions, for θ close to θj , the function (x, θ) → ψp0 (x, θ) is analytic in θ valued in the Zd -periodic, locally square integrable functions in x; hence, we compute 2 i··θ χ ˜ (θ)e (ψ (·, θ) − ψ (·, θ ))dθ p0 p0 j ∗ p0 2 d T L (R ) $ $2 $ $ ix·θ $ $ dx = χ ˜ (θ)e (ψ (x, θ) − ψ (x, θ ))dθ p0 p0 p0 j $ $ γ∈Zd
=
Cγ,0
γ∈Zd
C0,0
$ $ $ $
T∗
T∗
χ ˜p0 (θ)e
ixθ iγθ
e
$2 $ (ψp0 (x + γ, θ) − ψp0 (x + γ, θj ))dθ$$ dx
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In the first step, we rewrote the x-integral over Rd as the sum of the integrals over cubes covering Rd ; then, we shifted the argument x so as to center the cube at 0. In the next step, we use the periodicity of the periodic components, and Parseval’s formula to compute the θ-integral.
T∗
χ ˜p0 (θ)e
= C0,0
C0,0
= T∗
2 (ψp0 (·, θ) − ψp0 (·, θj ))dθ 2
$$ $ $
γ∈Zd
L (Rd )
T
$2 $ χ ˜p0 (θ)eixθ eiγθ (ψp0 (x, θ) − ψp0 (x, θj ))dθ$$ dx ∗ 2
=
i··θ
T∗
|χ ˜p0 (θ)(ψp0 (x, θ) − ψp0 (x, θj ))| dθdx
|χ ˜p0 (θ)|
2
2
|ψp0 (x, θ) − ψp0 (x, θj )| dx dθ ≤ C0,0
C . l2
To conclude, we used the analyticity of θ → ψp0 (·, θ). So we now are left with proving Lemma 2.1 when ϕj is replaced with the function x →
T∗
χ ˜p0 (θ)eix·θ dθ ψp0 (x, θj ).
This is an immediate consequence of Lemma 5.1 when one picks ε = 1/l and η = (2l + 1)/(πl). This completes the proof of Lemma 2.1.
3 The proof of Theorem 1.3 Theorem 1.3 is derived from Theorem 1.1 using the multiscale analysis done in [34, 9]. In order to apply the results of [34, 9], we have to check two assumptions: • the Wegner estimate; • the initial length scale estimate. We first state the Wegner estimate. This estimate is an estimate of the probability of presence of eigenvalues in a given interval. More precisely, we prove Proposition 3.1 There exists λ0 > 0 and c0 > 0 such that, for λ ∈ (0, λ0 ), one has, for n ≥ 1 and θ ∈ T∗2n+1 , for I, an interval in [E− (λ), E(λω) + c0 λ], one has & % & % n n (θ) has an eigenvalue in I} ≤ E #{eigenvalues of Hω,λ (θ) in I} P {Hω,λ ≤ Cλ−1 n2d |I|. (3.1)
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F. Klopp
Ann. Henri Poincar´e
Proposition 3.1 is proved in section 4. It does not imply Theorem 1.2. The volume dependence in (3.1) is not optimal. But it is sufficient for our purpose. In [12], the optimal volume dependence was obtained but for an approximation scheme different from the periodic one used in the present paper. We now state the initial length scale estimate; it is a consequence of Theorem 1.1. Proposition 3.2 Fix η ∈ (0, d/(4d + 4)) and ρ > d sufficiently large. Then, there exists a > 0, ε0 > 0 and λ0 > 0 such that, for 0 < λ < λ0 and 0 < ε < ε0 , for −ε −ε λ−ρ < n < eλ , with probability 1 − e−γ , for E ∈ Iη,λ and θ ∈ T∗2n+1 , one has √ n (θ))−1 1C0,n ≤ ea |E−λω|n . (3.2) 1C0,2n+1 \C0,2n−1 (E − Hω,λ Using Propositions 3.1 and 3.2 for θ = 0, Theorem 1.2 is a consequence of Theorem 12.2 and Proposition 13.1 of [34], and Theorem 3.8 of [9] modulo the following remark. In these results, the initial length scale needed to start the multi-scale analysis depends on λ. Though this dependence is not explicitly written down in [34, 9], following the proofs, one sees that the initial length scale grows polynomially in λ−1 . As Proposition 3.2 holds up to length scales to any polynomial order, we can indeed apply Theorem 12.2 and Proposition 13.1 of [34], and Theorem 3.8 of [9].
3.1
The proof of the initial length scale estimate
We now prove Proposition 3.2. Fix η, η as in Proposition 3.2 i.e. η < η < d/(4d+4) n and fix ρ > d. Define the event Ωγ,η ,n = {there exists an eigenvalue of Hω,λ in 1+η [E− (λ), E(λω) − λ ]}. Using Theorem 2.1 in conjunction with Theorem 1.1, we obtain that, there exists γη > 0 and ε > 0 such that, for 0 < λ < λη and −ε/2 λ−ρ ≤ n ≤ eλ , one has −ε/2
P(Ωλ,η ,n ) ≤ Cedλ
−ε
e−λ
1
−ε
≤ e− 2 λ .
By a Combes-Thomas estimate (see e.g. [11] and references therein), for E ∈ Iη,λ and ω ∈ Ωλ,η ,n , one has √ η +1 ||γ−γ |/C n (θ))−1 1Cγ ,0 ≤ Cλ−1−η e− |E−E(λω)+λ , (3.3) 1Cγ,0 (E − Hω,λ for some constant C independent of λ and η. Define δ(E) := |E − E(λω) + λ1+η |. We notice that, for E ∈ Iη,λ , E − E(λω) ≥ λ1+η ; as η < η , for λ sufficiently small, one has 1 δ(E) ≥ |E − E(λω)|. 2 Pick 0 < ε0 < ε. Summing (3.3) over γ ∈ C0,2n+1 \ C0,2n−1 and γ ∈ C0,n and −ε0 using the fact that λ−ρ < n < eλ , we get that, for ω ∈ Ωγ,η ,n and E ∈ Iη,λ , estimate (3.2) holds. This completes the proof of Theorem 3.2.
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4 Proof of the Wegner estimate The proof of Proposition 3.1 follows the philosophy introduced in [19] and is quite similar to the proof of Theorem 1.2 in [12]. For the reader’s convenience, we nevertheless reproduce the details here. As above, we fix a band edge E− (0) and θ ∈ T∗2n+1 . All the operators we now discuss are considered with θ-quasi-periodic boundary condition in the sense explained in section 2.2; in our notations, we forget about the parameter θ i.e. we write Ek instead of Ek (θ), H instead of H(θ), etc. Our statements are uniform in the parameter θ. Recall that the Floquet eigenvalues of H are denoted by (Ek )k≥1 and ordered increasingly. Define Π0 = Πn<E− (0)+1/2 and Π⊥ 0 = 1 − Π0 . One has n ⊥ ⊥ Π⊥ 0 Hω,λ Π0 ≥ (E− (0) + 1/2)Π0 .
Moreover, by the discussion of section 2.2, the projector Πn<E− (0)+1/2 is of rank at most Cnd . Using the decomposition 1 = Π0 +Π⊥ 0 , we see that E ≤ E− (0)+1/2 is an eigenvalue n if and only if it is an eigenvalue of the operator of Hω,λ n ⊥ n ⊥ −1 ⊥ n Π0 Hω,λ Π0 − λ2 Π0 Vωn Π⊥ Π0 Vω Π0 . 0 (E − Π0 Hω,λ Π0 )
Define
(4.1)
⊥ n ⊥ −1 ⊥ n Γ0 (ω, E) = Π0 Vωn Π⊥ Π0 Vω Π0 . 0 (E − Π0 Hω,λ Π0 )
The operator Γ0 (ω, E) and all its derivatives in ω and E are bounded for E ≤ E− (0) + 1/2 and ω bounded, and λ sufficiently small. By assumption (H2), there exists ω + ∈ supp(ω0 ) and C > 0 such that, for λ sufficiently small, E(λω) − E(λω + ) ≥ λ/C. (4.2) For more details, we refer to section 5.1. We rewrite Hω,λ = H λω+ + λVω˜ where (˜ ωγ )γ = (ωγ −ω + )γ . So, for λ sufficiently small, the operator H λω+ has no spectrum in the interval [(E− (0)+2E+ (0))/3, E(λω)+λ/C] (for some C > 0). Hence, for any n n ≥ 1, the operator H λω+ has no spectrum in [(E− (0) + 2E+ (0))/3, E(λω) + λ/C]. − − n We decompose Π0 = Π+ 0 + Π0 where Π0 = Π<E0 (where we recall that E0 = (E− (0) + E+ (0))/2). − − − − − For E ∈ [E0 , E(λω) + λ/(2C)], EΠ− 0 − Π0 Hω,λ Π0 = EΠ0 − Π0 Π0 Hω,λ Π0 Π0 is invertible and its inverse is bounded independently of ω and λ sufficiently small. Hence, E ∈ [E0 , E(λω) + λ/(2C)] is an eigenvalue of the operator defined in (4.1) if and only if E is an eigenvalue of + 0 + + n 2 0+ (ω, E) (4.3) Π+ 0 Hω,λ Π0 − λ Π0 Γ (ω, E)Π0 + Γ where
− n − −1 − n + n − Γ0+ (ω, E) = Π+ Π0 Vω Π0 . 0 Vω Π0 (E − Π0 Hω,λ Π0 )
730
F. Klopp
Ann. Henri Poincar´e
For E ∈ [E0 , E(λω) + λ/(2C)] and for ω bounded, the operator Γ0+ (ω, E) and all its derivatives in ω and E are bounded. By the Birman-Schwinger principle (see e.g. [18] and references therein), an energy E in the interval [E0 , E(λω) + λ/(2C)] is an eigenvalue (with multiplicity m) of the operator defined in (4.3) if and only if −1 is an eigenvalue (with multiplicity m) of the operator + n + n + −1/2 G(E, ω, λ) = λ(Π+ Π0 Vω˜ ,λ Π0 − 0 H λω + Π0 − E) & % + 0 n + 0+ −1/2 (ω, E) (Π+ . (4.4) λ Π0 Γ (ω, E)Π+ 0 +Γ 0 H λω + Π0 − E) Notice that, for E, E ∈ [E0 , E(λω) + λ/(2C)], the operator G(E, ω, λ) satisfies G(E, ω, λ) − G(E , ω, λ) ≤ C|E − E |λ−1 .
(4.5)
For E0 ∈ [E0 , E(λω) + λ/(2C)] and ε > 0 such that [E0 − ε, E0 + ε] ⊂ [E0 , E(λω) + λ/(2C)], if E is an eigenvalue in [E0 − ε, E0 + ε], then, by (4.5), G(E0 , ω, λ) has an eigenvalue in [−1 − Cελ−1 , −1 + Cελ−1 ]. This yields n in [E0 − ε, E0 + ε]} #{eigenvalues of Hω,λ
≤ #{eigenvalues of G(E0 , ω, λ) in [−1 − Cελ−1 , −1 + Cελ−1 ]}, the eigenvalues being counted with multiplicity. Hence, one has n in [E0 − ε, E0 + ε]}) E(#{eigenvalues of Hω,λ
≤ E(#{eigenvalues of G(E0 , ω, λ) in [−1 − Cελ−1 , −1 + Cελ−1 ]}). (4.6) To estimate the last expectation, we use the computations done in [12]. Define Iε = [−1−κ, −1+κ] where κ = Cελ−1 . Let ρ be a nonnegative, smooth, monotone decreasing function such that ρ(x) = 1, for x < −κ/2, and ρ(x) = 0, for x ≥ κ/2. We can assume that ρ has compact support since G(E0 , ω, λ) is lower semibounded independently of n. Then, one has E(#{eigenvalues of G(E0 , ω, λ) in [−1 − κ, −1 + κ]}) ≤ E{tr[ρ(λG(E0 , ω, λ) + 1 − 3κ/2) − ρ(G(E0 , ω, λ) + 1 + 3κ/2)]} ( ' 3κ/2 d ρ(G(E0 , ω, λ) + 1 − t) dt . ≤ E tr −3κ/2 dt In order to evaluate the ρ term in (4.7), define the vector field V=
γ∈Zd 2n+1
ω ˜ γ ∂ωγ
(4.7)
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and compute VG(E0 , ω, λ) = G(E0 , ω, λ) + λG1 (E0 , ω, λ) where G1 (E0 , ω, λ) is bounded uniformly in ω, λ and E0 ∈ [E0 , E(λω) + λ/(2C)]. n + n This follows from the fact that Π+ 0 H λω + Π0 does not depend on ω, that VVω ˜ ,λ = n 0 0+ Vω˜ ,λ , that VΓ (ω, E0 ) and VΓ (ω, E0 ) stay bounded, and that for E0 ∈ [E0 , E(λω) + λ/(2C)], dist(E, σ(H λω+ )) ≥ λ/C for some C > 1. We now write the ρ term in (4.7) as Vρ(G(E0 , ω, λ) + 1 − t) = ρ (G(E0 , ω, λ) + 1 − t)VG(E0 , ω, λ) = ρ (G(E0 , ω, λ) + 1 − t)(G(E0 , ω, λ) + O(λ)) We note that ρ ≤ 0, and that on suppρ , one has G(E0 , ω, λ) ≤ (−1 + 2κ). So, for λ sufficiently small, we obtain −ρ (G(E0 , ω, λ) + 1 − t) ≤ −
1 2(1 − 2κ)
ω∈Zd 2n+1
ω ˜γ
∂ρ (G(E0 , ω, λ) + 1 − t). (4.8) ∂ωγ
d ρ(x + 1 − t) = −ρ (x + 1 − t), the right With this estimate, and the fact that dt hand side of (4.7) is bounded from above by
−
1 2(1 − 2κ)
γ∈Zd 2n+1
3κ/2
−3κ/2
E{˜ ωγ
∂ tr[ρ(G(E0 , ω, λ) + 1 − t)]}dt. ∂ωγ
(4.9)
In order to evaluate the expectation, we select one random variable, say ωγ (γ ∈ Zd2n+1 ), and first integrate with respect to this variable using hypothesis (H3). Let h0 be the common density of the random variables (ωγ )γ∈Zd . By assumption (H3), −1 there is a decomposition supp(ω0 ) = ∪N l=0 (Ml , Ml+1 ) so that h0 is absolutely ˜ 0 be the function h ˜ 0 (x) := xh0 (x). As h ˜ 0 is continuous on each subinterval. Let h locally absolutely continuous, we can integrate by parts and obtain $ $ $ $ ˜ 0 (ωγ ) ∂ tr{ρ(G(E0 , ω, λ) + 1 − t) − ρ(G(E0 , ω, λ)M0 ,γ + 1 − t)}$ $ dωγ h $ $ ∂ωγ R $ −1 Ml+1 $N $ ˜ 0 (ωγ ) ∂ tr{ρ(G(E0 , ω, λ) + 1 − t) =$ dωγ h $ ∂ωγ l=0 Ml $ −ρ(G(E0 , ω, λ)M0 ,γ + 1 − t)}$ ˜ 0 ∞ |tr{ρ(G(E0 , ω, λ)MN ,γ + 1 − t) − ρ(G(E0 , ω, λ)M0 ,γ + 1 − t)}| ≤ h ˜ ∞ + h 0
sup |tr{ρ(G(E0 , ω, λ)x,γ + 1 − t) − ρ(G(E0 , ω, λ)M0 ,γ + 1 − t)}|,
x∈supp˜ ω
where G(E0 , ω, λ)x,γ is the operator G(E0 , ω, λ) with the coupling constant ωγ at the γ th -site fixed at the value ωγ = x. As G(E0 , ω, λ) is of rank at most Cnd , we
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get that, for some C > 0, one has $ $ $ $ ˜ 0 (ωγ ) ∂ tr{ρ(G(E0 , ω, λ) + 1 − t) − ρ(G(E0 , ω, λ)M0 ,γ + 1 − t)}$≤ Cnd . $ dωγ h $ $ ∂ωγ R
Pluging this in turn into (4.9), (4.8), (4.7), and finally (4.6), for λ sufficiently small, we obtain n ˜ 2d ελ−1 . E(#{eigenvalues of Hω,λ in [E0 − ε, E0 + ε]}) ≤ Cn2d κ = Cn
This completes the proof of Proposition 3.1.
5 Appendix 5.1
The dependence of Σλ on λ
In this section, we study the dependence in λ of Σλ (i.e. of E− (λ)) in a neighborhood of a gap of Σ0 . In the case of a nonnegative single site potential V , such a study was done in [17]. As E(0) is simple, there exists a unique Floquet eigenvalue of H, say En (·), such that, for some θ ∈ T∗ , E− (0) = En (θ). Let Z ⊂ T∗ be the set of points θ such that En (θ) = E− (0). This set is discrete by assumption (H1), thus, finite, as T∗ is compact. To fix ideas, set Z = {θj ; 1 ≤ j ≤ nz }. As E− (0) is simple, for 1 ≤ j ≤ nz , the Floquet eigenspace associated to E− (0) and θj is one-dimensional. Hence, by standard analytic perturbation theory (cf [14, 31]), for θ sufficiently close to θj , the Floquet eigenvalue En (θ) is analytic in θ, and one can find a Floquet eigenvector associated to En (θ), say ϕn (x, θ), that is normalized and analytic in θ. Applying now analytic perturbation theory to H λ (θ) (i.e. H λ with θ-periodic boundary conditions) for small λ, we obtain that, there exists δ > 0 and λ0 > 0 such that, for |λ| < λ0 and 1 ≤ j ≤ nz , • in the interval ]E− (0) − δ, E− (0) + δ[, H λ (θ) has a unique Floquet eigenvalue for the Floquet parameter |θ − θj | < δ; let En (θ, λ) be this eigenvalue; • En (θ, λ) is real analytic in (θ, λ) ∈ {|θ − θj | < δ}×]E− (0) − δ, E− (0) + δ[, and, there exists a Floquet eigenvector, say ϕn (x, θ, λ), that is normalized and analytic in (θ, λ); • the rest of the spectrum of H λ (θ) lies outside of ]E0 , E− (0) + δ[. This proves that E(λ) is equal to one of the numbers (En (θ˜j (λ), λ))1≤j≤nz where θ˜j (λ) is unique minimum of En (θ, λ) in {|θ−θj | < δ}. The points θ˜j (λ) are analytic in λ; they satisfy the equation ∇θ En (θ˜j (λ), λ) = 0 and the Hessian matrix of θ → En (θ, λ) is non degenerate in a neighborhood of θj for λ small. This is a
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consequence of perturbation theory as it holds for λ = 0 by assumption. One computes ∂λ En (θ˜j , λ)λ=0 = * ) ˜ λ)∂λ θ˜j (λ) V ϕn (x, θ˜j (λ), λ), ϕn (x, θ˜j (λ), λ) + ∇θ En (θ(λ), λ=0 = V (x)|ϕn (x, θj )|2 dx. Rd
(5.1) Hence, assumption (H2) is satisfied if and only if ∀1 ≤ j ≤ nz , V (x)|ϕn (x, θj )|2 dx = 0 Rd or ∃1 ≤ j < j ≤ nz , V (x)|ϕn (x, θj )|2 dx · Rd
Rd
V (x)|ϕn (x, θj )|2 dx < 0. (5.2)
For a given operator H, this condition is satisfied for a generic V . The computation (5.1) and the analyticity of the (En (θ˜j (λ), λ))1≤j≤nz implies that, for some 1 ≤ j ≤ nz , E(λ) = E− (0) + λ V (x)|ϕn (x, θj )|2 dx + O(λ2 ) (5.3) Rd
On the other hand, using the characterization of Σλ , the almost sure spectrum of Hω,λ in terms of admissible periodic spectra (see [15, 30]), we know that, for λ sufficiently small and t ∈ess-supp(ω0 ), we have 1 (E− (0) + E+ (0)) ≤ E− (λ) ≤ E(λt). 2 Let ω+ and ω− respectively be the essential supremum and infimum of the random variables (ωγ )γ∈Zd . As the random variables are not trivial, using (5.2) and the computation (5.1), one sees that, for some C > 0 and for λ sufficiently small |E(λω) − E(λω+ )| ≥ |λ|/C and |E(λω) − E(λω− )| ≥ |λ|/C. This implies that (0.2) holds for some C > 0 and λ sufficiently small.
5.2
A useful lemma
We prove Lemma 5.1 Pick ε > 0 and u ∈ L2 (Rd ) such that supp(u) ⊂ {|x| ≤ ε}. Then, for any η ∈ (0, 1), there exists u ˜ ∈ L2 (Rd ) such that
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• ˜ uL2 = uL2 , • u − u ˜L2 ≤ CηuL2 , η , γ ∈ Zd where • u ˆη is constant on each cube Cγ, 2ε
Cγ,r = {x = (x1 , . . . , xd ); −πr ≤ xi − 2πrγi < πr}. Lemma 5.1 is a very simple quantitative version of the Uncertainty Principle. It is the analogue of Lemma 6.2 in [21] developed for the discrete setting. Proof. Pick ε, η and u as in Lemma 5.1. Consider u as a (2ε/η)Zd -periodic function (continue it by 0 on the fundamental cell of (2ε/η)Zd and periodically to the rest of Rd ). Expand it in a Fourier series to get ) η *d πη πη u(x) = u ˆγ ei ε γx where uˆγ = u(x)e−i ε γx dx. 2ε Rd d γ∈Z
Parseval’s identity then reads 2ε d |ˆ uγ |2 . η d
u2L2 =
(5.4)
γ∈Z
Define the function v : Rd → Rd by v(ξ) =
2ε η
d
d η , γ ∈ Z . u ˆγ for ξ ∈ Cγ, 2ε
˜ be the inverse Fourier transform of v. Check that it By (5.4), v ∈ L2 (Rd ). Let u satisfies the properties stated in Lemma 5.1. First, (5.4) yields ˜ u2L2
1 = v2L2 = (2π)d
2ε2 πη 2
d γ∈Zd
C0,
|ˆ uγ |2 dξ = u2L2 . η 2ε
Let u ˆ be the Fourier transform of u; we compute Rd
2
|ˆ u(ξ) − v(ξ)| dξ =
γ∈Zd
C0,
η 2ε
$ d $$2 $ 2ε πη $ $ ˆ(ξ + γ ) − uˆγ $ dξ. $u $ $ ε η
On the other hand, we note u ˆ(ξ + γ
πη )− ε
2ε η
d
u ˆγ =
Rd
u(x)(eixξ − 1)e−i
πη ε γx
dx.
(5.5)
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As we did for u, we consider x → u(x)(eixξ − 1) as a (2ε/η)Zd -periodic function; then, Parseval’s identity yields $ d $$2 d $$ 2ε 2ε πη $ uˆγ $ = |u(x)(eixξ − 1)|2 dx $uˆ(ξ + γ ) − $ $ ε η η |x|≤ε d
γ∈Z
Substituting in (5.5), we obtain Rd
that is
2
|ˆ u(ξ) − v(ξ)| dξ = Rd
|x|≤ε
2
|u(x)|
|ˆ u(ξ) − v(ξ)|2 dξ ≤ Cη 2
This completes the proof of Lemma 5.1.
|x|≤ε
2ε η
d |e C0,
ixξ
2
− 1| dξ
dx
η 2ε
|u(x)|2 dx = Cη 2 u2L2 .
References [1] J. M. Barbaroux, J. M. Combes and P. D. Hislop, Localization near band edges for random Schr¨ odinger operators. Helv. Phys. Acta, 70(1-2), 16–43 (1997). Papers honouring the 60th birthday of Klaus Hepp and of Walter Hunziker, Part II (Z¨ urich, 1995). [2] M. Sh. Birman, Perturbations of periodic Schr¨ odinger operators. Lectures given at the Mittag-Leffler Insitute during the programm “Spectral Problems in Mathematical Physics”, 1992. [3] J. M. Combes, P. D. Hislop and E. Mourre, Spectral averaging, perturbation of singular spectra and localization, Transactions of the American Mathematical Society, 348, 4883–4895 (1996). [4] H.L. Cycon, R.G. Froese, W. Kirsch and B. Simon. Schr¨ odinger Operators. Springer Verlag, Berlin, 1987. [5] Y Colin de Verdi`ere, Sur les singularit´es de Van Hove g´en´eriques, in Analyse globale et physique math´ematique (Lyon, 1989), volume 46 of M´emoire de la Soci´et´e Math´ematique de France, pages 99–110, 1991. Colloquium en l’honneur d’E. Combet. [6] A. Dembo and O. Zeitouni, Large deviation techniques and applications. Jones and Bartlett Publi-shers, Boston, 1992. [7] J.-M. Deuschel and D. Stroock, Large deviations, volume 137 of Pure and applied Mathematics, Academic Press, 1989. [8] M. Eastham, The spectral theory of periodic differential operators. Scottish Academic Press, Edinburgh, 1973.
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[9] F. Germinet and A. Klein, Bootstrap multiscale analysis and Hilbert-Schmidt dynamical localization, Technical report, UCI, 2001. to appear in Comm. Math. Phys. [10] F. Ghribi. Asymptotique de Lifshitz pour des op´erateurs de Schr¨ odinger a ` champ magn´etique al´eatoire. PhD thesis, Universit´e Paris 13, Villetaneuse. en pr´eparation. [11] P. Hislop, Exponential decay of two-body eigenfunctions: A review. Available on mp-arc, 2001. [12] P. Hislop and F. Klopp. The integrated density of states for some random operators with nonsign definite potentials. To appear in Jour. Func. Anal. [13] D. Jerison and C. Kenig, Unique continuation and absence of positive eigenvalues for Schr¨ odinger operators. Annals of Mathematics 121, 463–494 (1984). [14] T. Kato, Perturbation Theory for Linear Operators. Springer Verlag, Berlin, 1980. [15] W. Kirsch and F. Martinelli, On the spectrum of Schr¨ odinger operators with a random potential, Communications in Mathematical Physics 85, 329–350 (1982). [16] W. Kirsch and B. Simon, Comparison theorems for the gap of Schr¨ odinger operators, J. Funct. Anal. 75(2), 396–410 (1987). [17] W. Kirsch, P. Stollmann and G. Stolz, Localization for random perturbations of periodic Schr¨ odinger operators, Random Oper. Stochastic Equations 6(3), 241–268 1998. [18] M. Klaus, Some applications of the Birman-Schwinger principle, Helv. Phys. Acta 55(1), 49–68 (1982/83). [19] F. Klopp, Localization for some continuous random Schr¨ odinger operators, Communications in Mathematical Physics 167, 553–570 (1995). [20] F. Klopp, Lifshitz tails for random perturbations of periodic schr¨ odinger operators, To appear in the proceedings of the conference “Schr¨ odinger operators”, Goa, Dec. 2000.. [21] F. Klopp, Weak disorder localization and Lifshitz tails, Technical report, Universit´e Paris-Nord, 2001. [22] F. Klopp and J. Ralston, Endpoints of the spectrum of periodic operators are generically simple, Methods and Applications of Analysis 7(3), 459–464 (2000). [23] F. Klopp, Internal Lifshits tails for random perturbations of periodic Schr¨ odinger operators, Duke Math. J. 98(2), 335–396 (1999). [24] F. Klopp, Internal Lifshitz tails for Schr¨ odinger operators with random potentials, To appear in Jour. Math. Phys.
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[25] F. Klopp and L. Pastur, Lifshitz tails for random Schr¨ odinger operators with negative singular Poisson potential, Comm. Math. Phys. 206(1), 57–103 (1999). [26] P. Kuchment, Floquet theory for partial differential equations, volume 60 of Operator Theory: Advances and Applications, Birkh¨auser, Basel, 1993. [27] I. M. Lifshitz, Structure of the energy spectrum of impurity bands in disordered solid solutions, Soviet Physics JETP 17, 1159–1170 (1963). [28] I.M. Lifshitz, S.A. Gredeskul and L.A. Pastur, Introduction to the theory of disordered systems, Wiley, New-York, 1988. [29] A. Outassourt, Comportement semi-classique pour l’op´erateur de Schr¨ odinger a potentiel p´eriodique, Journal of Functional Analysis 72, 65–93 (1987). ` [30] L. Pastur and A. Figotin, Spectra of Random and Almost-Periodic Operators, Springer Verlag, Berlin, 1992. [31] M. Reed and B. Simon, Methods of Modern Mathematical Physics, Vol IV: Analysis of Operators, Academic Press, New-York, 1978. [32] B. Simon, Semi-classical analysis of low lying eigenvalues III. width of the ground state band in strongly coupled solids, Annals of Physics 158, 415–420 (1984). [33] J. Sj¨ ostrand, Microlocal analysis for periodic magnetic Schr¨ odinger equation and related questions, In Microlocal analysis and applications, volume 1495 of Lecture Notes in Mathematics, Berlin, 1991. Springer Verlag. [34] P. Stollman, Caught by disorder. Birkh¨ auser, 2001. [35] E.C. Titschmarch, Eigenfunction expansions associated with second-order differential equations. Part II, Clarendon Press, Oxford, 1958. [36] I. Veselic, Localization for random perturbations of periodic Schr¨ odinger operators with regular Floquet eigenvalues, Technical report, Universt¨ at Bochum, 1998. Fr´ed´eric Klopp Universit´e de Paris-Nord D´epartement de Math´ematique Institut Galil´ee U.M.R. 7539 C.N.R.S 99 Avenue J.-B. Cl´ement F-93430 Villetaneuse France email: [email protected] Communicated by Bernard Helffer submitted 12/10/01, accepted 05/02/02
Ann. Henri Poincar´e 3 (2002) 757 – 772 c Birkh¨ auser Verlag, Basel, 2002 1424-0637/02/040757-16
Annales Henri Poincar´ e
Remarks on Nonlinear Schr¨ odinger Equations with Harmonic Potential R. Carles Abstract. Bose-Einstein condensation is usually modeled by nonlinear Schr¨ odinger equations with harmonic potential. We study the Cauchy problem for these equations. We show that the local problem can be treated as in the case with no potential. For the global problem, we establish an evolution law, which is the analogue of the pseudo-conformal conservation law for the nonlinear Schr¨ odinger equation. With this evolution law, we give wave collapse criteria, as well as an upper bound for the blow up time. Taking the physical scales into account, we finally give a lower bound for the breaking time. This study relies on two explicit operators, suited to nonlinear Schr¨ odinger equations with harmonic potential, already known in the linear setting.
1 Introduction This paper is devoted to existence and blow up results for the nonlinear Schr¨ odinger equation with isotropic harmonic potential, 2 2 i∂ u + ∆u = ω x2 u + λ|u |2σ u , t 2 2 u|t=0 = u0 ,
(t, x) ∈ R+ × Rn ,
(1.1)
where > 0, λ ∈ R, and ω, σ > 0. The notation x2 stands for |x|2 . Similar equations are considered for Bose-Einstein condensation (see for instance [8], [15], [16]), with σ = 1; the real λ may be positive or negative, according to the considered chemical element, and is proportional to 2 . With the operators used in [3] and [4] (see Eq. (1.3)), we prove existence results which are analogous to the wellknown results for the nonlinear Schr¨odinger equation with no potential (see for instance [7]). These operators simplify the proof of some results of [13], [15] and [16], and provide more general results (in particular, for the case of Bose-Einstein condensation in space dimension three). In addition, we state two evolution laws (Lemma 3.1), which can be considered as the analogue of the pseudo-conformal evolution law of the free nonlinear Schr¨ odinger field, and allow us to prove blow up results. Precisely, if we assume that λ is negative (attractive nonlinearity) and σ ≥ 2/n, then under the condition λ 1 ∇u0 2L2 + u 2σ+2 2σ+2 ≤ 0, 2 σ+1 0 L
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π the wave collapses at time t∗ ≤ 2ω (Prop. 3.2). In particular, blow up occurs for focusing cubic nonlinearities (λ < 0 and σ = 1) in space dimensions two and three, but not in space dimension one (for other reasons; see Sect. 2). Sect. 4 is devoted to estimates from below of the time t∗ , under the breaking 2 n assumption that the initial data u0 is bounded in u ∈ L (R ); xu,∇u ∈ L2 (Rn ) , uniformly with respect to ∈]0, 1] (in particular, this means that u0 is not oscillatory). We prove that if λ is negative and proportional to 2 , σ = 1 (the physical case), and n = 2 or 3, then the wave collapse time can be bounded from π − Λα , for some constant Λ and positive number α (Prop. 4.1). When below by 2ω n = 1, we consider the case of a quintic nonlinearity (σ = 2), which should be the right model for Bose-Einstein Condensation in low dimension (see [12]), and we π − Λ, for some constant Λ. Notice that all these results are prove that t∗ ≥ 2ω proved for fixed , with constants independent of ∈]0, 1]. The following quantities are formally independent of time,
N =u (t)2L2 , 1 ω2 λ xu (t)2L2 + u (t)2σ+2 E = ∇x u (t)2L2 + L2σ+2 . 2 2 σ+1
(1.2)
If N and E are defined at time t = 0, we prove that the solution u is defined locally in time, with the conservation of N and E , provided that σ < 2/(n − 2) when n ≥ 3. If λ ≥ 0, then the solution u is defined globally in time. If λ < 0, several cases occur. • If σ < 2/n, then the solution is defined globally in time. • If σ ≥ 2/n, then the solution is defined globally in time if u0 is sufficiently small. • If σ ≥ 2/n and E ≤
ω2 2 2 xu0 L2 ,
then the solution collapses at time t∗ ≤
π 2ω .
The operators on which our analysis relies are Jj (t) =
ω xj sin(ωt) − i cos(ωt)∂j ;
Hj (t) = ωxj cos(ωt) + i sin(ωt)∂j . (1.3)
We denote J (t) (resp. H (t)) the operator-valued vector with components Jj (t) (resp. Hj (t)). Lemma 1.1 J and H satisfy the following properties. • The commutation relation, ω2 2 ω2 2 2 2 x = H (t), i∂t + ∆ − x = 0. J (t), i∂t + ∆ − 2 2 2 2
(1.4)
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x2
• Denote M (t) = e−iω 2 tan(ωt) , and Q (t) = eiω 2 cot(ωt) , then J (t) = −i cos(ωt)M (t)∇x M (−t),
(1.5)
H (t) = i sin(ωt)Q (t)∇x Q (−t). • The modified Sobolev inequalities. For n ≥ 2, and 2 ≤ r < by 1 1 − . δ(r) ≡ n 2 r
2n n−2 ,
define δ(r) (1.6)
2n Let 2 ≤ r < n−2 (2 ≤ r ≤ ∞ if n = 1); there exists Cr independent of such that, for any ϕ ∈ Σ, 1−δ(r)
ϕLr ≤ Cr −δ(r) ϕL2
J (t)ϕ)L2 + H (t)ϕL2
δ(r)
.
(1.7)
• For any function F ∈ C 1 (C, C) of the form F (z) = zG(|z|2 ), we have, π Z, ω π π + Z. J (t)F (v) = ∂z F (v)J (t)v − ∂z¯F (v)J (t)v, ∀t ∈ 2ω ω
H (t)F (v) = ∂z F (v)H (t)v − ∂z¯F (v)H (t)v, ∀t ∈
(1.8)
Remark. Property (1.8) is a direct consequence of (1.5). Property (1.7) is a consequence of the usual Sobolev inequalities and (1.5). These operators are well-known in the linear theory (see e.g. [14] p. 108, [3]), they are the quantization of momentum and position, hence (1.4). Their action in the nonlinear setting, as stated in the above lemma, proves to be very efficient to analyze (1.1). Notations. We work with initial data which belong to the space Σ := u ∈ L2 (Rn ) ; xu, ∇u ∈ L2 (Rn ) . Notice that Σ = D( −∆ + |x|2 ): we work in the same space as in [13]. The notation r stands for the H¨older conjugate exponent of r. The paper is organized as follows. In Sect. 2, we study the local Cauchy problem for (1.1), and we give sufficient conditions for the solution of (1.1) to be defined globally in time. In Sect. 3, we give a sufficient condition under which the solution blows up in finite time, and provide an upper bound for the breaking time. In Sect. 4, we give a lower bound for the breaking time, that shows that the upper bound underscored in Sect. 3 is the physical breaking time in the semi-classical limit, provided that no rapid oscillation is present in the initial data. The results of Sections 2 and 3 were announced in [6].
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2 Existence results The solution of (1.1) with λ = 0 is given by Mehler’s formula (see e.g. [9]), x2 +y2 n/2
iω ω cos(ωt)−x.y 2 sin(ωt) e u0 (y)dy =: U (t)u0 (x). u (t, x) = 2iπ sin ωt Rn This formula defines a group U (t), unitary on L2 , for which Strichartz estimates are available, that is, mixed time-space estimates, which are exactly the same as t for U0 (t) = ei 2 ∆ . Recall the main properties from which such estimates stem (see [7], or [11] for a more general argument). • The group U (t) is unitary on L2 , U (t)L2 →L2 = 1. • For 0 < t ≤
π 2ω ,
the group is dispersive, with U (t)L1 →L∞ ≤ C|t|−n/2 .
We postpone the precise statement of Strichartz estimates to Sect. 4. Duhamel’s formula associated to (1.1) reads u (t, x) = U (t)u0 (x) − iλ−1
0
t
U (t − s) |u |2σ u (s, x)ds.
Replacing U (t) by U0 (t) yields Duhamel’s formula associated to 2 i∂ u + ∆u = λ|u |2σ u , t 2 u|t=0 = u0 .
(2.1)
The local Cauchy problem for this equation is now well-known in many cases (see for instance [7] for a review). In particular, the local well-posedness in Σ is established thanks to the operators ∇x and x/ + it∇x (Galilean operator). This result is proved thanks to Strichartz inequalities, and to the following properties. • The above two operators commute with i∂t +
2 2 ∆.
• They act on the nonlinearity |u |2σ u like derivatives. • Gagliardo-Nirenberg inequalities. From Lemma 1.1, the operators H and J meet all these requirements. Mimicking the classical proofs for (2.1) easily yields, Proposition 2.1 Let u0 ∈ Σ. If n ≥ 3, assume moreover σ < 2/(n − 2). Then there exists T > 0 such that (1.1) has a unique maximal solution u ∈ C([0, T [, Σ). u is maximal in the sense that if T is finite, then u (t)Σ → ∞ as t ↑ T . Moreover N and E defined by (1.2) are constant for t ∈ [0, T [.
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Remark. This result was proved in [13], for more general potentials. We want to underscore the fact that in the case of the harmonic potential, there is essentially nothing to prove, when using J and H . If λ > 0, the conservations of mass and energy provide a priori estimates on the Σ-norm of u (t), and prove global existence in Σ. If λ < 0 and σ < 2/n, then the energy E controls the Σ-norm of u (t). Indeed, from Gagliardo-Nirenberg inequalities (1.7), 1−δ(2σ+2)
u (t)L2σ+2 ≤ Cu (t)L2
δ(2σ+2) J (t)u L2 + H (t)u L2 .
Notice that the following identity holds point-wise, |ωxu (t, x)|2 + |∇x u (t, x)|2 = |J (t)u (t, x)|2 + |H (t)u (t, x)|2 , and one can rewrite the energy as E =
1 1 λ J (t)u 2L2 + H (t)u 2L2 + u (t)2σ+2 L2σ+2 . 2 2 σ+1
(2.2)
Therefore, using the conservation of mass N yields J (t)u 2L2 + H (t)u 2L2 ≤ 2E + C(J (t)u L2 + H (t)u L2 )nσ , and if σ < 2/n, then the quantity J (t)u 2L2 + H (t)u 2L2 remains bounded for all times (for any fixed ). Similarly, global existence can be proved for small data. Proposition 2.2 Let u0 ∈ Σ, and if n ≥ 3, assume σ < 2/(n − 2). Then u is defined globally in time and belongs to C([0, +∞[, Σ) in the following cases. • λ ≥ 0 (repulsive nonlinearity). • λ < 0 (attractive nonlinearity) and σ < 2/n. • λ < 0, σ ≥ 2/n and u0 Σ sufficiently small. Remark. In particular, in space dimension one, the solution u is always globally defined for cubic nonlinearities (σ = 1).
3 Wave collapse Split the energy E into E1 + E2 , with 1 λ J (t)u 2L2 + cos2 (ωt)u (t)2σ+2 L2σ+2 , 2 σ+1 1 λ sin2 (ωt)u (t)2σ+2 E2 (t) = H (t)u 2L2 + L2σ+2 . 2 σ+1
E1 (t) =
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Lemma 3.1 The quantities E1 and E2 satisfy the following evolution laws, ωλ dE1 = (nσ − 2) sin(2ωt)u (t)2σ+2 L2σ+2 , dt 2σ + 2 ωλ dE2 = (2 − nσ) sin(2ωt)u (t)2σ+2 L2σ+2 . dt 2σ + 2 Remark. This lemma can be regarded as the analogue of the pseudo-conformal conservation law, discovered by Ginibre and Velo ([10]) for the case with no potential (ω = 0). Sketch of the proof. Expanding |Jj (t)u (t, x)|2 yields, |Jj (t)u (t, x)|2 =ω 2 x2j sin2 (ωt)|u (t, x)|2 + 2 cos2 (ωt)|∂j u (t, x)|2 + ωxj sin(2ωt) Im(u∂j u). When differentiating the above relation with respect to time and integrating with respect to the space variable, one is led to computing the following quantities, ∂t |xj u (t, x)|2 dx =2 Im xj u ∂j u , ω2 λ Im xj u ∂j u − 2 Im ∂j2 u |u |2σ u , ∂t |∂j u (t, x)|2 dx = − 2 2 ω λ (3.1) ∂t Im (xj u ∂j u ) = |∇x u |2 + x2 |u |2 + |u |2σ+2 2 2 ω2 Re xj ∂j u x2 u − Re xj ∂j u ∆u + λ 2σ + 2 Re xj ∂j u |u | u . It follows, d dt
|J (t)u (t, x)|2 dx =
ωσλ sin(2ωt) σ+1
|u|2σ+2 2 − 2λ cos (ωt) Im ∂j2 u|u|2σ u.
Notice that it is sensible that the right hand side is zero when λ = 0; from the commutation relation (1.4), the L2 -norm of J (t)u is conserved when λ = 0, since odinger equation. J (t)u then solves a linear Schr¨ Finally, the first part of Lemma 3.1 follows from the identity, d u (t)2σ+2 = −(σ + 1) Im |u|2σ u∆u. L2σ+2 dt The second part of Lemma 3.1 follows from the relation E1 + E2 = E = cst. The justification of these formal computations relies on a regularizing technique, which can be found for instance in [7], Lemma 6.4.3.
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As an application of this lemma, we can prove wave collapse when E1 (0) ≤ 0. Proposition 3.2 Let u0 ∈ Σ be nonzero, and if n ≥ 3, assume σ < 2/(n − 2). Assume that the nonlinearity is attractive (λ < 0) and σ ≥ 2/n. Then under the condition 1 λ ∇u0 2L2 + u 2σ+2 2σ+2 ≤ 0, 2 σ+1 0 L u collapses at time t∗ ≤ π/2ω, π , lim ∇x u (t)L2 = ∞, ∃t∗ ≤ and lim u (t)L∞ = ∞. 2ω t→t t→t ∗ ∗ Proof. From our assumptions, if u ∈ C([0, T ]; Σ) with T ≤ π/2ω, 1 dE1 E1 (0) = E − ωxu0 2L2 ≤ 0, and ≤ 0, ∀t ∈ [0, T ]. 2 dt
(3.2)
So long as ∇x u remains bounded in L2 , so does xu . This follows from the conservations of mass and energy, along with Gagliardo-Nirenberg inequality. Assume u ∈ C([0, π/2ω]; Σ). Then letting t go to π/2ω yields
π 1
π 2 ≥ ωxu ,x , E1 2ω 2 2ω L2 which is impossible from (3.2) and the conservation of the L2 -norm of u . Thus, there exists t∗ ≤ π/2ω such that lim ∇x u (t)L2 = ∞.
t→t ∗
From the conservation of energy, lim u (t)2σ+2 L2σ+2 = ∞,
t→t ∗
and the last part of the proposition stems from the conservation of mass.
Remark. Notice that the blow up condition also reads ω2 xu0 2L2 . 2 In term of energy, this means that the blow up occurs for higher values of the Hamiltonian than in the case with no potential, where the similar condition reads E < 0. This condition was found independently by Zhang [16], in the particular case σ = 2/n. In particular, our approach can treat the case of Bose-Einstein condensation in space dimension three, where the cubic nonlinearity is supercritical (σ = 1 > 2/n = 2/3). E ≤
Corollary 3.3 Assume σ ≥ 2/n, λ < 0. Let v0 ∈ Σ. For k ∈ R, define u0 = kv0 . Then for |k| sufficiently large, u (t, x) collapses at time t∗ ≤ π/2ω, as in Prop. 3.2. Proof. For |k| large, E1 (0) becomes negative, and one can use the results of Prop. 3.2.
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4 Lower bound for the breaking time In this section, we specify the dependence of the coupling constant λ upon physical constants, and assume λ = a2 . We first assume that the nonlinearity is cubic, σ = 1. Physically, a is the s-wave scattering length. It is negative in the case of Bose-Einstein condensation for 7 Li system ([1], [2]). We prove that if the space dimension n is two or three, then the nonlinear term a2 |u |2 u in (1.1) is negligible in the semi-classical limit → 0, up to some time depending on . This will give us a lower bound for the breaking time t∗ when → 0, and prove that under the assumptions of Prop. 3.2, π . t∗ −→ →0 2ω As previously noticed, no blow up occurs for σ = 1 and n = 1, that is why we restrict our attention to n = 2 or 3. In the one-dimensional case, it has been proved in [12] that the right model for Bose-Einstein consists in replacing the cubic nonlinearity |u |2 u by the quintic nonlinearity |u |4 u . This case is critical for global existence issues (see Prop. 2.2, Prop. 3.2), and is treated at the end of this section. Define the function v as the solution of the linear Cauchy problem, 2 2 i∂ v + ∆v = ω x2 v , t 2 2 = u0 . v|t=0
4.1
(4.1)
The case n = 2 or 3
When n = 2 or 3, recall that we consider now the initial value problem for u , 2 2 i∂ u + ∆u = ω x2 u + a2 |u |2 u , t 2 2 (4.2) u|t=0 = u0 , where a is fixed. Our first result is independent of the sign of a. Proposition 4.1 Assume n = 2 or 3. Let u0 ∈ Σ be such that u0 L2 , ∇x u0 L2 and xu0 L2 are bounded, uniformly with ∈]0, 1]. Then there exist C, Λ, α > 0 and a finite real q such that the following holds. Let 0 > 0 be such that π/2ω − α Λα 0 > 0. Then for any ∈]0, 0 ], u is defined in Σ at least up to time π/2ω−Λ , and satisfies A (t)(u − v )(t) 2 ≤ C1/q , sup L 0≤t≤π/2ω−Λα
where A (t) can be either of the operators Id, J (t) or H (t). In particular, if a < 0 and u collapses at time t∗ , then π − Λα , ∀ ∈]0, 0 ]. t∗ ≥ 2ω
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Remark. Notice that the assumption ∇x u0 L2 be bounded uniformly with means that u0 has no -dependent oscillation. This is crucial, for quadratic oscilπ lations could lead for instance to t∗ = 4ω , see [5]. To prove Prop. 4.1, we first state precisely the Strichartz estimates we will use. Recall the classical definition (see e.g. [7]), 2n (resp. 2 ≤ r ≤ ∞ if n = 1, Definition 1 A pair (q, r) is admissible if 2 ≤ r < n−2 2 ≤ r < ∞ if n = 2) and 1 1 2 = δ(r) ≡ n − . q 2 r
Strichartz estimates provide mixed type estimates (that is, in spaces of the form Lqt (Lrx ), with (q, r) admissible) of quantities involving the unitary group t
U0 (t) = ei 2 ∆ . t
A simple scaling argument yields similar estimates when U0 is replaced with ei 2 ∆ , with precise dependence upon the parameter . As noticed in Sect. 2, the same t Strichartz estimates hold when ei 2 ∆ is replaced by U (t) (provided that only finite time intervals are involved). Proposition 4.2 Let I be a interval contained in [0, π/2ω]. For any admissible pair (q, r), there exists Cr such that for any f ∈ L2 , −1/q U (t)f q f L2 . r ≤ Cr L (I;L )
For any admissible pairs (q1 , r1 ) and (q2 , r2 ), there exists Cr1 ,r2 such that for F = F (t, x), U (t − s)F (s)ds ≤ Cr1 ,r2 −1/q1 −1/q2 F Lq2 (I;Lr2 ) . (4.3) q I∩{s≤t} r L
1 (I;L 1 )
The above constants are independent of I ⊂ [0, π/2ω] and ∈]0, 1]. We now state two technical lemmas on which the proof of Prop. 4.1 relies. The first one is easy, and we leave out the proof. Lemma 4.3 If n = 2 or 3, there exists q, r, s and k satisfying 1 1 2 r = r + s , 1 1 2 = + , q q k and the additional conditions:
(4.4)
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• The pair (q, r) is admissible, • 0<
1 k
< δ(s) < 1.
Remark. Notice that in particular, q is finite. Lemma 4.4 Assume n = 2 or 3, and let a ∈ C(0, T ; Σ) defined for some positive T , solution of 2 2 i∂ a + ∆a = ω x2 a + 2 F (a ) + 2 S , t 2 2 a|t=0 = 0. Assume that there exists C0 > 0 such that for any T < π/2ω, and any 0 ≤ t ≤ T , F (a )(t) r ≤
L
π 2ω
C0 2δ(s) a (t)Lr . −t
Then there exist C, Λ > 0 independent of ∈ [0, 1[ such that the following holds. Let 0 > 0 be such that π/2ω − Λα 0 > 0. Then for any ∈]0, 0 ], a (t)L2 ≤ C1−1/q S Lq (0,π/2ω−Λα ;Lr ) , sup π 0≤t≤ 2ω −Λα
where α =
1 kδ(s)−1 .
Proof of Lemma 4.4. From (4.3) with q1 = q2 = q, for any t < π/2ω, a Lq (0,t;Lr ) ≤ C1−2/q S Lq (0,t;Lr ) + C1−2/q F (a )Lq (0,t;Lr ) .
(4.5)
From our assumptions,
F (a )Lq (0,t;Lr )
≤
π 2ω
C0 r a (s) . 2δ(s) Lx q −s L (0,t)
Apply H¨ older’s inequality in time with (4.4), F (a )Lq (0,t;Lr ) ≤ C
≤ C
0
π 2ω
t
ds π 2ω
−t
−s
2/k kδ(s)
a Lq (0,t;Lr )
1 2δ(s)−2/k a Lq (0,t;Lr ) .
Plugging this estimate into (4.5) yields, for t ≤ π/2ω − Λα , a Lq (0,t;Lr ) ≤ C1−2/q S Lq (0,t;Lr ) + C1−2/q (Λα )2/k−2δ(s) a Lq (0,t;Lr ) .
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1 From (4.4), the power of in the last term is canceled for α = kδ(s)−1 . If in addition Λ is sufficiently large, the last term of the above estimate can be absorbed by the left hand side (up to doubling the constant C for instance),
a Lq (0,t;Lr ) ≤ C1−2/q S Lq (0,t;Lr ) . The last three estimates also imply, F (a )Lq (0,t;Lr ) ≤ CS Lq (0,t;Lr ) .
(4.6)
The lemma then follows from Prop. 4.2, (4.3), with this time q1 = ∞ and q2 = q, along with (4.6). Proof of Proposition 4.1. Denote w = u − v the remainder we want to assess. It solves the initial value problem, 2 2 i∂ w + ∆w = ω x2 w + a2 |u |2 u , t 2 2 (4.7) = 0. w|t=0 We first want to apply Lemma 4.4 with a = w . Since u = v + w , we can take F (w ) = a|u |2 w ,
S = a|u |2 v .
The point is now to control the Ls -norm of u . Notice that we can easily control the Ls -norm of v . Indeed, as we already emphasized, for any time t, v (t)L2 = u0 L2 ,
J (t)v L2 = ∇u0 L2 .
From Lemma 1.1, (1.5), and Gagliardo-Nirenberg inequality, we also have, C 1−δ(s) δ(s) v (t)L2 J (t)v L2 δ(s) | cos(ωt)| C 1−δ(s) δ(s) ≤
J (t)v L2 . δ(s) v (t)L2 π 2ω − t
v (t)Ls ≤
Therefore, the assumptions of Prop. 4.1 imply that there exists C0 > 0 independent of such that for any t < π/2ω, v (t)Ls ≤
C0 π 2ω
−t
δ(s) .
= 0 and we know from Prop. 2.1 that there exists T such that the Now w|t=0 Σ-norm of w is continuous on [0, T ]. In particular, there exists t > 0 such that the following inequality,
w (t)Ls ≤
C0 π 2ω
−t
δ(s) ,
(4.8)
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holds for t ∈ [0, t ]. So long as (4.8) holds, we have obviously u (t)Ls ≤
2C0 δ(s) . −t
π 2ω
This estimate allows us to apply Lemma 4.4, which yields, along with (4.4), and provided that t ≤ π/2ω − Λα , w L∞ (0,t;L2 ) ≤ C1−1/q |u |2 v Lq (0,t;Lr ) ≤ C1−1/q u 2Lk (0,t;Ls ) v Lq (0,t;Lr )
(4.9)
≤ CΛ−2/k 1/q . Now apply the operator J to (4.7). From Lemma 1.1, J w solves the same equation as w , with |u |2 u replaced by J (|u |2 u ). From (1.8), |J (t)(|u |2 u )(t, x)| ≤ 4|u (t, x)|2 |J (t)u (t, x)|. Writing J u = J v + J w and proceeding as above yields, so long as (4.8) holds, (4.10) J w L∞ (0,t;L2 ) ≤ CΛ−2/k 1/q . Combining (4.9) and (4.10), along with Gagliardo-Nirenberg inequality, yields, so long as (4.8) holds, w (t)Ls ≤ C
1 π 2ω
−t
δ(s) Λ
−2/k 1/q
.
(4.11)
Possibly enlarging the value of Λ, (4.11) shows that (4.8) remains valid up to time π/2ω − Λα . This proves Prop. 4.1 when A (t) = Id or J (t), from (4.9) and (4.10). The case A (t) = H (t) is then an easy by-product.
4.2
The case n = 1
We finally prove the analogue of the above results in space dimension one. When n = 1, one can do without Strichartz estimates, and simply use the Sobolev embedding H 1 ⊂ L∞ , 1/2 1/2 f L∞ ≤ Cf L2 ∂x f L2 . The wave u now solves 2 2 i∂ u + ∂ 2 u = ω x2 u + a2 |u |4 u , t 2 x 2 u|t=0 = u0 . We start with the analogue of Lemma 4.4.
(4.12)
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Lemma 4.5 Assume n = 1, and let a ∈ C(0, T ; Σ) defined for some positive T , solution of 2 2 i∂ a + ∂ 2 a = ω x2 a + 2 F (a ) + 2 S , t x 2 2 (4.13) a|t=0 = 0. Assume that there exists C0 > 0 such that for any T < π/2ω, and any 0 ≤ t ≤ T , F (a )(t) 2 ≤
L
C0 π 2ω
−t
2 a (t)L2 .
Then there exists C > 0 independent of ∈ [0, 1[ such that for any Λ ≥ 1, the following holds. Let 0 > 0 be such that π/2ω − Λ0 > 0. Then for any ∈]0, 0 ], π/2ω−Λ S (t) 2 dt. sup a (t)L2 ≤ C L π 0≤t≤ 2ω −Λ
0
Proof. Multiply (4.13) by a , integrate with respect to x, and take the imaginary part of the result. This yields, from Cauchy-Schwarz inequality, d a (t)L2 ≤ 2F (a )(t)L2 + 2S (t)L2 dt 2C0 a (t) 2 + 2S (t)L2 . ≤
2 L π 2ω − t The lemma then follows from the Gronwall lemma.
We can now prove the analogue of Prop. 4.1. Proposition 4.6 Assume n = 1. Let u0 ∈ Σ be such that u0 L2 , ∂x u0 L2 and xu0 L2 are bounded, uniformly with ∈]0, 1]. Then there exist C, Λ > 0 such that the following holds. Let 0 > 0 be such that π/2ω − Λ0 > 0. Then for any ∈]0, 0 ], u is defined in Σ at least up to time π/2ω − Λ, and satisfies A (t)(u − v )(t) 2 ≤ C, sup L 0≤t≤π/2ω−Λ
where A (t) can be either of the operators Id, J (t) or H (t).In particular, if a < 0 and u collapses at time t∗ , then π − Λ, ∀ ∈]0, 0 ]. t∗ ≥ 2ω Proof. The proof follows the proof of Prop. 4.1 very closely, if we take q = ∞, (s, k) = (∞, 4). Denote w = u − v the remainder we want to assess. It solves the initial value problem, 2 2 i∂ w + ∂ 2 w = ω x2 w + a2 |u |4 u , t x 2 2 w|t=0 = 0.
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We first want to apply the above lemma with a = w . Since u = v + w , we can take F (w ) = a|u |4 w , S = a|u |4 v . The point is now to control the L∞ -norm of u . Notice that we can easily control the L∞ -norm of v . Indeed, as we already emphasized, for any time t, v (t)L2 = u0 L2 ,
J (t)v L2 = ∂x u0 L2 .
From Lemma 1.1, (1.5), and Gagliardo-Nirenberg inequality, we also have, C 1/2 1/2 v (t)L2 J (t)v L2 | cos(ωt)|1/2 C 1/2 1/2 ≤
1/2 v (t)L2 J (t)v L2 . π 2ω − t
v (t)L∞ ≤
Therefore, the assumptions of Prop. 4.6 imply that there exists C0 > 0 independent of such that for any t < π/2ω, v (t)L∞ ≤
C0 π 2ω
−t
1/2 .
So long as w (t)L∞ ≤
C0 π 2ω
−t
1/2 ,
(4.14)
holds, we have obviously u (t)L∞ ≤
π 2ω
2C0 1/2 . −t
This estimate allows us to apply the above lemma, which yields, provided that t ≤ π/2ω − Λ, w L∞ (0,t;L2 ) ≤ C|u |4 v L∞ (0,t;L2 ) ≤ Cu 2L4 (0,t;L∞ ) v L∞ (0,t;L2 ) ≤ CΛ
−1
(4.15)
.
Similarly, applying the operator J to (4.7) yields, so long as (4.8) holds, J w L∞ (0,t;L2 ) ≤ CΛ−1 .
(4.16)
Combining (4.15) and (4.16), along with Gagliardo-Nirenberg inequality, yields, so long as (4.14) holds, w (t)L∞ ≤ C
1 π 2ω
−t
1/2 Λ
−1
.
(4.17)
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Taking Λ large enough, (4.17) shows that (4.14) remains valid up to time π/2ω−Λ. This proves Prop. 4.6 when A (t) = Id or J (t), from (4.15) and (4.16). The case A (t) = H (t) is then an easy by-product. Acknowledgement. The results in this paper were improved thanks to remarks made by T. Colin.
References [1] C. C. Bradley, C. A. Sackett, J. J. Tollett and R. G. Hulet, Evidence of Bose-Einstein Condensation in an Atomic Gas with Attractive Interactions, Phys. Rev. Lett. 75, 1687–1690 (1995). [2] C. C. Bradley, C. A. Sackett and R. G. Hulet, Bose-Einstein Condensation of Lithium: Observation of Limited Condensate Number, Phys. Rev. Lett. 78, 985–989 (1997). ´ [3] R. Carles, Equation de Schr¨ odinger semi-classique avec potentiel harmonique et perturbation non-lin´eaire, S´eminaire X-EDP, 2001–2002, Exp. No. III, ´ 12p., Ecole Polytechnique, Palaiseau, (2001). [4] R. Carles, Semi-classical Schr¨odinger equations with harmonic potential and nonlinear perturbation, preprint, to appear in Ann. Inst. H. Poincar´e Anal. Non Lin´eaire, 2001. [5] R. Carles, Critical nonlinear Schr¨ odinger equations with and without harmonic potential, to appear in Math. Mod. Meth. Appl. Sci., 2002. [6] R. Carles, Remarques sur l’´equation de Schr¨odinger non lin´eaire avec potentiel harmonique, Comptes Rendus de l’Acad´emie des Sciences. S´erie I. Math´ematique 334, 763–766 (2002). [7] T. Cazenave, An introduction to nonlinear Schr¨ odinger equations, Text. Met. Mat. 26, Univ. Fed. Rio de Jan., (1993). [8] C. Cohen-Tannoudji, Cours du Coll`ege de France, 1998–99, available at www.lkb.ens.fr/˜laloe/PHYS/cours/college-de-france/ . [9] R.P. Feynman and A.R. Hibbs, Quantum mechanics and path integrals (International Series in Pure and Applied Physics), Maidenhead, Berksh.: McGraw-Hill Publishing Company, Ltd., 365 p., 1965. [10] J. Ginibre and G. Velo, On a class of nonlinear Schr¨ odinger equations. II Scattering theory, general case, J. Funct. Anal. 32, 33–71 (1979). [11] M. Keel and T. Tao, Endpoint Strichartz Estimates, Amer. J. Math. 120, 5, 955–980 (1998).
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[12] E. B. Kolomeisky, T. J. Newman, J. P. Straley and X. Qi, Low-Dimensional Bose Liquids: Beyond the Gross-Pitaevskii Approximation, Phys. Rev. Lett. 85, 6, 1146–1149 (2000). [13] Yong-Geun Oh, Cauchy problem and Ehrenfest’s law of nonlinear Schr¨ odinger equations with potentials, J. Diff. Eq. 81, 2, 255–274 (1989). [14] Walter Thirring, A course in mathematical physics. Vol. 3, Springer-Verlag New York, 1981, Quantum mechanics of atoms and molecules, Translated from the German by Evans M. Harrell, Lecture Notes in Physics 141, MR 84m:81006. [15] Takeya Tsurumi and Miki Wadati, Stability of the D-dimensional nonlinear Schr¨ odinger equation under confined potential, J. Phys. Soc. Japan 68, 5, 1531–1536 (1999). [16] Jian Zhang, Stability of attractive Bose-Einstein condensates, J. Statist. Phys. 101, n 3-4, 731–746 (2000). R´emi Carles Math´ematiques Appliqu´ees de Bordeaux UMR 5466 CNRS 351 cours de la Lib´eration F-33405 Talence cedex France email: [email protected] Communicated by Rafael D. Benguria submitted 04/12/01, accepted 21/05/02
To access this journal online: http://www.birkhauser.ch
Ann. Henri Poincar´e 3 (2002) 773 – 792 c Birkh¨ auser Verlag, Basel, 2002 1424-0637/02/040773-20
Annales Henri Poincar´ e
The Wave Function of the Lyman–Alpha Photon Part I. The Wave Function in the Angular and Linear Momentum Representations H. E. Moses Abstract. To the best of the writer’s knowledge no one has given the wave function of a photon emitted in an atomic, molecular, or nuclear transition. In the present paper we derive the wave function in the angular momentum and linear momentum representations for the photon emitted by a non-relativistic hydrogen atom, when the electron of the atom falls from the first excited state to the ground state. This is the simplest transition which produces a photon. A two level model for the atom is used, in which the lower level (the ground state energy) is associated with a nondegenerate wave function and the upper level (the energy of the first excited state) is associated with wave functions corresponding to the four–fold degeneracy of the that state). We use a generalization of Dirac’s method for finding the eigenfunctions in resonance scattering. We find the exact solution of the two-level problem using the exact matrix elements of the interaction. The calculations are finite without renormalization. In the next paper we shall introduce the x–representation and thereby obtain the “position”, “shape”, and “trajectory” of the photon.
1 Introduction As far as the author is aware, no one has ever given the wave function of a photon emitted in an atomic transition or other process which releases a photon. Possibly part of the problem is the fact that the description of the photon, though known from Wigner’s monumental work [1], seems not to be used generally. We have introduced the wave function of the photon in an angular momentum basis [2, 3] to find the exact electromagnetic matrix elements for the hydrogen atom and to obtain from first principles the resonance scattering cross section of Lyman–α radiation from a hydrogen atom in the ground state. Since angular momentum is conserved in atomic transitions, this basis for the photon wave functions is a natural one for calculations. We use a generalization of the Dirac [4] procedure1 for finding the exact eigenvectors of the truncated Hamiltonian, which Dirac uses to describe resonance scattering. Friedrichs [5] has studied a simple model of the Dirac procedure in a mathematically rigorous fashion. He has obtained the Wigner–Weisskopf exponential decay as a limit of vanishing interaction. Closely related problems have been treated in [6]. The author is grateful to the referee for pointing out these papers. The approach in these papers is closer to 1 Dirac
uses a somewhat different language than ours to describe his procedure.
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the traditional one than ours. We believe our results are more explicit and avoid some hazards.
2 The Hamiltonian of the Hydrogen Atom Interacting with the Photon Field 2.1
The Full Hamiltonian
The full Hamiltonian of the non–relativistic atom interacting with the photon field x), which we take to be in the Coulomb gauge and whose vector potential is A( hence such that · A( x) = 0 , ∇ (1) is 1 H= 2m
2 ¯ h e e2 ∇ − A(x) − + Hp , i c r
(2)
x) 2 and where Hp is the Hamiltonian of the free photon field. On ignoring A( using Eq. (1) H
i¯h ¯ 2 2 e2 h ∇ − + Hp + A( x) · ∇ 2m r m = H0 + HI ,
= −
(3)
where H0 is the unperturbed Hamiltonian operating in the direct product space of the hydrogen Hamiltonian HH = −
¯2 2 h e2 ∇ − 2m r
(4)
and the free photon field whose Hamiltonian is Hp . Thus H 0 = HH + H p .
(5)
x) · ∇ is the interaction between the hydrogen atom The operator HI = (i¯ h/m)A( and the photon field. We shall simplify the interaction HI by restricting the number of matrix elements of HI between eigenstates of H0 . Unlike most workers in the field, we shall use the exact matrix elements of [2, 3] instead of the approximate elements of the long wave–length limit. The use of the exact matrix elements leads to finite results instead of the infinite results of the approximate elements. 2.1.1 The Free Photon Field. Questions of Gauge The free photon field2 , based on Wigner’s definition of a photon as a relativistic massless particle of spin 1, is treated in great detail [2, 3]. We shall briefly review 2 In our treatment, as in the treatments of atomic transitions by most other workers, we ignore the contribution to the photon field due to the electron as a source. This contribution leads to a self–energy problem whose contribution should be small.
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some properties of the quantized field in the angular momentum–energy basis. x; t) We shall work for the while in the Heisenberg picture where the operators E( x; t) are time–dependent Hermitian operators. We introduce the operator and H( G(E, j, m, λ) and its Hermitian adjoint G∗ (E, j, m, λ) as annihilation and creation operators which satisfy the usual commutation rules G(E, j, m, λ), G(E , j , m , λ ) = 0 ∗ G (E, j, m, λ), G∗ (E , j , m , λ ) = 0 (6) G(E, j, m, λ), G∗ (E , j , m , λ ) = E δ(E − E ) δj,j δm,m δλ,λ . The eigenvalue λ = ±1 describes the circular polarization of the photon: λ = −1 refers to a photon whose circular polarization is in the direction of propagation, while λ = 1 describes a photon whose circular polarization is opposite to the direction of propagation. The quantity |λ| = 1 is the spin of the photon. The eigenvalue E is in the continuous spectrum 0 < E < ∞ and gives the energy of the photon. The discrete variable j describes the total angular momentum of the photon, while m is the z– component of the angular momentum or “magnetic” quantum number of the photon. The ranges of these eigenvalues are: j = 1, 2, · · ·, and m = −j, −j + 1, −j + 2, · · · j − 1, j. x; t) and H( x; t) are required to transform in the usual The quantized fields E( relativistic fashion and satisfy Maxwell’s equations × H( x; t) = ∇ × E( x; t) = ∇ x; t) = 0 ∇ · H(
,
x; t) 1 ∂ E( c ∂t x; t) 1 ∂ H( − c ∂t x; t) = 0 . ∇ · E(
(7)
Then the only Hermitian solutions of Maxwell’s equations, which satisfy the usual commutation rules and the relativistic transformations rules, are obtained as follows. Define the operator Akmλ,n (r; t) by ∞ E 1 r e−i(Et/¯h) . Akmλ,n (r; t) = dE G(E, k, m, λ)jn (8) hc ¯ ¯hc 0
In Eq. (8) jn (r) is the usual spherical Bessel function. Let us now define the 1 (x; t) as follows: operator A 1 (x; t) A
∞ k
√ kkm (θ, φ) Akmλ,k (r; t) = − 2 ik Y
λ=±1 k=1 m=−k
k Yk,k+1,m (θ, φ) Akmλ,k+1 (r; t) − iλ 2k + 1 k+1 Yk,k−1,m (θ, φ)Akmλ,k−1 (r; t) . + iλ 2k + 1
(9)
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j,k,m (θ, φ) are the usual vector spherical harmonics in the In Eq. (9) the vectors Y notation of [7], for example. The angles θ and φ are the spherical polar angles of x as usually defined. x = r(sin θ cos φ, sin θ sin φ, cos θ) . 0 (x; t) is defined by The Hermitian operator A 0 (x; t) = A 1 (x; t) + A ∗1 (x; t). A
(10)
·A 0 (x; t) = 0. ∇
(11)
It is seen that Then the the only Hermitian operator solution of Maxwell’s equations which transform relativistically is (within unitary transformations) x; t) = E( x; t) = H(
0 (x; t) 1 ∂A c ∂t ×A 0 (x; t) . ∇ −
(12)
0 (x; t) is a vector potential in the Coulomb From Eq. (11) and (12) it is clear that A gauge. In this gauge the scalar potential is zero. It behooves us now to find the x; t) general vector and scalar (operator) potentials. The general vector potential A( and scalar potential V (x; t) are required to be Hermitian solutions of the equations x; t) = E( x; t) = H(
x; t) 1 ∂ A( (x; t) − ∇V c ∂t × A( x; t) , ∇ −
(13)
x; t) and H( x; t) satisfy Maxwell’s equations. The general vector and where E( scalar potentials are x; t) A( V (x; t)
0 (x; t) + ∇F (x; t) = A 1 ∂F (x; t) = − , c ∂t
(14)
where F (x; t) is any real function or Hermitian operator. It sets the gauge. The vector potential A0 is the minimal vector potential needed to obtain a solution of Maxwell’s equations. For the general Coulomb gauge, F (x; t) is independent of t and satisfies Laplace’s equation ∇2 F (x) = 0. If the vector and scalar potentials are to satisfy the Lorentz condition, then F (x; t) satisfies the wave equation ∇2 F (x; t) −
1 ∂ 2 F (x; t) = 0. c2 ∂t2
The vector and scalar potentials in any gauge can be represented as in Eq. (14). 0 (x; t). We shall show later that we need deal only with the vector potential A
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We shall now give the field Hamiltonian Hp . The classical Hamiltonian of the electromagnetic field is
1 x; t)2 . x; t)2 + H( (15) dx E( Hp = 8π As most other investigators do, we take this definition also for the quantized fields. x; t) and H( x; t), the field Hamiltonian Despite the appearance of the time t in E( operator Hp is independent of time. From Eq. (12)–(14), using normal re-ordering of annihilation and creation operators, we obtain the field Hamiltonian as Hp =
∞ j ∞
dE G∗ (E, j, m, λ)G(E, j, m, λ) .
(16)
λ=±1 j=1 m=−j 0
It will be convenient to use the symbol s for the set {E, j, m, λ}: s → {E, j, m, λ} , and
(17)
ds to denote the sums and integrals
∞ j ∞ dE ··· . ds · · · = E j=1 m=−j λ=±1
(18)
0
Later we shall use the symbol Ep for E when only one photon is present to distinguish the energy of the photon from the energy of the system. It is also useful to have a notation when there are sets of variables {Ei , ji , mi , λi } labeled by an index i. We use the symbol si to indicate the set:
si dsi · · ·
→ {Ei , ji , mi , λi } =
∞ ji ∞ dEi ··· . Ei m j
λi =±1
i=1
i=−ji
(19)
0
Moreover, if we are dealing with a function or operator f (Ei , ji , mi , λi ), we shall write
f (si ) ≡ f (Ei , ji , mi , λi ) dsi f (si ) ≡
∞
ji
∞
λi =±1 ji=1 mi =−ji 0
Thus Hp =
dEi f (Ei,ji ,mi ,λi ) , Ei
ds E G∗ (s)G(s) .
(20)
(21)
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We also introduce the (total) number operator N= ds G∗ (s)G(s) .
(22)
From the commutation rules Eq. (6), the expansion Eq. (9), Eq. (10), and Eq. (12), we can obtain the usual commutation rules for the components of the electric and magnetic fields and potentials [8]. 2.1.2 The Free Photon Field. The Hilbert Space We now specify the Hilbert space Hp upon which the operators G(s), Hp , N act. We shall use the Fock realization3 . One assumes the existence of a normalizable vacuum state |V > for which G(s)|V > = 0 , < V |V > = 1
(23)
N |V > = 0 , Hp |V > = 0 .
(24)
It follows that Thus |V > is an eigenstate of both N and Hp with the point eigenvalue 0. In addition there are non-normalizable n–particle states4 which with |V > span the space Hp . Such states are denoted by |E1 , j1 , m1 , λ1 ; E2 , j2 , m2 , λ2 ; · · · ; En , jn , mn , λn >≡ |s1 , s2 , · · · , sn > . They are defined as |s1 , s2 , · · · , sn >=
n
G∗ (si )|V > ,
n = 1, 2, · · · .
(25)
i=1
The asterisk now means Hermitian adjoint instead of complex conjugate, as before. The kets are symmetric in the arguments si . To show how the operators G(s) and G∗ (s) act in this basis, we shall define the symmetrization operator Sy acting on a function or operator f (s1 , s2 , · · · , sn ) by 1 Sy f (s1 , s1 , · · · , sn ) = P f (s1 , s2 , · · · , sn ) , (26) s 1 , s2 · · · , sn n! P
where the right– hand side of Eq. (26) means the sum over all permutations of the arguments s1 , s2 , · · · , sn . Then G∗ (s)|s1 , s2 , · · · , sn > G(s)|s1 , s2 , · · · , sn > 3 In
= |s1 , s2 , · · · , sn , s > Sy = E δ(s, sn )|s1 , s2 , · · · , sn−1 > , (27) s 1 , s2 , · · · , sn
K. O. Friedrichs, Mathematical Aspects of the Quantum Theory of Fields, Interscience, New York (1953), it is shown that the Fock realization is equivalent to the usual occupation number representation and oscillator representation even in a more general context than “quantization of fields in a box.” 4 These states are not in H . Suitable superpositions, however, are. p
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In Eq. (27)
779
δ(s, s ) = δ(E − E )δjj δmm δλλ .
(28)
Moreover the kets satisfy the completeness and orthonormality conditions |V >< V | +
∞
ds1
ds2 · · ·
dsn |s1 , s2 , · · · , sn >< s1 , s2 , · · · , sn | = Ip
n=1
and < V |s1 , s2 , · · · , sn > <
s1 , s2 , · · · , sm |s1 , s2 , · · · , sn
>
= 0 for any n = δn,m E1 E2 · · · En × δ(s1 , s1 )δ(s2 .s2 ) · · · δ(sn , sn ) .
(28a)
In Eq.(28a) Ei is the energy variable in si and Ip is the identity operator in photon space. One sees that N |s1 , s2 , · · · , sn > =
n|s1 , s2 , · · · , sn > n ( Ei )|s1 , s2 , · · · , sn > .
Hp |s1 , s2 , · · · , sn > =
(29)
i=1
Thus the kets |s1 , s2 , · · · , sn > are eigenkets of N and Hp . 2.1.3 Eigenstates of the Unperturbed Hydrogen Hamiltonian The unperturbed Hamiltonian HH for the hydrogen atom is given by Eq. (4). We shall denote the eigenstates of HH corresponding to point eigenvalues by |nH , jH , mH >, where nH , (nH = 1, 2 · · ·) is the principal quantum number, jH , (0 ≤ jH < nH ) is the quantum number describing the total angular momentum, and mH is the quantum number that describes the z–component of the angular momentum, (−jH ≤ mH ≤ jH ). These eigenfunctions are nomalizable and are usually normalized to unity.
< nH , jH , mH |nH , jH , mH >= δnH ,n δjH ,j δmH ,m . H
H
H
(30)
The energy associated with the principal quantum number nH will be denoted by E(nH ). Thus HH |nH , jH , mH >= E(nH )|nH , jH , mH > . (31) The continuous spectrum lies above the discrete spectrum. Its eigenstates are not normalizable. They are labeled by the energy EH , (EI < EH < ∞) where EI is the ionization energy, jH , (0 ≥ jH < ∞), mH , (−jH ≤ mH ≤ jH ) The (improper) eigenstates are denoted by |EH , jH , mH > they are orthogonal to the eigenstates corresponding to discrete values of the energy.
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2.1.4 Elimination of the Gauge Terms We now go back to the complete Hamiltonian for the hydrogen atom Eq. (2) and write the Schr¨ odinger equation for the combined hydrogen–photon system. The Schr¨ odinger equation is: 2 h ¯ e ¯h ∂ 1 ∇ − A(x; t) + eV ( ψ(x; t) , (32) q x; t) ψ(x; t) = − 2m i c i ∂t 0 and the gauge where the vector and scalar potentials are given in terms of A function F (x; t) by Eq. (14). Let us introduce the wave function Ψ(x; t) through ψ(x; t) = ei(e/¯hc)F (x;t) Ψ(x; t) .
(33)
Then Ψ(x; t) satisfies the Schr¨odinger equation Eq. (32) with V (x; t) ≡ 0 and A replaced by A0 . The gauge is eliminated by a unitary transformation. Thus we can work with the wave function Ψ(x; t) and, having solved for it, can find the solution in any other gauge5 . Henceforth, we shall assume that we are working in a gauge–free representation for which the Hamiltonian is given by Eq.(3)-(5) 0 (x))2 –term6 , the Hamiltonian H is given by which, after ignoring the (A H = H0 + H I .
(34)
The perturbation HI is given by HI =
i¯ h . A0 (x) · ∇ mc
(35)
3 The Eigenfunctions of the Hamiltonian H 3.1
The Eigenfunctions of H0
The eigenfunctions of the unperturbed Hamiltonian H0 are the direct product of the eigenfunctions of the hydrogen Hamiltonian HH and the unperturbed photon Hamiltonian Hp . Using Dirac’s notation for such products, we denote the eigenfunctions of H0 for which the hydrogen atom is in a bound state by |nH , jH , mH , s1 , s2 , · · · , sn >= |nH , jH , mH > |s1 , s2 , · · · , sn > ,
(36)
when there are n photons and |nH , jH , mH , V >= |nH , jH , mH > |V > ,
(37)
5 Possibly one can find the significance of the gauge by performing interference experiments when the gauge is not zero. 6 We are now working in the Schr¨ odinger picture so that operators are not time–dependent. 0. A 0 ( x; 0) We therefore drop the time t in the argument of A x)Schr¨ odinger = A0 ( Heisenberg .
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when there are no photons. When the energy of the atom is in the continuum, the analogous eigenfunctions of H0 are |EH , jH , mH , s1 , s2 , · · · , sn >= |EH , jH , mH > |s1 , s2 , · · · , sn > ,
(38)
|EH , jH , mH , V >= |EH , jH , mH > |V > .
(39)
It follows that: n Ei |nH , jH , mH , s1 , s2 , · · · , sn > , H0 |nH , jH , mH , s1 , s2 , · · · , sn > = E(nH ) +
H0 |EH , jH , mH , s1 , s2 , · · · , sn > = EH +
n
i=1
Ei |EH , jH , mH , s1 , s2 , · · · , sn > ,
i=1
H0 |nH , jH , mH , V > = E(nH )|nH , jH , mH , V > , H0 |EH , jH , mH , V > = EH |EH , jH , mH , V > .
3.2
(40)
The Splitting of the Hilbert Space
The Hermitian operators H0 and H are defined in the same Hilbert space H. The two–level assumption is that the perturbation HI acting on any eigenstate7 of H0 is zero except for the following eigenstates: |1, 0, 0, s >, |2, 0, 0, s >, |2, 1, mH , V >, |2, 1, mH , s >. (For a complete two–level system we should also include the eigenstates |1, 0, 0, V > and |2, 0, 0, V >. It can be shown, however, that these eigenstates do not contribute to the photon wave function [3] because of angular momentum conservation). The eigenstates which we use span a subspace of H which we shall call HD 8 . All the eigenstates of H0 except the four listed above are also eigenstates of H = H0 + HI . The interaction HI now denotes the truncated interaction. For simplicity and in order to be able to use the notation of [3] we rename the eigenstates of H0 which span HD in the following way: |1, s > |2, s > |2, M > |2, M, s >
≡ |1, 0, 0, Ep , j, m, λ > ≡ |2, 0, 0, Ep , j, m, λ > , ≡ |2, 1, M, V > ≡ |2, 1, M, Ep , j, m, λ >,
(41)
for M = 0, ±1 and s = {Ep , j, m, λ}. We denote by E1 and E2 the energies of the ground state and the first excited state, respectively9 . Then H0 |i, s > = 7 We
(Ei + Ep )|i, s >
use the terms “eigenstates”, “eigenkets”, and “eigenvectors” interchangeably. subscript D on HD refers to Dirac who uses this subspace in his resonance scattering theory. It is also the subspace used by Wigner and Weisskopf. 9 These definitions of E (i = 1, 2) represent a slight change of notation. Moreover, the helicity i variable β of [3] is called λ in the present paper. 8 The
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H0 |2, M > = H0 |2, M, s > =
Ann. Henri Poincar´e
E2 |2, M > , (E2 + Ep )|2, M, s > .
(42)
The eigenstates of H in HD have been calculated exactly in [3] using the exact matrix elements, not the long wave–length limit customarily used in such calculations. There are only eigenstates which correspond to eigenvalues E in the continuous spectrum of H0 + HI . There are no discrete eigenvalues [3, 5]. The point eigenvalue E1 of H0 is “swallowed” by the continuous spectrum, to use the language of [5]. We denote these eigenstates by |1, s)10 . We do not use perturbation theory to find it, as in the Wigner–Weisskopf theory. Instead we use Dirac’s method of solution which is an exact method for the truncated perturbation. The eigenket|1, s) is an eigenket of H with the eigenvalue E1 + Ep . H|1, s) = (E1 + Ep )|1, s) . Among unitarily equivalent sets, they have been specified by requiring11 H0 H t exp − i t |1, s) = lim exp i h ¯ ¯h
t→−∞
|1, s > .
(43)
It is shown in [3] that the eigenstates |1, s) span the space HD . Hence, the eigenstates of H0 which do not span HD together with the eigenstates |1, s) span the entire Hilbert space H. They are also eigenkets of the truncated Hamiltonian H. Hence if | > is any one of the eigenkets of H0 which does not span HD H0 H t exp − i t | >= | > . lim exp i h ¯ ¯h From scattering theory it is shown that t→±∞
H0 H lim exp i t exp − i t |1, s) = |1, s > −2π i δ(E − H0 )HI |1, s) . h ¯ h ¯
t→+∞
(44)
(45)
In the above equation E = E1 + Ep .
3.3
The Initial Value Problem and the Final Value Problem
Our objective is to start at time t = 0 in the eigenstate |2, M > of the Hamiltonian H0 (the first excited p–state of hydrogen with no photon present) and find the final state as t → ∞. We expect the final state to correspond to the direct product of the ground state of the hydrogen atom and the photon wave function for Lyman–α radiation12 . 10 We use the round bracket { ) } to distinguish it from the eigenstate of H , which uses the 0 angular bracket { > }. The notation used here differs from the notations of [3] but we do not think this will cause difficulty in referring to [3] for some of the results needed. 11 We are motivated by traditional quantum scattering theory, e.g. [9]. 12 If the initial state of hydrogen were the first excited S–state, one can show that in this approximation there would be no transition to the ground state. This result is, of course, also in the Wigner–Weisskopf treatment of transitions and is a consequence of the conservation of angular momentum.
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3.3.1 The Orthogonality and Completeness Relations. Matrix Elements of HI The kets of the H0 -basis in HD , namely |1, s >, |2, s >, and |2, M, s > satisfy the orthogonality relations < 1, s|1, s > < 2, s|2, s >
= =
Ep δ(s, s ) Ep δ(s, s )
< 2, M, s|2, M , s > < 1, s|2, s >
= =
Ep δ(s, s )δM,M < 2, s|1, s >= 0
< 1, s|2, M, s > < 2, s|2, M, s >
= =
< 2, M, s|1, s >= 0 < 2, M, s|2, s >= 0 .
(46)
In Eq. (45) Ep ∈ s. The identity operator ID in HD is ID = |1, s > ds < 1, s|+ |2, s > ds < 2, s|+ |2, M, s > ds < 2, M, s|. M=0,±1
(47) The kets |1, s) which are the complete set of eigenkets of H in HD satisfy the orthogonality relations (48) (1, s|1, s ) = Ep δ(s, s ) . This equation is proved from scattering theory (see [9]). The identity operator ID of Eq. (46) is also given by (49) ID = |1, s)ds(1, s| . The eigenkets |1, s) of H can be expanded in terms of the eigenkets |1, s >, |2, s >, |2, M, s >. |1, s) = = +
ID |1, s) |1, s > ds < 1, s |1, s) + |2, s > ds < 2, s |1, s) |2, M, s > ds < 2, M, s |1, s) .
(50)
M=0,±1
We shall now give the components of |1, s) from [3]. They will be expressed in terms of a set of functions which come from the matrix elements of HI . A(x) G1 (x)
x2 √ 2(x2 + 1)3 x 2 = − 2 3 [x + ( 32 )2 ]2 =
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Ann. Henri Poincar´e
x3 1 = −√ 2 12 (x + 1)3 ∞ dξ = [G1 (ξ)]2 γ− (x − ξ) ξ 0
I2 (x)
∞ dξ = [G2 (ξ)]2 γ− (x − ξ) ξ 0
I3 (x)
∞ dξ = [A(ξ)]2 γ− (x − ξ) . ξ
(51)
0
In the last three of Eq. (50) the function γ− (x) is the distribution defined by γ− (x) = lim
→0+
P 1 = −iπδ(x) + . x + i0 x
The real parts of the functions Ii (x) are13
5 5 1 5 4 2 6 π(− + x2 + x + x ) Re I1 (x) = 9 32 24 162 179 (x2 + 4 )4 8 64 7 2 11 4 2 + ( x + x3 + x5 + x ) + x log( |x|) 18 9 81 6561 3 3
1 25 2 1 25 4 25 6 π(− Re I2 (x) = − x − x + x (x2 + 1)6 2048 6144 1024 1024 1 25 8 1 10 1 1 x + x ) + (− x − x3 + x5 + 6144 2048 480 48 72 1 7 1 9 1 11 1 5 x + x + x ) + x log |x| + 24 96 720 12
105 2 105 4 1 35 6 7 − x + x + x Re I3 (x) = π(− 2 6 (x + 1) 1024 24 512 512 13 21 8 3 10 1 1 x + x ) + (− x + x3 + x5 + 1024 1024 20 48 2 1 7 1 1 1 x + x9 + x11 ) + x3 log |x| . + 4 12 80 2 The imaginary parts of Ii are Im I1 (x)
∞ dξ [G1 (x)]2 = −π [G1 (ξ)]2 δ(x − ξ) = −π H(x) ξ x 0
Im I2 (x)
∞ [G2 (x)]2 dξ = −π H(x) = −π [G2 (ξ)]2 δ(x − ξ) ξ x 0
13 These
results correct some misprints in [3].
(52)
(53)
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The Wave Function of the Lyman–Alpha Photon. Part I
Im I3 (x)
= −π
785
∞ [A(x)]2 dξ = −π H(x) . [A(ξ)]2 δ(x − ξ) ξ x
(54)
0
In Eq. (52) δ(x) is the usual Dirac δ–function and the symbol P means principal part when used in an integral. Also H(x) is the Heaviside function, defined by H(x)
= 1 if x ≥ 0 = 0 if x < 0 .
Furthermore we use α as the fine structure constant, a as the Bohr radius, and κ as the wave number of the Lyman–α radiation given by the Bohr formula: κ=
E2 − E1 3 α = . hc ¯ 8 a
Finally, we define functions γ1 (k), γ2 (k), γ3 (k), δ1 (k), δ2 (k), δ3 (k), δ(k) = γ(k) =
3 i=1
(55) 3 i=1
δi (k),
γi (k), which give the line shape and energy level shift, by14
δ1 (k)
=
δ2 (k)
=
δ3 (k)
=
γ1 (k)
=
γ2 (k)
=
γ3 (k)
=
γ4 (k)
=
2α3 Re I1 (ka) πa 2α3 Re I2 (k − κ)a πa 2α3 Re I3 (k − κ)a 3πa 2α3 − 2 |G1 (ka)|2 H(k) ka 2α3 − |G2 (k − κ)a |2 H(k − κ) (k − κ)a2 2α3 − |A (k − κ)a |2 H(k − κ) 3(k − κ)a2 γ1 (k) − γ2 (k) − γ3 (k) .
(56)
The matrix elements of the interaction HI which we shall need are < n, s|HI |2, M >
< n|HI |2, M, s >
= < 2, M |HI |n, s >∗ e2 α δM,m δj,1 Gn (ka) = −iλ a π = < 2, M, s|HI |n >∗
14 δ(k), as defined here, is not to be confused with the Dirac–δ. Context will make clear which δ is being used.
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e2 α δM,−m δj,1 Gn (ka) a π = < 2, M, s|HI ||2, M >∗ 2 α 1 1 1 M e = −(−1) δj,1 A(ka) . (57) a π m − M M = −(−1)m iλ
< 2, M |HI |2, M, s >
In Eq. (56) the quantity k in given in terms of the energy of the photon Ep contained in the set s by Ep . (58) k= hc ¯ Thus k is the wave number associated with the photon energy Ep . Then h ¯ k = Ep /c is the absolute value of the momentum p of the photon. The symbol j1 j2 j3 m1 m2 m3 is the usual symmetric Wigner form of Clebsch-Gordan coefficients used in discussions of the reduction of the direct product of irreducible representations of the rotation group [8]. The components of |1, s) in terms of the H0 representation (see Eq. (49)) are < 1, s |1, s) = kδ(k − k )δj,j δm,m δλ,λ + × < 2, M |1, s) = < 2, s |1, s) = < 2, M , s |1, s) = ×
3.4
1 δj,1 δj,j δm,m γ− (k − k ) 2π
λλ kk γ1 (k)γ1 (k ) k − κ − δ(k) − iγ(k) λ −(k/2π)γ1 (k) −iδj,1 δM,m k − κ − δ(k) − iγ(k) kk γ1 (k)γ2 (k + κ) 1 δj,1 δj,j δm,m λλ γ− (k − k − κ) 2π k − κ − δ(k) − iγ(k) 1 1 1 i m (−1) δj,1 δm,(m +M ) λ γ− (k − k − κ) m − m M 2π 3kk γ1 (k)γ3 (k + κ) . (59) k − κ − δ(k) − iγ(k)
Statement of the Problem and Its Solution
We are now in a position to state the problem in mathematical terms. As mentioned earlier, the system consisting of the hydrogen and photon field will initially be in an eigenstate of H0 , namely |2, M > which describes the atom being in the 2 − p state with no photons present. This state is in the Hilbert spaces H and HD . If there were no interaction HI , the system would remain in this eigenstate. The time variation of this eigenstate would be E2 H0 t |2, M >= exp − i t |2, M > . exp − i h ¯ ¯h
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The state remains the same with only a phase change dependent on time t as given above15 . However, with the interaction HI present, the state various in a more complicated way with time. Denoting the time–dependent state by |2, M ; t >, one has H (60) |2, M ; t >= exp − i t |2, M > . ¯h The time–dependent state is a time–varying superposition of eigenstates of H0 . Our objective is to find the eigenstate of H0 which |2, M ; t > approaches as t → ∞. Toward this end we use the tools of scattering theory. We evaluate H0 H t exp − i t |2, M >= |Φ > , lim exp i t→+∞ h ¯ ¯h
(61)
where the state |Φ > depends on the eigenvalue M . Eq. (61) means H0 H t |Φ > . exp − i t |2, M >≈ exp − i h ¯ ¯h That is, the solution of the perturbed time–dependent Schr¨ odinger equation approaches a solution of the unperturbed time–dependent Schr¨ odinger equation for large times. The quantity HH H0 Ep < 1, 0, 0, Ep, j, m, λ| exp i t exp − i t |Φ >= exp − i t ψ(Ep , j, m, λ) h ¯ ¯h ¯h (62) is the time-dependent wave function of the emitted photon in the energy–angular momentum representation16. We recall that < 1, s| ≡< 1, 0, 0, Ep , j, m, λ|. We shall now evaluate the state |Φ >. Using the resolution of the identity ID given by Eq. (49) H0 H t exp − i t |2, M > lim exp i t→+∞ h ¯ h ¯ H0 H = lim exp i t exp − i t ID |2, M > t→+∞ h ¯ ¯h ∞ +j ∞ H0 dEp H t exp − i t = lim exp i Ep t→+∞ ¯h ¯h λ =±1 j =1 m =−j 0 |1, 0, 0, Ep , j , m , λ )
(1, 0, 0, Ep , j , m , λ |2, M > .
In the above equation and later we shall show explicitly the variables which up to now have been designated collectively by the set s. We use Eq. (45) to evaluate H0 H t exp − i t |1, 0, 0, Ep , j , m , λ ). lim exp i t→+∞ h ¯ h ¯ 15 In
older quantum texts such a state would be described as a “standing wave.” are abridging a somewhat long argument.
16 We
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Thus the ket |Φ > of Eq. (61) is |Φ > = × ×
∞
j
∞
dEp
|1, 0, 0Ep , j , m , λ > Ep
λ =±1 j =1 m =−j 0 (1, 0, 0, Ep , j , m , λ |2, M
> −2πiδ(E1 + Ep − H0 )HI |1, 0, 0, Ep , j , m , λ )(1, 0, 0, Ep , j , m , λ |2, M > .
(63)
The wave function of the photon is given by Eq. (62). We therefore multiply both sides of Eq. (64) by Ep t < 1, 0, 0, Ep , j, m, λ| . exp − i h ¯ We note < 1, 0, 0, Ep, j, m, λ|1, 0, 0, Ep , j , m , λ >= Ep δ(Ep − Ep )δj,j δm,m δλ,λ , < 1, 0, 0, Ep , j, m, λ|δ(E1 + Ep − H0 ) = δ(Ep − Ep ) < 1, 0, 0, Ep , j, m, λ| . Then the wave function of the Lyman–α photon is
(64)
17
Ep Ep
t ψ(Ep , j, m, λ) = exp − i t (1, 0, 0, Ep , j, m, λ|2, M > exp − i h ¯ ¯h j ∞ 1 −2πi < 1, 0, 0, Ep , j, m, λ|HI |1, 0, 0, Ep , j , m , λ ) Ep j =1 m =−j λ=±1
× (1, 0, 0, Ep , j , m , λ |2, M > .
(65)
The quantity −
2πi < 1, 0, 0, Ep , j, m, λ|HI |1, 0, 0, Ep , j , m , λ ) Ep
is evaluated in Eq. (50) of [3]. Moreover, (1, 0, 0, Ep , j, m, λ|2, M > is just the complex conjugate of < 2, M |1, s) of Eq. (59). Thus we can evaluate the wave function explicitly in the energy–angular momentum representation.
4 The Wave Function of the Lyman–α Photon in the Energy–Angular Momentum Representation We are now able to give the wave function for the Lyman–α photon. Ep Ep t ψ(Ep , j, m, λ) = exp − i t λ δj,1 δm,M −(k/2π) γ1 (k) exp − i h ¯ ¯h 17 We
are departing from the bra and ket notation in describing the wave function of the photon.
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γ1 (k) i −2 2 2 k − κ − δ(k) + iγ(k) k − κ − δ(k) + γ(k)
Ep k − κ − δ(k) + iγ4 (k) t iλ δj,1 δm,M (k/2π) (−γ1 (k)) = exp − i 2 2 . h ¯ k − κ − δ(k) + γ(k) (66) In the above equation k = Ep /¯ hc, as before. Furthermore, since γ1 (k) < 0, the argument of the square root is positive as is the square root itself. As is usual in quantum mechanics, probability densities are constructed from wave functions. From the theory of scattering it can be shown that ∞
|ψ(Ep , 1, M, λ)|2
λ=±1 0
dEp = 1. Ep
(67)
We have verified Eq. (68) by numerical integration. The probability that the total angular momentum of the photon is given by j = 1 is 0. Likewise the probability that magnetic quantum number m does not equal M , the magnetic quantum number of the atom initially, is likewise 0. The probability that the photon has circular polarization λ = 1 is 12 . The probability that the photon has circular polarization λ = −1 is also 12 . The probability that the energy of the photon is in the interval E0 − ∆ < Ep < E0 + ∆ is
E 0 +∆
|ψ(Ep , 1, M, λ)|2
λ=±1 E −∆ 0
dEp . Ep
(68)
If we consider the “aperture” ∆ to be very small, the expression (69) becomes
|ψ(E0 , 1, M, λ)|2
λ=±1
∆ . E0
(69)
5 Examination of the Resonance We see from Eq. (67) that the probability reaches a maximum when k − κ − δ(k) is a minimum. Most workers using the resonance formula replace k in δi (k) and γi (k) by κ, reasoning that functions δi (k) and γi (k) are slowly varying functions of their arguments. We can plot both γ1 (κ)γ4 (κ) (70) v(k) = 2 2 (k − κ − δ(κ) + γ(κ) γ1 (k)γ4 (k) (71) w(k) = 2 2 (k − κ − δ(k) + γ(k)
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against k in the vicinity of the approximate resonance ≈ κ + δ(κ). We use a
=
5.29167 × 10−9 cm.
α =
7.29720 × 10−3
c
=
κ
=
2.997925 × 1010 cm./sec. 3 α = 5.17124 × 105 1/cm. 8 a
(72)
The curve for v(k) is very close to that for w(k) near the resonance18 . Thus the assumption is correct in the vicinity of the resonance and we shall replace δ(k) and γi (k) by δ = δ(κ), γi = γi (κ) there. For numerical values we have δ γ γ2
= −0.98498 1/cm. = γ1 = −0.0164536 1/cm = γ3 = 0.
(73)
Thus the wave function Eq. (67) simplifies considerably. We are now in a position to find the average energy of the photon Ep . We shall not use the approximation which leads to Eq. (74).
Ep
=
=
∞ j ∞
Ep |ψ(Ep , j, m, λ)|2
λ=±1 j=1 m=−j 0 ∞ k − γ1 (k) k hc ¯
π
0
dEp Ep
2 2 − κ − δ(k) + γ4 (k) dk . 2 2 2 k − κ − δ(k) + γ(k)
(74)
This integral diverges logarithmically for large wave number k if k is replaced by κ in γ1 (k), γ(k), γ4 (k) and δ(k)19 . On the other hand, if this approximation is not used, γ1 (k) provides a strong cut–off for large k and the integral converges strongly. We have evaluated the integral numerically. Our result is Ep = 1.63479 × 10−11 ergs.
(75)
This is also just the value of the energy associated with the resonance peak of the wave function. Hence Ep = h ¯ c[κ + δ(κ)] . This relation was proved numerically. However, it may be possible to prove it analytically. The sharpness of the resonance causes |ψ|2 to behave like a δ–function. 18 The
two curves agree to less than on part in 10−8 in the region. it possible that this approximation is responsible for some of the divergences found in quantum electrodynamics? 19 Is
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6 Linear Momentum Representation of the Wave Function of the Lyman–α Photon Let us define f ( p, M, λ) by f ( p, M, λ) =
c1/2 M,λ Y (θ, φ)ψ(cp, 1, M, λ) , p 1
(76)
where p = Ep /c = h ¯ k and Y1M,λ (θ, φ) is a generalized surface harmonic [10, 2]. The absolute value of the vector p is p and the direction of p is given by the usual spherical polar angles θ, φ. Explicitly the surface harmonics Y1M,λ are given by [11]. 1 3 1 3 2iφ 1,1 1,−1 (1 + cos θ), Y1 e (1 − cos θ) (θ, φ) = Y1 (θ, φ) = 4 π 4 π 3 −iφ 3 iφ Y10,1 (θ, φ) = − e e sin θ sin θ, Y10,−1 (θ, φ) = − 8π 8π 1 3 −2iφ 1 3 e (1 + cos θ) . (77) (1 − cos θ), Y1−1,−1 (θ, φ) = Y1−1,1 (θ, φ) = 4 π 4 π The function f ( p, M, λ) is the wave function in the linear momentum basis. It can be shown that d p = 1, (78) |f ( p, M, λ)|2 cp λ=±1
where the integration is taken over the entire 3-dimensional p–space. The quantity d p | f ( p, M, λ)|2 cp ∆V
gives the probability that the momentum vector p lies in the volume ∆V of p–space when the circular polarization is given by λ and the higher state of the atom is in the angular momentum state j = 1 with the magnetic quantum number given by M initially. For example, when M = 0 3 f ( p, 0, 1) = −i g(p/¯h) e−iφ sin θ . (79) 8π The direction–independent function g(k) is k − κ − δ(k) + iγ4 (k) g(k) = (k/2π) − γ1 (k) 2 . 2 (k − κ − δ(k) + γ(k)
(80)
This is the resonance factor first encountered in Eq. (67). The probability density for the state Eq. (77) when M = 0 is |f ( p, 0, λ)|2 =
3 |g(p/¯h)|2 sin2 θ . 8π
(81)
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The probability densities for M = ±1 are calculated similarly. Thus |f ( p, 1, 1)|2
=
|f ( p, −1, 1)|2
=
θ 3 |g(p/¯ h)|2 cos4 = |f ( p, −1, −1)|2 4π 2 θ 3 |g(p/¯ h)|2 sin4 = |f ( p, 1, −1)|2 . 4π 2
(82)
For a given value of p = | p| = Ep /c, i.e. the absolute value of the momentum, the probability that the photon will pass through the solid angle sin θ dθ dφ depends only on the angle θ. The “radiation pattern” for the various cases are readily understood and we shall not belabor the subject.
References [1] E. P. Wigner, Ann. Math., 40, 149 (1939). [2] H. E. Moses, Phys. Rev. A, 8, 1710 (1973). [3] H. E. Moses, Phys. Rev. A, 22, 2069 (1980). [4] P. A. M. Dirac, Principles of Quantum Mechanics, Oxford (1949), p. 199. [5] K. O. Friedrichs, Comm. Pure and App.Math., 1, 361 (1948). [6] G. Compagno, R. Passante and F. Persico, Jour. Mod Optics 37 1377 (1990) and L. Maiani and M. Testa ,Physics Letters B 356, 319 (1995). [7] A. R. Edmonds, Angular Momentum in Quantum Mechanics, Princeton University Press (1960). [8] H.E. Moses, Nuovo Cimento, Serie X, 42, 110 (1966). [9] H.E. Moses, Nuovo Cimento, 1, 103 (1955). [10] H.E. Moses, Ann. Phys., 41, 166 (1967). [11] H.E. Moses and A.F. Quesada.Archive Rat. Mech. and Anal., 50, 194 (1971).
Harry E. Moses Applimath Company 150 Tappan Street Brookline, MA 02445 USA email: [email protected] Communicated by Vincent Rivasseau submitted 28/07/01, accepted 26/11/01
Ann. Henri Poincar´e 3 (2002) 793 – 813 c Birkh¨ auser Verlag, Basel, 2002 1424-0637/02/040793-21
Annales Henri Poincar´ e
The Wave Function of the Lyman–Alpha Photon Part II. The Shape, Position, and Trajectory of the Photon H. E. Moses
Abstract. In Part I of the present paper we derived the wave function of the Lyman– α photon in both the linear and angular momentum bases using relativistic concepts for the photon wave function. In the present paper, Part II, we derive two X– representations. In the first we assume one–particle theory for the photon wave function and the usual commutation rules for the position operators Xi and linear momentum operators Pi . The second representation employs the quantized photon representation. The stress, energy tensor density is used to field to derive an X– provide a probability density in x–space which is relativistic. The two methods of defining x–space are compared. It is found in the present case that, despite the use of particle operators, the photon resembles a field far more than than it does a particle.
1 The Use of Commutation Rules to Define an x–Representation 1.1
The Coordinate Operators in a Linear Momentum Basis
We shall assume that Part I of the present paper is available to the reader. Equations in Part I which are referred to in Part II will have a prime (’) attached to the equation number. Likewise footnotes or references of Part I which are referred to will also have a prime attached to the footnote or reference number. The coordinate operators Xi are defined by the usual commutation relations with the linear momentum operators Pi , where the momentum operators operate on the photon wave functions f ( p, M, λ) of Eq. (77’) as follows: p, M, λ) = pi f ( p, M, λ) , Pi f (
(1)
where −∞ < pi < ∞ is the i th–component of p. The commutation rules are the same as for non-relativistic particle theory. They were first proposed for the Dirac electron [1]. The commutation rules are h ¯ [Xj , Pk ] = − δj,k , [Xj , Xk ] = 0 . i
(2)
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We require the operators Pi and Xi to be Hermitian operators with respect p, M, λ) and f ( p, M, λ) 1 . to the inner product of two photon wave functions f (1) ( (f (1) , f ) =
p, M, λ)f ( p, M, λ) f (1)∗ (
λ=±1
d p where p = | p| . cp
(3)
This prescription for obtaining coordinate operators is not really relativistic. For a relativistic treatment would require the existence of a time operator T which satisfies a commutation equation similar to Eq. (2) where Pi is replaced by the Hamiltonian operator H. However, von Neumann’s theorem says that if two operators Xi and Pi satisfy the commutation rules Eq. (2), each has a continuous spectrum ranging from −∞ to ∞. This condition on the spectra is acceptable and even expected for momentum and coordinate operators. If a time operator and the Hamiltonian operator satisfied a commutation relation such as Eq. (2), then each of them too would have a continuous spectrum ranging over the entire real axis. But we know that photons have a minimum energy of 0. Hence, a time operator, which is analogous to a coordinate operator, and the Hamiltonian cannot satisfy set of the commutation relation and the relativistic symmetry is broken2 . The X operators must be defined anew in each frame of reference. We have the following theorem: The coordinate operators Xi are given in the linear momentum representation by (p)
p, M, λ) = − Xj f (
¯ ∂ h 1 pj f ( p, M, λ) . − i ∂pj 2 p2
(4)
(p)
The superscript p on Xi means that the operator is expressed in the p− representation. This realization is unique (within unitary transformations) and the operators are Hermitian.
1.2
The Trajectory of the Photon
We are now able to give the trajectory of the photon. That is, we can calculate the mean value of the coordinate operators Xi as a function of time and quantum 1 We are assuming that the set of operators {X , P } are an irreducible set in the Hilbert space i i of photon wave functions. Roughly speaking, this means that any other operators in the space are functions of Xi , Pi . 2 Wigner in his treatment of relativistic particles Ref. (3’) avoids the use of the coordinate representation and coordinate operators and instead uses the 4–momentum and the 4 ×4 relativistic tensor which is a generalization of the angular momentum vector.
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number M of the 2–p state. We shall denote the mean values by X i (t, M ). Ep ∗ Ep d p X j (M ; t) = p, M, λ) Xj e−i h¯ t f ( p, M, λ) e−i h¯ t f ( cp λ=±1 ¯h ∂ p 1 pj = p, M, λ)ei h¯ ct − − f ∗ ( i ∂pj 2 p2 λ=±1 p
× e−i h¯ ct f ( p, M, λ)
d p . cp
(5)
Integrating over the angular variables we find X j (M ; t) = 0
(6)
for all M and t. The interpretation of this result is that for any given direction, indicated by a unit vector drawn from the origin, there are as many photons going the direction of the vector as those going in the opposite direction. This result is unsurprising.
1.3
The x–Representation E
The set of functions exp[−i h¯p t]f ( p, M, λ) ≡ f ( p, M, λ; t) can be considered as a p representation, that is, a representation in which the operators Pi are represented by multiplication by pi . We now look for a set of functions χ(x, M, λ; t) which are in a one–to–one correspondence with the functions f ( p, M, λ; t) such that (x)
Xj χ(x, M, λ; t) = xj χ(x, M, λ; t)
(7)
instead of Eq. (4) . The superscript x on the operator Xj indicates that the x– representation is being used. The consequences below result: p · x)] −i E t exp[ h¯i ( 1 e h¯ f ( χ(x, M, λ; t) = p, M, λ) . (8) d p √ 3 cp (2π¯ h) 2 √ cp i E t i h ¯ f ( p, M, λ) = p · x)]χ(x, M, λ; t) . (9) dx exp[− ( 3 e ¯h (2π¯ h) 2 d p |f ( p, M, λ)|2 = 1 . (10) dx |χ(x, M, λ; t)|2 = cp λ=±1
λ=±1
about x at time The probability that the photon is in the volume element dx t is given by P r(x; M ; t) dx where P r(x, M ; t) is the probability density given by |χ(x, M, λ; t)|2 . P r(x, M ; t) = λ=±1
(11)
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Thus P r(x, M ; t) gives the shape of the photon. The time dependence gives its trajectory. We shall now calculate χ(x, M, λ; t) from Eq. (8) and Eq. (77’). Actually, we shall consider the asymptotic wave function χasy (x, M, λ; t). This wave function is defined by lim
r→∞,r−ct=const.
r χ(x, M, λ; t) = g(r − ct, u, M, λ) .
(12)
In the above equation u = x/r and is the unit vector in the direction of x. We shall use methods that we have previously used in discussing wave propagation [2]. Indeed, the wave-function χ is a solution of the three–dimensional wave equation. The condition Eq. (12) states that χ(x, M, λ; t) ≈ χasy (x, M, λ; t) ≡
g(r − ct, u, M, λ) r
(13)
for large values of r and t. It is shown in [2] that the function g(w, u, M, λ) always exists. It is also shown that the exact function χ(x, M, λ; t) can be reconstructed from the asymptotic value. The space–time region in which Eq. (13) holds is called the wave zone. Eq. (12) and (13) state that, except for a factor 1/r, the wave function in x–space is a wave moving along each ray in x–space. The function g(w, u, M, λ) is found using the methods of [2]. g(w, u, M, λ)
λ M,λ ˆ ˆ (θ, φ) Y 2π 1 ∞ −γ1 (k) (k − κ − δ(k) + iγ4 (k)) ikw e dk . ×
2
2 k − κ − δ(k) + γ(k) =
(14)
0
In Eq. (14) θˆ and φˆ are the polar angles which fix the direction of u = x/r. The probability that a photon is lies between r and dr and in the solid angle r2 sin θˆ dθˆ dφˆ at time t is |g(r − ct, u, M, λ)|2 dr sin θˆ dθˆ dφˆ . The shape of the photon is given by |g(w, u, M, λ)|2 sin θˆ3 . It is seen from Eq. (13) and (14) that the probability density or shape function P r(r, u, M ; t) can also be written P r(r, u, M ; t) =
|Y M,λ (θ, ˆ φ)| ˆ 2 P (r − ct) 1 where r2
λ=±1
P (w)
=
∞ 2 −γ1 (k) (k − κ − δ(k) + iγ4 (k)) ikw 1 e dk . (15)
2
2 2 4π k − κ − δ(k) + γ(k) 0
3 The
factor sin θˆ which appears above is contained in the volume element ˆ d x ≡ r 2 sin θˆ dr dθˆ dφ.
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Thus the shape function is a wave which moves outward along all rays u with the velocity c. The amplitude of the wave varies along each ray as determined
of light ˆ φ)| ˆ 2 . The quantum number M and the angles θ and φ appear by |Y1M,λ (θ, λ=±1
in this factor and determine this factor as a function of θˆ 4 . Thus to finish the problem of finding the shape function of the photon requires us merely to evaluate the one–dimensional Fourier transform in Eq. (8) and (9). We shall first study the radiation pattern. Let us define the radiation pattern ˆ by function S(M ; θ) ˆ φ)| ˆ 2. ˆ = |Y1M,λ (θ, (16) S(M ; θ) λ=±1
Let us first consider the case where the atom is initially in the M = 0 state. Then from Eq(78’) ˆ = S(0; θ)
3 sin2 θˆ . 4π
(17)
When the magnetic quantum number M is 1 or -1 the radiation pattern functions are ˆ = 3 (1 + cos2 θ) ˆ . S(±1; θ) (18) 8π Thus the probability of finding the photon in the coordinate element dr dθˆ dφˆ is Q(r, u, M ; t)dr dθˆ dφˆ where Q(r, u, 0; t) = Q(r, u, ±1; t) =
3 ˆ (r − ct) sin3 θP 4π 3 ˆ + cos2 θ)P ˆ (r − ct) sin θ(1 8π
We have evaluated P numerically. The results are given graphically below. We have scaled the variable w = r − ct using the wave number κ = 2π/λα , whereλα is the wavelength of the Lyman-α radiation. Then the probability density ˆ The graph is depends upon r for a given time t and on the cone defined by θ. remarkable in that approximate causality is preserved in the quantum pictures of light. One might think that in Figure 1 the wriggles corresponding to wave propagation have been suppressed by the coarseness of the graphics. We shall show later that this suppression is not present. 4 The
ˆ use of the absolute value eliminates the φ–dependence.
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P 0.012 0.010 0.008 0.006 0.004 0.002 −1 × 107
−5 × 106
5 × 106
1 × 107
(r − ct) × κ
Figure 1. The Radial Shape of the Photon (Unscaled.) The probability has its peak for r slightly less than ct. This peak travels with the velocity of light outward along each radius. From exact causality one expects the probability to be 0 when r is greater than ct. Instead, the is a very small “precursor.” This precursor may be due to the approximations used in the analysis and that the starting points of r and t must be reset in the far field. Another possibility is that small cumulative errors in the numerical integrations are making themselves felt. Still another possibility is that Figure 1 is essentially correct and reflects the uncertainty principle in a particle picture of light. In any case the calculations must be almost correct to have causality so nearly satisfied. It will be noticed that we have not considered the regime for which (r−t)×κ < −107 . The reason for this is that our computer required an inordinate length of time to complete the calculations when longer intervals in this variable were considered5 .
1.4
The Electromagnetic Field Associated with the Lyman–α Photon
We are working with a single–photon theory. Equivalently, we are working with a first–quantized electromagnetic field (Ref 7’). In first quantized theory the operators G(E, j, m, λ) of Eq. (8’)6 are replaced by the wave function of the photon which in our case will be given by Eq. (67’), Then the electric vector x; t) = − 1 ∂ A 0 (x; t) E( c ∂t 5 Our
comments on the precursor and the range of variables also apply to later calculations. are replacing the variable k of Eq. (8’) by j so that we can use k as a wave number. Moreover, we are showing the velocity of light c explicitly. 6 We
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becomes7 x; t) E(
= −2
∞ ¯c h iλ Y1,1,M (θ, φ) Re dk k −k γ1 (k) π λ=±1
0
k − κ − δ(k) + iγ4 (k) −ikct ×
2
2 j1 (kr) e k − κ − δ(k) + γ(k) ∞ 1 Y1,2,M (θ, φ) + dk k −k γ1 (k) 3 0
k − κ − δ(k) + iγ4 (k) −ikct ×
2
2 j2 (kr) e k − κ − δ(k) + γ(k) ∞ 2 Y1,0,M (θ, φ) − dk k −k γ1 (k) 3 0
×
k − κ − δ(k) + iγ4 (k) −ikct .
2
2 j0 (kr) e k − κ − δ(k) + γ(k)
(19)
In Eq. (19) Re means the real part . The summation over λ gives zero for the first term of the expression in curly x; t) becomes brackets. The remaining two terms are doubled. The expression for E( x; t) = − E(
16¯ hc 1,2,M (θ, φ) Re Y 3π
∞ dk k
−k γ1 (k)
0
k − κ − δ(k) + iγ4 (k) −ikct ×
2
2 j2 (kr) e k − κ − δ(k) + γ(k) ∞ √ − 2 Y1,0,M (θ, φ) dk k −k γ1 (k) 0
×
k − κ − δ(k) + iγ4 (k) −ikct . j (kr) e 2 0
2
k − κ − δ(k) + γ(k)
(20)
Eq. (20) can be used to obtain the electric field of the photon for all values of x x; t). and time t. A similar expression can be used to find the magnetic field H( We shall find the fields for large values of the radius r and t. By “large values of the radius” we mean that r should be several times the wavelength of the Lyman– α radiation λα = 1.215 × 10−5 cm. Similarly, the length ct should also be several times the length λα . 7 We
ˆ now use θ and φ as the polar angles which give the direction of x, instead of θˆ and φ.
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x; t) In [2] it is shown that at such distances and time the electric field E( x; t) of any solution of Maxwell’s equations without sources and magnetic field H( must have the asymptotic forms − ct, u) G(r r − ct, u) G(r x; t) ≈ u × H( r x . u = r x; t) ≈ E(
(21)
In Eq. (21) G(w, u) is a real vector function of its arguments. which is orthogonal to u: u · G(w, u) = 0 . (22) We have called the space–time domain in which Eq. (2) holds “the wave zone” and the vector G(w, u) “the wave zone vector”. For a fixed direction u the asymptotic electromagnetic field satisfies the one–dimensional Maxwell’s equations if we ignore the factor r in the denominator. The wave zone vector for any solution of Maxwell’s equations without currents or sources is obtained from − ct, u) = G(r
lim
r→∞,r−ct=const.
x; t) . r E(
(23)
The vector G(w, u) always exists if the energy of the field is finite.8 We propose to take the solution of the electric field of the photon given by Eq. (20) and obtain the wave zone vector from Eq. (23). As a preliminary calculation we shall calculate limr→∞ r e−ikr jn (kr) for n = 0, 2. First we consider n = 2: 3 1 1 − 2ikr r e−ikr j2 (kr) = − k 3 r2 k 2i 3(1 + e−2ikr ) . (24) − k2 r As r → ∞ all the terms on the right hand side of Eq. (24) vanish except −(1 − e−2ikr )/2ik. However, as a distribution lim e−2ikr = 0 ,
r→∞
by the Riemann–Lebesgue theorem which says, roughly speaking, that if f (k) is a test function which is sufficiently smooth and integrable in the interval 0 < k < ∞, then ∞ dk f (k) e−2ikr = 0 . lim r→∞
8 In
0
the Gaussian cgs system it is measured in abvolts.
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(One could be more careful in taking the limits but we do not wish to belabor the point.) Thus 1 . (25) lim re−ikr j2 (kr) = − r→∞ 2ik Similarly, 1 . (26) lim re−ikr j0 (kr) = r→∞ 2ik From Eq. (20) lim
r→∞,r−ct=const.
x; t) = r E(
−
∞ 16¯h Re Y1,2,M (θ, φ) dk k 3π 0
×
k − κ − δ(k) + iγ4 (k) −k γ1 (k)
2
2 k − κ − δ(k) + γ(k)
×
eik(r−ct) lim r e−ikr j2 (kr)
−
∞ √ 2 Y1,0,M (θ, φ) dk k
r→∞
0
× ×
k − κ − δ(k) + iγ4 (k) −k γ1 (k)
2
2 k − κ − δ(k) + γ(k)
eik(r−ct) lim r e−ikr j0 (kr) . r→∞
(27)
On using Eq. (25) and (26) in Eq. (27) we obtain a relatively simple result for G(w, u). G(w, u) =
−2 ∞
×
dk 0
√ ¯c h 1,0,M (θ, φ) Re i Y1,2,M (θ, φ) + 2 Y 3π k − κ − δ(k) + iγ4 (k) ikw . −kγ1 (k)
2
2 e k − κ − δ(k) + γ(k)
(28)
The dependence on u is given by the factor which involves the angles and by a second factor which is a Fourier transform of the resonance term. The factor which gives the dependence on w requires numerical integration. However, the factor which depends on u can expressed as simple functions of the angles θ and φ. Let u, as before, be the unit vector in the direction of x and let aθ and aφ be the unit θ and φ, respectively. The components vectors in the√direction of increasing of Y1,2,M (θ, φ) + 2 Y1,0,M (θ, φ) can be obtained from Eq. (109)–(111) of [3] which give the components of the vector spherical harmonics in terms of spherical
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coordinates. One finds that √ 1,0,M (θ, φ) = 0 1,2,M (θ, φ) + 2 Y u · Y
(29)
as required. Also for M = 0 √ 1,2,0 (θ, φ) + 2 Y 1,0,0 (θ, φ) aθ · Y √ 1,2,0 (θ, φ) + 2 Y 1,0,0 (θ, φ) aφ · Y
= −
9 sin θ 8π
= 0.
(30)
For M = ±1, √ 1,2,M (θ, φ) + 2 Y 1,0,M (θ, φ) aθ · Y √ 1,2,M (θ, φ) + 2 Y 1,0,M (θ, φ) aφ · Y
3 = −M √ eiMφ cos θ 4 π 3 = i M √ eiMφ . 4 π
(31)
Thus for M = 0 there is only a θ–component:
G(w, u) = ×
∞ 3¯ hc k − κ − δ(k) + iγ4 (k) sin θ aθ Re i dk −kγ1 (k)
2 2
2 2π k − κ − δ(k) + γ(k) 0 ikw . (32) e
On the other hand, for M = ±1, G(w, u) has both θ and φ components.Then 3¯ hc G(w, u) = − M Re i eiMφ iaθ cos θ − aφ 2 4π ∞ k − κ − δ(k) + iγ4 (k) ikw × . (33) dk −kγ1 (k)
2
2 e k − κ − δ(k) + γ(k) 0
Or on using Gθ (w, u) and Gφ (w, u) as the θ and φ components of G(w, u) we have for M = 0 Gθ (w, u)
= −
∞ 3¯ hc k − κ − δ(k) sin θ dk −kγ1 (k)
2
2 2 2π k − κ − δ(k) + γ(k) 0
∞ ×
sin kw +
dk 0
Gφ
≡ 0.
γ4 (k) −kγ1 (k)
2
2 cos kw k − κ − δ(k) + γ(k) (34)
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For M = ±1 Gθ (w, u) =
∞ 3¯ hc M cos θ cos φ dk −kγ1 (k) 2π 2 0
×
k − κ − δ(k) 2 2 cos kw k − κ − δ(k) + γ(k) ∞ dk
+
−kγ1 (k)
0
−
∞ dk
M sin φ 0
∞ +
dk
0
Gφ (w, u) = ×
−
k − κ − δ(k) −kγ1 (k) 2 2 sin kw k − κ − δ(k) + γ(k)
γ4 (k) −kγ1 (k) 2 2 cos kw , k − κ − δ(k) + γ(k)
3¯ hc M cos φ 2π 2
∞ dk
(35)
−kγ1 (k)
0
k − κ − δ(k) 2 sin kw 2 k − κ − δ(k) + γ(k) ∞
+
dk
−kγ1 (k)
0
∞ +
γ4 (k) 2 sin kw 2 k − κ − δ(k) + γ(k)
M sin φ
dk 0
∞ +
dk
k − κ − δ(k) −kγ1 (k) 2 2 cos kw k − κ − δ(k) + γ(k)
−kγ1 (k)
0
γ(k) 2 2 cos kw k − κ − δ(k) + γ(k)
γ(k) 2 sin kw . 2 k − κ − δ(k) + γ(k)
(36)
We have computed the θ–component9 of the electric field for the case that M = 0 and θ = π/2. The electric field has a wave–group–like structure with wavelength roughly equal to that of Lyman-α radiation. The shape of the envelope is similar to that of the probability density P . The radiation appears as a long pulse moving outward 9 It
is the only component.
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107 × Gθ
−1 × 107
−5 × 106
5 × 106
1 × 107
(r − ct) × κ
Figure 2. Gθ Component of Wave Zone Vector for M = 0 and θ = π/2. −107 < (x − ct) × κ < 107
107 × Gθ 4 2 −10
−5
5
10
(r − ct) × κ
2 4
Figure 3. Within the Wave Zone Vector −10 < (x − ct) × κ < 10. along the radius with the velocity of light. For a given time t the envelope has its maximum near r = ct. The computations are clearly trying to indicate a wave–like structure of the pulse, unlike the situation for Figure 1 which is free of the highfrequency component. To see more clearly the wave – Figure 3 shows the form of the wave zone vector within the pulse of Figure 2 we have expanded the horizontal axis by a factor of 106 . For a general value of θ the wave zone vector components of Figures 2 and 3 are multiplied by sin θ. For θ = π/2, i.e. on the “equator” of the sphere surrounding
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the atom, where the field is a maximum, the electric field is polarized parallel to the z–axis. On the poles of the sphere there is no field. The electric fields for M = ±1 can be evaluated similarly.
2 Energy Density Definition of Probability in x–Space. Shape of the Photon 2.1
The Reason for Second Quantization
Since the introduction of the coordinate operators is fraught with contradictions with respect to the requirements of relativity, we are obliged to compare the consequences of alternative definitions of probability in x–space with the results just obtained. As explained in Part I, a candidate for an alternative definition of probability is the normalized energy density P r1 (x; t) =
x; t)]2 + [H( x; t)]2 1 [E( . 8π Ep
(37)
In Eq. (38) E p is the expectation value of the energy of the photon Eq. (75’) and (76’)10 . A reason to define this as a probability density in x–space is that if the energy density had a sharp peak at x0 , it would be reasonable to say that the photon is at x0 . The four–vector {x, ct} is now a label instead of a set of spectral variables. Thus we have no need to define Xj operators as we did earlier. Moreover, the energy density is part of the stress–energy tensor and the total energy is the time–component of the energy-momentum four-vector, each of whose transformation properties are relativistic11 . x; t) and H( x; t) We shall use second quantization of the fields. In this case E( are Hermitian operators. We shall first derive some results using first quantization. We consider Eq. (9’) –(12’) in first quantized theory where G(E, j, m, λ) is a general photon wave function in first quantization. We rewrite Eq. (9’) as12 ∞ j ∞ Et 2 j A1 (x; t) = − i dE G(E, j, m, λ) exp[−i ] hc ¯ ¯h m=−j −∞ λ=±1 j=1
E j j,j+1,m (θ, φ) jjm (θ, φ)jj r − iλ Y × Y hc ¯ 2j + 1
E
E j+1 Yj,j−1,m (θ, φ) jj−1 × jj+1 r + iλ r . (38) hc ¯ 2j + 1 ¯hc R 1 x; t)]2 + expectation value is equal to the total energy of the field 8π { d x [E( 2 H( x; t)] }.(See [2]) . 11 The transformation properties of the photon wave function are given in [4]. 12 We hope that no one will confuse the spherical Bessel function j with the angular momentum n quantum number also designated by j. 10 This
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1 (x; t) which we shall denote by A (w) (x; t). We want the wave zone form of A 1 By definition (w)
1 (x; t) = r A rA 1
lim
r→∞,r−ct=const.
(x; t) ≡ G1 (r − ct, u)
(39)
0 (x; t), E( x; t), From G1 (r − ct, u) we shall be able to get the wave zone forms of A x; t). and H( lim
0 (x; t) = r A (w) (x; t) ≡ G0 (r − ct, u) , rA 0
(40)
x; t) = r E (w) (x; t) ≡ G(r − ct, u) , r E(
(41)
x; t) = r H (w) (x; t) ≡ u × G(r − ct, u) . r H(
(42)
r→∞,r−ct=const.
lim
r→∞,r−ct=const.
lim
r→∞,r−ct=const.
As before, u = xr . From G1 (w, u) one can obtain G0 (w, u): G0 (w, u) = G1 (w, u) + G1∗ (w, u)
(43)
∂ G0 (w, u) . G(w, u) = ∂w
(44)
and G(w, u):
We shall first derive an expression for the wave zone vector G1 , since the other are derived from it. The generalization of Eq. (24) wave zone vectors G0 and G and (26) is (−i)(n+1) (45) lim r e−ikr jn (kr) = r→∞ 2k as a distribution. On substituting Eq. (45) into Eq. (38) we obtain G1 (w, u)) as G1 (w, u) =
−
j ∞ ¯c h dE i G(E, j, m, λ) ei kw 2 E j=1 m=−j ∞
λ=±1
× j,m,λ (u) = O +
0
j,m,λ (u) where k = E , O ¯hc j j,j,m (θ, φ) + λ Yj,j+1,m (θ, φ) −Y 2j + 1 j+1 λ Yj,j−1,m (θ, φ) . 2j + 1
(46)
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Thus G0 (w, u)
= −i
j ∞ ¯c h dE G(E, j, m, λ) ei kw 2 E j=1 m=−j ∞
λ=±1
0
∞ j ∞ dE ∗ × Oj,m,λ (u) − G (E, j, m, λ) e−i kw E λ=±1 j=1 m=−j 0 ∗ × O u) . j,m,λ (
(47)
Moreover, G(w, u) =
j ∞ 1 ∞ dE G(E, j, m, λ) ei kw 2¯ hc j=1 m=−j λ=±1
×
j,m,λ (u) + O
×
∗ O u) j,m,λ (
0
∞ j ∞
dE G∗ (E, j, m, λ) e−i kw
λ=±1 j=1 m=−j 0
.
(48)
u · G(w, u) = 0 .
(49)
It should be noted that This result is a general form for the wave zone vector in terms of photon wave functions in an energy–angular momentum representation. The total energy of the photon field is Ep =
j ∞ λ=±1 j=1 m=−j
∞
dE G∗ (E, j, m, λ)G(E, j, m, λ) .
0
The energy density in the wave zone is 2 (w) 1 (w) (x; t) 2 E (x; t) + H 8π 2 1 = G(r − ct, u) 2 4πr ∞ j j ∞ ∞ 1 = dE 8π¯ hc r2 j=1 m=−j
D(x; t) =
λ=±1
∞ 0
0
λ =±1 j =1 m =−j
dE G(E, j, m, λ)G(E , j , m , λ )
(50)
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j ,m ,λ (u) j,m,λ (u)O × ei(r−ct)(k+k ) O
+ G∗ (E, j, m, λ)G∗ (E , j , m , λ )e−i(r−ct)(k+k ) ∗ ∗ (u) + 2G∗ (E, j, m, λ)G(E , j , m , λ )e−i(r−ct)(k−k ) ×O u)O j,m,λ ( j ,m ,λ ∗ j ,m ,λ (u) . ×O u)O (51) j,m,λ ( It can be shown that13
dx D(x; t) = E p
2.2
(52)
The Effect of Second Quantization
To second-quantize the theory we replace the wave functions G(E, j, m, λ) by annihilation operators and its complex conjugate G∗ (E, j, m, λ) by creation operators whose actions are described in Part I of the present paper. We use the same symbol for the operators as the wave functions which they replace. Now E p and D(x; t) are operators in the Heisenberg picture. The wave functions in the second quantized theory will be independent of time in this picture. Let < E p > , < D(x; t) > be the expectation values of these operators between the wave function of the photon which is emitted from the atom. If in Dirac bra and ket notation |χ(M ) > is that state, then
Of course,
< Ep >
=
< χ(M )|E p |χ(M ) > ,
< D(x; t) >
=
< χ(M )|D(x; t)|χ(M ) > .
(53)
dx < D(x; t) >=< E p > .
(54)
We propose to define a relative probability density for the position of the photon as14 Prel (x; t) =< D(x; t) > . (55) As in the first quantized theory, the quantity Prel is a truly relativistic quantity. It transforms as a component of the expectation relativistic stress– energy density. Now, the wave function of the photon |χ(M ) > in the second quantized 13 It is perhaps a surprising fact that the total energy of the exact fields equals the integrated energy density constructed from the asymptotic fields in the wave zone. 14 The probability density will be a function of M the magnetic quantum number of the initial p-state of the hydrogen atom.
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formalism is ∞ |χ(M ) > = 0
×
j ∞ dE |E, j, m, λ > E j=1 m=−j λ= pm1
ψ(E, j, m, λ) .
(56)
In Eq. (56) k = E/¯ hc and ψ(E, j, m, λ) is the wave function of the photon as given by Eq. (67’). Moreover, the ket |E, j, m, λ > is the one–particle ket |s > of Eq. (25’) with s written in expanded form s → {E, j, m, λ}. The wave function of the photon represented as a ket has neither a vacuum component |V > nor components |s1 , s2 · · · , sn > for n > 1. From Eq. (67’) ∞ |χ(M ) >
=i 0
× ×
dE λ|E, 1, M, λ > E λ=±1
k/2π − γ1 (k) k − κ − δ(k) + iγ4 (k)
2
2 . k − κ − δ(k) + γ(k)
(57)
The expectation value < D(x; t) > contains (see Eq. 51) terms of the form < χ(M )|G(E, j, m, λ)G(E , j , m , λ )|χ(M ) > . But these terms are zero because G(E , j , m λ )|χ(M ) > is proportional to the vacuum state of the photon and G(E, j, m, λ) acting on the vacuum state gives zero. Similarly, < χ(M )|G∗ (E, j, m, λ)G∗ (E , j , m , λ )|χ(M ) >= 0. Thus < D(x; t) >
=
×
1 2 4π¯ hcr
λ=±1 λ =±1 λ =±1 λ =±1 ∞ ∞ j j ∞ ∞
dE
j=1 j =1 m=−j m =−j 0
× ×
∗ O u) j,m,λ (
dE
0
∞ 0
dE E
∞ 0
dE E
−i(r−ct)(k−k )
j ,m ,λ (u)e ·O λ λ k /2π k /2π γ1 (k )γ1 (k )
×
k − κ − δ(k ) − iγ4 (k ) k − κ − δ(k ) + iγ4 (k )
2
2
2
2 k − κ − δ(k ) + γ(k ) k − κ − δ(k ) + γ(k )
×
< E , 1, M, λ |G∗ (E, j, m, λ)G(E , j , m , λ )|E , 1, M, λ > . (58)
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In Eq. (64), k = E /¯ hc, k = E /¯ hc, k = E /¯hc. We first evaluate < E , 1, M, λ |G∗ (E, j, m, λ)G(E , j , m , λ )|E , 1, M, λ > using Eq. (27’), (28’ ) and (28a’). We find that < E , 1, M, λ |G∗ (E, j, m, λ)G(E , j , m , λ )|E , 1, M, λ > =
EE δ(E − E )δ(E − E )δj ,1 δj,1 δm ,M δm,M δλ ,λ δλ,λ .
(59)
We substitute into Eq. (58) and sum and integrate over the δ–functions. Thus15 < D(x; t) > = ×
2 ∞ 1 1,M,λ λ O dE −kγ1 (k) 2 2 4π ¯ hcr 0 λ=±1 2 k − κ − δ(k) − iγ4 (k) ik(r−ct) e .
2
2 k − κ − δ(k) + γ(k)
(60)
But dE = h ¯ c dk and
2 2 √ 1,2,M (θ, φ) + 2 Y1,0,M (θ, φ) . 1,M,λ (u) = 2 Y λO 3
(61)
λ=±1
The quantity in square brackets is given by Eq. (30)-(32) in terms of components. For M = 0 2 1,0,λ (u) = 3 sin2 θ , λO (62) 4π λ=±1
while for M = ±1 2 1,M,λ (u) = 3 (1 + cos2 θ) . λO 8π
(63)
λ=±1
The “radiation pattern” is identical to that when the coordinate operators are used to define “position” (Eq. (17) and (18)). For M = 0 3¯ hc sin2 θ P0 (r − ct) (64) < D(x; t) >= 16π 3 r2 and for M = ±1 < D(x; t) >= 15 If
3¯ hc (1 + cos2 θ) P0 (r − ct) . 32π 3 r2
a is a vector with complex components, we define |a|2 to be a∗ · a.
(65)
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In Eq. (64) and (65) the function P0 (w) is given by P0 (w)
=
∞ dk −kγ1 (k) 0
×
2 k − κ − δ(k) + iγ4 (k) ikw .
2
2 e k − κ − δ(k) + γ(k)
(66)
The function < D(x; t) > gives the relative probability density of finding a photon at x, t. The function P0 (w) gives the radial distribution.
Energy Density
200000 150000 100000 50000
−1 × 107
−5 × 106
5 × 106
1 × 107
(r − ct) × κ
Figure 4. The Radial Probability Density Function P0 (Unscaled) The radial probability density function P0 is very similar to the radial probability density P of Eq. (15) obtained by assigning coordinate operators to the one particle photon wave function. From Eq.(64) it is seen that the density and hence probability of finding a photon is greatest at θ = π/2. or at the equator of the sphere whose north pole is the angle θ = 0 and where the probability for finding a photon is 0. Thus for the transition from the atomic state for which M = 0 the maximum probability for finding a photon is at the equator of an expanding sphere, just inside the radius r = ct. There are fainter rings of probability for smaller values of r, which correspond to local maxima in Figure 4. These remarks also hold for the coordinate definition of probability. When the transition from the excited state for which M = ±1 causes a photon to be emitted, one has the same expanding sphere, which has a non– zero probability over the entire surface. The probability is greatest at the north
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and south poles of the sphere, θ = 0 , π. The probability is a minimum at the equator, θ = 0. The graph Figure 4 shows some wriggles. One must ask the question whether the graph hides fine wriggles as in Figure 3. In Figure 5 we expand the horizontal axis.
Energy Density
52815.5 52815 52814.5 52814 52813.5 −10
−5
5
10
(r − ct) × κ
52812.5 Figure 5. The Radial Probability Density Function P0 on an Expanded Horizontal Axis (Unscaled) The probability density P0 shows none of the structure (wave forms) shown by the wave zone vector in Figure 3. The probability density P obtained by using do not show this structure either. Our notion of the the coordinate operators X photon shape as being a particle which carries a wave form which reproduces the wave form of the radiated field is not valid. The photon is a rather smooth particle which expands like a non-spherical but highly symmetric balloon enclosed by a “soft” layer (the precursor) on the a hard layer near r = ct. On the other hand, the field does have the wave-like structure after the hard layer of the balloon has passed beyond the region of observation.
3 Conclusion The notion of photon as a particle of light having momentum and energy, and, presumably, having position and shape, was introduced by Einstein almost a hundred years ago. Our computation using quantum mechanical ideas and relativistic electromagnetic fields, shows that the field associated with a photon emitted by an atom leads to results much more in consonance with field notions, despite the
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introduction of position variables and other dynamical variables associated with particles. This result appears to be true for any photon emitted by an atom, molecule, or excited nucleus16 . Perhaps this result explains a paradox that when one uses classical tools, such as gratings and prisms, and classical wave theory to measure radiations due to quantum transitions, one gets very good results, despite the inconsistency of the procedure. However, the photon shape and trajectory are quite different than those which arise from our notion of what a particle should be.
References [1] H.L. Pryce, Proc. Roy. Soc. Lond., A 150, 165 (1935). [2] H.E. Moses and R.T. Prosser, Proc. Roy. Soc. Lond. A, 422, 351 (1989). [3] H. E. Moses, SIAM Jour. App. Math., 21, 114 (1971). [4] H.E. Moses, Annals of Physics, 41, 158 (1967).
Harry E. Moses Applimath Company 150 Tappan Street Brookline, MA 02445 USA email: [email protected] Communicated by Vincent Rivasseau submitted 28/07/01, accepted 26/11/01
To access this journal online: http://www.birkhauser.ch
16 The
sharpness of the resonance is responsible.
Ann. Henri Poincar´e 3 (2002) c Birkh¨ auser Verlag, Basel, 2002
Annales Henri Poincar´ e
Editorial Note
In view of some recent public exchanges, the Editors of Annales Henri Poincar´e wish to reaffirm the journal’s steadfast commitment to the inclusive exchange of scientific ideas, filtered by merit, and to the spirit of nurturing science and its institutions and culture everywhere in the world.
The Editorial Board and the Editor in Chief
Ann. Henri Poincar´e 3 (2002) 817 – 845 c Birkh¨ auser Verlag, Basel, 2002 1424-0637/02/050817-29
Annales Henri Poincar´ e
Hierarchy of (2 + 1)-Dimensional Nonlinear Schr¨ odinger Equation, Self-Dual Yang-Mills Equation, and Toroidal Lie Algebras S. Kakei, T. Ikeda and K. Takasaki Abstract. The hierarchy structure associated with a (2 + 1)-dimensional Nonlinear Schr¨ odinger equation is discussed as an extension of the theory of the KP hierarchy. Several methods to construct special solutions are given. The relation between the hierarchy and a representation of toroidal Lie algebras are established by using the language of free fermions. A relation to the self-dual Yang-Mills equation is also discussed.
1 Introduction There have been many studies on multi-dimensional integrable evolution equations. An example of such equations was given by Calogero [C], which is a (2 + 1)dimensional extension of the Korteweg-de Vries equation, x 1 1 uy dx. (1.1) ut = uxxy + uuy + ux 4 2 Bogoyavlensky [Bo1] showed that there is a hierarchy of higher-order integrable equations associated with (1.1). In the previous paper [IT1], two of the present authors generalized the Bogoyavlensky’s hierarchy based on the Sato theory of the Kadomtsev-Petviashvili (KP) hierarchy [Sa, SS, DJKM, JM, JMD, UT], and discussed the relationship to toroidal Lie algebras. We note that the relation between integrable hierarchies and toroidal algebras has been discussed also by Billig [Bi], Iohara, Saito and Wakimoto [ISW1, ISW2] by using vertex operator representations. In this paper, we shall consider a (2 + 1)-dimensional extension of the nonlinear Schr¨ odinger (NLS) equation [Bo2, Sc, St1, St2], X (|u|2 )Y dX = 0, (1.2) iuT + uXY + 2u and the hierarchy associated with this equation. In the case X = Y , this equation is reduced to iuT + uXX + 2|u|2 u = 0, (1.3) which is the celebrated Nonlinear Schr¨ odinger (NLS) equation. Equation (1.2) is related to the self-dual Yang-Mills (SDYM) equation, and has been studied
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by several researchers from various viewpoints: Lax pairs [Bo2, Sc, St1], Hirota bilinear method [SOM, St2], twistor approach [St2], Painlev´e analysis [JB], and so on. Strachan [St2] pointed out that (1.2) is transformed to Hirota-type equations, (iDT + DX DY )G · F = 0,
2 ¯ DX F · F = 2GG,
(1.4)
with the transformation u = G/F . Here we have used the Hirota’s D-operators, Dxm · · · Dyn f · g = (∂x − ∂x )m · · · (∂y − ∂y )n f (x, . . . , y)g(x , . . . , y )|x =x,y =y , (1.5) and the bar ¯· denotes complex conjugation. Based on the bilinear equations (1.4), Sasa, Ohta and Matsukidaira [SOM] constructed determinant-type solutions. Their work strongly suggests that equation (1.2) may be related to the KP hierarchy. The main purpose of the present paper is to generalize the results of the previous work [IT1] so that we can treat equation (1.2) and the SDYM equation. We shall use the language of formal pseudo-differential operators (PsDO for short) that have matrix coefficients. In other words, we will generalize the theory of the multi-component KP hierarchy [Di, Sa, UT] to the (2 + 1)-dimensional NLS hierarchy. We will also use the free fermion operators [DJKM, JM, JMD] to clarify the relation to the toroidal Lie algebras. This paper is organized as follows: In Section 2, we introduce SDYM-type time evolutions to the 2-component KP hierarchy and show that the resulting hierarchy contains the (2 + 1)-dimensional NLS equation (1.2). We also discuss bilinear identity for the τ -functions, and relation to the SDYM equation. In Section 3, we present two ways to construct special solutions. Relation to toroidal Lie algebras is explained in Section 4. Based on the Fock space representation, we derive the bilinear identities from the representation-theoretical viewpoint. Section 5 is devoted to the concluding remarks.
2 Formulation of the (2 + 1)-dimensional NLS hierarchy 2.1
2-component KP hierarchy
We first review the theory of the multi-component KP hierarchy [Di, Sa, UT] in the language of formal pseudo-differential operators with (N × N )-matrix coefficients. Let ∂x denote the derivation ∂/∂x. A formal PsDO is a formal linear combiˆ = an ∂ n , of integer powers of ∂x with matrix coefficients an = an (x) nation, A x n that depend on x. The index n ranges over all integers with an upper bound. The least upper bound is called the order of this PsDO. The first non-vanishing coefficient aN is called the leading coefficient. If the leading coefficient is equal to I, the unit matrix, the PsDO is said to be monic. It is convenient to use the following notation: ˆ <0 def ˆ k def ˆ ≥0 def = an ∂xn , [A] = an ∂xn , (A) = ak . (2.1) [A] n≥0
n<0
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Addition and multiplication (or composition) of two PsDO’s are defined as follows. Addition of two PsDO’s is an obvious operation, namely, the termwise sum of the coefficients. Multiplication is defined by extrapolating the Leibniz rule n n ∂x ◦ f = (2.2) f (k) ∂xn−k , k k≥0
to the case where n is negative. Here the circle “◦” stands for composition of two operators, and f (k) the k-th derivative ∂ k f /∂xk of f . More explicitly, the product ˆ = bn ∂ n is given by ˆ =A ˆ ◦B ˆ of two PsDO’s A ˆ = an ∂ n and B C x x n n ˆ = C
m m+n−k a(k) m bn ∂x k
(2.3)
m,n,k
ˆ n is the sum of a finite number of Note that the n-th order coefficient cn = (C) n ˆ terms. Any PsDO A = n≤N an ∂x with an invertible leading coefficient aN has an inverse PsDO. In particular, any monic PsDO is invertible. We shall frequently ˆB ˆ rather than A ˆ ◦B ˆ if it does not cause confusion. One can make sense of write A λx the action of PsDO’s on e by simply extrapolating the derivation rule ∂xn eλx = λn eλx to negative powers of ∂x . Hereafter we consider only the 2-component case since it is sufficient for our purpose. However it is easy to generalize the results below to higher-component case. Let us introduce the 2-component version of the Sato-Wilson operator, ˆ def W = I+
∞
wn ∂x−n ,
(2.4)
n=1
where w j = wj (x, x(1) , x(2) ) denote the (2 × 2)-matrix-valued functions that de(1) (1) (2) (2) pend on infinitely many variables (x, x(1) , x(2) ) = (x, x1 , x2 , . . . , x1 , x2 , . . .). The 2-component KP hierarchy is defined by the Sato equation, ˆ ∂W (α)
∂xn
ˆ (α) = W ˆ −1 ˆ E α ∂ nW B x n
ˆ (α) W ˆ −W ˆ Eα ∂ n, =B x n
≥0
,
(2.5)
for n = 1, 2, . . . , α = 1, 2, with E α = (δiα δjα )i,j=1,2 .
2.2
From the 2-component KP hierarchy to the (2 + 1)-dimensional NLS hierarchy
ˆ ∂x W ˆ −1 We impose the constraint W
<0
= 0, which means that
ˆ ∂x W ˆ −1 = I∂x , W
(2.6)
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or equivalently,
ˆ ∂W = 0, ∂x
∂
∂
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ˆ = 0. + (2) W (1) ∂xn ∂xn ˆ satisfies the condition [∂x , A] ˆ = 0, the correspondence If a PsDO A ˆ = A an ∂xn ←→ A(λ) = an λn n∈Z
(2.7)
(2.8)
n∈Z
preserves sums, products and commutators. Here λ is used as a formal indeterminate (spectral parameter). Under this constraint, we can use the correspondence def
(1)
(2)
(2.8) and the remaining time evolutions are those of xn = xn − xn . The evolution equations with respect to xn are obtained from (2.5): ∂W (λ) = B n (λ)W (λ) − λn W (λ)Q, ∂xn n def (2) −1 B n (λ) = B (1) , n (λ) − B n (λ) = λ W (λ)QW (λ) ≥0
(2.9)
with Q = E 1 − E 2 . For example, the explicit form of B 1 (λ) and B 2 (λ) are given by B 1 (λ) = λQ + w 1 Q − Qw1 , B 2 (λ) = λ2 Q + λ(w 1 Q − Qw1 ) + (w 2 Q − Qw2 ) − (w1 Q − Qw 1 )w 1 .
(2.10)
We now introduce a new set of infinite time variables y = (y0 , y1 , . . .). Since the first one y0 plays a special role, we will use the notation yˇ = (y1 , y2 . . .). The time evolutions with respect to yˇ are defined as ∂W (λ) ∂W (λ) = C n (λ)W (λ) + λn , ∂yn ∂y0
∂W (λ) C n (λ) = − λn W (λ)−1 . ∂y0 ≥0
(2.11)
We remark that the variables y are essentially the same as those appeared in the works on hierarchy structure of the SDYM equation [N, T1, T2, T3, T4]. Define a formal series Ψ(λ) (called the formal Baker-Akhiezer function) as
∞ def ˆ −n Ψ(λ) = W Ψ0 (λ) = I + wn λ (2.12) Ψ0 (λ), n=1
ξ(x;λ)+µy +µξ(ˇy;λ) 0 e def Ψ0 (λ) = 0 where ξ(x; λ) is given by def
ξ(x; λ) =
∞ n=1
0 e−ξ(x;λ)+νy0 +νξ(ˇy;λ)
xn λn .
,
(2.13)
(2.14)
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Unlike the ordinary NLS case, the spectral parameter λ = λ(y) may depend on the variables y as ∂λ ∂λ = λn . (2.15) ∂yn ∂y0 Both of the additional spectral parameters µ, ν are constants with respect to x and y. Note that Ψ0 (λ) obeys linear differential equations, ∂ ∂ Ψ0 (λ) = λn Ψ0 (λ). ∂yn ∂y0
∂ Ψ0 (λ) = λn QΨ0 (λ), ∂xn
(2.16)
In terms of Ψ(λ), the evolutions equations (2.9) and (2.11) are rewritten as ∂ Ψ(λ) = B n (λ)Ψ(λ), ∂xn ∂ n ∂ Ψ(λ) = C n (λ) + λ Ψ(λ). ∂yn ∂y0
(2.17) (2.18)
The compatibility condition for (2.17) with n = 1, 2 gives (12)
2
∂w1 ∂x2
(21)
∂w −2 1 ∂x2
=
(12)
(12) 2 (21) + 8 w1 w1 ,
(21)
(21) 2 (12) + 8 w1 w1 ,
∂ 2 w1 ∂x21
∂ 2 w1 = ∂x21
(ij)
(2.19)
(12)
where w1 denotes the (i, j)-element of the matrix w1 . If we set u = −2w1 , (21) u = 2w1 , t1 = ix1 and t2 = 2ix2 , then equations (2.19) are reduced to the NLS equation (1.3). In the case of n = 1, equations (2.17), (2.18) can be written explicitly as ∂ Ψ(λ) = (λQ + w 1 Q − Qw1 )Ψ(λ), ∂x1 ∂ ∂w1 ∂ Ψ(λ). Ψ(λ) = λ − ∂y1 ∂y0 ∂y0
(2.20) (2.21)
The compatibility condition for (2.20) and (2.21) is reduced to the following nonlinear coupled equations: (11) (12) (12) (12) (22) −∂y1 w1 = −∂x1 ∂y0 w1 + w1 · ∂y0 w1 − w1 , (11) (21) (21) (21) (22) (2.22) , ∂y1 w1 = −∂x1 ∂y0 w1 + w1 · ∂y0 w1 − w1 (11) (22) (12) (21) = −2 w1 w1 . ∂x1 w1 − w1 If we impose the conditions (21)
w1
(12)
= −w1
,
(22)
w1
(11)
= w1
,
xj ∈ iR,
yj ∈ R,
(2.23)
equations (2.22) yield the (2 + 1)-dimensional NLS equation (1.2) by setting u = (12) w1 , x = ix1 , y = y0 , t = −y1 . In this sense, the evolution equations (2.5) and
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(2.11), with the reduction condition (2.6), give a hierarchy of integrable equations associated with the (2+1)-dimensional NLS equation (1.2). We note that the linear equations (2.20), (2.21) are the Lax pair that has been given in several preceding works [Bo2, Sc, St1, St2].
2.3
Relation to the self-dual Yang-Mills equation
We first briefly review the classical theory of the self-dual gauge fields [Pr, Y]. Let Au = Au (y, z, y¯, z¯) (u = y, z, y¯, z¯) be matrix-valued functions. Here the suffix does not denotes the differentiation. Define the field strength F uv (u, v = y, z, y¯, z¯) as F uv = ∂u Av − ∂v Au + [Au , Av ].
(2.24)
The self-dual Yang-Mills (SDYM) equations is formulated as F yz = F y¯z¯ = 0,
F yy¯ + F zz¯ = 0,
(2.25)
which is invariant under the gauge-transformation ˜ u = G−1 Au G + G−1 (∂u G). Au → A
(2.26)
Under the suitable choice of G of (2.26), we can take Ay = Az = 0 and the self-duality equations (2.25) is reduced to ∂y¯Az¯ − ∂z¯Ay¯ + [Ay¯, Az¯] = 0,
∂y Ay¯ + ∂z Az¯ = 0.
(2.27)
The second equation ensures the existence of the potential w such that Ay¯ = −∂z w,
Az¯ = ∂y w.
(2.28)
In terms of w, we can rewrite (2.27) as (∂y ∂y¯ + ∂z ∂z¯)w + [∂y w, ∂z w] = 0.
(2.29)
We note that this equation appeared in several works on the SDYM [BLR, LM, Pa] and is associated with a cubic action [LM, Pa]. The nonlinear equations (2.29) can be obtained as the compatibility condition for the following linear equations: (∂z¯ − λ∂y + ∂y w)Ψ = 0,
(∂y¯ + λ∂z − ∂z w)Ψ = 0.
(2.30)
These equations are of the same form as (2.21). To treat these equations simultaneously, we introduce another set of variables z = (z0 , z1 , z2 , . . .), which play the same role as y, i.e., ∂ ˆ n (λ) + λn ∂ Ψ(λ) = C Ψ(λ). (2.31) ∂zn ∂z0 In particular, the evolution equation with respect to z1 is ∂ ∂ ∂w1 Ψ(λ). Ψ(λ) = λ − ∂z1 ∂z0 ∂z0
(2.32)
Setting y0 = y, y1 = z¯, z0 = z, z1 = −¯ y, and w 1 = w, we can identify (2.21) and (2.32) with the linear equations (2.30) for the SDYM.
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Bilinear identity
Theorem 1 The formal Baker-Akhiezer functions Ψ(λ; x, y) satisfy the bilinear equation, dλ k λ Ψ(λ; x, y0 − ξ(ˇb, λ), yˇ + ˇb)Ψ(λ; x , y0 − ξ(ˇ c, λ), yˇ + cˇ)−1 = 0, (2.33) 2πi for k ≥ 0. Here x, x , y, ˇb and cˇ are understood to be independent variables. The contour integral is understood symbolically, namely, just to extract the coefficient of λ−1 : λn dλ/(2πi) = δn,−1 . Proof. In the case of x = x, ˇb = cˇ = 0, it is clear that dλ k λ Ψ(λ; x, y)Ψ(λ; x, y)−1 = 0, 2πi
(2.34)
for k ≥ 0. Iteration of the evolution equation of Ψ(λ; x, y) gives rise to higher order equations of the form, ∂xα11 ∂xα22 . . . Ψ(λ; x, y) = B α1 ,α2 ,... (λ)Ψ(λ; x, y),
(2.35)
for k, α1 , α2 , . . . ≥ 0, and B α1 ,α2 ,... (λ) being a polynomial in λ. Combining these equations with (2.34), we obtain the bilinear equations, dλ k α1 α2 λ ∂x1 ∂x2 . . . Ψ(λ; x, y) Ψ(λ; x, y)−1 = 0, (2.36) 2πi which can be cast into a single equation, dλ k λ Ψ(λ; x, y)Ψ(λ; x , y)−1 = 0. 2πi
(2.37)
Next we use (2.18) to obtain (∂y1 − λ∂y0 )β1 (∂y2 − λ2 ∂y0 )β2 . . . Ψ(λ; x, y) = C β1 ,β2 ,... (λ)Ψ(λ; x, y)
(2.38)
for β1 , β2 , . . . ≥ 0, C β1 ,β2 ,... (λ) being a polynomial in λ. This yields dλ k λ (∂y1 − λ∂y0 )β1 (∂y2 − λ2 ∂y0 )β2 . . . Ψ(λ; x, y) Ψ(λ; x , y)−1 = 0, (2.39) 2πi and we have
dλ k λ Ψ(λ; x, y0 − ξ(ˇb, λ), yˇ + ˇb)Ψ(λ; x , y0 , yˇ)−1 = 0. 2πi
(2.40)
Similar discussion with the differential equations for Ψ(λ)−1 gives the desirous result.
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Now we derive the bilinear identity for τ -functions of the (2 + 1)-dimensional NLS hierarchy. In the 2-component case [Di, JM, UT], we need three τ -functions ˜ y) that are consistently introduced by F (x, y), G(x, y) and G(x, 1 (2.41) F (x, y) F (x(1) − [λ−1 ], x(2) , y) λ−1 G(x(1) , x(2) − [λ−1 ], y) × Ψ0 (λ), ˜ (1) − [λ−1 ], x(2) , y) λ−1 G(x F (x(1) , x(2) − [λ−1 ], y)
Ψ(λ) =
1 Ψ0 (λ)−1 F (x, y) F (x(1) + [λ−1 ], x(2) , y) × ˜ (1) , x(2) + [λ−1 ], y) −λ−1 G(x
Ψ(λ)−1 =
(2.42) −λ−1 G(x(1) + [λ−1 ], x(2) , y) , F (x(1) , x(2) + [λ−1 ], y)
def where we have used the notation [λ−1 ] = 1/λ, 1/2λ2, 1/3λ3 , . . . . For the mo˜ is not assumed to be the complex ment, we will forget the complex structure, i.e., G (1) (2) ˜ conjugate of G. Note that F , G and G depend only on xn = xn − xn under the condition (2.6). The denominator of the integral of (2.33) is of the form F (x, y0 − ξ(ˇb, λ), yˇ + ˇb)F (x , y0 − ξ(ˇ c, λ), yˇ + cˇ),
(2.43)
which is a power series. According to Theorem 1, one can insert any power series of λ in (2.33). If we insert (2.43) itself therein, the denominator cancels out, so that we obtain the following identities for the (2 + 1)-dimensional NLS hierarchy: ˜ satisfy the Corollary 1 For any non-negative integer k, the functions F , G and G bilinear equations,
dλ k ξ((x−x )/2,λ) λ e F (x − [λ−1 ], y + bλ )F (x + [λ−1 ], y + cλ ) 2πi ˜ − [λ−1 ], y + c ) = 0, (2.44) − λk−2 eξ((x −x)/2,λ) G(x + [λ−1 ], y + bλ )G(x λ dλ k−1 ξ((x−x )/2,λ) λ F (x − [λ−1 ], y + bλ )G(x + [λ−1 ], y + cλ ) e 2πi − eξ((x −x)/2,λ) G(x + [λ−1 ], y + bλ )F (x − [λ−1 ], y + cλ ) = 0, (2.45) dλ k−1 ξ((x−x )/2,λ) ˜ G(x − [λ−1 ], y + bλ )F (x + [λ−1 ], y + cλ ) e λ 2πi ˜ − [λ−1 ], y + cλ ) = 0, (2.46) − eξ((x −x)/2,λ) F (x + [λ−1 ], y + bλ )G(x
where bλ denotes (b0 , b1 , b2 , . . .) with the constraint b0 = −ξ(ˇb, λ).
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The bilinear identities (2.44)–(2.46) can be rewritten into a series of Hirotatype differential equations. The simplest examples are ˜ = 0, Dx2 1 F · F − 2GG (Dx1 Dy0 − Dy1 )G · F = 0, ˜ · F = 0. (Dx1 Dy0 + Dy1 )G
(2.47) (2.48) (2.49)
˜ = −G, ¯ x1 = −iX, y0 = Y , and These equations coincide with (1.4) if we set G y1 = T .
3 Special solutions of the hierarchy 3.1
Double-Wronskian solutions
We first apply the method due to one of the authors [T1, T2, T3] to construct a special class of solutions for the (2 + 1)-dimensional NLS hierarchy, which we shall seek in the form Ψ(λ) = IλN + w1 λN −1 + · · · + w N Ψ0 ,
(3.1)
with wn = w n (x, y) being unknown functions. As the data for the solution constructed below, let us consider a formal series Ξ(λ) = j∈Z ξ j λ−j where ξ j = ξ j (x, y) are (2 × 2N )-matrix-valued functions of the form,
(j) (j) f1 (x, y) · · · f2N (x, y) ξ j (x, y) = . (3.2) (j) (j) g1 (x, y) · · · g2N (x, y) Here we assume
(0)
f 1. det ..
(0) f2N
(N )
(0)
··· .. .
f1 .. .
g1 .. .
··· .. .
···
(N ) f2N
(0) g2N
···
(N ) g1 .. 0. . = (N ) g2N
(3.3)
We furthermore impose the following conditions for Ξ(λ): • • •
∂ Ξ(λ) = λn QΞ(λ) + Ξ(λ)αn (n = 1, 2, . . .), ∂xn ∂ ∂ Ξ(λ) = λn Ξ(λ) + Ξ(λ)β n (n = 1, 2, . . .), ∂yn ∂y0 λΞ(λ) = Ξ(λ)γ,
where αn , β n , γ are (2N × 2N )-matrices.
(3.4) (3.5) (3.6)
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We now consider a monic polynomial W N (λ) of the form W N (λ) = IλN + w1 λN −1 + · · · + w N ,
(3.7)
which is characterized uniquely by the linear equation dλ W N (λ)Ξ(λ) = 0. 2πiλ
(3.8)
Solving equation (3.8) explicitly by the Cram´er formula, we have for example (12)
w1
(21)
w1
|0, 1, . . . , N ; 0, 1, . . . , N − 2| , |0, 1, . . . , N − 1; 0, 1, . . . , N − 1| |0, 1, . . . , N − 2; 0, 1, . . . , N | , = (−1)N +1 |0, 1, . . . , N − 1; 0, 1, . . . , N − 1| = (−1)N
where we have used the notation due (k1 ) f1 def . |k1 , . . . , km ; l1 , . . . , ln | = .. (k1 ) f2N
(3.9)
to Freeman and Nimmo [F]: (km )
··· .. .
f1
···
(l )
.. .
g1 1 .. .
··· .. .
(k ) f2Nm
(l ) g2N1
···
. (ln ) g2N (l )
g1 n .. .
(3.10)
Proposition 1 The monic polynomial W N (λ) characterized by (3.8) solves (2.5) and (2.11) simultaneously. Proof. ¿From (3.6), we obtain
dλ n λ W N (λ)Ξ(λ) = 0, 2πiλ
(3.11)
for any non-negative integer n. Differentiating (3.8) with respect to xn and applying (3.4), we have ∂W N (λ) dλ + λn W N (λ)Q Ξ(λ) = 0. (3.12) 2πiλ ∂xn There exist polynomials B n (λ) and R(λ) such that ∂W N (λ) + λn W N (λ)Q = B n (λ)W N (λ) + R(λ), ∂xn
(3.13)
where the degree of R(λ) is at most N − 1. In view of (3.11) and (3.12), we obtain R(λ)Ξ(λ)dλ = 0. The condition (3.3) implies R(λ) = 0 and that W N (λ) satisfies (2.9). Differentiating (3.8) with respect to yn and applying (3.5), we have ∂W N (λ) ∂ dλ + λn W N (λ) Ξ(λ) = 0. (3.14) 2πiλ ∂yn ∂y0
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We can rewrite the second term in the left hand side as follows: W N (λ) Thus we obtain
dλ 2πiλ
∂ ∂ ∂W N (λ) = ◦ W N (λ) − . ∂y0 ∂y0 ∂y0 ∂W N (λ) ∂W N (λ) − λn ∂yn ∂y0
(3.15)
Ξ(λ) = 0.
(3.16)
Since the expression in parentheses is an polynomial in λ, we can apply exactly the same argument above to get the unique polynomial C n (λ) and show that W N (λ) satisfies (2.11). Note that equations (3.4)–(3.6) are invariant under the transformations Ξ(λ) αn βn γ
→ Ξ(λ)H ,
(3.17)
∂H → H −1 αn H + H −1 , ∂xn ∂H ∂H → H −1 β n H + H −1 − H −1 γ n , ∂yn ∂y0 → H −1 γH,
(3.18) (3.19) (3.20)
where H = H(x, y) is an (2N × 2N )-invertible matrix. These formulas are a generalization of the transformations (2.22) of [T2]. As discussed by one of the authors [T2], this invariance property shows that the manifold from which the unknown functions {w1 , . . . , wN } take values is essentially a Grassmann manifold. We now consider the reality condition (2.23): Proposition 2 Let P 1 be a (2 × 2)-matrix and P 2 a (2N × 2N )-matrix, both of which are invertible. If Ξj satisfies the condition, Ξj = P 1 Ξj P 2 ,
(3.21)
then the corresponding W N satisfies P −1 1 WNP 1 = WN. In particular, if P 1 is of the form
0 1 , −1 0
P = (ab)
the coefficients w j = (wj
which agree with (2.23).
(3.23)
)a,b=1,2 satisfies
(22)
wj
(3.22)
(11)
= wj
,
(21)
wj
(12)
= −wj
,
(3.24)
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Proof. Taking the complex conjugation of (3.8), we have dλ (P −1 W N (λ)P 1 )Ξ(λ) = 0. 2πiλ 1
(3.25)
Then we find that two monic polynomials W N (λ) and P −1 1 W N (λ)P 1 are characterized by the same data Ξ(λ). This proves the results. We shall give an example that corresponds to soliton-type solutions. For the purpose, we choose the form of fj and gj of (3.2) as ∞ ∞ (j) j n n fk (x, y) = ak pk exp pk xn + rk y0 + rk pk yn , (j) gk (x, y)
=
bk pjk
n=1 ∞
exp −
pnk xn
+
rk y0
n=1 ∞
+
n=1
rk pnk yn
(3.26) ,
n=1
where ri , ri (i = 1, . . . , N ) are arbitrary complex numbers, and pi = pi (y), ai = ai (y), bi = bi (y) (i = 1, . . . , N ) are arbitrary (local) solution of the equations ∂pi ∂pi = pni , ∂yn ∂y0
∂ai ∂ai = pni , ∂yn ∂y0
∂bi ∂bi = pni ∂yn ∂y0
(n = 1, 2, . . . ).
(3.27)
Moreover, pi (y) (i = 1, . . . , N ) are assumed to be pairwise distinct. Then Ξ(λ) satisfies the linear equations (3.4)–(3.6). Furthermore, if we impose the condition b2j = a2j−1 ,
a2j = −b2j−1 ,
r2j = r2j−1 ,
r2j
x(2) n =
(1) xn ,
= r2j−1
yn ∈ R
p2j = p2j−1 ,
(j = 1, . . . , N ),
(3.28)
(n = 1, 2 . . .),
then the corresponding Ξj satisfies (3.21) with P 1 of (3.23). We conclude that the polynomial W N (λ) constructed from the data above gives a solution of the (2 + 1)-dimensional NLS hierarchy. Especially for equations (2.22), the solution is given by quotient of the “double Wronskian” (3.9).
3.2
Application of the Riemann-Hilbert problem
In case of the SDYM hierarchy, the Riemann-Hilbert problem plays an important role [CFYG, T4, UN, W]. We shall show how to apply this problem to the (2 + 1)dimensional NLS hierarchy. We first consider two solutions Ψ and Φ of (2.18), which are analytic functions on |λ| > 1 − (including λ = ∞) and |λ| < 1 + respectively. Here is a constant and 0 < < 1. We further assume that both Ψ and Φ are invertible. If we define g(λ) as (3.29) g(λ) = Ψ(λ)−1 Φ(λ),
Vol. 3, 2002
Hierarchy of (2 + 1)-Dimensional Nonlinear Schr¨ odinger Equation
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then g(λ) is holomorphic on 1 − < |λ| < 1 + and satisfies ∂g(λ) ∂g(λ) = λn . ∂yn ∂y0
(3.30)
In other words, g = g(λ; y) is invariant under the translation, g(λ; y) = g(λ; y + bλ ) = g(λ; y0 − ξ(ˇb, λ), yˇ + ˇb) .
(3.31)
On the contrary, starting from g with the property (3.31), we can reconstruct Ψ and Φ that satisfy the analyticity requirements (the Riemann-Hilbert problem). Hereafter we assume that ξ(ˇb, λ) and ξ(ˇ c, λ) are analytic functions on |λ| < 1 + . This is a growth condition on the behavior of bn and cn as n → ∞. So far we have not included the NLS-type time evolutions x. To this aim, we ˜=g ˜ (λ; x, y) as define g ˜ (λ; x, y) = exp[ξ(x, λ)Q]g(λ; y) exp[−ξ(x, λ)Q], g
(3.32)
˜ , we where we assume g(λ; y) enjoys the invariance (3.31). Starting from this g ˜ such that consider the Riemann-Hilbert decomposition of the matrix g ˜ (λ; x, y)−1 V˜ (λ; x, y), g˜(λ; x, y) = W
(3.33)
˜ (λ) and V˜ (λ) are analytic functions on |λ| > 1 − and |λ| < 1 +
where W respectively. ˜ Proposition 3 If we define Ψ(λ) as def ˜ ˜ Ψ(λ) = W (λ) exp[ξ(x, λ)Q],
(3.34)
˜ then Ψ(λ) solves the bilinear identity (2.33). ˜ (λ; x, y + bλ ) = g ˜ (λ; x, y + cλ ) reads Proof. The translational invariance g ˜ (λ; x, y + b ) · W ˜ (λ; x, y + c )−1 = V˜ (λ; x, y + b ) · V˜ (λ; x, y + c )−1 . (3.35) W λ λ λ λ Since the right-hand-side is analytic on |λ| < 1 + , we have dλ k ˜ ˜ (x, y + c )−1 = 0, λ W (λ; x, y + bλ ) · W λ 2πi
(3.36)
where the contour is taken as the unit circle with the center at λ = 0. On the other hand, the function g˜ (λ; x, y) satisfies the differential equations ∂˜ g (λ; x, y) = λn [Q, g˜ (λ; x, y)] ∂xn
(3.37)
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for n = 1, 2, . . ., which entails ˜ ˜ ˜ (λ)QW ˜ (λ)−1 + ∂ W (λ) W ˜ (λ)−1 = λn V˜ (λ)QV˜ (λ)−1 + ∂ V (λ) V˜ (λ)−1 . λn W ∂xn ∂xn (3.38) ¿From the analyticity requirements, it follows that the left-hand-side is a polynomial of degree at most n, which we denote B n (λ). It is straightforward to shows that ˜ ∂ Ψ(λ) ˜ = B n (λ)Ψ(λ), (3.39) ∂xn which are nothing but the evolution equations (2.17). With these equations, we can apply the same argument as the proof of Theorem 1. The resulting equation coincides with (2.33). Sasa et al. constructed a class of solutions of the (2 + 1)-dimensional NLS equation (1.2) that are expressed in terms of two-directional Wronskians [SOM]. In the case of the SDYM, this class of solutions has been discussed by Corrigan et al. [CFYG] based on the Atiyah-Ward ansatz [W], N λ (λ; y) . (3.40) g(λ; y) = 0 λ−N Substituting this g for (3.32), we know that g˜ is of the same form; N λ ˜(λ; x, y) ˜ (λ; x, y) = g , ˜(λ; x, y) = (λ; y) exp[2ξ(x, λ)]. 0 λ−N
(3.41)
Applying the same argument as that of Corrigan et al. [CFYG], we can obtain a class of solutions to the (2 + 1)-dimensional NLS hierarchy, which is an extension of the solutions of Sasa et al.
4 Relation to the toroidal Lie algebras 4.1
Definitions and a class of representations
We start with the definitions of the (M + 1)-toroidal Lie algebra, which is the universal central extension of the (M + 1)-fold loop algebra [Kas, MEY]. Let g be a finite-dimensional simple Lie algebra over C. Let R be the ring of Laurent ±1 polynomials of M + 1 variables C[s±1 , t±1 1 , . . . , tM ]. Also assume M ≥ 0. The module of K¨ ahler differentials ΩR of R is defined with the canonical derivation d : R → ΩR . As an R-module, ΩR is freely generated by ds, dt1 , . . . , dtM . Let · : ΩR → ΩR /dR be the canonical projection. Let K denote ΩR /dR. Let (·|·) be the normalized Killing form [Kac1] on g. We define the Lie algebra structure on def
gtor = g ⊗ R ⊕ K by [X ⊗ f, Y ⊗ g] = [X, Y ] ⊗ f g + (X|Y )(df )g,
[K, gtor ] = 0.
This bracket defines a universal central extension of g ⊗ R [Kas, MEY].
(4.1)
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We have, for u = s, t1 , . . . , tM , the Lie subalgebras gu = g ⊗ C[u±1 ] ⊕ C d log u, def
(4.2)
with the brackets given by [X ⊗ um , Y ⊗ un ] = [X, Y ] ⊗ um+n + mδm+n,0 (X|Y ) Ku ,
(4.3)
which are isomorphic to the affine Lie algebra g with the canonical central element def Ku = d log u. In terms of the generating series, def X(z) = X ⊗ un · z −n−1 , (4.4) n∈Z
the relation (4.3) is equivalent to the following operator product expansion (OPE, in short. See, for example, [Kac2]) : X(z)Y (w) ∼
1 1 [X, Y ](w) + (X|Y )Ku . z−w (z − w)2
We prepare the generating series of gtor as follows: def A ⊗ sn tm · z −n−1 , Am (z) =
(4.5)
(4.6)
n∈Z s Km (z) =
def
sn tm d log s · z −n ,
(4.7)
sn tm d log tk · z −n−1 ,
(4.8)
n∈Z tk Km (z) =
def
n∈Z mM 1 where A ∈ g, m = (m1 , . . . , mM ) ∈ ZM , tm = tm 1 · · · tM , and k = 1, . . . , M . The m relation d(sn t ) = 0 can be neatly expressed by these generating series as M
∂ s t Km (z) = mk K m (z), ∂z
(4.9)
k=1
and the bracket (4.1) as Xm (z)Yn (w)
∼
1 1 s [X, Y ]m+n (w) + (X|Y )Km+n (w) z−w (z − w)2 +
M mk tk (X|Y )Km+n (w). z−w
(4.10)
k=1
To construct a class of representations of gtor , we consider the space of polynomials, (k) def M (k) (4.11) Fy = ⊗ C[yj , j ∈ N] ⊗ C[e±y0 ] . k=1
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We define the generating series ϕ
(k)
def
(z) =
nyn(k) z n−1 ,
M
def
Vm (y; z) =
n∈N
exp mk
yn(k) z n
,
(4.12)
n∈N
k=1
for each k = 1, . . . , M , m ∈ ZM . Proposition 4 Let (V, π) be a representation of gs such that d log s → c · idV for c ∈ C. Then we can define the representation π tor of gtor on V ⊗ Fy such that Xm (z) → X π (z) ⊗ Vm (z),
(4.13)
s Km (z)
→ c · idV ⊗ Vm (z),
(4.14)
tk Km (z)
→ c · idV ⊗ ϕ
(4.15)
(k)
def
where X ∈ g, m ∈ Z and X π (z) =
n∈Z
(z)Vm (z),
π(X ⊗ sn )z −n−1 .
Proof. By the OPE (4.5) and the property Vm (z)Vn (z) = Vm+n (z), we obtain X(z) ⊗ Vm (z) Y (w) ⊗ Vn (w) ! " 1 c ∼ [X, Y ](w) + (X|Y ) z−w (z − w)2 " ! ∂Vm (w) (z − w) Vn (w) ⊗ Vm (w) + ∂w 1 c [X, Y ](w) ⊗ Vm+n (w) + ∼ (X|Y ) ⊗ Vm+n (w) z−w (z − w)2 +
M mk c (X|Y )ϕ(k) (w)Vm+n (w). z−w
(4.16)
k=1
Comparing the last line to (4.10), we have the desirous result. Remark. In the preceding works [BB, IT1, ISW1, ISW2], a much bigger Lie algebra that includes the derivations to gtor is considered. Here we do not consider the derivations since those are not needed for our purpose, i.e., treating the (2 + 1)dimensional NLS hierarchy. Hereafter we consider only the sltor 2 -case to treat the (2 + 1)-dimensional NLS hierarchy. The generators of sl2 is denoted by E, F and H as usual: [E, F ] = H,
[H, E] = 2E,
[H, F ] = −2F.
(4.17)
We prepare the language of the 2-component free fermions [JM]. Note that the notation we use below is that of [JM] and slightly different from that of [IT1, JMD].
Vol. 3, 2002
Hierarchy of (2 + 1)-Dimensional Nonlinear Schr¨ odinger Equation (α)
(α)∗
Let A be the associative C-algebra generated by ψj , ψj the relations, (α)
(β)∗
[ψi , ψj
]+ = δij δαβ ,
(α)
(β)
(α)∗
[ψi , ψj ]+ = [ψi
n∈Z
(j ∈ Z, α = 1, 2) with
(β)∗
, ψj
In terms of the generating series defined as ψn(α) λn , ψ (α)∗ (λ) = ψn(α)∗ λ−n ψ (α) (λ) =
833
]+ = 0.
(4.18)
(α = 1, 2),
(4.19)
n∈Z
the relation (4.18) are rewritten as ψ (α) (λ), ψ (β)∗ (µ) = δαβ δ(λ/µ), + (α) (β) ψ (λ), ψ (µ) = ψ (α)∗ (λ), ψ (β)∗ (µ) = 0, +
(4.20)
+
def where δ(λ) = n∈Z λn is the formal delta-function. Consider a left A-module with a cyclic vector |vac satisfying (α)
ψj |vac = 0
(j < 0),
(α)∗
ψj
|vac = 0
(j ≥ 0).
(4.21)
This A-module A|vac is called the fermionic Fock space, which we denote by F . We also consider a right A-module (the dual Fock space F ∗ ) with a cyclic vector
vac| satisfying (α)
vac|ψj
=0
(j ≥ 0),
(α)∗
vac|ψj
=0
(j < 0).
(4.22)
We further define the generalized vacuum vectors as def
(1) |s2 , s1 = Ψ(2) s2 Ψs1 |vac ,
(α)∗ (α)∗ ψs · · · ψ−1 def = 1 (α) (α) ψs−1 · · · ψ0
(4.23)
(α) (α) ψ−1 · · · ψs def = 1 (α)∗ (α)∗ ψ0 · · · ψs−1
(s < 0), (s = 0), (s > 0). (4.24) There exists a unique linear map (the vacuum expectation value),
Ψ(α) s
(s < 0), (s = 0), (s > 0),
def
(2)∗
s1 , s2 | = vac|Ψ(1)∗ s1 Ψs2 ,
Ψ(α)∗ s
F ∗ ⊗A F −→ C
(4.25)
such that vac| ⊗ |vac → 1. For a ∈ A we denote by vac|a|vac the vacuum expectation value of the vector vac|a ⊗ |vac (= vac| ⊗ a|vac ) in F ∗ ⊗A F . Using the expectation value, we prepare another important notion of the normal ordering: (α) (β)∗ def (α) (β)∗ (α) (β)∗ : ψi ψj : = ψi ψj − vac|ψi ψj |vac .
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Lemma 1 ([DJKM, JM, JMD]) The operators E(z) = ψ (1) (z)ψ (2)∗ (z), F (z) = ψ (2) (z)ψ (1)∗ (z), H(z) = : ψ (1) (z)ψ (1)∗ (z) : − : ψ (2) (z)ψ (2)∗ (z) : ,
(4.26)
2 on the fermionic satisfy the OPE (4.5) with c = 1, i.e., give a representation of sl Fock space F . ¿From Lemma 1 and Proposition 4, we have a representation of gtor on F ⊗ Fy . We will use this representation in what follows to derive bilinear identities. Note that the operators E(z), F (z) and H(z) are invariant under the following automorphism of fermions: def Fytor =
(a)
(a)
(a)∗
ι(ψj ) = ψj+1 ,
4.2
ι(ψj
(a)∗
) = ψj+1
(j ∈ Z, a = 1, 2).
(4.27)
Derivation of the bilinear identity from representation theory
We first introduce the following operator acting on Fytor ⊗ Fytor : Ω
tor def
=
m∈ZM α=1,2
dλ (α) ψ (λ)Vm (y; λ) ⊗ ψ (α)∗ (λ)V−m (y ; λ). 2πiλ
(4.28)
Lemma 2 The operator Ωtor enjoys the following properties: (i) (ii)
tor [Ωtor , sltor 2 ⊗ 1 + 1 ⊗ sl2 ] = 0,
Ω
tor
⊗2
(|s2 , s1 ⊗ 1)
= 0.
(4.29) (4.30)
Proof. Since the representation of sltor under consideration is constructed from 2 Lemma 1, it is enough to show Ωtor , ψ (α) (p)ψ (β)∗ (p)Vn (y; p) ⊗ 1 + 1 ⊗ ψ (α) (p)ψ (β)∗ (p)Vn (y ; p) = 0, (4.31) for α, β = 1, 2 and n ∈ ZM . From (4.20), we have ψ (α) (p)ψ (β)∗ (q), ψ (γ) (λ) = δβγ δ(q/λ)ψ (α) (p), ψ (α) (p)ψ (β)∗ (q), ψ (γ)∗ (λ) = −δαγ δ(p/λ)ψ (β) (q).
(4.32)
These equations and the relation Vm (y; λ)Vn (y; λ) = Vm+n (y; λ) give the commutativity above.
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Hierarchy of (2 + 1)-Dimensional Nonlinear Schr¨ odinger Equation
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If we translate Lemma 2 into bosonic language, then it comes out a hierarchy of Hirota bilinear equations. To do this, we present a summary of the boson(α) fermion correspondence in the 2-component case. Define the operators Hn as def (α) (α) (α)∗ Hn = j∈Z ψj ψj+n for n = 1, 2, . . . , α = 1, 2, which obey the canonical (α)
(β)
(α)
commutation relation [Hm , Hn ] = mδm+n,0 δαβ · 1. The operators Hn generate the Heisenberg subalgebra (free bosons) of A, which is isomorphic to the algebra (α) (α) with the basis {nxn , ∂/∂xn (α = 1, 2, n = 1, 2, . . . )}. Lemma 3 ([DJKM, JM, JMD]) For any |ν ∈ F and s1 , s2 ∈ Z, we have the following formulas, ( ' (1) (2) s1 , s2 |eH(x ,x ) ψ (1) (λ)|ν (1)
=
(−)s2 λs1 −1 eξ(x
=
(−)s2 λ−s1 e−ξ(x
,λ)
' ( (1) (2) s1 , s2 |eH(x ,x ) ψ (1)∗ (λ)|ν (1)
' ( (1) −1 (2) s1 − 1, s2 |eH(x −[λ ],x ) |ν ,
,λ)
'
s1 + 1, s2 |eH(x
(1)
+[λ−1 ],x(2) )
( ' (1) (2) s1 , s2 |eH(x ,x ) ψ (2) (λ)|ν ' ( (2) (1) (2) −1 = λs2 −1 eξ(x ,λ) s1 , s2 − 1|eH(x ,x −[λ ]) |ν , ( ' (1) (2) s1 , s2 |eH(x ,x ) ψ (2)∗ (λ)|ν ' ( (2) (1) (2) −1 = λ−s2 e−ξ(x ,λ) s1 , s2 + 1|eH(x ,x +[λ ]) |ν ,
( |ν ,
(4.33)
(4.34)
(4.35)
(4.36)
where the “Hamiltonian” H(x(1) , x(2) ) is defined as H(x
(1)
,x
(2)
def
) =
∞
(α) x(α) n Hn .
(4.37)
α=1,2 n=1
We prepare one more lemma due to Billig [Bi]. nj Pj , where Pj Lemma 4 ([Bi], Proposition 3. See also [ISW2]) Let P (n) = j≥0 n are differential operators that may not depend on z. If n∈Z z P (n)f (z) = 0 for some function f (z), then P ( − z∂z )f (z)|z=1 = 0 as a polynomial in . Now we are in position to state the bilinear identity for the (2+1)-dimensional denote a group of invertible linear transformations on NLS hierarchy. Let SLtor 2 Fytor generated by the exponential action of the elements in sl2 ⊗ R acting locally nilpotently. Define the τ -function associated with g ∈ SLtor 2 as s ,s
def tor
τs21,s12 (x(1) , x(2) , y) = def
s1 , s2 |eH(x
(1)
def
,x(2) )
g(y)|s2 , s1 tor ,
(4.38)
where |s2 , s1 tor = |s2 , s1 ⊗ 1 and tor s1 , s2 | = s1 , s2 | ⊗ 1. Hereafter we shall omit the superscripts “tor” if it does not cause confusion. Since g ∈ SLtor 2 , the
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Ann. Henri Poincar´e
τ -function (4.38) have the following properties [JM]: ++1,s2 −+1 +,s2 − τss21+1,s = (−1) τss21,s , 1 1 +1
∂ ∂ s ,s + (2) τs21,s12 = 0, (1) ∂xj ∂xj def
(1)
(4.39) (4.40)
(2)
i.e., the τ -function depends only on {xj = xj − xj } and {yj }. Proposition 5 For non-negative integers k, l1 and l2 , the τ -functions satisfy dλ s1 −s1 +k−2 ξ((x−x )/2,λ) λ (−1)s2 +s2 e 2πi s −1,s
s +1,s 2
(x + [λ−1 ], y + bλ )
s ,s −1
s ,s +1
(x − [λ−1 ], y + bλ ) = 0.
× τs 1+l ,s2 +l (x − [λ−1 ], y − bλ )τs21,s1 2 2 1 1 dλ s2 −s2 +k−2 ξ((x −x)/2,λ) λ e + 2πi
× τs21+l22 ,s1 +l1 (x + [λ−1 ], y − bλ )τs21,s12
(4.41)
Proof. This is the direct consequence of Lemmas 2, 3, 4. Setting 0,0 F = τ0,0 ,
1,−1 G = τ0,0 ,
˜ = −τ −1,1 , G 0,0
(4.42)
one can show that (4.41) contains the bilinear equations (2.44)–(2.46) with the condition ˇb + cˇ = 0. We now turn to the 2-dimensional derivative NLS (DNLS) equation [St1],
X 2 iuT + uXY + 2i u (|u| )Y dX = 0. (4.43) X
This equation can also be treated in terms of the bilinear formulation [SOM]. Following Sasa et al., we set u=
fg , f˜2
u=−
f˜g˜ , f2
(4.44)
where we have assumed f = f˜,
g = −˜ g.
(4.45)
The validity of this assumption will be discussed in the next section. These u and u solve (4.43) if the variables f and g obey the Hirota equations, (iDX DY − DT )f · g = 0, (iDX DY + DT )˜ g · f˜ = 0,
(4.46)
(iDX DY + 2DT )f · f˜ = DY g˜ · g, iDX f · f˜ = g˜ g.
(4.48)
(4.47) (4.49)
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837
We note that our bilinearization is slightly different from that of Sasa et al. The first two equations can be obtained from (4.41) by making the change of the variables, X = ix1 , f=
1,0 τ0,1 ,
g=
Y = y0 , T = y1 , f˜ = τ 0,0 , g˜ = τ 1,−1 .
0,1 τ0,1 ,
0,0
0,0
(4.50)
The bilinear identity including the rest two can be obtained in the same way as Proposition 5: Proposition 6 For non-negative integers k, the τ -functions satisfy dλ s1 −s1 +k−2 ξ((x−x )/2,λ) s2 +s 2 (−1) λ e 2πi s −1,s
s +1,s
s ,s −1
(x + [λ−1 ], y − bλ )τs21,s12+1 (x − [λ−1 ], y + bλ )
× τs21,s1 2 (x − [λ−1 ], y − bλ )τs21,s1 +12 (x + [λ−1 ], y + bλ ) dλ s2 −s2 +k−2 ξ((x −x)/2,λ) + λ e 2πi × τs21,s12
s ,s +1
s ,s s ,s = τs21,s12+1 (x, y0 , yˇ − ˇb)τs21,s12 (x , y0 , yˇ + ˇb).
(4.51)
Proof. Using Ωtor |s2 , s1 tor ⊗ |s2 , s1 + 1 tor = (|s2 , s1 + 1 ⊗ emy0 ) ⊗ |s2 , s1 ⊗ e−my0 (4.52) instead of (4.30), we can derive the desirous result. Expanding (4.51), we can obtain Hirota-type differential equations including the following ones: 1,0 0,0 1,−1 0,1 (Dx1 Dy0 − 2Dy1 )τ0,1 · τ0,0 = Dy0 τ0,0 · τ0,1 , 1,0 Dx1 τ0,1
·
0,0 τ0,0
+
0,1 1,−1 τ0,1 τ0,0
= 0.
(4.53) (4.54)
These equations agree with (4.48) and (4.49).
4.3
Reality conditions and soliton-type solutions
In this section, we consider an algebraic meaning of the reality condition (3.21). To this aim, we introduce an automorphism ρ of the fermion algebra as (α)∗ ρ ψn(α) = ψ−n−1 ,
(α) ρ ψn(α)∗ = ψ−n−1
which have the following properties:
(n ∈ Z, α = 1, 2),
(4.55)
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• ρ2 = id, (α) (α) • ρ Hn = −Hn
Ann. Henri Poincar´e
(α = 1, 2), ∀
• vac|ρ(g)|vac = vac|g|vac ,
g ∈ SLtor 2 .
We note that the similar automorphism have been discussed by Jaulent, Manna and Martinez-Alonso [JMM]. Assuming the conditions x(i) n ∈ iR (n ∈ N, i = 1, 2),
ρ(g) = g,
(4.56)
we find that ' ( (1) (2)
0, 0|eH(x(1) ,x(2) ) g|0, 0 = 0, 0|eH(x ,x ) g|0, 0 , ' ( (1) (2)
1, −1|eH(x(1) ,x(2) ) g|0, 0 = − 1, 1|eH(x ,x ) g|0, 0 .
(4.57) (4.58)
˜ of (4.42) satisfy the reality conUnder these conditions, the τ -functions F , G, G dition, ˜ F = F, G = −G. (4.59) Next we introduce another automorphism σ to treat the (2 + 1)-dimensional DNLS equation (4.43): σ(ψn(1) ) = ψn(2) , σ(ψn(1)∗ ) = ψn(2)∗ , (1)
(1)∗
σ(ψn(2) ) = ψn+1 , σ(ψn(2)∗ ) = ψn+1 ,
(4.60)
which have the following properties, • If ι(g) = g, then σ 2 (g) = g, (1) (2) • σ Hn = H n ,
(2) (1) σ Hn = H n ,
• 1, 0|σ(g)|0, 1 = vac|g|vac ,
∀
g ∈ SLtor 2 .
Imposing the conditions (2)
x(1) n = xn
(n ∈ N),
σ(g) = g,
(4.61)
we find that ' ( (1) (2)
0, 0|eH(x(1) ,x(2) ) g|0, 0 = 1, 0|eH(x ,x ) g|0, 1 , ' ( (1) (2)
1, −1|eH(x(1) ,x(2) ) g|0, 0 = − 0, 1|eH(x ,x ) g|0, 1 .
(4.62) (4.63)
In this case, the τ -functions f , g, f˜, g˜ of (4.50) satisfy the reality condition (4.45).
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As an example of special solutions, we consider soliton-type solutions given by ' ( (1) (2) s ,s (4.64) τs21,s12 x(1) , x(2) = s1 , s2 |eH(x ,x ) g N (y)|s2 , s1 , def
N
g N (y) =
exp aj ψ (1) (pj )ψ (2)∗ (pj )Vmj (y; pj )
j=1
+bj ψ
(2)
(qj )ψ
(1)∗
(qj )Vnj (y; qj ) .
(4.65)
In the NLS case ρ(g N ) = g N , the parameters should obey the conditions, qj = pj ,
bj = −aj
(j = 1, . . . , N ).
(4.66)
(j = 1, . . . , N ).
(4.67)
In the DNLS case σ(g N ) = g N , we have qj = pj ,
b j = aj p j
We conclude that the NLS case and the DNLS case have different complex structure that correspond to different real forms of the toroidal Lie algebra sltor 2 .
4.4
Bilinear identity for the SDYM hierarchy
The SDYM equation can also be treated also by the Hirota’s bilinear method [SOM]. Toward this aim, we shall take so-called “Yang’s R-gauge” defined as fol¯ such that lows: Due to (2.25), there exist matrix-valued functions G and G ) ) ¯ = GA ¯ y¯, ∂y¯G ∂y G = GAy , (4.68) ¯ ¯ z¯. ∂z G = GAz , ∂z¯G = GA def ¯ −1 , the self-duality equation (2.25) takes If we define the matrix J as J = GG the form (4.69) ∂y¯ J −1 ∂y J + ∂z¯ J −1 ∂z J = 0.
We then consider the gauge field J of the form, 1 1 τ2 τ1 −g J= , e= , f = , 2 − eg e f f τ5 τ5
g=
τ3 . τ5
(4.70)
The gauge field J of (4.70) solves (4.69) if the τ -functions satisfy the following seven Hirota-type equations [SOM], τ52 + τ2 τ8 − τ4 τ6 = 0, Dy τ1 · τ5 = Dz¯τ4 · τ2 ,
(4.71) (4.72)
Dy τ2 · τ6 = Dz¯τ5 · τ3 ,
(4.73)
Dy τ4 · τ8 = Dz¯τ5 · τ7 , Dz τ1 · τ5 = Dy¯τ2 · τ4 ,
(4.74) (4.75)
Dz τ2 · τ6 = Dy¯τ3 · τ5 , Dz τ4 · τ8 = Dy¯τ7 · τ5 ,
(4.76) (4.77)
where we have introduced auxiliary dependent variables τ4 , τ6 , τ7 , τ8 .
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The bilinear identity associated with these equations is given as follows: Proposition 7 For non-negative integers k, the τ -functions satisfy dλ s1 −s1 +k−2 ξ((x−x )/2,λ) (−1)s2 +s2 λ e 2πi s −1,s
s +1,s
× τs21,s1 2 (x − [λ−1 ], y − bλ )τs21+1,s12+1 (x + [λ−1 ], y + bλ ) dλ s2 −s2 +k−2 ξ((x −x)/2,λ) + λ e 2πi s ,s2 +1 s ,s −1 x + [λ−1 ], y − bλ τs21+1,s x − [λ−1 ], y + bλ × τs21,s12 1 +1 s ,s
s ,s
2 = τs21+1,s (x, y0 , yˇ − ˇb)τs21,s12+1 (x , y0 , yˇ + ˇb) 1
s ,s s ,s2 − τs21,s12+1 (x, y0 , yˇ − ˇb)τs21+1,s (x , y0 , yˇ + ˇb). 1
(4.78)
Proof. This can be proved in the same fashion as Proposition 5; Use Ωtor |s1 , s2 tor ⊗ |s1 + 1, s2 + 1 tor = (|s1 + 1, s2 ⊗ emy0 ) ⊗ |s1 , s2 + 1 ⊗ e−my0 − (|s1 , s2 + 1 ⊗ emy0 ) ⊗ |s1 + 1, s2 ⊗ e−my0
(4.79)
instead of (4.30). Expanding (4.78) and applying (4.39), we can obtain the following Hirotatype equations, +1,s2 s1 ,s2 +1 +1,s2 s1 ,s2 +1 ,s2 2 ) + τss21+1,s τ − τss21,s τ = 0, (τss21,s 1 1 s2 ,s1 +1 1 +1 s2 +1,s1 +1,s2 −1 Dy0 τss21,s 1
·
,s2 τss21,s 1
=
−1,s2 +1 ,s2 · τss21,s = Dy0 τss21,s 1 1
+1,s2 Dy1 τss21,s 1 +1 s1 ,s2 +1 Dy1 τs2 ,s1 +1
· ·
+1,s2 τss21+1,s , 1 ,s2 +1 τss21+1,s , 1
(4.80) (4.81) (4.82)
which agree with (4.71)–(4.74) if we set y¯ = y0 , 0,0 τ1 = τ1,−1 , 0,0 τ5 = τ0,0 ,
0,1 τ2 = iτ1,0 , 0,1 τ6 = iτ0,1 ,
z = y1 , −1,1 τ3 = τ0,0 ,
τ7 =
1,−1 τ0,0 ,
1,0 τ4 = iτ1,0 ,
τ8 =
(4.83)
1,0 iτ0,1 .
If we introduce another set of variables {zj (j = 0, 1, . . .)} that play the same role as {yj } and set z¯ = z0 , y = −z1 , the corresponding τ -functions solve (4.71)–(4.77) simultaneously. We remark that the introduction of the variables {zj } corresponds to the symmetry of the 3-toroidal Lie algebra as mentioned in Section 4.1. To consider the reality condition for the SU (2)-gauge fields, we introduce an anti-automorphism κ as (4.84) κ ψn(α) = ψn(α)∗ , κ ψn(α)∗ = ψn(α) (n ∈ Z, α = 1, 2), which have the following properties:
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• κ2 = id, • vac|κ(g)|vac = vac|g|vac ,
∀
g ∈ SLtor 2 .
Using κ, we impose the following condition on g = g(y, z): κ(g(y, z)) = g(y, z).
(4.85)
Then we find that the τ -function (4.38) with x(1) = x(2) = 0 obeys
s1 , s2 |g(y, z)|s2 , s1 = s1 , s2 |g(y, z)|s2 , s1 ,
(4.86)
and that e, f and g of (4.70) satisfies f = −f, ˜ as If we define J ˜= J
ω 0
0 ω J ω −1 0
e = g. 0 , ω −1
(4.87)
1+i ω= √ , 2
(4.88)
˜ = tJ ˜ (See, for example, [Pr]). ˜ satisfies (4.69) and the reality condition J then J
5 Concluding remarks We have described the hierarchy structure associated with the (2 + 1)-dimensional NLS equation (1.2) based on the theory of the KP hierarchy, and discussed several methods to construct special solutions. Using the language of the free fermions, we have obtained the bilinear identities from the representation of the toroidal Lie algebras. The solutions constructed explicitly in this paper are limited in the class of soliton-type. In case of the hierarchy of the (2 + 1)-dimensional KdV equation (1.1), an algebro-geometric construction of the Baker-Akhiezer function is indeed possible [IT2]. It may be also possible to discuss algebro-geometric (“finite-band”) solutions for the (2 + 1)-dimensional NLS hierarchy by extending our construction of the soliton-type solutions. Furthermore, by extending our theory, it may be possible to consider (2 + 1)dimensional generalizations of other soliton equations, such as the sine-Gordon equation, the Toda lattice, and so on. We will discuss the subjects elsewhere.
Acknowledgments The authors would like to thank Dr. Yasuhiro Ohta, Dr. Yoshihisa Saito, Dr. Narimasa Sasa for their interests and discussions. The first author is partially supported by Waseda University Grant for Special Research Project 2000A-155, and the Grant-in-Aid for Scientific Research (No. 12740115) from the Ministry of Education, Culture, Sports, Science and Technology.
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Vol. 3, 2002
Hierarchy of (2 + 1)-Dimensional Nonlinear Schr¨ odinger Equation
Saburo Kakei Department of Mathematical Sciences School of Science and Engineering Waseda University Ohkubo 3-8-1, Shinjyuku-ku Tokyo 169-8555 Japan (Present address: Department of Mathematics Rikkyo University Nishi-ikebukuro 3-34-1, Toshima-ku Tokyo 171-8501 Japan email: [email protected]) Takeshi Ikeda Department of Applied Mathematics Okayama University of Science Ridaicho 1-1 Okayama 700-0005 Japan email: [email protected] Kanehisa Takasaki Department of Fundamental Sciences Faculty of Integrated Human Studies Kyoto University Yoshida, Sakyo-ku Kyoto 606-8501 Japan email: [email protected] Communicated by Tetsuji Miwa submitted 30/07/01, accepted 29/04/02
To access this journal online: http://www.birkhauser.ch
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Annales Henri Poincar´ e
Boundary WZW, G/H, G/G and CS theories K. Gaw¸edzki
Abstract. We extend the analysis [19] of the canonical structure of the Wess-ZuminoWitten theory to the bulk and boundary coset G/H models. The phase spaces of the coset theories in the closed and in the open geometry appear to coincide with those of a double Chern-Simons theory on two different 3-manifolds. In particular, we obtain an explicit description of the canonical structure of the boundary G/G coset theory. The latter may be easily quantized leading to an example of a twodimensional topological boundary field theory.
1 Introduction Bidimensional boundary conformal field theory is a subject under intense study in view of its applications to boundary phenomena in 1+1- or two-dimensional critical systems and to the brane physics in string theory. Although much progress has been achieved in understanding boundary CFT’s since the seminal paper of Cardy [6], much more remains to be done. The structure involved in the boundary theories is richer than in the bulk ones and their classification program involves new notions and an interphase with sophisticated mathematics [28][30]. One approach that offered a conceptual insight into the properties of correlation functions of boundary conformal models consisted of relating them to boundary states in threedimensional topological field theories [12][13]. In the simplest case of the boundary Wess-Zumino-Witten (WZW) models (conformal sigma models with a group G as a target [34]), the topological three-dimensional model appears to be the group G Chern-Simons (CS) gauge theory [32][35]. In [19] it has been shown how the relation between the boundary WZW model and the CS theory arises in the canonical approach. The purpose of the present paper is to extend the analysis of [19] to the case of the coset G/H models of conformal field theory obtained by gauging in the group G WZW model the adjoint action of a subgroup H ⊂ G. In the WZW model the simplest class of boundary conditions is obtained by restricting the boundary values of the classical G-valued field g to fixed conjugacy classes in the group labeled by weights of the Lie algebra g of G. Such boundary conditions reduce to the Dirichlet conditions for toroidal targets. It was shown in [19] that the phase space of the WZW model on a strip with such boundary conditions is isomorphic to the phase space of the CS theory on the time-line R times a disc D with two time-like Wilson lines. In the coset models we shall use more general boundary conditions requiring that field g belongs on the boundary components to
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pointwise product of group G and subgroup H conjugacy classes. The phase space of the coset theory on a strip with such boundary conditions becomes isomorphic to the phase space of the double CS theory on R × D with group G and group H gauge fields, both coupled to two time-like Wilson lines. In particular, the phase space of the boundary G/G coset model1 becomes isomorphic to the moduli space of flat connections on the 2-sphere with four punctures. The latter case lends itself easily to quantization giving rise to an example of the two-dimensional boundary topological field theory, a structure that promises to play the role of the K-theory of loop spaces [28]. Much of the motivation for the present work stemmed from interaction with Volker Schomerus who generously shared his insights with the author. The discussions with Laurent Freidel are also greatfully acknowledged. Special thanks are due to the Erwin Schr¨ odinger Institute in Vienna were this work was started. After the present work was finished, we have received the paper [9] which discussed the same boundary conditions in the coset theories that the ones proposed here.
2 Action functionals of the WZW and coset theories The Wess-Zumino-Witten model is a specific two-dimensional sigma model with a group manifold G as the target. For simplicity, we shall assume that G is compact connected and simply connected. We shall denote by g the Lie algebra of G. The G-valued fields of the WZW model are defined on two-dimensional surfaces Σ (the “worldsheets”) that we shall take oriented and equipped with a conformal or pseudo-conformal structure. The action of the model in the Euclidean signature is the sum of two terms: k ¯ + S W Z (g) . (2.1) S(g) = 4πi tr (g −1 ∂g)(g −1 ∂g) Σ
Above, tr stands for the Killing form normalized so that the long coroots have length squared equal to 2. The second (Wess-Zumino) term in the action is related to the canonical closed 3-form χ(g) = 13 tr (g −1 dg)3 on G. Informally, it may be written as k WZ (g) = 4πi g ∗ ω (2.2) S Σ
where ω is a 2-form on G such that dω = χ. This definition is, however, problematic since there is no global ω with the last property. If Σ has no boundary then the 1 As discussed in detail in [11], there are other ways to impose boundary conditions in the G/G theory relating it to the boundary topological Poisson sigma models of [7]
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problem may be solved by setting [34] S W Z (g) =
k 4πi
849
g∗ χ ,
(2.3)
B
where B is a 3-manifold such that ∂B = Σ and g extends g to B. It is well known that this determines S W Z (g) modulo 2πik so that the amplitudes exp[−S W Z (g)] are well defined if k is integer. By the Stokes Theorem, the definition (2.3) reduces to the naive expression (2.2) whenever g maps into the domain of a local form ω. The variation δS W Z (g) involves only the 3-form χ so that the classical equations are determined unambiguously. For surfaces with boundary, one should impose proper boundary conditions on fields g. Let ∂Σ = Sn1 where Sn1 are disjoint circles always considered with the orientation inherited from Σ. Let, for µ in the Cartan subalgebra of g, Cµ denote the conjugacy class {γ e2πiµ γ −1 | γ ∈ G}. We shall require that g(Sn1 ) ⊂ Cµn .
(2.4)
These are the so called fully symmetric conformal boundary conditions. When restricted to a conjugacy class Cµ , the 3-form χ becomes exact. In particular, the 2-form ωµ (g) = tr (γ −1 dγ) e2πiµ (γ −1 dγ) e−2πiµ
(2.5)
on Cµ satisfies dωµ = χ|Cµ . Let Σ = Σ#(Dn ) be the surface without boundary obtained from Σ by gluing discs Dn to the boundary components Sn1 of Σ, see Fig. 1.
Σ
Σ’ Fig. 1
Each field g satisfying the boundary conditions (2.4) may be extended to g : Σ → G in such a way that g (Dn ) ⊂ Cµn (the conjugacy classes are simply connected). Following [3][16], we shall define the WZ-action of the field g satisfying the boundary conditions (2.4) by setting ∗ k WZ WZ (g) = S (g ) − 4πi g ωµn . (2.6) S n D n
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This again reduces to the naive definition whenever g maps into the domain of a form ω such that dω = χ, provided that the restrictions of ω to Cµn coincide with ωµn . A different choice of the restrictions would change the boundary contributions to the classical equations. As explained in [16], (for k = 0) the right hand side of (2.6) is well defined modulo 2πi iff k is an integer and Cµn = Cλn /k for integrable weights2 λn . The boundary conditions are thus labeled by the same set as the bulk primary fields of the current algebra also corresponding to integrable weights. This is an illustration of the Cardy’s theory of boundary conditions [6]. The G/H coset theories [21] may be realized as the versions of the group G WZW theory where the adjoint action of the subgroup H ⊂ G has been gauged [5][17][23][18]. Let A denote a 1-form with values in ih, where h stands for the Lie algebra of H. For ∂Σ = ∅, the action of the theory coupled to the gauge field A = A10 + A01 is S(g, A) = S(g) −
k 2πi
¯ tr (g∂g −1 )A01 + A10 (g −1 ∂g)
Σ
+gA10 g −1A01 − A10 A01 .
(2.7)
In fact, getting rid of the so called “fixed point problem” [31][14] (that obstructs factorization properties of the theory) requires considering the WZW theory coupled to gauge fields in non-trivial H/Z-bundles, where Z is the intersection of the center of G with H [29][22]. For simplicity, we shall not do it here. For the surfaces with boundary ∂Σ = Sn1 , we shall use the same formula (2.7) to include the coupling to the gauge field, but we shall admit more general boundary conditions for the field g than the ones considered before. Namely, we shall assume that g|Sn1 = gn h−1 n
with gn : Sn1 → CµGn ,
hn : Sn1 → CνHn .
(2.8)
In other words, we shall admit fields g that, on each boundary component, are a pointwise product of loops in conjugacy classes of, respectively, group G and group H, keeping also track of the decomposition factors3 . We shall label such conditions by pairs (µn , νn ) ≡ Mn . For νn = 0, they reduce to the conditions considered in the previous section. We still need to generalize the definition of the Wess-Zumino term of the action to fields g satisfying (2.8). Such fields may be extended to maps g : Σ → G in such a way that g |Dn = gn hn−1 and 2 The
3 The
with
gn (Dn ) ⊂ CµGn ,
g |Sn1 = gn ,
hn |Sn1 = hn
hn (Dn ) ⊂ CνHn .
weights integrable at level k are the ones lying in the positive Weyl alcove inflated by k decomposition of elements of the pointwise product CµGn (CνHn )−1 might not be unique.
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We shall define then S W Z (g) = S W Z (g ) −
k 4πi
n
∗
∗
gn ωµGn − hn ωνHn
−1 −1 + tr (gn dgn )(hn dhn ) . Dn
(2.9)
−1
The form in the brackets has (gn hn )∗ χ as the exterior derivative which assures invariance of the right hand side under continuous deformations of gn and hn inside discs Dn . If k = 0 and H is simply connected then a slight extension of the argument in [16] shows that S W Z (g) given by (2.9) is well defined modulo 2πi iff k is integer and CµGn = CλGn /k , CνHn = CηHn /k , where λn and ηn are integrable weights of g and h, respectively. We shall use (2.1), (2.9) and (2.7) to define the complete gauged action S(g, A). With the above choices of the boundary labels, the gauge invariance exp[−S(hgh−1 , hAh−1 + hdh−1 )] = exp[−S(g, A)]
(2.10)
holds for h : Σ → H. If H has an abelian factor, then the same selection of boundary conditions is imposed if we add to the demand that exp[−S W Z (g)] be well defined the requirement of the gauge-invariance (2.10). For example, for the parafermionic SU (2)/U (1) coset theory, this restricts the boundary labels to pairs (λn , ηn ) = (jn σ3 , mn σ3 ) with jn = 0, 12 , . . . , k2 and mn = 0, 12 , . . . , k − 12 . The labels of the parafermionic primary states (j, m) have additional selection rule j = m mod 1 and the identification (j, m) ∼ ( k2 − j, m + k2 mod k). For the boundary labels, the first may be imposed by requiring the gauge invariance with respect to h : Σ → U (1)/Z2 and the second by identifying the decompositions −1 gn h−1 . Similarly, in the general case we may impose the local n and (−gn )(−hn ) H/Z gauge invariance and identify the decompositions differing by an element in Z [20]. Such restrictions lead to the same labeling of the boundary conditions and of the primary fields, but is not obligatory if we ignore the fixed point problem. Since the gauge field A enters quadratically into the action (2.7), it may be eliminated classically (and also quantum mechanically) from the equations of motion. What results is a sigma model with the space G/Ad(H) of the orbits of the adjoint action of H on G as the target. The target space G/Ad(H) (that may be singular) comes equipped with a specific metric, a non-meric volume form (“dilaton field”) and a 2-form. Let [g] denote the projection of g to G/Ad(H). The boundary conditions (2.8) restrict the boundary values of [g] to the projection to G/Ad(H) of the rotated G-group conjugacy class CµGn e−2πiνn (but contain more data if the decomposition gn h−1 n is not unique). For example, for G = SU (2) with elements ( z¯z
−z z¯
), where |z|2 + |z |2 = 1, and for H = U (1), the coset space
G/Ad(H) may be identified with the unit disc D = {z | |z| ≤ 1}. The boundary conditions (2.8) with (λn , νn ) corresponding to (jn , mn ) restrict the boundary
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values of [g] to the intervals [e 2πi(j−m)/k , e−2πi(j+m)/k ] ⊂ D with 2k end-points on the disc boundary. Since the conjugacy classes of U (1) are composed of single points, the decomposition gn h−1 n is unique in this case, given the conjugacy class labels. Imposing the restriction j = m mod 1 eliminates half of the interval endpoints [26].
3 Canonical structure of the WZW and coset theories The classical field theory studies the solutions of the variational problem δS = 0 determined by the action functional S. The space of solutions on a worldsheet with the product structure R × N and Minkowski signature admits a canonical closed 2-form Ω, see e.g. [15] or [19]. If the latter is degenerate (a situation in gauge theories, where the degenerate directions correspond to local gauge transformations), one passes to the space of leaves of the degeneration distribution. By definition, the resulting space is the phase space of the theory and it carries the canonical symplectic structure4 .
3.1
Bulk WZW model
Let us start with the well known case of the WZW model on the cylinder Σ = R × S 1 = {(t, x mod 2π)}. The variational equation δS = 0 becomes a non-linear version of the wave equation ∂+ (g −1 ∂− g) = 0 ,
(3.1)
where ∂± = ∂x± with x± = x±t. The solutions may be labeled by the Cauchy data g(t, · ) and (g −1 ∂t g)(t, · ). The space of solutions forms the phase space P W ZW of the bulk WZW model. Its canonical symplectic form is given by the expression [15] Ω
WZ
=
k 4π
2π tr − δ(g −1 ∂t g) g −1 δg 0
+ 2 (g −1 ∂+ g) (g −1 δg)2 (t, x) dx
(3.2)
which is t-independent5 . Similarly as for the wave equation, the general solution of (3.1) may be decomposed as g(t, x) = g (x+ ) gr (x− )−1 .
(3.3)
The left-right movers g,r : R → G are not necessarily periodic but satisfy g,r (y + 2π) = g,r (y) γ for the same γ ∈ G. They are determined uniquely up 4 We ignore here the eventual problems with the infinite-dimensional character of the spaces and singularities that may be usually dealt with in concrete situations 5 We use the symbol δ for the exterior derivative on the space of classical solutions
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to the simultaneous right multiplication by an element of G. The expression of the symplectic form in terms of the left-right movers is described in Appendix A. The currents J = ik g ∂+ g −1 = ik g ∂+ g−1 ,
Jr = ik g −1 ∂− g = ik gr ∂+ gr−1
(3.4)
generate the current algebra symmetries of the theory. The conformal symmetries are generated by the components T =
1 2k
tr J2 ,
Tr =
1 2k
tr Jr2
(3.5)
of the energy momentum tensor.
3.2
Bulk G/H model
In the same cylindrical worldsheet geometry Σ = R × S 1 , the classical equations for the coset G/H model take the form: D+ (g −1 D− g) = 0 , E g −1 D− g = 0 = E g D+ g −1 , F (A) = 0 ,
(3.6)
where D± = ∂± + [A± , · ] are the light-cone covariant derivatives, E is the orthogonal projection of g onto h and F (A) = dA + A2 is the curvature of A. The equations are preserved by the H-valued gauge transformations of the fields. The gauge transformations provide for the degeneration of the canonical closed 2-form on the space of solutions so that the phase space P G/H of the bulk coset theory is composed of the gauge-orbits of the solutions of the classical equations (3.6). The gauge-orbits of solutions may be parametrized in a more effective way. The flat gauge field A may be expressed as h−1 dh for h : R2 → H such that h(t, x + 2π) = ρ−1 h(t, x) for some ρ ∈ H. The map h is determined uniquely up to the left multiplication by an element of H. Let us set g = hgh−1 . Note that g : R2 → G with g(t, y + 2π) = ρ−1 g(t, y) ρ. In terms of field g, the classical equations reduce to ∂+ ( g −1 ∂− g) = 0 ,
E g−1 ∂− g = 0 = E g ∂+ g−1 .
(3.7)
The gauge-orbits of the classical solutions of (3.6) are in ono-to-one correspondence with the orbits of pairs ( g , ρ) under the simultaneous conjugation by elements of H. In terms of these data, the canonical symplectic form on P G/H , obtained following the general prescriptions of [15], is given by Ω
G/H
=
k 4π
2π tr − δ( g −1 ∂t g) g−1 δ g + 2 ( g −1 ∂+ g) ( g −1 δ g )2 (t, x) dx 0
k g(t, 0)−1 (δρ)ρ−1 + 4π tr (δρ)ρ−1 g(t, 0)
g)(t, 0) − ((δ g ) g −1 )(t, 0) − ( g −1 δ
(3.8)
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for any fixed t. The solutions of the classical equations (3.7) may be expressed again by the left-right movers: g(t, x) = g (x+ )gr (x− )−1 , where g,r : R → G are such that −1 = 0 E g,r ∂y g,r
and
g,r (y + 2π) = ρ−1 g,r (y) γ .
(3.9)
Given g , the one-dimensional fields g,r are determined up to the simultaneous right multiplication by an element of G. The expression for the symplectic form ΩG/H in terms of the left-right movers is given in Appendix A. The left-right components of the energy-momentum tensor T = − 2 tr (g D+ g −1 )2 = − 2 tr (g ∂+ g−1 )2 , k
k
(3.10) Tr =
k −2
tr (g
−1
2
D− g)
=
k −2
tr (gr ∂+ gr−1 )2
generate the conformal symmetries of the bulk coset model.
3.3
Bulk G/G model
For the topological coset G/G theory, the classical equations (3.7) reduce to g−1 d g = 0, i.e. g is constant and it commutes with the monodromy ρ. The phase space P G/G may be identified with the space of commuting pairs ( g , ρ) in G modulo simultaneous conjugations. It is finite-dimensional, in agreement with the topological character of the theory. It comes equipped with the symplectic form k (3.11) g − g −1 δ g − (δ g ) g −1 . ΩG/G = 4π tr (δρ)ρ−1 g−1 (δρ)ρ−1 Up to a simultaneous conjugation, g = e 2πi µ and ρ = e 2πi ν for µ and ν in the Cartan algebra and the symplectic form becomes a a constant form on the product of two copies of the Cartan algebra. ΩG/G = 2π k tr [dν dµ] .
(3.12)
In particular, conjugation-invariant functions of g Poisson-commute and so do those of ρ.
3.4
Boundary WZW model
The canonical treatment of the boundary theories is quite analogous to that of the bulk ones, except for the necessity to treat the boundary contributions. We consider the strip geometry Σ = R × [0, π] with Minkowski signature and impose on the field g : Σ → G of the WZW model the boundary conditions discussed in Sect. 2: g(t, 0) ∈ Cµ0 ,
g(t, π) ∈ Cµπ .
(3.13)
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For variations δg tangent to the space of fields respecting conditions (3.13), the classical equations δS(g) = 0 reduce to the bulk equation (3.1) supplemented with the boundary equations g −1 ∂− g + g ∂+ g −1 = 0
for x = 0, π .
(3.14)
The classical solutions obeying (3.13) form the phase space PµW0 µZπ of the boundary WZW model. Its symplectic form is given by [19] Z ΩW µ0 µπ =
π
k 4π
tr − δ(g −1 ∂t g) g −1 δg + 2(g −1 ∂+ g) (g −1 δg)2 (t, x) dx
0
+
k 4π
ωµ0 (g(t, 0)) − ωµπ (g(t, π))
(3.15)
for any fixed t. As in the bulk, the classical equations may be solved explicitly, as was described in [19]. We have6 g(t, x) = g (x+ ) m0 g (−x− )−1 = g (x+ ) mπ g (2π − x− )−1 ,
(3.16)
where m0 ∈ Cµ0 , mπ ∈ Cµπ and g : R → G satisfy g (y + 2π) = g (y) γ
for
γ = m−1 0 mπ .
(3.17)
Note that the boundary conditions (3.13) are fulfilled. The orbits of the triples (g , m0 , mπ ) under the right multiplication of g by elements of G accompanied by the inverse adjoint action on m0 and mπ are in one-to-one correspondence with the classical solutions. The expression of the symplectic form in terms of these data is given in Appendix A. The boundary WZW theory has a single current 1 tr J 2 . J = ik g ∂+ g−1 with the corresponding energy-momentum tensor T = 2k
3.5
Boundary G/H model
For the boundary coset G/H model with the G-valued field g and ih-valued gaugefield A defined on the strip R × [0, π], we shall impose the boundary conditions g(t, 0) = g0 (t) h0 (t)−1 ,
g(t, π) = gπ (t) hπ (t)−1
(3.18)
with g0 , h0 , gπ and hπ mapping the boundary lines into the conjugacy classes CµG0 , CνH0 , CµGπ and CνHπ , respectively, see (2.8). The gauge fields A will not be restricted. We shall label such boundary conditions by the pairs (M0 , Mπ ), where M0 = (µ0 , ν0 ) and Mπ = (µπ , νπ ). The variational equations δS(g, A) = 0 reduce now to the bulk equations (3.6) supplemented with the boundary equations (g
−1
−1 h−1 0 D t h0 = 0 = hπ D t hπ , −1 D− g)( · , 0) + h0 (g D+ g )( · , 0) h−1 = 0, 0
(g −1 D− g)( · , π) + hπ (g D+ g −1 )( · , π) h−1 = 0, π where Dt = D+ − D− is the covariant derivative along the boundary. 6 We
use here a slightly different parametrization of the solutions than in [19].
(3.19) (3.20) (3.21)
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The flat gauge field A may be gauged away by representing it as h−1 dh for h mapping the strip to G. Setting as in the bulk geometry g = hgh−1 and, on the boundary components, g0 = hg0 h−1 , h0 = hh0 h−1 , and similarly for gπ amd hπ , we reduce the bulk equations to (3.7) and the boundary equations to g)( · , 0) + n0 ( g ∂+ g−1 )( · , 0) n−1 = 0, ( g −1 ∂− 0
(3.22)
g ∂+ g−1 )( · , π) n−1 = 0, ( g −1 ∂− g)( · , π) + nπ ( π
(3.23)
with h0 and hπ equal, respectively, to constant elements n0 ∈ CνH0 and nπ ∈ CνHπ . The boundary conditions (3.18) become: g(t, 0) = g0 (t) n−1 0
g(t, π) = gπ (t) n−1 π
(3.24)
for g0 mapping the line into CµG0 and gπ into CµGπ . As in the bulk case, the G/H phase space PM of the G/H coset model with the boundary conditions (3.18) is 0 Mπ composed of the gauge-orbits of the classical solutions. The latter are in one-to-one correspondence with the orbits of the triples ( g, n0 , nπ ) under the simultaneous conjugation by elements of H. In this parametrization, the symplectic form of the boundary theory is given by G/H
ΩM0 Mπ =
k 4πi
+
k 4πi
−
k 4πi
π
tr − δ( g −1 ∂t g) g−1 δ g + 2( g −1 ∂+ g) ( g −1 δ g )2 (t, x) dx
0 ωµG0 ( g0 (t)) − ωνH0 (n0 ) + tr ( g0−1 δ g0 )(t) n−1 0 δn0 ωµGπ ( gπ (t)) − ωνHπ (nπ ) + tr ( gπ−1 δ gπ )(t) n−1 π δnπ
(3.25)
for any fixed t. Similarly as in the case of the WZW model, see (3.16) and (3.17), the twodimensional fields g satisfying the classical equations (3.22), (3.23) and the boundary conditions (3.24) may be rewritten in terms of a one-dimensional field g as + − −1 −1 nπ g(t, x) = g (x+ ) m0 g (−x− )−1 n−1 0 = g (x ) mπ g (2π − x )
(3.26)
with m0 ∈ CµG0 , mπ ∈ CµGπ and g : R → G satisfying E g ∂y g−1 = 0
and
g (y + 2π) = ρ−1 g (y) γ
−1 for ρ = n−1 0 n π , γ = m0 mπ .
(3.27)
Given ( g , n0 , nπ ), the triple (g , m0 , mπ ) is determined up to the right multiplication of g by an element of G accompanied by the adjoint action of its inverse may be rewritten in terms of the on m0 and mπ . The symplectic form ΩG/H M0 Mπ data (g , m0 , n0 , mπ , m0 , nπ ). The result is given by formula (A.2) in Apendix A. The single energy-momentum component of the boundary G/H model is T = − k2 tr (g ∂+ g−1 )2 .
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Boundary G/G model
In the boundary topological coset G/G theory, the classical equations imply that h−1 h−1 g0 , h0 , gπ , hπ and g = g0 gπ are all constant so that the phase space π 0 = G/G PM0 Mπ of the boundary G/G theory is composed of the orbits under simultaneous conjugations of the quadruples h0 , gπ , hπ ) ∈ Cµ0 × Cν0 × Cµπ × Cνπ ⊂ G4 ( g0 , with
h−1 h−1 g0 = gπ π . 0
(3.28)
The symplectic form is given by k = 4πi ωµ0 ( g0 ) − ων0 ( h0 ) + tr ( g0−1 δ g0 ) h−1 ΩG/G 0 δ h0 M0 Mπ k − 4πi ωµπ ( gπ ) − ωνπ ( hπ ) + tr ( gπ−1 δ gπ ) h−1 π δ hπ .
(3.29)
Using this expression, one may check that conjugation invariant functions of g Poisson commute. Below, we shall find a more transparent interpretation for the phase spaces of the two-dimensional theories described above, including the last example.
4 Canonical structure of the CS theory The classical Chern-Simons theory [32, 35] is determined by the action functional of ig-valued 1-forms A (gauge fields, connections) on an oriented 3-manifold M k 2 CS S (A) = − 4π tr A dA + 3 A3 (4.1) M
that does not require a metric on M for its definition.
4.1
Case without boundary
If M has no boundary then, under the G-valued gauge transformations g : M→ G k g∗χ . (4.2) S CS (gAg −1 + g dg −1 ) = S CS (A) − 4π M
In particular, the action is invariant under gauge transformations homotopic to CS 1 and, for integer k, e−S (A) is invariant under all gauge transformations. The classical equations δS CS = 0 are well known to require that F (A) = 0 with the solutions corresponding to flat connections. In the cylindrical geometry M = R × Σ with ∂Σ = ∅, the canonical closed 2-form on the space of solutions is
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degenerate along the gauge directions. Writing A = A + A0 dt where A is tangent to Σ and t is the coordinate of R, we may use the gauge freedom to impose the condition A0 = 0. In this gauge, the classical equations reduce to ∂t A = 0 ,
F (A) = 0
(4.3)
with the solutions given by static flat connections on the surface Σ. The canonical closed 2-form on the space of solutions becomes k tr (δA)2 . (4.4) ΩCS = 4π Σ
Its degeneration is given by the static gauge transformations A → gAg −1+ g dg −1 . The phase space P CS of the CS theory is then composed of the gauge-orbits of flat gauge fields A. In other words, P CS coincides with the moduli space of flat connections on Σ. Formula (4.4) defines the canonical symplectic structure on P CS . Below, we shall need several refinements of the above well known scheme.
4.2
Wilson lines
First of all, the CS theory may be coupled to a Wilson line C ⊂ M marked with a label µ belonging to the Cartan subalgebra of g. Let γ be a G-valued map defined on the line C. In the presence of these data, the action functional is modified to CS CS S (A, γ) = S (A) + ik tr µ γ −1 (d + A)γ C
= S CS (gAg −1 + g dg −1 , gγ) for g homotopic to 1. The corresponding classical equations read F (A) = 2πi γµγ −1 C , d(γµγ −1 ) + [A, γµγ −1 ] C = 0 ,
(4.5)
(4.6)
where C is viewed as a singular current. They imply that A is a flat connection with a singularity on C. In the cylindrical geometry M = R × Σ with the Wilson line R × {ξ} we may still go to the A0 = 0 gauge in which the classical equations reduce to ∂t A = 0 ,
∂t (γµγ −1 ) = 0 ,
F (A) = 2πi γµγ −1 δξ .
(4.7)
The canonical 2-form ΩCS on the space of solutions has now the form µ = ΩCS − ik tr µ(γ −1 dγ)2 ΩCS µ
(4.8)
where the last term is the Kirillov-Kostant 2-form on the (co)adjoint orbit Oµ in g passing through µ. The orbits of pairs (A, γµγ −1 ) solving (4.7) under the time-
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independent gauge transformations7 form the phase space PµCS of the theory. ΩCS µ defines on PµCS the canonical symplectic structure. Of course, one may consider the CS theory with several Wilson lines.
4.3
Boundaries
If the 3-manifold M has a boundary then one needs to impose boundary conditions on the gauge fields A. In the cylindrical geometry M = R × Σ where ∂Σ = ∅, we may require that A0 = 0
on
R × ∂Σ .
(4.9)
The classical equations are still given by F (A) = 0 and the closed 2-form by (4.4). The only modification is that the degeneration of the latter is given now by the gauge transformations equal to 1 on ∂Σ. The same remarks pertain to the case with time-like Wilson lines.
4.4
Double CS theory
The last modification of the CS theory on a 3-manifold M = R × Σ that we shall need is the double theory [29] with a pair (A, B) of the, respectively, group G and group H ⊂ G gauge fields. The action functional of the double theory is the difference of the CS actions for group G and H : S 2CS (A, B) = S S (A) − S CS (B) .
(4.10)
On the boundary R × ∂Σ we shall impose the boundary conditions (1 − E) A0 = 0 ,
E A0 = B0 ,
E Aτ = Bτ ,
(4.11)
where Aτ denotes the component of A tangent to ∂Σ. The phase space of the double theory P 2CS is composed of the pairs (A, B) of flat connections on Σ satisfying the last condition of (4.11), modulo G-valued gauge transformations of A and H-valued ones of B that coincide on the boundary of Σ. The symplectic form k 2CS (4.12) = 4π tr (δA)2 − (δB)2 . Ω Σ
Clearly, both gauge fields may be coupled to time-like Wilson lines with labels in the Cartan subalgebras of g and h, respectively. In the particular case when with H = G, the double CS theory reduces to the single one on the space R × Σ = Σ#(−Σ) obtained by gluing Σ along the boundary to its the double surface Σ copy with reversed orientation. The phase spaces reduce accordingly. 7 In fact, the singular terms in (4.7) require some care. A possible way is to consider only solutions of (4.7) that around ξ are of the form A = i γµγ −1 dϕ , where ϕ denotes the argument of a local complex parameter and to admit the gauge transformations that are constant around ξ. Different choices of local parameters lead then to canonically isomorphic phases spaces.
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5 Symplectic relations between the WZW, coset and CS theories The symplectic structure of the phase spaces of the WZW and coset theories is given by the complicated expressions, see (3.2), (3.8), (3.11), (3.15), (3.25), (3.29), (A.1) and (A.2). Although obtained by applying the general procedures of [15][19], these expressions are far from being transparent. On the other hand, the interpretation of the symplectic structure of the phase spaces of the CS theory determined by the standard constant symplectic form on the space of two-dimensional gauge fields and by the Kirillov-Kostant form on the coadjoint orbits, see (4.4), (4.8) or (4.12), has a clear interpretation. In the present section, we shall describe symplectic isomorphisms between the phase spaces of the WZW and coset theories and those of the CS theory, elucidating this way the canonical structure of the first ones. The existence of such isomorphisms for the bulk WZW and coset theories has been known for long time, see [35][8][29]. We only give their slightly more explicit realization. The isomorphism of the boundary WZW theory phase space with a moduli space of flat connections on a twice punctured disc has been first described in [19]. It represents another aspect of the relations between the boundary conformal theories and the topological three-dimensional theories developed in [12][13].
5.1
Bulk WZW model
The bulk WZW model on R × S 1 corresponds to the CS theory on R × Z, where Z = {z | 12 ≤ |z| ≤ 1}, with the boundary condition A0 = 0. The isomorphism I between the phase spaces P CS and P W Z of the two theories is defined by the formula giving the classical solution of the WZW model on R × S 1 in terms of a flat connection on Z : A
g(t, x) = P e x,t
,
(5.1)
···
where P e stands for the path-ordered (from left to right) exponential and x,t is an appropriate contour, see Fig. 2. In particular, x,0 is a radial segment from eix to 12 eix and x,t is obtained from x,0 by rotating continuously the beginning of the segment by angle t and its end by angle −t. It is not difficult to see that I is a symplectic isomorphism, see Appendix B. In terms of the CS gauge field A, + − the currents (3.4) become J (x+ ) = ik Aϕ (eix ) and Jr (x− ) = −ik Aϕ ( 12 eix ), where Aϕ denotes the angular component of A.
5.2
Bulk G/H model
For the bulk G/H coset model, the corresponding CS theory is the double one on R × Z. Recall that the phase space of the latter is formed by the gauge-orbits of pairs (A, B) of, respectively, group G and group H flat connections on Z whose components tangent to the boundary are related by EAϕ = Bϕ . Choose a base
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ei(x+t)
1 2
1
1ei(x−t) 2
l x,t
Fig. 2 point 1 ∈ Z. We may consider w =
1 i
ln z =
z 1
dz iz
as the coordinate on the
covering space Z of Z. Let us inroduce two maps gA and hB from Z to G and H , respectively, 1 gA (w) = P e
z
−1
1
A
,
hB (w) = P e
z
B
.
(5.2)
−1
Clearly A = gA dgA and B = hB dhB . We shall set g(t, x) = hB (x + t)−1 gA (x + t) gA (x − t + w0 )−1 hB (x − t + w0 ) for w0 = i ln 2. Note that g(t, x + 2π) = ρ ρ = P e
−1
(5.3)
g (t, x) ρ where
B
(5.4)
C
with C the clock-wise contour around the unit circle from 1 to 1. It is straightforward to check that g satisfies the classical equations (3.7) of the bulk G/H coset model. Under the G-valued gauge transformations of A and H-valued ones of B that agree on the boundary of Z, the pair ( g , ρ) undergoes a simultaneous conjugation by a fixed element of H. We infer that (5.3) defines an injective map I from the phase space P 2CS of the double CS theory on R × Z to the phase space P G/H of the bulk coset G/H model. Using the parametrization of the solutions g of the coset model by the left-right movers g (y) = h−1 (y)gA (y) , B
gr (y) = hB (y + w0 )−1 gA (y + w0 ) ,
satisfying (3.9) for ρ given by (5.4) and γ = P e
C
(5.5)
A
,
(5.6)
it is easy to see that the map I is also onto. The main result is that it defines a symplectic isomorphism, see Appendix B.
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Bulk G/G model
In the special case of H = G where the phase space P 2CS reduces to that of the gauge-orbits of flat connections on the torus represented as the double surface Z#(−Z), the field g of (5.3) is (t, x)-independent and it describes the parallel transport around the a-cycle, see Fig. 3.
1
a b
Fig. 3 Similarly, the monodromy ρ, commuting with g, describes the parallel transport around the b-cycle. Equations (3.11) and (3.12) express then the symplectic form (4.4) in terms of the holonomy of the gauge field and is a special case of the result of [2]. Note that conjugation-invariant functions of the holonomy around a fixed cycle on the torus Z#(−Z) Poisson-commute.
5.4
Boundary WZW model
The case of the boundary WZW model with the boundary conditions (3.13) has been analyzed in [19]. The corresponding CS theory is the one on the solid cylinder R × D where D is the unit disc in the complex plane, with two time-like Wilson lines, say R × { 21 } with label µ0 and R × {− 21 } with label −µπ , see Fig. 4. i(x+t) e
l x,t
1 2
lπ
l0
1 2
i(t−x) e
Fig. 4
Vol. 3, 2002
Boundary WZW, G/H, G/G and CS Theories
863
Let A be a flat connection on D with F (A) = 2πi γ0 µ0 γ0−1 δ 12 − 2πi γπ µπ γπ−1 δ− 12 .
(5.7)
Its holonomy around the contour 0 of Fig. 4 lies then in the conjugacy class Cµ0 and around π in Cµπ . To each such connection, we may associate a classical solution of the WZW theory on a strip R × [0, π] by setting A
g(t, x) = P e x,t
,
(5.8)
with the contour x,t as in Fig. 4. In particular, for t = 0, x,0 goes from eix to e−ix crossing once the interval (− 12 , 12 ). For other times, x,t is obtained by rotating both ends of x,0 by the angle t. Note how the boundary conditions (3.13) are assured. The bulk equations (3.1) and the boundary ones (3.14) are also satisfied. The right hand side of (5.8) is clearly invariant under the gauge transformations of A equal to 1 on the boundary of the disc. The decomposition (3.16) of the solution in terms of the one-dimensional field g is obtained by setting A
g (y) = P e y
A
,
m0 = P e 0
A
mπ = P e π
,
.
(5.9)
Here for y ∈ [0, π], the contour y coincides with the interval [eiy , 0] and it is deformed continuously for other values of y. Contours 0 and π are as in Fig. 4. We obtain this way an isomorphism Iµ0 µπ from the phase space PµCS(−µπ ) of the CS 0 of the theory on R×D with two time-like Wilson lines onto the phase space PµW0 µZW π boundary WZW theory. As was explained in [19], Iµ0 µπ preserves the symplectic structure. We sketch in Appendix B the idea of the proof. In [19], this result was used to quantize the boundary WZW theory.
5.5
Boundary G/H model
Finally, let us consider the coset G/H theory on the strip R × [0, π] with the (M0 , M1 ) boundary conditions (3.18). It corresponds to the double CS theory on R×D coupled to Wilson lines. The group G gauge field is coupled to lines R×{ 21 } and R × {− 21 } with labels µ0 and −µπ and the group H gauge field to the same lines with labels ν0 and −νπ , respectively. Let us define A
gA (y) = P e y
B
,
hB (y) = P e y
with the contour y as in (5.9), and A
m0 = P e
0
A
, mπ = P e
π
, n0 = P e
0
,
(5.10)
B
, nπ = P e
π
B
.
(5.11)
864
K. Gaw¸edzki
Ann. Henri Poincar´e
The monodromy of gA and hB is given by: gA (y + 2π) = gA (y) γ
for
γ = m−1 0 mπ ,
hB (y + 2π) = hB (y) ρ
for
ρ = n−1 0 nπ .
(5.12)
Setting g (y) = hB (y)−1 gA (y) we obtain a one-dimensional field satisfying (3.27) and describing via (3.26) a classical solution g(t, x) of the boundary G/H coset theory with the (M0 , Mπ ) boundary conditions. Clearly, g is invariant under the gauge transformations of A and B equal on the boundary of the disc. We obtain this 2CS between the phase space PM of the double CS way an isomorphism IM (−Mπ ) 0 Mπ 0
G/H of the theory on R × D with two pairs of Wilson lines and the phase space PM 0 Mπ boundary coset model. The proof that IM0 Mπ preserves the symplectic structure is similar to the one in the case of the boundary WZW model, see Appendix B.
5.6
Boundary G/G model
In the special case H = G, the phase space of the double CS theory reduces to that of the single theory on the 2-sphere S 2 = D#(−D) with four Wilson lines: R × { 21 } and R × {− 21 } in R × D with labels µ0 and −µπ and their images in h0 ∈ Cν0 , hπ ∈ Cνπ R × (−D) with labels −ν0 and νπ . The group elements and −1 −1 and g = g0 h0 = gπ hπ , see (3.28), are given by the contour integrals A
h0 = P e 0
A
,
hπ = P e π
A
,
g = P e
,
(5.13)
where 0 and π are the copies in −D of 0 and π , see Fig. 4, and is the closed contour as in Fig. 5 starting and ending at the center of −D, with the broken
l 1 2
1 2
Fig. 5 pieces contained in −D and the solid ones in D. The equality of the symplectic form on the moduli space of flat connections A on S 2 with four punctures to the form of (3.29) is essentially again a special case of the result of [2].
Vol. 3, 2002
Boundary WZW, G/H, G/G and CS Theories
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6 Quantization of the boundary G/G coset model The description of the canonical structure of the two-dimensional WZW and coset theories in terms of the moduli spaces of flat connections on surfaces with boundary may be further reduced by the “topological fusion” to the case of a disc with a single insertion and of a closed surface with multiple insertions. This provides a good starting point for the quantization of the theory. Indeed, quantization of the moduli spaces on a disc with a single puncture is an example of the orbit method in the representation theory [25]. It gives rise to the highest weight representations of the current algebra [1][8]. On the other hand, quantization of the moduli spaces of flat connections on closed surfaces leads to the finite-dimensional spaces of conformal blocks of the WZW theories [35] that may be also viewed as spaces of invariant tensors of the iπ quantum group Uq (g) for q = e k+g∨ (g ∨ stands for the dual Coxeter number of G). This was described in detail for the boundary WZW theory with G = SU (2) in [19]. Here we shall carry out the quantization program for the boundary coset theory G/G. Recall that the canonical structure of the G/G model has been described directly in terms of the moduli spaces of flat connections on closed surfaces so that no topological fusion will be needed. As a result, we shall obtain an example of a two-dimensional boundary topological field theory, a structure that has recently attracted some attention [24][27][28]. Let us start from the well known case of the bulk G/G coset model. As explained above, the phase space of the theory coincides with the moduli space of flat connections on the torus Z#(−Z). The quantization of this moduli space gives rise to the space of the conformal blocks H of the the WZW theory on the torus8 . The space is spanned by the affine characters χkλ of level k of the associated to Lie algebra g or by the characters χλ of irreducible current algebra g representations of group G of integrable highest weights λ restricted to the points9 ζ+ρ ζ ≡ 2π k+gW∨ for ζ running through the integrable weights and ρW standing for the Weyl vector. As is well known, the restricted characters induce under pointwise multiplication a commutative ring Rk , the fusion ring of the WZW theory. This way, the space H of the conformal blocks on the torus becomes a commutative algebra with unity H ∼ = Rk ⊗ C. The unit element is given by the character χ0 ≡ 1 of the trivial representation. H may be viewed as the algebra of functions on the discrete set {ζ | ζ integrable} = Spec(H). The operator of multiplication g ) on the by the restricted character χλ is the quantizations of the function χλ ( phase space P G/G . We may equip H with a non-degenerate symmetric bilinear 8 One way to proceed is to use the natural K¨ ahler structure of the moduli space given by its identification with the moduli space of holomorphic vector bundles, see e.g. [10] 9 We view the characters as functions on the Cartan algebra identified with functions on the group via the exponential map
866
K. Gaw¸edzki
Ann. Henri Poincar´e
a1 a2 , a3 = a1 , a2 a3
(6.1)
form such that10 χλ , χλ = δλ¯ λ satisfying 1, 1 = 1 ,
for ai ∈ H. These are the data of a two-dimensional bulk topological field theory [4]. Such a theory assigns to each compact, not-necessarily connected, oriented surface Σ with the boundary ∂Σ = Sn1 a linear functional IΣ : ⊗ H → C, the n
amplitude of Σ. This assignment is supposed to have three properties. First, I is supposed to be multiplicative under disjoint products: IΣ1 Σ2 = IΣ1 ⊗ IΣ2 .
(6.2)
Second, it should be be covariant with respect to surface homeomorphisms preserving orientation. This means that IΣ = IΣ π for any permutation π of factors in ⊗ H within the connected components of Σ so that IΣ depends only on the coln
lection of the numbers of boundary circles and handles in each component. Third, I is required to be consistent with the gluing of boundary components: IΣnm = IΣ Pnm
(6.3)
if Σnm is obtained from Σ by gluing together the n-th and the m-th boundary components of opposite orientation. Pnm = id ⊗ . . . ⊗ pγ ⊗ . . . ⊗ pγ . . . ⊗ id γ n m where pγ ⊗ pγ ∈ H ⊗ H (6.4) P = γ
is the dual of the bilinear form · , · (we may take pγ = χλ , pγ = χλ¯ ). It is easy to see that if Σ is connected and has g handles then
(6.5) an , P g . IΣ ⊗ an = n
n
In particular, the surfaces of Fig. 6 (with the orientation inherited from the plane) correspond to the amplitudes a → a, 1 ,
a1 ⊗ a2 → a1 , a2 ,
a1 ⊗ a2 ⊗ a3 → a1 a2 , a3 .
(6.6)
They permit to reconstruct the amplitudes for all surfaces. For the bulk G/G theory and an = χλn , expression (6.5) is equal to an integer N(λn ) (g), the Verlinde dimension of the space of conformal blocks on Σ with insertions of the primary fields with labels λn . Explicitly [33], ζ 2−2g ζ ζ N(λn ) (g) = (S0 ) (Sλn /S0 ) , (6.7) ζ
n
10 λ ¯ denotes the highest weight of the representation of G complex conjugate to the one with the highest weight λ.
Vol. 3, 2002
Boundary WZW, G/H, G/G and CS Theories
867
Fig. 6 ζ
λ
where S λ = S ζ = S ζλ¯ are the elements of the matrix giving the modular transformation of the affine characters χkλ . For closed surfaces, IΣ is an integer N (g). It is equal to 1 for the sphere and to dim(H), i.e. the number of integrable weights, for the torus. In short, the bulk topological G/G coset theory is the theory of the Verlinde dimensions. They may all be obtained from the fusion coefficients
ζ ζ Nλη = Nληζ¯ (0) which define the product in H: χλ χη = Nλη χζ . We would like to extend this structure to the case of the boundary G/G coset theory with the (M0 , Mπ ) boundary conditions where M0 = (µ0 , ν0 ) and G/G Mπ = (µπ , νπ ). Recall that we have identified the phase space PM of this 0 M1 2CS 2 of flat connections on S = D#(−D) theory with the moduli space PM 0 (−Mπ ) with four punctures labeled by µ0 , −ν0 , −µπ and νπ , all in the Cartan algebra of G. For k a positive integer and µ0 = λ0 /k, ν0 = η0 /k, µπ = λπ /k, νπ = ηπ /k, 2CS where λ0 , η0 , λπ , ηπ are integrable weights, the phase space PM gives upon 0 (−Mπ ) quantization the space of conformal blocks of the WZW theory on D#(−D) with ¯ π in D and by η¯0 and ηπ in insertions of the primary fields labeled by λ0 and λ −D. We shall denote this space by HL0 Lπ with L0 = (λ0 , η0 ) and Lπ = (λπ , ηπ ). By the factorization properties of the spaces of conformal blocks, HL0 Lπ ∼ = ⊕ Hom(Hλπ η¯π ζ , Hλ0 η¯0 ζ )
(6.8)
ζ
where Hληζ denotes the space of conformal blocks on S 2 with insertions of three primary fields labeled by the integrable weights λ, η and ζ. In particular, HLL is an associative (in general, non-commutative) algebra with unity, a direct sum of matrix algebras. More generally, there is a natural bilinear product HL1 L2 × HL2 L3 → HL1 L3 defined by composition of homomorphisms in each ζ-component. It gives HL1 L2 the structure of a left HL1 L1-module and of a right HL2 L2-module. It is useful to consider the direct sum of the boundary spaces Hb = ⊕ HL1 L2 . L1 ,L2
The product in Hb defined by ab = =
L
aL1L bLL2
(6.9)
868
K. Gaw¸edzki
Ann. Henri Poincar´e
for a = (aL1 L2 ) and b = (bL1 L2 ) makes Hb an associative algebra with unity 1 = (δL1 L2 ). Each space HL1 L2 is, additionally, a module of the commutative algebra H with the character χλ ∈ Rk acting diagonally in the decomposition (6.8) as the This action quantizes the classical observables χ ( multiplication by χλ (ζ). g ), λ where, in the CS description, g is is the holonomy around the countour on Fig. 5, see (5.13). The induced structure of the H-module on Hb satisfies a (b c) = (a b) c = b (a c)
(6.10)
for a ∈ H and b, c ∈ Hb which is equivalent to the statement that a b = (a 1) b and that elements a1 are in the center of Hb . We shall equip Hb with a non-degenerate symmetric bilinear form · , · b with the only non-vanishing matrix elements between subspaces with permuted boundary labels, i.e. such that a, b b = aL1 L2 , bL2 L1 b . (6.11) L1 ,L2
Explicitly, we shall set: aL1 L2 , bL2 L1 b =
ζ
(±S 0 ) tr [ aL1 L2 (ζ) aL2 L1 (ζ) ] ,
(6.12)
ζ
where the sign is fixed once for all. It is easy to see that the bilinear form · , · b satisfies a b, c b = a, b c b .
(6.13)
The last relation, together with the symmetry of the form implies the cyclic symmetry a b, c b = b c, a b = c a, b b . Let PL1 L2 =
A
(6.14)
A
pL1 L2 ⊗ pL2 L1 ∈ HL1 L2 ⊗ HL2 L1
(6.15)
A
be the dual of the bilinear form (6.12) on HL1 L2×HL2 L1 . We may take A = (ζ, i, j) and A
ζ
−1 2
pL1 L2 = (±S 0 )
eiL
1ζ
A
e∗j , L ζ
ζ
−1 2
pL2 L1 = (±S 0 )
2
ejL
2ζ
e∗i , L ζ 1
(6.16)
where, for L = (λ, η), (eiLζ ) is a basis of Hληζ and (e∗i ) is the dual basis. The ¯ Lζ bilinear forms on Hb and on H are tied together by the relation A A aL1 L1 pL1 L2 bL2 L2 , pL2 L1 b = aL1 L1 , pγ 1 b pγ 1 , bL2 L2 b . (6.17) A
γ
Vol. 3, 2002
Boundary WZW, G/H, G/G and CS Theories
Indeed, with the use of (6.16), the left hand side may be rewritten as tr [ aL1 L1 (ζ)] tr [ bL2 L2 (ζ)]
869
(6.18)
ζ
and the right hand side is ζ S ζ χ ¯ (ζ ) tr [ a S 0 χλ (ζ) (ζ)] tr [ bL2 L2 (ζ )] . L1 L1 λ 0
(6.19)
λ,ζ,ζ
Note that the sign ambiguity in the definition (6.12) of the bilinear form on Hb disappears from both expressions. The equality of the two sides is inferred by using ζ = S ζ , S ζ χ ¯ (ζ) = S ζ and the unitarity of the modular the relations S 0 χλ (ζ) λ 0 λ λ ζ matrix (Sλ ). We may abstract from the above construction an algebraic structure H, · , · , Hb , · , · b (6.20) such that 1. H is a finite-dimensional associative commutative algebra with unity equipped with the non-degenerate symmetric bilinear form · , · , 2. Hb = ⊕HL1 L2 is a finite-dimensional associative algebra with unity (in general non-commutative) equipped with the non-degenerate symmetric bilinear form · , · b , 3. Hb is an H-module with each HL1 L2 being a submodule, 4. relations (6.1), (6.9), (6.11), (6.13), (6.10) and (6.17) hold. Such a structure defines a boundary two-dimensional topological field theory [24] [27] [28]. The amplitudes of such a theory correspond11 to compact oriented surfaces Σ with boundary where the boundary components Sn1 may contain distinguished closed disjoint subintervals (possibly the whole component) marked with labels L of the boundary conditions, see Fig. 7. Let, for each Sn1 with labeled subintervals, (Ins ) be the collection of the remaining subintervals of Sn1 . The amplitude assigned to such a labeled surface is a linear functional (6.21) IΣ : ⊗ H ⊗ ⊗ Hb → C , n
n,s
where the first tensor product is over the boundary components without labeled subintervals. IΣ is required to vanish on the all the components HL1 L2 of Hb except those with (L1 , L2 ) given by the labels of the intervals adjacent to Ins . For example, the amplitude of the labeled surface of Fig. 8 is a linear functional on 11 There are minor differences between our formulation and that of the above references, mostly a matter of convenience. In particular, we consider only boundary orientations induced from the bulk.
870
K. Gaw¸edzki
Ann. Henri Poincar´e
L3 L1
Σ
L2
Fig. 7 L4
..
.
L3
LS
L2 L1
Fig. 8 H ⊗ HL1 L2⊗ HL2 L3⊗ . . . ⊗ HL L1 . The amplitude assignment I is still required S to obey (6.2) and to be covariant under orientation and label preserving homeomorphisms. The latter means that IΣ = IΣ π for cyclic permutations of boundary intervals and their labels within boundary circles. The amplitudes depend this way on the collections of boundary labels with the cyclic order within each boundary circle (including the empty collection). The consistency with gluing (6.3) is now generalized to include the gluing along two unlabeled intervals of opposite orientation and permuted labels of the adjacent intervals as in Fig. 9. In the latter case the dual bilinear form P ∈ H ⊗ H should be replaced in (6.3) by the dual form PL1 L2 ∈ HL1 L2 ⊗ HL2 L1 inserted in the appropriate factors of the tensor product ⊗ Hb . n,s
It is not difficult to construct the amplitudes IΣ from the data (6.20). For completeness, we shall describe the argument. First, besides the bulk amplitudes already discussed, it is enough to know only the amplitudes corresponding to labeled surfaces of Fig. 10 aLL → aLL , 1 b ,
aL1 L2 ⊗ bL2 L1 → aL1 L2 , bL2 L1 b ,
a ⊗ aL1 L2 ⊗ bL2 L3 ⊗ cL3 L1 → aL1 L2 bL2 L3 , a cL3 L1 b .
Vol. 3, 2002
Boundary WZW, G/H, G/G and CS Theories
... L 2
...
L1
L2 . . .
...
...
L2
L1
...
L1 . . .
871
...
Fig. 9
L
L3
1
L
L2
1
L
L2 Fig. 10
Gluing the unlabeled disc to the inner boundary of the annulus gives the disc with three labeled boundary intervals and the amplitude aL1 L2 ⊗ bL2 L3 ⊗ cL3 L1 → aL1 L3 bL3 L2 , cL2 L1 b .
(6.22)
We may subsequently glue such a disc to the annulus as in Fig. 11 to obtain the amplitude a ⊗ aL1 L2 ⊗ bL2 L3 ⊗ cL3 L4 ⊗ dL4 L1 A A
→ aL1 L2 bL2 L3 , a pL3 L1 b pL1 L3 cL3 L4 , dL4 L1 b A
= aL1 L2 bL2 L3 cL3 L4 , a dL4 L1 b ,
(6.23)
where the last equality follows from the trivial identity A
A
A
aL1 L2 pL2 L1 b pL1 L2 bL2 L1 b = aL1 L2 , bL2 L1 b .
(6.24)
872
K. Gaw¸edzki
Ann. Henri Poincar´e
L4
L1
L3
L2 Fig. 11 One obtains similarly the amplitudes of the general annuli of Fig. 8. They are given by the linear functionals 1
S
a ⊗ aL 1 L 2 ⊗ · · · ⊗ aL
S
L1
→
1
S−1
aL 1 L 2 · · · aL 1
S−1
S
L
S
, a aL
S
L1
b
(6.25)
S
invariant under the cyclic permutations of aL1 L2 · · · aL L1 due to (6.10) and (6.14). S The formula extends to the cases with S = 1 and S = 0 corresponding to the surfaces depicted in Fig. 12 if we interpret it as giving the linear maps a ⊗ aLL →
L
L
Fig. 12 aLL , a 1LL and a → 1LL , a 1LL b , respectively, where 1LL stands for the unity of HLL . For a general surface, we may obtain its amplitude by first cutting off the labeled boundary circles around nearby unlabeled ones as in Fig. 13, and then composing the amplitude from that of the annuli of Fig. 8 and of the ones for the surface with unlabeled boundary. It is easy to show that the resulting amplitudes are consistent with the gluing of surfaces. For unlabeled surfaces, this is a well known fact. For surfaces glued along two unlabeled boundary intervals in boundary components with labels, there are two different cases.
Vol. 3, 2002
Boundary WZW, G/H, G/G and CS Theories
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Σ L2
L3
L5
L4
L1 Fig. 13 If the glued intervals are in two different boundary components, then the consistency boils down to the case of Fig. 14 and it follows with the use of (6.24).
L4
L3
L’1
=
L’3
L1
L’1
... ...
L1
L’2
L2
...
...
L2
L3
...
L4
L’2
L’4
L’3
L’4
Fig. 14 The case when one glues two intervals in the same boundary components may be similarly reduced to the check that the gluings of Fig. 15 give the same result. The first one leads to the amplitude 1
S
S
1
→ a ⊗ aL 1 L 2 ⊗ · · · ⊗ aL L 1 ⊗ b L L ⊗ · · · ⊗ b L L S 1 2 S 1 1 S A 1 S aL 1 L 2 · · · aL L p L L b L L · · · b L L , a p A
b . L L S
A
1
1
1
1
S
2
1
1
1
(6.26)
The second one results in 1
S
S
1
a ⊗ aL 1 L 2 ⊗ · · · ⊗ aL L 1 ⊗ b L L ⊗ · · · ⊗ b L L
→ S 1 2 S 1 1 S−1 S 1 S −1 S aL1 L2 · · · aL L , pγ aL L b · bL L · · · , bL , pγ a bL L γ
S−1
S
S
1
1
2
S −1
S
S
L 1
b . (6.27)
874
K. Gaw¸edzki
L3’
L3
L’S’ L’1
L6
=
L2
LS
L4
L3’
...
...
L’2
L’4
...
L6
L5
L4
L’2
L’4 L’1
LS
L1
L3
’ LS’
...
L5
Ann. Henri Poincar´e
L2 L1
Fig. 15
=
Fig. 16
The equality of both expressions follows from (6.17). Conversely, the amplitudes (6.21) of a two-dimensional boundary topological field theory define the data (6.20). First, the amplitudes of Fig. 6 determine the unity, the bilinear form and the product in H. The commutativity of the latter follows from the homeomorhism covariance of the amplitudes that allows to permute the two inner discs of the third surface of Fig. 6. The associativity of the product in H results from the equality of the two ways to glue the amplitudes for the sphere without four discs presented in Fig. 16. The first of the relations (6.1) is equivalent to the normalization of the amplitude of the sphere S 2 to 1 and the second follows again from the homeomorphismcovariance of the amplitudes. Similarly, the amplitudes of the first two surfaces of Fig. 10 give the unit elements 1LL ∈ HLL and the bilinear form pairing HL1 L2 and HL2 L1 . The amplitude of the third surface applied to 1 ∈ H, together with the bilinear form · , · b , determine the product HL1 L2 × HL2 L3 → HL1 L3 in such a way that the cyclic invariance (6.14) holds. The associativity is proved similarly as before by equating two ways of gluing a disc with four labeled boundary intervals from pairs of discs with three labeled boundary intervals, see Fig. 17. The action of the elements of the bulk space H on the boundary space Hb is obtained from the amplitude of the annulus of Fig. 18 with the use of the bilinear
Vol. 3, 2002
Boundary WZW, G/H, G/G and CS Theories
L
875
L
4
4
L
L
1
3
=
L
L3
1
L2
L2 Fig. 17
L1
L2
Fig. 18 form · , · b . By definition, this action preserves the subspaces HL1 L2 ⊂ Hb . The proof that it defines a representation of the commutative algebra H in Hb follows from Fig. 19. Similarly, relations (6.10) follow from Fig. 20 and (6.17) from Fig. 15 with S = S = 1. We obtain this way the algebraic structure (6.20) possessing all the four properties listed. We shall call a two-dimensional topological field theory unitary if there exist anti-linear involutions C : H → H and Cb : Hb → Hb with Cb (HL1 L2 ) = HL2 L1 such that the sesqui-linear forms C · , · and Cb · , · b define scalar products on H and Hb and that 5. C(ab) = (Ca)(Cb) , Cb (a b) = (Cb b)(Cb a) , Cb (a b) = (Ca)(Cb b) for a, b ∈ H and a, b ∈ Hb . The last three properties guarantee that I−Σ = IΣ
⊗ C ⊗ ⊗ Cb , n
n,s
(6.28)
where −Σ denotes the surface with the reversed orientation and, conversely, they follow from (6.28). For the G/G theory, one may take for C the complex conjugation of functions of integrable weights and for Cb the hermitian conjugation of
876
K. Gaw¸edzki
L
L2
1
Ann. Henri Poincar´e
L1
L2
=
Fig. 19
L3
L2
L1
L3
L2
L1
L2
L3
L1
Fig. 20 linear transformations in HL0 Lπ , see (6.8), relative to some scalar product in the spaces Hληζ of three-point conformal blocks. One obtains then a unitary topologζ ical field theory provided the sign in (6.12) is chosen so that ±S 0 > 0. Recall that, due to (6.10), the elements a1LL for a ∈ H are in the center of HLL . Following [24], we shall call the boundary condition L irreducible if all the elements of the center of HLL are of this form. This is the case in the G/G theory. To each boundary condition L one may associate a state aL ∈ H using the amplitude of the second surface of Fig. 12 and the bilinear form on H. Explicitly, aL is defined by demanding that 1LL , a 1LL b = aL , a
(6.29)
for all a ∈ H. We shall call the family of boundary conditions (L) complete if the states (aL ) span H. In the G/G theory, for L = (λ, η), = N (S ) aL (ζ) 0 λη ¯ ζ
ζ
−1
(6.30)
and the completeness is easy to see by taking, for example, the conditions with L = (λ, 0). On the other hand, the diagonal subfamily of boundary conditions corresponding to L = (λ, λ) is, in general, not complete since not all integrable weights appear in the fusion of pairs of complex conjugate weights (e.g. for G = SU (2), a( j,j) (j ) vanishes for half-integer spins j ).
Vol. 3, 2002
Boundary WZW, G/H, G/G and CS Theories
877
The bulk topological theories may be perturbed by “massive” topological perturbations. For example, in the SU (2)/SU (2) model such perturbations permit to establish a relation with twisted minimal N = 2 topological theories. One of the interesting open problems for future research is how to extend such relations to the case of the boundary G/G theory.
A
Appendix
When expressed in terms of the left and right movers, the symplectic form of the bulk G/H coset theory becomes:
Ω
G/H
k 4π
=
2π tr (g−1 δg ) ∂y (g−1 δg ) − (gr−1 δgr ) ∂y (gr−1 δgr ) dy tr (δρ)ρ−1 ((δg )g−1 )(0) − ((δgr )gr−1 )(0)
0
−
k 4π
+ g (0)−1 (δρ)ρ−1 g (0) − gr (0)−1 (δρ)ρ−1 gr (0) − (g−1 δg )(0) + (gr−1 δgr )(0) (δγ)γ −1 .
(A.1)
The expression for the bulk WZW model symplectic form ΩW ZW may be obtained from the latter by setting ρ identically to 1. Similarly, the expression in terms of the left-mover g for the boundary G/H model symplectic form becomes: G/H
ΩM0 Mπ =
k 4π
2π tr (g−1 δg ) ∂y (g−1 δg ) dy +
k 4π
−1 tr (δn0 )n−1 0 (δnπ )nπ
0
−1 −1 − (δm0 )m−1 ((δg ) g−1 )(0) 0 (δmπ )mπ − (δρ)ρ
+ (g−1 δg )(0) (δγ)γ −1 − (δρ)ρ−1 g (0) (δγ)γ −1 g (0)−1 + ωµG0 (m0 ) − ωνH0 (n0 ) − ωµGπ (mπ ) + ωνHπ (nπ )
(A.2)
ZW is and the expression for the boundary WZW model symplectic form ΩW µ0 µπ obtained by setting ρ, n0 and nπ identically to 1.
B Appendix Our proof of the fact that the isomorphisms I , I , Iµ0 µπ and IM between the 0 Mπ WZW and G/H phase spaces and the CS ones preserve the symplectic structure is based on a direct calculation of the form tr (δA)2 and of its counterpart for the Σ
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gauge field B, very much in the spirit of a similar calculation [2] for closed surfaces. Consider first the bulk case that is somewhat simpler. It is enough to examine the case of the G/H coset theory which for H = {1} reduces to the WZW model. With gA given by (5.2), we have (B.1) tr (δA)2 = d tr (gA−1 δgA ) d(gA−1 δgA ) . Integrating the last expression over the annulus Z cut along the interval [ 12 , 1],
z
z
1 2
1
1 2
0
1 2
1
Fig. 21 see Fig. 21, and using the Stokes theorem, we infer that k 4π
2
tr (δA) Z
=
k 4π
2π tr (gA−1 δgA )(y) ∂y (gA−1 δgA )(y)
− (gA δgA )(y + w0 ) ∂y (gA−1 δgA )(y + w0 ) dy 0
−1
k − 4π tr (δγ)γ −1 (gA−1 δgA )(0) − (gA−1 δgA )(w0 ) ,
(B.2)
where the second line is the contribution from the integrals along the cut. Similar expression holds for B, hB and ρ replacing A, gA and γ, respectively. Subtracting both formulae, we obtain an expression for the symplectic form of the double CS theory on Z which may be shown to coincide with the right hand side of (A.1) by using (5.5) and the second equality of (3.9). The case of the boundary G/H coset model may be treated similarly. We define for z in the unit disc D cut along the sub-interval [− 21 , 1] of the real axis
+i0
gA (z) = P e
z
+i0
A
,
hB (z) = P e
z
B
.
(B.3)
hB (eiy ) = Note that for y ∈ (0, 2π) we have the equalities gA (eiy ) = gA (y) and hB (y) for gA and hB given by (5.10).
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Similarly as before, −1 tr (δA)2 = lim tr ( gA δ gA ) d( gA−1 δ gA , D
→0 ∂D
879
(B.4)
where D is the cut unit disc without -discs around ± 12 , see Fig. 21. A tedious but straightforward calculation results in the formula for the symplectic structure of the double CS phase space that coincides with equation (A.2). We leave the details to the reader just stressing that a more direct and conceptual proof of equality between the canonical structures of two-dimensional CFT’s and three-dimensional topological field theories would be welcome.
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A. Yu. Alekseev and V. Schomerus, D-branes in the WZW model, Phys. Rev. D 60, R061901-R061902 (1999).
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M. F. Atiyah, Topological quantum field theory, Publ. IHES 68, 175–186 (1989).
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S. Elitzur, G. Moore, A. Schwimmer and N. Seiberg, Remarks on the canonical quantization of the Chern-Simons-Witten theory, Nucl. Phys. B 326, 104–134 (1989).
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S. Elitzur and G. Sarkissian, D-branes on a gauged WZW model, Nucl. Phys. B 625, 166–178 (2002).
[10] F. Falceto and K. Gaw¸edzki, Chern-Simons States at Genus One, Commun. Math. Phys. 159, 549–579 (1994). [11] F. Falceto and K. Gaw¸edzki, Boundary G/G theory and topological PoissonLie model, Lett. Math. Phys. 59, 61–79 (2002).
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[12] G. Felder, J. Fr¨ ohlich, J. Fuchs and C. Schweigert, Conformal boundary conditions and three-dimensional topological field theory, Phys. Rev. Lett. 84, 1659–1662 (2000). [13] G. Felder, J. Fr¨ ohlich, J. Fuchs and C. Schweigert, Correlation functions and boundary conditions in RCFT and three-dimensional topology, Compos. Math. 131, 189–237 (2002). [14] J. Fuchs, B. Schellekens and C. Schweigert, The resolution of field identification fixed points in diagonal coset theories, Nucl.Phys. B 461, 371–406 (1996). [15] K. Gaw¸edzki, Classical origin of quantum group symmetries in Wess-ZuminoWitten conformal field theory, Commun. Math. Phys. 139, 201–213 (1991). [16] K. Gaw¸edzki, Conformal field theory: a case study. In: Conformal Field Theory, Frontiers in Physics 102, eds. Nutku, Y., Sa¸clioglu, C., Turgut, T., Perseus Publishing, Cambridge Ma. 2000, pp. 1–55. [17] K. Gaw¸edzki and A. Kupiainen, G/H conformal field theory from gauged WZW model, Phys. Lett. B 215, 119–123 (1988). [18] K. Gaw¸edzki and A. Kupiainen, Coset construction from functional integrals, Nucl. Phys. B 320, 625–668 (1989). [19] K. Gaw¸edzki, I. Todorov and P. Tran-Ngoc-Bich, Canonical quantization of the boundary Wess-Zumino-Witten model, arXiv:hep-th/0101170. [20] K. Gaw¸edzki and N. Reis, in preparation. [21] P. Goddard, A. Kent and D. Olive, Unitary representations of the Virasoro and super-Virasoro algebras, Commun. Math. Phys. 103, 105–119 (1986). [22] K. Hori, Global aspects of gauged Wess-Zumino-Witten models, Commun. Math. Phys. 182, 1–32 (1996). [23] D. Karabali, Q. Park, H. J. Schnitzer and Z. Yang, A GKO construction based on a path integral formulation of gauged Wess-Zumino-Witten actions, Phys. Lett. B 216, 307–312 (1989). [24] C. I. Lazaroiu, On the structure of open-closed topological field theory in two-dimensions, Nucl. Phys. B 603, 497–530 (2001). [25] A. Kirillov, Elements of the Theory of Representations, Berlin, Heidelberg, New York, Springer 1975. [26] J. Maldacena, G. Moore and N. Seiberg, Geometrical interpretation of Dbranes in gauged WZW models, arXiv:hep-th/0105038.
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[27] G. Moore, Some Comments on Branes, G-flux, and K-theory, Int. J. Mod. Phys. A 16, 936–944 (2001). [28] G. Moore, Santa Barbara lectures, http://online.itp.ucsb.edu/online /mp01. [29] G. Moore and N. Seiberg, Taming the conformal Zoo, Phys. Lett. B 220, 422–430 (1989). [30] V. B. Petkova and J.-B. Zuber, Conformal boundary conditions and what they teach us, arXiv:hep-th/0103007. [31] A. N. Schellekens and S. Yankielowicz, Field identification fixed points in the coset construction, Nucl. Phys. B 334, 67–102 (1990). [32] A. Schwarz, The partition function of degenerate quadratic functional and Ray-Singer invariants, Lett. Math. Phys. 2, 247–252 (1978). [33] E. Verlinde, Fusion rules and modular transformations in 2D conformal field theory, Nucl. Phys. B 300 [FS 22], 360–376 (1988). [34] E. Witten, Non-abelian bosonization in two dimensions, Commun. Math. Phys. 92, 455–472 (1984). [35] E. Witten, Quantum field theory and the Jones polynomial, Commun. Math. Phys. 121, 351–399 (1989). Krzysztof Gaw¸edzki C.N.R.S. Laboratoire de Physique ENS-Lyon 46, All´ee d’Italie F-69364 Lyon France email: [email protected] Communicated by Jean-Bernard Zuber submitted 03/12/01, accepted 08/04/02
To access this journal online: http://www.birkhauser.ch
Ann. Henri Poincar´e 3 (2002) 883 – 894 c Birkh¨ auser Verlag, Basel, 2002 1424-0637/02/050883-12
Annales Henri Poincar´ e
Sign of the Monodromy for Liouville Integrable Systems R. Cushman and V˜ u Ngoc San Abstract. In this note we show that the monodromy of a two degree of freedom integrable Hamiltonian system has a universal sign in the case of a focus-focus singularity. We also show how to extend the monodromy index to several focusfocus fibers when the integrable system has an S 1 symmetry.
1 Introduction The Hamiltonian monodromy of integrable systems has a surprisingly recent history dating back to Duistermaat’s 1980 article [8]. Its application to quantum spectra was suggested in 1988 [5]. But it was not before 1998 – with the rigorous quantum formulation [17] and several examples [3], [7], [14], [10] (and others) – that it became a common tool for the analysis of spectra of many mathematically and physically relevant models (eg. [19]). (Quantum) Hamiltonian monodromy is usually used to demonstrate the nonexistence of global action variables (or good quantum numbers). This can be detected by a sort of “point defect” in the lattice of joint eigenvalues. The goal of our note is to sharpen this analysis by showing that this point defect can be attributed a sign, and in the generic case this sign is always positive (theorem 1). Moreover, as a first step in the study of systems with several isolated singularities, in theorem 3 we show how to compute the global monodromy in case of an S 1 symmetry (ie. one global action). A consequence of this sign for general systems without S 1 symmetry is that the global monodromy can cancel only for systems with complicated topology (proposition 5). We apply our results to a simple example with two points of monodromy: the quadratic spherical pendulum, for which we have also numerically computed the joint spectrum.
2 General Setup Let M be a 4-dimensional connected symplectic manifold with symplectic form ω, let B be a 2-dimensional manifold, and let F : M → B be a smooth proper surjective Lagrangian fibration with singularities which has connected fibers. We
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assume that the set of critical values ci of F is discrete and that each critical point of F in F −1 (ci ) is a focus-focus singularity. Recall that a point m ∈ M with dF (m) = 0 is called a focus-focus singularity if there exist local canonical coordinates (x, y, ξ, η) ∈ (T ∗ R2 , ω = dξ ∧ dx + dη ∧ dy) near m and a local chart of B at F (m) such that the vector space spanned by the Hessians D2 F1 (m) and D2 F2 (m) (where (F1 , F2 ) are the components of F ) is generated by the standard focus-focus quadratic forms (q1 , q2 ): q1 = xξ + yη
q2 = xη − yξ.
Recall that any critical point of F of Morse-Bott type (=“non-degenerate” in the sense of [9]) whose critical value is isolated in B is of focus-focus type. We are mainly interested in the case where F comes from a Liouville integrable system. Here B is a connected subset of R2 and F = (H1 , H2 ), where Hi are Poisson commuting Hamiltonians. Typically, M is a connected open subset of where F may have non focus-focus critical points, see a symplectic manifold M [9].
3 Monodromy Let Br = B \ {ci } be the set of regular values of F and denote by Fr the restriction of F to Mr = F −1 (Br ). Then Fr is a regular Lagrangian fibration over Br with compact connected fibers. In a local chart of Br the fibration Fr = (H1 , H2 ) is a Liouville integrable system. By the Arnold-Liouville theorem, the fibers of F are affine 2-torii on which the flows of the Hamiltonian vector fields XH1 and XH2 define a linear action of T2 . The 2-torus bundle Fr : Mr → Br obtained this way is locally trivial. In fact it is locally a principal 2-torus bundle. The obstruction for it to be globally a principal bundle is the monodromy µ. More precisely, monodromy is the holonomy of a Z2 -bundle over Br whose fiber is the lattice of 2π-periodic vector fields, which in a local chart on Br about c are given by linear combinations of XH1 and XH2 whose flow on F −1 (c) is 2π-periodic. For more details, see [8], [4, Appendix D]. Let P → Br be this bundle of period lattices. Then the monodromy µ ∈ Hom(π1 (Br ), Aut(P)). Given a point c ∈ Br , a period lattice Pc with basis {X1 , X2 } and a loop γ in Br passing through c, the monodromy µc (γ) is a matrix in Gl(2, Z), whose conjugacy class in Gl(2, Z) is invariant under a change of basis. If γ encircles a single critical value c of Fr , then there is a basis B such that the monodromy is the unipotent matrix 1 0 , (1) k 1 see [20], [6]. Here k is a nonzero integer called the monodromy index of γ relative to the basis B. The absolute value |k| is invariant under conjugation by elements
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of Gl(2, Z) and hence is independent of the choice of basis B. We call |k| the absolute monodromy index. In [2], [12] and [20] it was shown that this latter index is precisely the number of focus-focus critical points in F −1 ( c). Moreover, F −1 ( c) is homeomorphic to a |k|-pinched 2-torus.
4 Oriented monodromy Suppose now that Br is oriented, which is indeed the case when Br is an open subset of R2 . Then there is a induced orientation on the Liouville torii and hence on the bundle of period lattices P. This induced orientation is determined as follows. Let {α1 , α2 } be a positively oriented ordered basis of Tc∗ Br , which is dual to a positively oriented basis of Tc Br . Then the ordered basis of tangent vectors to F −1 (c) given by the set of vector fields {ω (F ∗ (α1 ), ω (F ∗ (α2 )} is said to be positively oriented. In the case of our two degree of freedom Liouville integrable system, if we use the standard orientation on R2 , then {XH1 F −1 (c) , XH2 F −1 (c) } gives the induced positive orientation for the 2-torus F −1 (c). We define the oriented monodromy index of the oriented loop γ in Br around the focus-focus critical value c to be the integer k in (1) when the basis chosen to compute it is positively oriented. The number k is invariant under conjugation by orientation preserving automorphisms. When referring to the oriented monodromy index of a focus-focus critical value c we assume that γ is positively oriented. Remark In this article we use the convention of (1) to write the monodromy matrix as a lower triangular matrix (instead of an upper triangular one), which amounts to a sign convention for k. Theorem 1 The oriented monodromy index of a focus-focus critical value is positive and hence is equal to the number of focus-focus critical points in the critical fiber. Proof. Using Eliasson’s theorem [9] one can find a chart near a focus-focus critical point (which corresponds to the critical value 0) so that F = g(q1 , q2 ), where g is a local diffeomorphism of R2 , and q1 = xξ + yη, q2 = xη − yξ, where (x, ξ, y, η) are coordinates for R4 with symplectic form dx ∧ dξ + dy ∧ dη. Using the symplectomorphism (x, ξ, y, η) → (−x, −ξ, y, η) one may change the sign of q2 , if necessary, to ensure that the ordered basis {Xq1 , Xq2 } is positively oriented. In other words, we can ensure that the local diffeomorphism g is orientation preserving, that is, det Dg(0) > 0. Following [18] we can choose a point c near the critical value 0 and an ordered basis B of the form {α Xq1 + β Xq2 , Xq2 }, where α, β > 0, for which the monodromy matrix is
1 1
0 1
.
Since B has the same orientation as the ordered basis {Xq1 , Xq2 }
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and hence as the ordered basis {XH1 , XH2 }, we see that the monodromy index is positive. Note that theorem 1 is purely local, since a small enough neighborhood of a focus-focus critical value is always orientable. Making no orientability assumptions on Br , theorem 1 can be phrased as follows. Theorem 1 (bis) The monodromy index k of a loop in Br around a single focusfocus critical value is positive if and only if the loop and the basis chosen to compute k have the same orientation.
5 Parallel transport The fibration Fr : Mr → Br endows Br with an integral affine structure, whose charts are the action coordinates. This affine structure induces a parallel transport on T Br , whose holonomy is the contragredient of the holonomy of the 2-torus bundle P → Br , that is, the monodromy. For more details see [1]. Suppose that c1 and c2 are two critical values of F that can be joined by a path Γ : [0, 1] → B such that Γ : (0, 1) → Br . Assume that a neighborhood of Γ ci in the positive sense. in B is orientable and fix a small loop γi which encircles We obtain Corollary 2 The monodromy index of γ1 with respect to some basis B has the same sign as the monodromy index of γ2 computed with respect to a basis obtained by parallel transport of B. Proof. The holonomy of the affine manifold B being dual to the monodromy, has determinant 1. Hence parallel transport is orientation preserving.
6 Case of S 1 symmetry Locally, a focus-focus singularity always admits an S 1 symmetry. However this symmetry does not in general extend globally, in particular when several critical fibers are present. This issue will be discussed in section 7. We show in this section how to extend the oriented monodromy index to several focus-focus points when the fibration F has a global S 1 symmetry. Here B is oriented an connected. Let G be the monodromy group of the regular fibration (= the image under µ of the fundamental group π1 (Br )). For any c ∈ Br , G acts on the lattice H1 (F −1 (c), Z) Z2 .
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Theorem 3 Suppose that B is oriented, connected and simply connected. Then the following properties are equivalent 1. each element of G has a non-trivial fixed point in H1 (F −1 (c), Z); 2. there is a non-trivial X ∈ H1 (F −1 (c), Z) that is fixed by G; 3. G is Abelian; 4. there is a symplectic S 1 action on (M, ω) that preserves the fibration F ; 5. there is a Hamiltonian S 1 action on (M, ω) that preserves the fibration F ; 6. there is a unique group homomorphism µ ¯ : π1 (Br ) → Z such that for any γ ∈ π1 (Br ) the monodromy µ(γ)with respect to a positively oriented basis is 1 k
conjugate in Sl(2, Z) to
0 1
with k = µ ¯(γ).
Proof. First note that properties 1 and 2 are of course independent of the choice of the base point c. We choose an oriented basis of H1 (F −1 (c), Z), which allows 2 us to identify G with a subgroup of Sl(2, Z) acting on Z . In this proof we shall denote by Mk the matrix
1 k
0 1
.
The first three assertions are simple properties of Sl(2, Z). Proof of 1=⇒2. Let g0 be a non-trivial element of G, and g be any element of G. Since g0 , g and g0 g have all 1 in their spectrum, they all have trace equal to 2. Because we can find an integral eigenvector of g0 , there is an integral basis of Z2 in which g0 = Mk (k = 0) and g = ac db . But Tr(g0 g) = a + kb + d = 2 + kb
which implies b = 0. Then g must have the form Mc . In other words the second element of our basis is necessarily a common eigenvector for all g ∈ G. Proof of 2=⇒3. Complete X into an integral basis of Z2 . Then all g ∈ G have the form Mk(g) in this basis. Hence they commute, by virtue of the formula Mk Mk = Mk+k .
(2)
Proof of 3=⇒1. The fundamental group π1 (Br ) is generated by the set γ1 , . . . , γn , where γi is a small loop around a single focus-focus critical value. Since Br is connected these loops can be deformed in Br to pass through the point c. Hence the corresponding monodromy transformations µi = µ(γi ) generate G. Since they are all trigonalizable (they are conjugate to Mk for some k) and G is Abelian, they are simultaneously trigonalizable. Now the product law (2) implies property 1. Proof of 2=⇒4. Recall that H1 (F −1 (c), Z) is isomorphic to the period lattice Pc : in a local chart of Br where F = (H1 , H2 ), the periodic vector fields on the torus F −1 (c) of the form xXH1 + yXH2 for constant x and y are determined uniquely by the homology class of any of their orbits.
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Thus we identify X with its representant in Pc . By parallel transport it locally extends to a flat local section of P, that is, a 2π-periodic vector field X on F −1 (U ), where U is a small neighborhood of c ∈ Br . The 1-form iX ω is invariant under the joint flow of F , and hence is of the form F ∗ β, for a 1-form β on U . By Liouville-Arnold theorem, dβ = 0, hence X is symplectic. Since by hypothesis the action of the monodromy group G on X is trivial, X can be extended to a global section of the bundle of period lattices P over Br . The 2π-flow of this vector field defines a symplectic S 1 action on F −1 (Br ) preserving the fibration. At the focus-focus singularity m, the period lattice no longer exists. However near m there is a unique 2π-periodic Hamiltonian vector field (with prescribed orientation) that is tangent to the Lagrangian foliation (see for instance [17]). Hence the above S 1 action extends uniquely to a global S 1 action on M preserving the fibration F . Note that this shows that the 1-form iX ω is the pull-back by F of a global closed 1-form β on B. Proof of 4=⇒5. Let Φ be the symplectic S 1 action and let X be the infinitesimal generator of Φ. Since Φ is symplectic, X is locally Hamiltonian. Since F is preserved by Φ, X is locally constant on the leaves in any action-angle coordinates. Hence X is actually a section of P above Br . Hence we are in the situation of the proof above, and there is a closed 1-form β on B such that iX ω = F ∗ β. Since H 1 (B) = 0, β is exact, namely β = dL. Hence X = XF ∗ L is a Hamiltonian vector field on (M, ω). Thus the S 1 action Φ is Hamiltonian on (M, ω) with momentum map L ◦ F . Proof of 5=⇒6. As in the proof above, we let L be a smooth function on B such that X = XL◦F , where X is the generator of the S 1 action. Since L is a global action, the 1-form dL is invariant under parallel transport on T ∗ Br defined by the integral affine structure on Br . Thus X is fixed by the monodromy group G: hence the hypotheses of assertion 2 are satisfied. Recall the choice of generators γi in the proof of 3=⇒1. Then X can be completed to an integral basis of Pc in which for all i, µ(γi ) = Mki for some ki ∈ Z. We define µ ¯ to be the homomorphism that ¯ realizes assigns to a loop γ = γi1 · · · γip the integer k = ki1 + · · · + kip . Note that µ an isomorphism between G and d Z where d is the gcd of (k1 , . . . , kn ). Proof of 6=⇒1. Obvious, since any matrix of the form Mk has a fixed point. Corollary 4 Suppose that there is a global Hamiltonian S 1 action on (M, ω) preserving F . Then the monodromy index along an embedded, positively oriented loop γ in Br increases with the number of focus-focus critical values inside γ. In particular it can never cancel out. Proof. Each each focus-focus critical value adds a positive integer to the global monodromy index.
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7 Vanishing of the monodromy Some integrable systems do not have an S 1 action. For instance if B is a sphere this would contradict corollary 4, since a loop around all focus-focus critical values would be contractible. However, even without an S 1 action, it is not easy to have the monodromy cancel along an embedded loop, as shown in the following proposition. Proposition 5 Assume that B is oriented, connected and simply connected. Let γ be an embedded loop in Br such that the monodromy along γ is trivial. Let n be the number of focus-focus critical values inside γ, and suppose that they are all simple: their index is 1. Then n is a multiple of 12. Proof. This is a consequence of the following lemma. See also Moishezon [13, p.179]. Lemma 6 Suppose that there are matrices A1 , A2 , . . . , An in Sl(2, Z) such that n
where F =
1 1
0 1
(Ai F A−1 i ) = id,
(3)
i=1
. Then n is a multiple of 12.
Proof. (In order to stick to the usual conventions for the modular group, we shall use T = tF instead of F . The result follows by transposing (3).) It is well known (see [15]) that the modular group G = Sl(2, Z)/{±I} admits the following presentation G = S, T ; S 2 = (ST )3 = I, where S =
0 1
−1 0
. From this it easily follows that Sl(2, Z) admits the following
presentation Sl(2, Z) = S, T ;
S 4 = I, S 2 = (ST )3 .
Therefore the abelianization K of Sl(2, Z) is the group K = S, T ;
S 4 = I, S 2 = (ST )3 , ST = T S,
which yields K = S, T ; T 12 = I, S = T −3 . Hence K Z/12Z and T is a generator of K. The image of the formula (3) in K gives T n = I, which implies that n is a multiple of 12. As pointed to us by V. Matveev and O. Khomenko [11], from the data in the hypothesis of lemma 6, one can construct an integrable system with 12k focusfocus fibers and whose local monodromy around each critical value ci is equal in
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some fixed basis to Ai F A−1 i . Hence the monodromy around all critical values is the identity. This is done by pasting together a chain of fibrations with one focus-focus fiber, where the gluing maps between two tori are given by the Ai ’s. We therefore obtain a singular torus fibration over an open disc in R2 that cannot admit any S 1 symmetry, due to corollary 4. To have an example of a sequence of matrices in Sl(2, Z) satisfying the hy−1 potheses of lemma 6, take A2j = I and A2j+1 = S. Then the product T ST S = 0 −1
1 1
is of order 6 in Sl(2, Z).
If one constructs a singular Lagrangian fibration over the open disc in R2 using these gluing matrices Aj , with 12 focus-focus critical values, we see that we can obtain as monodromy matrices of oriented loops the following ones: T (loop around one critical value), T −1 (because of (3)), S −1 (which is obtained by looping around the first three critical values, since T ST S −1T = S −1 ), and finally S (again because of (3)). Therefore by arbitrarily composing the corresponding loops together, we obtain any matrix of Sl(2, Z). When B is a Riemann surface, one can show further that the number of focus-focus points (if they are all simple) is equal to 12k, where k is the Euler characteristic of B. See [16] for more details. For example in [21] Tien Zung constructs an integrable system on a K3 surface which yields a singular Lagrangian fibration over S 2 with 24 simple focus-focus points.
8 Example with S1 symmetry Consider the quadratic spherical pendulum. This is a Hamiltonian system on T S 2 ⊆ T R3 (with coordinates (x, ξ)) defined by x, x = 1 and x, ξ = 0, where , is the usual Euclidean inner product. The symplectic form on T S 2 is 3 the restriction of i=1 dxi ∧ dξi to T S 2 . The Hamiltonian is H(x, ξ) =
1 2
ξ, ξ + V (x3 ),
where V (x3 ) = 2(x3 − α)2 with α ∈ (0, 1). H is invariant under the lift of rotation around the x3 axis to T S 2 . Hence H Poisson commutes with the angular momentum K(x, ξ) = ξ × x, e3 . Thus the quadratic spherical pendulum is Liouville integrable with energy momentum mapping F : T S 2 → R2 : (x, ξ) → (H(x, ξ), K(x, ξ)), that is, F = (H, K). The set of critical values of F (see figure 1) is composed of two points A = (2(1 − α)2 , 0) and B = (2(1 + α)2 , 0) and a smooth parabola-like
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curve parametrized by h = 2z −1 (α − z)(1 + zα − 2z 2 ) k = ±2(1 − z 2 ) α/z − 1
891
for z ∈ (0, α].
K z→0 2
1
z=
1 4 0
A 1
B 2
H
3
4
–1
–2
z→0
Figure 1: critical values of the momentum map F . Here α = 1/4. It is straightforward to check that each point on the above curve corresponds to a relative equilibrium of the quadratic spherical pendulum, whose image under the tangent bundle projection is a horizontal circle on S 2 with x3 = ±z. The isolated points are unstable equilibria namely, the poles of S 2 , which are of focusfocus type. Since the fibers F −1 (A) and F −1 (B) contain each a single critical point, both A and B have oriented monodromy index 1. Hence the global index around both points is 2.
9 Semiclassical quantization The constancy of the sign of the monodromy is easily seen on a semiclassical joint spectrum. The latter has a local lattice structure admitting a discrete parallel transport, which is an asymptotic version of the integral affine structure on Br . For more details see [17]. This shows Theorem 7 Let a positively oriented basis B of the quantum lattice around a focusfocus point evolve in the positive sense. Then we obtain a final basis by applying 1 0 to B a 2 × 2 matrix which is conjugate in Sl(2, Z) to k 1 with k ≥ 0.
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We illustrate theorem 7 with the quantum quadratic spherical pendulum. Let ˆ and K ˆ be the self-adjoint operators acting on L2 (S 2 ) defined as follows: H ˆ H ˆ K
= =
2 2 2 ∆S ∂ − i ∂θ ,
+ V (x3 )
where ∆S 2 is the Laplace-Beltrami operator on S 2 (with positive eigenvalues), ˆ et K ˆ V = 2(x3 − α)2 and θ is the polar angle around the vertical axis (Ox3 ). H ˆ K] ˆ = 0 and hence define a quantum are -differential operators that commute: [H, integrable system. Their classical limit is given by the principal symbols H and K in C ∞ (T ∗ S 2 ), which are of course the Hamiltonians of section 8. K
2
1
A 0
1
B 2
3
4
H
−1
−2
Figure 2: Joint spectrum for the quadratic spherical pendulum. The quantum monodromy is represented by the deformation of a small cell of the asymptotic lattice. ˆ and K ˆ for α = 1/4 and = 0.1. For Figure 2 shows the joint spectrum of H such “large” values of the easiest way to compute the spectrum globally is to ˆ in the basis of standard spherical harmonics express the matrix associated to H ˆ (they are also eigenfunctions of K). The action of the potential V is obtained from the recurrence relation of the Legendre polynomials. This matrix can be cut to a finite size without any important loss in the accuracy of the computation, due to the fact that the modes we are looking at are microlocalized in a region of bounded energy H ≤ Hmax , which is compact. We have used this method of calculation to produce figure 2.
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The best way to have precise results near critical values for small would be to use the singular Bohr-Sommerfeld rules of [18].
Acknowledgments The authors wish to thank Prof. B. Zhilinskii and Dr. D. Sadovskii of the Universit´e du Littoral in Dunkerque for telling us their monodromy conjectures which resulted in corollary 4 and assertion 6 in theorem 3. We also wish to thank Drs. Nguyˆen Tiˆen Zung and V.S. Matveev of the Universit´e de Montpellier and the Universit¨ at Freiburg, respectively, for their valuable remarks and criticisms of an early draft of this note. Finally, we acknowledge interesting remarks by the referee. The authors were (partially) supported by European Commission funding for the Research Training Network “Mechanics and Symmetry in Europe” (MASIE), Contract No. HPRN-CT-2000-00113.
References [1] L. Bates, Monodromy in the Champagne bottle, Z. Angew. Math. Phys. 42, 837–847 (1991). [2] A.V. Bolsinov and A.T. Fomenko, Application of classification theory for integrable hamiltonian systems to geodesic flows on 2-sphere and 2-torus and to the description of the topological structure of momentum mappings near singular points, J. Math. Sci. (Dynamical systems, 1) 78, 542–555 (1996). [3] M.S. Child, Quantum states in a Champagne bottle, J. Phys. A. 31, 657–670 (1998). [4] R. Cushman and L. Bates, Global aspects of classical integrable systems, Birkh¨auser, 1997. [5] R. Cushman and J.J. Duistermaat, The quantum spherical pendulum, Bull. Amer. Math. Soc. (N.S.) 19, 475–479 (1988). [6]
, Non-hamiltonian monodromy, J. Differential Equations 172, 42–58 (2001).
[7] R. Cushman and D.A. Sadovski´ı, Monodromy in the hydrogen atom in crossed fields, Phys. D 142, no. 1–2, 166–196 (2000). [8] J.J. Duistermaat, On global action-angle variables, Comm. Pure Appl. Math. 33, 687–706 (1980). [9] L.H. Eliasson, Hamiltonian systems with Poisson commuting integrals, Ph.D. thesis, University of Stockholm, 1984.
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[10] L. Grondin, D. Sadovskii and B. Zhilinski´ı, Monodromy in systems with coupled angular momenta and rearrangement of bands in quantum spectra, Phys. Rev. A (3) 65, no. 012105, 1–15 (2001). [11] V. Matveev, Private communication, email 5 November 2001. [12]
, Integrable hamiltonian systems with two degrees of freedom. topological structure of saturated neighborhoods of saddle-saddle and focus points, Mat. Sb. 187, 29–58 (1996).
[13] B. Moishezon, Complex surfaces and connected sums of complex projective planes, Lecture Notes in Mathematics, no. 603, Springer-Verlag, 1977. [14] D.A. Sadovski´ı and B.I. Zhilinski´ı, Monodromy, diabolic points, and angular momentum coupling, Phys. Lett. A 256, no. 4, 235–244 (1999). [15] J.-P. Serre, Cours d’arithm´etique, Presses Universitaires de France, 1970. [16] M. Symington, in preparation. [17] S. V˜ u Ngo.c, Quantum monodromy in integrable systems, Commun. Math. Phys. 203, no. 2, 465–479 (1999). [18]
, Bohr-Sommerfeld conditions for integrable systems with critical manifolds of focus-focus type, Comm. Pure Appl. Math. 53, no. 2, 143–217 (2000).
[19] H. Waalkens and H.R. Dullin, Quantum monodromy in prolate ellipsoidal billiards, Ann. Physics 295, 81–112 (2002). [20] Nguyˆen Tiˆen Zung, A note on focus-focus singularities, Diff. Geom. Appl. 7, no. 2, 123–130 (1997). [21]
, Symplectic topology of integrable hamiltonian systems, II: Topological classification, Preprint Univ. Montpellier, math.DG/0010181, 2000.
Richard Cushman Mathematics Institute University of Utrecht 3508TA Utrecht The Netherlands email: [email protected] Communicated by Eduard Zehnder submitted 27/12/01, accepted 24/05/02
V˜ u Ngoc San Institut Fourier Universit´e Joseph Fourier, BP 74 38402-Saint Martin d’H`eres Cedex France email: [email protected]
Ann. Henri Poincar´e 3 (2002) 895 – 920 c Birkh¨ auser Verlag, Basel, 2002 1424-0637/02/050895-26
Annales Henri Poincar´ e
Some Upper Bounds on the Number of Resonances for Manifolds with Infinite Cylindrical Ends T. Christiansen Abstract. We prove some sharp upper bounds on the number of resonances associated with the Laplacian, or Laplacian plus potential, on a manifold with infinite cylindrical ends.
The purpose of this note is to bound the number of resonances, poles of the meromorphic continuation of the resolvent, associated to a manifold with infinite cylindrical ends. These manifolds have an infinity which is in some sense onedimensional, even though the manifold is n-dimensional. The bounds we obtain on the resonances reflect this dichotomy- we obtain one bound like that for onedimensional scattering, and other bounds of the type expected for n-dimensional manifolds. As part of our study of resonances, we relate poles of the resolvent to L2 eigenvalues and to poles of appropriately defined “scattering matrices.”
1 Introduction A smooth Riemannian manifold X is said to be a manifold with cylindrical ends if it can be decomposed as X = Xcomp ∪ X∞ , where Xcomp is a compact manifold with boundary Y , X∞ [a, ∞)t × Y , (Y, g) is a compact Riemannian manifold (possibly disconnected), and the metric on X∞ is (dt)2 + g. We may also allow X itself to be a manifold with boundary, and then the ends take the form [a, ∞)t × Y , with Y a smooth, compact manifold with boundary. In this case, we require that the boundary of Xcomp be compact and smooth except for a finite number of corners corresponding to a × ∂Y . An example of such a manifold is a waveguide, a domain with smooth boundary in the plane, with one or more infinite straight ends. If we allow X to have a boundary, we will consider the Laplacian with Dirichlet or Neumann boundary conditions. Let ∆Y be the Laplacian on (Y, g). Let {σj2 }, σ12 ≤ σ22 ≤ σ32 ≤ · · · be the set of all eigenvalues of ∆Y , repeated according to their multiplicity, and let ν12 < ν22 < ν32 < · · · be the distinct eigenvalues of ∆Y . Then the resolvent of the Laplacian ∆ on X, or of ∆ + V , for V ∈ L∞ comp (X) real-valued, has a meromorphic continuation ˆ to the Riemann surface Z on which (z − νj2 )1/2 is a single-valued function for all j ([11, 16]). Thus, the resonances, poles of the meromorphic continuation of the ˆ The complicated nature of this Riemann resolvent, are associated to points in Z.
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surface makes more difficult the question of bounding the number of resonances, and makes necessary the restrictions we place on {νj2 }. Part of our study of resonances is to understand the relationship between poles of the resolvent and their multiplicities and poles of a “scattering matrix.” For a manifold with cylindrical ends, there are several reasonable objects to call the scattering matrix. One is an infinite dimensional matrix. In Section 3 we define this matrix and associated finite-dimensional matrices which contain the information about poles which we desire. We show that if z0 ∈ Zˆ is not a ramification point of Zˆ and is a pole of the resolvent, then there is an associated L2 eigenfunction with an appropriate expansion at infinity or there is a pole of a “scattering matrix” at z0 . We make this precise and address the issue of multiplicities in Theorem 3.1 and Proposition 3.3. In [2] such a relationship is noted, though to our knowledge no proof appears in the literature, so we include it here for completeness. We will often require that there exists an α > 0 such that 2 2 − νm−1 ≥ ανm νm
(H1)
for all sufficiently large m. Examples of cross-sectional manifolds that satisfy such requirements are spheres, an interval with Dirichlet or Neumann boundary conditions, and projective space. We can easily construct a manifold that satisfies (H1) with m ends by taking Y to be the disjoint union of m copies of the same manifold from the previous list. Other situations are possible as well, of course. If n > 2, then assumption (H1) is not a generic condition. It would be interesting to know if results analogous to our Theorems 1.1 and 1.2 hold in greater generality. Let P be the operator ∆, the Laplacian, or ∆ + V , for real-valued V ∈ 2 −1 is bounded on L2 (X) L∞ comp (X). Then, for z ∈ C \ [ν1 , ∞), R(z) = (P − z) except, perhaps, for a finite number of z. It has a meromorphic continuation to the Riemann surface Zˆ described earlier. We bound the number of resonances in ˆ In doing so, we take the view that resonances near the physical certain regions of Z. sheet, the sheet of Zˆ on which the resolvent is bounded, are more interesting, as they have greater physical relevance. Let rj (z) = (z − νj2 )1/2 . Assuming the hypothesis (H1), we simplify the study of the resonances somewhat and are then able to better bound them. Theorem 1.1 Assume X satisfies the hypothesis (H1) and let β < 1. Then, in the √ connected components of {z ∈ Zˆ : |rm (z)| < β ανm } that meet the physical sheet, n−1 there are at most Oβ (m ) resonances. We remark that we count all poles with their multiplicities and that the poles of the resolvent include eigenvalues. In [5, 18] there is an example of a family of manifolds that has lim inf λ→∞ N (λ)λ−n > 0, where N (λ) is the number of eigenvalues of the Laplacian with norm less than λ2 . Since the cross-sectional manifolds in the example can be taken to be n − 1-dimensional unit spheres, this shows that the order appearing in this theorem is optimal. This type of bound is indicative of the n-dimensional nature of the manifold. In a simpler case, we can say a bit more.
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Theorem 1.2 Let X = R × Y and suppose X satisfies (H1), and let ρ > 0 be fixed. Consider the operator ∆ + V , for real-valued V ∈ L∞ comp (X). Then, on the ˆ connected components of {z ∈ Z : |rm (z)| < ρ} that meet the physical sheet, the number of poles is bounded by C(1 + mn−2 ). In case V = V (t), t ∈ R, this theorem is easy to prove, and this example shows that the bound is of optimal order. Our proofs of these two theorems involve an adaptation of techniques developed by Melrose ([15]), Zworski ([24, 25]), and Vodev ([21]) to bound the number of poles. (See [12] and [22] for further references and results.) It involves constructing an approximation to the resolvent to find a holomorphic function whose zeros include the poles of the resolvent. Then we bound the function and apply Jensen’s theorem. This requires some knowledge of the function at a “base point.” For us, that will mean a lower bound. Since we will be changing the base point, we need some kind of uniform lower bound and that is different from these other applications of this technique. In order to do this, we will construct approximations of the resolvent especially well-suited to the regions where we work. The following theorem does not require the hypothesis (H1), and its proof uses a different technique. ˆ and let {zk } be the resonances of P on this sheet. Theorem 1.3 Fix a sheet of Z, Then | Im r1 (zk )| < ∞. |r1 (zk )|2 This theorem is an analogue of what one finds in one-dimensional scattering theory (see [9, 23]), where the natural variable to consider is λ = z 1/2 . The problem of obtaining upper bounds on the resonance-counting function has been widely studied for Euclidean (e.g. [9, 15, 21, 23, 25]) and hyperbolic scattering (e.g. [12, 13, 17, 19]). For a survey and further references, see [22] or [26]. In this paper we use results of [16], which studied the Laplacian on compact manifolds with boundary and exact b-metrics. Under a change of variable, a special case of such manifolds is the class of manifolds considered here (see also [11]). The papers [5, 18] independently obtained that the number of eigenvalues less than λ2 of the Laplacian grows at most like cλn . The existence of eigenvalues or complex resonances has been studied in, for example, [1, 2, 3, 5, 6, 7, 18] and references. In finishing the paper, the author received a copy of [8]. There Edward obtains a result similar to our Theorem 1.1 for the Laplacian on waveguides, that is, domains in the plane which outside of a compact set coincide with (−∞, ∞) × π, and thus fall in the category of manifolds which we consider. The waveguides 2 are very regularly distributed for either the satisfy hypothesis (H1) and their νm Dirichlet or Neumann Laplacian. Use the metric induced on Zˆ by the pull-back of 2 ˆ Then summing over the νm the metric on C to define distance on Z. ≤ r we can
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obtain from our Theorem 1.1 an upper bound #{zj : zj is a resonance of ∆, dist(zj , physical sheet) < c1
|π(zj )|, |π(zj )| < r} = O(r)
for some c1 > 0, where π : Zˆ → Z is projection. This may be compared to Theorem 2 of [8], where for c1 = 1/2, Edward obtains a bound O(r3+ ), any > 0. Remark. We note that the “black-box” formalism of [20] can be adapted to this situation. Thus we could replace P by P˜ , a more general self-adjoint, compactly supported perturbation of the Laplacian than the ones considered here. If P˜ satisfies the assumptions of [20], properly interpreted for this setting, and is bounded below, then Theorem 1.3 will hold for P˜ . Let P˜ # = P˜|{t
2 Preliminaries Let rj (z) = (z−νj2 )1/2 and identify the physical sheet of Zˆ as being the part of Zˆ on which Im rj (z) > 0 for all j and all z and on which R(z) is bounded. Other sheets will be identified, when necessary, by indicating for which values of j Im rj (z) < 0. Each sheet can be identified with C \ [ν12 , ∞). With this convention, there are points in Zˆ which belong to no sheet but which belong to the boundary of the closure of two sheets, and the ramification points, which correspond to {νj2 } and belong to the closure of four sheets (except for ramification points corresponding to ν12 ). We note that sheets that meet the physical sheet are characterized by the existence of a J ∈ N such that Im rj (z) < 0 for all z on that sheet if and only if j ≤ J. Let {φj } be an orthonormal set of eigenfunctions of ∆Y associated with {σj2 }. On an end, we use the coordinates (t, y), with t ∈ (a, ∞) and y ∈ Y . Let r˜l (z) = rj (z) if σl2 = νj2 . We define an operator on [0, ∞)t × Yy . Using the same notation for an operator and its Schwartz kernel, let Rel (z) =
i (ei|t−t |˜rl (z) − ei|t+t |˜rl (z) )φl (y)φl (y ). 2˜ rl (z)
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Then, for z ∈ C \ [σ12 , ∞), RY (z) =
Rel (z)
899
(1)
l
is the resolvent of the Dirichlet Laplacian on [0, ∞) × Y . As an operator from 2 ([0, ∞) × Y ) it has a holomorphic continuation to the L2comp([0, ∞) × Y ) to Hloc ˆ Riemann surface Z and we will see that it is this that determines the surface to which the resolvent of P has a meromorphic continuation. In general, we shall use z to stand for a point in Zˆ and π(z) to represent its projection to C. For w ∈ Rm , w = (1 + |w|2 )1/2 .
3 The Scattering Matrix and Poles of the Resolvent We emphasize that the results of this section, about the correspondence between the poles of the resolvent and the poles of the scattering matrix, do not require hypothesis (H1). There are several reasonable definitions of the scattering matrix in this setting. We recall some results of [4, 16]. Fix a coordinate t on the ends, so that t = 0 lies on the ends. Then, for all but a finite number of z in the physical space, there are functions Φj (z, p) with (P − π(z))Φj (z, p) = 0
(2)
and, on the ends, Φj (z, t, y) = e−i˜rj (z)t φj (y) +
Smj (z)ei˜rm (z)t φm (y).
(3)
The Φj have a meromorphic continuation to all of Zˆ and thus, so do the Smj . The Smj (z) depend on the choice of the coordinate t in a fairly straightforward way (see [5]). This dependence is not important here as it does not change the location of the scattering poles, so we ignore it but we do consider the coordinate t to be fixed throughout, and chosen so that {p ∈ X : t = 0} ⊂ X∞ . There are several reasonable choices of objects to call the scattering matrix. One possibility is the infinite matrix of the Sij (z), as in [16]. Another, which is well-defined for z on the boundary of the physical sheet, is a normalized, finitedimensional matrix of the Sij (z), where the dimension changes as z crosses a νl2 . This is used in [4] and has the advantage of being unitary (though we note that the variable used in [4] is λ = z 1/2 ). Here, however, this is unnecessarily complicated as it requires the introduction of (z − νi2 )1/4 . We shall work with finite-dimensional matrices, of the form (Sij (z))i,j∈E˜ for some set E˜ ⊂ N, where E˜ is chosen, depending on z, to be most helpful for our purposes.
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Definitions and Some Properties
In order to describe the poles and their multiplicities of the resolvent and matrices of the Sij (z), we introduce some notation. Definition 3.1 We define the multiplicity of a pole of the resolvent R(z) at z0 to be mz0 (R) = dim Image Ξz0 (R), where Ξz0 (R) is the singular part of R at the point z0 . ˆ then It follows as in [13, Lemma 2.4] that, if z0 is not a ramification point of Z, mz0 (R) is also the rank of the residue of R at z0 . In order to define the multiplicity of the pole of a matrix, we shall use the following lemma. Though it may be well known, we include it and a proof for the convenience of the reader. Lemma 3.1 Suppose A(z) is a d × d-dimensional meromorphic matrix, invertible for some value of z. Then, near z0 , it can be put into the form p p A(z) = E(z) (z − z0 )−kj Pj + (z − z0 )lj Pj + P0 F (z) j=1
j=p+1
where E(z), F (z), and their inverses are holomorphic near z0 , and Pi Pj = δij Pi , tr P0 = d − p , tr Pi = 1, i = 1, . . . , p . The kj and lj are, up to rearrangement, uniquely determined. Proof. We outline a proof. Without loss of generality we may assume that z0 = 0. First we show the existence of such a decomposition. Choose k such that z k A(z) = B(z) is holomorphic at 0. Now the proof of the existence of such a decomposition follows much like a proof from [10, Section VI.2.4]. Let B(z) = (bij (z)). Choose an element bij (z) that vanishes to the lowest order at 0, and by permuting rows and columns make this element b11 (z). By subtracting from the kth row the first row multiplied by bk1 (z)/b11 (z) (which is holomorphic near z = 0), we can make all the entries in the first column, other than the first one, zero. Similar column operations reduce B(z) to the form
b11 (z) 0 ··· 0 c22 (z) · · · .. .. . . ··· 0 cd2 (z) · · ·
0 c2d (z) .. . cdd (z)
.
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Repeating the procedure on the (d − 1) × (d − 1) dimensional matrix of the cij , we obtain a matrix of the form b11 (z) 0 0 ··· 0 0 c22 (z) 0 ··· 0 0 0 α33 (z) · · · α3d (z) .. .. .. .. . . . ... . 0
0
αd3 (z)
···
αdd (z)
.
Repeating this procedure a finite number of times, we obtain a matrix with only the diagonal entries, hii (z), nonzero. Since hii (z) = z mi gi (z), with gi (0) = 0 and gi (z) holomorphic near 0, by multiplying by a diagonal matrix with nonzero entries 1/gi (z) we obtain a diagonal matrix with diagonal entries of the form z mi . Finally, multiplying the whole thing by z −k I we obtain a matrix equivalent to A(z) near z = 0. We note that the construction ensures that E and F are holomorphic near 0. Moreover, since each of E and F is the product of elementary matrices and one diagonal matrix (holomorphic near 0, with nonzero determinant), E and F are both invertible near 0. To see the uniqueness of the ki , li , we prove it for B(z) = z k A(z), where B is holomorphic at 0, and use the straightforward relationship between A and B. Suppose there are two such decompositions: B(z) = E1 (z)D1 (z)F1 (z) and B(z) = E2 (z)D2 (z)F2 (z), where Ei (z) and Fi (z) are as in the statement of the lemma and the Di are diagonal matrices with nonzero entries dmm,i = z lm,i . By row and column operations we can make 0 ≤ l1,i ≤ l2,i ≤ · · · ≤ ld,i . We have E(z)D1 (z) = D2 (z)F (z)
(4)
for new matrices E(z), F (z), holomorphic and invertible near 0. Thus it is easy to see that rank(D1 (0)) = rank(D2 (0)) ≡ r0 . Using (4) and the fact that lj,i = 0 if and only if j ≤ r0 , we obtain that, if E(z) = (eij (z)), F (z) = (fij (z)), eij (0) = 0 = fji (0) if j ≤ r0 and i > r0 .
(5) (j)
We finish the proof of the uniqueness by induction. Let rj,i = rank(D i (0)), and notice it suffices to prove that rj,1 = rj,2 for all j, and that we have j rj,i = d. Suppose we have shown that rq,1 = rq,2 ≡ rq for q ≤ N , and that, if RN = r0 + r1 + · · · + rN , eij (0) = 0 = fji (0) for j ≤ RN , i > RN .
(6)
If v = (v1 , v2 , . . . , vd )t , let Πs v = (0, . . . , 0, vs+1 , . . . , vd )t for s ∈ N, s ≤ d. Then rank(ΠRN E(0)ΠRN ) = d − RN = rank(ΠRN F (0)ΠRN ),
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using (6) and the fact that E(0) and F (0) are invertible. Since, for j ≤ N , (j) (j) D1 (0)ΠRN = 0 and ΠRN D2 (0) = 0, we get
dN +1 (N +1) (Π E(z)D (z)Π ) E(0)D (0)Π = rank Π RN 1 RN |z=0 RN RN 1 dz N +1
dN +1 (N +1) = rank D1 (0) = rank (ΠRN D2 (z)F (z)ΠRN )|z=0 dz N +1
(N +1) (0) = rank D2
rank
and thus r1,N +1 = r2,N +1 ≡ rN +1 . If RN + rN +1 < d, from (4) we obtain eij (0) = 0 = fji (0) for j ≤ RN + rN +1 , i > RN + rN +1 . The induction need only continue until RN +1 = d, finishing the proof.
Definition 3.2 Let A(z) be a meromorphic matrix, invertible for some values of z, and let k1 , k2 , . . . , kp , lp+1 , lp+2 , . . . , lp be as in Lemma 3.1. Set µmz0 (A) =
p
kj ,
j=1
the “maximal multiplicity” of the pole of A at z0 . Set µdz0 (A) =
p j=1
kj −
p
lj ,
j=p+1
the “determinantal multiplicity” of the pole of A at z0 . We note that µdz0 (A) = min{j ∈ Z : (z − z0 )j det A(z) is regular at z0 }. Each of these measures of multiplicity will be useful. Suppose E ⊂ N is a finite subset. Let E˜ = {l ∈ N : σl2 = νj2 for some j ∈ E}. Define the matrix SE (z) = (Sij (z))i,j∈E˜. ˆ let For z ∈ Z,
Ez = {j ∈ N : Im rj (z) ≤ 0}
and Jz = {j ∈ N : j ≤ max Ez }.
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If E ⊂ N is a finite set, define wE : Zˆ → Zˆ as follows. To z we may associate the set of square roots {rj (z)}. Then wE (z) may be determined by saying it is the element of Zˆ associated to the set {rj (wE (z))}, with −rj (z), if j ∈ E rj (wE (z)) = if j ∈ E. rj (z), ˆ where For J ∈ N, define a similar operator wJ : Zˆ → Z, −rj (z), if j ≤ J rj (wJ (z)) = rj (z), if j > J. This operator appears in the relations (14) and (15). The following Proposition generalizes similar results of [16] and [4]. ˜ Proposition 3.1 Let E ⊂ N be a finite set. For j ∈ E, Φj (wE (z)) = Skj (wE (z))Φk (z) k∈E˜
and
(SE (z))−1 = SE (wE (z)).
˜ consider Proof. For j ∈ E, Ψj (z) = Φj (wE (z)) −
Skj (wE (z))Φk (z).
k∈E˜
Then (P − π(z))Ψj (z) = 0 and, on the ends, Ψj has an expansion Ψj (z, t, y) = ei˜rj (z)t φj (y) −
+
Skj (wE (z))Slk (z)ei˜rl (z)t φl (y)
k,l∈E˜
Slj (wE (z)) −
l∈E˜
Skj (wE (z))Slk (z) ei˜rl (z)t φl (y).
(7)
k∈E˜
Since, for z on the physical sheet, Im r˜l (z) > 0 for all l, Ψj (z) ∈ L2 (X) there and thus Ψj (z) ≡ 0 for all z in the physical space. By analytic continuation, Ψj (z) ≡ 0 ˆ This proves the first part of the Proposition. It also shows that, for for all z ∈ Z. ˜ j, l ∈ E, Skj (wE (z))Slk (z) = δjl ; k∈E˜
that is, (SE (z))−1 = SE (wE (z)).
904
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Relation between Poles of the Resolvent and Poles of SJ z0 (z)
First, we recall some results of [16, Sections 6.7, 6.8] on the nature of poles of the resolvent on the boundary of the physical sheet. Proposition 3.2 Suppose z lies on the boundary of the physical sheet. Then, if z ˆ the multiplicity of z as a pole of the resolvent is not a ramification point of Z, is equal to the dimension of the L2 null space of P − π(z). If z is a ramification point, then the resolvent has a double pole at z, with coefficient the projection onto the L2 null space of P − π(z), if there is any. The residue has rank equal to the dimension of {f : (P − π(z))f = 0, f ∈ L2 (X),
∂ f ∈ L2 (X)}. ∂t
Since Φj (z, p) = χ(t)e−i˜rj (z)t φj − (P − z)−1 (P − π(z))χ(t)e−i˜rj (z)t φj ,
(8)
where χ(t) ∈ C ∞ (R) is supported in t > a and is one in a neighborhood of infinity, it is clear by their definition that the Sij (z) cannot have a pole unless R(z) has a pole. More can be said, and we begin our study of the relationship between the poles of the resolvent and poles of the scattering matrix with the following ˆ and z0 is not a ramification point. Then Theorem 3.1 Suppose z0 ∈ Z, mz0 (R) = µmz0 (SJz0 ) + dim{f ∈ L2 (X) : (P − π(z0 ))f = 0, f ∼
cj ei˜rj (z0 )t φj }
j∈J˜z0
where the expansion must be valid on any end. The notation f∼
cj ei˜rj (z0 )t φj
j∈E˜
means that f|t>0 (t, y) =
cj ei˜rj (z0 )t φj (y).
j∈E˜
In proving this theorem, we will use some techniques from [13, Section 2]. We will call {f : (P − π(z0 ))f = 0, f ∼ cj ei˜rj (z0 )t φj }
(9)
j∈J˜z0
the set of eigenfunctions of type I. This depends on z0 of course, and we will note the dependence in cases of possible confusion. We first show that
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ˆ then Lemma 3.2 If z0 is not a ramification point of Z, µmz0 (SJz0 ) ≤ mz0 (R) − dim{f ∈ L2 (X) : (P − π(z0 ))f = 0, f ∼
cj ei˜rj (z0 )t φj }.
j∈J˜z0
Proof. Just as in Lemma 2.4 of [13], whose notation we use, one can prove that if ˆ then near z0 R(z) has a pole at z0 , for z0 not a ramification point of Z, R(z) =
p Ak (z0 ) + H(z0 , z) (z − z0 )k
(10)
k=1
where H(z0 , z) is holomorphic near z0 , q
Ak (z0 ) =
alm k (z0 )ϕl ⊗ ϕm
l,m=1
and (ϕl ⊗ ϕm )f (p) = ϕl (p)
p ∈X
ϕm (p )f (p )dvX .
As in [13], we have (P − π(z0 ))Ak (z0 ) = Ak+1 (z0 ) = Ak (z0 )(P − π(z0 )). If ak (z0 ) = (alm k (z0 ))1≤l,m≤q , then a1 (z0 ) is symmetric with rank q, d(z0 ) = a1 (z0 )−1 a2 (z0 ) is nilpotent, and ak (z0 ) = a1 (z0 )dk−1 (z0 ), k > 1. Moreover, ϕl has an expansion on the ends of the form ϕl (t, y)|t>a = cljm tm ei˜rj (z0 )t φj (y) (11) j
m≤p
([16]) and the ϕl are linearly independent. Let B(z) be the matrix B(z) = (blm (z))l,m≤q with blm (z) =
p alm k (z0 ) . (z − z0 )k
k=1
Then B(z) can be written, as in Proposition 2.11 of [13], as p B(z) = E # (z) (z − z0 )−kj Pj + P0 F # (z)
(12)
j=1
where E # (z), F # (z) and their inverses are holomorphic near z0 , Pi Pj = δij Pi , tr Pi = 1, i = 0, tr P0 = q − p , and k1 + k2 + · · · + kp = q.
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First assume that there are no type I eigenfunctions of P with eigenvalue π(z0 ). We recall that we can construct the generalized eigenfunctions Φj , j ∈ J˜z0 , as indicated in (8). We then obtain the entries of SJz0 (z) by first restricting Φj to t = 0 and then extracting the coefficients of φk . This, taken with the representation (12), shows that the singular part of SJz0 (z) near z0 will be given by p E (z) (z − z0 )−kj Pj + P0 F (z) j=1
where E (z), F (z) are holomorphic matrices near z0 which may not be invertible, or even square, and this proves the lemma in this special case. To handle the case where π(z0 ) is an eigenvalue of P with ne (z0 ) = dim{f ∈ L2 (X) : (P − π(z0 ))f = 0, f ∼ cj ei˜rj (z0 )t φj } > 0, (13) j∈J˜z0
ˆ R(z)ψ we need to be a bit more careful. We first remark that for a general z ∈ Z, is well-defined if ψ ∈ e−t max(0,− Im rj (z)) L2 (X). This can be seen from the construction of the analytic continuation of the resolvent in [16]. Now suppose ψ is an L2 eigenfunction of P . Then it is exponentially decreasing, and, for J ∈ N, if z and wJ (z) both lie on the boundary of the physical space, (14) (R(z) − R(wJ (z))) ψ = 0. By analytic continuation, if z, wJ (z) are such that ψ ∈ e−t max(0,− Im rj (z),− Im rj (wJ (z))) L2 (X), then (14) holds. By repeatedly applying (14) and using our knowledge of the structure of the resolvent on the closure of the physical space, we obtain that if ψ ∈ {f ∈ L2 (X) : (P − π(z0 ))f = 0, f ∼ cj ei˜rj (z0 )t φj }, j∈J˜z0
then there is a neighborhood of z0 so that in this neighborhood R(z) =
1 ψ ⊗ ψ + B1 (z, z0 , ψ) (π(z0 ) − π(z))ψ2
with B1 (z, z0 , ψ)ψ = 0. That is, each L2 eigenfunction with an expansion at infinity of this type contributes to the singularities of R(z) in the expected way. Using the fact that if j ∈ Jz0 ψ(P − π(z0 ))χ(t)e−i˜rj (z0 )t φj (y) = 0, X
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we see that the singularities of R(z) at z0 corresponding to L2 eigenfunctions of type I do not contribute to the singularities of SJz0 (z) at z0 . After making this observation, the proof for the case ne (z0 ) > 0 follows just as in the case ne (z0 ) = 0. The proof of Theorem 3.1 will be completed by ˆ then Lemma 3.3 If z0 is not a ramification point of Z, µmz0 (SJz0 ) + dim{f ∈ L2 (X) : (P − π(z0 ))f = 0, f ∼
cj ei˜rj (z0 )t φj }
j∈J˜z0
≥ mz0 (R). Proof. We use, for J ∈ N, R(z) − R(wJ (z)) =
i 1 Φj (z) ⊗ Φj (wJ (z)) . 2 2 2 r˜j (z)
(15)
σj ≤νJ
Equation (15) holds, by Stone’s formula, [16, Section 6.8], and [4, Section 2.2], for 2 z on the boundary of the physical space, with νJ2 < π(z) < νJ+1 , and then holds on the rest of Zˆ by analytic continuation. Using Proposition 3.1, we may write (15) as R(z) − R(wJ (z)) =
i 1 . Smj (z)Φm (wJ (z)) ⊗ Φj (wJ (z)) 2 2 2 2 2 r˜j (z)
(16)
σj ≤νJ σm ≤νJ
Recalling equations (10) and (11), we see that if ψ is in the image of the singular part of R(z) at z0 , then ψ is a linear combination of eigenfunctions of type I (see (9)) and {f : (P − π(z0 ))k f = 0 for some k ∈ N; f ∼
j
bjm tm ei˜rj (z0 )t φj (y),
m≤k
bjm = 0 for some j ∈ J˜z0 , some m}. (17) We call functions of the form (17) type II. Now take J = max Ez0 . If g is in the image of the singular parts of R(z) at both z0 and at wJ (z0 ), then it must be of type I. It is the appearance of the type II functions which we must understand. Suppose g is in the image of the residue of R(z) at z0 and is of type II. Then, since it is not in the image of the residue of R(z) at wJ (z0 ), it must be in the image of the residue of 1 (18) Smj (z)Φm (wJ (z)) ⊗ Φj (wJ (z)) r ˜ j (z) 2 2 2 2 σj ≤νJ σm ≤νJ
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at z = z0 . First assume that Φj (wJ (z)), j ∈ J˜z0 has no poles at z0 . Recalling that we may write SJz0 (z) near z0 as in Lemma 3.1, we can write (18) as p
(z − z0 )−kj Ψ1j (z) ⊗ Ψ2j (z) + H(z, z0 )
(19)
j=1
where k1 + · · · + kp = µmz0 (SJz0 (z)), and Ψ1j (z), Ψ2j (z), and H(z, z0 ) are holomorphic near z0 . Then it is easy to see that the rank of the image of the residue of (19) cannot exceed µmz0 (SJz0 (z)), and the total rank of the residue of R(z) at z0 cannot exceed µmz0 (SJz0 ) + dim{f ∈ L2 (X) : (P − π(z0 ))f = 0, f ∼ cj ei˜rj (z0 )t φj }. j∈J˜z0
To finish the proof, we need only understand what happens if Φj (wJ (z)) has a pole at z0 for some j ∈ J˜z0 . However, a pole of Φj (wJ (z)) cannot contribute to the singularity of R(z) at z0 because the expansion on the ends of the singular part is of the wrong form. Therefore, the proof of this case follows much as the proof of the previous one. The following proposition will also be useful. Proposition 3.3 Suppose z0 ∈ Zˆ is such that π(z0 ) is not in the spectrum of P . Then mz0 (R) = µdz0 (SEz0 ). Proof. We sketch the proof of this proposition, as it is very similar to the proof of Theorem 3.1. Just as in Lemma 3.2, one can show that µmz0 (SEz0 ) ≤ mz0 (R). Here, of course, dim{f ∈ L2 (X) : (P − π(z0 ))f = 0, f ∼ cj ei˜rj (z0 )t φj } = 0, j∈J˜z0
by our assumption on π(z0 ). Moreover, SEz0 (z) has no zeros at z0 as a zero would imply the existence of an L2 eigenfunction, and thus µmz0 (SEz0 ) = µdz0 (SEz0 ). Finishing the proof thus requires showing that mz0 (R) ≤ µmz0 (SEz0 ). This can be done as in Lemma 3.3, first noting that, if ψl ≡ 0 is in the image of the singular part of R at z0 , then on the ends ψl (t, y) = bljm tm ei˜rj (z0 )t φj (y) (20) j
m≤mj
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where bljm = 0 for some j ∈ Ez0 and some m. In fact, because π(z0 ) is not an eigenvalue of P , the linear independence of ψ1 , ψ2 , . . . , ψq in the image of the singular part of R(z) at z0 is equivalent to the linear independence of bljm tm ei˜rj (z0 )t φj (y), j∈E˜z0 m≤mj
l = 1, . . . , q. Using this and the fact that Ξz0 (R) is symmetric, where Ξz0 is the singular part at z0 , we obtain that dim Image Ξz0 (R)
≤ dim Image Ξz0
j,m∈E˜z0
1 . Smj (z)Φm (wJ (z)) ⊗ Φj (wJ (z)) r˜j (z)
Then, if Φj (wJ (z)) is regular at z0 for all j ∈ J˜z0 , it is easy to see that mz0 (R) ≤ µmz0 (SEz0 ). Again, if Φj (wJ (z)) has a pole at z0 , it does not contribute to the singularities of R(z) at z0 but corresponds instead to a singularity of R(z) at wJ (z0 ).
4 Proof of Theorem 4.1 In this section we bound the number of poles of the resolvent in neighborhoods of the ramification points on the boundary of the physical sheet, with the size of the neighborhoods increasing. We will use the fact that if χ = 1 for t < max(a, 0), then dim Image Ξz0 (R) = dim Image Ξz0 (Rχ), where Ξz0 (T ) is the singular part of T at z0 . We recall Theorem 1.1 Theorem. Assume X satisfies the hypothesis (H1) and let β < 1. Then, in the √ connected components of {z ∈ Zˆ : |rm (z)| < β ανm } that meet the physical sheet, there are at most Oβ (mn−1 ) resonances. A corollary to this is Theorem 4.1 Assume X satisfies the hypothesis (H1) and let β < 1. Then, on the closure of the sheet with Im rj (z) < 0 if and only if j ≤ m, there are at most √ Oβ (mn−1 ) poles of the resolvent of P with |rm (z)| < β ανm . We shall use the Fredholm determinant method used in, for example, [12, # 15, 21, 24, 25]. We will find a trace class operator Km (z) so that, in the desired # (z), and thus region, the poles of the resolvent are contained in the zeros of I + Km # # in the zeros of det(I + Km (z)). In addition, Km (z0 ) = 0, where z0 ∈ Zˆ is a “base point” which depends on m. To do this, we first construct an approximation of the
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T. Christiansen
Ann. Henri Poincar´e
resolvent adapted to this problem and valid in this region, obtaining a compact operator Km (z) so that the poles of the resolvent in this region are contained in the zeros of I + Km (z). Further manipulations simplify I + Km (z). For m > 1, there are two ramification points on the boundary of the physical 2 , and thus, for large m, two connected comsheet of Zˆ which correspond to νm √ ˆ ponents of {z ∈ Z : |rm (z)| < β ανm } which meet the physical sheet. We shall work near the ramification point which is reached by taking a limit as Im π(z) ↓ 0, 2 , with z in the physical sheet. We will designate this ramification Re π(z) → νm 2 2 point by (νm )+ . A similar analysis works near the other point, (νm )− , obtained 2 by taking a limit for z in the physical sheet, Im π(z) ↑ 0, Re π(z) → νm . We assume m > 1. For c > max(a, 0), let Xc = Xcomp ∪ (X∞ ∩ [a, c] × Y ). For ζ ∈ C, let Rc (ζ) = (P|Xc − ζ)−1 be the resolvent of P on Xc with Dirichlet boundary conditions. (If X has a boundary and we are considering Neumann boundary conditions on X, we use Neumann boundary conditions on Xc here.) Recall that RY (z) = (Dt2 + ∆Y − z)−1 is defined by (1). For i = 1, 2, 3, choose χi ∈ Cc∞ (X) so that χi χi+1 = χi , i = 1, 2, χ1 ≡ 1 on Xmax(a,0) , ∇χi only depends on t, and |∇χi | < γ, |∆χi | < γ. It suffices to take −1 γ ≤ α(1 − β)2 (1000(1 + α)) . Choose c0 so that the support of χ3 is properly contained in Xc0 . 2 + 14 α(1 − β)2 νm i and let Choose z0 in the physical plane so that π(z0 ) = νm Em (z) = χ3 Rc0 (π(z))Πm χ2 +χ3 Rc0 (π(z0 ))(1−Πm )χ2 +(1−χ1 )RY (z)(1−χ2 ). 2 Here Πm projects off of the eigenfunctions of P|Xc0 with eigenvalues in (νm − 2 5ανm , νm + 5ανm ). This of course depends on c0 , but we omit this dependence in our notation. Then ˜ m (z), (21) (P − π(z))Em (z) = I + K
where ˜ m (z) = [P, χ3 ] (Rc0 (π(z))Πm + Rc0 (π(z0 ))(1 − Πm )) χ2 K − χ3 (π(z) − π(z0 ))Rc0 (π(z0 ))(1 − Πm )χ2 − [P, χ1 ]RY (z)(1 − χ2 ), ˜ m (z) is meromorphic on Z, ˆ with and, if the domain is restricted to L2comp (X), K the poles corresponding to poles of Rc0 (π(z))Πm . Let χ ∈ Cc∞ (X) be 1 on the support of χ3 . Then ˜ m (z)χ) = χ(I + Km (z)) (P − π(z))Em (z)χ = χ(I + K and Km (z) is compact.
(22)
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Our choice of z0 and γ guarantee that I + Km (z0 ) is invertible, with norm bounded by 2. Then, by analytic Fredholm theory, I + Km (z) is invertible at all ˆ The points where it is not invertible correspond to but a discrete set of points of Z. zeros of I + Km (z), and the poles of the cut-off resolvent (P − z)−1 χ are included in the union of the zeros of I + Km (z) and the poles of Em (z). However, we will restrict our attention to a region that does not include any poles of Em (z). Using the fact that I + Km (z0 ) is invertible, with norm bounded by 2, we obtain that in the region in question, the poles of the resolvent are contained in the zeros of (23) I + (I + Km (z0 ))−1 (Km (z) − Km (z0 )). √ Now we restrict our attention to a region where |rm (z) − rm (z0 )| < ρ ανm , where ρ = (β 2 /4 + 3/4)1/2 2 and z lies on one of the four sheets that meet the ramification point (νm )+ , with m large. In this region, we have
(I + Km (z0 ))−1 [P, χ3 ] (Rc0 (π(z)) − Rc0 (π(z0 )) Πm χ2 ≤
1 , 3
and ((I + Km (z0 ))−1 [P, χ1 ]
(Rej (z) − Rej (z0 )) (χ − χ2 ) ≤
2 σj2 >νm+1
1 . 3
(24)
For (24), we are using the fact that [P, χ1 ] depends only on t. Therefore, in this region the poles of the resolvent are contained in the zeros of # (z), I + Km
where
(25)
# Km (z) = Lm (z) (K1m (z) + K2m (z))
with K1m (z) = −χ3 (π(z) − π(z0 ))Rc0 (π(z0 ))(1 − Πm )χ2 , K2m (z) = −[P, χ1 ] (Rej (z) − Rej (z0 )) (χ − χ2 ),
(26) (27)
2 σj2 ≤νm+1
and the norm of Lm (z) is bounded, independent of m and z, as long as we stay in the region described. # (z) is trace class. We consider the function Now Km # (z)), h(z) = det(I + Km
and note that h(z0 ) = 1 and that h is holomorphic on Zˆ when |rm (z) − rm (z0 )| < √ 2 ανm . We will apply Jensen’s theorem to h to obtain an upper bound on the number of zeros of the resolvent in this region, and to do this we need the following lemma.
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√ Lemma 4.1 If |rm (z)− rm (z0 )| ≤ ρ ανm and z lies in the connected component of n−1 √ 2 {z ∈ Zˆ : |rm (z )−rm (z0 )| ≤ ρ ανm } containing (νm )+ , then |h(z)| ≤ CeCm . Proof. We remark that because of Weyl’s law and the hypothesis (H1), for large m, νm is bounded above and below by constant multiples of m. We use the property that | det(I + A + B)| ≤ det(I + |A|)2 det(I + |B|)2
(28)
(e.g. [12, Lemma 6.1]). Since (I − Πm )χ2 has rank bounded by Cmn−1 and √ Rc0 (π(z0 )) ≤ νCm , we obtain, in the region with |rj (z) − rj (z0 )| ≤ ρ ανm , and thus |π(z) − π(z0 )| ≤ C νm , | det(1 + |Lm (z)K1m (z)|)| ≤ CeCm
n−1
.
Consider next Lm (z)K2m (z) = Lm (z)[P, χ1 ]
(Rej (z) − Rej (z0 )) (χ − χ2 )
2 σj2 ≤νm+1
= Lm (z)[P, χ1 ]
m+1
i ei|t−t |rj (z) − ei|t+t |rj (z) − ei|t−t |rj (z0 ) + ei|t+t |rj (z0 )
j=1
2rj (z)
×
φl (y)φl (y )(χ − χ2 ).
σl2 =νj2
Since [P, χ1 ] and χ − χ2 have disjoint supports,
[P, χ1 ] ei|t−t |rj (z) − ei|t+t |rj (z) − ei|t−t |rj (z0 ) + ei|t+t |rj (z0 ) φl (y)φl (y )(χ − χ2 ) is a rank four operator with norm bounded by CeC max(0,− Im rj (z)) . We need, therefore, to bound m
2 | Im rj (z)|#{σl2 : σl2 = νj2 } + #{σl2 : σl2 = νm+1 }.
(29)
1
We have
#{σl2 : σl2 = νj2 } ≤ Cj n−2 .
2 − νj2 |)−1/2 , if On the region of interest, | Im rj (z)| can be bounded by 2ανm (|νm √ j < m, and ανm , if j = m. Therefore, m
2 | Im rj (z)|#{σl2 : σl2 = νj2 } + #{σl2 : σl2 = νm+1 }
1
=
m−2 1
| Im rj (z)|#{σl2 : σl2 = νj2 } + O(mn−1 ). (30)
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To bound (29), then, it suffices to bound νm−1 2 −1/2 n−2 2ανm νm − l2 l dl 0
n−1 = 2ανm
νm−1 /νm
0
n−1 (1 − s2 )−1/2 sn−2 ds ≤ αCνm . (31)
Proof of Theorem 1.1. The poles of the resolvent in the region in question are 2 )+ , we use rm (z) as contained in the zeros of h(z). Working near the point (νm a coordinate, which we may do as long as we keep away from other ramification points or regions where rm (z) fails to be one-to-one. We apply Jensen’s Theorem to √ h, using a circle centered at z0 and having radius ρ ανm . Then the theorem follows √ because the disk |rm (z)| ≤ β ανm is properly contained in the disk |rm (z) − √ rm (z0 )| ≤ ρ ανm , with the ratio of the distance between the two boundaries and the radius of the larger disk bounded from below, independent of m.
5 Proof of Theorem 1.2 We recall the statement of Theorem 1.2: Theorem. Let X = R × Y and suppose X satisfies (H1), and let ρ > 0 be fixed. Consider the operator ∆ + V , for real-valued V ∈ L∞ comp (X). Then, on the conˆ nected components of {z ∈ Z : |rm (z)| < ρ} that meet the physical sheet, the number of poles is bounded by C(1 + mn−2 ). In this section we continue to use the {σj2 } to denote the eigenvalues of ∆Y , repeated according to their multiplicity, though our manifold X actually has two ends isomorphic to (1, ∞) × Y . We use the other notation introduced earlier as well. We prove this theorem by the Fredholm determinant method as in the previous theorem. We assume that ρ > 1. Let R0 (z) = (∆ − z)−1 ; its Schwartz kernel is given by R0 (z) =
∞ l=1
i ei|t−t |˜rl (z) φl (y)φl (y ), 2˜ rl (z)
(32)
and let RV (z) = (∆ + V − z)−1 . We have (∆ + V − π(z))R0 (z) = I + V R0 (z).
(33)
Let χ(t) ∈ Cc∞ (X) be one on the support of V , with |χ| ≤ 1. Then, if z0 ∈ Zˆ is not a ramification point and RV (z)χ has a pole at z0 , then I + V R0 (z0 )χ has nontrivial null space. For r > 0, m ∈ N, set Bm,r to be the connected components of {z ∈ Zˆ : |rm (z)| < r} that meet the physical sheet.
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1/2
Let ρ˜ = max(ρ, 2V ∞ ), and restrict z to Bm,4ρ˜. Take m sufficiently large 2 that the only ramification points in Bm,4ρ˜ correspond to νm . Then I + V R0 (z)χ = (I + K1 (z))(I + (I + K1 (z))−1 K2 (z)) where K1 (z) has Schwartz kernel i ei|t−t |rj (z) K1 (z) = V (t, y) φl (y)φl (y )χ(t ) 2rj (z) 2 2 j=m
σl =νj
and K2 (z) K2 (z) = V (t, y)
i
2rm (z)
ei|t−t |rm (z)
φl (y)φl (y )χ(t ).
2 σl2 =νm
By choosing m sufficiently large, so that
A1/2 16ρ˜2 1 , exp A √ < ανm 2V |ανm − 16ρ˜2 |1/2 where A = maxt,t ∈supp χ |t − t |, we have (I + K1 (z))−1 ≤ 2 on Bm,4ρ˜. The poles of RV (z) in Bm,4ρ˜, other than the ramification point, correspond to values of z for which I + (I + K1 (z))−1 K2 (z) has non-trivial null space. We remark that 2 2 R0 (z) has a pole of rank MY (νm ) at the ramification point, where MY (νm ) is the 2 multiplicity of νm as an eigenvalue of ∆Y , and this can contribute a pole of up to the same multiplicity to RV (z), even if I + (I + K1 (z))−1 K2 (z) is invertible there. Let K3 (z) = (I + K1 (z))−1 K2 (z). The poles of RV (z) in Bm,4ρ˜ are contained in the zeros of I + K3 (z) in the same region, except, possibly, at the ramification points, as discussed above. Now we 2 )+ as in the proof of the previous theorem, as a similar analysis shall work near (νm will work for the other connected component of Bm,4ρ˜. Choose z0 in the physical 2 space with π(z0 ) = νm + 4iV ∞ . Then K2 (z0 ) ≤ 1/4, and −1 I + (I + K1 (z0 ))−1 K2 (z0 ) = (I + K3 (z0 ))−1 ≤ 2. Let
h(z) =
rm (z) rm (z0 )
2 2MY (νm )
det I + (I + K3 (z0 ))−1 (K3 (z) − K3 (z0 )) .
Then h(z0 ) = 1, and, except, possibly, for some at the ramification point, the poles 2 of RV (z) in the connected component of Bm,4ρ˜ that includes (νm )+ , are contained 2 n−2 in the zeros of h(z) in the same region. This misses at most 2MY (νm ) = O(νm ) poles of RV at the ramification point. An application of Jensen’s theorem on a circle centered at z0 and with |rm (z) − rm (z0 )| ≤ 3ρ˜ will then finish the proof, after we have proved the following lemma.
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2 Lemma 5.1 On the connected component of Bm,4ρ˜ that contains (νm )+ ,
|h(z)| ≤ exp(C(1 + |m|n−2 )). Proof. Let Rems (z) be the operator with Schwartz kernel i
2rm (z)
χ(t)(ei|t−t |rm (z) − 1)
and let Rem0 (z0 ) be the operator with Schwartz kernel i χ(t)ei|t−t |rm (z0 ) . 2rm (z0 )
Let A1 (z) = (I + K1 (z))−1 V
i 2rm (z)
φl ⊗ φl χ.
2 σl2 =νm
Then K3 (z) − K3 (z0 ) = (I + K1 (z))−1 V Rems (z) − (I + K1 (z0 ))−1 V Rem0 (z0 ) φl ⊗ φl χ + A1 . × 2 σl2 =νm
Now we shall use (28) and | det(I + |BT |)| ≤ det(I + B|T |) (e.g. [12, Lemma 6.1]). Then | det(I + (I + K3 (z0 ))−1 (K3 (z) − K3 (z0 )))|
≤ det(I + (I + K3 (z0 ))−1 (I + K1 (z))−1 V |Rems (z)
φl ⊗ φl χ|)4
2 σl2 =νm
× det(I + (I + K3 (z0 ))−1 (I + K1 (z0 ))−1 V |Rem0 (z0 )
φl ⊗ φl χ|)4
2 σl2 =νm
× det(I + |(I + K3 (z0 ))−1 A1 |)2 . On Bm,4ρ˜, (I + K3 (z0 ))−1 (I + K1 (z))−1 V ≤ C. For a compact operator A, let µ1 (A) ≥ µ2 (A) ≥ µ3 (A) ≥ · · · be the characteristic values of A; that is, the eigenvalues of |A∗ A|1/2 . Then, if A is trace class, det(I + |A|) = (I + µj (A)). For p = 0, 1, 2, . . . , 2 )+j (Rems (z) µpMY (νm
2 σl2 =νm
2 φl ⊗ φl χ) = µ ˜p (rm (z)), j = 1, . . . , MY (νm ).
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Here we use the fact that Rems (z) depends on z only through rm (z). The µ ˜p (w) are independent of m. Therefore,
| det(I + (I + K3 (z0 ))−1 (I + K1 (z))−1 V |Rems (z)
φl ⊗ φl χ|)|
2 σl2 =νm 2
≤ C MY (νm ) ≤ C m
n−2
.
The same argument gives | det(I + (I + K3 (z0 ))−1 (I + K1 (z0 ))−1 V |Rem0 (z0 )
φl ⊗ φl χ|)| ≤ C m
n−2
.
2 σl2 =νm
A similar argument bounds 2
rm (z)2MY (νm ) det(I + |(I + K3 (z0 ))−1 A1 |)2 . 2 The rank of (I + K3 (z0 ))−1 A1 is MY (νm ), and (I + K3 (z0 ))−1 A1 is bounded for 4ρ˜ ≥ |rm (z)| > c > 0. Since rm (z)(I + K3 (z0 ))−1 A1 is bounded on Bm,4ρ˜, 2 M (νm )
rm Y
2
det(I + |(I + K3 (z0 ))−1 A1 |) ≤ C MY (νm ) ≤ C m
n−2
on Bm,4ρ˜. A consequence of Theorem 1.2 is
Corollary 5.1 Let X = R × Y and suppose X satisfies (H1). Let V ∈ L∞ comp (X) be real-valued, and let N (λ) = #{λ2j ≤ λ2 : λ2j is an eigenvalue of ∆ + V }. Then N (λ) = O(λn−1 ). Proof. Suppose τ ∈ R+ . Then, using (32) and (33), we see that if 1 V (length supp V + 1) ≤ minj |rj (τ )| 2 then τ cannot be an eigenvalue of ∆ + V . Here length supp V =
max
a,b∈supp V
|a − b|.
This means that any eigenvalue must lie within a fixed distance of some νj2 . Theorem 1.2 provides a bound on the number of eigenvalues within such a ball; summing over the νj2 we obtain the corollary.
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6 Proof of Theorem 1.3 We recall Theorem 1.3: ˆ and let {zk } be the resonances of P on this sheet. Then Theorem. Fix a sheet of Z, | Im(r1 (zk ))| |r1 (zk )|2
k
< ∞.
In the proof of this theorem we shall use Proposition 3.3 and Carleman’s Theorem, which we recall (e.g. [14, Section V.i]). Theorem (Carleman). If F (ζ) is a holomorphic function in the region Im ζ ≥ 0, F (0) = 1, and if ak = rk eiθk (k = 1, 2, . . . ) are its zeros in this region, then
1 rk 1 ( − 2 ) sin θk = rk R πR
rk ≤R
+
1 2π
π
0
0
R
ln |F (Reiθ )| sin θdθ
1 1 − 2 x2 R
ln |F (x)F (−x)|dx +
1 Im F (0). 2
We note that Carleman’s Theorem also holds for a function F (ζ) which is holomorphic in Im ζ > 0 and continuous in Im ζ ≥ 0. In order to see this, apply Carleman’s Theorem to F (ζ) = F (ζ + i )/F (i ), > 0. Then, since both sides of the equation are continuous in for small ≥ 0, the theorem holds in this case as well. Proof of Theorem 1.3. ˆ and let Fix a sheet of Z, E = {j ∈ N : Im rj (z) < 0 on this sheet}. By Proposition 3.3, the poles of the resolvent on this sheet (but not on its boundary) correspond, with multiplicity, to the poles of det SE (z) on this sheet. We have, by Proposition 3.1, SE−1 (z) = SE (wE (z)), and, if z lies on the sheet with Im rj (z) < 0 if and only if j ∈ E, then wE (z) lies on the physical sheet. Therefore, we reduce the problem to a question about the zeros of det SE (z) for z on the physical sheet. It is helpful to identify the physical sheet with the upper half plane using the variable ζ = r1 (z). Let Ψij (ζ) = Sij (z(ζ)) and Ψ(ζ) = SE (z(ζ)). Using the fact that SE (z) is meromorphic on Zˆ we can extend Ψ to the closed upper half plane by continuity, except, perhaps, for a finite number of points corresponding
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to poles of SE (z). We shall also call these points poles of Ψ(ζ). The matrix Ψ(ζ) has at most a finite number of poles in the closed upper half plane. To prove the theorem, we shall use Carleman’s theorem applied to a multiple of det Ψ(ζ), chosen so that the product is holomorphic in the upper half plane and continuous on its closure. In order to do this, we need bounds on det Ψ. Let J = max{j ∈ E}. If ζ ∈ R, ζ 2 + ν12 > νJ2 , then |Ψij | ≤ C ([4]). To bound |Ψij (ζ)| away from the real axis, we need to bound |Sij (z)|, where Sij is determined by the expansion on the ends of Φj , see (2) and (3). We recall that Φj (z(ζ)) = χ(t)e−i˜rj (z(ζ))t φj − (P − z(ζ))−1 (P − ζ 2 − ν12 )χ(t)e−i˜rj (z(ζ))t φj (34) where χ(t) ∈ C ∞ (R) is supported in t > a and is one in a neighborhood of infinity. ˜ We note that for j ∈ E, (P − ζ 2 − ν12 )χ(t)e−i˜rj (z(ζ))t φj ≤ C ζ 2 + ν12 − σj2 1/2 eC| Im r˜j (z(ζ))| ≤ C ζ 2 + ν12 − σj2 1/2 eCIm ζ
ei˜rl (z(ζ))t φl
and Since
|t>a
= (2| Im r˜l (z(ζ))|)−1/2 e−a Im r˜l (z(ζ)) .
(35) (36)
(P − ζ 2 − ν12 )−1 ≤ (dist(ζ 2 + ν12 , σ(P )))−1 ,
we have, using (3), (34), (35) and (36), |Ψlj (ζ)| ≤
C ζ 2 + ν12 − σl2 1/2 CIm ζ e dist(ζ 2 + ν12 , σ(P ))
when |ζ| is large. This proves | det Ψ(ζ)| ≤ CeCIm ζ if Im ζ > > 0 and |ζ| large. To obtain a bound in the closure of the upper half-plane, apply the Phragmen-Lindel¨ of theorem to h(ζ) =
k0 1
ζ − ζj det Ψ(ζ) ζ − ζj + M i
where ζ1 , ζ2 , . . . , ζk0 are the poles of det Ψ(ζ) in the closed upper half plane and M is chosen sufficiently large that M > Im kj , j = 1,. . . ,k0 . Then h(ζ) is holomorphic in the upper half plane, continuous in the closed upper half plane, and |h(ζ)| ≤ CeCIm ζ in the closed upper half plane. An application of Carleman’s Theorem to h(ζ) finishes the proof. Acknowledgments. It is a pleasure to thank Maciej Zworski for suggesting the problem of counting resonances in this setting. I am grateful to him and to Dan Edidin for helpful discussions. Thanks to Julian Edward and the referee for helpful comments.
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References [1] A. Aslanyan and E.B. Davies, Separation of variables in perturbed cylinders, preprint. [2] A. Aslanyan, L. Parnovski and D. Vassiliev, Complex resonances in acoustic waveguides, Quart. J. Mech. Appl. Math. 53, no. 3, 429–447 (2000). [3] W. Bulla, F. Gesztesy, W. Renger and B. Simon, Weakly coupled bound states in quantum waveguides, Proc. Amer. Math. Soc. 125, no. 5, 1487–1495 (1997). [4] T. Christiansen, Scattering theory for manifolds with asymptotically cylindrical ends, J. Funct. Anal. 131, 2, 499–530 (1995). [5] T. Christiansen and M. Zworski, Spectral asymptotics for manifolds with cylindrical ends, Ann. Inst. Fourier 45, 1, 251–263 (1995). [6] E.B. Davies and L. Parnovski, Trapped modes in acoustic waveguides, Quart. J. Mech. Appl. Math. 51, no. 3, 477–492 (1998). [7] P. Duclos, P. Exner and B. Meller, Exponential bounds on curvature-induced resonances in a two-dimensional Dirichlet tube, Helv. Phys. Acta 71, no. 2, 133–162 (1998). [8] J. Edward, On the resonances of the Laplacian on waveguides. To appear, Journ. Math. Anal. and Appl. [9] R. Froese, Asymptotic distribution of resonances in one dimension, J. Differential Equations 137, no. 2, 251–272 (1997). [10] F.R. Gantmacher, The Theory of Matrices, Volume I. Chelsea Publishing Company, New York, 1959. [11] L. Guillop´e, Th´eorie spectrale de quelques vari´et´es `a bouts, Ann. Scient. Ec. Norm. Sup. 22, 4, 137–160 (1989). [12] L. Guillop´e and M. Zworski, Upper bounds on the number of resonances for non-compact Riemann surfaces, J. Functional Analysis 129, No. 2, 364–389 (1995). [13] L. Guillop´e and M. Zworski, Scattering asymptotics for Riemann surfaces, Annals of Mathematics 145, 597-660 (1997). [14] B. Ja. Levin, Distribution of zeros of entire functions, American Mathematical Society, Providence, R.I. 1964 viii+493 pp. [15] R.B. Melrose, ‘Polynomial bounds on the distribution of poles in scattering ´ by an obstacle’, Journ´ees “Equations aux D´eriv´ees partielles” Saint-Jean-deMonts, 1984.
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[16] R.B. Melrose, The Atiyah-Patodi-Singer Index Theorem, A.K. Peters, Wellesley, MA 1993. [17] W. M¨ uller, Spectral geometry and scattering theory for certain complete surfaces of finite volume, Invent. Math. 109, 265–305 (1992). [18] L. Parnovski, Spectral asymptotics of the Laplace operator on manifolds with cylindrical ends, Int. J. Math. 6, 911-920 (1995). [19] A. Selberg, G¨ ottingen lectures in: Collected Works, Vol. I, 626–674, SpringerVerlag, Berlin, 1989. [20] J. Sj¨ ostrand and M. Zworski, Complex scaling and the distribution of scattering poles, J. of the AMS, 4, No. 4, 729–769 (1991). [21] G. Vodev, Sharp bounds on the number of scattering poles for perturbations of the Laplacian, Comm. Math. Phys. 146, 205–216 (1992). [22] G. Vodev, Resonances in the Euclidean scattering, Cubo Matem´ atica Educacional 3 no. 1, 317–360 (2001). [23] M. Zworski, Distribution of poles for scattering on the real line, J. Funct. Anal. 73 (2), 277–296 (1987). [24] M. Zworski, Sharp polynomial bounds on the number of scattering poles of radial potentials, J. Funct. Anal. 82, 370–403 (1989). [25] M. Zworski, Sharp polynomial bounds on the number of scattering poles, Duke Math. Jour. 59 (2), 311–323 (1989). [26] M. Zworski, Counting scattering poles, Spectral and scattering theory (Sanda, 1992), 301–331, Lecture Notes in Pure and Appl. Math. 161, Dekker, New York, (1994). T. Christiansen1 Department of Mathematics University of Missouri Columbia, Missouri 65211 USA email: [email protected] Communicated by Bernard Helffer submitted 21/12/02, accepted 18/05/02
1 Partially
supported by the NSF grant DMS 0088922.
Ann. Henri Poincar´e 3 (2002) 921 – 938 c Birkh¨ auser Verlag, Basel, 2002 1424-0637/02/050921-18
Annales Henri Poincar´ e
Analyticity of the Gibbs State for a Quantum Anharmonic Crystal: No Order Parameter R. A. Minlos, E. A. Pechersky and V. A. Zagrebnov Abstract. We prove that for a ν-dimensional quantum crystal model of interacting anharmonic oscillators of mass m there exists m0 such that in the light-mass domain 0 < m < m0 the corresponding Gibbs state is analytic with respect to external field (conjugate to site displacements) for all temperatures T ≥ 0, i.e. including the ground state. This means that for the model with harmonic interaction and a symmetric double-well one-site potential, the light-mass quantum fluctuations suppress the symmetry breaking structural phase transition known in this model for ν ≥ 3 and m > M0 ≥ m0 , where M0 is large enough.
1 Introduction In the paper [1] the existence of Gibbs states for a ν-dimensional quantum anharmonic crystal model was proved for all temperatures T : 0 ≤ T < ∞, and for small masses m < m0 of one-dimensional oscillators. This result has been recently extended to the multi-dimensional oscillators in [2]. The proof in [1] is based on a reduction of this quantum model to a classic one for ensemble of continuous trajectories (paths, or ”spins”) as a configuration space with help of the FeynmanKac formula. Using this reduction the cluster expansion method was developed in [1] for the random field of paths to construct the corresponding infinite-volume quantum Gibbs state. The standard question of Statistical Mechanics is whether this state is unique? In this particular case it splits into two questions : the first concerns the uniqueness of the classical counterpart (field of paths), whereas the second question concerns the uniqueness of the quantum state. It is known that in our model the later is not unique at low temperatures as soon as ν ≥ 3, and the oscillator mass m is sufficiently large : m > M0 (”classical” limit), see [3] - [6]. For a symmetric double-well anharmonic oscillators this nonuniqueness is due to a symmetry breaking structural phase transition with a nonzero displacement order parameter at low temperatures T < Tc (m). On the other hand, one can prove that there exists m0 > 0 such that for m < m0 (”quantum” limit) this phase transition is suppressed for all temperatures T ≥ 0 by microscopic quantum fluctuations (tunneling effect) [7]. In the conjecture concerning our model it is crucial that on the phase diagram (m, T ) the line of the critical temperatures Tc (m) decreases for decreasing m and becomes Tc (m0 ) = 0 for some non-zero threshold mass m0 > 0. This conjecture about the critical line Tc (m), and the
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phenomenon of suppression of the phase transition by quantum fluctuations can be checked explicitly for the spherical approximation of this quantum anharmonic crystal model [8], [9]. There it was shown that (m = m0 > 0, T = Tc (m0 ) = 0) is indeed the critical point of a quantum phase transition from non-zero to zero displacement order parameter when decreasing mass crosses m0 > 0 along the line T = 0. Similar phenomenon is known in the quantum X − Y (or the Ising) model in a transverse magnetic field, see e.g.[10]. Moreover, it is also known that with decreasing mass m the macroscopic quantum fluctuations change their properties. In the ”quantum” limit they all become normal, see [11]-[13]. Whereas in the phase transition regime (i.e. on the critical line Tc (m) ≥ 0) they could be abnormal or even supernormal (squeezed), which is known for the moment only for the corresponding spherical approximation, see [8],[9]. It is noteworthy that the uniqueness problem of the classical counterpart of our model can be posed in the spirit of the DLR-theory [14],[15]-[18], since definition of the classical ensemble of paths introduces specifications, which are a set of consistent measures [1]. Hence, uniqueness for the classical counterpart has a clear sense and means the existence only one Gibbs random field of paths, see [15]-[18], where it is proved in the light-mass limit for temperatures strictly separated from zero. By virtue of noncompactness of the path (”spin”) variables, one has to take a precaution to consider only tempered Gibbs measures [19]. We have to mention that in fact, the paper [1] proves the uniqueness of the Gibbs state of the ν-dimensional quantum crystal model for all temperatures including zero 0 ≤ T < ∞, in the following restricted sense : Let be boundary condition fixed by the ”freezing” of particle positions on the surface of the crystal. This corresponds to the straight-line configurations of quantum paths on the whole space-time boundary. If these positions belong to a fixed bounded interval, then the corresponding infinite random Gibbs field of paths is unique, i.e. independent of these boundary conditions, in the class of tempered Gibbs measures in the light-mass domain, i.e., as soon as m < m0 . The same is true if we fix boundary conditions by the ”freezing” of the surface particle quantum paths in such a way that they would have some uniformly bounded amplitudes. This manner to fix boundary conditions (i.e. quantum Gibbs measure) seems not to be acceptable from the physical point of view, since we manipulate with the particles as with classical objects. More acceptable is to fix the Gibbs state by a (conjugate to displacements) external field. We shall follow this strategy below . As far as concerns the uniqueness problem of quantum states, at the present time there is no conventional setting of the question that would have a clear physical sense, except the KMS uniqueness [20]. In the present paper we prove (Theorem 1 ) analytic properties of the model with respect to the external field in the light-mass domain m < m0 for any T ≥ 0. This means in particular that for symmetric double-well anharmonic potential we can rule out (Theorem 2 ) symmetry breaking phase transitions in ”direction” of the external field for all temperatures including T = 0, cf. [7]. It is relevant to em-
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phasize that similar to [15]-[18], our method does not solve the uniqueness problem in the DLR-sense in the light-mass domain at T = 0, since it does not prove the uniqueness of the Gibbs field of paths for this temperature, if the boundary trajectories do not have limited amplitudes. The main obstacle to making this conclusion is the noncompactness of the path (”spin”) variables. However, since our Theorem 2 shows that in the light-mass domain m < m0 there is no order parameter corresponding to the symmetry breaking structural phase transition that exists for m > M0 , one could anticipate in this domain the uniqueness of the quantum Gibbs state for all temperatures including T = 0. For the proof in the case of the compact ”spins” see [21]. Notice that this makes a striking difference between the quantum model and its classical analog corresponding formally to m → ∞ (or the Planck constant → 0): in the quantum case the suppression of any ordering is expected even at zero temperature, due to the tunneling microscopic quantum fluctuations for sufficiently light masses m < m0 .
2 The model and main results We study a quantum model of interacting one-dimensional anharmonic oscillators on the lattice Zν , ν ≥ 1. For any finite Λ ⊂ Zν the Hamiltonian of the model is the operator: HΛh = −
2 ∂ 2 + 2m ∂qx2 x∈Λ
(qx − qy )2 +
V (qx ) − h
x∈Λ
x,y∈Λ,x−y=1
qx
(1)
x∈Λ
which acts in the Hilbert space HΛ = L2 (RΛ , dΛ q) of functions defined on the set of oscillator configurations QΛ = {qx ∈ R1 : x ∈ Λ} ∈ RΛ . In (1) > 0 is a parameter of the nearest-neighbor harmonic interaction, and V is a one-site potential , which is a real polynomial function of the form V (q) = a0 q 2s + a1 q 2s−1 + · · · + a2s−2 q 2 ,
(2)
where s > 1 and a0 > 0. This means that below we consider empty boundary conditions. It is semibounded from below and For real h the operator HΛh isself-adjoint. h −βHΛ . it generates the Gibbs semigroup e β≥0
Remark 1 It is worth to stress that below we consider the integer s > 1, that leads s−1 to a small scaled interaction = m s+1 in the light-mass limit. This is a key to the cluster expansion constructed in [1]. On the other hand the case s ≤ 1 leads in the light-mass limit to harmonic-term domination, see discussion in [7]. We extracted the last (linear) term in (1) from V in order to find an explicit expression of cluster coefficients as functions of the external field h conjugated to
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displacements {qx }x∈Λ . Let B(HΛ ) be algebra of bounded operators on HΛ . The finite-volume Gibbs state on B(HΛ ) is the continuous linear functional AhΛ,β =
h 1 Tr(e−βHΛ A), A ∈ B(HΛ ), h ZΛ,β
(3) h
where 0 < β < ∞ is the inverse temperature : β −1 = kB T . Since e−βHΛ is a traceh h class (nuclear) operator for β > 0, the partition function ZΛ,β = Tre−βHΛ exists. Therefore, the functional (3) is well-defined on B(HΛ ) for 0 < β < ∞. Moreover, there exists the extension of ·hΛ,β to the case β = ∞. Namely, h h , ψΛ,0 ), AhΛ,∞ = lim AhΛ,β = (AψΛ,0
(4)
β→∞
h is the ground-state eigenvector corresponding to the minimal eigenvalue where ψΛ,0 h
of HΛh . Notice that for any bounded Λ ⊂ Zν the Gibbs semigroup {e−βHΛ }β≥0 is h defined for any complex h ∈ C , and e−βHΛ ∈ T r − class(HΛ ) of operators on HΛ h for β > 0. Moreover, ZΛ,β is a holomorphic function of h ∈ C , see e.g. [22] and h=0 [23]. Since ZΛ,β > 0 , there exists a real h0 (m, Λ) > 0 such that the states (3) and (4) are defined on B(HΛ ) for complex h : |h| < h0 (m, Λ). In fact, we shall show that h0 does not depend on Λ in the light-mass domain m < m0 . ν Let Λ1 ⊂ Λ2 ⊂ · · · and ∪∞ n=1 Λn = Z , and let A∞ be the closure of the 0 inductive limit A∞ of increasing the subalgebras B(HΛ1 ) ⊂ B(HΛ2 ) ⊂ · · · , where the embedding B(HΛn ) ⊂ B(HΛn+1 ) is defined by the representation HΛn+1 = HΛn ⊗ HΛn+1 \Λn . If for any A ∈ A0∞ there exists the limit Ahβ = lim AhΛn ,β , 0 ≤ β −1 < ∞, n→∞
(5)
then ·hβ is called a (limiting) Gibbs state. Since the state (5) is norm-continuous, it can be extended to a state on A∞ . Recall that 1 ln ZΛn (β, h) f (β, h) = lim − (6) n→∞ β|Λn | is the thermodynamic limit of the free-energy density. The following theorem is the first main result of the present paper : Theorem 1 There exists a mass m0 > 0 such that for all 0 < m < m0 : (a) There is h0 (m) = inf Λ h0 (m, Λ) > 0 such that the limits (5) and (6) exist in the domain {0 ≤ β −1 < ∞} × {h ∈ C : |h| < h0 (m)}.
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(b) The limit f (β, h) is an analytic function in the circle {h ∈ C : |h| < h0 (m)} for any β −1 ≥ 0. (c) For any β −1 ≥ 0 and A ∈ A0∞ the function Ahβ is analytic in the circle {h ∈ C : |h| < h0 (m)}. Remark 2 It is an open question, whether h0 (m) does depend on m in the light mass domain 0 < m < m0 . Remark 3 Again it is unknown, whether (c) is true for the state ·hβ . The positive answer would follow from the uniform estimate: Ahβ ≤ C A , for A ∈ A0∞ and {0 ≤ β −1 < ∞} × {h ∈ C : |h| < h0 (m)}. In our estimates below (see (29)) the coefficient C = CA is A-dependent.
3 Proofs The Hamiltonian (1) can be represented as HΛh = −
1 ∂2 + 2m ∂qx2 x∈Λ
(qx − qy )2 +
Vh (qx )
x∈Λ
x,y∈Λ,x−y=1
where Vh (q) := V (q) − hq, and we put for simplicity = 1. Therefore, the Hamiltonian HΛh satisfies all conditions required in [1] for real h. Notice that in [1] the proof of existence of the state is based on cluster expansions. However, to study the analyticity of the state (5), or the limiting free-energy (6) for complex h, we need another form of the cluster expansions, which explicitly takes into account the cluster coefficient dependence on h. To this end we first modify the technique h of [1] by giving a new form of the cluster expansion of the partition function ZΛ,β for non-zero temperatures β < ∞ as well as for the case β = ∞, which requires some special attention. With the new cluster expansion in hands one can apply the construction of [1] to obtain the free-energy density and the limiting Gibbs states, when Λ ↑ Zν . Therefore, below we sketch a particular generalization of [1] to Vh stressing in the proofs only new aspects related to h−analyticity, and omitting details which the reader can find in [1].
3.1
Dilatation and reduction to classical ensemble of trajectories
Let U : L2 (RΛ , dΛ q)→L2 (RΛ , dΛ q) be the unitary mapping |Λ|/(4s+4) 1 f ({qx }x∈Λ ) (U f )({ qx }x∈Λ ) = m generated by the dilatation: qx =
1 m
1/(2s+2) qx .
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Then taking into account the transformations 2s−1
β = m s+1 β,
h = m 2s+2 h
s
we obtain for the state (3) the following representation: 1
h
·Λ,β =
Zh Λ,β
h
Tr{e−βHΛ (·)}.
(7)
Here Λh = − 1 H 2 where
∂2 + ∂ qx2
x∈Λ
( qx − qy )2 +
V ( qx ) − h
x∈Λ
x,y∈Λ,x−y=1
1 2 V ( q ) = a0 q2s + a1 m 2s+2 q2s−1 + a2 m 2s+1 q2s−2 + · · · ,
=m
s−1 s+1
.
qx ,
(8)
x∈Λ
(9) (10)
Remark 4 Since below we consider exclusively the operators and the states (7) after dilatation U , the hat ” ” will be systematically omitted. Let the self-adjoint one-dimensional Schr¨ odinger operator with the polynomial potential (2) 1 ∂2 + V (q) h=− 2 ∂q 2 be defined in L2 (R1 , dq). Let ψ0 be its ground state, i.e. the unique positive normalized eigenvector corresponding to the minimal eigenvalue E0 . Then using the kernel of the Gibbs semigroup generated by h: Gt (q1 , q2 ) = e−th (q1 , q2 ), t ≥ 0, one can define a stationary diffusion process ξ with the invariant measure dν(q) = ψ02 (q)dq and with the transition probability density (with respect to dν) equal to : ρt (q1 /q2 ) =
Gt (q1 , q2 ) . ψ0 (q1 )ψ0 (q2 )e−E0 t
It is known that (with the probability 1) the sample paths of this process are continuous. Notice that for s > 1 (2) the transition probability density ρt (q1 /q2 ) → 1, as t → ∞, uniformly and at exponential rate, see [1, Corollary 5.1] or [2, Lemma 3.1] for a simpler proof. Let µβ be a restriction of the measure of the process ξ on the set C[−β, β] (see [1] for details). Let Cper [−β, β] = {ω(·) ∈ C[−β, β] : ω(−β) = ω(β)}, and let µper be the conditional measure of µβ conditioned by ω(−β) = ω(β). Notice β per that µβ can be interpreted as a measure corresponding to the stationary (with
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respect to rotations) diffusive process ξβ on the circle Oβ of the length 2β. By dνβ we denote a related to the process ξβ invariant measure, which has density ρ2β (q/q) with respect to the measure dν. The process ξβ is Markovian: if one fixes it in two points ξt1 = x1 and ξt2 = x2 on the circle, 0 < t1 − t2 < 2β, then its values on two complementary arcs are independent. ν Zν Let Mβ = µZβ and Mβper = (µper be product-measures on the spaces β ) ν ν (C[−β, β])Z and (Cper [−β, β])Z correspondingly. Then we can represent the partition function of the model with the Hamiltonian (8) using the Feynman–Kac formula: h,per h = aΛ ZΛ,β , (11) ZΛ,β where h,per := Z Λ,β (Cper [−β,β])Λ
(12) e−
x,y∈Λ,x−y=1
β
−β
(ωx (t)−ωy (t))2 dt+h
x∈Λ
β
−β
ωx (t)dt
Mβper (dωΛ ),
Here ωΛ = {ωx ∈ Cper [−β, β], x ∈ Λ} and aΛ = (Tre−βh )|Λ| . Notice that by virtue of our definition of the process ξ the potential V does not appear in the formal Hamiltonian corresponding to ensemble of trajectories ωΛ in (12). Since we are going to consider below the low temperature regime, it is convenient to use another representation of (12). Following [1] we make a partition of the interval [−β, β] into subintervals having the length a > 0 such that β = aN with an integer N . As in [1] we put a = a() to be a function of the parameter involved in the interaction term of (8), (12). Similar to [1] we shall choose the value of a() in dependence on properties of the stationary process ξ , see Remark 6 below. Notice that for any trajectory ωx = {ωx (t), −β ≤ t ≤ β} ∈ Cper [−β, β], x ∈ Zν , one can define restrictions : ω(x,k) := {ω(x,k) (t) = ωx (t + ka), 0 ≤ t ≤ a}
(13)
on the intervals [ka, (k + 1)a], k ∈ {−N, −N + 1, . . . , N − 1} := [−N, N − 1] of the partition of the interval [−β, β] . For those k s and any x ∈ Zν the restrictions verify the continuity conditions : q(x,k+1) := ω(x,k+1) (0) = ω(x,k) (a).
(14)
In the case of periodic trajectories we consider the residue group z2N , i.e. k ∈ z2N . Then we have q(x,N ) = q(x,−N ) . (15) Therefore, in fact the configurations {ωx ∈ Cper [−β, β], x ∈ Zν } coincides with a collection of ”elementary” trajectories defined by the restrictions (13) satisfying conditions (14) and (15). We denote this collection by ωβ := {ω(x,k) , x ∈ Zν , k ∈ z2N } ∈ Ωβ ,
(16)
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Ann. Henri Poincar´e
where Ωβ is the set of those trajectories, and the conditions (14) and (15) are assumed. Let Λ ⊆ Zν and U = Λ × z2N . Then we define on the configurations of ”elementary” trajectories (16) the interaction Hamiltonian by a HU,h (ωβ ) := (ω(x,k) (t) − ω(y,k) (t))2 dt (17) 0
(x,k),(y,k)∈U x−y=1 a
−h
ω(x,k) (t)dt.
(x,k)∈U
0
Since the product measure Mβper is also defined on the space Ωβ of the ”elementary”
,h,per trajectories (16), the corresponding to HΛ,h (ωβ ) perturbed Gibbs measure Mβ,Λ is defined by the Radon-Nikodim derivative: ,h,per dMβ,Λ
dMβper
,h
e−HU (ωβ ) (ωβ ) = . Zh,per
(18)
Λ,β
Let Q = {q(x,k) ∈ R1 : (x, k) ∈ Zν × z2N }. Then by ·Q we denote the conditional expectation with respect to Mβper given the configuration Q. Then for the partition function (12) we get the following representation : h,per = Z Λ,β
= RU
a
2 exp − (ω(x,k) (t) − ω(y,k) (t)) dt ×
0
(x,k),(y,k)∈U, x−y=1
(x,k)∈U
exp h
(x,k),(x,k+1)∈U
×
ω(x,k) (t)dt
0
a
Q
ρa (q(x,k+1) /q(x,k) )
ν(dq(x,k) ). (19)
(x,k)∈U
Notice that ·Q depends only on QU = {q(x,k) ∈ R1 : (x, k) ∈ U = Λ × z2N }. As above ρa (q /q) denotes the transition probability density for the time-interval a with respect to the invariant measure dν(q) = ψ02 (q)dq of the stationary diffusion process ξβ on the circle Oβ . The similar formulae hold for the case of the measure Mβ , i.e. nonperiodic trajectories, when U = Λ × [−N, N − 1] and the condition (15) is not required.
3.2
Cluster expansion
Notice that in [1] we first obtained cluster expansion for the partition function for the nonperiodic measure Mβ . Then we extended it to the case of the periodic
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measure Mβper via some routine transformation, see [1, Section 5.3] for details. Below we proceed in the similar way. Having the cluster expansion for the partition function one can obtain then the corresponding expansions for observables using a standard method of [19], see also [1]. h for the Therefore, our next step is to obtain the cluster expansion for ZΛ,β nonperiodic measure Mβ . To this end we start (cf.(19)) with partition function ZUh (Q) conditioned by the configuration Q:
a
h ZU (Q) := exp − (ω(x,k) (t) − ω(y,k) (t))2 dt × (x,k),(y,k)∈U,x−y=1
0
exp h
(x,k)∈U
a
ω(x,k) (t)dt
0
. (20) Q
In fact this expression is conditioned only by the subconfiguration QU = {q(x,k) ∈ R1 : (x, k) ∈ U = Λ × [−N, N − 1]}. By virtue of the Markovian property of the process ξ one gets from (20) that
a
h ZU (Q) = exp − (ω(x,k) (t) − ω(y,k) (t))2 dt × −N ≤k
x,y∈Λ,x−y=1
0
x∈Λ
exp h 0
a
ω(x,k) (t)dt
. (21) Q
Now we introduce some useful for below notations and definitions. Let x, y ∈ Zν be such that x − y = 1 and let k ∈ [−N, N − 1]. Then the set of the four sites on the lattice Zν × [−N, N − 1] : = {(x, k), (x, k + 1), (y, k), (y, k + 1)} is called a plaquette. The number L() = k is called the level of the plaquette , whereas B() = x, y will denote its basis . The two-point set = {(x, k), (x, k + 1)} we call time-edge or simply edge. The number L() = k we call the level, and B() = x the basis of the edge . We shall also treat the plaquette = {(x, k), (x, k + 1), (y, k), (y, k + 1)} as a set of two time-edges = {1 () = {(x, k), (x, k + 1)}, 2 () = {(y, k), (y, k + 1)}} Let C = {1 , . . . , l , 1 , . . . , s } be a collection (cluster) of elementary sets: the plaquettes and the triangles. Then the union of these sets : ∂C = (∪li=1 i ) ∪ (∪sj=1 j ), coincides with the set of sites of the cluster C. We say that the cluster C is disconnected, if it can be presented as a union : C = C1 ∪ C2 of non-empty clusters such that ∂C1 ∩ ∂C2 = ∅; otherwise it is connected. Thus each non-empty cluster C is a union of connected components. Connected collection γ = {1 , . . . , p , 1 , . . . , r } of different plaquettes and edges having the same level k we call a contour, L(γ) = k is the level of the
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contour. For the contours B(i ) ∩ B(j ) = ∅ implies i ∩ j = ∅. A connected collection τ = {1 , . . . , q } of different edges, i.e. the edges with the same basis x, is called a series. Let p = p(γ) and r = r(γ) be correspondingly the number of plaquettes and the number of edges of the contour γ. Similarly, q(τ ) denotes the number of edges in the series τ . By ∂γ we denote the set of sites in γ, and by ∂τ the set of sites in τ . 0 Let Γ be a connected collection composed by contours {γi }li=1 and series l1 {τj }j=1 : Γ = {γ1 , . . . , γl0 , τ1 , . . . , τl1 }. (22) Γ is called an aggregate, and ∂Γ will denote the set of sites in Γ. The aggregate satisfies the following properties : - For any pair of different contours γi and γj ∈ Γ such that k(γi ) = k(γj ) we have ∂γi ∩ ∂γj = ∅. - One has ∂τi ∩ ∂τj = ∅ for any different series τi and τj of the aggregate Γ. Let = {(x, k), (y, k)(x, k + 1), (y, k + 1)}, x − y = 1 be a plaquette. Then we use the notations ω1 () (t), ω2 () (t) for the pair of paths ω(x,k) , ω(y,k) . Besides, if = {(x, k), (x, k + 1)}, then we denote the path between those points 1 2 by ω (t) = ω(x,k) (t) and q = ω (0), q = ω (a). Now we can rewrite (20), (21) identically in the following form:
ZUh (Q)
=
⊂U
a 2 (ω1 () (t) − ω2 () (t)) dt − 1 + 1 × exp − 0
exp h
⊂U N −1 k=−N
a
ω (t)dt − 1 + 1
0
1 +
= Q
γi (Q) , K
s≥1 {γ1 ,...,γs :k(γi )=k,i=1,...,s} 1≤i≤s
where γ (Q) := K
∈γ
a (ω1 () (t) − ω2 () (t))2 dt − 1 × exp − 0
exp h ∈γ
0
a
ω (t)dt − 1
Q
. (23)
γ (Q) is conditioned by the two-level configuration {q(x,k) ∈ R1 : Notice that K (x, k) ∈ U = Λ × [k, k + 1]}. Since the transition probability density ρt (q1 /q2 ) → 1, as t → ∞, uniformly and at exponential rate, we can use [1] to obtain the cluster expansion coefficients for the partition function in the presence of the external
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Analyticity of the Gibbs State for a Quantum Anharmonic Crystal
field h. Let
KΓ (h) :=
γ (Q) K
γ∈Γ
931
2 1 [ρa (q /q )
− 1]
τ ∈Γ ∈τ
,
(24)
ν
where ·ν means expectation with respect to the product-measure ν(dq(x,k) ). Then (cf.(19)) we get : h ZΛ,β =1+
s
(x,k)∈U
KΓi (h).
(25)
s≥1 Γ1 ,...,Γs i=1
Let AL ∈ A0,comm ⊂ A0∞ . Here A0,comm is a subalgebra of multiplication op∞ ∞ erators generated by the local bounded functions : AL := AL (QL ), L ⊂ Zν , |L| < := {(x, 0) : x ∈ L} ⊆ Zν+1 . Then the set of expectations AL h ∞, where L β defines a state on A0,comm . (Notice that the Gibbs reconstruction of the state on ∞ the whole algebra A0∞ can be realized by the same manner as it is done in [1].) Since by definition qx = q(x,k=0) = ω(x,k=0) (t = 0), any observable AL is in fact a function of paths (13), similar to [1, Section 6.1],[19, Ch.III] one gets for AL hβ the representation AL hβ = Kξh (AL )D∂ξ∪L (η) KΓ (h), (26) ξ,η
Γ∈η
with some coefficients DR (η). = ∅, Here ξ = {Γ1 , . . . , Γs } are such that ∂Γi ∩ ∂Γj = ∅ for i = j and ∂Γi ∩ L whereas η = {Γ1 , . . . , Γp } is a collection of aggregates (possibly duplicated) such = ∅. Here that the set ∂Γ1 , . . . , ∂Γp is connected, but ∂η ∩ (∂ξ ∪ L)
Kξh (AL )
=
AL (QL )
γ (Q) K
Γ∈ξ γ∈Γ
2 1 [ρa (q /q )
− 1]
.
(27)
τ ∈Γ ∈τ
Notice that (26) is defined for any set of the weights KΓ (h) and Kξh (AL ) satisfying the estimates of Proposition 1 we are going to prove below. The same estimates show that coefficients KΓ (h) and Kξh (AL ) are analytic functions in the circle {h ∈ C : |h| < h0 } for some small h0 . Therefore, AL hβ is, in turn, an analytic (in this . circle) function for each local operator AL ∈ A0,comm ∞ Proposition 1 There are 0 > 0 and h0 > 0 such that for || < 0 , |h| < h0 , and Γ = (γ1 , . . . , γs , τ1 , . . . , τl ) one has the estimate: |KΓ (h)| < C0
γ∈Γ
p(γ) r(γ)
λ()
h
τ ∈Γ
λ()
q(τ )
.
(28)
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Ann. Henri Poincar´e
Here λ() is such that lim→0 λ() = 0. Similarly one has : h K (AL ) < AL |KΓ (h)|, ξ
(29)
Γ∈ξ
where AL = supR|L| {|AL ({qx }x∈L )|}. The both KΓ (h) and Kξh (AL ) are analytic functions of h in the circle {h ∈ C : |h| < h0 }. Remark 5 The coefficients DB (η) are such that if (28) holds, then |DB (η)| |KΓ (h)| < C |B| , η: ∂η∩B =∅
(30)
Γ∈η
where B ⊂ Zν+1 and C > 0 (see [19]). Proof of Proposition 1. To get the estimates (28) and (29) we follow the method of [1]. Our main tool will be Lemma 5.2 [1]. For reader’s convenience we formulate it below : Lemma 1 Let {(Ex , µx )}x∈T be a family of probability spaces indexed by a finite space T. LetS = {S} be a family of subsets of T. For any S ∈ S let functions fS : ET = x∈T Ex → R be such that fS depends only on variables of Ex for x ∈ S. At last let nS be such that for every x ∈ T 1 ≤1 nS
S: x∈S
Then
n1 S nS fS dµx ≤ |fS | dµx ET ET S
x∈T
S
x∈T
γ (Q), see (23). To this end we use With help of Lemma 1 we first estimate K Lemma 1 by taking all objects as follows: T is the set of time-edges included in γ with taking in account their possible multiplicity. S = { : = {1 (), 2 ()} ∈ γ} ∪ { : ∈ γ}. The probability 1 2 spaces are E = {ω ∈ C[0, a] : q = q(x,k) , q = q(x,k+1) } with a measure µ which is the conditional measure on E generated by µβ under boundary 1 2 conditions q = q(x,k) , q = q(x,k+1) . We put a
f (ω1 , ω2 ) = exp − (ω1 () (t) − ω2 () (t))2 dt − 1, (31) 0
for ∈ S , and
f (ω ) = exp h 0
a
ω (t)dt − 1,
(32)
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933
for ∈ S . Let integers n and n be attributed correspondingly to all plaquettes and time-edges in such a way that 2ν 1 + ≤ 1. n n
(33)
Then Lemma 1 implies for (23) that F (Q ) F (Q ), Kγ (Q) ≤ ∈γ
(34)
∈γ
where
n (n )−1 a
exp − F (Q ) = (ω1 () (t) − ω2 () (t))2 dt − 1 , Q
0
(35) and
n (n )−1
exp h ω (t)dt − 1 F (Q ) = .
(36)
Q
Here Q and Q are values conditioned by the configuration Q at the end-points of and correspondingly. Now, to estimate KΓ (h) (24) we again apply Lemma 1. We put T = ∂Γ, and S= [{ : ∈ γ} ∪ { : ∈ γ}] { : ∈ τ }. γ∈Γ
τ ∈Γ
The sign means that the edge will be counted twice, if it belongs to the both sets on the left- and right-hand sides of . The probability spaces are E(x,k) = R with the measures µ(x,k) = ν. We have three kinds of functions with supports on the end-points of plaquettes and edges : F (Q ), F (Q) for ∈ γ, and 2 1 /q ) − 1|, for ∈ τ . Then we choose the integer n , n satisG (Q) = |ρa (q fying 4ν 4 + ≤ 1, n n that implies (33). Then applying Lemma 1 we obtain the bound (n )−1 F (Q )n |KΓ (h)| ≤ ν(dq(x,k) ) γ∈Γ ∈γ (x,k)∈() (n )−1 1 2 × )ν(dq ) F (Q )n ν(dq ∈γ
×
τ ∈Γ ∈τ
n
G (Q )
(n )−1 1 2 ν(dq )ν(dq )
.
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Ann. Henri Poincar´e
Remark 6 Let a = a() := −A ln , where A > 0. Then following the same line reasoning as in [1, Section 5.2] the integrals F (Q )n
(n )−1
ν(dq(x,k) )
and
(n )−1 1 2 Gn ν(dq )ν(dq )
(x,k)∈()
can be estimated by λ() = R2/3 for n = 8ν and n = 8. Therefore, it rests only to estimate the integral: 1 2 F (Q )n ν(dq )ν(dq ), which gives dependence on the external field h. Our estimate below is based on Lemma 5.3 of [1]. Since distribution density of the random variable q 1 = ω (t = 0) with respect to ν is unit, this lemma claims that there exist b > 0 such that ρa (q 1 , q 2 ) = ρa (q 2 /q 1 ) > b for any q 1 , q 2 , and a ≥ a0 (). Here ρa (q 1 , q 2 ) is the density of the stationary distribution of the pair of random variables (ω (0), ω (a)) with the respect of the measure ν × ν. Then 1 2 )ν(dq )= F (Q )n ν(dq n a
1 2 )ν(dq ) ν(dq 2 1 exp h ≤ ω (t)dt − 1 ρ (q /q ) a 2 1 ρa (q /q ) 1 ,q 2 ) 0 R2 (q n a
1 µβ (dω ) =: J, exp h ω (t)dt − 1 ∆ b 0 where µβ is the measure corresponding to the process ξ. The representation a 1 a
a exp h = |h| ω (t)dt − 1 ω (t)dt ehs 0 ω (t)dt ds 0
0
0
gives the bound n
J ≤ |h|
1 b
a 0
n a ω (t)dt en |h| 0 |ω (t)|dt
(37)
µβ
Using results of [1, Section 5.2] we conclude that the expectation with respect to µβ in the right-hand side of (37) is bounded by some C > 0 as soon as |h| < h0 = h0 () and a < a(). Therefore we obtain (28). The estimate of coefficients Kξh (AL ) goes through verbatim of the above arguments, that gives (29). Since the same line of
Vol. 3, 2002
Analyticity of the Gibbs State for a Quantum Anharmonic Crystal
935
reasoning as above yields similar estimates for derivatives ∂h KΓ (h) and ∂h Kξh (AL ) in the circle {h ∈ C : |h| < h0 ()}, the functions KΓ (h) and Kξh (AL ) are analytic in this circle. This finishes the proof of Proposition 1. Proof of Theorem 1. Now the statements (a),(b) and (c) of Theorem 1 follow from the Proposition 1, representations (25),(26), and relation (10) between ( ) and m.
3.3
Order parameter
We recall that the order parameter for the model (1) is defined by : σ(β) := lim lim h→0 n→∞
1 qx hΛn ,β . |Λn |
(38)
x∈Λn
If the one-site potential V (q) in (1) is double-well and symmetric, then for ν ≥ 3 (see [3]-[6]) there is M0 such that for m > M0 there exists a critical temperature Tc (m) > 0. At Tc (m) one has a symmetry breaking phase transition with a nonzero displacement order parameter at low temperatures: σ± (β > βc (m)) := lim lim h→±0 n→∞
1 qx hΛn ,β = ±σ+ (β > βc (m)), |Λn |
(39)
x∈Λn
and one gets zero order parameter for T ≥ Tc (m) : σ± (β ≤ βc (m)) = 0.
(40)
Now we can proof our second main result : Theorem 2 Let V (q) be a double-well symmetric one-site potential with s > 1. Then there exists a mass m0 > 0 such that for any 0 < m ≤ m0 and for all temperatures, including β = ∞, the order parameter is trivial for empty boundary conditions: (41) σ± (β) = 0. Proof. This statement is in fact a corollary of some general results of [19] and assertion (b) of our Theorem 1. Indeed, one has that: σΛn (β, h) :=
1 qx hΛn ,β = ∂h fΛn (β, h), |Λn |
(42)
x∈Λn
where fΛn (β, h) := −
1 ln ZΛn (β, h) β|Λn |
(43)
is the finite-volume free-energy density, cf (6). By virtue of cluster representation of partition function (25) one gets that ZΛn (β, h) = 0, the function fΛn (β, h) is
936
R. A. Minlos, E. A. Pechersky and V. A. Zagrebnov
Ann. Henri Poincar´e
analytic in h in some (independent of the volume) circle in the complex plane. Moreover, the limit (6) exists and it is analytic in this circle , see [19, Ch.III,§4]. Then by Theorem 1(b) the limit σ(β, h) := limn→∞ σΛn (β, h) is an analytic function in the circle {h ∈ C : |h| < h0 (m)}. Since by symmetry of the one-site potential we have σΛn (β, h = 0) = 0, one gets limh→±0 σ(β, h) = 0 for any temperature as soon as 0 < m < m0 .
Acknowledgments This paper was originated during E.A.P.’s stay at CPT-CNRS-Luminy and V.A.Z.’s stay at IITP-Moscow funded by the French-Russian Project 7787. The manuscript reached its final form during R.A.M.’s visit to CPT-CNRS-Luminy funded by the same Project and grants of RFFI (Grant 99-01-00284) and CRDF (Grant RM1-2085). E.A.P. also would like to acknowledge the latter Grant as well as RFFI (Grant 99-01-00003) and A.N.Lyapunov Center at MSU (Project 98-02). We thank the referee for numerous useful remarks.
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[9] A. Verbeure and V.A. Zagrebnov, Dynamics of Quantum Fluctuations Operators in an Anharmonic Crystal Model, J. Stat. Phys. 79, 377–393 (1995). [10] P. Pfeuty and R.J. Elliott, The Ising Model with a Transverse Field - II Ground State Properties, J. Phys. C 4, 2370–2385 (1971). [11] S. Albeverio, Yu. G. Kondratiev,and Yu. Kozitsky, Suppression of Critical Fluctuations by Strong Quantum Effects in Quantum Lattice Systems, Commun. Math. Phys. 194, 493–512 (1998). [12] Yu. Kozitsky, Quantum Effects in a Lattice Model of Anharmonic Vector Oscillators, Lett. Math. Phys. 51, 71–81 (2000). [13] Yu. Kozitsky, Scalar Domination and Normal Fluctuations in N -vector Quantum Anharmonic Crystal, Lett. Math. Phys. 53, 289–303 (2000). [14] R.L. Dobrushin, Prescribing a system of random variables by conditional distributions, Teor. Veroatn. Primen. 15, 458–486 (1970) (transl. from the Russian). [15] S. Albeverio, Yu. G. Kondratiev, M. R¨ ockner and T.V. Tsikalenko, Uniqueness of Gibbs States for Quantum Lattice Systems, Prob. Theory Relat. Fields 108, 193–218 (1997). [16] S. Albeverio, Yu. G. Kondratiev, M. R¨ ockner and T.V. Tsikalenko, Dobrushin’s Uniqueness for Quantum Lattice Systems with Nonlocal Interaction, Commun. Math. Phys. 189, 621–630 (1997). [17] S. Albeverio, Yu. Kondratiev, Yu. Kozitsky and M. R¨ ockner, Uniqueness for Gibbs Measures of Quantum Lattices in Small Mass Regime, Ann. Inst. H. Poincar´e: Probab.Statist. 137, 43–69 (2001). [18] S. Albeverio, Yu. G. Kondratiev, R.A. Minlos and A.L. Rebenko, Small-Mass Behavior of Quantum Gibbs States for Lattice Models with Unbounded Spins, J. Stat. Phys. 92, 1153–1172 (1998). [19] V.A. Malyshev and R.A. Minlos, Gibbs Random Fields, Cluster Expansions, (Kluwer Acad.Publ., Dordrecht, 1991). [20] O. Bratteli and D.W. Robinson, Operator Algebras and Quantum Statistical Mechanics,Vol.2, Second edition (Springer, Berlin, 1996). [21] S. Albeverio, Yu. G. Kondratiev, R.A.Minlos and G.V.Shchepan’uk, Uniqueness Problem for Quantum Lattice Systems Models with Compact Spins, Lett. Math. Phys. 52, 185–195 (2000). [22] H.D. Maison, Analyticity of the Partition Function for Finite Quantum Systems, Commun. Math. Phys. 22, 166–172 (1971).
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[23] V.A. Zagrebnov, Perturbations of Gibbs Semigroups, Commun. Math. Phys. 120, 653–664 (1989). Robert A. Minlos and Eugene A. Pechersky Institute for Information Transmissions Problems Bolshoj Karetny per.19, GSP-4 Moscow 101447 Russia Valentin A. Zagrebnov Universit´e Aix-Marseille II Centre de Physique Th´eorique CNRS-Luminy-Case 907 F-13288 Marseille Cedex 09 France email: [email protected] Communicated by Joel Feldman submitted 30/10/01, accepted 24/05/02
To access this journal online: http://www.birkhauser.ch
Ann. Henri Poincar´e 3 (2002) 939 – 965 c Birkh¨ auser Verlag, Basel, 2002 1424-0637/02/050939-27
Annales Henri Poincar´ e
Effective N -Body Dynamics for the Massless Nelson Model and Adiabatic Decoupling without Spectral Gap S. Teufel Abstract. The Schr¨ odinger equation for N particles interacting through effective pair potentials is derived from the massless Nelson model with ultraviolet cutoffs. We consider a scaling limit where the particles are slow and heavy, but, in contrast to earlier work [7], no “weak coupling” is assumed. To this end we prove a spaceadiabatic theorem without gap condition which gives, in particular, control on the rate of convergence in the adiabatic limit.
1 Introduction The physical picture underlying nonrelativistic quantum electrodynamics is that of charged particles which interact through the exchange of photons and dissipate energy through emission of photons. In situations where the velocities of the particles are small compared to the propagation speed of the photons the interaction is given through effective, instantaneous pair potentials. If, in addition, also accelerations are small, then dissipation through radiation can be neglected in good approximation. Instead of full nonrelativistic QED we consider the massless Nelson model. This model describes N spinless particles coupled to a scalar Bose field of zero mass. The content of this work is a mathematical derivation of the time-dependent Schr¨ odinger equation for N particles with Coulombic pair potentials from the massless Nelson model with ultraviolet cutoffs. The key mechanism in our derivation is adiabatic decoupling without a spectral gap. Before we turn to a more careful discussion of the type of scaling we shall consider, notice that the coupling of N noninteracting particles to the radiation field has three effects. • The effective mass, or more precisely, the effective dispersion relation of the particles is modified. The term “effective” refers to the reaction of the particles to weak external forces. The physical picture is that each particle now carries a cloud of photons with it, which makes it heavier. • The particles feel an interaction mediated through the field. If the propagation speed of the particles is small compared to the one of the photons, then retardation effects should be negligible and the interaction between the particles can be described in good approximation by instantaneous pair potentials.
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• Energy is dissipated through photons moving freely to infinity. The motion of the particles is, in general, no longer of Hamiltonian type. The rate of energy emitted as photons is proportional to the acceleration of a particle squared. The scaling to be studied is most conveniently explained on the classical level. The classical equations of motion for N particles with positions qj , masses mj and rigid “charge” distributions ρj coupled to the scalar field φ(x, t) with propagation speed c are 1 ¨ φ(x, t) c2
=
mj q¨j (t)
=
∆x φ(x, t) − −
R3
N
ρj (x − qj (t))
(1)
j=1
dx (∇x φ)(x, t) ρj (x − qj (t)) ,
1≤j≤N.
(2)
One can think of ρj (x) = ej ϕ(x) as a smeared out point charge ej with a form factor ϕ ≥ 0 satisfying R3 dx ϕ(x) = 1. Taking the limit c → ∞ in (1) yields the Poisson equation for the field and thus, after elimination of the field, (2) describes N particles interacting through smeared Coulomb potentials. Mass renormalization for the particles in not visible at leading order. Instead of taking c → ∞ one can as well explore for which scaling of the particle properties one obtains analogous effective equations. Since retardation effects should be negligible, the initial velocities of the particles are now assumed to be O(ε) compared to the fixed propagation speed c = 1 of the field, ε 1. In order to see motion of the particles over finite distances, we have to follow this dynamics at least over times of order O(ε−1 ). To make sure that the velocities are still of order O(ε) after times of order O(ε−1 ), the accelerations must be at most of order O(ε2 ). The last constraint also guarantees that the energy dissipated over times of order O(ε−1 ) is at most of order O(ε3 ). The natural procedure would now be to consider such initial data, for which the velocities stay of order O(ε) over sufficiently long times. The problem simplifies if we assume, as we shall do in this work, that the mass of a particle is of order O(ε−2 ). As a consequence accelerations are and stay of order O(ε2 ) uniformly for all initial conditions. In this scaling limit mass renormalization is not visible at leading order. Indeed, if we substitute t = εt and mj = ε2 mj in (1) and (2), we find that the limit ε → 0 is equivalent to the limit c → ∞. After quantization, however, the two limiting procedures are no longer equivalent. The limit c → ∞ for the Nelson model was analyzed by Davies [7] and later also by Hiroshima [10], who removed the ultraviolet cutoff. A comparison of their results with ours can be found at the end of this introduction. We will adopt the point of view that it is more natural to explore the regime of particle properties which gives rise to effective equations than to take the limit c → ∞. The deeper reason for our choice is that the more natural procedure of restricting to appropriate initial conditions gives rise to a similar mathematical struc-
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ture. If the bare mass of the particles of order O(1), then the proper scaling which yields effective equations with renormalized masses was introduced and analyzed for the classical Abraham model by Kunze and Spohn, see [12, 21] and references therein. Denoting again the ratio of the velocities of the particles and the field as ε, they consider charges initially separated by distances of order O(ε−2 ) in units of their diameter. Hence the forces are O(ε4 ) initially. For times up to order O(ε−3 ) and for appropriate initial conditions – excluding head on collisions – the separation of the particles remains of order O(ε−2 ) and thus the velocities remain of order O(ε). In particular, the rescaled macroscopic position q (t ) = ε−2 q(t /ε3 ) satisfies (d/dt )2 q (t ) = O(ε4 ), which matches the order of the forces. As a consequence one obtains a sensible limiting dynamics for the macroscopic variables. One would expect that the same scaling limit applied to the quantum mechanical model yields in a similar fashion effective dynamics with renormalized dispersion. However, inserting this scaling into the massless Nelson model, one faces mathematical problems beyond those in the simpler m = O(ε−2 ) scaling. Without going into details we remark that the main problem is that for massless bosons the Hamiltonian at fixed total momentum does not have a ground state in Fock space, cf. [8, 6]. (As a consequence it is not even clear how to translate the result in [23] for a single quantum particle coupled to a massive quantized scalar field and subject to weak external forces to the massless case.) Nevertheless, the simpler scaling with m = O(ε−2 ) provides at least a first step in the right direction, since the mechanism of adiabatic decoupling without gap will certainly play a crucial role also in a more refined analysis. In the remainder of the introduction we briefly present the massless Nelson model, explain our main result and compare it to Davies’ “weak coupling limit” [7]. Up to a modified dispersion for the particles, the following model is obtained through canonical quantization of the classical system (1) and (2). The state space for N spinless particles is L2 (R3N ) and as Hamiltonian we take Hp =
N −c2max ∆xj + c4max m2 ,
(3)
j=1
where cmax is the maximally attainable speed of the particles and m their mass, = 1. As explained before, we consider the scaling limit ε 1 with
cmax = O(ε)
and m = O(ε−2 ) .
(4)
It might seem somewhat artificial to have a relativistic dispersion relation for the particles which does not contain the speed of light, but some other maximal speed cmax . This is done only for the sake of simple presentation. We could as well 1 consider the quadratic dispersion Hp = − N j=1 2m ∆xj for the particles. However, there would be no maximal speed and we would be forced to either introduce a cutoff for large momenta or to change the topology in (17). While both strategies
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are technically straightforward by using exactly the same methods as in [22] in the context of Born-Oppenheimer approximation, they would obscure the simple structure of our result. We insert the scaling (4) into (3) and change units such that the particle Hamiltonian is now given through Hpε =
N −ε2 ∆xj + 1 .
(5)
j=1
The particles are coupled to a scalar field whose state is an element of the bosonic Fock space over L2 (R3 ) given as m 2 3 F = ⊕∞ m=0 ⊗(s) L (R ) ,
(6)
0 2 3 where ⊗m (s) is the m-times symmetric tensor product and ⊗(s) L (R ) := C. The Hamiltonian for the free bosonic field is
Hf = dΓ(|k|) ,
(7)
where k is the boson momentum. In our units the propagation speed of the bosons is equal to one. The reader who is not familiar with the notation is asked to consult the beginning of Section 3, where the model is introduced in full detail. In the standard Nelson model the coupling between the j th particle and the field is given through HI,j =
R3
dy φ(y) ρj (y − xj ) ,
(8)
where φ is the field operator in position representation and xj the position of the j th particle. The charge density ρj ∈ L1 (R3 )∩L2 (R3 ) of the j th particle is assumed to be spherically symmetric and its Fourier transform is denoted by ρˆj . For the moment we also assume an infrared condition, namely that N ρˆj (k) ∈ L2 (R3 ) . 3/2 |k| j=1
(9)
Condition (9) constrains the total charge of the system but not that of an individual particle to zero. The state of the combined particles + field system is an element of H = L2 (R3N ) ⊗ F and its time evolution is generated by the Hamiltonian H ε = Hpε ⊗ 1 + 1 ⊗ dΓ(|k|) +
N j=1
HI,j .
(10)
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σ( H (x) ) 0
x
e0 E (x) 0
Figure 1: The spectrum of H0 (x) for N = 2. The thick line indicates the eigenvalue E0 (x) sitting at the bottom of continuous spectrum. Note that H contains no terms which directly couple different particles. All interactions between the particles must be mediated through the boson field. Our goal is the construction of approximate solutions of the time dependent Schr¨ odinger equation iε
d Ψ(t) = H ε Ψ(t) , dt
Ψ(0) = Ψ0 ∈ H
(11)
from solutions of an effective Schr¨ odinger equation iε
d ε ψ(t) = Heff ψ(t) , dt
ψ(0) = ψ0 ∈ L2 (R3N )
(12)
for the particles only. Notice the factor ε in front of the time derivative in (11) and (12), which means that we switched to a time scale of order ε−1 in microscopic units. As explained before, this is necessary in order to see nontrivial dynamics of the particles, since their speed is O(ε). We remark that the scaling (4) coincides with the one in time-dependent Born-Oppenheimer approximation, where m = O(ε−2 ) is the mass of the nuclei and where, at fixed kinetic energy, the velocities of the nuclei are also of order O(ε). The Hamiltonian (10) has the same structure as the molecular Hamiltonian and the role of the electrons in the Born-Oppenheimer approximation is now played by the bosons. The key observation for the following is that the interaction Hamiltonian depends only on the configuration x of the particles and that the operator H0 (x) = dΓ(|k|) +
N j=1
HI,j (x) ,
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which acts on F for fixed x ∈ R3N , has a unique ground state Ω(x) with ground state energy j−1 N E0 (x) = Vij (xi − xj ) + e0 , (13) j=2 i=1
where Vij (z) = − and
dv dw R3 ×R3
N
e0 = −
1 2 j=1
ρi (v − z)ρj (w) 4π|v − w|
(14)
ρj (v)ρj (w) . 4π|v − w|
(15)
dv dw R3 ×R3
Vij (z) is the electrostatic interaction energy of the charge distributions ρi and ρj at distance z, however, with the “wrong” sign. It is a peculiarity of the electromagnetic field that the interaction between charges with equal sign is repulsive. e0 is the sum of all self energies. The remainder of the spectrum is purely absolutely continuous and the ground state eigenvalue is not isolated, cf. Figure 1. Let P∗ (x) = |Ω(x) Ω(x)|, then the states in Ran P∗ =
⊕ R3N
dx ψ(x)Ω(x) : ψ ∈ L (R 2
3N
) ⊂H
(16)
correspond to wave packets without free bosons. These states are sections of ground states, not sections of vacua, and as such contain what is called “virtual” bosons in physics. States in RanP∗ could be called dressed many-particle states, since the particles carry a cloud of virtual bosons. If the particles are moving at small speeds and if the accelerations are also small, one expects that no free bosons are created, i.e. that RanP∗ is approximately invariant under the dynamics generated by H ε . Moreover the wave function ψ(x) of the particles should approximately be governed by the effective Schr¨ odinger equation (12) with ε Heff =
j−1 N N −ε2 ∆xj + 1 + Vij (xi − xj ) . j=1
j=2 i=1
⊕ dx ψ0 (x)Ω(x) ∈ RanP∗ we Our main result, Theorem 7, states that for Ψ0 = have ⊕
ε −iH ε t/ε Ψ0 − dx e−iHeff t/ε ψ0 (x) e−ie0 t/ε Ω(x) (17) e 3N R = O(ε ln(1/ε)) (1 + |t|) Ψ0 . Notice that in the approximate solution of the full Schr¨ odinger equation the state of the field is, up to a fast oscillating global phase e−ie0 t/ε , adiabatically following
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the motion of the particles. In particular, there are no bosons traveling back and forth between the particles and the phrase that the particles “interact through the exchange of bosons”, which comes from perturbation theory, should not be taken literally in the present setting. Let us stress the physical relevance of the allowed initial conditions in our result. Heuristically one would expect that any initial state for the Nelson model radiates its free bosons to infinity and thus effectively becomes a state in RanP∗ . To make even precise this statement is not completely trivial and it is a hard problem in scattering theory to prove it. In case of the Nelson model for a single particle and with an infrared cutoff in the interaction, asymptotic completeness for Compton scattering was proven only recently in [9]. In particular, for a single particle the heuristic expectation formulated above holds true and every initial state, after a sufficiently long time, follows the effective dynamics. As mentioned before, there is a strong similarity to the time-dependent BornOppenheimer approximation, where one obtains an effective Schr¨ odinger equation for the nuclei in a molecule with an effective potential generated by the electrons [22]. In both cases the physical mechanism which leads to the approximate invariance of the subspace RanP∗ is adiabatic decoupling. I.e. the separation of time scales for the motion of the different parts of the system lets the fast degrees of freedom, in our case the bosons, instantaneously adjust to the motion of the slow degrees of freedom, the particles. However, for massless bosons – in contrast to the Born-Oppenheimer approximation – there is no spectral gap which pointwise separates the energy band E0 = {(x, E0 (x)) : x ∈ R3N } from the remainder of the spectrum of H0 (x), but E0 lies at the bottom of continuous spectrum. Hence we need a space-adiabatic theorem, cf. [22, 23, 17], without gap condition. The prefix space in space-adiabatic is used to distinguish this type of result from the standard adiabatic theorem of quantum mechanics, to which we refer as the time-adiabatic theorem. In the timeadiabatic theorem one considers Hamiltonians with an explicit time-dependence on a slow time scale, while in the space-adiabatic setting the separation of time scales has a dynamical origin. Only recently time-adiabatic theorems without a gap condition were established in [2, 5, 24]. In Section 2 a general space-adiabatic theorem without gap condition is formulated and proved. The proof is based on ideas developed in [24] and our approach gives, in particular, good control on the rate of convergence in the adiabatic limit. As an application of the result from Section 2 we consider in Section 3 the scaling limit ε → 0 of the massless Nelson model as described above. We emphasize at this point that, in view of the missing gap condition, the rate of convergence O(ε ln(1/ε)) in (17) is surprisingly fast, since it is almost as good as in the case with a gap. Moreover, if all particles have individually total charge equal to zero, then the rate is exactly O(ε) as in the case with a gap. Hence, the logarithmic correction must be attributed to the Coulombic long range character of the interparticle interaction. In the situation with gap it is known [15, 17] that the wave function stays in a subspace RanP∗ε which is ε-close to the band subspace RanP∗
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up to an error of order O(ε∞ ). However, in the situation without gap, we expect that a piece of order εα , α < ∞, of the wave function is “leaking out” of RanP∗ . Physical considerations suggest that α = 3/2 for the present problem, see Remark 9. As a consequence, the ε2 corrections to the effective Hamiltonian are still dominating dissipation and can be formally derived from the results in [17]. The effective Hamiltonian (55) then contains a renormalized mass term and the momentum dependent Darwin interaction. In Section 3 we also consider an infrared-renormalized model suggested in [1] and [14], which allows us to do without the global infrared condition (9). The results are exactly the same as in the standard Nelson model. We remark that in [3] the time-adiabatic theorem without gap condition was applied to the Dicke model with a constant magnetic field whose direction changes slowly in time. The Dicke model is a simplified version of the spin-boson model, i.e. a two-level system coupled to the quantized massless scalar field. In the Dicke model one drops the anti-resonant terms in the interaction and can, as a consequence, explicitly calculate the ground state as a function of time. This is very similar to our setting, where we will obtain a rather explicit expression for the ground state of H0 (x) as a function of x ∈ R3N . In [3] the order of the error in the adiabatic limit is O(ε ln(1/ε)), exactly as in (17). As to be explained at the end of Section 2.1, in both cases the specific form of the error can be traced back to the spectral density of the massless scalar field at zero energy. Finally let us compare our results to those obtained by Davies [7], who considers the limit c → ∞ for the Hamiltonian √ H c = Hp ⊗ 1 + 1 ⊗ dΓ(c|k|) + cHI . (18) Notice that H c is obtained through canonical quantization of (1) and (2) if one does not set c = 1 as we did before. Davies proves that for all t ∈ R c
s − lim e−iH t (ψ ⊗ Ω) = (e−i(Heff +e0 )t ψ) ⊗ Ω , c→∞
where Ω = {1, 0, 0, . . .} denotes the Fock vacuum and Hp := Hpε=1 and Heff := ε=1 . This shows that although the limit c → ∞ is equivalent to our scaling on Heff the classical level, the results for the quantum model differ qualitatively. While we obtain effective dynamics for states which contain a nonzero number of bosons independent of ε, cf. (17), the c → ∞ limit yields effective dynamics for states which contain no bosons at all. Furthermore, the limit ε → 0 is a singular limit as no limiting dynamics for ε = 0 exists.
2 A space-adiabatic theorem without gap condition Generalizing from the time-adiabatic theorem of quantum mechanics [11], we consider perturbations of self-adjoint operators H0 , which are fibered over the base space Rn , where, for better readability, we use M := Rn to denote this base space.
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Let Hf be a separable Hilbert space and let dx denote Lebesgue measure. Recall that H0 acting on H = L2 (M, dx) ⊗ Hf = L2 (M, dx; Hf ) is called fibered, cf. [20], if there is a measurable map M x → H0 (x) with values in the self-adjoint operators on Hf such that H0 =
⊕
M
dx H0 (x) .
There seems to be no standard name for the set
Σ = (x, s) ∈ M × R, s ∈ σ(H0 (x)) and we propose to call it the fibered spectrum of H0 . Let σ∗ ⊂ Σ be such that x → P∗ (x) is measurable, where P∗ (x) denotes ⊕ the spectral projection of H0 (x) associated with σ∗ (x). Then P∗ = M dx P∗ (x) is an orthogonal projection which commutes with H0 , but which is in general not a spectral projection of H0 . We consider perturbations of H0 , which mix the fibers, in a sense, slowly. As a prototype consider for a sufficiently regular real valued function h on “momentum space” Rn the self-adjoint operator hε = h(−iε∇x ) on L2 (M ). Here ε > 0 is the adiabatic parameter and [hε ⊗ 1, A] = O(ε) for any operator A which is fibered over M . Let H ε = H0 + h ε ⊗ 1 , then the invariant subspaces for H0 constructed above are still “approximately” ε invariant for H ε with ε small, since [H ε , P∗ ] = O(ε) and thus [e−iH s , P∗ ] = ε O(ε|s|). But the relevant time scale for the dynamics generated by h is t/ε with ε t = O(1). Thus the unitary group of interest is e−iH t/ε . However, according to −iH ε t/ε , P∗ ] = O(|t|) and the subspaces RanP∗ seem to be the naive argument, [e not even approximately invariant as ε → 0. It is well known [22, 23] that the failure of the naive argument can be cured if σ∗ is separated by a gap from the remainder of the fibered spectrum Σ. Then ε [e−iH t/ε , P∗ ] = O(ε) (1+|t|), a result that was baptized space-adiabatic theorem in [22]. The object of this section is to establish an analogous result without assuming a gap condition. We remark that the general setup for space-adiabatic theory are Hamiltonians which are “fibered” over phase space, in the sense that they can be written as quantizations of operator valued symbols [17].
2.1
Assumptions and results
Let H0 (x), x ∈ M , be a family of self-adjoint operators on some common dense domain D ⊂ Hf , Hf a separable Hilbert space. Let · H0 (x) denote the graph norm of H0 (x) on D, i.e., for ψ ∈ D, ψH0 (x) = H0 (x)ψ + ψ. We assume
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that all the H0 (x)-norms are equivalent in the sense that there is an x0 ∈ M and constants C1 , C2 < ∞ such that C1 ψH0 (x0 ) ≤ ψH0 (x) ≤ C2 ψH0 (x0 ) . Then H0 =
⊕
M
dx H0 (x)
with domain D(H0 ) = L2 (M ) ⊗ D is self-adjoint, where here and in the following D resp. D(H0 ) are understood to be equipped with the · H0 (x0 ) resp. · H0 norm. For k ∈ N0 and E some Banach space let k n k n α n Cb (R , E) = f ∈ C (R , E) : sup ∂x f (x)E < ∞ ∀ α ∈ N with |α| ≤ k . x∈Rn
L(H1 , H2 ) denotes the space of bounded linear operators from H1 to H2 and Lsa (H) denotes the set of bounded self-adjoint operators on H. Let | · | be the Euclidean norm on Rn and denote the Hessian of a function A on Rn by ∇(2) A(x). For the resolvent we write Rλ (A) = (A − λ)−1 . Let m ≥ 2. m Assumption Hm 0 . Let H0 (·) ∈ Cb (M, L(D, Hf )) and for all x ∈ M let P∗ (x) be an orthogonal projection such that H0 (x) P∗ (x) = E(x) P∗ (x) with P∗ (·) ∈ Cbm+1 (M, L(Hf )) and E(·) ∈ Cbm (M, R). In addition one of the following assertions holds:
(i) For 1 ≤ j ≤ n lim ess sup δ RE(x)−iδ (H0 (x)) (∂xj P∗ )(x)P∗ (x)L(Hf ) = 0 .
δ→0
x∈M
(19)
(ii) There is a constant δ0 > 0 and a function η : [0, δ0 ] → [0, δ0 ] with η(δ) ≥ δ and a constant C < ∞ such that for δ ∈ (0, δ0 ] and 1 ≤ j ≤ n ess sup RE(x)−iδ (H0 (x)) (∂xj P∗ )(x)P∗ (x)L(Hf ) ≤ C δ −1 η(δ) . x∈M
(iii) In addition to (20) for 1 ≤ k, j ≤ n also
ess sup ∂xk RE(x)−iδ (H0 (x))(∂xj P∗ )(x)P∗ (x) x∈M
L(Hf )
(20)
≤ C δ −1 η(δ) (21)
holds. A few remarks concerning Assumption Hm 0 are in order: • It is not assumed that P∗ (x) is the spectral projection of H0 (x) corresponding to the eigenvalue E(x). However, (19) holds pointwise in x whenever P∗ (x) is the spectral projection and has finite rank, cf. Proposition 2.
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• Inequality (20) is always satisfied with η(δ) = 1. For Assumption Hm 0 (ii) and (iii) to have nontrivial consequences on the rate of convergence in the adiabatic theorem, η(δ) must satisfy limδ→0 η(δ) = 0. These assumptions might look rather artificial at first sight, but turn out to be very natural in the proof and also in our application. For the simpler time-adiabatic setting, which gives rise to similar conditions, we refer the reader to [24]. • The regularity of P∗ (x) has to be assumed, since it does not follow from the regularity of H0 (x) without the gap condition, even if P∗ (x) is spectral. The regularity of E(x) follows from the one of H0 (x) and P∗ (x) whenever P∗ (x) has finite rank, as can be seen by writing E(x) = tr(H0 (x)P∗ (x))/trP∗ (x). ⊕ The “band subspace” RanP∗ defined through P∗ = M dx P∗ (x) is invariant under the dynamics generated by H0 , since [H0 , P∗ ] = 0 holds by construction. We will consider perturbations hε of H0 satisfying Assumption hm . For ε ∈ (0, 1] let hε be a self-adjoint operator with domain D(h) ⊂ H independent of ε such that H0 + hε is essentially self-adjoint on D(h) ∩ D(H0 ). There exists an operator (Dh)ε ∈ Lsa (H)⊕n with sup | (Dh)ε | L(H) < ∞
ε∈(0,1]
satisfying: (i) There is a constant C < ∞ such that for each A ∈ Cbm (M, L(Hf )) m
[hε , A] + i ε ∇x A · (Dh)ε L(H) ≤ C
j=2
εj
sup x∈M, |α|=j
∂xα A(x)L(Hf ) .
(ii) There is a constant C < ∞ such that | [(Dh)ε , H0 ] | L(D(H0 ),H) + | [(Dh)ε , hε ] | L(H) ≤ ε C . By assumption, H ε = H0 + hε is essentially self-adjoint on D(h) ∩ D(H0 ) and we use its closure, again denoted by H ε , to define for t ∈ R U ε (t) = e−iH
ε
t/ε
.
Since, according to Assumption hm (i), [H ε , P∗ ] = [hε , P∗ ] = O(ε), the naive argument gives [U ε (t), P∗ ] = |t|O(1). Indeed, our aim is to cure the failure of the naive argument and to show that RanP∗ is invariant for U ε (t) in the limit ε → 0. To this end we will compare U ε (t) with the unitary group generated by ε = H0 + P∗ hε P∗ + P∗⊥ hε P∗⊥ . Hdiag
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ε Also Hdiag is self-adjoint on D(H ε ) since P∗ (·) ∈ Cbm (M, L(Hf )) and thus H ε − ε ⊥ ε Hdiag = P∗ [h , P∗ ]P∗ − P∗ [hε , P∗ ]P∗⊥ is bounded according to hm (i). Again we abbreviate for t ∈ R ε ε (t) = e−iHdiag t/ε , Udiag
and we have by construction that ε (t)] = 0 , P∗ , Udiag i.e. RanP∗ and RanP∗⊥ are invariant subspaces for the dynamics generated by ε Hdiag . m Theorem 1. Assume Hm for some m ≥ 2. Let ε ∈ (0, δ0 ], then 0 and h
• Hm 0 (i) implies that for t ∈ R ε lim U ε (t) − Udiag (t) ε→0
L(H)
= 0,
(22)
• Hm 0 (ii) implies that for some constant C < ∞ and all t ∈ R 1 ε ε (t) ≤ C η(ε 2 ) (1 + |t|) , U (t) − Udiag
(23)
• Hm 0 (iii) implies that for some constant C < ∞ and all t ∈ R ε ε (t) ≤ C η(ε) (1 + |t|) . U (t) − Udiag
(24)
L(H)
L(H)
Note that in Theorem 1 the whole spectrum of possible rates of convergence between o(1) and O(ε) as in the case with gap is covered. The estimates for the massless Nelson model as an application of Theorem 1 will show that, in principle, all rates can occur. The following proposition shows that, assuming the first part of Hm 0 but (i) always holds pointwise in x if neither (i), (ii) or (iii), then Assumption Hm 0 P∗ (x) is the spectral projection and has finite rank. The proof is analogous to the one of Lemma 4 in [2]. Proposition 2. Assume H10 without (i), (ii) or (iii). If P∗ (x) is the spectral projection of H0 (x) corresponding to the eigenvalue E(x) and has finite rank, then lim δ RE(x)−iδ (H0 (x)) (∇x P∗ )(x)P∗ (x)L(Hf ) = 0 .
δ→0
(25)
Proof. Since P∗ (x) has finite rank, the uniform statement (25) follows if we can show that limδ→0 δ RE(x)−iδ (H0 (x)) ψ = 0 for all ψ ∈ Ran(∇x P∗ )(x)P∗ (x). We have δ2 lim i δRE(x)−iδ (H0 (x)) ψ2Hf = lim µψ (dλ) = µψ (E(x)) , δ→0 δ→0 R (λ − E(x))2 + δ 2 (26)
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where µψ denotes the spectral measure of H0 (x) for ψ. Since P∗ (x) is the spectral projection on {E(x)} and since, according to (29), Ran(∇x P∗ )(x)P∗ (x) ⊂ RanP∗⊥ (x) we have µψ (E(x)) = 0 and thus (25). It is clear from (26) that additional information on the regularity of the spectral measure µψ provides some control on the rate of convergence in (26). E.g., if µψ (dλ) = ρψ (λ)dλ with ρψ ∈ L∞ (R, dλ), then δ2 δ2 µψ (dλ) ≤ ρ dλ = O(δ) ψ ∞ (λ − E(x))2 + δ 2 (λ − E(x))2 + δ 2 R R and hence (20) would hold pointwise in x with η(δ) = δ 1/2 . In a sense, the rate O(ε ln(1/ε)) for the massless Nelson model (17) is a consequence of the relevant spectral measure having a density ρ(λ) ∼ λ − E(x) for 0 < λ − E(x) 1. This explains why in [3] for the Dicke model in R3 the same error estimates are obtained: the same spectral density enters the proof. Finally we emphasize that (25) for all x ∈ M does not imply Hm 0 (i), even in the case of compact M . This is because for pointwise convergence to imply uniform convergence one would need uniform equicontinuity of a sequence of functions. However, in the time-adiabatic setting it is indeed sufficient to have (25) for almost all x ∈ I, where I ⊂ R is the relevant time interval, see [24].
2.2
Proof of Theorem 1
We start with the standard argument and find that on D(H ε ) t d ε ε ε (t) = − U ε (t) ds (s) U ε (t) − Udiag U (−s) Udiag ds 0 t ε i ε ε = − U (t) Udiag (s) , ds U ε (−s) H ε − Hdiag ε 0 where
ε H ε − Hdiag = P∗⊥ hε P∗ + P∗ hε P∗⊥ = P∗⊥ hε , P∗ P∗ + adj. .
(27)
In (27) and in the following “± adj.” means that the adjoint operator of the first term in a sum is added resp. subtracted. Inserting hm (i) into (27) and the result back into (27) one obtains ε ε U (t) − Udiag (t)L(H) = (28)
0
t
ε ds U ε (−s) P∗⊥ (∇x P∗ ) P∗ · (Dh)ε P∗ + adj. Udiag (s)
L(H)
+ O(ε)|t| .
In (28) we also used that (∇x P∗ )(x) = P∗⊥ (x)(∇x P∗ )(x)P∗ (x) + adj. ,
(29)
which follows from (∇x P∗ )(x) = (∇x P∗2 )(x) = (∇x P∗ )(x)P∗ (x) + P∗ (x)(∇x P∗ )(x).
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The nontrivial part in adiabatic theorems is to show that also the remaining term on the right hand side of (28) vanishes as ε → 0. Assuming a gap condition, the basic idea is to express the integrand, which is O(1), as the time-derivative of a function that is O(ε) plus a remainder that is O(ε) and integrate, cf. [23, 22]. The key ingredient in this case would be the operator F (x) = RE(x) (H0 (x)) (∇x P∗ )(x) P∗ (x) ,
(30)
which is, according to (29), well defined and bounded if the eigenvalue E(x) is separated from the rest of the spectrum of H0 (x) by a gapand ifP∗ (x) is spectral. The definition (30) is made to give [H0 , F ] = P∗⊥ ∇x P∗ P∗ . However, in absence of a gap (30) is not well defined as an operator on Hf and, following [24], we shift the resolvent into the complex plane and define Fδ (x) = RE(x)−iδ (H0 (x)) P∗⊥ (x) (∇x P∗ )(x) P∗ (x) . One now obtains H0 (x) , Fδ (x) = P∗⊥ (x) ∇x P∗ (x) P∗ (x) + Yδ (x)
(31)
with Yδ (x) = − i δRE(x)−iδ (H0 (x)) (∇x P∗ )(x) P∗ (x) .
(32)
Hm 0
(i), (ii) and (iii) each imply that limδ→0 Yδ L(H) = 0. To see Assumptions this recall that for A(·) ∈ L∞ (M, L(Hf )) one has AL(H) = ess sup A(x)L(Hf ) . x∈Rn
Note that for better readability we omit the Euclidean norm | . . . | in the notation and understand that A always includes also the Euclidean norm if A is an operator with n components. Thus with (32) we can make the remainder in (31) arbitrarily small by choosing δ small enough. However, for the time being we let δ > 0 but carefully keep track of the dependence of all errors on δ. By assumption H0 (·) ∈ Cbm (M, L(D, Hf )) and P∗ (·) ∈ Cbm+1 (M, L(Hf )), which implies Fδ (·) ∈ Cbm (M, L(Hf )⊕n ) and hence, according to hm (i), ε h , Fδ
L(H)
≤ C
m j=1
εj sup ∂xα Fδ L(H) =: f1 (ε, δ) .
(33)
|α|=j
Combining (31) and (33) we obtain ε H , Fδ = P∗⊥ ∇x P∗ P∗ + O(Yδ , f1 (ε, δ)) ,
(34)
where in (34) and in the following O(a, b, c, . . .) stands for a sum of operators whose norm in L(H) is bounded by a constant times a + b + c + . . .. Defining Bδ = Fδ · (Dh)ε P∗ − adj. ,
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one finds with hm (ii) and f2 (δ) = Fδ L(H,D(H0 )) that ε H , Bδ = H ε , Fδ · (Dh)ε P∗ + Fδ · H ε , (Dh)ε P∗ (35) + Fδ · (Dh)ε H ε , P∗ + adj. ⊥ ε = P∗ ∇x P∗ P∗ · (Dh) + adj. + O(ε, Yδ , f1 (ε, δ), εf2 (δ)) . Now the integrand in (28) can be written as the time-derivative of Aδ (s) = − i ε U ε (−s) Bδ U ε (s) , plus a remainder: d Aδ (s) = ds =
U ε (−s) [H ε , Bδ ] U ε (s) U ε (−s) P∗⊥ ∇x P∗ P∗ · (Dh)ε P∗ + adj. U ε (s) + O(ε, Yδ , f1 (ε, δ), εf2 (δ)) .
(36)
Inserting (36) into (28) enables us to do integration by parts, ε U (t) − U ε (t) ≤ diag L(H) t d ε ε ≤ A ds (s) U (−s) U (s) δ diag ds 0 L(H)
+ |t| O ε, Yδ , f1 (ε, δ), εf2 (δ) ≤ Aδ (t)L(H) + Aδ (0)L(H) t d ε ε U + ds A (s) (−s) U (s) δ diag ds 0 L(H)
+ |t| O ε, Yδ , f1 (ε, δ), εf2 (δ)
≤ C ε (2 + |t|) Fδ L(H) + |t| O ε, Yδ , f1 (ε, δ), εf2 (δ) .
(37)
For the last inequality in (37) we used that Aδ (t)L(H) ≤ C εFδ L(H) uniformly for t ∈ R and that ε i d ε ε ε U (t0 , s) Udiag (s, t0 ) = − U ε (t0 , s) H ε (s) − Hdiag (s) Udiag (s, t0 ) ds ε is bounded uniformly, according to (27) and hm (i). Writing out the various terms in (37) explicitly, we conclude that there is a constant C < ∞ such that ε U (t) − U ε (t) ≤ C ε F + C |t| ε + Yδ L(H) δ L(H) diag L(H) +ε Fδ L(H,D(H0 )) + ε Fδ L(H) +
m j=1
εj sup ∂xα Fδ L(H) . |α|=j
(38)
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Hence we are left to establish bounds on Fδ , on its derivatives and on Yδ in terms of δ, which is the content of the following Lemma. m ⊕n ) and there is a constant Lemma 3. Assume Hm 0 , then Fδ (·) ∈ Cb (M, L(Hf ) C < ∞ such that for δ ∈ (0, δ0 ]
Fδ L(H,D(H0 )) ≤
C η(δ) , δ
C η(δ) δ j+1
sup ∂xα Fδ L(H) ≤
|α|=j
for 1 ≤ j ≤ m .
(39) (40)
m In case Hm 0 (i) holds, we have (39) and (40) with η(δ) = 1. Furthermore, if H0 m (i) holds, then limδ→0 Yδ L(H) = 0 and if H0 (ii) or (iii) holds, then Yδ L(H) ≤ C η(δ). If Hm 0 (iii) holds, then (40) can be improved to
sup ∂xα Fδ L(H) ≤
|α|=j
C η(δ) δj
for 1 ≤ j ≤ m .
(41)
Before we turn to the proof of Lemma 3 we finish the proof of Theorem 1. Assuming Hm 0 (i), (22) follows by inserting the bounds from Lemma 3 into (38) and choosing δ = δ(ε) such that limε→0 δ(ε) = 0 and limε→0 ε/δ(ε)2 = 0. If Hm 0 (ii) holds, then the bounds (39) and (40) inserted into (38) yield m ε η(δ) j η(δ)
η(δ) ε U (t) − Udiag + C ε + η(δ) + ε + (t)L(H) ≤ C ε ε j+1 |t| . (42) δ δ δ j=1 1
In (42) the optimal choice is δ(ε) = ε 2 , which gives (23). Finally, the bounds (39) and (41) inserted into (38) yield m ε η(δ) j η(δ)
η(δ) U (t) − U ε (t) |t| , + C + ε + η(δ) + ε ≤ C ε ε diag L(H) δ δ δj j=1
where the optimal choice δ(ε) = ε gives (24). Proof of Lemma 3. We abbreviate RE(x)−iδ (H0 (x)) as R(δ, x) in this proof and note that R(δ, ·) ∈ Cbm (M, L(Hf )) and thus Fδ (·) ∈ Cbm (M, L(Hf )⊕n ) follow from H0 (·) ∈ Cbm (M, L(D, Hf )) together with P∗ (·) ∈ Cbm+1 (M, L(Hf )). We start with the case Hm 0 (ii), where (i) is included by making the obvious changes for η(δ) = 1. Assumption Hm 0 (ii) immediately yields Fδ L(H) ≤
C η(δ) δ
(43)
and the bound on Yδ . (39) follows from H0 (x)R(δ, x) = 1 + (E(x) − iδ)R(δ, x) and (43) together with the assumption that E(x) is uniformly bounded.
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For (40) we start by observing that ∇x Fδ (x)
= R(δ, x) ∇x (∇x P∗ )(x)P∗ (x) + ∇x R(δ, x) (∇x P∗ )(x)P∗ (x) (44)
= R(δ, x)∇x (∇x P∗ )(x)P∗ (x) − R(δ, x)(∇x H0 (x) − ∇x E(x))Fδ (x).
Using (43) and the fact that |∇x E(x)| is uniformly bounded by assumption, we infer from (44) that |∇x Fδ | L(H) = sup |∇x Fδ (x)| L(Hf ) ≤ C(δ −1 + δ −2 η(δ)) . x∈M
Hence (40) follows for j = 1, since δ ≤ η(δ) by assumption. By differentiating (44) again, we find, using a reduced notation with obvious meaning, that
∇(2) Fδ = −2Rδ (∇H0 − ∇E) ∇Fδ + Rδ ∇(2) (∇P∗ )P∗ − Rδ (∇(2) H0 − ∇(2) E) Fδ . (45) Hence ∇(2) Fδ L(H) ≤ C(δ −3 η(δ) + δ −1 + δ −2 η(δ)) which proves (40) for j = 2. By repeated differentiation one finds inductively (40) for j ≤ m. To show (41) assuming Hm 0 (iii), note that (41) holds by assumption for j = 1 and inserted into (45) it gives ∇(2) Fδ L(H) ≤ C(δ −2 η(δ) + δ −1 + δ −2 η(δ)). Analogously the estimates for all larger j ≤ m are improved by a factor of δ.
3 Effective dynamics for the Nelson model As explained in the introduction, we consider N spinless particles coupled to a scalar, massless, Bose field with an ultraviolet regularization in the interaction. This class of models is nowadays called Nelson’s model [1, 4, 13, 14] after E. Nelson [16], who studied the ultraviolet problem. We briefly complete the introduction of the model and collect some basic, well known facts. A point in the configuration space R3N of the particles is denoted by x = (x1 , . . . , xN ) and the Hamiltonian Hpε for the particles is defined in (5). Hpε is self-adjoint on the domain H 1 (R3N ), the first Sobolev space. 2 3 The Hilbert space √ for the scalar field is the bosonic Fock space over L (R ) defined in (6). On D( N ), N the number operator, the annihilation operator a(f ) acts for f ∈ L2 (R3 ) as √ (m) (a(f )ψ) (k1 , . . . , km ) = m + 1 dk f¯(k) ψ (m+1) (k, k1 , . . . , km ) , R3
√ (m) 2 only if ∞ < ∞. where ψ = (ψ (0) , ψ (1) , ψ (2) , . . .) ∈ D( N ) if and m=0 mψ √ ∗ The adjoint a (f ), which is also defined on D( N ), is the creation operator and
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for f, g ∈ L2 (R3 ) the operators a(f ) and a∗ (g) obey the canonical commutation relations (CCRs) dk f¯(k) g(k) =: f, g , [a(f ), a(g)] = [a∗ (f ), a∗ (g)] = 0 . [a(f ), a∗ (g)] = R3
(46) It is common to write a(f ) = R3 dk f¯(k) a(k). The Hamiltonian of the field as defined in (7) can formally be written as dk |k| a∗ (k) a(k) . Hf = R3
More on the m-particle sector the action of Hf is (Hf ψ)(m) (k1 , . . . , km ) = m explicitly, (m) (k1 , . . . , km ), and Hf is self-adjoint on its maximal domain. For f ∈ j=1 |kj |ψ L2 (R3 ) the Segal field operator
1 Φ(f ) = √ a(f ) + a∗ (f ) 2 √ is essentially self-adjoint on D( N ). The field operator φ as used in (8) is related to Φ through φ(f ) = Φ(f / |k|). For the following it turns out to be more convenient to write the interaction Hamiltonian in terms of Φ, where
HI = Φ |k| v(x, k) acts on the Hilbert space H = L2 (R3N ) ⊗ F of the full system. We will consider two different choices for v(x, k) in more detail. For the standard Nelson model (SN), as discussed in the introduction, one has vSN (x, k) =
N j=1
eik·xj
ρˆj (k) . |k|3/2
(47)
For the infrared-renormalized models (IR), as considered by Arai [1] and, more generally, by L¨ orinczi, Minlos and Spohn [14], one has vIR (x, k) =
N ik·xj ρˆj (k) e −1 . |k|3/2 j=1
(48)
In both cases, the charge distribution ρj ∈ L1 (R3 ) of the j th particle is assumed to be real-valued and spherically symmetric. As to be discussed below, cf. Remarks 4 and 6, we have to assume the infrared condition (9) for the (SN) model, but not for the (IR) model. The infrared condition implies, in particular, that the total charge of the system of N particles must be zero. The full Hamiltonian is given as the sum H ε = Hpε ⊗ 1 + 1 ⊗ Hf + HI + VIR ⊗ 1
(49)
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and is essentially self-adjoint on D(Hp ⊗1)∩D(1⊗Hf ) if supx |k|s v(x, k) L2 (R3 ) < ∞ for s ∈ { 21 , 1}. Only in the (IR) model a potential VIR is added, which acts as multiplication with the bounded, real-valued function VIR (x) =
N j,i=1
ρi (k)∗ −ik·xj 1 ρˆj (k)ˆ dk e + |k|2 2 R3
2 N j=1 ρˆj (k) dk . |k| R3
(50)
Remark 4. If the charge distributions satisfy the infrared condition (9), then the (SN) Hamiltonian and the (IR) Hamiltonian are related by the unitary transformation N ρˆj∗ (k) , UG = exp −iΦ i 3/2 |k| j=1 cf. [1], which is related to the Gross transformation [16] for x = 0. If the infrared condition is not satisfied, the (SN) model and the (IR) model carry two inequivalent representations of the CCRs for the field operators. Physically speaking, the transformation UG removes the mean field that the N charges would generate, if all of them would be moved to the origin. The vacuum in the (IR) representation corresponds to this removed mean field in the original representation, a fact which has to be taken care of in the interaction: for each particle the interaction term is now evaluated relative to the interaction at x = 0, cf. (48), which makes also necessary the counter terms VIR . If the total charge of the system is different from zero, then the mean field is long range and, as a consequence, the corresponding transformation is no longer unitarily implementable. Indeed, it was shown that the (SN) Hamiltonian with confining potential does not have a ground state, cf. [13], while the (IR) Hamiltonian with the same confining potential does have a ground state, cf. [1]. ♦ In order to apply Theorem 1 we observe that HI (x) acts for fixed x ∈ R3N (∼ = M ) on F (∼ = Hf ) and with H0 (x) = Hf + HI (x) + VIR (x) we have ε
H =
Hpε
⊗1+
⊕ M
dx H0 (x)
∼ = h ε + H0 .
The following proposition collects some results about H0 (x) and its ground state. Its proof is postponed to after the presentation of the the main theorem. Proposition 5. Assume that v(x, ·) ∈ L2 (R3 ) for all x ∈ R3N and that for some n≥1 (i) | · |∂xα v(x, ·) ∈ L2 (R3 ) for all x ∈ R3N and 0 ≤ |α| ≤ n, (ii) supx∈R3N | · | ∂xα v(x, ·) L2 (R3 ) < ∞ for 0 ≤ |α| ≤ n, (iii) supx∈R3N ∂xα v(x, ·) L2 (R3 ) < ∞ for 1 ≤ |α| ≤ n.
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Let Im v(x, ·), ∇x v(x, ·) L2 (R3 ) = 0 for all x ∈ R3N and VIR (·) ∈ Cb3 (R3N ). Then 1. H0 (x) is self adjoint on D = D(Hf ) for all x ∈ R3N and H0 (·) ∈ Cbn (R3N , L(D, F )), where D is equipped with the graph-norm of Hf . 2. H0 (x) has a unique ground state Ω(x) for all x ∈ R3N and, in particular, H0 (x)Ω(x) = E(x)Ω(x) for 1 E(x) = − dk |k| |v(x, k)|2 + VIR (x). (51) 2 R3 Furthermore Ω(·) ∈ Cbn (R3N , F ). It is straightforward to check that Im v(x, ·), ∇x v(x, ·) L2 (R3 ) = 0 for vSN defined in (47) and vIR defined in (48). For the (SN) model as well as for the (IR) model (51) is easily evaluated and one finds E(x) = E0 (x) as given in (13). Remark 6. For the (SN) model the assumptions made on v(x, k) in Proposition 5 are satisfied, if ρˆj (k) decays sufficiently fast for large |k| and each j = 1, . . . , N , or, equivalently, if ρj (x) is sufficiently smooth. This is an ultraviolet condition individually for each particle. But v(x, ·) ∈ L2 (R3 ) follows from v(0, ·) ∈ L2 (R3 ), which is exactly the global infrared condition (9). While the necessity for an ultraviolet regularization remains in the (IR) − 32 ik·xj model, the infrared condition is replaced − 1) ∈ L2 (R3 ), by j |k| ρj (k)(e which can be satisfied without having j ρj (0) = 0. Thus the (IR) model allows us to consider particles with total charge different from zero. ♦ Let P∗ (x) = |Ω(x) Ω(x)|, then P∗ (·) ∈ Cbn (R3N , L(F )) and RanP∗ is a candidate for an adiabatically decoupled subspace. Indeed, we will show that hε = ε m ε Hp ⊗ 1 satisfies Assumption h with (Dh)xj = −iε∇xj / −ε2 ∆xj + 1 and that H0 and P∗ satisfy Assumption Hm 0 (iii) with η(δ) = δ ln(1/δ) if particles with charges different from zero are present and η(δ) = δ if all particles have total charge zero. Hence we can apply Theorem 1 to conclude that for some constant C<∞ ε −iH ε t/ε (52) − e−iHdiag t/ε ≤ C η(ε) (1 + |t|) , e ε with Hdiag = P∗ H ε P∗ + (1 − P∗ ) H ε (1 − P∗ ). Next observe that the ground state band subspace RanP∗ is unitarily equivalent to the Hilbert space L2 (R3N ) of the N particles in a natural way. Let
U : RanP∗ → L2 (R3N ) ,
ψ → (Uψ)(x) = Ω(x), ψ(x) F ,
then it is easily checked that for ϕ ∈ L2 (R3N ) ∗
U ϕ=U
−1
⊕
ϕ=
dx ϕ(x) Ω(x) . R3N
(53)
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ε Hence the part of Hdiag acting on RanP∗ is unitarily equivalent to ε eff H = U P∗ H ε P∗ U ∗
acting on the Hilbert space of the N particles only. The following theorem shows ε has, at leading order, exactly the form expected from the heuristic that H eff “Peierls substitution” argument. Theorem 7. Let H ε be defined as in (49) with v(x, k) either vSN (x, k) as in (47) or vIR (x, k) as in (48). For 1 ≤ j ≤ N let ρj ∈ L1 (R3 ) such that ρˆj satisfies |k|s ρj (k) ∈ L2 (R3 ) for s ∈ {−1, 4}. For the (SN) model assume, in addition, the infrared condition (9). Let ε Heff
=
Hpε
+
j−1 N
Vij (xi − xj ) + e0 ,
j=2 i=1
with Vij (z) and e0 as in (14) and (15). Then there is a constant C < ∞ such that
ε −iH ε t/ε (54) − U ∗ e−iHeff t/ε U P∗ ≤ C η(ε) (1 + |t|) , e where η(ε) = ε ln(1/ε). If all charges satisfy the infrared condition individually, i.e. if ρˆj (k)/|k|3/2 ∈ L2 (R3 ) for all j = 1, . . . , N , then (54) holds with η(ε) = ε. For sake of better readability the global energy shift e0 was, as opposed to ε (17), absorbed into the definition of Heff . Remark 8. The unitary U intertwines the position operator x ⊗ 1 on L2 (R3N ) ⊗ F with the position operator x on L2 (R3N ) exactly and the momentum operator −iε∇x ⊗ 1 on L2 (R3N ) ⊗ F with −iε∇x on L2 (R3N ) up to an error of order ε. Thus one can directly read off the position distribution and approximately also ε the momentum distribution of the particles from the solution ψ(t) = e−iHeff t/ε ψ0 of the effective dynamics. It is not necessary to transform back to the full Hilbert ♦ space using U ∗ . Remark 9. From the discussion of the introduction one expects, on physical grounds, that the energy lost through radiation is of order O(ε3 ) after times of −1 order O(ε ). Hence the error of order O(ε ln 1/ε) in (54) is not optimal in the sense that the error does not correspond to emission of free bosons and thus to dissipation. Indeed, we expect that the situation is similar to adiabatic perturbation theory with gap, cf. [17]. There should be a subspace RanP∗ε which is ε-close to RanP∗ and for which the analogous expression to (52) holds with an error of 3 order O(ε 2 ), possibly with a logarithmic correction. The corresponding effective Hamiltonian would then contain two additional terms of order ε2 , which, for the case of quadratic dispersion Hpε = −
N ε2 j=1
2
∆xj
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for the particles, can be calculated using formula (47) in [17]. As result we obtain for the Weyl symbol of the effective Hamiltonian including the momentum dependent Darwin term Heff (p, q) =
N 1 2 ε2 (pj · κ)(pi · κ) −ik·(qj −qi ) ∗ p + E(q) + dk e ρˆj (k)ˆ ρi (k) ε j 2m 2 |k|2 3 j j=1 j
with mεj = 1/(1 +
ε2 2 ej )
and 1 ej = 4π
dv dw R3 ×R3
ρj (v)ρj (w) |v − w|
the electromagnetic mass. As explained above, for the rigorous justification of (55) a space-adiabatic theorem without gap but for rotated subspaces P∗ε is needed and thus it is beyond the scope of the present paper. ♦ Before proving Theorem 7 we make up for the Proof of Proposition 5. A standard estimate (cf. e.g. [4] Proposition 1.3.8) shows that for f ∈ L2 (R3 ) and any a > 0 f / | · | 4L2 (R3 ) 2 2 Φ(f )ψF ≤ aHf ψF + + 2 ψ2F . (56) a Hence Φ(f ) is infinitesimally Hf -bounded whenever f L2 (R3 ) + f / | · |L2 (R3 ) < ∞. Then Kato-Rellich implies that H0 (x) is self-adjoint on D(Hf ), since by as sumption | · |v(x, ·) ∈ L2 . Using (i), (ii) and VIR (·) ∈ Cbn (R3N ), we obtain from (56) that (57) ∂xα H0 (x) = Φ (|k| ∂xα v(x, k)) + ∂xα VIR (x) is relatively bounded with respect to Hf for |α| ≤ n. Moreover, (ii), (56) and (57) imply that H0 (·) ∈ Cbn (R3N , L(D, F )). To compute the ground state energy E(x) observe that from “completing the square” one finds v(x, k)
v ∗ (k, x) a(k) + √ dk |k| a∗ (k) + √ H0 (x) = 2 2 R3 1 2 dk |k| |v(x, k)| + VIR (x) . − 2 R3 √ It is well known that the map → a(k) + v(x, k)/ 2 comes from the unitary a(k)
transformation U (x) = exp iΦ iv(x, ·) , i.e. U (x) a(f ) U ∗ (x) = a(f ) + f, v(x, ·) ,
(58)
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whenever v(x, ·) ∈ L2 . Equation (58) follows from the fact that [A, [A, B]] = 0 implies [exp(iA), B] = exp(iA)[iA, B] and the CCRs. Transformations of the form (58) are called Bogoliubov transformations. Therefore H0 (x) = U (x) Hf U ∗ (x) + E(x) with 1 E(x) = − dk |k| |v(x, k)|2 + VIR (x) . 2 R3 Since Hf Ω0 = 0 for the unique ground state Ω0 = (1, 0, 0, . . .) ∈ F, we find H0 (x)Ω(x) = E(x)Ω(x)
with Ω(x) = U (x)Ω0 .
Next we need to take derivatives of U (x) with respect to x. It follows from the CCRs (46) that for f, g ∈ L2 (R3 ) Φ(f ), Φ(g) = i Im f, g , and thus, by assumption, that ∇x Φ(iv(·, √ x)) = Φ(i∇x v(·, x)) commutes with Φ(iv(·, x)). Hence we obtain that on D( N ) ∇x U (x) = U (x) iΦ(i∇x v(·, x)) = iΦ(i∇x v(·, x)) U (x) .
(59)
By further differentiating (59) we can get up to nth derivatives since ∂xα v(x, k) ∈ L2 (R3 ) for |α| ≤ n. In particular we find with (iii) that Ω(x) = U (x)Ω0 ∈ Cbn (R3N , F ). Proof of Theorem 7. We start by showing that the assumptions of Theorem 1 are indeed satisfied and thus (52) follows. It is straightforward to check that the assumptions on ρˆj imply the assumptions of Proposition 5 for n = 5. Hence the first part of H40 follows with P∗ (x) = |Ω(x) Ω(x)|. For H40 (iii) observe that, using (59), ∇x Ω(x) = ∇x U (x)Ω0 = U (x) iΦ(i∇x v(·, x))Ω0 , and thus Ω(x), ∇x Ω(x) F = Ω0 , iΦ(i∇x v(·, x))Ω0 F = 0 .
(60)
As a consequence, (∇x P∗ )(x)P∗ (x) = |∇x Ω(x) Ω(x)|. Hence we obtain for 1 ≤ j≤N R(δ, x)(∇xj P∗ )(x)P∗ (x) = i |U (x) Rf (δ) Φ(i∇xj v(·, x))Ω0 Ω(x)| ,
(61)
where Rf (δ) = (Hf − iδ)−1 . For (20) one therefore finds R(δ, x)(∇xj P∗ )(x)P∗ (x)2L(F )
= =
Rf (δ)Φ(i∇xj v(x, ·))Ω0 2F 3 2 (|k| − iδ)−1 k eik·xj ρˆj (k)|k|− 2 2
(62)
L (R3 )
Whenever ρj satisfies the infrared condition ρˆj (k)|k|− 2 ∈ L2 (R3 ), (62) is bounded uniformly in δ since |k|/(|k| − iδ) ≤ 1. In general we assume that ρˆj (k)|k|−1 ∈ 3
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L2 (R3 ). Using that ρˆj (k) is bounded uniformly according to Riemann-Lebesgue, (62) becomes 1 2 |k| −1 ik·xj − 32 ρˆj (k)|k| ≤ C1 d|k| 2 + C2 ≤ C ln 1/δ , (|k| − iδ) k e 2 3 |k| + δ 2 L (R ) 0 (63) for δ ∈ (0, 12 ]. To obtain the estimate (21), we differentiate (61) and find
∇xi R(δ, x)(∇xj P∗ )(x)P∗ (x) = i |U (x) Rf (δ)Φ(i∇xj v(x, ·))Ω0 ∇xi Ω(x)| − |U (x) Φ(i∇xi v(x, ·)) Rf (δ)Φ(i∇xj v(x, ·))Ω0 Ω(x)| (2)
+ i |U (x) Rf (δ)Φ(i∇ij v(x, ·))Ω0 Ω(x)| .
(64)
All three terms in (64) can be bounded by using the same type of arguments as in (62) and (63): For the first term use in addition that ∇xi Ω(x)Hf < C and for the second term one has to estimate the components in the 0-boson sector and in 4 the 2-boson sector separately. In summary we showed that H0 (iii) is satisfied with η(δ) = δ ln(1/δ), and with η(δ) = δ if all charges satisfy the infrared condition individually. We are left to check for Assumption h4 . Let hε = Hpε ⊗1 = j h(−iε∇xj )⊗1, with h(p) = p2 + 1. Essential self-adjointness of hε + H0 on D(Hp ⊗ 1) ∩ D(H0 ) is a standard result, cf. Proposition 2.1 in [1]. We define (Dh)εj = (∇h)(−iε∇xj ) ⊗ 1, with (Dh)ε L(H)⊕ 3N ≤ 1, and postpone the technical proof of the following Lemma to the end of this section. Lemma 10. hε and (Dh)ε satisfy h4 (i) and (ii). We conclude that all assumptions of Theorem 1 are satisfied for the (SN) and the (IR) model and thus (52) holds. However, (54) follows from (52) by the following Lemma and an argument like (27). Lemma 11. There is a constant C < ∞ such that for ε > 0 sufficiently small ε ∗ ε H P∗ L(H) ≤ ε2 C . diag − U Heff U Proof. In order to apply h4 (i) to U(x), we have to extend U(x) : RanP∗ (x) → C = defined in (53) to a map U(·) ∈ Cb4 (R3N , L(F )) first. To this end let U(x) ∗ |Ω0 Ω(x)| and note that U U = P∗ . With this definition one finds ε Hdiag P∗
=
H0 P∗ + P∗ hε P∗ = E P∗ + P∗ U∗ hε U P∗ + P∗ [ hε , U∗ ] U P∗
=
ε U ∗ Heff U P∗ + P∗ [ hε , U∗ ] U P∗ ,
and we are left to show that P∗ [ hε , U∗ ] U P∗ = O(ε2 ). Using h4 (i) with A = U∗ we find that P∗ [ hε , U∗ ] U P∗ = −iε P∗ (∇x U∗ ) · (Dh)ε U P∗ + O(ε2 ) .
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However, according to (60) P∗ (x)(∇x U∗ )(x) = |Ω(x) Ω(x), ∇x Ω(x) Ω0 | = 0 , and thus the desired result follows. Proof of Lemma 10. Heuristically h4 (i) and (ii) hold, because they are just special cases of the expansion of a commutator of pseudodifferential operators. However, since hε is unbounded and A is only 4-times differentiable, we need to check the estimates “by hand”. For notational simplicity we restrict ourselves to the case N = 1, from which the general case follows immediately. Let g(p) = 1/ p2 + 1, g ε = g(−iε∇x ) ⊗ 1 and A ∈ Cb4 (R3 , Hf ), then | · |s g ∈ L1 (R3 ) for s ∈ {0, 4} and thus for ψ ∈ S
ε
g Aψ (x) = dy g(y) A(x − εy) ψ(x − εy) 1 2 (2) ds y, ∇ A(x − sεy) y ψ(x − εy) = dy g(y) A(x) − εy · ∇A(x) + ε 0
= Ag ε ψ (x) − i ε ∇A · ∇g ε ψ (x) 1 ds y, ∇(2) A(x − sεy) y ψ(x − εy) . (65) + dy g(y) ε2 0
From (65) one concludes after a lengthy but straightforward computation involving several integrations by parts that ε2 ∆x [g ε , A] = −i ε∇A · (∇g)ε (ε2 ∆x ) + R with R ≤ C
4 j=2
εj
sup x∈R3N , |α|=j
∂xα A(x)L(Hf ) .
Hence we find [hε , A] = [(1 − ε2 ∆x )g ε , A] = (1 − ε2 ∆x )[g ε , A]− [ε2 ∆x , A]g ε = −i ε∇A·(Dh)ε + R with R ≤ C
4 j=2
εj
sup x∈R3N , |α|=j
∂xα A(x)L(Hf ) .
This proves h4 (i). By the same type of arguments one shows also h4 (ii).
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Acknowledgments I am grateful to Herbert Spohn for suggesting the massless Nelson model as an application for a space-adiabatic theorem without gap, as well as for numerous valuable discussions, remarks and hints concerning the literature. Parts of this work developed during a stay of the author at the Universit´e de Lille and I thank Stephan De Bi`evre and Laurent Bruneau for hospitality and for a helpful introduction to Reference [1]. For critical remarks which lead to an improved presentation I thank Detlef D¨ urr and the referee.
References [1] A. Arai, Ground state of the massless Nelson model without infrared cutoff in a non-Fock representation, Rev. Math. Phys. 13, 1075–1094 (2001). [2] J. E. Avron and A. Elgart, Adiabatic theorem without a gap condition, Commun. Math. Phys. 203, 445–463 (1999). [3] J. E. Avron and A. Elgart, Adiabatic theorem without a gap condition: Twolevel system coupled to quantized radiation field, Phys. Rev. A 58, 4300 (1998). [4] V. Betz, Gibbs measures relative to Brownian motion and Nelson’s model, Dissertation, TU M¨ unchen (2002). [5] F. Bornemann, Homogenization in time of singularly perturbed mechanical systems, Lecture Notes in Mathematics 1687, Springer, Heidelberg, 1998. [6] T. Chen, Operator-theoretic infrared renormalization and construction of dressed 1-particle states in non-relativistic QED, Dissertation, ETH Z¨ urich No. 14203 (2001). [7] E. B. Davies, Particle-boson interactions and the weak coupling limit, J. Math. Phys. 20, 345–351 (1979). [8] J. Fr¨ ohlich, On the infrared problem in a model of scalar electrons and massless scalar bosons, Ann. Inst. Henri Poincar´e 19, 1–103 (1973). [9] J. Fr¨ ohlich, M. Griesemer and B. Schlein, Asymptotic completeness for Compton scattering, preprint, mp arc 01-420 (2001). [10] F. Hiroshima, Weak coupling limit with a removal of an ultraviolet cutoff for a Hamiltonian of particles interacting with a massive scalar field, Inf. Dim. Anal., Quant. Prob. and Related Topics 1, 407–423 (1998). [11] T. Kato, On the adiabatic theorem of quantum mechanics, Phys. Soc. Jap. 5, 435–439 (1958).
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[12] M. Kunze and H. Spohn, Slow motion of charges interacting through the Maxwell field, Commun. Math. Phys. 203, 1–19 (2000). [13] J. L¨ orinczi, R. A. Minlos and H. Spohn, The infrared behavior in Nelson’s model of a quantum particle coupled to a massless scalar field, Ann. Henri Poincar´e 3, 1–28 (2002). ˙ Minlos and H. Spohn, Infrared regular representation of [14] J. L¨orinczi, R.A. the three dimensional massless Nelson model, to appear in Lett. Math. Phys. (2001). [15] A. Martinez and V. Sordoni, On the time-dependent Born-Oppenheimer approximation with smooth potential, Comptes Rendus Acad. Sci Paris 334, 185–188 (2002). [16] E. Nelson, Interaction of nonrelativistic particles with a quantized scalar field, Jour. Math. Phys. 5, 1190–1197 (1964). [17] G. Panati, H. Spohn and S. Teufel, Space-adiabatic perturbation theory, ePrint ArXive math-ph/0201055 (2002). [18] G. Panati, H. Spohn and S. Teufel, Space-adiabatic perturbation theory in quantum dynamics, Phys. Rev. Lett. 88, 250405 (2002). [19] M. Reed and B. Simon, Methods of modern mathematical physics II, Academic Press (1975). [20] M. Reed and B. Simon, Methods of modern mathematical physics IV, Academic Press (1978). [21] H. Spohn, Dynamics of charged particles and their radiation field, in preparation. [22] H. Spohn and S. Teufel, Adiabatic decoupling and time-dependent BornOppenheimer theory, Commun. Math. Phys. 224, 113–132 (2001). [23] S. Teufel and H. Spohn, Semiclassical motion of dressed electrons, Rev. Math. Phys. 4, 1–28 (2002). [24] S. Teufel, A note on the adiabatic theorem without gap condition, Lett. Math. Phys. 58, 261–266 (2001). Stephan Teufel Zentrum Mathematik Technische Universit¨ at M¨ unchen 80290 M¨ unchen Germany email: [email protected] Communicated by Gian Michele Graf submitted 25/03/02, accepted 17/06/02
Ann. Henri Poincar´e 3 (2002) 967 – 981 c Birkh¨ auser Verlag, Basel, 2002 1424-0637/02/050967-15
Annales Henri Poincar´ e
Curvature-Induced Bound States for a δ Interaction Supported by a Curve in R3 P. Exner and S. Kondej Abstract. We study the Laplacian in L2 (Ê3 ) perturbed on an infinite curve Γ by a δ interaction defined through boundary conditions which relate the corresponding generalized boundary values. We show that if Γ is smooth and not a straight line but it is asymptotically straight in a suitable sense, and if the interaction does not vary along the curve, the perturbed operator has at least one isolated eigenvalue below the threshold of the essential spectrum.
1 Introduction Relations between the geometry and spectral properties are one of the vintage topics of mathematical physics. In the last decade they attracted attention also in the context of quantum mechanics. A prominent example is the curvature-induced ˇ GJ, DE, RB]. This effect appears to be binding in infinite tube like regions [ES, a robust one: it has been demonstrated recently that bends can produce localized states not only if the transverse confinement is hard, i.e. realized by a Dirichlet condition, but also when it is weaker corresponding to a potential well or a δ interaction [EI]. The result is appealing, not only because it concerns an interesting mathematical problem, but also in view of applications in mesoscopic physics where such operators are used as a natural model for semiconductor “quantum wires”. Since in the latter electrons are trapped due to interfaces between two different materials representing finite potential jumps, by tunneling effect they can be found outside the wire, albeit not too far because the exterior is (for the energies in question) the classically forbidden region. The main result of the paper [EI] concerns nontriviality of the discrete spectrum for a class of operators in L2 (R2 ) which can be formally written as −∆ − αδ(x − Γ) with α > 0, where Γ is a curve which is not a straight line but it is asymptotically straight in a suitable sense. A question naturally arises whether a similar result is valid for a curve in R3 . Such an extension is not trivial, because ˇ representing in a the argument in [EI] relies on the resolvent formula of [BEKS] sense a generalization of the Birman-Schwinger theory. The said formula is valid for singular perturbations of the Laplacian which can be treated by means of a quadratic-form sum, i.e. as long as the codimension of the manifold supporting the perturbation is one. Thus if we want to address the stated question, we are forced to look for
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other tools. One possibility is to employ the resolvent formula for a curve in R3 derived in [Ku]. However, since it uses rather strong regularity hypotheses about the curve we take another route and begin instead with an abstract formula for strongly singular perturbations due to A. Posilicano [Po1]. When it is specified to our particular case, it contains again an embedding operator into a space of functions supported on the curve Γ, however, this time it is not the “naive” L2 but rather a suitable element from the scale of Sobolev spaces. Of course, one can regard it as a generalization of Krein’s formula; recall that such a way of expressing the resolvent can be used not only to describe δ interaction perturbations but also more general dynamics supported by zero measure sets [Ka, KK, Ko]. Another aspect of the absence of a description in terms of the quadratic-form sum concerns the very definition of the operator we want to study. We have to employ boundary conditions which relate the corresponding generalized boundary values in the normal plane to the curve modeled after the usual two-dimensional δ interaction [AGHH], which requires us to impose stronger regularity conditions on Γ. Furthermore, a modification of the Birman-Schwinger technique used in [EI] demands stronger restrictions on the regularity of the curve. On the other hand, apart of these technical hypotheses our main result – stated in Theorem 5.6 below – is analogous to that of [EI], namely that for any curve which is asymptotically straight but not a straight line the corresponding operator has at least one isolated eigenvalue. This conclusion is by no means obvious having in mind how different are the point interactions in one and two dimensions.
2 The resolvent formula As a preliminary let us show how self-adjoint extensions of symmetric operators can be characterized in terms of a Krein-type formula derived in [Po1]; we refer to this paper for the proof and a more detailed discussion. With a later purpose on mind we do not strive for generality and restrict ourselves to the case of the Hilbert space H := L2 (R3 ) ≡ L2 and the Laplace operator, −∆ : D(∆) → L2 , which is well known to be self-adjoint on the domain D(∆) which coincides with the usual Sobolev space H 2 (R3 ) ≡ H 2 . For any z belonging to the resolvent set (−∆) = C \ [0, ∞) we define the resolvent as the bounded operator Rz := (−∆ − z)−1 : L2 → H 2 . Consider a bounded operator τ : H2 → X into a complex Banach space X and its adjoint in the dual space X . Recall that for a closed linear operator A : X → Y the adjoint is defined by (A∗ l)(x) = l(Ax) for all x ∈ D(A) and l ∈ D(A∗ ) ⊆ Y . Then we can introduce the operators Rτz = τ Rz : L2 → X ,
˘ τz = (Rτz¯ )∗ : X → L2 , R
which are obviously bounded too. Let Z be an open subset of (−∆) symmetric w.r.t. the real axis, i.e. such that z ∈ Z implies z¯ ∈ Z. Suppose that for any
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z ∈ Z there exists a closed operator Qz : D ⊆ X → X satisfying the following conditions, ˘ τz , Qz − Qw = (z −w)Rτw R z¯ l1 (Q l2 ) = l2 (Qz l1 ) . ∀l1 , l2 ∈ D ,
(2.1) (2.2)
It will be used to construct a family of self-adjoint operators which coincide with −∆ when restricted to ker τ . They can be parametrized by symmetric operators Θ : D(Θ) ⊆ X → X . To this end, we define QzΘ = Θ + Qz : D(Θ) ∩ D ⊆ X → X , ¯ −1 ) exist and are bounded } . ZΘ := { z ∈ ρ(−∆) : (QzΘ )−1 , (QzΘ With this notation we can state the result we want to borrow from [Po1]. Theorem 2.1 Assume that the conditions
and
ZΘ = ∅
(2.3)
Ran τ ∗ ∩ L2 = {0}
(2.4)
are satisfied. Then the bounded operator z ˘ z (Qz )−1 Rz , Rτ,Θ := Rz − R τ Θ τ
z ∈ ZΘ ,
is the resolvent of the self-adjoint operator −∆τ,Θ defined by ˘ z (Qz )−1 τ fz , fz ∈ D(∆) } , D(∆τ,Θ ) = { f ∈ L2 : f = fz − R τ Θ (−∆τ,Θ − z)f := (−∆ − z)fz , which coincides with −∆ on the ker τ . z by The above formula allows us to study the singularities of the resolvent Rτ,Θ z −1 means of those of (QΘ ) in full analogy with the usual Birman-Schwinger method. Indeed, using the argument of [Po2] one derives easily the equivalence
z ∈ σdisc (AΘ ) ⇔ 0 ∈ σdisc (QzΘ ) .
3 Singular perturbation on a curve in R3 Henceforth, we will be interested in a specific class of perturbations of the Laplacian on H = L2 (R3 ). The free resolvent Rz = (−∆ − z)−1 : L2 (R3 ) → H 2 (R3 ) ,
z ∈ (−∆) ,
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is an integral operator with the kernel √
ei z|x−y| . G (x−y) = 4π |x−y| z
Let Γ ⊂ R3 be a curve defined as a graph of a continuous function which is assumed to be piecewise C 1 . Recall that Γ admits a natural parametrization by the arc length which is unique up to a choice if the reference point; we denote the parameter as s and use the symbol γ(s) : R → R3 for the corresponding function. Then we have (3.1) |γ(s)−γ(s)| ≤ |s−s | . To specify further the family of curves which we will consider, we introduce for any ω ˜ ∈ (0, 1) and ε˜ > 0 the set s Sω˜ ,˜ε := (s, s ) : ω ˜< <ω ˜ −1 if |s+s| > ξ(˜ ω )˜ ε, s and |s−s | < ε˜ if |s+s| < ξ(˜ ω )˜ ε , where ξ(˜ ω ) :=
1+˜ ω 1−˜ ω
. We adopt the following assumptions:
(a1) there exists a c ∈ (0, 1) such that |γ(s)−γ(s )| ≥ c |s−s |, (a2) there are ω ∈ (0, 1), µ ≥ 0 and positive ε, d such that the inequality 1−
|γ(s)−γ(s )| |s−s | ≤ d |s−s | (|s−s | + 1)(1 + (s2 +s2 )µ )1/2
holds for all (s, s ) ∈ Sω,ε . The first condition means, in particular, that Γ has no cusps and self-intersections. The second assumption is basically a requirement of asymptotic straightness (see Remark 5.7), but in contrast to [EI] it restricts also the behaviour of |γ(s)−γ(s )| at small distances; it is straightforward to check that the bound cannot be satisfied unless Γ is C 1 -smooth. To make use of Theorem 2.1 we take X = L2 (R) and denote the corresponding scalar product by (·, ·)l (see also Remark 3.1 below). The operator τ : H 2 (R3 ) → L2 (R) which we will employ in our construction is a trace map defined in the following way: τ φ(s) := φ(γ(s)) ; it is a standard matter to check that the definition makes sense and the operator τ is bounded [BN]. The adjoint operator τ ∗ : L2 (R) → H −2 (R3 ) is determined by the relation τ ∗ h, ω = (h, τ ω)l ,
h ∈ L2 (R) ,
ω ∈ H −2 (R3 ) ,
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where ·, · stands for the duality between H −2 (R3 ) and H 2 (R3 ), in other words, we can write τ ∗ h = hδΓ , where δΓ is the Dirac measure supported by Γ. Since δΓ ∈ / L2 (R3 ) we get Ran τ ∗ ∩ L2 (R3 ) = {0} , so condition (2.4) is satisfied. Remark 3.1 Notice that the map τ as introduced above is not surjective. Indeed, since γ(s) is a Lipschitz function we have Ran τ = H 1 (R) – cf. [BN]. However, we lose nothing by keeping X = L2 (R) in the further discussion. The problem at hand is to define an operator Qz : D ⊆ L2 (R) → L2 (R) satisfying the conditions (2.1) and (2.2). To this end some preliminaries are needed. Since our considerations concern spectral properties at the negative halfline, it suffices for further discussion to restrict ourselves to z = −κ2 with κ > 0. In such a case it is convenient to modify slightly the used notation by introducing 2
Qκ := Q−κ ,
2
Rκτ := Rτ−κ ,
˘ −κ2 . ˘ κτ := R R τ
and similarly e−κ|s−s | , 4π |s−s |
Gκ (s−s ) :=
e−κ|γ(s)−γ(s )| . 4π |γ(s)−γ(s )|
Gκ (γ(s)−γ(s )) =
The difference of these two kernels, Bκ (s, s ) = Gκ (γ(s)−γ(s )) − Gκ (s−s ) , defines the integral operator Bκ : D(Bκ ) → L2 (R) with the domain D(Bκ ) = {f ∈ L2 (R) : Bκ f ∈ L2 (R)}. A key observation is that this operator has a definite sign: −κξ in view of (3.1) and of the fact that the function ξ → e ξ decreases monotonically for κ, ξ positive, we have (3.2) Bκ (s, s ) ≥ 0 . The operator Bκ is related obviously with the deviation of Γ from a straight line; below we shall demonstrate that properties for a curve satisfying the assumptions (a1) and (a2) with any µ ≥ 0 is bounded (see Remark 5.4). Next we need to show how the free resolvent kernel behaves when one of the three dimensions is integrated out. By a direct computation one can show that for all κ, κ > 0 and f1 , f2 ∈ L2 (R) the following relation, f1 (s)f2 (s ) [Gκ (s−s ) − Gκ (s−s )] ds ds R2 = f1 (s)f2 (s ) [Tˇκ (s−s ) − Tˇκ (s−s )] ds ds , R2
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is valid, where Tˇκ (s−s ) := −
1 (2π)2
1/2 ip(s−s ) ln p2 +κ2 e dp .
This result means, in particular, that κ f1 (s)f2 (s ) G (s−s ) ds ds − R2
(3.3)
R
R2
f1 (s)f2 (s ) Tˇκ (s−s ) ds ds
(3.4)
is κ-independent. Let Tκ : D(Tκ) → L2 (R) be the integral operator with the domain D(Tκ ) = {f ∈ L2 (R) : R Tˇκ (s − s )f (s ) ds ∈ L2 (R)} and the kernel 1 Tκ (s−s ) := Tˇκ (s−s ) + 2π (ln 2 + ψ(1)) where −ψ(1) ≈ 0.577 is Euler’s number. Then Tk is self-adjoint and we can define the operator Qκ f = (Tκ +Bκ )f : D ≡ D(Tκ ) → L2 (R) , which is also self-adjoint and has the needed properties: 2
Lemma 3.2 The operators Q−κ ≡ Qκ satisfy the conditions (2.1), (2.2). Proof. Let f1 , f2 ∈ D, then a direct computation yields ˘ κτ f2 )l (κ2 − κ2 )(f1 , Rκτ R = f1 (s)f2 (s ) [Gκ (γ(s)−γ(s )) − Gκ (γ(s)−γ(s ))] ds ds .
R2
On the other hand, by definition of Qκ and the κ-independence of the expression (3.4) we find that (f1 , (Qκ − Qκ )f2 )l is also given by the right-hand side of the last formula, which proves (2.1). Since Qκ is self-adjoint, the condition (2.2) is satisfied too. The operator Θ : L2 (R) → L2 (R) appearing in Theorem 2.1 will be identified here with the multiplication by a real number, Θf = −αf with α ∈ R and the sign convention made with a later purpose on mind. Then the operator QκΘ = Θ + Qκ : D → L2 (R) is self-adjoint for any κ > 0. For simplicity we identify in the following the symbols of the operators τ, Θ with γ, α, respectively. In this notation Theorem 2.1 says the following: if κ ∈ Zα , i.e. if the operator (Qκα )−1 = (Qκ−α)−1 : L2 (R) → L2 (R) exists and is bounded, then ˘ κγ (Qκ −α)−1 Rκγ Rκγ,α = Rκ − R
(3.5)
is the resolvent of a self-adjoint operator which we denote as −∆γ,α . In Section 5 below we will show that the real part of Zα is non-empty being equal to the interval
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(sκ , ∞) with the exception of a discrete set, thus verifying a posteriori that the assumption (2.3) is satisfied. Furthermore, the operator −∆γ,α coincides with −∆ on ker τ = {g ∈ H 2 (R3 ) : g(x) = 0, x ∈ Γ} and ˘ κγ (Qκ −α)−1 τ fk , fκ ∈ D(∆)} , D(−∆γ,α ) = {f ∈ L2 : f = fκ − R (−∆γ,α + κ2 )f = (−∆ + κ2 )fκ .
4 The interaction in terms of boundary conditions To proceed further we have to impose slightly stronger regularity requirement on the curve Γ . Specifically, we assume that it is given by a function γ(s) : R → R3 which is C 1 everywhere and piecewise C 2 , and satisfies the condition (a1). Then we can introduce, apart of a discrete set, the Frenet’s frame for Γ, i.e. the triple (t(s), b(s), n(s)) of the tangent, binormal and normal vectors, which are by assumption piecewise continuous functions of s. Given ξ, η ∈ R we denote r = (ξ 2 +η 2 )1/2 and define the set the “shifted” curve ξη Γr ≡ Γξη r := { γr (s) ≡ γr (s) := γ(s) + ξb(s) + ηn(s) } .
It follows from the smoothness of γ in combination with (a1) that there exists an r0 > 0 such that Γr ∩ Γ = ∅ holds for each r < r0 . 2 (R3 \ Γ) is continuous on R3 \ Γ its restriction to Since any function f ∈ Hloc Γr , r < r0 is well defined; we denote it as f Γr (s). In fact, we can regard f Γr (s) as a distribution from D (R) with the parameter r. We shall say that a function 2 f ∈ Hloc (R3 \ Γ) ∩ L2 (R3 ) belongs to Υ if the following limits Ξ(f )(s) := − lim
r→0
1 f (s) , ln r Γr
Ω(f )(s) := lim f Γr (s) + Ξ(f )(s) ln r , r→0
exist a.e. in R, are independent of the direction 1r (ξ, η), and define functions from L2 (R). The limits here are understood in the sense of the D (R) topology. With these prerequisites we are able now to characterize the operator −∆γ,α discussed above in terms of (generalized) boundary conditions, postponing the proof to the appendix. Theorem 4.1 With the assumption stated above we have D(−∆γ,α ) = Υα := { g ∈ Υ : 2παΞ(g)(s) = Ω(g)(s) } , −∆γ,α f = −∆f
for
x ∈ R3 \ Γ .
(4.1)
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5 Curvature-induced bound states Let us first find the spectrum of −∆γ0 ,α where γ0 is a linear function describing a straight line. Since Bκ = 0 holds in this case we have Qκ = Tκ . Then the resolvent formula (3.5) yields σ(−∆γ0 ,α ) = { −κ2 : α ∈ σ(Tκ ) = σac (Tκ ) } . Using the momentum representation of Tκ we immediately get σac (Tκ ) = (−∞, sκ ] , where sκ :=
1 2π (ψ(1)
− ln(κ/2)). Hence the spectrum of −∆γ0 ,α is given by σ(−∆γ0 ,α ) = σac (−∆γ0 ,α ) = [ζ0 , ∞) ,
where ζ0 = −4e2(−2πα+ψ(1)) as we expect with the spectrum of a two-dimensional δ interaction [AGHH] and the natural separation of variables in mind. To find the spectrum of −∆γ,α for a non-straight curve we treat the respective operator Qκ as a perturbation of the one corresponding to a straight line. First we have to localize the essential spectrum. Following step by step the argument given in the proof of Proposition 5.1 of Ref. [EI] we get Lemma 5.1 Let Γ be a curve given by a function γ(s) satisfying (a1) and (a2) with µ > 1/2. Then σess (−∆γ,α ) = [ζ0 , ∞). Next we observe that a nontrivial bending pushes the upper bound of the spectrum of Qκ up. Lemma 5.2 If Γ is not a straight line we have sup σ(Qκ ) > sκ .
(5.1)
Proof. Let φ be a non-negative function from C0∞ (R) such that φ(0) = 0. Given λ > 0 we set φλ (s) := λ1/2 φ(λs). To show (5.1) it suffices to check the following inequality (Qκ φλ , φλ )l − sκ (φλ , φλ )l > 0 , which is easily seen to be equivalent to 1 − 2π
λu ln 1+ κ R
2 1/2
ˆ 2
φ(u) du + λ
R2
Bκ (s, s )φ(λs)φ(λs ) ds ds > 0 ,
(5.2) ˆ where φ stands for the Fourier transform of φ. The first term in the last expression can expanded as
2
1 λ
ˆ 2 − u2 φ(u)
du + O(λ4 ) . 4π κ R
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Since Γ is not straight by assumption the inequality (3.2) is sharp in an open subset of R2 , so there is D > 0 such that λ R2 Bκ (s, s )φ(λs)φ(λs ) ds ds ≥ Dλ as λ → 0. Consequently, for all sufficiently small λ the inequality (5.2) is satisfied. On the other hand, the part of the spectrum in (sκ , ∞) added in this way is at most discrete provided the curve has the asymptotic straightness properties expressed by the assumption (a2) with µ large enough. Lemma 5.3 If µ > 1/2 then Bk are Hilbert-Schmidt operators. Moreover, norms Bκ HS are uniformly bounded with respect κ ≥ κ0 = |ζ0 |1/2 . Proof. Denote ρ ≡ ρ(s, s ) := |γ(s)−γ(s )| and σ ≡ σ(s, s ) := |s−s |. In this notation the assumptions (a1), (a2) can be written as (a1) there is a c ∈ (0, 1) such that ρ(s, s ) ≥ cσ(s, s ), (a2) there are ω ∈ (0, 1), µ ≥ 0 and ε, d > 0 s.t. for all (s, s ) ∈ Sω,ε we have 1−
dσ(s, s ) ρ(s, s ) ≤ . σ(s, s ) (σ(s, s )+1)(1 + (s2 +s2 )µ )1/2
Next we notice that the perturbation kernel is monotonous with respect to the spectral parameter, Bκ (s, s ) ≤ Bκ (s, s )
for κ < κ ,
thus to prove lemma it suffices to show that Bκ0 is a Hilbert-Schmidt operator. −κ0 υ Since the function υ → e υ is strictly decreasing and convex in (0, ∞), we have the following estimate, −κ0 σc e−κ0 σ e e−κ0 ρ − ≤− (σ − ρ) , 0≤ ρ σ σc where σc := cσ and c is the constant appearing in (a1). Thus we get 0≤
e−κ0 σ σ−ρ e−κ0 ρ − ≤ (κ0 σc + 1) 2 e−κ0 σc , ρ σ σc
and moreover, the assumption (a1) gives the bound σ−ρ ≤ 1 − c. σ In view of (a2), there exists a positive c˜ such that σ(s, s ) ≥ c˜ .
(5.3)
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holds for any (s, s ) ∈ R2 \ Sω,ε . Combining the last three inequalities we have in R2 \ Sω,ε the estimate e−κ0 σ 1 e−κ0 ρ − (5.4) ≤ M1 e−κ0 σc 4π ρ σ with M1 := (4π)−1 (1−c) c−2(κ0 c+ c˜−1 ). On the other hand using (5.3) and (a2) we get 1 e−κ0 ρ e−κ0 σ 1 − (5.5) ≤ M2 e−κ0 σc 2 4π ρ σ (1 + (s +s2 )µ )1/2 for (s, s ) ∈ Sω,ε , where M2 := (4π)−1 dc−2 max{1, κ0 c}. Putting now the estimates (5.4), (5.5) together we find Bk (s, s )2 dsds R2
≤ M12 ≤
R2 \Sω,ε
1 2 1+ω M 2 1 1−ω
e−2κ0 c|s−s | ds ds + M22
0
∞
e−κ0 cu u du + M22
Sω,ε
Sω,ε
e−2κ0 c|s−s | ds ds 1 + (s2 +s2 )µ
e−2κ0 c|s−s | ds ds < ∞ , 1 + (s2 +s2 )µ
which proves the result because the last integral converges for µ > 1/2.
Remark 5.4 As we have said, the assumption (a2) includes a decay of the quantity characterizing the non-straightness at large distances within Sω,ε as well as a restriction for s close to s . The latter (which is independent of µ) ensures the boundedness of Bκ uniformly w.r.t. κ. As in the proof of the above lemma the uniformity is easy; it suffices to check that Bκ0 is bounded. To this end we employ the Schur-Holmgren bound: we have Bκ0 l ≤ Bκ0 SH , where the right-hand side of the last inequality is for integral operators with symmetric positive kernels defined as Bκ0 SH = sup Bκ0 (s, s ) ds . s∈R
R
Let us use the notation from the previous proof. If σ ≤ ε, then by assumption (a2) there exists for any µ ≥ 0 a C1 > 0 such that Bκ0 (s, s ) ≤ C1 . On the other hand, if σ > ε then by (5.3) we can find C2 > 0 such that Bκ0 (s, s ) ≤ C2 e−κ0 σc . Combining the above two inequalities we get the following estimate,
s+ε ∞ e−κ0 cε Bκ0 (s, s )ds ≤ C1 ds + 2C2 e−κ0 c|s−s | ds = 2 C1 ε + C2 , κ0 c s−ε s+ε R which shows that Bκ0 SH is finite.
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Lemma 5.5 Let Γ be defined as before. Then the function κ → Qκ is continuous in the norm operator in (κ0 , ∞), and moreover, lim sup σ(Qκ ) = −∞ .
κ→∞
(5.6)
Proof. First we observe that the function κ → Tκ is continuous in the norm operator. Indeed, for any f ∈ D we have 2
1 p 2 + κ2 = |fˆ(p)|2 dp ln 2 4(2π)3 R p + κ2 1 κ 2 ln f 2l → 0 ≤ 4(2π)3 κ
(Tκ −Tκ )f l
(5.7)
as κ → κ. On the other hand, in analogy with [EI] we can estimate |(Bκ −Bκ )(s, s )| ≤ 2(Bκ (s, s )2 +Bκ (s, s )2 ) ≤ 4Bκ˜ (s, s )2 , 2
where κ ˜ := min{κ, κ } arriving therefore at lim Bκ −Bκ HS → 0 ;
κ →κ
(5.8)
from (5.7) and (5.8) we get the norm-operator continuity. Let further f ∈ D. The limiting relation (5.6) follows directly from the bound 2 1 κ 2 1/2 (Q f, f )l = − ln p +κ + ln 2 + ψ(1) |fˆ(p)|2 dp (2π)3/2 R 1 κ 2 2 +(Bκ f, f )l ≤ (− ln + ψ(1)) f l + S f l , 2 (2π)3/2 where S := supκ≥κ0 Bk l < ∞.
Now we are in position to state and prove our main result. Theorem 5.6 Let Γ be a curve determined by a function γ : R → R3 which is C 1 and piecewise C 2 , and satisfies the conditions (a1), (a2) with µ > 1/2. Then the operator −∆γ,α has at least one isolated eigenvalue in (−∞, ζ0 ). Proof. By Lemma 5.2 we have sup σ(Qκ ) > sκ , while by Lemma 5.3 this operator has only isolated eigenvalues of a finite multiplicity in (sκ , ∞). Let λ(κ) be such an eigenvalue of Qκ . Using then Lemma 5.5 we conclude that the function λ(·) is 1/2 such continuous and λ(κ) → −∞ as κ → ∞. Consequently, there is κ ˜ > |ζ0 | that λ(˜ κ) = α. From the resolvent formula (3.5) we then infer that −˜ κ2 ∈ (−∞, ζ0 ) is an eigenvalue of −∆γ,α .
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Remarks 5.7 (a) It is clear that the claim holds without the C 2 assumption, however, the latter is needed if we want to interpret the δ interaction on the curve in the spirit of Theorem 4.1. Furthermore, we see that any deviation from a straight Γ pushes the spectrum threshold below the value ζ0 but without the assumption (a2) we cannot be sure about the nature of this added part of the spectrum. (b) One may ask what the requirement of asymptotic straightness expressed by (a2) means. Suppose that γ is C 2 smooth. Then the curvature of Γ is every1/2 3 2 where defined and can expressed as k(s) = , where ki (s) := i=1 ki (s)
εijk γj (s)γk (s) with the summation convention for the indices of the Levi-Civita tensor. It allows us estimate the distance between γ(s) and γ(s ) in the following way, 1
s1 2 1/2 3 s πν |γ(s)−γ(s )| = cos ki (s2 ) ds2 + ds1 i=1 s 2 s ν=0 2
s1 3 s 1 ≥ ki (s2 ) ds2 1− ds1 , i=1 s 2 s
where we assume without loss of generality that s > s . Suppose that there are positive β, ci such that |ki (s)| ≤ ci |s|−β . Then |k(s)| ≤ 3c |s|−β , where c = maxi {ci } and one can estimate 2 s s1 |γ(s)−γ(s )| 1 1− ≤ k(s ) ds ds1 2 2 |s−s | 2 |s−s | s s s 2 1 1 3c2 c2 |s −s| c2 2 ≤ |s −s | ds ≤ ≤ . 1 1 2 |s−s | |s |2β s 2 |s |2β 2 |s |2β−2 Thus the conclusion is the same as in the two-dimensional case discussed in [EI]: the assumption (a2) with µ > 1/2 is satisfied if β > 5/4.
Appendix: proof of Theorem 4.1 First we check the inclusion D(−∆γ,α ) ⊆ Υα . Suppose that f ∈ D(−∆γ,α ), i.e. that there is fκ ∈ D(∆) such that ˘ κγ (Qκ −α)−1 τ fκ . f = fκ − R
(A.1)
Denote h := (Qκ −α)−1 τ fκ ∈ L2 (R), so f = fκ − Rκ τ ∗ h. Since fκ ∈ H 2 (R3 ) and 2 τ ∗ h ∈ H −2 (R3 ) is a measure supported by Γ we can conclude that f ∈ Hloc (R3 \Γ) – see [RS]. Using properties of the Macdonald function K0 (ς) and the following relation 2 2 1/2 1 e−κ(r +(s−s ) ) 1 = K0 ((p21 + κ2 )1/2 r) eip1 (s−s ) dp1 4π (r2 +(s−s)2 )1/2 (2π)2 R
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we can check that −κ(r2 +(s−s )2 )1/2 1 1 e lim ln rh(s) = Tκ h(s). h(s )ds + r→0 4π R (r2 +(s−s)2 )1/2 2π Now it is easy to demonstrate that the function ˘ κγ h)(x) = (R
1 4π
R
e−κ|x−γ(s)| h(s) ds |x−γ(s)|
satisfies the limiting relation ˘ κ h) Γr (s) + 1 ln rh(s) = Tκ h(s) + Bκ h(s) lim (R r→0 2π
(A.2)
with respect to families of “shifted” curves described in Sec. 4. The above limits are understood in distributional sense. It follows from (A.1) and (A.2) that Ξ(f )(s) = −
1 h(s). 2π
(A.3)
On the other hand, since fκ ∈ D(∆) = H 2 (R3 ) the same relations (A.1) and (A.2) yield Ω(f )(s) = (τ fκ )(s) − (Qκ h)(s) = −αh(s).
(A.4)
Combining (A.3) and (A.4) we obtain that f ∈ Υ and 2παΞ(f )(s) = Ω(f )(s). Conversely, one can show by analogous considerations that any function from Υα ˘ κγ (Qκ − α)−1 τ fκ with fκ ∈ D(∆), so can be represented in the form f = fκ − R D(−∆γ,α ) = Υα . Moreover, since (−∆γ,α + κ2 )f = (−∆ + κ2 )fκ and τ ∗ h ∈ H −2 (R3 ) is a measure supported by Γ we infer that −∆γ,α f (x) = −∆f (x),
x ∈ R3 \Γ .
This completes the proof.
Acknowledgments The work was supported by GAAS under the contract #1048101. The authors are obliged to A. Posilicano for making his results available to them prior to publication and to the referee for useful comments.
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References [AGHH] S. Albeverio, F. Gesztesy, R. Høegh-Krohn and H. Holden, Solvable Models in Quantum Mechanics, Springer, Heidelberg 1988. [BN]
O.V. Besov, V.P. Il’in and S.M. Nikol’skii, Integral Representations of Functions and Imbedding Theorems, Winston & Sons, Washington 1978–9.
ˇ J.F. Brasche, P. Exner, Yu.A. Kuperin and P. Seba, ˇ [BEKS] Schr¨ odinger operators with singular interactions, J. Math. Anal. Appl. 184 (1994), 112–139. [DE]
P. Duclos and P. Exner, Curvature-induced bound states in quantum waveguides in two and three dimensions, Rev. Math. Phys. 7 (1995), 73– 102.
[EI]
P. Exner and T. Ichinose, Geometrically induced spectrum in curved leaky wires, J. Phys. A34 (2001), 1439–1450.
ˇ [ES]
ˇ P. Exner and P. Seba, Bound states in curved quantum waveguides, J.Math. Phys. 30 (1989), 2574–2580.
[GJ]
J. Goldstone and R.L. Jaffe, Bound states in twisting tubes, Phys. Rev. B45 (1992), 14100–14107.
[Ka]
W. Karwowski, Hamiltonians with additional kinetic energy terms on hypersurfaces, in Applications of Self-Adjoint Extensions in Quantum Physics, Springer, LNP 324, Berlin 1989; pp. 203–217.
[KK]
W. Karwowski and V. Koshmanenko, Schr¨odinger operator perturbed by dynamics of lower dimension, in Differential Equations and Mathematical Physics, American Math. Society, Providence, R.I., 2000; pp. 249–257.
[Ko]
S. Kondej, Perturbation of the dynamics by objects supported by small sets, PhD Thesis, Wroclaw 2001.
[Ku]
Y.V. Kurylev, Boundary condition a curve for a three-dimensional Laplace operator, J. Sov. Math. 22 (1983), 1072–1082.
[Po1]
A. Posilicano, A Krein-like formula for singular perturbations of selfadjoint operators and applications, J. Funct. Anal. 183 (2001), 109–147.
[Po2]
A. Posilicano, Self-adjoint extensions by additive perturbations, submitted for publication
[RS]
M. Reed and B. Simon, Methods of Modern Mathematical Physics, II. Fourier Analisis, Self-adjointness, Academic Press, New York 1978.
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[RB]
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W. Renger and W. Bulla, Existence of bound states in quantum waveguides under weak conditions, Lett. Math. Phys. 35 (1995), 1–12.
P. Exner and S. Kondej Department of Theoretical Physics Nuclear Physics Institute Academy of Sciences ˇ z near Prague 25068 Reˇ Czech Republic email: [email protected], [email protected] Communicated by Gian Michele Graf submitted 19/03/02, accepted 17/05/02
To access this journal online: http://www.birkhauser.ch
Ann. Henri Poincar´e 3 (2002) 983 – 1002 c Birkh¨ auser Verlag, Basel, 2002 1424-0637/02/050983-20
Annales Henri Poincar´ e
Relations between the Hepp-Lieb and the Alli-Sewell Laser Models F. Bagarello Abstract. In this paper we show that the dissipative version of the laser model proposed by Alli and Sewell can be obtained by considering the stochastic limit of the (open system) hamiltonian introduced by Hepp and Lieb in their seminal work. We also prove that the Dicke-Haken-Lax hamiltonian produces, after the stochastic limit is considered, the generator of a semigroup with equations of motion very similar to those of Alli-Sewell, and coinciding with these under suitable conditions.
I Introduction In two recent papers, [1, 2], a dissipative laser model has been introduced and analyzed in some details. In particular in [1] (AS in the following) the rigorous definition of the unbounded generator of the model, which consists of a sum of a free radiation and a free matter generator plus a matter-radiation term, is given and the existence of the thermodynamical limit of the dynamics of some macroscopic observables is deduced. Moreover, the analysis of this dynamics shows that two phase transitions occur in the model, depending on the value of a certain pumping strenght. In [2] the analysis has been continued paying particular attention to the existence of the dynamics of the microscopic observables, which are only the ones of the matter since, in the thermodynamical limit, we proved that the field of the radiation becames classical. Also, the existence of a transient has been proved and an entropy principle has been deduced. On the other hand, in a series of papers [3, 4] culminating with the fundamental work by Hepp and Lieb [5] (HL in the following) many conservative models of matter interacting with radiation were proposed. In particular, in [5] the authors have introduced a model of an open system of matter and of a single mode of radiation interacting among them and with their (bosonic) reservoirs, but, to simplify the treatment, they have considered a simplified version in which the matter bosonic reservoir is replaced by a fermionic one. In this way they avoid dealing with unbounded operators. This is what they call the Dicke-Haken-Lax model (DHL model in the following). In [1, 2] the relation between the AS model and a many mode version of the HL model is claimed: of course, since no reservoir appear in the semigroup formulation as given by [1], this claim is reasonable but it is not clear the explicit way in which HL should be related to AS. In this paper we will prove that the relation between the two models is provided by (a slightly modified version of)
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the stochastic limit (SL), [6] and reference therein. In particular, if we start with the physical AS system (radiation and matter) and we introduce in a natural way two reservoirs (one is not enough!) for the matter and another reservoir for the radiation, then the SL of the hamiltonian for the new system constructed in this way returns back the original AS generator, under very reasonable hypotheses. Moreover, the model which we have constructed ad hoc to get this generator surprisingly coincides with the HL laser model, [5]. This is the content of Section 3, which follows a section where we introduce all the models we will deal with, to keep the paper self-contained. In Section 4 we will consider the SL of the fermionic version of the HL model, known as the DHL model, [5, 7]. We will find that, even if the form of the generator apparently differs from the one by AS, under certain conditions on the coefficients which define the model, the equations of motion for the observables of the matterradiation system coincide with the ones given in AS. In Section 5 we give our conclusions while the Appendix is devoted to summarize few results on SL which are used everywhere in this paper. Before concluding this section we wish to remark that we call the procedure proposed here stochastic limit even if a minor difference exists between the original approach, [6], and the one we will use here, namely the appearance of different powers of the over-all coupling constant λ which appear in our hamiltonian operators. The final remark concerns our notation which we try to keep as simple as possible by neglecting the symbol of tensor product (almost) everywhere in the paper.
II The Physicals Models In this section we will discuss the main characteristics of the three physical models which will be considered in this paper. In particular, we will only give the definition of the hamiltonians for the HL and the DHL models and the expression of the generator for the AS model, without even mentioning mathematical details like, for instance, those related to the domain problem intrinsic with all these models due to the presence of bosonic operators. We refer to the original papers for these and further details which are not relevant in this work. We begin with the AS model. This model is a dissipative quantum system, Σ(N ) , consisting of a chain of 2N + 1 identical two-level atoms interacting with an n−mode radiation field, n fixed and finite. We build the model from its constituent parts starting with the single atom. This is assumed to be a two-state atom or spin, Σat . Its algebra of observables, Aat , is that of the two-by-two matrices, spanned by the Pauli matrices (σx , σy , σz ) and the identity, I. They satisfy the relations σx2 = σy2 = σz2 = I; σx σy = iσz , etc.
(2.1)
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We define the spin raising and lowering operators σ± =
1 (σx ±iσy ). 2
(2.2)
We assume that the atom is coupled to a pump and a sink, and that its dynamics is given by a one-parameter semigroup {Tat (t)|t∈R+ } of completely positive, identity preserving contractions of Aat , whose generator, Lat , is of the following form. Lat σ± = −(γ1 ∓i)σ± ; Lat σz = −γ2 (σz − ηI),
(2.3)
where (> 0) is the energy difference between the ground and excited states of the atom, and the γ’s and η are constants whose values are determined by the atomic coupling to the energy source and sink, and are subjected to the restrictions that 0 < γ2 ≤2γ1 ; −1≤η≤1.
(2.4)
The matter consists of 2N + 1 non-interacting copies of Σat , located at the sites r = −N, . ., N of the one-dimensional lattice Z. Thus, at each site, r, there is a copy, Σr , of Σat , whose algebra of observables, Ar , and dynamical semigroup, Tr , are isomorphic with Aat and Tat , respectively. We denote by σr,u the copy of σu at r, for u = x, y, z, ±. We define the algebra of observables, A(N ) , and the dynamical semigroup, N (N ) of the matter to be ⊗N is the r=−N Ar and ⊗r=−N Tr , respectively. Thus, A algebra of linear transformations of C4N +2 . We identify elements Ar of Ar with those of A(N ) given by their tensor products with the identity operators attached to the remaining sites. Under this identification, the commutant, Ar , of Ar is the tensor product ⊗s=r As . The same identification will be implicitly assumed for the other models. (N ) Tmat ,
(N )
(N )
It follows from these specifications that the generator, Lmat , of Tmat is given by the formula (N ) Lmat = Ll , (2.5) l∈IN
where IN = {−N, . . . , −1, 0, 1, . . . , N }. Here Lr σr,± = −(γ1 ∓i)σr,± ; Lr σr,z = −γ2 (σr,z − ηI); and Lr (Ar Ar ) = (Lr Ar )Ar ∀Ar ∈Ar , Ar ∈Ar
(2.6)
We assume, furthermore, that the radiation field consists of n(< ∞) modes, represented by creation and destruction operators {al , al |l = 0, . ., n − 1} in a Fock-Hilbert space Hrad as defined by the standard specifications that (a) these operators satisfy the CCR, [al , am ] = δlm I; [al , am ] = 0,
(2.7)
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and (b) Hrad contains a (vacuum) vector Φ, that is annihilated by each of the a’s and is cyclic w.r.t. the algebra of polynomials in the a ’s. The formal generator of the semigroup Trad of the radiation is n−1 iωl [al al , .] + 2κl al (.)al − κl {al al , .} , (2.8) Lrad = l=0
where {., .} denotes anticommutator, and the frequencies, ωl , and the damping constants, κl , are positive. We refer to [1] for a rigorous definition of Lrad . The composite (finite) system is simply the coupled system, Σ(N ) , comprising the matter and the radiation. We assume that its algebra of observables, B (N ) , is the tensor product A(N ) ⊗R, where R is the −algebra of polynomials in the a’s, a ’s and the Weyl operators. Thus, B (N ) , like R, is an algebra of both bounded and unbounded operators in the Hilbert space H(N ) := C4N +2 ⊗Hrad . We shall identify elements A, R, of A(N ) , R, with A⊗Irad and Imat ⊗R, respectively. We assume that the matter-radiation coupling is dipolar and is given by the interaction Hamiltonian (N ) ) Hint = (σr,+ φ(N + h.c.), (2.9) r r∈IN
where we have introduced the so-called radiation field, φ(N ) , whose value at the site r is n−1 ) = −i(2N + 1)−1/2 λl al exp(2πilr/n). (2.10) φ(N r l=0
Here the λ’s are real-valued, N −independent coupling constants. Among the other results contained in [1], one of the most relevant is that the map
(N )
(N )
L(N ) = Lmat + Lrad + i[Hint , .] is really the generator of a N -depending semigroup, T (N ) , regardless of the un(N ) bounded nature of both Lrad and Hint . This is the starting point for a successive analysis, see [1, 2]. Now we introduce the HL model, changing a little bit the notations with respect to the original paper, [5], and introducing n modes for the radiation instead of the only one considered by HL. The HL hamiltonian for the 2N + 1 atoms and for the n modes of the radiation can be written as follows: H = H (S) + H (R) ,
(2.11)
where ”S” refers to the system (radiation+matter) and ”R” to the reservoir. The hamiltonian of the system is H (S) = ωR
n−1 j=0
a†j aj + µ
l∈IN
σl,z
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n−1 α +√ (σl,+ aj e2πijl/n + σl,− a†j e−2πijl/n ) + 2N + 1 j=0 l∈I N
β +√ 2N + 1
n−1
(σl,+ a†j e−2πijl/n + σl,− aj e2πijl/n ),
(2.12)
j=0 l∈IN
which differs from the one in [5] for the phases introduced in the last two terms, phases which are related to the presence of many modes in this hamiltonian with respect to the original one. Notice that the presence of β means that we are not restricting our model to the rotating wave approximation, (RWA). The hamiltonian for the reservoir contains two main contributions, one related to the two reservoirs of the matter and one to the reservoir of the radiation. We have: (A) H (R) = H (P ) + Hl , (2.13) l∈IN
where H
(P )
=
n−1
† √ n−1 (rj (g j )aj + rj (gj )a†j ), dk ωr,j (k)rj (k) rj (k) + α †
j=0
(2.14)
j=0
and (A)
Hl
=
2 s=1
dk ωms (k)m†s,l (k)ms,l (k) +
√ α(m†1,l (h1 )σl,− + h.c.)
√ + α(m†2,l (h2 )σl,+ + h.c.)
(2.15)
Few comments are necessary in order to clarify the formulas above. 1) first of all we are using the notation: rj (gj ) = dk rj (k)gj (k) and rj† (g j ) = dk rj† (k)g j (k). Here dk is a shortcut notation for dk 3 . 2) the functions gj and h1,2 are introduced by HL to regularize the bosonic fields rj (k) and m(1,2),l (k). 3) we notice that in this model two indipendent reservoirs, m1,l (k) and m2,l (k), are introduced for (each atom of) the matter, while only one, rj (k), is used for (each mode of) the radiation. This result will be recovered also in our approach. 4) it should be pointed out that the hamiltonian above is really only one of the possible extentions of the HL original one to the n-modes situation, and, in fact, is quite a reasonable extension. In particular we are introducing different dispersion laws and different regularizing functions hj for each mode of the radiation, while we use the same ω and the same h for the atoms localized in different lattice sites. This ”non-symmetrical” choice is motivated by the
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AS model itself, where we see easily that the free evolution of the matter observables does not dipend on the lattice site, while in the form of the generator (2.8) a difference is introduced within the different modes of the radiation. The reason for that is, of course, the mean field approximation which is being used to deal with the model. √ 5) the coupling constant α is written explicitly for later convenience. The role of each term of the hamiltonian above is evident. Instead of using the above expression for H, where we explicitly consider the effect of the system and the effect of the reservoirs, we divide H as a free and an interaction part, in the following way: √ (2.16) H = H0 + αHI , where H0 = ω R
n−1
a†j aj + µ
j=0
σl,z +
2
dkωms (k)m†s,l (k)ms,l (k)
l∈IN s=1
l∈IN
+
n−1
dkωr,j (k)rj (k)† rj (k)
(2.17)
j=0
and HI =
n−1
(rj† (g j )aj +rj (gj )a†j )
j=0
+
[(m†1,l (h1 )σl,− + h.c.) + (m†2,l (h2 )σl,+ + h.c.)] +
l∈IN
√ n−1 α (σl,+ aj e2πijl/n + σl,− a†j e−2πijl/n ) + +√ 2N + 1 j=0 l∈I N
β + α(2N + 1)
n−1
(σl,+ a†j e−2πijl/n + σl,− aj e2πijl/n ). (2.18)
j=0 l∈IN
The only non trivial commutation relations, which are different from the ones already given in (2.1,2.7), are: [rj (k), rl (k )† ] = δj,l δ(k − k ),
[ms,l (k), m†s ,l (k )] = δs,s δl,l δ(k − k )
(2.19)
We end this section by introducing the DHL model. The main difference, which is introduced to avoid dealing with unbounded operators, consists in the use of a fermionic reservoir for the matter, and for this reason the Pauli matrices of both AS and HL are replaced by fermionic operators as described in details, for instance, in [7].
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The idea for introducing these operators is quite simple: since we are considering only two-levels atoms (this is the reason why Pauli matrices appear!) a possible description of one such atom could consist in using two pairs of independent fermi operators, (b− , b†− ) which annihilates and creates one electron in the lowest energy level Ψ− , with energy E− , and (b+ , b†+ ) which annihilates and creates one electron in the upper energy level Ψ+ , with energy E+ . If we restrict the Hilbert space of the single atom to the states in which exactly one electron is present, in the lower or in the upper level, it is clear that b†+ b− behaves like σ+ , that is when it acts on a vector with one electron in the lowest state (b†− Ψ0 , Ψ0 being the state with no electrons), it returns a state with an electron in the upper level (b†+ Ψ0 ), and so on. Moreover b†+ b+ − b†− b− has eigenvectors b†± Ψ0 with eigenvalues ±1, so that it can be identified with σz . Going back to the finite system we put σ+,l = b†+,l b−,l ,
σ−,l = b†−,l b+,l ,
σz,l = b†+,l b+,l − b†−,l b−,l .
(2.20)
where l ∈ IN . The only non trivial anti-commutation relations for operators localized at the same lattice site are: {b±,l , b†±,l } = 1.
(2.21)
Moreover, see [7], two such operators commute if they are localized at different lattice site. For instance we have [b±,l , b†±,s ] = 0 if l = s. Since the number of the atomic operators is now doubled with respect to the HL model, it is not surprising that also the number of the matter reservoir operators is doubled as well: from 2 × (2N + 1) we get 4 × (2N + 1) operators, each one coupled with a b±,l 1 operator. On the other hand, the part of the radiation is not modified passing from the HL to the DHL model. Let us write the hamiltonian for√ the open system in the form which is more convenient for us and using λ instead of α. We have H = H0 + λHI ,
(2.22)
where H0 = ω R
n−1
a†j aj + µ
j=0
+
l∈IN
(b†+,l b+,l − b†−,l b−,l ) +
n−1
dk ωr,j (k)rj (k)† rj (k) +
j=0
† † (k)Bs,l (k) + Cs,l (k)Cs,l (k)) dk (k)(Bs,l
(2.23)
l∈IN s=± 1 We use here x to indicate one of the two possibilities: x or x† , x being a generic operator of the physical system.
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and HI =
n−1
(rj† (g j )a + rj (gj )a†j ) + λ
j=0
+
l∈IN
(N ) † b+,l b−,l
(φl
+ h.c.) +
† † [b†s,l (Bs,l (gBs ) + Cs,l (gCs )) + (Bs,l (gBs ) + Cs,l (gCs ))bs,l ]. (2.24)
l∈IN s=±
Here gB± and gC± are real function. The commutation rules for the radiation operators (system and reservoir) coincide with the ones of the HL model. For what concerns the matter operators (system and reservoirs) the first remark is that any two operators localized at different lattice sites commutes, as well as any operator of the radiation with any observable of the matter. As for operators localized at the same lattice site, the only non trivial anticommutators are † † (k )} = {C±,l (k), C±,l (k )} = δ(k − k ), {B±,l (k), B±,l
(2.25)
together with the (2.21), while all the others are zero. Finally, to clarify the different roles between the B and the C fields it is enough to consider their action on the ground state of the reservoir ϕ0 : † rj (k)ϕ0 = B±,l (k)ϕ0 = C±,l (k)ϕ0 = 0.
(2.26)
These equations, together with what has been discussed, for instance, in [7], show that B is responsible for the dissipation, while C is the pump. Again in reference [7] it is discussed which kind of approximations, other than using a fermionic reservoir, are introduced to move from ”real life” to the DHL model. Here we mention only a few: the atom is considered as a two level system; only n modes of radiation are considered (n=1 in the original model, [5]); the electromagnetic interaction is written in the dipolar approximation and within the RWA; the model is mean field, etc. It is worth remarking that since all the contributions in H0 above are quadratic in the various creation and annihilation operators, they all commute among them. This fact will be used in the computation of the SL of this model.
III Alli-Sewell versus Hepp-Lieb We begin this section with a pedagogical note on the single-mode single-atom version of the AS model. This will be useful in order to show that two reservoirs must be introduced to deal properly with the matter. After that we will consider the full AS model and we will show that the hamiltonian which produces the AS generator after considering its SL is nothing but the HL hamiltonian in the RWA. We will conclude this section proving that adding the counter-rotating term (the one proportional to β in (2.12)) does not affect this result, since its contribution disappear rigorously after the SL.
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The starting point is given by the set of equations (2.3)-(2.10) restricted to n = 1 and N = 0, which means only one mode of radiation and a single atom. (N ) With this choice the phases in φl disappear so that the interaction hamiltonian (2.9) reduces to Hint = i(σ− a† − h.c.), (3.1) and the total generator is L = Lmat + Lrad + i[Hint , .]. Let us suppose that the atom is coupled not only to the radiation by means of Hint , but also to a bosonic background m(k) with the easiest possible dipolar interaction: (3.2) HMm = σ+ m(h) + h.c. Of course this background must have a free dynamics and the natural choice is (3.3) H0,m = dk ωm (k)m† (k)m(k). For what concerns the radiation background the situation is completely analogous: HR,r = ar† (g) + h.c. (3.4) H0,r = dk ωr (k)r† (k)r(k), are respectively the free hamiltonian and the radiation-reservoir interaction. We take the complete hamiltonian as simply the sum of all these contributions, with the coupling constant λ introduced as below: † H = H0 + λHI = {µσz + ωR a a + dk ωm (k)m† (k)m(k) + dk ωr (k)r† (k)r(k)} +λ{(ar† (g) + h.c.) + (σ+ m(h) + h.c.) + λi(σ− a† − h.c.)}.
(3.5)
Taking the SL of this model simply means, first of all, considering the free evolution of the interaction hamiltonian, HI (t) = eiH0 t HI e−iH0 t , see Appendix and reference [6]. It is a simple computation to obtain that, if ωR = 2µ, †
i(ωr −ωR )t
(3.6)
i(2µ−ωm )t
†
) + h.c.) + λi(σ− a − h.c.). (3.7) In this case the SL produces, see Appendix, the following effective time-depending interaction hamiltonian: HI (t) = (ar (ge
(sl)
HI
) + h.c.) + (σ+ m(he
(t) = (arg† (t) + h.c.) + (σ+ mh (t) + h.c.) + i(σ− a† − h.c.),
(3.8)
where the dependence on λ disappears and the operators rg (t), mh (t) and their hermitian conjugates satisfy the following commutation relations for t ≥ t , (g)
[rg (t), rg† (t )] = Γ− δ(t − t ),
(h)
[mh (t), m†h (t )] = Γ− δ(t − t ).
(3.9)
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Here we have defined the following complex quantities:
(g)
Γ− = (h) Γ−
0
dk|g(k)|2 e−i(ωr (k)−ωR )τ ,
dτ −∞
0
=
dk|h(k)|2 e−i(2µ−ωm (k))τ .
dτ
(3.10)
−∞
We want to stress that the restriction t > t does not prevent to deduce the commutation rules (3.13) below, which are the main ingredient to compute the SL. However, the extension to t < t can be easily obtained as discussed in [6]. It is clear now what should be the main analytical requirement for the regularizing functions h and g: they must be such that the integrals above exist finite! In order to obtain the generator of the model we introduce the wave operator Ut (in the interaction representation) which satisfy the following operator differential equation: (sl)
∂t Ut = −iHI
(t)Ut , with
U0 = 1 .
(3.11)
In [6], and reference therein, it is proven that for a large class of quantum mechanical models, the equation above can be obtained as a suitable limit of differential equations for a λ-depending wave operator. Analogously, rg (t) and mh (t) can be considered as the limit (in the sense of the correlators) of the rescaled operators 2 1 −i(ωr −ωR )t/λ2 ) and λ1 m(hei(2µ−ωm )t/λ ). It is not surprising, therefore, that λ r(ge not only the operators but also the vectors of the Hilbert space of the theory are affected by the limiting procedure λ → 0. In particular, the vacuum η0 for the operators rg and mh , mh (t)η0 = rg (t)η0 = 0, does not coincide with the vacuum ϕ0 for m(k) and r(k), r(k)ϕ0 = m(k)ϕ0 = 0, see [6] for more details. Equation (3.11) above can be rewritten in the more convenient form Ut = 1 − i
0
t
HIsl (t )Ut dt ,
(3.12)
which is used, together with the time consecutive principle, see Appendix and [6], and with equation (3.9), to obtain the following useful commutation rules (g)
(h)
[rg (t), Ut ] = −iΓ− aUt ,
[mh (t), Ut ] = −iΓ− σ− Ut .
(3.13)
If we define the flow of a given observabe X of the system as jt (X) = Ut† XUt , the generator is simply obtained by considering the expectation value of ∂t jt (X) (ξ) on a vector state η0 = η0 ⊗ ξ, where ξ is a generic state of the system. Using formulas (3.11,3.13) and their hermitian conjugates, together with the properties of the vacuum η0 , the expression for the generator follows by identifying L in the equation < ∂t jt (X) >η(ξ) =< jt (L(X)) >η(ξ) . 0
0
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The result is L(X) = L1 (X) + L2 (X) + L3 (X), (g)
(g)
L1 (X) = Γ− [a† , X]a − Γ− a† [a, X], (h)
(h)
L2 (X) = Γ− [σ+ , X]σ− − Γ− σ+ [σ− , X], L3 (X) = i2 [σ− a† − σ+ a, X]
(3.14)
It is evident that both L1 and L3 can be rewritten in the same form of the radiation and interaction terms of the AS generator but this is not so, in general, for L2 which has the form of the AS radiation generator only if the pumping parameter η is equal to −1. This is not very satisfactory and, how we will show in the following, is a consequence of having introduced a single reservoir for the atom. We will show that the existence of a second reservoir allows for the removal of the constraint η = −1 we have obtained in the simplified model above. With all of this in mind it is not difficult to produce an hamiltonian which should produce the full AS generator for the physical system with 2N + 1 atoms and n modes of radiation. With respect to the one discussed above, it is enough to double the number of reservoirs for the matter and to sum over l ∈ IN for the matter and over j = 0, 1, . . . , n − 1 for the radiation. The resulting hamiltonian is therefore necessarily very close to the HL one: H = H0 + λHI ,
(3.15)
with H0 = ω R
n−1
a†j aj
j=0
+
+µ
n−1
σl,z +
l∈IN
2
dk ωms (k)m†s,l (k)ms,l (k)
l∈IN s=1
dk ωr,j (k)rj (k)† rj (k)
(3.16)
j=0
and HI =
n−1
(rj† (g j )aj + rj (gj )a†j ) +
j=0
+(m†2,l (h2 )σl,+
+ h.c.)] + λ
[(m†1,l (h1 )σl,− + h.c.)
l∈IN (N )
(φl
σl,+ + h.c.),
(3.17)
l∈IN
where the radiation field has been introduced in (2.10). It is clear that, but for the RWA which we are assuming here, there are not many other differences between this hamiltonian and the one in (2.11)-(2.15). It is worth mentioning that λ appears both as an overall coupling constant, see (3.15), and as a multiplying factor of
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√ + h.c.) and plays the same role as α in the HL hamiltonian. As for the commutation rules they are quite natural: but for the spin operators, which satisfy their own algebra, all the others operators satisfy the CCR and commute whenever they refer to different subsystems. In particular, for instance, all the m1,l (k) commute with all the m2,l (k ), for all k, k and l, l . The procedure to obtain the generator is the same as before: we first compute HI (t) = eiH0 t HI e−iH0 t , which enters in the differential equation for the wave operator. Taking the limit λ → 0 of the mean value in the vector state defined by (ξ) ϕ0 = ϕ0 ⊗ ξ of the first non trivial approximation of the rescaled version of Ut (sl) we deduce the form of an effective hamiltonian, HI (t), which is simply (N ) σl,+ l∈IN (φl
(sl)
HI
(t) =
n−1
j=0
l∈IN
† (aj rg,j (t) + h.c.) +
+
(σl,+ m1,l (t) + h.c.)
(σl,− m2,l (t) + h.c.) +
l∈IN
(N )
(φl
σl,+ + h.c.).
(3.18)
l∈IN
Again, we are assuming that ωR = 2µ, which is crucial in order not to have a time (sl) dependence in the last term of HI (t) in (3.18). The only non trivial commutation rules for t > t for the new operators are: (g)
† [rg,j (t), rg,j (t )] = Γ−,j δj,j δ(t − t ), (h )
[m1,l (t), m†1,l (t )] = Γ− 1 δl,l δ(t − t ), [m2,l (t), m†2,l (t )]
=
(h ) Γ− 2 δl,l δ(t
(3.19)
− t ),
where we have defined the following complex quantities:
(g)
Γ−,j = (h ) Γ− 1 (h )
0
dτ −∞ 0
=
Γ− 2 =
dτ
−∞ 0
dτ
−∞
dk|gj (k)|2 ei(ωr,j (k)−ωR )τ , dk|h1 (k)|2 ei(ωm1 (k)−ωR )τ , dk|h2 (k)|2 ei(ωm2 (k)+ωR )τ .
(3.20)
The above commutators, given for t > t , are sufficient to compute the commutation relations between the fields of the reservoir and the wave operator (sl) Ut = 1 − i HI (t )Ut dt , as it is obtained after the SL. We get (g)
[rg,j (t), Ut ] = −iΓ−,j aj Ut , (h )
[m1,l (t), Ut ] = −iΓ− 1 σl,− Ut , (h )
[m2,l (t), Ut ] = −iΓ− 2 σl,+ Ut .
(3.21)
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The expression for the generator can be obtained as for the N = 0, n = 1 model described before, that is computing the mean value < ∂t jt (X) >η(ξ) . Here, as 0
before, jt (X) is the flux of the system observable X, jt (X) = Ut† XUt , and η0 is the vacuum of the operators rg,j (t) and ms,l (t), s = 1, 2. The computation gives the following result, which slightly generalize the one in (3.14): L(X) = L1 (X) + L2 (X) + L3 (X), n−1 (g) † (g) L1 (X) = (Γ−,j [aj , X]aj − Γ−,j a†j [aj , X]), j=0
L2 (X) =
(h )
(h1 )
(h )
(h2 )
(Γ− 1 [σ+,l , X]σ−,l − Γ− σ+,l [σ−,l , X]
l∈IN
+Γ− 2 [σ−,l , X]σ+,l − Γ− σ−,l [σ+,l , X], (N ) [(φl σ+,l + h.c.), X]. L3 (X) = i
(3.22)
l∈IN
It is not difficult to compare this generator with the one proposed by AS, see formulas ((2.3),(2.10)), and the conclusion is that the two generators are exactly the same provided that the following equalities are satisfied: (g)
(g)
(h )
(h )
(h )
(h )
Γ−,j = ωj , Γ−,j = kj , (Γ− 1 + Γ− 2 ) = γ1 , (Γ− 1 − Γ− 2 ) = 1 1 (h ) (h ) Γ− 1 = γ2 (1 − η), Γ− 2 = γ2 (1 + η). (3.23) 4 4 Here the lhs all contain variables of the hamiltonian model while the rhs are related to the AS generator. Due to the fact that the hamiltonian in (3.15)-(3.17) essentially coincides with the one in (2.11)-(2.15) with β = 0, that is in the RWA, we can conclude that if we start with the HL hamiltonian, choosing the regularizing functions in such a way that the equalities (3.23) are satisfied, the SL produces a generator of the model which is exactly the one proposed in [1, 2], with the only minor constraint γ2 = 2γ1 , which is a direct consequence of (3.23). From a physical point of view the implications of this result are quite interesting: the original model, [5], was not (easily) solvable and for this reason a certain number of approximations were introduced. Among these, the crucial ones are the replacement of the original reservoir with what HL call a singular reservoir which, moreover, is made of fermions. Under these assumptions the model can be discussed in some details, and this was done in [8], where the thermodynamic limit for the intensive and the fluctuation observables was discussed. What we have shown here is that all these approximations can be avoided using another kind of perturbative approach, that is the one provided by the SL. The resulting model is exactly the one proposed and studied in [1, 2]. The role of the singular reservoir, or the need for a fermionic reservoir, is therefore not crucial and can be avoided. However, we will consider the DHL model in the next section in order to complete our analysis.
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We conclude this section with a remark concerning the role of the RWA and its relation with the SL. In particular, this is a very good approximation after the SL is taken. To show why, we first notice that adding a counter-rotating term (extending the one in (2.18)) to the interaction hamiltonian HI in (3.17), considering the same coupling constant for both the rotating and the counterrotating term (β = α), simply means to add to HI in (3.17) a contribution like (N ) λ l∈IN (φl σl,− + h.c.). While the rotating term, if 2µ = ωR , does not evolve freely, the free time evolution of this other term is not trivial. However, the differences with respect to the previous situation all disappear rigorously after the SL, because these extra contributions to the mean value of the wave operator go to zero with λ, so that at the end the expression for the generator is unchanged. This allows us to conclude that the full HL hamiltonian is equivalent to the AS generator, where the equivalence relation is provided by the SL. We want to end this section with a final remark concerning the different number of phase transitions in the two different situations, 1 for the HL and 2 for the AS model. In view of the above conclusion, we can guess that the SL procedure produces some loss of information and, as a consequence, a difference between the original and the approximated system. This is not so surprising since, thought being a powerful tool, nevertheless the SL is nothing but a perturbative method!
IV The SL of the DHL model In this section we consider the SL of the DHL model as introduced in Section 2. In particular we find the expression of the generator and we show that, under some conditions on the quantities defining the model, the equations of motion do not differ from the ones in AS. The free evolved interaction hamiltonian HI in (2.24) is, HI (t) = eiH0 t HI e−iH0 t = +λ
n−1
(aj rj† (g j ei(ωr,j −ωR )t ) + h.c.)
j=0 (N ) (φl b†+,l b−,l l∈IN
+C+,l (gC+ e
it(µ− )
)) +
+ h.c.) +
[b†+,l (B+,l (gB+ eit(µ− ) )
l∈IN
† (B+,l (gB+ e−it(µ− ) )
† + C+,l (gC+ e−it(µ− ) ))b+,l
+b†−,l (B−,l (gB− e−it(µ+ ) ) + C−,l (gC− e−it(µ+ ) )) † † +(B−,l (gB− eit(µ+ ) ) + C−,l (gC− eit(µ+ ) ))b−,l ].
(4.1)
Following the usual strategy discussed in the Appendix and in [6], we conclude t that (the rescaled version of) the wave operator Uλ (t) = 1 − iλ 0 HI (t )Uλ (t )dt converges for λ → 0 to another operator, which we still call the wave operator,
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satisfying the equation t (ls) (ls) Ut = 1 − i HI (t )Ut dt , or, equivalently ∂t Ut = −iHI (t)Ut , with U0 = 1. 0
Here
(ls) HI (t) (ls)
HI
(4.2) is an effective time dependent hamiltonian defined as (t)
=
n−1
† (aj rg,j (t) + h.c.) +
j=0
+
l∈IN
(N ) † b+,l b−,l
(φl
+ h.c.)
† † [b†+,l (β+,l (t) + γ+,l (t)) + (β+,l (t) + γ+,l (t))b+,l
l∈IN † † +b†−,l (β−,l (t) + γ−,l (t)) + (β−,l (t) + γ−,l (t))b−,l ].
(4.3)
(ls)
The operators of the reservoir which appear in HI are the stochastic limit of the original (rescaled) time evoluted operators of the reservoir and satisfy (anti)commutation relations which are related to those of the original ones. In particular, after the SL, any two operators of the matter (system and reservoirs) localized at different lattice sites commutes, as well as any operator of the radiation with any observable of the matter. As for operators localized at the same lattice site, the only non trivial anticommutators are (B±)
† (t )} = δ(t − t )Γ− {β±,l (t), β±,l
(C±)
† {γ±,l (t), γ±,l (t )} = δ(t − t )Γ−
,
,
(4.4)
which should be added to (g)
[rg,j (t), rg,j (t )] = δj,j δ(t − t )Γ−,j .
(4.5)
In all these formulas the time ordering t > t has to be understood and the following quantities are defined: 0 (g) dτ dk |gj (k)|2 ei(ωr,j (k)−ωR )τ , Γ−,j = (B±) Γ− (C±) Γ−
−∞ 0
=
dτ
−∞ 0
=
dτ
−∞
dk (gB± (k))2 ei( (k)∓µ)τ , dk (gC± (k))2 e−i( (k)∓µ)τ .
(4.6)
We call now η0 the vacuum of these limiting operators. We have † rg,j (t)η0 = β±,l (t)η0 = γ±,l (t)η0 = 0.
(4.7)
Paying a little attention to the fact that here commutators and anti-commutators simultaneously appear, we can compute the commutators between the operators
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rg,j (t) , γ±,l (t), β±,l (t) and the wave operator Ut by making use of ((4.4),(4.5)). We give here only those commutation rules which are used in the computation of the generator: (g)
[rg,j (t), Ut ] = −iΓ−,j aj Ut , (B±)
[β±,l (t), Ut ] = −iΓ−
b±,l Ut ,
(C±) † b±,l Ut .
† (t), Ut ] = iΓ− [γ±,l
(4.8)
We are now ready to compute the generator following exactly the same strategy as in the previous section and in Appendix. However, we must remark that, due to the presence of fermionic operators, to make the computation simpler, we will focus on system operators X which are quadratic (or quartic, . . . ) in the matter † b†−,l , operators localized in a given lattice site (e.g. X = b+,l b†−,l , X = b†+,l β+,l γ−,l . . . ) in such a way to ensure the commutativity between X and any of the matter (ls) operators entering in the operator HI (t), (4.3). We get for the full generator the following sum of three different contributions: L(X) = L1 (X) + L2 (X) + L3 (X), L1 (X) =
n−1
(g)
(g)
(Γ−,j [a†j , X]aj − Γ−,j a†j [aj , X]),
j=0
L2 (X) =
(B+)
(Γ−
[b†+,l , X]b+,l
l∈IN (B+) −Γ− b†+,l [b+,l , X] + (B−)
+Γ−
(C+)
Γ−
(C+)
[b+,l , X]b†+,l − Γ−
(B−) † b−,l [b−,l , X]
[b†−,l , X]b−,l − Γ−
b+,l [b†+,l , X]
(C−)
+ Γ−
[b−,l , X]b†−,l
(C−)
b−,l [b†−,l , X]), (N ) † L3 (X) = i [(φl b+,l b−,l + h.c.), X].
−Γ−
(4.9)
l∈IN
We see that the first and the last terms exactly coincide with the analogous contributions of the AS generator, but for a purely formal difference which is due to the different matter variables which are used in the two models. The second contribution, on the other hand, cannot be easily compared with the free AS matter generator. What is convenient, and sufficient, to get full insight about L2 , is to compute its action on a basis of the local algebra, that is on b†+,l b−,l (≡ σ+,l ) and
on b†+,l b+,l − b†−,l b−,l (≡ σz,l ), all the others being trivial or an easy consequence of these ones. It is not hard to find the result: (B+)
L2 (b†+,l b−,l ) = −b†+,l b−,l ( [Γ− (B+) −i [Γ−
−
(B−) Γ−
−
(C+) Γ−
+
(B−)
+ Γ−
(C−) Γ− ]),
(C+)
+ Γ−
(C−)
+ Γ−
]−
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Relations between the Hepp-Lieb and the Alli-Sewell Laser Models (B+)
L2 (b†+,l b+,l − b†−,l b−,l ) = 2(−b†+,l b+,l ( [Γ− (B−) +b†−,l b−,l ( [Γ−
+
(C−) Γ− )
−
(C+)
+ Γ−
(C−) Γ− ).
(C+)
) + Γ−
999
+ (4.10)
The equation for σ+,l is recovered without any problem, modulo some identification (B+) (B−) (C+) (C−) ( [Γ− + Γ− + Γ− + Γ− ] = γ1 , . . . ), while to recover the equation for σz,l it is necessary to choose properly the regularizing functions which define the different Γ− . In particular we need to have the following equality fulfilled: (B+)
(Γ−
(C+)
+ Γ−
(B−)
) = (Γ−
(C−)
+ Γ−
).
(4.11)
Under this condition we can conclude that the SL of the DHL model produces the same differential equations as the AS generator, as for the HL model. It is also easy to check that, as a consequence of our approach, we must have γ1 = γ2 in the generator we obtain. Of course this result is not surprising since already in the HL paper, [5], the fact that the two models are quite close (under some aspects) was pointed out. Here we have learned also that the SL of both these models, at least under some conditions, give rise to the same dynamical behaviour.
V
Outcome and Future Projects
We have proved that the relation between the HL and the AS model, whose existence is claimed in [1], is provided by the SL. This result is quite interesting since it shows that the approximations introduced by HL in their paper [5], in particular the use of the fermionic reservoir for the matter which produces the DHL model, together with the so called singular reservoir approximation, can be avoided by using the original HL model with no approximation, taking its SL and finally using the results in [1, 2] to analyze, e.g., the thermodynamical limit of the model. It is interesting to remark that while in the AS model two phase transitions occur, in the HL model we only have one. This could be a consequence of the SL procedure, which is nothing but a perturbative approach simplifying the study of the quantum dynamics, so that some of the original features of the model can be lost after the approximation. We want to conclude this paper by remarking that this is not the first time the HL model is associated to a dissipative system, as in the AS formulation. A similar strategy was discussed by Gorini and Kossakowski already in 1976, [9]. It would be interesting to study their generator again in the connection with the SL to see if any relation between their generator and the HL original hamiltonian appears.
Acknowledgments I am indebted with Prof. Lu for a suggestion which is at the basis of this paper. I also would like to aknowledge financial support by the Murst, within the
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project Problemi Matematici Non Lineari di Propagazione e Stabilit` a nei Modelli del Continuo, coordinated by Prof. T. Ruggeri.
A
Appendix: Few results on the stochastic limit
In this Appendix we will briefly summarize some of the basic facts and properties concerning the SL which are used all throughout the paper. We refer to [6] and references therein for more details. Given an open system S+R we write its hamiltonian H as the sum of two contributions, the free part H0 and the interaction λHI . Here λ is a (sort of) coupling constant, H0 contains the free evolution of both the system and the reservoir, while HI contains the interaction between the system and the reservoir and, for composite systems, amoung the different buildind blocks of the whole physical system. Working in the interaction picture, we define HI (t) = eiH0 t HI e−iH0 t and the so called wave operator Uλ (t) which satisfies the following differential equation ∂t Uλ (t) = −iλHI (t)Uλ (t),
(A.1)
with the initial condition Uλ (0) = 1. Using the van-Hove rescaling t → λt2 , see [7, 6] for instance, we can rewrite the same equation in a form which is more convenient for our perturbative approach, that is ∂t Uλ (
t i t t ) = − HI ( 2 )Uλ ( 2 ), λ2 λ λ λ
with the same initial condition as before. Its integral counterpart is t i t t t HI ( 2 )Uλ ( 2 )dt , Uλ ( 2 ) = 1 − λ λ 0 λ λ
(A.2)
(A.3)
which is the starting point for a perturbative expansion, which works in the following way: let ϕ0 be the ground state of the reservoir and ξ a generic vector of (ξ) the system. Then we put ϕ0 = ϕ0 ⊗ ξ. We want to compute the limit, for λ going to 0, of the first non trivial order of the mean value of the perturbative expansion (ξ) of Uλ (t/λ2 ) above in ϕ0 , that is the limit of t t1 i t1 t2 Iλ (t) = (− )2 dt1 dt2 < HI ( 2 )HI ( 2 ) >ϕ(ξ) , (A.4) 0 λ λ λ 0 0 for λ → 0. Under some regularity conditions on the functions which are used to smear out the (typically) bosonic fields of the reservoir, this limit is shown to exist for many relevant physical models, see [6] and [10] for a recent application to many body theory. At this stage all the complex quantities like the various Γ− we have introduced in the main body of this paper appear. We call I(t) the limit limλ→0 Iλ (t). In the same sense of the convergence of the (rescaled) wave operator Uλ ( λt2 ) (the convergence in the sense of correlators), it is possible to check that also
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the (rescaled) reservoir operators converge and define new operators which do not satisfy canonical commutation relations but a modified version of these. Moreover, these limiting operators depend explicitly on time and they live in a Hilbert space which is different from the original one. In particular, they annihilate a vacuum vector, η0 , which is no longer the original one, ϕ0 . It is not difficult to write down, as we have done several times in this paper, (ls) the form of a time dependent self-adjoint operator HI (t), which depends on the system operators and on the limiting operators of the reservoir, such that the first t (ls) non trivial order of the mean value of the expansion of Ut = 1 − i 0 HI (t )Ut dt (ξ) on the state η0 = η0 ⊗ ξ coincides with I(t). The operator Ut defined by this integral equation is called again the wave operator. The form of the generator follows now from an operation of normal ordering. ˜ = More in details, we start defining the flux of an observable of the system X ˜ t . Then, using ˜ = Ut† XU X ⊗1 1res , where 1res is the identity of the reservoir, as jt (X) ˜ = iU † [H (ls) (t), X]U ˜ t. the equation of motion for Ut and Ut† , we find that ∂t jt (X) t I (ξ) In order to compute the mean value of this equation on the state η0 , so to get rid of the reservoir operators, it is convenient to compute first the commutation relations between Ut and the limiting operators of the reservoir. At this stage the so called time consecutive principle is used in a very heavy way to simplify the computation. This principle, which has been checked for many classes of physical models, and certainly holds in our case where all the interactions are dipolar, states that, if β(t) is any of these limiting operators of the reservoir, then [β(t), Ut ] = 0, for all t > t .
(A.5)
Using this principle and recalling that η0 is annihiled by the limiting annihilation operators of the reservoir, it is now a technical exercise to compute < ∂t jt (X) >η(ξ) 0 and, by means of the equation < ∂t jt (X) >η(ξ) =< jt (L(X)) >η(ξ) , to identify the 0 0 form of the generator of the physical system.
References [1] G. Alli and G. L. Sewell, New methods and structures in the theory of the multi-mode Dicke laser model, J. Math. Phys. 36, 5598 (1995). [2] F. Bagarello and G. L. Sewell, New Structures in the Theory of the Laser Model II: Microscopic Dynamics and a Non-Equilibrim Entropy Principle, J. Math. Phys. 39, 2730–2747 (1998). [3] R. H. Dicke, Coherence in Spontaneous Radiation Processes, Phys. Rev. 93, 99–110 (1954.) [4] R. Graham and H. Haken, Laserlight- First example of a second order phase transition far away from thermal equilibrium, Z. Phys. 237, 31 (1970); and
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H. Haken, Handbuch der Physik, Bd. XXV/2C, Springer, Heidelberg, Berlin, New York, 1970. [5] K. Hepp and E. H. Lieb, Phase transitions in reservoir-driven open systes with applications to lasers and superconductors, Helv. Phys. Acta 46, 573 (1973). [6] L. Accardi, Y. G. Lu and I. Volovich, Quantum Theory and its Stochastic Limit, Springer (2001). [7] P. A. Martin, Mod`eles en M´ecanique Statistique des Processus Irr´eversibles, Lecture Notes in Physics 103, Springer-Verlag, Berlin. [8] K. Hepp and E. H. Lieb, in Constructive Quantum Field Theory, G. Velo and A. S. Wightman Eds., Lect. Notes in Phys. 25, Springer (1973). [9] V. Gorini and A. Kossakowski, N-level system in contact with a singular reservoir, J. Math. Phys., 17, 1298–1305 (1976). [10] L. Accardi and F. Bagarello, The stochastic limit of the Fr¨ ohlich Hamiltonian: relations with the quantum Hall effect, submitted to Int. Jour. Phys., Preprint N. 443 del Centro Vito Volterra. Fabio Bagarello Dipartimento di Matematica ed Applicazioni Fac. Ingegneria, Universit` a di Palermo Viale delle Scienze I-90128 Palermo Italy email: [email protected] Communicated by Gian Michele Graf submitted 5/02/02, accepted 16/04/02
To access this journal online: http://www.birkhauser.ch
Ann. Henri Poincar´e 3 (2002) 1003 – 1018 c Birkh¨ auser Verlag, Basel, 2002 1424-0637/02/051003-16
Annales Henri Poincar´ e
Pure Point Dynamical and Diffraction Spectra J.-Y. Lee, R. V. Moody∗ and B. Solomyak†
Abstract. We show that for multi-colored Delone point sets with finite local complexity and uniform cluster frequencies the notions of pure point diffraction and pure point dynamical spectrum are equivalent.
1 Introduction The notion of pure pointedness appears in the theory of aperiodic systems in two different forms: pure point dynamical spectrum and pure point diffraction spectrum. The objective of this paper is to show that these two widely used notions are equivalent under a type of statistical condition known as the existence of uniform cluster frequencies. The basic objects of study here are Delone point sets of Rd . The points of these sets are permitted to be multi-colored, the colors coming from a finite set of colors. We also assume that our point sets Λ have the property of finite local complexity (FLC), which informally means that there are only finitely many translational classes of clusters of Λ with any given size. Under these circumstances, the orbit of Λ under translation gives rise, via completion in the standard Radin-Wolff type topology, to a compact space XΛ . With the obvious action of Rd , we obtain a dynamical system (XΛ , Rd ). The dynamical spectrum refers to the spectrum of this dynamical system, that is to say, the spectrum of the unitary operators Ux arising from the translational action on the space of L2 -functions on XΛ . On the other hand, the diffraction spectrum (which is the idealized mathematical interpretation of the diffraction pattern of a physical experiment) is obtained by first assigning weights to the various colors of the multiset and then determining the autocorrelation, if it exists, of this weighted multiset. The Fourier transform of the autocorrelation is the diffraction measure whose pure pointedness is the question. There is a well known argument of S. Dworkin ([4], [6]) that shows how to deduce pure pointedness of the diffractive spectrum from pure pointedness of the dynamical system. Our main result (Theorem 3.2) shows that, under the additional assumption that Λ has uniform cluster frequencies (or equivalently, that ∗ RVM acknowledges on-going support from the Natural Sciences and Engineering Research Council of Canada. † BS acknowledges support from NSF grants DMS 9800786 and DMS 0099814.
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the dynamical system XΛ is uniquely ergodic), the process can be reversed, so in fact the two notions of pure pointedness are equivalent. The present understanding of diffractive point sets is very limited. One of the important consequences of this result is that it allows the introduction of powerful spectral theorems in the study of such sets. Our forthcoming paper [8] on diffractive substitution systems makes extensive use of this connection. In the last section we discuss what can be salvaged when there are no uniform cluster frequencies. Then the equivalence of pure point dynamical and pure point diffraction spectra still holds – perhaps, not for the original Delone set Λ, but for almost every Delone set in XΛ , with respect to an ergodic invariant measure. The prototype of the dynamical system (XΛ , Rd ) is a symbolic dynamical system, that is, the Z-action by shifts on a space of bi-infinite sequences. In the symbolic setting, the equivalence of pure point dynamical and diffraction spectra has been established by Queffelec [11, Prop. IV.21], and our proof is largely a generalization of her argument. When the dynamical spectrum is not pure point, its relation to the diffraction spectrum is not completely understood. It follows from [4] that the latter is essentially a “part” of the former. So, for instance, if the dynamical spectrum is pure singular/absolutely continuous, then the diffraction spectrum is pure singular/absolutely continuous (apart from the trivial constant eigenfunction which corresponds to a delta function at 0). However, the other direction is more delicate: Van Enter and Mi¸ekisz [5] have pointed out that, in the case of mixed spectrum, the non-trivial pure point component may be “lost” when passing from dynamical spectrum to diffraction spectrum. The presentation below contains a number of results that are essentially wellknown, though not always quite in the form needed here. For the convenience of the reader we have attempted to make the paper largely self-contained.
2 Multisets, dynamical systems, and uniform cluster frequencies A multiset or m-multiset in Rd is a subset Λ = Λ1 × · · · × Λm ⊂ Rd × · · · × Rd (m copies) where Λi ⊂ Rd . We also write Λ = (Λ1 , . . . , Λm ) = (Λi )i≤m . We say that Λ = (Λi )i≤m is a Delone multiset in Rd if each Λi is Delone and supp(Λ) := m d i=1 Λi ⊂ R is Delone. Although Λ is a product of sets, it is convenient to think of it as a set with types or colors, i being the color of points in Λi . A cluster of Λ is, by definition, a m-multiset P = (Pi )i≤m where Pi ⊂ Λi is finite for all i ≤ m. The cluster P is non-empty if supp(P) is non-empty. Many of the clusters that we consider have the form A ∩ Λ := (A ∩ Λi )i≤m , for a bounded set A ⊂ Rd . There is a natural translation Rd -action on the set of Delone multisets and their clusters in Rd . The translate of a cluster P by x ∈ Rd is x + P = (x + Pi )i≤m . We say that two clusters P and P are translationally equivalent if P = x + P for some x ∈ Rd .
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We write BR (y) for the closed ball of radius R centered at y and use also BR for BR (0). Definition 2.1 The Delone multiset Λ has finite local complexity (FLC) if for every m R > 0 there exists a finite set Y ⊂ supp(Λ) = i=1 Λi such that ∀x ∈ supp(Λ), ∃ y ∈ Y : BR (x) ∩ Λ = (BR (y) ∩ Λ) + (x − y). In plain language, for each radius R > 0 there are only finitely many translational classes of clusters whose support lies in some ball of radius R. In this paper we will usually assume that our Delone multisets have FLC. Let Λ be a Delone multiset and X be the collection of all Delone multisets each of whose clusters is a translate of a Λ-cluster. We introduce a metric on Delone multisets in a simple variation of the standard way: for Delone multisets Λ1 , Λ2 ∈ X, ˜ 1 , Λ2 ), 2−1/2 } , d(Λ1 , Λ2 ) := min{d(Λ
(2.1)
where ˜ 1 , Λ2 ) = d(Λ
inf{ε > 0 : ∃ x, y ∈ Bε (0), B1/ε (0) ∩ (−x + Λ1 ) = B1/ε (0) ∩ (−y + Λ2 )} .
Let us indicate why this is a metric. Clearly, the only issue is the triangle inequality. Suppose that d(Λ1 , Λ2 ) ≤ ε1 , d(Λ2 , Λ3 ) ≤ ε2 ; we want to show that d(Λ1 , Λ3 ) ≤ ε1 + ε2 . We can assume that ε1 , ε2 < 2−1/2 , otherwise the claim is obvious. Then (−x1 + Λ1 ) ∩ B1/ε1 (0) = (−x2 + Λ2 ) ∩ B1/ε1 (0) for some x1 , x2 ∈ Bε1 (0), (−x2 + Λ2 ) ∩ B1/ε2 (0) = (−x3 + Λ3 ) ∩ B1/ε2 (0) for some x2 , x3 ∈ Bε2 (0). It follows that (−x1 − x2 + Λ1 ) ∩ B1/ε1 (−x2 ) = (−x2 − x2 + Λ2 ) ∩ B1/ε1 (−x2 ). Since B1/ε1 (−x2 ) ⊃ B(1/ε1 )−ε2 (0), this implies (−x1 − x2 + Λ1 ) ∩ B(1/ε1 )−ε2 (0) = (−x2 − x2 + Λ2 ) ∩ B(1/ε1 )−ε2 (0).
(2.2)
Similarly, (−x2 − x2 + Λ2 ) ∩ B(1/ε2 )−ε1 (0) = (−x2 − x3 + Λ3 ) ∩ B(1/ε2 )−ε1 (0). A simple computation shows that ε11 − ε2 ≥ ε1 , ε2 < 2−1/2 , so by (2.2) and (2.3),
1 ε1 +ε2
and
1 ε2
− ε1 ≥
1 ε1 +ε2
(−x1 − x2 + Λ1 ) ∩ B1/(ε1 +ε2 ) (0) = (−x2 − x3 + Λ3 ) ∩ B1/(ε1 +ε2 ) (0), hence d(Λ1 , Λ3 ) ≤ ε1 + ε2 .
(2.3) when
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We define XΛ := {−h + Λ : h ∈ Rd } with the metric d. In spite of the special role played by 0 in the definition of d, any other point of Rd may be used as a reference point, leading to an equivalent metric and more importantly the same topology on XΛ . The following lemma is standard. Lemma 2.2 ([12], [13]) If Λ has FLC, then the metric space XΛ is compact. The group Rd acts on XΛ by translations which are obviously homeomorphisms, and we get a topological dynamical system (XΛ , Rd ). Definition 2.3 Let P be a non-empty cluster of Λ or some translate of Λ, and let V ⊂ Rd be a Borel set. Define the cylinder set XP,V ⊂ XΛ by XP,V := {Λ ∈ XΛ : −g + P ⊂ Λ for some g ∈ V }. Let η(Λ) > 0 be chosen so that every ball of radius η(2Λ) contains at most one point of supp(Λ), and let b(Λ) > 0 be such that every ball of radius b(2Λ) contains at least a point in supp(Λ). These exist by the Delone set property. The following technical result will be quite useful. Lemma 2.4 Let Λ be a Delone multiset with FLC. For any R ≥ b(2Λ) and 0 < δ < η(Λ), there exist Delone multisets Γj ∈ XΛ and Borel sets Vj with diam(Vj ) < δ, Vol(∂Vj ) = 0, 1 ≤ j ≤ N , such that XΛ =
N
XPj ,Vj
j=1
is a disjoint union, where Pj = BR (0) ∩ Γj . Proof. For any R ≥ b(2Λ) consider the clusters {BR (0) ∩ Γ : Γ ∈ XΛ }. They are non-empty, by the definition of b(Λ). By FLC, there are finitely many such clusters up to translations. This means that there exist Γ1 , . . . , ΓK ∈ XΛ such that for any Γ ∈ XΛ there are unique n = n(Γ) ≤ K and u = u(Γ) ∈ Rd satisfying BR (0) ∩ Γ = −u + (BR (0) ∩ Γn ). For j = 1, . . . , K let Wj = {u(Γ) : Γ ∈ XΛ such that n(Γ) = j}. By construction, XΛ = K j=1 XPj ,Wj , and this is a disjoint union. Next we show that the sets Wj are sufficiently “nice,” so that they can be obtained from a finite number of closed balls using operations of complementation, intersection, and union.
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Let b = b(Λ) and fix j. Since every ball of radius b/2 contains a point in supp(Λ), we have that Wj ⊂ Bb (0). Indeed, shifting a cluster of points in BR (0) by more than b would move at least one point out of BR (0). Let Pj := BR (0) ∩ Γj . The set Wj consists of vectors u such that −u + Pj is a BR (0)-cluster for some Delone multiset in XΛ . Thus u ∈ Wj if and only if the following two conditions are met. The first condition is that for each x ∈ supp(Pj ), we have −u + x ∈ BR (0). The second condition is that no points of Γj outside of BR (0) move inside after the translation by −u. Since Wj ⊂ Bb (0), only the points in BR+b (0) have a chance of moving into BR (0). Thus we need to consider the BR+b (0) extensions of Pj . By FLC, in the space XΛ there are finitely many BR+b (0)-clusters that extend the cluster Pj . Denote these clusters by Q1 , . . . , QL . Summarizing this discussion we obtain Wj = (−BR (0) + x) ∩ (−(Rd \ BR (0)) + x) . x∈supp(Pj )
i≤L
x∈supp(Qi )\BR (0)
This implies that Wj is a Borel set, with Vol(∂Wj ) = 0. nj It remains to partition each Wj such that Wj = k=1 Vjk , where diam(Vjk ) ≤ δ, 0 < δ < η(Λ). To this end, consider, for example, a decomposition of the cube [−b, b]d into a disjoint union of (half-open and closed) grid boxes of diameter less than δ < η(Λ). Let Q denote the (finite) collection of all these grid boxes. Then Wj =
(Wj ∩ D) =
D∈Q
nj
Vjk ,
k=1
where Vjk ’s are disjoint and Vol(∂Vjk ) = 0. Note that the union XPj ,Wj = nj k=1 XPj ,Vjk is disjoint, from the definition of Wj and diam(Vjk ) < η(Λ) for all k ≤ nj . So the lemma is proved. For a non-empty cluster P and a bounded set A ⊂ Rd denote LP (A) = {x ∈ Rd : x + P ⊂ A ∩ Λ}, where means the cardinality. In plain language, LP (A) is the number of translates of P contained in A, which is clearly finite. For a bounded set F ⊂ Rd and r > 0, let F +r := {x ∈ Rd : dist(x, F ) ≤ r}, F −r := {x ∈ F : dist(x, ∂F ) ≥ r} ⊃ F \ (∂F )+r . A van Hove sequence for Rd is a sequence F = {Fn }n≥1 of bounded measurable subsets of Rd satisfying lim Vol((∂Fn )+r )/Vol(Fn ) = 0, for all r > 0.
n→∞
(2.4)
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Definition 2.5 Let {Fn }n≥1 be a van Hove sequence. The Delone multiset Λ has uniform cluster frequencies (UCF) (relative to {Fn }n≥1 ) if for any non-empty cluster P, the limit freq(P, Λ) = lim
n→∞
LP (x + Fn ) ≥0 Vol(Fn )
exists uniformly in x ∈ Rd . Recall that a topological dynamical system is uniquely ergodic if there is a unique invariant probability measure. Theorem 2.6 Let Λ be a Delone multiset with FLC and {Fn }n≥1 be a van Hove sequence. The system (XΛ , Rd ) is uniquely ergodic if and only if for all continuous functions f : XΛ → C (f ∈ C(XΛ )), 1 (In )(Γ, f ) := f (−g + Γ) dg → const, n → ∞, (2.5) Vol(Fn ) Fn uniformly in Γ ∈ XΛ , with the constant depending on f . This is a standard fact (both directions, see e.g. [16, Th. 6.19], [3, (5.15)], or [11, Th. IV.13] for the case of Z-actions); we include a (well-known) elementary proof of the needed direction for the reader’s convenience. Proof of sufficiency in Theorem 2.6. For any invariant measure µ, exchanging the order of integration yields In (Γ, f ) dµ(Γ) = f dµ, XΛ
XΛ
so by the Dominated Convergence Theorem, the constant in (2.5) is X f dµ. If Λ there is another invariant measure ν, then X f dµ = X f dν for all f ∈ C(XΛ ), Λ Λ hence µ = ν. Now we prove that FLC and UCF imply unique ergodicity of the system (XΛ , Rd ). This is also a standard fact, see e.g. [11, Cor. IV.14(a)] for the case of Z-actions. Theorem 2.7 Let Λ be a Delone multiset with FLC. Then the dynamical system (XΛ , Rd ) is uniquely ergodic if and only if Λ has UCF. Proof. Let XP,V be a cylinder set with diam(V ) ≤ η(Λ) and f be the characteristic function of XP,V . Then we have by the definition of the cylinder set: f (−x − h + Λ) dx Jn (h, f ) := Fn
=
Vol{x ∈ Fn : −x − h + Λ ∈ XP,V }
=
Vol{x ∈ h + Fn : −y + P ⊂ −x + Λ for some y ∈ V }
((h + Fn ) ∩ (xν + V )) Vol
=
ν
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where xν are all the vectors such that xν + P ⊂ Λ. It is clear that the distance between any two vectors xν is at least η(Λ), so the sets xν + V are disjoint. Let r = max{|y| : y ∈ V } + max{|x| : x ∈ supp(P)}. Then Vol(V )LP (h + Fn−r ) ≤ Jn (h, f ) ≤ Vol(V )LP (h + Fn+r ).
(2.6)
Note that LP (h + Fn+r ) − LP (h + Fn−r ) ≤ LP (h + ∂Fn +2r ) ≤
Vol(∂Fn +2r ) . Vol(B η(Λ) ) 2
So
lim
n→∞
Jn (h, f ) Vol(V ) · LP (h + Fn ) − Vol(Fn ) Vol(Fn )
= 0 uniformly in h ∈ Rd .
(2.7)
If (XΛ , Rd ) is uniquely ergodic, Jn (h, f ) exists uniformly in h ∈ Rd n→∞ Vol(Fn ) for continuous functions f approximating the characteristic function f of the cylinder set. Thus for any cluster P, lim
lim
n→∞
LP (h + Fn ) exists uniformly in h ∈ Rd , Vol(Fn )
i.e. Λ has UCF. On the other hand, we assume that Λ has UCF. By Lemma 2.4, f ∈ C(XΛ ) can be approximated in the supremum norm by linear combinations of characteristic functions of cylinder sets XP,V . Thus, it is enough to check (2.5) for f the characteristic function of XP,V with diam(V ) < η(Λ). We can see in the above (2.7) that (2.5) holds for all −h + Λ uniformly in h ∈ Rd under the assumption that Λ has UCF. Then we can approximate the orbit of Γ ∈ XΛ on Fn by −hn +Λ as closely as we want, since the orbit {−h + Λ : h ∈ Rd } is dense in XΛ by the definition of XΛ . So we compute all those integrals (2.5) of −hn + Λ over Fn and use the fact that independent of hn they are going to a constant. Since each of these is uniformly close to (In )(Γ, f ) in (2.5), we get that (In )(Γ, f ) too goes to a constant. Therefore (XΛ , Rd ) is uniquely ergodic. Denote by µ the unique invariant probability measure on XΛ . As already mentioned, the constant in (2.5) must be X f dµ. Thus, the proof of unique Λ ergodicity yields the following result. Corollary 2.8 Let Λ be a Delone multiset with FLC and UCF. Then for any Λcluster P and any Borel set V with diam(V ) < η(Λ), we have µ(XP,V ) = Vol(V ) · freq(P, Λ).
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3 Pure-pointedness and Diffraction 3.1
Dynamical spectrum and diffraction spectrum
Suppose that Λ = (Λi )i≤m is a Delone multiset with FLC and UCF. There are two notions of pure pointedness that appear in this context. Although they are defined very differently, they are in fact equivalent. Given a translation-bounded measure ν on Rd , let γ(ν) denote its autocorrelation (assuming it is unique), that is, the vague limit 1 γ(ν) = lim (3.1) (ν|Fn ∗ ν|Fn ) , n→∞ Vol(Fn ) where {Fn }n≥1 is a van Hove sequence 1 . In particular, for the Delone multiset Λ we see that the autocorrelation is unique for any measure of the form ai δΛi , where δΛi = δx and ai ∈ C . (3.2) ν= i≤m
x∈Λi
Indeed, a simple computation shows γ(ν) =
m i,j=1
ai aj
freq((y, z), Λ)δy−z .
(3.3)
y∈Λi ,z∈Λj
Here (y, z) stands for a cluster consisting of two points y ∈ Λi , z ∈ Λj . The measure is γ(ν) is positive definite, so by Bochner’s Theorem the Fourier transform γ(ν) a positive measure on Rd , called the diffraction measure for ν. We say that the is a pure point or discrete measure ν has pure point diffraction spectrum if γ(ν) 2 measure . On the other hand, we also have the measure-preserving system (XΛ , µ, Rd ) associated with Λ. Consider the associated group of unitary operators {Ux }x∈Rd on L2 (XΛ , µ): Ux f (Λ ) = f (−x + Λ ). Every f ∈ L2 (XΛ , µ) defines a function on Rd by x → (Ux f, f ). This function is positive definite on Rd , so its Fourier transform is a positive measure σf on Rd called the spectral measure corresponding to f . We say that the Delone multiset Λ has pure point dynamical spectrum if σf is pure point for every f ∈ L2 (XΛ , µ). We recall that f ∈ L2 (XΛ , µ) is an eigenfunction for the Rd -action if for some α = (α1 , . . . , αd ) ∈ Rd , Ux f = e2πix·α f,
for all x ∈ Rd ,
where · is the standard inner product on Rd . 1 Recall
that if f is a function in d , then f˜ is defined by f˜(x) = f (−x). If µ is a measure, µ ˜ is defined by µ ˜(f ) = µ(f˜) for all f ∈ C0 ( d ). In particular for ν in (3.2), ν˜ = i≤m ai δ−Λi . 2 We
also say that Λi (resp Λ) has pure point diffraction spectrum if γ(δ Λi ) (resp each
γ(δ Λi ), i = 1, . . . , m) is a pure point measure.
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Theorem 3.1 σf is pure point for every f ∈ L2 (XΛ , µ) if and only if the eigenfunctions for the Rd -action span a dense subspace of L2 (XΛ , µ). This is a straightforward consequence of the Spectral Theorem, see e.g. Theorem 7.27 and §7.6 in [17] for the case d = 1. The Spectral Theorem for unitary representations of arbitrary locally compact Abelian groups, including Rd , is discussed in [10, §6].
3.2
An equivalence theorem for pure pointedness
In this section we prove the following theorem. Theorem 3.2 Suppose that a Delone multiset Λ has FLC and UCF. Then the following are equivalent: (i) Λ has pure point dynamical spectrum; (ii) The measure ν = i≤m ai δΛi has pure point diffraction spectrum, for any choice of complex numbers (ai )i≤m ; (iii) The measures δΛi have pure point diffraction spectrum, for i ≤ m. The theorem is proved after a sequence of auxiliary lemmas. Fix complex numbers (ai )i≤m and let ν = i≤m ai δΛi . For Λ = (Λi )i≤m ∈ XΛ let νΛ = ai δΛi , i≤m
so that ν = νΛ . To relate the autocorrelation of ν to spectral measures we need to do some “smoothing.” Let ω ∈ C0 (Rd ) (that is, ω is continuous and has compact support). Denote ρω,Λ := ω ∗ νΛ and let
fω (Λ ) := ρω,Λ (0) for Λ ∈ XΛ .
Lemma 3.3 fω ∈ C(XΛ ). Proof. We have fω (Λ ) =
ω(−x) dνΛ (x) =
i≤m
ai
ω(−x).
x∈−supp(ω)∩Λi
The continuity of fω follows from the continuity of ω and the definition of topology on XΛ . Denote by γω,Λ the autocorrelation of ρω,Λ . Since under our assumptions there is a unique autocorrelation measure γ = γ(ν), see (3.1) and (3.2), we have ) ∗ γ. γω,Λ = (ω ∗ ω
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Lemma 3.4 ([4], see also [6]) σfω = γ ω,Λ . Proof. We provide a proof for completeness, following [6]. By definition, fω (−x + Λ) = ρω,Λ (x). Therefore, γω,Λ (x)
=
1 ρω,Λ (x + y)ρω,Λ (y) dy n→∞ Vol(Fn ) F n 1 fω (−x − y + Λ)fω (−y + Λ) dy lim n→∞ Vol(Fn ) F n fω (−x + Λ )fω (Λ ) dµ(Λ )
=
(Ux fω , fω ) ,
= =
lim
XΛ
(3.4)
where {Fn }n≥1 is a van Hove sequence. Here the third equality is the main step; it follows from unique ergodicity and the continuity of fω , see Theorem 2.6. Thus, γ ω,Λ = (U(·) fω , fω ) = σfω ,
and the proof is finished.
The introduction of the function fω and the series of equations (3.4) is often called Dworkin’s argument. Fix ε with 0 < ε < b(1Λ) . Consider all the non-empty clusters of diameter ≤ 1/ε in Γ ∈ XΛ . There are finitely many such clusters up to translation, by FLC. Thus, there exists 0 < θ1 = θ1 (ε) < 1 such that if P, P are two such clusters, then ρH (P, P ) ≤ θ1 ⇒ P = −x + P Here
for some x ∈ Rd .
(3.5)
ρH (P, P ) = max{ρH (Pi , Pi ) : i ≤ m},
where ρH (Pi , Pi )
=
max{dist(x, Pi ), dist(y, Pi ) : x ∈ Pi , y ∈ Pi }, 1, if Pi = ∅ and Pi = ∅ (or vice versa),
if Pi , Pi = ∅;
with P = (Pi )i≤m and P = (Pi )i≤m . Let
θ = θ(ε) := min{ε, θ1 , η(Λ)}
(3.6)
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and fi,ω (Λ ) = (ω ∗ δΛi )(0)
for Λ = (Λi )i≤m ∈ XΛ .
Denote by Ei the cluster consisting of a single point of type i at the origin; formally, Ei = (∅, . . . , ∅, {0} , ∅, . . . , ∅). i
Let χEi ,V be the characteristic function for the cylinder set XEi ,V . Lemma 3.5 Let V ⊂ Rd be a bounded set with diam(V ) < θ, where θ is defined by (3.6), and 0 < ζ < θ/2. Let ω ∈ C0 (Rd ) be such that x ∈ V −ζ ; ω(x) = 1, ω(x) = 0, x ∈ Rd \ V ; 0 ≤ ω(x) ≤ 1, x ∈ V \ V −ζ . Then fi,ω − χEi ,V 22 ≤ freq(Ei , Λ) · Vol((∂V )+ζ ). Proof. We have by the definition of Ei and Definition 2.3: 1, if Λi ∩ (−V ) = ∅; χEi ,V (Λ ) = where Λ ∈ XΛ . 0, otherwise, On the other hand, since ω is supported in V and there is at most one point of Λi in V , ω(−x), if ∃ x ∈ Λi ∩ (−V ); fi,ω (Λ ) = ω(−x) dδΛi (x) = 0, otherwise. It follows that fi,ω (Λ ) − χEi ,V (Λ ) = 0 if
Λi ∩ (−V −ζ ) = ∅.
Thus, fi,ω − χEi ,V
22
|fi,ω (Λ ) − 1|2 dµ(Λ )
≤ XE
i
,V \V −ζ
≤ µ(XEi ,V \V −ζ ) = freq(Ei , Λ) · Vol(V \ V −ζ ) ≤ freq(Ei , Λ) · Vol((∂V )+ζ ), as desired.
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Lemma 3.6 Let P = (Pi )i≤m = B1/ε (0) ∩ Γ with Γ ∈ XΛ , and diam(V ) < θ, where θ is defined by (3.6). Then the characteristic function χP,V of XP,V can be expressed as χP,V = χx+Ei ,V . i≤m x∈Pi
Proof. We just have to prove that
XP,V =
Xx+Ei ,V .
i≤m x∈Pi
A Delone multiset Γ is in the left-hand side whenever −v + P ⊂ Γ for some v ∈ V . A Delone multiset Γ is in the right-hand side whenever for each i ≤ m and each x ∈ Pi there is a vector v(x) ∈ V such that −v(x) + x ⊂ Γ, where x = (∅, . . . , ∅, {x} , ∅, . . . , ∅) stands for a single element cluster. Thus, “⊂” is trivial. i
The inclusion “⊃” follows from the fact that diam(V ) < θ, see (3.6) and (3.5). Denote by Hpp the closed linear span in L2 (XΛ , µ) of the eigenfunctions for the dynamical system (XΛ , µ, Rd ). The following lemma is certainly standard, but since we do not know a ready reference, a short proof is provided. Lemma 3.7 If φ and ψ are both in L∞ (XΛ , µ) ∩ Hpp , then their product φψ is in L∞ (XΛ , µ) ∩ Hpp as well. Proof. Fix arbitrary > 0. Since φ ∈ Hpp , we can find a finite linear combination of eigenfunctions φ = ai fi such that 2< φ − φ
. ψ∞
Since the dynamical system is ergodic, the eigenfunctions have constant modulus, hence φ ∈ L∞ . Thus, we can find another finite linear combination of eigenfunctions ψ = bj fj such that 2< ψ − ψ
. φ∞
Then 2 φψ − φψ
− ψ) 2 + (φ − φ)ψ ≤ φ(ψ 2 2 ≤ φ∞ ψ − ψ2 + ψ∞ φ − φ ≤ 2.
It remains to note that φψ ∈ Hpp since the product of eigenfunctions for a dynamical system is an eigenfunction. Since is arbitrarily small, φψ ∈ Hpp , and the lemma is proved.
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Proof of Theorem 3.2. (i) ⇒ (ii) This is essentially proved by Dworkin in [4], see also [6] and [1]. By Lemma 3.4, pure point dynamical spectrum implies that γ ω,Λ is pure point for any ω ∈ C0 (Rd ). Note that ω |2 γ. γω,Λ = |
(3.7)
Choosing a sequence ωn ∈ C0 (Rd ) converging to the delta measure δ0 in the vague topology, we can conclude that γ is pure point as well, as desired. (This approximation step requires some care; it is explained in detail in [1].) (ii) ⇒ (iii) obvious. (iii) ⇒ (i) This is relatively new, although it is largely a generalization of Queffelec [11, Prop. IV.21]. We are given that δΛi has pure point diffraction spectrum, that is, γi := γ(δ Λi ) is pure point, for all i ≤ m. In view of (3.7) and Lemma 3.4, we obtain that σfi,ω is pure point for all i ≤ m and all ω ∈ C0 (Rd ). So fi,ω ∈ Hpp for all i ≤ m and all ω ∈ C0 (Rd ). Fix ε > 0 and let V be a bounded set with diam(V ) < θ = θ(ε), where θ is defined by (3.6), and Vol(∂V ) = 0. Find ω ∈ C0 (Rd ) as in Lemma 3.5. Since Vol((∂V )+ζ ) → Vol(∂V ) = 0 in Lemma 3.5, as ζ → 0, we obtain that χEi ,V ∈ Hpp . Therefore, also Ux χEi ,V = χx+Ei ,V ∈ Hpp . Then it follows from Lemma 3.6 and Lemma 3.7 that χP,V ∈ Hpp where P = B1/ε (0) ∩ Γ for any Γ ∈ XΛ , diam(V ) < θ, and Vol(∂V ) = 0. Our goal is to show that Hpp = L2 (XΛ , µ). Since (XΛ , µ) is a regular measure space, C(XΛ ) is dense in L2 (XΛ , µ). Thus, it is enough to show that all continuous functions on XΛ belong to Hpp . Fix f ∈ C(XΛ ). Using the decomposition XΛ = N j=1 XPj ,Vj from Lemma 2.4 we can approximate f by linear combinations of characteristic functions of cylinder sets XPj ,Vj . So it suffices to show that these characteristic functions are in Hpp , which was proved above. This concludes the proof of Theorem 3.2.
4 Concluding remarks: what if the UCF fails? Here we present a version of the main theorem for Delone multisets which do not necessarily have uniform cluster frequencies. For this we must assume that in addition to the van Hove property (2.4) our averaging sequence {Fn } is a sequence of compact neighbourhoods of 0 satisfying the Tempel’man condition: (i) ∪Fn = Rd (ii) ∃ K ≥ 1 so that Vol(Fn − Fn ) ≤ K · Vol(Fn ) for all n.
(4.1)
Let Λ be a Delone multiset with FLC in Rd . Consider the topological dynamical system (XΛ , Rd ) and an ergodic invariant Borel probability measure µ (such measures always exist). The ergodic measure µ will be fixed throughout the section. Theorem 4.1 Suppose that a Delone multiset Λ has FLC. Then the following are equivalent:
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(i) The measure-preserving dynamical system (XΛ , µ, Rd ) has pure point spectrum; (ii) For µ-a.e. Γ ∈ XΛ , the measure ν = i≤m ai δΓi has pure point diffraction spectrum, for any choice of complex numbers (ai )i≤m ; (iii) For µ-a.e. Γ ∈ XΛ , the measures δΓi have pure point diffraction spectrum, for i ≤ m. In fact, this formulation is closer to the work of Dworkin [4] who did not assume unique ergodicity. The proof is similar to that of Theorem 3.2, except that we have to use the Pointwise Ergodic Theorem instead of the uniform convergence of averages in the uniquely ergodic case (2.5). Theorem 4.2 (Pointwise Ergodic Theorem for Rd -actions (see, e.g. [15], [2]) 3 ) Suppose that a Delone multiset Λ has FLC and {Fn } is a van Hove sequence satisfying (4.1). Then for any f ∈ L1 (XΛ , µ), 1 f (−x + Γ) dx → f (Λ ) dµ(Λ ), as n → ∞, (4.2) Vol(Fn ) Fn for µ-a.e. Γ ∈ XΛ . For a cluster P ⊂ Λ, a bounded set A ⊂ Rd , and a Delone set Γ ∈ XΛ , denote LP (A, Γ) = #{x ∈ Rd : x + P ⊂ A ∩ Γ}. Lemma 4.3 For µ-a.e. Γ ∈ XΛ and for any cluster P ⊂ Λ, freq (P, Γ) := lim
n→∞
LP (Fn , Γ) , Vol(Fn )
(4.3)
exists for µ-a.e. Γ ∈ XΛ . Moreover, if diam(V ) < η(Λ), then the cylinder set XP,V satisfies, for µ-a.e. Γ ∈ XΛ : µ(XP,V ) = Vol(V ) · freq (P, Γ).
(4.4)
Note that we no longer can claim uniformity of the convergence with respect to translation of Γ. Sketch of the proof. Fix a cluster P ⊂ Λ and let XP,V be a cylinder set, with diam(V ) < η(Λ). Applying (4.2) to the characteristic function of XP,V and arguing as in the proof of Theorem 2.7 (with −h + Λ replaced by Γ), we obtain (4.3) and (4.4) for µ-a.e. Γ. Since there are countably many clusters P ⊂ Λ, we can find a set of full µ-measure on which (4.3) and (4.4) hold for all P. 3 For recent developments of this theorem in the direction of general locally compact amenable groups, see [9].
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For a Delone set Γ = (Γi )i≤m , let ν = m i=1 ai δΓi . Then, for µ-a.e. Γ, the autocorrelation γ(ν) exists as the vague limit of measures Vol1(Fn ) (ν|Fn ∗ ν|Fn ), and γ(ν) =
m i,j=1
ai aj
freq ((y, z), Γ)δy−z ,
y∈Γi ,z∈Γj
is a positive measure, called for µ-a.e. Γ, which is the analogue of (3.3). Again, γ(ν) the diffraction measure, giving the meaning to the words “pure point diffraction spectrum” in Theorem 4.1. Sketch of the proof of Theorem 4.1. For µ-a.e. Γ ∈ XΛ , the Pointwise Ergodic Theorem 4.2 holds for all functions f ∈ C(XΛ ) (since the space of continuous functions on XΛ is separable). The νΛ , ρω,Λ , and fω are defined the same way as in Section 3. Lemma 3.3 applies to our situation. Next we can show that σfω = γ ω,Γ
(4.5)
for µ-a.e. Γ. This is proved by the same chain of equalities as in (3.4), except that we average over Fn defined in (4.1) and use Theorem 4.2 instead of Theorem 2.7. Lemma 3.5 goes through, after we replace freq(Ei , Λ) by freq (Ei , Γ), for µ-a.e. Γ. There are no changes in Lemmas 3.6 and 3.7, since we did not use UCF or unique ergodicity in them. The proof of Theorem 4.1 now follows the scheme of the proof of Theorem 3.2. We only need to replace Λ by µ-a.e. Γ, for which hold all the “typical” properties discussed above.
References [1] M. Baake and R. V. Moody, Weighted Dirac combs with pure point diffraction, math.MG/0203030, Preprint, 2002. [2] J. Chatard, Sur une g´en´eralisation du th´eor`eme de Birkhoff, C.R. Acad. Sc. Paris, t.275, Serie A, 1135–1138 (1972). [3] M. Denker, C. Grillenberger and K. Sigmund, Ergodic Theory on Compact Spaces, Springer Lecture Notes in Math. 527, Springer, 1976. [4] S. Dworkin, Spectral theory and X-ray diffraction, J. Math. Phys. 34, 2965– 2967 (1993). [5] A. C. D. van Enter and J. Mi¸ekisz, How should one define a (weak) crystal? J. Stat. Phys. 66, 1147–1153 (1992). [6] A. Hof, Diffraction by aperiodic structures, in The Mathematics of LongRange Aperiodic Order, (R. V. Moody, ed.), 239–268, Kluwer, 1997.
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[7] J.-Y. Lee and R. V. Moody, Lattice Substitution Systems and Model Sets. Discrete and Computational Geometry 25, 173–201 (2001). [8] J.-Y. Lee, R. V. Moody and B. Solomyak, Consequences of Pure Point Diffraction Spectra for Multiset Substitution Systems, Discrete and Computational Geometry (to appear), 2002. [9] E. Lindenstrauss, Pointwise theorems for amenable groups, Invent. Math. 146, 259-295 (2001). [10] G. W. Mackey, Unitary Group Representations in Physics, Probability, and Number Theory, Addison-Wesley, 1978,1989. [11] M. Queffelec, Substitution dynamical systems – spectral analysis, Springer Lecture Notes in Math. 1294, Springer, 1987. [12] C. Radin and M. Wolff, Space tilings and local isomorphism, Geometriae Dedicata 42, 355–360 (1992). [13] M. Schlottmann, Generalized model sets and dynamical systems, in: Directions in Mathematical Quasicrystals, eds. M. Baake and R. V. Moody, CRM Monograph series, AMS, Providence RI (2000), 143–159. [14] B. Solomyak, Dynamics of self-similar tilings, Ergodic Th. Dynam. Sys. 17, 695–738 (1997). [15] A. A. Tempel’man, Ergodic theorems for general dynamical systems, Dokl. Akad. Nauk SSSR, vol. 176 (1967), no. 4, 790–793 (English translation: Soviet Math. Dokl., vol. 8 (1967), no. 5, 1213–1216). [16] P. Walters, An introduction to ergodic theory, Springer Graduate Texts in Mathematics, Springer-Verlag, New York, 1982. [17] J. Weidmann, Linear Operators in Hilbert Space, Springer Graduate Texts in Mathematics, Springer-Verlag, New York, 1980. Jeong-Yup Lee and Robert V. Moody Department of Mathematical and Statistical Sciences University of Alberta Edmonton, Alberta T6G 2G1 Canada email: [email protected] email: [email protected] Communicated by Jean Bellissard submitted 03/01/02, accepted 04/04/02
Boris Solomyak Department of Mathematics University of Washington Seattle, WA 98195 USA email: [email protected]
Ann. Henri Poincar´e 3 (2002) 1019 – 1047 c Birkh¨ auser Verlag, Basel, 2002 1424-0637/02/051019-29
Annales Henri Poincar´ e
Linearly Recurrent Circle Map Subshifts and an Application to Schr¨ odinger Operators B. Adamczewski and D. Damanik∗
Abstract. We discuss circle map sequences and subshifts generated by them. We give a characterization of those sequences among them which are linearly recurrent. As an application we deduce zero-measure spectrum for a class of discrete onedimensional Schr¨ odinger operators with potentials generated by circle maps.
1 Introduction and Results 1.1
Introduction
The concept of linear recurrence or linear repetitivity, LR in short, has been recently discussed and investigated by quite a number of researchers within various frameworks. For example, the articles [15, 17, 19] study the LR property from the point of view of combinatorics on words, whereas [14, 32, 38] discuss its implications within the theory of tilings. In both cases one considers structures (e.g., an infinite word or a tiling of Euclidean space), or families of structures (e.g., a subshift or a family of tilings), and their local patterns (e.g., subwords or patches occurring in the given tiling) which are equivalence classes modulo translations. Fixing such a local pattern, one may look at the set of occurrences of the pattern in the structure and compare the distance between two “consecutive” occurrences with the size of the pattern. If the distance is bounded by a fixed linear function of the size, the structure is said to have the LR property. Although the concepts are the same in spirit, applied to words it is usually referred to as linear recurrence, whereas among tiling theorists this concept is usually called linear repetitivity. Since this article will be concerned with a class of words and subshifts, we will henceforth use the term linear recurrence. The usefulness of the LR property has been independently realized by numerous people who had quite different applications in mind. LR has been shown to have consequences in mathematical disciplines as diverse as combinatorics [15, 19], ergodic theory [14, 32, 34], and spectral theory of Schr¨ odinger operators [35]. ∗ D. D. was supported in part by the National Science Foundation through Grant DMS– 0010101.
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Our present study is motivated by the paper [35]. Consider discrete onedimensional Schr¨ odinger operators (Hψ)(n) = ψ(n + 1) + ψ(n − 1) + V (n)ψ(n)
(1)
in 2 (Z), where the potential V : Z → R is given by V (n) = λχ[0,β) (nα + θ
mod 1).
(2)
Here, λ = 0 is the coupling constant, α ∈ (0, 1) irrational is the rotation number, and β ∈ (0, 1) and θ ∈ [0, 1) are arbitrary numbers. These potentials are called circle map potentials in the mathematical physics community (cf. [23, 24, 25]) and codings of rotations by people working in combinatorics on words or symbolic dynamics. The operator (1) with potential (2) has been studied in many papers; for example, [3, 4, 6, 10, 11, 12, 16, 23, 24, 25, 26, 27, 28, 29, 39, 40]. One expects the following picture to be true (cf. [9]): The operator H has purely singular continuous spectrum which is supported on a Cantor set of Lebesgue measure zero. To establish this, one has to prove the following three properties of H: (i) The spectrum σ(H) of H has Lebesgue measure zero. (ii) The absolutely continuous spectrum σac (H) of H is empty. (iii) The point spectrum σpp (H) of H is empty. Actually, it is easy to see that (i) implies (ii). However, (ii) is known in great generality while (i) is not. Namely, it follows from Kotani [31] and Last and Simon [33] that for all parameter values allowed above (recall λ = 0 and α irrational), (ii) holds. Moreover, (iii) is known in many cases. For example, Delyon and Petritis showed that the point spectrum is empty for every λ and β, almost every α, and almost every θ [16]. Hof et al., on the other hand, prove (iii) for every λ, α, and β, and generic θ (i.e., for a dense Gδ set) [24]. Thus, properties (ii) and (iii) are well understood. This is not the case for property (i). Until very recently, there was only one approach to (i). This approach is based on trace maps and it allowed Bellissard et al. to prove the zero measure property in the case where α = β, that is, in the Sturmian case [3] (see also S¨ ut˝ o [40] for the Fibonacci case). Their results were extended to the quasi-Sturmian case in [13]. (A quasi-Sturmian sequence is essentially a morphic image of a Sturmian sequence.) In the non-(quasi-)Sturmian case, very little is known. The only result, due to H¨ornquist and Johansson [25], concerns a small class which can be shown to be generated by substitutions so that the adaptation [5] of [3] to potentials generated by substitutions applies. Essentially, the absence of a trace map is the reason that no other results are known for the non-Sturmian case. A new approach to zero-measure Cantor spectrum, which is not based on trace maps, was recently developed by Lenz [35]. It is therefore natural, and was in fact suggested in [35], to try to apply this new approach to the potentials in (2). This new approach shows that linear recurrence allows one to deduce (i). Thus, we are led to the following question: For which
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choices of parameter values is V in (2) linearly recurrent? It is the aim of this paper to answer this question. In fact, we shall characterize this set of parameter values. We note that the examples considered by H¨ornquist and Johansson are linearly recurrent so that our result contains theirs. For convenience, we will slightly change the setting from individual sequences to subshifts. However, at the end of Section 5 we shall clearly state for which parameter values we get property (i). The organization of the article is as follows. In the remainder of this section we will recall some key notions and state our main result which provides a characterization of the circle map sequences/subshifts which are linearly recurrent. In Section 2 we will develop the general setup and in particular recall the connection between LR subshifts and primitive S-adic subshifts. The link between circle map sequences and interval exchange transformations, and particularly the results of [1] which will be crucial to our paper, will be explained in Section 3. Section 4 contains the proof of our main result. The application of this theorem to Schr¨ odinger operators is discussed in Section 5. Appendix A explains how to prove a finite index for some circle map sequences which are not LR. Finally, in Appendix B we discuss possible generalizations of the approach presented in this paper. Acknowledgments. We would like to thank Julien Cassaigne for useful discussions and particularly for his contributions to what is presented in Appendix A. Moreover, D. D. would also like to express his gratitude for the hospitality at CPT and IML at CNRS, Luminy where this work was initiated.
1.2
Circle maps
Definition 1 Let (α, β) ∈ (0, 1)2 . The circle map corresponding to the parameters (α, β) is the symbolic sequence U = (un )n≥0 defined over the binary alphabet {0, 1} by: 1 if {nα} ∈ [0, β[, un = 0 else. We will restrict our attention to circle maps where α is irrational and β ∈ Z + αZ. The case α rational is not interesting since the associated circle map is periodic (and hence, in this case, the corresponding Schr¨ odinger operator has purely absolutely continuous spectrum which is supported on a finite union of closed intervals). The case β = α gives a Sturmian sequence and, more generally, the case β ∈ Z + αZ corresponds to quasi-Sturmian sequences and will be not considered in this paper (see [7, 37]). (Zero-measure spectrum for chr¨ odinger operators with quasi-Sturmian potentials was shown in [13]). Definition 2 A circle map is called nondegenerate if its parameters satisfy: • α is irrational, • β ∈ Z + αZ. Such a circle map is called admissible if in addition we have α < β.
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The D-expansion
In a previous paper [1], one of us has investigated the links between nondegenerate circle maps and three-interval exchange transformations. An algorithm was introduced which can be regarded as a speed-up of the Rauzy induction for threeinterval exchange transformations. This algorithm is also a generalization of the classical continued fraction algorithm. Let us introduce a map D : [0, 1[×R∗+ −→ [0, 1[×R∗+ given by
{ x } y−1 1 , 1 x 1 x y−1 − y−1 y−1 − y−1 y x x (x, y) −→ { 1−y }, 1−y − 1−y (0, 1)
if
y > 1,
if
y < 1,
if
y = 1.
Definition 3 Given an admissible circle map with parameters (α, β), the associated D-expansion is given by the sequence (an , in )n∈N which is defined as follows:
an
=
in
=
xn yn − 1 1 if yn < 1 0 if yn > 1
where α−β 1 − 1−β α α , . (xn , yn ) = Dn (x0 , y0 ) and (x0 , y0 ) = + 1 α 1 − 1−β +1 α 1 − 1−β α α
For a circle map corresponding to (α, β) ∈ [0, 1[2 which is nondegenerate and not admissible (i.e., α > β), its D-expansion is given by the D-expansion associated with the admissible circle map corresponding to (1 − α, 1 − β). Conversely, for any sequence (an , in )n∈N with (an )n∈N not ultimately vanishing and (in )n∈N not ultimately constant, and any k ∈ N, there is exactly one nondegenerate pair (α, β) such that 1−β α = k and the corresponding circle map sequence has D-expansion (an , in )n∈N . We also want to mention the recent paper [22] of Ferenczi, Holton, and Zamboni which introduces a generalized continued fraction algorithm for three-interval exchange transformations which is based on a different induction process.
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Results
Our main result is Theorem 4 which gives a characterization of linearly recurrent nondegenerate circle map subshifts. Theorem 4 A nondegenerate circle map subshift is linearly recurrent if and only if its D-expansion (an , in )n∈N satisfies the following: there exists an integer M such that for every integer n, (i) an ≤ M , (ii) in = in+1 = . . . = in+t ⇒ t ≤ M , (iii) an = an+1 = . . . = an+t = 0 ⇒ t ≤ M . In the following, we will call this condition the (∗)-condition. In particular, the class of LR nondegenerate circle map subshifts contains, but is not equal to, the circle map subshifts corresponding to parameters (α, β), where α and β lie in the same quadratic field. This follows directly from the fact proved in [17] that a primitive substitutive subshift is linearly recurrent and Theorem 5 (Adamczewski [1]) For a subshift associated with a nondegenerate circle map corresponding to parameters (α, β), the following are equivalent: (i) It is primitive substitutive, that is, it can be generated by the morphic image of a fixed point of a primitive substitution. (ii) The associated D-expansion is ultimately periodic. (iii) α and β lie in the same quadratic field. In terms of interval exchange transformations, Theorem 4 is a full geometric generalization of the following theorem. Theorem 6 (Durand [20]) A Sturmian subshift associated with an irrational number α is linearly recurrent if and only if the coefficients of the continued fraction expansion of α are bounded.
2 Definitions and Background 2.1
Symbolic sequences and substitutions
A finite and nonempty set A is called alphabet. The elements of A are called letters. A finite word on A is a finite sequence of letters and an infinite word on A is a sequence of letters indexed by N. The length of a finite word ω, denoted by |ω|, is the number of letters it is built from. The empty word, ε, is the unique
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word of length 0. We denote by A∗ the set of finite words on A and by AN the set of sequences over A. Let U = (uk )k∈N be a symbolic sequence defined over the alphabet A. A factor of U is a finite word of the form ui ui+1 . . . uj , 0 ≤ i ≤ j. If ω is a factor of U and a a letter, then |w|a is the number of occurrences of the letter a in the word ω. We denote by L(U ) the set of all the factors of the sequence U , L(U ) is called the language of U . A sequence in which all the factors have an infinite number of occurrences is called recurrent. When these occurrences have bounded gaps, the sequence is called uniformly recurrent. A sequence U is called K-power free if uK ∈ L(U ) implies u = ε. A sequence U is called power free if there exists an integer K such that U is K-power free. Endowed with concatenation, the set A∗ is a free monoid with unit element ε. A map from A to A∗ \{ε}, called substitution on A, can be extended by concatenation to an endomorphism of the free monoid A∗ and then to a map from AN to itself. Given a substitution σ defined on A, we call the matrix Mσ = (|σ(j)|i )(i,j)∈A2 the incidence matrix associated with σ. The composition of substitutions corresponds to the multiplication of incidence matrices. A substitution is called primitive if there exists a power of its incidence matrix for which all the entries are positive.
2.2
Return words
We present here the main definitions concerning the notion of return words introduced in [18]. Let U be a uniformly recurrent sequence over the alphabet A and let u = u1 u2 . . . un be a nonempty prefix of U . A return word to u of U is a factor u[i,j−1] (= ui ui+1 . . . uj−1 ) of U such that i and j are two consecutive occurrences of u. The sequence U can be written in a unique way as a concatenation of return words to u. Let RU,u be the set of return words to u in U . Then U = ω0 ω1 . . . ωi . . ., where ωi ∈ RU,u . The fact that U is uniformly recurrent implies that RU,u is a finite set. We can therefore consider a bijective map ΛU,u from RU,u to the finite set {1, 2, . . . , Card(RU,u )} = AU,u , where, for definiteness, the return words are ordered according to their first occurrence (i.e., Λ−1 U,u (1) is the first return word ω0 , −1 ΛU,u (2) is the first ωi which is different from ω0 , and so on). The derived sequence of U on u is the sequence with values in the alphabet AU,u given by Du (U ) = ΛU,u (ω0 )ΛU,u (ω1 ) . . . ΛU,u (ωi ) . . . . To such a sequence we can associate a morphism ΘU,u from AU,u to A∗ defined by: ΘU,u (i) = ωi . We obtain ΘU,u (Du (U )) = U . The morphism ΘU,u is called the return morphism to u of U . When AU,u = A, we will call it return substitution to u of U . When it
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does not create confusion, we will suppress the “U ” in the symbols RU,u , ΘU,u , and AU,u . Proposition 7 (Durand [18]) Let u be a nonempty prefix of U . Then the following holds. (i) The set Ru is a code and the map Θu is one to one. (ii) Let v be a nonempty prefix of Du (U ). Then there exists a nonempty prefix w of U such that Dv (Du (U )) = Dw (U ). Moreover, we have Θu ◦ Θv = Θw . A derived sequence of a derived sequence is hence a derived sequence. Definition 8 Let U be a symbolic sequence defined over the alphabet A starting with the symbol 1 ∈ A. We introduce the following notation: D(0) (U ) = U and, for n ∈ N, D(n+1) (U ) = D1 (D(n) (U )); Θ0 is the identity map and, for n ∈ N, Θn+1 = Θn ◦ ΘD(n) (U),1 . Remark 9 According to Proposition 7, we obtain that (D(n) )n∈N is a sequence of derived sequences of U and (Θn )n∈N is a sequence of return morphisms of U .
2.3
LR sequences
Definition 10 Let A be an alphabet, K a positive integer, and U a sequence over A. The sequence U is called K-linearly recurrent (K-LR) if it is uniformly recurrent and for all ω ∈ Ru , we have |ω| ≤ K|u|. A sequence is called linearly recurrent (LR) if it is K-LR for some K. Proposition 11 (DHS [17]) Let U be an aperiodic K-LR sequence over an alphabet A. Then: 1. For every n, each factor of U of length n has at least one occurrence in each factor of U of length (K + 1)n. 2. U is (K + 1)-power free. 3. For every nonempty prefix u of U and for all ω ∈ Ru , we have
2.4
1 K |u|
< |ω|.
Subshifts and LR subshifts
Let A be an alphabet. The topology of AN is given by the product of the discrete topologies on A. We denote by T the standard shift transformation which associates to each symbolic sequence U = (uk )k≥0 the sequence T (U ) = (uk )k≥1 . To a sequence U in AN we associate the dynamical system (O(U ), T ), where O(U ) is the closure of the orbit of U under the shift. This dynamical system is called the subshift associated with U . A dynamical system is minimal if it has no nontrivial invariant closed set. For a subshift associated with a sequence U , minimality of the subshift is equivalent to uniform recurrence of U .
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Definition 12 A subshift is called primitive substitutive if it contains a primitive substitutive sequence (i.e., a sequence which is the morphic image of a fixed point of a primitive substitution). A minimal subshift associated with a sequence U is called linearly recurrent (LR) if and only if U is LR.
2.5
S-adic sequences and S-adic subshifts
Let A be an alphabet, a a letter of A, and S a finite set of substitutions from A to A∗ . We will say that a sequence U ∈ AN is an S-adic sequence (generated by (σn )n∈N ∈ S N and a) if there exists a sequence (σn )n∈N ∈ S N such that U = limn→∞ σ0 σ1 . . . σn (aa . . .). Let U be such a sequence. If there exists an integer s0 such that for all b ∈ A and all c ∈ A, the letter b has an occurrence in σr+1 σr+2 . . . σr+s0 (c), then U is called a primitive S-adic sequence (with constant s0 ). The subshift associated with an S-adic sequence (resp., a primitive S-adic sequence) is called an S-adic subshift (resp., a primitive S-adic subshift). These notions were introduced by S. Ferenczi in [21] and by F. Durand in [19]. It was claimed in [19] that a subshift is LR if and only if it is primitive S-adic. In [20], Durand provides a counterexample and exhibits a primitive S-adic subshift which is not LR. However, LR does imply primitive S-adic and with an additional condition we can obtain a partial converse given in Proposition 14 below. Definition 13 Let A be an alphabet and σ a substitution on A. The substitution σ is called (b, c)-proper if for any letter i in A, σ(i) begins with b and ends with c. An S-adic sequence is called proper if there exist two letters b and c in A such that any substitution in S is a (b, c)-proper substitution. A subshift which contains a proper and primitive S-adic sequence is called a proper primitive S-adic subshift. Proposition 14 (Durand [20]) A subshift (X, T ) is LR if and only if it is a proper primitive S-adic subshift.
2.6
Interval exchange transformations
Interval exchange transformations are classical examples of dynamical systems. Definition 15 Let s ∈ N, s ≥ 2. Let σ be a permutation of the set {1, 2, . . . , s} and let λ = (λ1 , λ2 , . . . , λs ) be a vector in Rs with strictly positive entries. Let I = [0, |λ|[, where |λ| =
s i=1
λi and for 1 ≤ i ≤ s, Ii = λj , λj . j
j≤i
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The interval exchange transformation associated with (λ, σ) is the map E from I into itself, defined as the piecewise isometry which arises from ordering the intervals Ii with respect to σ. More precisely, if x ∈ Ii , E(x) = x + ai , where ai = λσk − λk . k<σ−1 (i)
k
We can introduce a natural coding of the orbit of a point under the action of an interval exchange transformation by assigning to each element of this orbit the number of the interval which contains it. Remark 16 Let us consider an interval exchange transformation E, and U the natural coding of the orbit of the point 0 under E. The natural coding of the orbit of the point 0 under the action of the induced map of E on its first interval is the derived sequence on the letter 1 of U . Moreover, the associated induced substitution corresponds to the return substitution to 1 of U . In the case of the Rauzy induction, one does not induce on the first interval but on an interval which is larger. However, the induction on the first interval can be decomposed into several steps of the Rauzy induction. We refer the reader to [36] for information on the useful notion of Rauzy induction for interval exchange transformations.
3 A Geometric Interpretation In this section, we investigate the geometric link between Theorems 4 and 6. The symmetric Rauzy induction for two-interval exchange transformations is introduced in [2]. From the study of this induction process, the authors of [2] obtain an S-adic expression for Sturmian subshifts. Let τ1 and τ2 be substitutions on {0, 1} defined as follows: τ1 (0) = 01 and τ2 (0) = 0 τ2 (1) = 10. τ1 (1) = 1 Proposition 17 Let α ∈ (0, 1) be an irrational number. The Sturmian subshift associated with α is generated by the sequence lim τ i1 τ i2 τ i3 τ i4 n→∞ 2 1 2 1
i
. . . τ22n−1 τ1i2n (0),
where [0; i1 + 1, i2 , i3 , i4 , . . .] is the continued fraction expansion of α. The decomposition of the two-interval exchange transformation associated to α under the symmetric Rauzy induction is symbolized in Figure 1. The fact that α is irrational implies that this two-interval exchange transformation satisfies the well-known I.D.O.C. condition (short for Infinite and Disjoint Orbit Condition)
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τ1
Ann. Henri Poincar´e
τ2
21
Figure 1: The symmetric Rauzy induction graph for two-interval exchange transformations. G1 :
τ1
21
G2 :
,
τ2
21
Figure 2: The primitivity subgraphs for two-interval exchange transformations. introduced in [30]. It also implies that an orbit under the symmetric Rauzy induction does not ultimately remain in one of the primitivity subgraphs G1 or G2 represented in Figure 2. Moreover, an I.D.O.C. two-interval exchange is LR if and only if its orbit under the symmetric Rauzy induction can stay in any of the primitivity subgraphs G1 and G2 only for a bounded number of consecutive induction steps. This last remark provides a geometric interpretation of Theorem 6. We present now an analogous study in the case of nondegenerate circle map subshifts. Let us introduce the following four substitutions, defined over the alphabet {1, 2, 3}, given by: σ1 1 − → 13 2 − → 2 3 − → 3
1 2 3
σ2 → 1 − −→ 2 −→ 23
1 2 3
σ3 −→ 1 −→ 23 −→ 3
σ4 1 − → 1 2 − → 13 3 − → 2
For each integer k, we also consider the following morphism: Φk : {1, 2, 3}∗ 1 2 3
−→ −→ −→ −→
{1, 0}∗ 1, 10k+1 , 10k .
If (Un )n∈N ∈ {0, 1}N, the sequence (Un )n∈N is defined by 1 if Un = 0, Un = 0 if Un = 1. Having fixed the above notation, we can give the following S-adic expression for nondegenerate circle map subshifts. Theorem 18 (Adamczewski [1]) Let us consider nondegenerate parameters (α, β) ∈ (0, 1) and let (an , in )n∈N be the D-expansion associated with (α, β). The circle
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map subshift associated with parameters (α, β) is generated by the sequence n i 1−i a a j j lim Φ 1−β σ1 σ2 j σ3 ◦ σ4 σ1 j σ4 (1) n→∞
if α < β and by
α
lim 1T Φ
n→∞
j=0
β 1−α
n
j=0
ij 1−ij a a σ1 σ2 j σ3 ◦ σ4 σ1 j σ4 (1)
if α > β. The proof is based on a study of an induction process for three-interval exchange transformations close to that of Rauzy. We also obtain an analog to Proposition 17 in the case of nondegenerate circle map subshifts. Figure 3 is the analog of Figure 1 and Figure 4 is the analog of Figure 2. To a nondegenerate circle map we can associate an I.D.O.C. three-interval exchange transformation. The orbit of such an interval exchange transformation under the Rauzy induction does not ultimately remain in one of the primitivity subgraphs G1 , G2 , or G3 represented in Figure 4. σ4
σ1
σ2 312
321 σ3
σ1 231
σ4
Figure 3: The Rauzy induction graph for three-interval exchange transformations. Moreover, an I.D.O.C. three-interval exchange transformation is LR if and only if its orbit under the Rauzy induction can stay in any of the primitivity subgraphs G1 , G2 , and G3 only for a bounded number of consecutive induction steps. This last remark provides a geometric interpretation of the (∗)-condition in Theorem 4 and will be proved in Section 4. A similar study could clearly be carried out in the general case of an I.D.O.C. interval exchange transformation. However, the results quickly become hard to read since the complexity of the equivalent to the (∗)-condition increases rapidly (cf. Appendix B). In this section we have exhibited some similarities between the Sturmian and the circle map cases. On the other hand, some aspects of the two cases do not have mutual counterparts. The strategy used to prove Theorem 6 is the following: • Exhibit a primitive S-adic expression for Sturmian subshifts generated by an irrational α when the coefficients of the continued fraction expansion of α are bounded and use this to establish linear recurrence in this case.
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G1 :
312
321
, G2 :
312
321
231
, G3:
Ann. Henri Poincar´e
321
231
Figure 4: The primitivity subgraphs for three-interval exchange transformations. • Show that otherwise a Sturmian sequence contains arbitrarily high powers. We thus obtain that a Sturmian sequence is LR if and only if it is power free. However, such an equivalence does not hold for circle maps. We can therefore not mimic the strategy used in the Sturmian case. For example, the circle map sequences with D-expansion (an , in )n∈N , where (in )n∈N is the periodic sequence (10)ω (i.e., in = 0 if n is even and in = 1 if n is odd), an = 1 if n is a power of 2 and 0 otherwise, are both non-LR and power free (see Appendix A).
4 Proof of Theorem 4 The proof of Theorem 4 is based on Theorem 18 and Proposition 14 which states that a proper primitive S-adic subshift is LR. Our strategy to prove this theorem is the following: • We exhibit a proper primitive S-adic expression for three-interval exchanges associated with circle maps whose D-expansion satisfies the (∗)-condition (Proposition 20). • We prove the existence of a uniform upper bound of the gaps between successive occurrences of letters in the different derived sequences of an LRsequence (Lemma 24). • Finally, we show that such a uniform bound does not exist for a circle map whose D-expansion does not satisfy the (∗)-condition (Proposition 23). For i ∈ {1, 2, 3, 4}, let Ai denote the incidence matrix of the substitution σi which has been defined in the previous section. For every integer k, we write Fk = (σ1 σ2k σ3 ) and Gk = (σ4 σ1k σ4 ),
(3)
and for the associated incidence matrices, we write Bk = (A1 Ak2 A3 ) and Ck = (A4 Ak1 A4 ).
(4)
Definition 19 Let (C, D) ∈ M3 (R)2 , C = (ci,j ), D = (di,j ). We say that C ≥ D if ci,j ≥ di,j , ∀(i, j) ∈ {1, 2, 3}2. Similarly, we say that C > D holds if ci,j > di,j , ∀(i, j) ∈ {1, 2, 3}2. Proposition 20 A nondegenerate circle map whose D-expansion satisfies the (∗)condition is the image by a morphism of a proper primitive S-adic sequence.
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Lemma 21 If C is a nonnegative matrix in M3 (Z), then for every integer k, the following four inequalities hold: Bk C ≥ C, CBk ≥ C, Ck C ≥ C, and CCk ≥ C. Proof. This follows directly from Bk = I3 + A k with A k ≥ 0 and Ck = I3 + Bk with Bk ≥ 0. Lemma 22 Let (an , in )n∈N be a D-expansion satisfying the (∗)-condition with an ij 1−ij (k+1)M0 integer M0 and let S = , k ∈ N . Then S is a finite set of j=kM0 Faj ◦ Gaj substitutions and each of its element is (1, 3)-proper. Proof. The set S is clearly finite because the sequence (an )n∈N is bounded by M0 . In view of (3), we obtain for every integer k 1 2 3
Fk → 13 − −→ 2k+1 3 −→ 2k 3
1 2 3
Gk −→ 12k −→ 12k+1 −→ 13
Let k be an integer and i ∈ {1, 2, 3}. Then Fk (i) ends with 3 and Fk (1) begins with 1. Moreover Gk (i) begins with 1 and Gk (1) ends with 3. It follows thus that each composition of substitutions of types Fk and Gk in which the two types both appear is (1, 3)-proper. Part (ii) of the (∗)-condition allows us to conclude. Proof of Proposition 20. Let us consider a circle map U whose D-expansion satisfies the (∗)-condition with some integer M0 . Theorem 18 provides us with an S-adic expression for this circle map. Our goal is now to prove that we can extract a proper primitive S-adic expression for U from this representation. We can suppose that U is admissible in order to simplify the notation. We have n ij 1−ij a a j j σ1 ◦ σ2 ◦ σ3 ◦ σ4 ◦ σ1 ◦ σ4 (1) . U = lim Φ 1−β
n→∞
α
j=0
Let
n ij 1−ij a a V = lim σ1 ◦ σ2 j ◦ σ3 ◦ σ4 ◦ σ1 j ◦ σ4 (1) . n→∞
(5)
j=0
Thus, U = Φ 1−β (V ) α
(6)
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and
V = lim n→∞
We have then
V = lim n→∞
Let S=
n
n
j=0
j (1). Faijj ◦ Ga1−i j
j (1). Faijj ◦ Ga1−i j
(7)
j=kM0
(k+1)M 0
(k+1)M0
k=0
Ann. Henri Poincar´e
j Faijj ◦ Ga1−i , k∈N j
j=kM0
.
Then Lemma 22 implies that (7) gives us a proper S-adic representation of V . We have now to prove that this representation is primitive or more precisely that there exists an integer s0 such that for every integer r and all b ∈ {1, 2, 3} and c ∈ {1, 2, 3}, the letter b has an occurrence in k+s 0 (r+1)M 0 j (c). Faijj ◦ Ga1−i j r=k
j=rM0
Or similarly, we have to show that the corresponding product of matrices k+s 0 (r+1)M 0 Baij ◦ Ca1−ij j
r=k
j
j=rM0
is positive, where the matrices Bl and Cl are defined in (4). Let us consider the matrix (r+1)M0 j Mr = Baijj ◦ Ca1−i . j j=rM0
By the fact that the D-expansion associated with U satisfies the (∗)-condition with the integer M0 , we get ∃j1 ∈ {1, 2, . . . , l} such that ij1 = 0, ∃j2 ∈ {1, 2, . . . , l} such that ij2 = 1, ∃j3 ∈ {1, 2, . . . , l} such that aj3 ≥ 1. The previous remark and Lemma 21 show that at least one of the following inequalities holds: Mr ≥ B 0 C1 , Mr ≥ B 1 C0 , Mr ≥ C1 B 0 , Mr ≥ C0 B 1 .
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Now we just have to remark that each element of {B0 C1 , B1 C0 , C1 B0 , C0 B1 }2 is positive. Therefore, we obtain primitive S-adicity of our representation with constant s0 = 2. We therefore obtain that U is the image under the morphism Φ 1−β of α the proper primitive S-adic sequence V , concluding the proof. Proposition 23 A nondegenerate circle map subshift whose D-expansion does not satisfy the (∗)-condition is not linearly recurrent. Since we will work with the derived sequences of a given circle map sequence in our proof of Proposition 23, we start off by discussing LR properties of derived sequences of an LR sequence. Lemma 24 Let U be a K-linearly recurrent sequence defined over an alphabet A and let ω be a nonempty prefix of U . Then every word of length at least K 2 (K + 1) in Dω (U ) contains all the elements of Aω . Proof. Let ω be a factor of U and i ∈ Aω = {1, 2, . . . , d}. Then there exists a unique word ωi such that Θω (i) = ωi . By definition we have ∀j ∈ Aω , |ωj | ≤ K|ω|. This inequality implies that ωi appears in each word of length at least (K + 1)(K|ω|), in view of Proposition 11. Moreover, again by Proposition 11, we have ∀j ∈ Aω ,
1 |ω| ≤ |ωj | ≤ K|ω|. K
The set Rω is a code. We thus obtain that the letter i occurs in each word of length at least K 2 (K + 1) in Dω (U ). Lemma 25 Let U be a K-linearly recurrent sequence. Then, for every integer n, we have ∀i ∈ An , |Θn (i)| ≤ K 2 (K + 1), where the maps Θn are introduced in Definition 8. Proof. Let i be an element of An and Θn (i) = ωi . By definition of the return words and the sequence D(n) , the letter 1 has just one occurrence in ωi and 1 is the first letter of ωi . Then, 1 does not appear in the maximal proper suffix of ωi . Lemma 24 implies that the length of this suffix is at most K 2 (K + 1) − 1. Lemma 26 Let U be a K-linearly recurrent sequence defined over an alphabet A and let ω be a nonempty prefix of U . Then the sequence Dω (U ) is K 3 -linearly recurrent.
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Proof. This statement and its proof are very similar in spirit to the previous two lemmas. Let x be a factor of Dω (U ). Consider any occurrence of x in Dω (U ) and the length of the corresponding return word to x in Dω (U ) (i.e., the length of the gap between this occurrence of x and the next, plus the length of x). We use again that Rω is a code. Namely, to this occurrence of x in Dω (U ) corresponds a word of length at most K · |ω| · |x| in U whose return words have length at most K 2 · |ω| · |x|. Choose the one that corresponds to this particular occurrence and go back via ΛU,ω to factors of Dω (U ). We conclude that the length of the return word to x in question is bounded by K 3 · |x|. In the previous steps, we have made repeated use of Proposition 11. This shows that Dω (U ) is K 3 -linearly recurrent since x and its occurrence were arbitrary. Lemma 27 Let r be a positive integer. Then for every (i1 , i2 , . . . , ir ) ∈ {1, 2, 3, 4}r, we have |σi1 ◦ σi2 ◦ · · · ◦ σir (123)| ≥ σi1 ◦ σi2 ◦ · · · ◦ σir−1 (123) + 1. Proof. We just have to remark that for each k ∈ {1, 2, 3, 4}, there exists a letter b ∈ {1, 2, 3} such that |σk (b)| ≥ 2 and that 1, 2, and 3 occur in σk (123). Lemma 28 Let r be a positive integer and (i1 , i2 , . . . , i3r+1 ) ∈ {1, 2, 3, 4}3r+1. Then there exists at least one letter b ∈ {1, 2, 3} such that σi1 ◦ σi2 ◦ · · · ◦ σi3r+1 (b) > r. Proof. According to Lemma 27, it follows by induction that σi1 ◦ σi2 ◦ · · · ◦ σi3r+1 (123) ≥ 3r + 1. The assertion follows immediately.
Lemma 29 Let n be an integer, (m0 , m1 , . . . , mn ) ∈ Nn , and (l0 , l1 , . . . , ln ) ∈ {0, 1}n. Then, for each b ∈ {1, 2, 3}, we have n m (i) j=0 σ1 σ2 j σ3 (b) ≤ 1, 1 n l 1−l (ii) j=0 (σ1 σ3 ) j ◦ (σ4 σ4 ) j (b) ≤ 1, 2 n mj (iii) j=0 σ4 σ1 σ4 (b) ≤ 1. 3
Here, |w|i denotes the number of occurrences of the symbol i in the word w. n m Proof. (i) The incidence matrix associated with the substitution j=0 σ1 σ2 j σ3 n is j=0 Bmj , where the matrices Bmj are defined in (4). For each integer k, the matrix Bk is of the form 1 0 0 × × × , × × ×
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and so the matrix nj=0 Bmj is of course of the same form. Then, the definition of the incidence matrix allows us to conclude. n l 1−l (ii) The incidence matrix associated with j=0 (σ1 σ3 ) j ◦ (σ4 σ4 ) j is equal to n lj 1−lj . The matrices B0 and C0 are of the form j=0 B0 C0
× 0 × and so the matrix
n
j=0
× × 1 0 , × ×
l 1−l B0j C0 j is of the same form.
n m (iii) The incidence matrix associated with the substitution j=0 σ4 σ1 j σ4 is n equal to j=0 Cmj , where the matrices Cmj are defined in (4). For each integer k, the matrix Ck is of the form × × × × × × , 0 0 1 and so the matrix
n
j=0
Cmj is of the same form, concluding the proof.
Proof of Proposition 23. Let U be a circle map whose D-expansion (an , in )n∈N does not satisfy the (∗)-condition. Let V be as in (5) so that we have (6). Let us assume for the moment that 1 − β > α so that V is the derived sequence corresponding to the prefix 1 of U . We will comment later on the case 1 − β < α. Now assume there exists an integer K such that U is K-LR. We consider four cases. (i) Let us suppose that the sequence (an )n∈N is unbounded. Then a direct consequence of the fact that σ2an (3) = 2an 3, σ1an (1) = 13an , and that powers propagate by substitution is that U cannot be (K +1)-power free. Proposition 11 thus yields a contradiction. (ii) Let us suppose that the sequence (in )n∈N contains arbitrarily long blocks of 1’s. In particular, there exists an integer n0 such that in0 = in0 +1 = . . . = in0 +12K 2 (K+1) = 1.
(8)
We recall that there exists an increasing sequence of integers (kN )N ∈N such that kN ij 1−ij a a σ1 σ2 j σ3 ◦ σ4 σ1 j σ4 , ΘN = j=kN −1 +1
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where ΘN is introduced in Definition 8. This follows from Remark 16 and the fact that, as was already observed in [1], certain steps of our induction process correspond to induction on the first interval of three-interval exchange transformations associated with U . According to Lemmas 25 and 28, the fact that U is K-LR implies that for each integer N , kN +1 − kN < 3K 2 (K + 1) + 1.
(9)
Now, let us consider two particular elements of the sequence (kN )N ∈N : ! kN1 = min kN , n0 ≤ kN ≤ n0 + 12K 2(K + 1) and
! kN2 = max kN , n0 ≤ kN ≤ n0 + 12K 2 (K + 1) .
By the inequality (9), we obtain that kN1 and kN2 are well-defined and kN2 − kN1 ≥ 6K 2 (K + 1) + 1.
(10)
Let us introduce the substitution Θ = ΘN1 +1 ΘN1 +2 . . . ΘN2 . Then, kN2
Θ=
ij 1−ij a a σ1 σ2 j σ3 ◦ σ4 σ1 j σ4 .
j=kN1 +1
More precisely, using condition (8), we have kN2
Θ=
a σ1 σ2 j σ3 .
(11)
j=kN1 +1
Proposition 7 implies that Θ is a return substitution for U since it is a composition of return substitutions. Thus there exists a nonempty prefix ω of U such that Θ = ΘU,ω . According to the inequality (10) and Lemma 28, we obtain that there exists a letter b in the alphabet {1, 2, 3} such that |Θ(b)| ≥
kN2 − kN1 > 2K 2 (K + 1), 3
and it follows from the equality (11) and Lemma 29 that |Θ(b)|1 ≤ 1. But Θ(b) is necessarily a factor of Dω (U ). Hence there exists a factor of Θ(b) of length greater or equal than K 2 (K + 1) in which the letter 1 does not occur. We obtain finally that there exists a factor of Dω (U ) of length greater than or equal to K 2 (K + 1) in which the letter 1 does not occur. This last remark is in contradiction with the K-LR property of U in view of Lemma 24.
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(iii) Let us suppose that the sequence (in )n∈N contains arbitrarily long blocks of 0’s. Then, we just have to mimic the above arguments in order to find a return substitution Θ for U and a letter b in {1, 2, 3} such that |Θ (b)|3 ≤ 1 and |Θ (b)| > 2K 2 (K + 1). We thus obtain a nonempty prefix ω of U such that Dω (U ) contains a factor of length greater than or equal to K 2 (K + 1) in which the letter 3 does not occur. (iv) Let us suppose that the sequence (an )n∈N contains arbitrarily long blocks of 0. Then, analogous reasoning gives a return substitution Θ for U and a letter b in {1, 2, 3} such that: |Θ (b)|2 ≤ 1 and |Θ (b)| > 2K 2 (K + 1). We find a nonempty prefix ω of U such that Dω (U ) contains a factor of length greater than or equal to K 2 (K + 1) in which the letter 2 does not occur. Thus we arrive at a contradiction in each case. Recall that we assumed 1−β > α at the beginning of the proof. Let us now discuss the case where 1 − β < α. In this case V in (5) is not the derived sequence corresponding to the prefix 1 of U , that is, D1 (U ) = V . In fact, V takes three values, while 1 has only two return words, 1 and 10. However, for sufficiently large n, it is relatively easy to see that D(n) (U ) is one of the sequences obtained in the induction process of [1] (leading to the representation (6)) and hence there is a morphism Ψ such that V = Ψ(D(n) (U )). If we now again assume that U is LR, then so is D(n) (U ), by Lemma 26, and hence we get that V is LR. Now we can derive a contradiction following the steps given above. Proof of Theorem 4. In view of Proposition 14, Theorem 4 follows directly from Propositions 20 and 23.
5 Application of Theorem 4 to Schr¨ odinger Operators In this section we apply our main result, Theorem 4, to discrete one-dimensional Schr¨ odinger operators with potentials given by circle maps. As explained in the introduction, this is in part motivated by previous results on their Sturmian counterparts and a recent result of Lenz which relates aspects of their spectral theory to LR properties. A discrete one-dimensional Schr¨ odinger operator acts in the Hilbert space 2 (Z). If φ ∈ 2 (Z), then Hφ is given by (Hφ)(n) = φ(n + 1) + φ(n − 1) + V (n)φ(n), where V : Z → R. The map V is called the potential.
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If A is an alphabet, T : Z → Z is the standard shift, Ω ⊆ AZ is T -invariant (i.e., T Ω = Ω) and closed (discrete topology on A and product topology on AZ ), then Ω is called a (two-sided) subshift. Given such a subshift and a function f : A → R, we define, for ω ∈ Ω, a potential V = Vω by Vω (n) = f (ωn ) and an operator Hω (as above, with this particular potential). It is a standard result that if Ω is minimal, then the spectrum of Hω does not depend on ω, that is, there is a set Σ ⊆ R such that σ(Hω ) = Σ for every ω ∈ Ω (see, e.g., [9]). A special case of a recent result of Lenz is given in the following theorem. Theorem 30 (Lenz [35]) If Ω is a linearly recurrent subshift and Ω and f are such that the resulting potentials Vω are aperiodic, then Σ has Lebesgue measure zero. Note in particular that the result is essentially independent of the function f . Moreover, it suffices that at least one Vω is aperiodic. This implies that all Vω are aperiodic. Our goal is to apply this theorem to circle map subshifts. A circle map generates a two-sided subshift as follows. If u ∈ {0, 1}N is a circle map corresponding to parameters (α, β), the associated subshift is given by Ω = Ωα,β = {ω ∈ {0, 1}Z : every factor of ω is a factor of u}. If we restrict the sequences in Ω to the right half-line, we get exactly the one-sided subshift that was introduced and discussed above. By recurrence, the languages associated with the one-sided and two-sided subshifts are the same. In particular, LR-properties are the same for both subshifts. Combining our Theorem 4 and the theorem of Lenz, we obtain the following result. Theorem 31 Suppose that u is a nondegenerate circle map corresponding to parameters (α, β) whose D-expansion (an , in )n∈N satisfies the (∗)-condition. Consider the associated subshift Ω = Ωα,β and, for a nonconstant function f : {0, 1} → R, the operators (Hω )ω∈Ω . Then we have that for every ω ∈ Ω, the spectrum of Hω has Lebesgue measure zero. It is easy to see that for every θ, the sequence ωn = χ[0,β) (nα + θ mod 1) is an element of Ωα,β . In other words, Theorem 31 says that if α, β are such that their D-expansion (an , in )n∈N satisfies the (∗)-condition, then the potential V in (2) is linearly recurrent for every choice of θ and λ = 0, and in this case, the operator H satisfies property (i) from the introduction.
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Appendix A In this section we give a proof (and a little bit more) of the power freeness of the sequence we consider in the end of the Section 3. This proof was suggested by J. Cassaigne [8]. Let us introduce the following two substitutions, defined over {1, 2, 3}, given by: g = σ1 σ2 σ3 σ4 σ4 f = σ1 σ3 σ4 σ4 1 2 3
−→ −→ −→
13 1323 133
1 2 3
−→ −→ −→
13 13223 1323
where the substitutions σi are defined in Section 3. We denote by F the largest language defined over the alphabet {1, 2, 3} which satisfies the following three conditions: • ∀ω ∈ {1, 2, 3}∗, ω 4 ∈ F ⇒ ω = ε, • ∀ω ∈ {1, 2, 3}∗ and ∀z ∈ {1, 2, 3}, (ωz)3 ω ∈ F, • 11 ∈ F. The language F is naturally obtained as the union of all the languages defined over the alphabet {1, 2, 3} which satisfy these three conditions. Lemma 32 If ω ∈ F, then f (ω) and g(ω) are two elements of F . Proof. Let ω be an element of F . We consider three cases to prove that f (ω) ∈ F. 1. Assume there exists a nonempty word M such that M 4 is a factor of f (ω). Then, M could be decomposed in a unique way in xf (v)y, where (x, v, y) ∈ {ε, 3, 23, 33, 323} × {1, 2, 3}∗ × {ε, 1, 13, 132} and the length of v is maximal with the convention that if v ends with the letter 1, then y = ε. We consider two subcases. (a) Let us suppose that v = ε. Then M = xy and thus M ∈ {3, 33, 3313, 32313} ∪ {31, 313, 3132, 2313, 331, 3231} ∪ {23, 33132, 323, 323132} ∪ {23132} ∪ {231}. But M ∈ {3, 33, 3313, 32313} because 33 is always followed by a 1 in f (ω). If M ∈ {31, 313, 3132, 2313, 331, 3231}, we obtain that there exists a letter z ∈ {1, 2, 3} such that z 3 is a factor of ω. This gives a contradiction because ω ∈ F. The word M cannot belong to the set {23, 33132, 323, 323132} because 23 is always followed by a 1 in f (ω). M cannot belong to {23132} because the letter 2 is always followed by a 3 in f (ω). Finally, M cannot belong to {231} because the letter 1 is never followed by a 2 in f (ω).
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(b) Let us suppose that v = ε. Then M 4 = xf (v)yxf (v)yxf (v)yxf (v)y and necessarily yx = f (z) with z ∈ {ε, 1, 2, 3}. If z = ε, then M 4 = f (v 4 ). The fact that v does not end with a 1 allows us to infer that v 4 is a factor of ω. We obtain a contradiction because ω ∈ F. If z is a letter, then f ((vz)3 v) is a factor of f (ω). The fact that v does not end with a 1 shows that (vz)3 v is a factor of ω. We obtain a contradiction because ω ∈ F. 2. Let us suppose that there exist a word M and a letter z such that (M z)3 M is a factor of f (ω). Then, M can be decomposed in a unique way in xf (v)y, where (x, v, y) ∈ {ε, 3, 23, 33, 323} × {1, 2, 3}∗ × {ε, 1, 13, 132} and the length of v is maximal with the convention that if v ends with the letter 1, then y = ε. We obtain that (M z)3 M = xf (v)yzxf (v)yzxf (v)yzxf (v)y, and necessarily xzy = f (m) with m ∈ {1, 2, 3} and |m| ≤ 2 because |xzy|1 ≤ 2 and the letter 1 has exactly one occurrence in the image of each letter. Again we consider two subcases. (a) Let us suppose that |m| = 2. Then there exist two letters a and b such that yzx = f (ab). But |y|1 ≤ 1 and |x|1 = 0 imply that y = f (a) and z = 1. We get (M z)3 M = xf ((vab)3 va). If a = 1, then (vab)3 va is a factor of f (ω) and we obtain a contradiction because ω ∈ F. If a = 1, xf ((v1b)3 v) is a factor of f (ω). We recall that zx = 1x = f (b). It follows that if b = 2 or b = 3, then x = 323 or x = 33 and thus x is always preceded by the letter 1 in f (ω). This implies that 1xf ((v1b)3 v) is a factor of f (ω). But since 1xf ((v1b)3 v) = f ((bv1)3 bv) and v does not end with the letter 1, it follows that (bv1)3 bv is a factor of ω. This is in contradiction with ω ∈ F. Finally, if b = 1, then f ((v11)3 v) is a factor of f (ω). The fact that v does not end with the letter 1 gives that (v11)3 v is a factor of ω and thus 11 is a factor of ω. We get a contradiction since 11 ∈ F. (b) Let us suppose that |m| = 1, then (M z)3 M = xf ((vm)3 v)y. In particular, f ((vm)3 v) is a factor of f (ω). But since v does not end with the letter 1, (vm)3 v is a factor of ω. We obtain a contradiction because m is a letter and ω ∈ F. 3. Let us suppose that 11 is a factor of f (ω). This yields a contradiction immediately because the letter 1 is always followed by a 3 in f (ω) by definition of f. The proof for g is exactly the same.
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Proposition 33 A circle map whose D-expansion (an , in )n∈N satisfies • (in )n∈N = (10)ω , • in = 0 implies an = 0, and • in = 1 implies an ∈ {0, 1} is power free. Proof. Let U be such a circle map and V be the natural coding of the three-interval exchange transformation associated with U . Theorem 18 says that there exists a sequence of integers (bn )n∈N such that U = lim Φ 1−β f b0 g b1 f b2 g b3 . . . f b2n (1) n→∞
and thus
α
V = lim f b0 g b1 f b2 g b3 . . . f b2n (1). n→∞
With the previous notation, 1 ∈ F. Then Lemma 32 implies that f b0 g b1 f b2 g b3 . . . f b2n (1) ∈ F for every integer n. We thus obtain L(V ) ⊂ F. This implies that V is 4-power free. Then, in view of the definition of the morphisms Φk , U is clearly power free if 1−β α > 0 (i.e., 1 − β > α). In the case where 1 − β < α, we can use an argument similar to the one used in the proof of Proposition 23. It is relatively easy to see that if a sequence is not power free, then all of its derived sequences are not power free, either. We have already noticed at the end of the proof of Proposition 23 that for sufficiently large n, there is a morphism Ψ such that V = Ψ(D(n) (U )). Now, if we assume that U is not power free, then D(n) (U ) is not power free and hence V is not power free because morphisms propagate powers. We therefore obtain a contradiction to the 4-power freeness of V obtained above. In particular, we obtain the power freeness of the sequences mentioned in Section 3. These sequences are of course not LR in view of Theorem 4 and hence they are both power free and not LR. To the best of our knowledge, these are the first examples of sequences with these two properties. We end this appendix with the following conjecture concerning the power freeness of circle maps. Conjecture. A nondegenerate circle map is power free if and only if its D-expansion (an , in )n∈N satisfies the following: there exists an integer M such that for every integer n, we have • an ≤ M , • in = in+1 = . . . = in+M ⇒ ∃k, n ≤ k ≤ n + M such that ak = 0.
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Appendix B We present here what would be the analog of the geometric considerations of Section 3 in the case of I.D.O.C. four-interval exchange transformations which lie in the Rauzy class of (4321). The notion of Rauzy class for an interval exchange transformation was introduced in [36]. Let us introduce the following substitutions, defined on the alphabet {1, 2, 3, 4}, given by 1 2 3 4
σ1 −→ −→ −→ −→
1 14 2 3
1 2 3 4
σ2 −→ −→ −→ −→
14 2 3 4
1 2 3 4
σ3 −→ −→ −→ −→
1 2 3 34
1 2 3 4
σ4 −→ −→ −→ −→
1 2 34 4
1 2 3 4
σ5 −→ −→ −→ −→
1 2 24 3
1 2 3 4
σ6 −→ −→ −→ −→
1 24 3 4
The Rauzy induction graph for the Rauzy class of (4321) is given in Figure 5. The orbit of an I.D.O.C. four-interval exchange transformation in the Rauzy class of (4321) under the Rauzy induction cannot be ultimately confined to one of its primitivity subgraphs G1 , G2 , G3 or G4 represented in Figures 6, 7, 8, and 9, respectively. Moreover, an I.D.O.C. four-interval exchange in the Rauzy class of (4321) is LR if and only if its orbit under the Rauzy induction can stay in any of the primitivity subgraphs G1 , G2 , G3 , and G4 only for a bounded number of consecutive induction steps. 2431
4132
σ5 σ6
σ2
σ2
σ1 σ3
σ5
σ4
3142
σ4 4213
σ1
4321
σ6
2413
σ1
3241
σ3
σ2
Figure 5: The Rauzy induction graph for the Rauzy class of (4321).
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4132
σ5
σ2
σ6
σ5 3142
4321
σ6
σ4 4213
σ3
Figure 6: The primitivity subgraph G1 for the Rauzy class of (4321).
2431
4132
σ2
σ2
σ1 σ4
σ4 4213
σ1
4321
σ6
2413
σ1
3241
σ3
Figure 7: The primitivity subgraph G2 for the Rauzy class of (4321).
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2431
4132
σ5 σ6
σ2
σ1
σ5 3142
σ1
4321
σ6
σ1
σ4 4213
3241
σ2
Figure 8: The primitivity subgraph G3 for the Rauzy class of (4321).
2431
σ2
σ1 σ3 σ4 σ1
4321
2413
σ1
3241
σ2
Figure 9: The primitivity subgraph G4 for the Rauzy class of (4321).
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References [1] B. Adamczewski, Codages de rotations et ph´enom`enes d’autosimilarit´e, to appear in J. Th´eor. Nombres Bordeaux. [2] P. Arnoux, S. Ferenczi and P. Hubert, Trajectories of rotations, Acta Arith. 87, 209–217 (1999). [3] J. Bellissard, B. Iochum, E. Scoppola and D. Testard, Spectral properties of one-dimensional quasi-crystals, Commun. Math. Phys. 125, 527–543 (1989). [4] J. Bellissard, B. Iochum and D. Testard, Continuity properties of the electronic spectrum of 1D quasicrystals, Commun. Math. Phys. 141, 353–380 (1991). [5] A. Bovier and J.-M. Ghez, Spectral properties of one-dimensional Schr¨ odinger operators with potentials generated by substitutions, Commun. Math. Phys. 158, 45–66 (1993); Erratum: Commun. Math. Phys. 166, 431–432 (1994). [6] M. Casdagli, Symbolic dynamics for the renormalization group of a quasiperiodic Schr¨ odinger equation, Commun. Math. Phys. 107, 295–318 (1986). [7] J. Cassaigne, Sequences with grouped factors, Developments in Language Theory III, Aristotle University of Thessaloniki, 211–222 (1998). [8] J. Cassaigne, private communication [9] D. Damanik, Gordon-type arguments in the spectral theory of one-dimensional quasicrystals, in Directions in Mathematical Quasicrystals, M. Baake, R. V. Moody, eds., CRM Monograph Series 13, AMS, Providence, RI, 277–305 (2000). [10] D. Damanik, R. Killip and D. Lenz, Uniform spectral properties of onedimensional quasicrystals, III. α-continuity, Commun. Math. Phys. 212, 191– 204 (2000). [11] D. Damanik and D. Lenz, Uniform spectral properties of one-dimensional quasicrystals, I. Absence of eigenvalues, Commun. Math. Phys. 207, 687–696 (1999). [12] D. Damanik and D. Lenz, Uniform spectral properties of one-dimensional quasicrystals, II. The Lyapunov exponent, Lett. Math. Phys. 50, 245–257 (1999). [13] D. Damanik and D. Lenz, Uniform spectral properties of one-dimensional quasicrystals, IV. Quasi-Sturmian potentials, preprint, Caltech and TU Chemnitz (2001). [14] D. Damanik and D. Lenz, Linear repetitivity, I. Uniform subadditive ergodic theorems and applications, Discrete Comput. Geom. 26, 411–428 (2001).
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[15] D. Damanik and D. Zare, Palindrome complexity bounds for primitive substitution sequences, Discrete Math. 222, 259–267 (2000). [16] F. Delyon and D. Petritis, Absence of localization in a class of Schr¨ odinger operators with quasiperiodic potential, Commun. Math. Phys. 103, 441–444 (1986). [17] F. Durand, B. Host and C. Skau, Substitutional dynamical systems, Bratteli diagrams and dimension groups, Ergodic Theory Dynam. Systems 19, 953–993 (1999). [18] F. Durand, A characterization of substitutive sequences using return words, Discrete Math. 179, 89–101 (1998). [19] F. Durand, Linearly recurrent subshifts have a finite number of non-periodic subshift factors, Ergodic Theory Dynam. Systems 20, 1061–1078 (2000). [20] F. Durand, Corrigendum and appendum to: Linearly recurrent subshifts have a finite number of non-periodic subshift factors, preprint, LAMFA, Facult´e de Math´ematiques et d’Informatique, Universit´e de Picardie Jules Verne (2001). [21] S. Ferenczi, Rank and symbolic complexity, Ergodic Theory Dynam. Systems 16, 663–682 (1996). [22] S. Ferenczi, C. Holton and L. Zamboni, Structure of three interval exchange transformations. I. An arithmetic study, Ann. Inst. Fourier (Grenoble) 51, 861–901 (2001). [23] A. Hof, Some remarks on discrete aperiodic Schr¨ odinger operators, J. Statist. Phys. 72, 1353–1374 (1993). [24] A. Hof, O. Knill and B. Simon, Singular continuous spectrum for palindromic Schr¨ odinger operators, Commun. Math. Phys. 174, 149–159 (1995). [25] M. H¨ ornquist and M. Johansson, Singular continuous electron spectrum for a class of circle sequences, J. Phys. A 28, 479–495 (1995). [26] B. Iochum, L. Raymond and D. Testard, Resistance of one-dimensional quasicrystals, Physica A 187, 353–368 (1992). [27] B. Iochum and D. Testard, Power law growth for the resistance in the Fibonacci model, J. Stat. Phys. 65, 715–723 (1991). [28] S. Jitomirskaya, Singular spectral properties of a one-dimensional discrete Schr¨ odinger operator with quasiperiodic potential, Adv. Sov. Math. 3, 215– 254 (1991). [29] M. Kaminaga, Absence of point spectrum for a class of discrete Schr¨ odinger operators with quasiperiodic potential, Forum Math. 8, 63–69 (1996).
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[30] M. Keane, Interval exchange transformations, Math. Z. 141, 25–31 (1975). [31] S. Kotani, Jacobi matrices with random potentials taking finitely many values, Rev. Math. Phys. 1, 129–133 (1989). [32] J. C. Lagarias and P. A. B. Pleasants, Repetitive Delone sets and quasicrystals, to appear in Ergodic Theory Dynam. Systems [33] Y. Last and B. Simon, Eigenfunctions, transfer matrices and absolutely continuous spectrum of one-dimensional Schr¨ odinger operators, Invent. Math. 135, 329–367 (1999). [34] D. Lenz, Uniform ergodic theorems on subshifts over a finite alphabet, Ergodic Theory Dynam. Systems 22, 245–255 (2002). [35] D. Lenz, Singular spectrum of Lebesgue measure zero for quasicrystals, Commun. Math. Phys. 227, 119–130 (2002). ´ [36] G. Rauzy, Echanges d’intervalles et transformations induites, Acta Arith. 34, 315–328 (1979). [37] G. Rote, Sequences with subword complexity 2n, J. Number Theory 46, 196– 213 (1994). [38] B. Solomyak, Nonperiodicity implies unique composition for self-similar translationally finite tilings, Discrete Comput. Geom. 20, 265–279 (1998). [39] A. S¨ ut˝ o, The spectrum of a quasiperiodic Schr¨ odinger operator, Commun. Math. Phys. 111, 409–415 (1987). [40] A. S¨ ut˝ o, Singular continuous spectrum on a Cantor set of zero Lebesgue measure for the Fibonacci Hamiltonian, J. Stat. Phys. 56, 525–531 (1989).
Boris Adamczewski Institut de Mathmatique de Luminy CNRS UPR 9016 13288 Marseille cedex 09 France email: [email protected]
Communicated by Jean Bellissard submitted 31/05/02, accepted 11/07/02
David Damanik Department of Mathematics 253–37 California Institute of Technology Pasadena, CA 91125 USA email: [email protected]
Ann. Henri Poincar´e 3 (2002) 1049 – 1111 c Birkh¨ auser Verlag, Basel, 2002 1424-0637/02/0601049-63
Annales Henri Poincar´ e
Kasner-Like Behaviour for Subcritical Einstein-Matter Systems T. Damour, M. Henneaux, A.D. Rendall and M. Weaver Abstract. Confirming previous heuristic analyses ` a la Belinskii-Khalatnikov-Lifshitz, it is rigorously proven that certain “subcritical” Einstein-matter systems exhibit a monotone, generalized Kasner behaviour in the vicinity of a spacelike singularity. The D-dimensional coupled Einstein-dilaton-p-form system is subcritical if the dilaton couplings of the p-forms belong to some dimension-dependent open neighborhood of zero [1], while pure gravity is subcritical if D ≥ 11 [13]. Our proof relies, like the recent Theorem [15] dealing with the (always subcritical [14]) Einstein-dilaton system, on the use of Fuchsian techniques, which enable one to construct local, analytic solutions to the full set of equations of motion. The solutions constructed are “general” in the sense that they depend on the maximal expected number of free functions.
1 Introduction 1.1
The problem
In recent papers [1, 2, 3], the dynamics of the coupled Einstein-dilaton-p-form system in D spacetime dimensions, with action (in units where 8πG = 1), (j)
S[gαβ , φ, Aγ1 ···γnj ] = SE [gαβ ] + Sφ [gαβ , φ] +
k
(j)
Sj [gαβ , φ, Aγ1 ···γnj ] + “more”,
(1.1)
j=1
√ (1.2) R −g dD x, √ 1 (1.3) Sφ [gαβ , φ] = − ∂µ φ ∂ µ φ −g dD x, 2 √ 1 (j) (j) Fµ1 ···µnj +1 F (j) µ1 ···µnj +1 eλj φ −g dD x, Sj [gαβ , φ, Aγ1 ···γnj ] = − 2(nj + 1)! (1.4) SE [gαβ ] =
1 2
was investigated in the vicinity of a spacelike (“cosmological”) singularity along the lines initiated by Belinskii, Khalatnikov and Lifshitz (BKL) [4]. In (1.1), gαβ is the spacetime metric, φ is a massless scalar field known as the “dilaton”, while the (j) Aγ1 ···γnj are a collection of k exterior form gauge fields (j = 1, . . . , k), with exponential couplings to the dilaton, each coupling being characterized by an individual
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constant λj (“dilaton coupling constant”). The F (j) ’s are the exterior derivatives F (j) = dA(j) , whereas “more” stands for possible coupling terms among the pforms which can be either of the Yang-Mills type (1-forms), Chern-Simons type [5] or Chapline-Manton type [6, 7]. The degrees of the p-forms are restricted to be smaller than or equal to D − 2 since a (D − 1)-form (or D-form) gauge field carries no local degree of freedom. In particular, scalars (nj = 0) are allowed among the A(j) ’s but we then require that the corresponding dilaton coupling λj be non-zero, so that there is only one “dilaton”. Similarly, we require λj = 0 for the (D − 2)forms (if any), since these are “dual”1 to scalars. This restriction to a single dilaton is mostly done for notational convenience: if there were other dilatons among the 0-forms, then, these must be explicitly treated on the same footing as φ and separated off from the p-forms because they play a distinct rˆ ole. In particular, they would appear explicitly in the generalized Kasner conditions given below and in the determination of what we call the subcritical domain. The discussion would proceed otherwise in the same qualitative way. The main motivation for studying actions of the class (1.1) is that these arise as bosonic sectors of supergravity theories related to superstring or M-theory. In fact, in view of various no-go theorems, p-form gauge fields appear to be the only massless, higher spin fields that can be consistently coupled to gravity. Furthermore, there can be only one type of graviton [8]. With this observation in mind, the Action (1.1) is actually quite general. The only restriction concerns the scalar sector: we assume the coupling to the dilaton to be exponential because this corresponds to the tree-level couplings of the dilaton field of string theory. Note, however, that string-loop effects are expected to generate more general couplings exp(λφ) → B(φ) which can exhibit interesting “attractor” behaviours [9]. We also restrict ourselves by not including scalar potentials; see, however, the end of the article for some remarks on the addition of a potential for the dilaton, which can be treated by our methods. Two possible general, “competing” behaviours of the fields in the vicinity of the spacelike singularity have been identified2 : 1. The simplest is the “generalized Kasner behaviour”, in which the spatial scale factors and the field exp(φ) behave at each spatial point in a monotone, power-law fashion in terms of the proper time as one approaches the singularity, while the effect of the p-form fields A(j) ’s on the evolution of gµν and φ can be asymptotically neglected. In that regime the spatial curvature terms can be also neglected with respect to the leading order part of the extrinsic curvature terms. In other words, as emphasized by BKL, time derivatives asymptotically dominate over space derivatives so that one sometimes uses the terminology “velocity-dominated” behaviour [11], instead of “general1 We
recall that the Hodge duality between a (nj + 1)-form and a (D − nj − 1)-form allows one to replace (locally) a nj -form potential A(j) by a (D − nj − 2)-form potential A(j ) (with dilaton coupling λj = −λj ). 2 For a recent extension of these ideas to the brane-worlds scenarios, see [10].
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ized Kasner behaviour”. We shall use both terminologies indifferently in this paper, recalling that in the presence of p-forms, which act as potentials for the evolution of the spatial metric and the dilaton (as do the spatial curvature terms), “velocity-dominance” means not only that the spatial curvature terms can be neglected, but also that the p-forms can be neglected in the Einstein-dilaton evolution equations.3 2. The second regime, known as “oscillatory” [4], or “generalized mixmaster” [12] behaviour, is more complicated. It can be described as the succession of an infinite number of increasingly shorter Kasner regimes as one goes to the singularity, one following the other according to a well-defined “collision” law. This asymptotic evolution is presumably strongly chaotic. It is expected that, at each spatial point, the scale factors of a general inhomogeneous solution essentially behave as in certain homogeneous models. For instance, for D = 4 pure gravity this guiding homogeneous model is the Bianchi IX model [4, 12], while for D = 11 supergravity it is its naive one-dimensional reduction involving space-independent metric and three-forms [2]. Whether it is the first or the second behaviour that is relevant depends on: (i) the spacetime dimension D, (ii) the field content (presence or absence of the dilaton, types of p-forms), and (iii) the values of the various dilaton couplings λj . Previous work reached the following conclusions: • The oscillatory behaviour is general for pure gravity in spacetime dimension 4 [4], in fact, in all spacetime dimensions 4 ≤ D ≤ 10, but is replaced by a Kasner-like behaviour in spacetime dimensions D ≥ 11 [13]. (The sense in which we use “general” will be made precise below.) • The Kasner-like behaviour is general for the gravity-dilaton system in all spacetime dimensions D ≥ 3 (see [14, 15] for D = 4). • The oscillatory behaviour is general for gravity coupled to p-forms, in absence of a dilaton or of a dual (D − 2)-form (0 < p < D − 2) [2]. In particular, the bosonic sector of 11-dimensional supergravity is oscillatory [1]. Particular instances of this case have been studied in [16, 17, 18]. • The case of the gravity-dilaton-p-form system is more complicated to discuss because its behaviour depends on a combination of several factors, namely the dimension D, the menu of p-forms, and the numerical values of the dilaton couplings. For a given D and a given menu of p-forms there exists a “subcritical” domain D (an open neighborhood of the origin λj = 0 for all j’s) such that: (i) when the λj belong to D the general behaviour is Kasner3 The
Kasner solution is generalized in two ways: first, the original Kasner exponents include a dilaton exponent (if there is a dilaton), which appears in the Kasner conditions; second, the exponents are not assumed to be constant in space. We shall shorten “exhibits generalized Kasner behaviour” to Kasner-like. We stress that we do not use this term to indicate that the solution becomes asymptotically homogeneous in space.
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like, but (ii) when the λj do not belong to D the behaviour is oscillatory. Note that D is open. Indeed, the behaviour is oscillatory when the λj are on the boundary of D, as happens for instance for the low-energy bosonic sectors of type I or heterotic superstrings [1]. For a single p-form, the subcritical domain D takes the simple form |λj | < λcj , where λcj depends on the formdegree and the spacetime dimension. (λcj can be infinite.) For a collection of p-forms, D is more complicated and not just given by the Cartesian product of the subcritical intervals associated with each individual p-form. The above statements were derived by adopting a line of thought analogous to that followed by BKL. Now, as understood by BKL themselves, these arguments, although quite convincing, are somewhat heuristic. It is true that the original arguments have received since then a considerable amount of both numerical and analytical support [19, 20, 21, 22, 23]. Yet, they still await a complete proof. One notable exception is the four-dimensional gravity-dilaton system, which has been rigorously demonstrated in [15] to be indeed Kasner-like, confirming the original analysis [14]. Using Fuchsian techniques, the authors of [15] have proven the existence of a local (analytic) Kasner-like solution to the Einstein-dilaton equations in four dimensions that contains as many arbitrary, physically relevant functions of space as there are local degrees of freedom, namely 6 (counting q and q˙ independently). To our knowledge, this was the first construction, in a rigorous mathematical sense, of a general singular solution for a coupled Einstein-matter system. Note in this respect several previous works in which formal solutions had been constructed near (Kasner-like) cosmological singularities by explicit perturbative methods, to all orders of perturbation theory [24, 25]. The situation concerning the more complicated (and in some sense more interesting) generalized mixmaster regime is unfortunately – and perhaps not surprisingly – not so well developed. Rigorous results are scarce (note [26]) and even in the case of the spatially homogeneous Bianchi IX model only partial results exist in the literature [27]. The purpose of this paper is to extend the Fuchsian approach of [15] to the more complicated class of models described by the Action (1.1) and to prove that those among the above models that were predicted in [13, 1, 2] to be Kasner-like are indeed so. This provides many new instances where one can rigorously construct a general singular solution for a coupled Einstein-matter (or pure Einstein, in D ≥ 11) system. In fact, our (Fuchsian-system-based) results prove that the formal perturbative solutions that can be explicitly built for these models do converge to exact solutions. This provides a further confirmation of the general validity of the BKL ideas. We shall also explicitly determine the subcritical domain D for a few illustrative models. For all the relevant systems, we construct local (near the singularity) analytic solutions, which are “general” in the sense that they contain the right number of freely adjustable arbitrary functions of space (in particular, these solutions have generically no isometries), and which exhibit the generalized (monotone) Kasner time dependence.
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Strategy and outline of the paper
Our approach is the same as in [15], and results from that work will be used frequently here without restating the arguments. Here is an outline of the key steps. A d + 1 decomposition is used, for d spatial dimensions, d = D − 1. A Gaussian time coordinate, t, is chosen such that the singularity occurs at t = 0. The first step in the argument consists of identifying the leading terms for all the variables. This is accomplished by writing down a set of evolution equations which is obtained by truncating the full evolution equations, and then solving this simpler set of evolution equations. This simpler evolution system is called the Kasnerlike4 evolution system (or, alternatively, the velocity-dominated system). It is a system of ordinary differential equations with respect to time (one at each spatial point) which coincides with the system that arises when investigating metricdilaton solutions that depend only on time. The precise truncation rules are given in Subsection 2.2 below. The second step is to write down constraint equations for the Kasner-like system (called “velocity-dominated” constraints) and to show that these constraints propagate, i.e., that if they are satisfied by a solution to the Kasner-like evolution equations at some time t0 > 0, then they are satisfied for all time t > 0. In the set of Kasner-like solutions, one expects that there is a subset, denoted by V , of solutions which have the property of being asymptotic to solutions of the complete Einstein-dilaton-p-form equations as t → 0, i.e., as one goes to the singularity. This subset is characterized by inequalities on some of the initial data, which, however, are not always consistent. The existence of a non-empty V requires the dilaton couplings to belong to some range, the “subcritical range”. When V is non-empty and open, the solutions in V involve as many arbitrary functions of space as a “general solution” of the full Einstein equations should. On the other hand, if V is empty the construction given in this paper breaks down and the dynamical system is expected to be not Kasner-like but rather oscillatory. To show that indeed, the solutions in V (when it is non-empty) are asymptotic to true solutions, the third step is to identify decaying quantities such that these decaying quantities along with the leading terms mentioned above uniquely determine the variables, and to write down a Fuchsian system for the decaying quantities which is equivalent to the Einstein-matter evolution system. As the use of Fuchsian systems is central to our work let us briefly recall what a Fuchsian system is and how such a system is related to familiar iterative methods. For a more detailed introduction to Fuchsian techniques see [15, 28, 29, 30] and references therein. Note that we shall everywhere restrict ourselves to the analytic case. We expect that our results extend to the C ∞ case, but it is a non-trivial task to prove that they do. 4 Note that we use the terms “Kasner-like solutions” to label both exact solutions of the truncated system and solutions of the full system that are asymptotic to such solutions. Which meaning is relevant should be clear from the context.
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The general form of a Fuchsian system for a vector-valued unknown function u is t ∂t u + A(x) u = f (t, x, u, ux),
(1.5)
where the matrix A(x) is required to satisfy some positivity condition (see below), while the “source term” f on the right-hand side is required to be “regular.” (See [15] for precise criteria allowing one to check when the positivity assumption on A(x) is satisfied and when f is regular.) A key point is that f is required to be bounded by terms of order O(tδ ) (with t → 0, δ > 0) as soon as u and their space derivatives ux are in a bounded set (a simple, concrete example of a source term satisfying this condition is f = tδ1 + tδ2 u + tδ3 ux , with δi ’s larger than δ). A convenient form of positivity condition to be satisfied by the matrix A(x) is that the operator norm of τ A(x) be bounded when 0 < τ < 1 (and when x varies in any open set). Essentially this condition restricts the eigenvalues of the matrix A(x) to have positive real parts. The basic property of Fuchsian systems that we shall use is that there is a unique solution to the Fuchsian equation which vanishes as t tends to zero [28]. One can understand this theorem as a mathematically rigorous version of the recursive method for solving the Equation (1.5). Indeed, when confronted with Equation (1.5), it is natural to construct a solution by an iterative process, starting with the zeroth order approximation u0 = 0 (which is the unique solution of (1.5) with f ≡ 0 that tends to zero as t → 0), and solving (n−1) a sequence of equations of the form t∂t u(n) + A(x)u(n) = f (t, x, u(n−1) , ux ). At each step in this iterative process the source term is a known function which essentially behaves (modulo logarithms) like a sum of powers of t (with spacedependent coefficients). The crucial step in the iteration is then to solve equations of the type t∂t u + A(x)u = C(x)tδ(x) . The positivity condition on A(x) guarantees the absence of homogeneous solutions remaining bounded as t → 0, and ensures the absence of “small denominators” in the (unique bounded) inhomogeneous solution generated by each partial source term: uinhom = (δ + A)−1 Ctδ . (See, e.g., [25] for a concrete iterative construction of a Kasner-like solution and the proof that it extends to all orders.) This link between Fuchsian systems and “good systems” that can be solved to all orders in a formal iteration makes it a priori probable that all cases which the heuristic approach `a la BKL has shown to be asymptotic to a Kasner-like solution (by checking that the leading “post-Kasner” contribution is asymptotically sub-dominant) can be cast in a Fuchsian form. The main technical burden of the present work will indeed be to show in detail how this can be carried out for the evolution systems corresponding to all the sub-critical (i.e., non-oscillatory) Einstein-matter systems. Our Fuchsian formulation proves that (in the analytic case) the formal all-orders iterative solutions for the models we consider do actually converge to the unique, exact solution having a given leading Kasner asymptotic behaviour as t → 0. Finally, the fourth step of our strategy is to prove that the constructed solution does satisfy also all the Einstein and Gauss-like constraints so that it is
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a solution of the full set of Einstein-matter equations. We shall deal successively with the matter (Gauss-like) constraints, and the Einstein constraints. Our paper is organized as follows. In Section 2, we first consider the paradigmatic example of gravity coupled to a massless scalar field and to a Maxwell field in 4 spacetime dimensions. The Action (1.1) reads in this case S[gαβ , φ, Aγ ] =
1 2
√ 1 {R − ∂µ φ ∂ µ φ − Fµν F µν eλφ } −g d4 x. 2
(1.6)
For this simple example, we shall explicitly determine the subcritical domain D, i.e., the critical value λc such that the system is Kasner-like when −λc < λ < λc . Because this case is exemplary of the general situation, while still being technically rather simple to handle, we shall describe in some detail the explicit steps of the Fuchsian approach. In Section 3, vacuum solutions governed by the pure Einstein Action (1.2) with D ≥ 11 are considered. This system was argued in [13] to be Kasner-like and we show here how this rigorously follows from the Fuchsian approach. Note that, contrary to what happens when a dilaton is present, Fuchsian techniques apply here even though not all Kasner exponents can be positive. In Sections 4–8, the results of the previous sections are generalized to the wider class of systems (1.1). First, in Section 4, to solutions of Einstein’s equation with spacetime dimension D ≥ 3 and a matter source consisting of a massless scalar field, governed by the action SE [gαβ ] + Sφ [gαβ , φ]. This is the generalization to any D ≥ 3 of the case D = 4 treated in [15]. In Section 5, we turn to the general situation described by the Action (1.1), without, however, including the additional terms represented there by “more”. We then give some general rules for computing the subcritical domain of the dilaton couplings guaranteeing velocity-dominance (Section 6). The inclusion of interaction terms is considered in the last sections. It is shown that they do not affect the asymptotic analysis. This is done first for the Chern-Simons and Chapline-Manton interactions in Section 7, and next, in Section 8, for the Yang-Mills couplings (for some gauge group G), for which the action reads √ 1 1 {R − ∂µ φ ∂ µ φ − Fµν · F µν eλφ } −g dD x. (1.7) S[gαβ , φ, Aγ ] = 2 2 Here the dot product, F · F , is a time-independent, Ad-invariant, non-degenerate scalar product on the Lie algebra of G (such a scalar product exists if the algebra is compact, or semi-simple). Contrary to what is done in Sections 2, 5 and 7, we must work now with the vector potential (and not just with the field strength), since it appears explicitly in the coupling terms. In Section 9 we show that self-interactions of a rather general type for the scalar field can be included without changing the asymptotics of the solutions.
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Explicitly, we add a (nonlinear) potential term, √ SNL [gαβ , φ] = − V (φ) −g dD x,
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(1.8)
to the Action (1.1), where V (φ) must fulfill some assumptions given in Section 9. V (φ) may, for example, be an exponential function of φ, a constant, or a suitable power of φ. Similar forms for V (φ) were considered with D = 4 in [31]. Finally, in Section 10, we state two theorems that summarize the main results of the paper and give concluding remarks.
1.3
On the generality of our construction
As we shall see the number of arbitrary functions contained in solutions to the velocity-dominated constraint equations is equal to the number of arbitrary functions for solutions to the Einstein-matter constraints. In this function-counting sense, our construction describes what is customarily called a “general” solution of the system. Intuitively speaking, our construction concerns some “open set” of the set of all solutions (indeed, the Kasner-like behaviour of the solution is unchanged under arbitrary, small perturbations of the initial data, because this simply amounts to changing the integration functions). Note that, in the physics literature, such a “general” solution is often referred to as being a “generic” solution. However, in the mathematics literature the word “generic” is restricted to describing either an open dense subset of the set of all solutions, or (when this can be defined) a subset of measure unity of the set of all solutions. In this work we shall stick to the mathematical terminology. We shall have nothing rigorous to say about whether our general solution is also generic. However, we wish to emphasize the following points. First, let us mention that the set V of solutions to the velocity-dominated equations that are asymptotic to solutions of the complete equations is not identical to the set U of all solutions to the velocity-dominated constraint equations. The subset V ⊂ U is defined by imposing some inequalities on the free data. These inequalities do not change the number of free functions. Therefore the solutions in V are still “general”. One can wonder whether there could be a co-existing general behaviour, corresponding to initial data that do not fulfill the inequalities. For instance, could such “bad” initial data lead to a generalized mixmaster regime? This is a difficult question and we shall only summarize here what is the existing evidence. There are heuristic arguments, supported by numerical study, [14, 32, 33, 34] that suggest that if one starts with initial data that do not fulfill the inequalities, one ends up, after a finite transient period (with a finite number of “collisions” with potential walls), with a solution that is asymptotically velocity-dominated, for which the inequalities are fulfilled almost everywhere. In that sense, the inequalities would not represent a real restriction since there is a dynamical mechanism that drives the solution to the regime where they are
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satisfied. For the subcritical values of the dilaton couplings that make the inequalities defining V consistent, there is thus no evidence for an alternative oscillatory regime corresponding to a different (open) region in the space of initial data5 . It has indeed been shown that the inequalities defining V are no restriction in a large spatially homogeneous class [27]. Such rigorous results are, however, lacking in the inhomogeneous case. In fact, an interesting subtlety might take place in the inhomogeneous case. The heuristic arguments and numerical studies of [33, 34] suggest the possibility that the mechanism driving the system to V may be suppressed at exceptional spatial points in general spacetimes, with the result that the asymptotic data at the exceptional spatial points are not consistent with the inequalities we assume and lead to so-called “spikes”. This picture has been given a firm basis in a scalar field model with symmetries [35] but the status of the spikes in a general context remains unclear. Finally, since we only deal with spacelike singularities, the classes of solutions we consider do not contain all solutions governed by the Action (1.1). Other types of singularities (e.g. timelike or null ones) are known to exist. Whether these other types of singularities are general is, however, an open question.
1.4
Billiard picture
At each spatial point, the solution of the coupled Einstein-matter system can be pictured, in the vicinity of a spacelike singularity, as a billiard motion in a region of hyperbolic space [36, 37, 3, 38]. Hyperbolic billiards are chaotic when they have finite volume and non chaotic otherwise. In this latter case, the “billiard ball” generically escapes freely to infinity after a finite number of collisions with the bounding walls. Subcritical Einstein-matter systems define infinite-volume billiards. The velocity-dominated solutions correspond precisely to the last (as t → 0) free motion (after all collisions have taken place), in which the billiard ball moves to infinity in hyperbolic space.
1.5
Conventions
We adopt a “mostly plus” signature (− + + + . . .). The spacetime dimension is D ≡ d + 1. Greek indices range from 0 to d, while Latin indices ∈ {1, . . . , d}. The spatial Ricci tensor is labeled R and the spacetime Ricci tensor is labeled (D) R. Our curvature conventions are such that the Ricci tensor of a sphere is positive definite. Einstein’s equations read Gαβ = Tαβ , where Gαβ = Rαβ − Rgαβ /2 denotes the√Einstein tensor and Tαβ denotes the matter stress-energy tensor, Tαβ = −(2/ −g)δSmatter /δg αβ , and units such that 8πG = 1. The spatial metric compatible covariant derivative is labeled ∇a and the spacetime metric compatible covariant derivative is labeled (D) ∇α . The velocity-dominated metric compatible 5 The oscillatory regime may however be present for peculiar initial data, presumably forming a set of zero measure. For instance, gravity + dilaton is generically Kasner-like, but exhibits an oscillatory behaviour for initial data with φ = 0 (in D < 11 spacetime dimensions).
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covariant derivative is labeled 0 ∇a . According to the context, g denotes the (positive) determinant of gab in d + 1-decomposed expressions, and the (negative) determinant of gµν in spacetime expressions. Whenever tδ or t−δ appears, δ denotes a strictly positive number, arbitrarily small. We use Einstein’s summation convention on repeated tensor indices of different variances. (When the need arises to suspend the summation conventions for some non-tensorial indices, we shall explicitly mention it.) In expressions where there is a sum that the indices do not indicate, all sums in the expression are indicated explicitly by a summation symbol. Indices on the velocity-dominated metric and the velocity-dominated extrinsic curvature are raised and lowered with the velocity-dominated metric.
1.6
d + 1 decomposition
Consider a solution to the Einstein’s equations following from (1.1), consisting of a Lorentz metric and matter fields on a D-dimensional manifold M which is diffeomorphic to (0, T ) × Σ for a d-dimensional manifold, Σ, such that the metric induced on each t = constant hypersurface is Riemannian, for t ∈ (0, T ). Here D is an integer strictly greater than two. Furthermore, consider a d + 1 decomposition of the Einstein tensor, Gαβ , and the stress-energy tensor, Tαβ , with a Gaussian time coordinate, t ∈ (0, T ), and a local frame {ea } on Σ. Note that the frame ea = eia (x)∂i is time-independent. The spacetime metric reads ds2 = −dt2 + gab (t, x)ea eb , where ea = eai (x)dxi (with eai eib = δba ) is the co-frame. Let ρ = T00 , ja = −T0a and Sab = Tab . Define C
=
2G00 − 2T00
=
−k
a
b
k
b
a
(1.9) 2
+ (tr k) + R − 2ρ.
C = 0 is the Hamiltonian constraint. Similarly, Ca = 0 is the momentum constraint, where Ca
= −G0a + T0a = ∇b k
b
a
(1.10)
− ∇a (tr k) − ja .
In Gaussian coordinates, the relation between the metric and the extrinsic curvature is ∂t gab = −2kab . (1.11) The evolution equation for the extrinsic curvature is obtained by setting E a b = 0, with 1 T δab (1.12) E a b = (D) Ra b − T a b + (D − 2) (1.13) ⇒ ∂t k a b = Ra b + (tr k) k a b − M a b . Here M ab = Sab −
1 ((tr S) − ρ)δ a b . D−2
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2 Scalar and Maxwell fields in four dimensions 2.1
Equations of motion
As said above, let us start by considering in detail, as archetypal system, the system defined by the Action (1.6), i.e., the spacetime dimension is D = 4 and the matter fields are a massless scalar field exponentially coupled to a Maxwell field, with the magnitude of the dilaton coupling constant smaller in magnitude than some positive real number determined below, 0 ≤ |λ| < λc . The stress-energy tensor of the matter fields is 1 1 Tµν = (4) ∇µ φ (4) ∇ν φ − gµν (4) ∇α φ (4) ∇α φ + [Fµα Fν α − gµν Fαβ F αβ ]eλφ . 2 4 The matter fields satisfy the following equations. (4) (4)
∇α (4) ∇α φ =
∇µ (F µν eλφ ) = (4)
∇[α Fβγ]
=
λ Fαβ F αβ eλφ , 4 0, 0.
The 3+1 decomposition of the stress-energy tensor is best expressed in terms of the √ electric spatial vector density E a = g F 0a eλφ and the magnetic antisymmetric spatial tensor Fab . 1 1 1 {(∂t φ)2 + g ab ea (φ)eb (φ) + gab E a E b e−λφ + g ab g ch Fac Fbh eλφ }, 2 g 2 1 ja = −∂t φ ea (φ) + √ E b Fab , g 1 1 M a b = g ac eb (φ) ec (φ) − {gbc E a E c − δ a b gch E c E h }e−λφ g 2 1 a ch ij ac hi +{g g Fch Fbi − δ b g g Fci Fhj }eλφ . (2.1) 4 The matter constraint equations are ρ
=
b ea (E a ) + fba Ea h Fc]h e[a (Fbc] ) + f[ab
= =
0 0.
(2.2) (2.3)
c are the (time-independent) structure functions of the frame, [ea , eb ] = Here fab c fab ec . The matter evolution equations are
λ λ gab E a E b e−λφ − g ab g ch Fac Fbh eλφ , 2g 4 √ 1 a ic √ bh i ac ∂t E a = eb ( gg ac g bh Fch eλφ ) + (fib g + fbi g ) gg Fch eλφ , 2 1 c 1 ∂t Fab = −2e[a ( √ gb]c E c e−λφ ) + fab √ gch E h e−λφ . g g
∂t2 φ − (trk)∂t φ = g ab ∇a ∇b φ +
(2.4) (2.5) (2.6)
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Ann. Henri Poincar´e
Velocity-dominated evolution equations and solution
The Kasner-like, or velocity-dominated, evolution equations are obtained from the full evolution equations by: (i) dropping the spatial derivatives from the rightc -terms count as derivatives hand sides of (1.13), (2.4), (2.5) and (2.6) (note that fab and that we keep the time derivatives of the magnetic field in (2.6) even though Fab = ∂a Ab − ∂b Aa ); and (ii) dropping the p-form terms in both the Einstein and dilaton evolution equations. This is a general rule and yields in this case ∂t 0 gab ∂t 0 k a b
= =
∂t2 0 φ − (tr 0 k) ∂t 0 φ = 0 a
∂t E ∂t 0 Fab
= =
−2 0 kab , (tr 0 k) 0 k a b ,
(2.7) (2.8)
0,
(2.9)
0, 0.
(2.10) (2.11)
(As we shall see below, interaction terms of Yang-Mills or other types – if any – should also be dropped.) It is easy to find the general analytic solution of the evolution system (2.7)– (2.11) since the equations are the same as for “Bianchi type I” homogeneous models (one such set of equations per spatial point). Taking the trace of (2.8) shows that −1/tr 0 k = t+C(x). By a suitable redefinition of the time variable one can set C(x) to zero. Then (2.8) shows that −t 0 k a b ≡ K a b is a constant matrix (which must satisfy trK = K a a = 1, and be such that 0 gac (t0 )K c b is symmetric in a and b), 0 a
k b (t) = −t−1 K a b .
(2.12)
Injecting this information into (2.7) leads to a linear evolution system for 0 gab : t ∂t 0 gab = 2 0 gac K c b , which is solved by exponentiation, c 2K t 0 0 gab (t) = gac (t0 ) . (2.13) t0 b
The other evolution equations are also easy to solve, 0
φ(t) E (t)
= =
Fab (t)
=
0 a 0
A ln t + B, E
0 a 0
Fab .
(2.14) (2.15) (2.16)
In (2.13) (t/t0 )2K denotes the exponentiation of a matrix. Quantities on the lefthand side of (2.12)–(2.16) may be functions of both time and space, while all the time dependence of the right-hand side is made explicit. For instance, (2.16) is saying that the spacetime dependence of the general magnetic field 0 Fab (t, x) (solution of the velocity-dominated evolution system) is reduced to a simple space dependence 0 Fab (x) (where 0 Fab is an antisymmetric spatial tensor). Let pa denote the eigenvalues of K a b , ordered such that p1 ≤ p2 ≤ p3 . Since trK = 1, we have
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the constraint p1 + p2 + p3 = 1.
(2.17)
In the works of BKL the matrix Solution (2.13) is simplified by using a special frame {ea } with respect to which the matrices 0 gab (t0 ) and K a b are diagonal. However, as emphasized in [15], this choice can not necessarily be made analytically on neighborhoods where the number of distinct eigenvalues of K a b is not constant. To obtain an analytic solution, while still controlling the relation of the solution to the eigenvalues of K a b , a special construction was introduced in [15]. This construction is based on some (possibly small) neighborhood U0 of an arbitrary spatial point x0 ∈ Σ and uses a set of auxiliary exponents qa (x). These auxiliary exponents remain numerically close to the exact “Kasner exponents” pa (x), are analytic and enable one to define an analytic frame (see below). To construct the auxiliary exponents qa (x) one distinguishes three cases: Case I (near isotropic): If all three eigenvalues are equal at x0 , choose a number > 0 so that for x ∈ U0 , maxa,b |pa (x) − pb (x)| < /2. In this case define qa = 1/3 on U0 , a = 1, 2, 3. Case II (near double eigenvalue): If the number of distinct eigenvalues at x0 is two, choose > 0 so that for x ∈ U0 , maxa,b |pa − pb | > /2, and |pa − pb | < /2 for some pair, a , b , a = b , shrinking U0 if necessary. Denote by p⊥ the distinguished exponent not equal to pa , pb . In this case define q⊥ = p⊥ and qa = qb = (1 − q⊥ )/2 on U0 . Case III (near diagonalizable): If all eigenvalues are distinct at x0 , choose > 0 so that for x ∈ U0 , min a,b |pa (x) − pb (x)| > /2, shrinking U0 if necessary. a=b
In this case define qa = pa on U0 . The frame {ea }, called the adapted frame, is required to be such that the related (time-dependent) frame {˜ ea (t) ≡ t−qa ea } is orthonormal with respect to the a velocity-dominated metric at some time t0 > 0, i.e., such that 0 gab (t0 ) = t2q 0 δab . (Here and in the rest of the paper, the Einstein summation convention does not apply to indices on qa and pa . These indices should be ignored when determining sums. Furthermore, quantities with a tilde will refer to the frame {˜ ea (t)}.) In addition, in Case II it is required that e⊥ be an eigenvector of K corresponding to q⊥ and that ea , eb span the eigenspace of K corresponding to the eigenvalues pa , pb . In case III it is required that the ea be eigenvectors of K corresponding to the eigenvalues qa (≡ pa ). In all cases it is required that {ea } be analytic. The auxiliary exponents, qa , are analytic, satisfy the Kasner rela tion qa = 1, are ordered (q1 ≤ q2 ≤ q3 ), and satisfy q1 ≥ p1 , q3 ≤ p3 and maxa |qa − pa | < /2. If qa = qb , then 0 gab , 0 g ab , 0 g˜ab and 0 g˜ab all vanish, and the same is true with g replaced by k. Equations (2.12)–(2.16), with the form of gab (t0 ) and K a b specialized as given just above, are the general analytic solution to the velocity-dominated evolution equations in the sense that any analytic solution to the velocity-dominated evolution equations takes this form near any x0 ∈ Σ by choice of (global) time coordinate and (local) spatial frame.
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Velocity-dominated constraint equations
When written in terms of the velocity-dominated variables, the velocity-dominated constraints take the same form as the full constraint equations, except the Hamiltonian constraint, which is obtained by dropping spatial gradients and electromagnetic contributions to the energy-density. This is a general rule, valid also for the more general models considered below. Thus, if we define 0 0
ρ
ja
=
1 (∂t 0 φ)2 , 2
1 = −∂t 0 φ ea ( 0 φ) +
0g
0 b 0
E
Fab ,
we get 0 C = 0 and 0 Ca = 0 for the velocity-dominated constraints corresponding to the Hamiltonian and momentum constraints, with 0 0
C
=
Ca
=
− 0 k a b 0 k b a + (tr 0 k)2 − 2 0 ρ, 0
∇b 0 k b a − ea (tr 0 k) − 0 ja .
(2.18) (2.19)
The velocity-dominated matter constraint equations read b 0 a ea ( 0 E a ) + fba E
=
0,
h 0 e[a ( 0 Fbc] ) + f[ab Fc]h
=
0.
For the Solution (2.12)–(2.14) the velocity-dominated Hamiltonian constraint equation is equivalent to (2.20) pa 2 + A2 = 1. The conditions (2.17) and (2.20) are the famous Kasner conditions when the dilaton is present. While p1 is necessarily non-positive when A = 0, this is no longer the case when the dilaton is nontrivial (A = 0): all pa ’s can then be positive. This is the major feature associated with the presence of the dilaton, which turns the mixmaster behaviour of (4-dimensional) vacuum gravity into the velocity-dominated behaviour. We shall call (pa , A) the Kasner exponents (because they are the exponents of the proper time in the solution for the scale factors and exp φ) and refer to (2.17) and (2.20) as the Kasner conditions (note that A is often denoted pφ to emphasize its relation to the kinetic energy of φ, and its similarity with the other exponents). A straightforward calculation shows that ∂t 0 C − 2(tr 0 k) 0 C
= 0,
(2.21)
1 = − ea ( 0 C). (2.22) 2 Thus if the velocity-dominated Hamiltonian and momentum constraints are satisfied at some t0 > 0, then they are satisfied for all t > 0. Similarly, since 0 E a and 0 Fab are independent of time, if the matter constraints are satisfied at some time t0 > 0, then they are clearly satisfied for all time. ∂t 0 Ca − (tr 0 k) 0 Ca
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Critical value of dilaton coupling λc
Our ultimate goal is to show that the velocity-dominated solutions asymptotically approach (as t → 0) solutions of the original system of equations. We shall prove that this is the case provided the Kasner exponents pi , A, subject to the Kasner conditions 2 pa 2 − pa = 1 (2.23) pa + A2 = 0, obey additional restrictions. These restrictions are inequalities on the Kasner exponents and read explicitly 2p1 − λA > 0,
p1 > 0,
2p1 + λA > 0.
(2.24)
As explained in [1], and rigorously checked below, these restrictions are necessary and sufficient to ensure that the terms that are dropped when replacing the full Einstein-dilaton-Maxwell equations by the velocity-dominated equations become indeed negligible as t → 0. More precisely, the first condition (respectively the third) among (2.24) guarantees that one can neglect the electric (respectively, magnetic) part of the energy-momentum tensor of the electromagnetic field in the Einstein equations, whereas the condition p1 > 0 is necessary for the spatial curvature terms to be asymptotically negligible. The conditions (2.24) define the set V of velocity-dominated solutions referred to in the introduction. It is clear that if |λ| is small enough – in particular, if λ = 0 – the Inequalities (2.24) can be fulfilled since the Kasner exponents can be all positive when the dilaton is included. But if |λ| is greater that some critical value λc , it is impossible to fulfill simultaneously the Kasner conditions (2.23) and the Inequalities (2.24), because one of the terms ±λA becomes more negative than 2p1 is positive. In that case, the set V is empty and our construction breaks down. For |λ| < λc , however, the set V is non-empty and, in fact, stable under small perturbations of the Kasner exponents since (2.24) defines an open region on the Kasner sphere. We determine in this subsection the critical value λc such that (2.23) and (2.24) are compatible whenever |λ| < λc . To that end, we follow the geometric approach of [3, 39]. In the 4-dimensional space of the Kasner exponents (pa , A), we consider the “wall chamber” W defined to be the conical domain where p1 ≤ p2 ≤ p3 , 2pa − λA ≥ 0, pa ≥ 0, 2pa + λA ≥ 0.
(2.25)
These inequalities are not all independent since the four conditions p1 ≤ p2 ≤ p3 , 2p1 − λA ≥ 0, 2p1 + λA ≥ 0
(2.26)
imply all others. The quadratic Kasner condition (2.23) can be rewritten Gµν pµ pν = 0,
(pµ ) ≡ (pa , A)
where Gµν defines a metric in “Kasner-exponent space”
2 dS 2 = Gµν dpµ dpν = dpa 2 − dpa + (dA)2
(2.27)
(2.28)
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Ann. Henri Poincar´e
The metric (2.28) has Minkowskian signature (−, +, +, +). An example of timelike direction is given by p1 = p2 = p3 , A = 0. Inside or on the light cone, the function pa does not vanish. The upper light cone (in the space of the Kasner exponents) is conventionally defined by (2.27) and the extra condition pa > 0. It is clear from our discussion that the Kasner conditions (2.23) and the Inequalities (2.24) are compatible if and only if there are light like directions in the interior of the pa = 1 wall chamber W (by rescaling pµ → αpµ , α > 0, one can always make for any point in the interior of the wall chamber so that this condition does not bring a restriction). The problem amounts accordingly to determining the relative position of the light cone (2.27) and the wall chamber (2.26). This is most easily done by computing the edges of (2.26), i.e., the onedimensional intersections of three faces among the four faces (2.26) of W. There are four of them: (i) p1 = p2 = A = 0, p3 = α; (ii) p1 = A = 0; p2 = p3 = α; (iii) 2p1 = 2p2 = 2p3 = λA = α; and (iv) 2p1 = 2p2 = 2p3 = −λA = α, where in each case, α ≥ 0 is a parameter along the edge (α = 0 being the origin). The vectors eµA (A = 1, 2, 3, 4) along the edges corresponding to α = 1, namely (0, 0, 1, 0), (0, 1, 1, 0), (1/2, 1/2, 1/2, 1/λ) and (1/2, 1/2, 1/2, −1/λ) form a basis in Kasner-exponent space. Any vector v µ can thus be expanded along the eµA , v µ = v A eµA . A point P in Kasner-exponent space is on or inside the wall chamber W if and only if its coordinates pA in this basis fulfill pA ≥ 0 with P inside when pA > 0 for all A s. Thus, if all the edge vectors eµA are timelike or lightlike, the Kasner conditions are incompatible with the Inequalities (2.24) since any linear combination of causal vectors with non-negative coefficients is on or inside the forward light cone (the eµA ’s are future-directed since p1 + p2 + p3 > 0 for all of them). If, however, one (or more) of the edge vectors lies outside the light cone, then, the Kasner conditions and the Inequalities (2.24) are compatible. The nature of some of the edge vectors depends on the value of the dilaton coupling λ: while the first one is always lightlike and the second one always timelike, the squared norm of the last two is −3/2 + 1/λ2 = (2 − 3λ2 )/(2λ2 ). This determines the critical value 2 (2.29) λc = 3 such that the edge vectors are timelike or null (incompatible inequalities) if |λ| ≥ λc , but spacelike (compatible inequalities) if |λ| < λc . Note that the value of λ that arises from dimensionally reducing 5-dimensional vacuum gravity down to 4 √ dimensions is λ = 6 and exceeds the critical value. This “explains” the conclusion reached in [14] that the gravity-dilaton-Maxwell system obtained by Kaluza-Klein reduction of 5-dimensional gravity is oscillatory. We shall assume from now on that |λ| < λc and that the Kasner exponents fulfill the above inequalities. For later use, we choose a number σ > 0 so that, for all x ∈ U0 , σ < 2p1 −λA, σ < 2p1 +λA and σ < p1 /2. Reduce if necessary so that < σ/7. If is reduced, it may be necessary to shrink U0 so that the conditions imposed in Section 2.2 remain satisfied. In Section 2.5 it is assumed that and U0
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are such that the conditions imposed in Section 2.2 and the conditions imposed in this paragraph are all satisfied.
2.5
Fuchsian system which is equivalent to the Einstein-matter evolution equations
2.5.1 Rewriting of equations Theorem 3 in [15] (Theorem 4.2 in preprint version), on which we rely for our result, states that a Fuchsian equation (i.e., as we mentioned above, an equation of the form (1.5) where A satisfies a positivity condition and f is regular, which includes a boundedness property) has a unique solution u that vanishes as t ↓ 0, and furthermore spatial derivatives of u of any order vanish as t ↓ 0, as shown in [28]. Our goal is to recast the Einstein-matter evolution equations as a Fuchsian equation for the deviations from the velocity-dominated solutions. Thus, we denote the unknown vector u as u = (γ a b , λa bc , κa b , ψ, ωa , χ, ξ a , ϕab )
(2.30)
where the variables γ a b etc. are related to the Einstein-matter variables by gab
=
ec (γ a b ) = kab = φ
=
ea (ψ) t ∂t ψ + β ψ
= =
Ea Fab
= = c
0
gab + 0 gac tα b γ c b , c
t−ζ λa bc , c gac ( 0 k c b + t−1+α b κc b ), 0
φ + tβ ψ,
−ζ
t ωa , χ, 0 a
E + tβ ξ a , 0 Fab + tβ ϕab .
(2.31) (2.32) (2.33) (2.34) (2.35) (2.36) (2.37) (2.38)
In the first of these equations tα b is not the exponentiation of a matrix with c components αc b such as occurs in (2.13). The expression tα b is for each fixed value of c and b the number which is t raised to the power given by the number αc b (defined below). In Equations (2.31) and (2.33) there is no summation on the index b (but there is a summation on c). In (2.38) ϕab is a totally antisymmetric spatial tensor, which contributes three independent components to u. This assumption is consistent with the form of the evolution equation for ϕab , Equation (2.46) below. The exponents appearing in (2.31)–(2.38) are as follows. Define α0 = 4, β = /100 and ζ = /200 (where is the same (small) quantity which entered the definition of the auxiliary exponents qa in Section 2.2 and which was further restricted at the end of Section 2.4). All of these quantities are independent of t and x. Finally define αa b = 2 max(qb − qa , 0) + α0 = 2qmax{a,b} − 2qa + α0 .
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Ann. Henri Poincar´e
Note that the numbers αa b are all strictly positive. In the second definition of αa b we have used the fact that the qa ’s are ordered. The role of αa b is to shift the spectrum of the Fuchsian-system matrix A, in Equation (1.5), to be positive. It is not clear to what extent the choice of αa b is fixed by the requirement of getting a Fuchsian system. It seems that the (triangle-like) Inequality (42) of [15] (Inequality (5.9) in preprint version) is a key property of these coefficients. We shall further comment below on the specific choice of αa b and its link with the BKL-type approach to the cosmological behaviour near t = 0. When writing the first-order evolution system for u we momentarily abandon the restriction that gab and kab be symmetric, as in [15]. Thus we need to define g ab , and we do so by requiring that gab g bc = δa c . This implies that g ab gbc = δ a c . We lower indices on tensors by contraction with the second index of gab and also raise indices on tensors by contraction with the second index of g ab . This choice is so that raising and then lowering a given index results in the original tensor, and the same for lowering and then raising an index. The position of the indices on quantities appearing in u and other such quantities is fixed. Repeated indices on these quantities imply a summation. On the other hand, as we already mentioned above, one qualifies the summation convention by insisting that indices repeated only because of their occurrence on pa , qa , αa b and other such nontensorial quantities should be ignored when determining sums. Substituting (2.31)–(2.38) in the evolution equations yields equations of motion for u of the form (1.5) t ∂t γ a b + αa b γ a b + 2κa b − 2(t 0 k a c )γ c b + 2γ a c (t 0 k c b ) = a
−2 tα
c a c +α b −α b
γ a c κc b
(2.39)
t ∂t λa bc = tζ ec (t ∂t γ a b ) + ζ tζ ec (γ a b ) t ∂t κ
a
b
a
+ α bκ
a
b
0 a
(2.40)
− (t k b )(trκ) = t (trκ)κ α0
a
b
+t
2−αa b S
( R
a
b
− M b ) (2.41) a
t ∂t ψ + βψ − χ = 0
(2.42)
t ∂t ωa = t {ea (χ) + (ζ − β)ea (ψ)} ζ
t ∂t χ + βχ = t
α0 −β
β
(tr κ)(A + t χ) + t +t2−β {
(2.43) 2−β ab
0
g ∇a ∇b φ + t
2−ζ
∇ ωa a
λ λ gab E a E b e−λφ − g ab g ch Fac Fbh eλφ } 2g 4
(2.44)
√ t ∂t ξ a + βξ a = t1−β {eb ( gg ac g bh Fch eλφ )
t ∂t ϕab + βϕab
1 a ic √ bh i ac +(fib g + fbi g ) gg Fch eλφ } 2 1 c 1 = t1−β {−2e[a ( √ gb]c E c e−λφ ) + fab √ gch E h e−λφ } g g
(2.45) (2.46)
All the quantities entering these equations have been defined, except S Ra b . This is done by taking the Ricci tensor of the symmetric part g(ab) of gab [15]. More
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explicitly, S Ra b = g ac S Rcb , with S
Rab
˜ ahb h = tqa +qb S R (2.47) qa +qb S ˜h S ˜h S ˜i S ˜h S ˜i S ˜h i S ˜ h }, {˜ eh ( Γab ) − e˜a ( Γhb ) + Γab Γhi − Γhb Γai + f˜ah Γ = t ib
and the connection coefficients in the frame {˜ ea },
1 1 S ˜c i i c Γab = S g˜ch e˜a (˜ + f˜ab g(bh) ) + e˜b (˜ g(ha) ) − e˜h (˜ g(ab) ) − g˜(ia) f˜bh − g˜(bi) f˜ah . 2 2 (2.48) Here, S g˜ab is defined as the inverse of g˜(ab) . Once it is shown that the tensor gab in Equation (2.31) is symmetric, then it follows that S Ra b = Ra b and that Equations (2.39)–(2.46) are equivalent to the Einstein-matter equations. 2.5.2 The system (2.39)–(2.46) is Fuchsian A good deal of the work needed to show that Equation (1.5) (as written out in Equations (2.39)–(2.46)) is Fuchsian was done in [15], in the massless scalar field case considered there. The form of the velocity-dominated evolution and the form of the function u are the same in the two cases except for the crucial addition of new source terms and new evolution equations involving the Maxwell field. The presence of the new components does not alter already existing parts of the matrix A, nor already existing terms in f . The difference between A here and A in the massless scalar field case considered in [15] is that here there are additional rows and columns, such that the only non-vanishing new entries are on the diagonal and strictly positive. Therefore the argument in [15] that their A satisfies the appropriate positivity condition implies that our A satisfies the appropriate positivity condition. On the other hand, it is crucial to control in detail the new source terms in f , connected to the Maxwell field, which were absent in [15]. It is for the study of these terms that the results of [1], and in particular the Inequalities (2.24) which were shown there to guarantee that Maxwell source terms become asymptotically subdominant near the singularity, become important. Recall that the crucial criterion for the source f (t, x, u, ux) is that it be O(tδ ), for some strictly positive δ. In regard to this estimate, we use the notation “big O,” “ ” and “small o” as follows. Given two functions F (t, x, u, ux ) and G(t, x, u, ux ) we use the notation F G, to denote that, for every compact set K, there exists a constant C and a number t0 > 0 such that |F (t, x, u, ux )| ≤ C|G(t, x, u, ux )| when (x, u, ux ) ∈ K and 0 < t ≤ t0 (see Definition 1 in [15]). If G is a function only of t (e.g. a power of t), then we replace F G with F = O(G). If f (t, x, u, ux ) = O(tδ ), then by reducing the value of δ (keeping it positive) we have that f (t, x, u, ux ) = o(tδ ) with a “small o” which denotes that f /tδ tends to zero uniformly on compact sets K as t → 0. The new source terms involving the Maxwell field are: the last four terms in M a b (see Equation (2.1)), the last two terms on the right-hand side of Equation (2.44) and the terms of the right-hand sides of Equations (2.45) and (2.46).
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Ann. Henri Poincar´e
The calculation of the estimates starts in the frame, {˜ ea }, defined in Section 2.2. For more details concerning the basic estimates, we refer the reader to [15]. In the frame {˜ ea } the Kasner-like metric is (cf. (2.13)) 2(K−Q) c t 0 g˜ab = 0 g˜ac (t0 ) , (2.49) t0 b −2(K−Q) a t 0 ab 0 cb g˜ = g˜ (t0 ), (2.50) t0 c
where the matrix Q is the diagonal matrix Q b ≡ qa δ a b which commutes with K. With our choice of frame, 0 g˜ab (t0 ) = δab and 0 g˜ab (t0 ) = δ ab . In Lemma 2 in [15] (Lemma 5.1 in preprint version), the form of (2.49) and (2.50) is considered and it is shown that 0 g˜ab = O(t− ) and 0 g˜ab = O(t− ). It is useful to write down expressions for the proposed metric and extrinsic curvature in the frame {˜ ea }. The components in terms of this frame are a
g˜ab ˜ kab
= =
0
g˜ab + 0 g˜ac tα˜ b γ c b , c g˜ac ( 0 k˜c + t−1+α˜ b κc b ). c
b
Here, α ˜ b = α b + qa − qb = |qa − qb | + α0 is symmetric in a, b, α ˜ab = α ˜b a . To get 0 cb an estimate for the inverse metric, we note first that the inverse of gac g is given by g ca0 gcb . Thus it is possible to express the latter quantity algebraically in terms of 0 gab and γ a b . Now define a
a
γ¯a b = −t−α˜ b (δba − g˜ac0 g˜cb ), a
(2.51)
which, from what we just observed, can be expressed algebraically in terms of known quantities and γ a b . Then one has g˜ab =0 g˜ab + tα˜ c γ¯ ac 0 g˜cb . a
(2.52)
As a consequence of an argument given in [15] which uses the (triangle-like) Inequality (42) of that paper ((5.9) in preprint version) and the matrix identity preceding it, this exhibits γ¯a b as a regular function of γ a b . In particular, if it is known that γ a b is o(1) then the same is true of γ¯a b . To better grasp the usefulness of the introduction of the exponents αab and a α ˜ b , and the link of the Fuchsian estimates with the approximate estimates used in the BKL-like works, let us consider more closely the simple case where all the Kasner exponents are distinct (Case III). In this case pa = qa and one can diagonalize the Kasner-metric, so that, in the rescaled frame e˜a , we have simply (for all t ≤ t0 ) 0 g˜ab (t) = δab . In such a case, the BKL-type estimates would be obtained (in the time-dependent rescaled frame e˜a ) by approximating the exact BKL metric by its Kasner limit, i.e., simply g˜ab (t) = δab . By contrast, the estimates a of the Fuchsian analysis are made with the exact metric, g˜ab (t) = δab + tα˜ b γ ab ,
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in which γ ab , being part of u, is considered to be in a compact set and hence is bounded. As the diagonal α ˜aa = α0 > 0, we see that (in the frame e˜a ) the diagonal components of the “Fuchsian” metric asymptote those of the “BKL” metric, and that both are close to one. Concerning the non-diagonal components (in the frame e˜a ) of the “Fuchsian” metric we see that they are constrained, by construction (i.e., by the choice α ˜ ab = |qa − qb | + α0 ), to tend to zero faster than t|qa −qb | . This closeness between the metrics used in the two types of estimates explains the parallelism between the rigorous results derived here and the heuristic estimates used in BKL-type works. If we come back to the general case where the Kasner metric cannot be diagonalized in an analytic fashion, the optimal estimates become worse by a negative power of t (coming from the estimate of the matrix difference 2(K − Q) in Equations (2.49), (2.50) above). The proposed metric in the frame {˜ ea } satisfies then g˜ab t|qa −qb |−
g˜ab t|qa −qb |− .
and
The proposed inverse metric in the adapted frame is g ab = 0 g ab + tα c γ¯ ac 0 g cb . a
The proposed metric in the adapted frame satisfies gab t2qmax{a,b} −
and
g ab t−2qmin{a,b} − .
(2.53)
Estimates of spatial derivatives of the proposed metric are also needed. ec (˜ gab ) t|qa −qb |−δ−
and
ec (˜ g ab ) t|qa −qb |−δ− ,
ec (gab ) t2qmax{a,b} −δ−
and
ec (g ab ) t−2qmin{a,b} −δ− (2.54)
for some strictly positive δ. The determinant of the proposed metric also appears in some of the new source terms. From (2.13), the form of 0 gab (t0 ) and tr K = 1, one gets 0 g = t2 . From (2.49) and 0 g˜ab (t0 ) = δab one gets 0 g˜ = 1. The expression for the determinant is a sum of terms of the form gab gcd gef , such that in each term, each index, 1, 2, 3, occurs exactly twice. From the Kasner relation for the qa ’s and the relation between the two frames, it follows that g = t2 g˜. Considering the form of the various √ √ expressions, one then obtains 1/g = O(t−2 ), g = O(t), 1/ g = O(t−1 ), and √ 1/ g−1/ 0 g = O(t−1+α0 −3 ) = O(t−1+ ). Spatial derivatives of the determinant also appear in f . Considering the form of g˜ − 0 g˜ and that ea ( 0 g˜) = 0, it follows that ea (˜ g ) = O(tα0 −δ−3 ), and ea (g) = O(t2+α0 −δ−3 ). Finally, ea (g −1/2 ) = −
ea (g) = O(t−1+α0 −δ−3 ). 2g 3/2
(2.55)
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Let us now consider the new source terms in f , beginning with the last four a terms of t2−α b M a b . To estimate the contributions of E a and Fab it is sufficient to note from (2.37) and (2.38) that E a = O(1) and Fab = O(1). Then we get, using the definition of αa b and (2.53), t2−α
a
b
1 1 {gbc E a E c − δ a b gch E c E h }e−λφ g 2 t−2qmax{a,b} +2qa +2qmax{b,c} −λA−α0 − + t+2qmax{c,h} −λA−α0 − c
t
c,h
2q1 −λA−α0 −
= O(t
−α0 − +σ
δ
) = O(t ),
a 1 t2−α b {g ac g hi Fch Fbi − δ a b g ch g ij Fci Fhj }eλφ 4 t2−2qmax{a,b} +2qa −2qmin{a,c} −2qmin{h,i} +λA−α0 −2
c,h=c,i=b
+
t2−2qmin{c,h} −2qmin{i,j} +λA−α0 −2
c,h,i=c,j=h
t2q1 +λA−α0 −2 = O(t−α0 −2 +σ ) = O(tδ ) for some strictly positive δ. The crucial inputs in getting these estimates are the Inequalities (2.24). We recall also that the quantity σ (linked to (2.24) being satisfied) was introduced at the end of Subsection 2.4. The estimate of the last two terms on the right-hand side of (2.44) is 1 t2−β gab E a E b e−λφ g
= O(t−β− +σ ) = O(tδ ),
t2−β g ab g ch Fac Fbh eλφ
= O(t−β−2 +σ ) = O(tδ )
The right-hand side of (2.45) is O(tα0 −β−δ−5 +σ ) = O(tδ ). The right-hand side of (2.46) is O(tα0 −β−δ−4 +σ ) = O(tδ ). The other terms which occur in f were estimated in [15], resulting in f = O(tδ ). To show that we indeed have a Fuchsian equation, we need to check not only that f = O(tδ ), but also that ∂u f = O(tδ ) and ∂ux f = O(tδ ), along with other regularity conditions [15, 28]. In [15] it is shown that f is regular with Equation (31) in that paper and the remarks following Equation (31). In our case there is a factor involving the determinant of the metric in various of the terms in f which are not present in the case considered in [15]. The discussion surrounding Equation (31) in [15] applies to our case as well, even for terms in f containing g ±1/2 . The Kasner-like contribution is the leading term, and this function of t and x can be factored out. What is left is of the form w(t, x, u, ux )(1 + h(t, x, u))±1/2 , which is analytic in h at h = 0. The conditions listed following Equation (31) hold. Thus we conclude that (1.5) as written out in (2.39)–(2.46) is a Fuchsian equation.
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2.5.3 Symmetry of metric It remains to show that gab is symmetric, so that Equation (1.5) as written out in (2.39)–(2.46) is equivalent to the Einstein-matter evolution equations. The structure of the argument is the same in any dimension and so it will be written down for general d6 . The number of distinct eigenvalues of K a b is maximal almost everywhere. Thus it is enough to show that g[ab] and k[ab] vanish in the case that the Kasner-like metric is diagonal, since then by analytic continuation they vanish on the entire domain. We therefore consider the case that the Kasner-like metric is diagonal. The redefinitions (2.31), (2.33) from the variables gab , kab to the variables γ a b , a κ b were viewed in the previous subsections as a change between variables with no particular symmetry properties in their indices (18 on each side). One can, however enforce g[ab] = 0 by assuming that γ a b is symmetric and vice versa. Indeed, under a our diagonality assumption for the Kasner-like metric, g˜ab (t) = δab +tα˜ b γ ab where α ˜ ab = |qa − qb | + α0 is symmetric in (a, b). Accordingly, imposing the symmetry γ a b = γ b a algebraically ensures the symmetry of gab . Similarly, one can enforce k[ab] to vanish by imposing consistent constraints on κa b : inserting (2.31) into (2.33) (with the velocity-dominated solution diagonal) and writing out the constraint kab − kba = 0 gives the following condition on κa b κa b − κb a − γ a b pb + γ b a pa + tα(ab)c (γ a c κc b − γ b c κc a ) = 0,
(2.56)
with α(ab)c = 2pmax{a,c} +2pmax{b,c} −2pmax{a,b} −2pc +α0 . These conditions show that there are only six independent components among the κa b , which can be taken to be those with a ≤ b. This is because, the relation (2.56) can be solved uniquely for the components κa b with a > b, given the other ones, at least for t small. That this is true can be seen as follows. Rearrange the Equations (2.56) so that the terms containing κa b with a > b are on the left-hand side and all other terms are on the right-hand side. The result is an inhomogeneous linear system of the form A(t, x)v(t, x) = w(t, x) where A(t, x) and w(t, x) are known quantities and v denotes the components κa b with a > b which we want to determine. Furthermore A(t, x) = I + o(1), where I denotes the identity matrix. It follows that A(t) is invertible for t small, which is what we wanted to show. The solution κa b (a > b) remains moreover bounded when γ b a and κa b are in a compact set. We shall assume from now on that γ a b is symmetric and κa b constrained by (2.56), so that symmetry of the metric is automatic. The redefinitions (2.31), (2.33) from gab , kab to γ a b , κa b can now be viewed as an invertible change of variables, from 12 6 The argument for the symmetry of the metric in [15] is not valid as written since some terms were omitted in the evolution equation for the antisymmetric part of the extrinsic curvature. The correct equation is
∂t (kab − kba ) = (trk)(kab − kba ) − 2(kac k c b − kbc k c a ). In the following a proof of the symmetry of the tensors gab and kab is supplied with the help of a different method.
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independent variables to 12 independent variables. We can also clearly assume λa bc in (2.32) to be symmetric in a, b. With these conventions, there are less components in u than in the previous subsections. The independent components can be taken to be γ a b , κa b and λa bc with a ≤ b, together with the matter variables. An independent system of evolution equations is given by (2.39)–(2.41) with a ≤ b for the gravitational variables, and the same evolution equations as before for the matter variables. These evolution equations are equivalent to all the original evolution equations, since the Equations (2.39)–(2.41) with a > b are then automatically fulfilled, as can be shown using the fact that the Einstein tensor and the stress-energy tensor are symmetric for symmetric metrics. To see this it must be shown that given a symmetric tensor Sab , the vanishing of S a b = g ac Scb for a ≤ b implies that Sab = 0. Consider the linear map which takes a symmetric tensor Sab , raises an index, and keeps the components of the result with a ≤ b. This is a mapping between vector spaces of dimension d(d+ 1)/2 and can be shown to be an isomorphism by elementary linear algebra. This proves the desired result. Now, this reduced evolution system is also Fuchsian. This follows from the same reasoning as above, which still holds because all components of u, including the non-independent ones, can still be assumed to be bounded. Therefore, there is a unique u that goes to zero, which must be equal to the one considered in the previous subsections. The metric considered previously is thus indeed symmetric. 2.5.4 Unique solution on a neighborhood of the singularity Given an analytic solution to the velocity-dominated evolution equations on (0, ∞) × Σ, such that Inequalities (2.24) are satisfied, we now have a solution u to a Fuchsian equation (and a corresponding solution to the Einstein-matter evolution equations) in the intersection of a neighborhood of the singularity with (0, ∞) × U0 where U0 is a neighborhood of an arbitrary point on Σ. These local solutions can be patched together to get a solution to the Einstein-matter evolution equations everywhere in space near the singularity. It may seem like there could be a problem patching together the solutions obtained on distinct neighborhoods with non-empty intersection because the Fuchsian equation is not the same for different allowed choices of and adapted local frame. The construction is possible because different allowed choices of and local frame result in a well-defined relationship between the different solutions u which are obtained, such that the corresponding Einstein-matter variables agree on the intersection (up to change of basis). It therefore follows that given an analytic solution to the velocity-dominated evolution equations on (0, ∞) × Σ, such that Inequalities (2.24) are satisfied, our construction uniquely determines a solution to the Einstein-matter evolution equations everywhere in space, near the singularity.
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2.6
Kasner-Like Behaviour for Subcritical Einstein-Matter Systems
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Einstein-matter constraints
2.6.1 Matter constraints The time derivative of the matter constraint quantities (the left-hand side of Equations (2.2) and (2.3)) vanishes. If the velocity-dominated matter constraints are satisfied, the matter constraint quantities are o(1). A quantity which is constant in time and o(1) must vanish. Therefore the matter constraints are satisfied. 2.6.2 Diagonal Kasner metrics It remains to show that the Hamiltonian and momentum constraints are satisfied, that C and Ca , defined in (1.9) and (1.10), vanish. Since we now have a metric, gµν , it follows that ∇µ Gµν = 0. Since the matter evolution and constraint equations are satisfied, it follows that ∇µ T µν = 0. From the vanishing of the right-hand side of (1.12) and the vanishing of the covariant divergence of both the Einstein tensor and the stress-energy tensor, it follows that ∂t C
=
∂t Ca
=
2(trk)C − 2∇a Ca 1 (trk) Ca − ∇a C. 2
(2.57) (2.58)
Now define C¯ = t2−η1 C and C¯a = t1−η2 Ca , with 0 < η2 < η1 < β. t ∂t C¯ + η1 C¯
=
t ∂t C¯a + η2 C¯a
=
2(1 + t tr k)C¯ − 2t2−η1 +η2 ∇a C¯a 1 (1 + t tr k)C¯a − tη1 −η2 ∇a C¯ 2
(2.59) (2.60)
On the right-hand side of (2.59) and (2.60) C¯ and C¯a are to be considered as ¯ C¯a ). If it is shown that (2.59) and (2.60) is a Fuchsian components of u = (C, system, then there is a unique solution u such that u = o(1). It is clear that u = 0 is a solution to (2.59) and (2.60). If it is shown that C¯ = o(1) and C¯a = o(1), (i.e., that C = o(t−2+η1 ) and Ca = o(t−1+η2 )), then they must be this unique solution. Furthermore, it is sufficient to consider the case that the Kasner-like metric is diagonal, since the number of distinct eigenvalues of K a b is maximal on an open set of Σ. If the constraints vanish on an open set of their domain, then by analytic continuation they vanish everywhere on their domain. Therefore we consider the case that the Kasner-like metric is diagonal and show first that 1 + t tr k ∇a C¯a
= =
O(tδ ) O(t
−2+δ+η1 −η2
(2.61) )
(2.62)
(when C¯a is bounded) so that the system (2.59), (2.60) is Fuchsian (the complete regularity of f (t, x, u, ux) defined by (2.59) and (2.60) can be easily verified); and
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second, that = o(t−2+η1 ) = o(t−1+η2 ).
C Ca
(2.63) (2.64)
Some facts which will be used to show this follow. Consider indices a ∈ {1, 2, 3}. The following inequalities hold for some positive integer n and for real numbers, qa , ordered such that if a < b, then qa ≤ qb . (In later sections we define ordered auxiliary exponents, qa , for a ∈ {1, . . . , d}, for arbitrary fixed d ≥ 2. Then (2.65)–(2.67) hold more generally for indices in {1, . . . , d}.) n
|qai−1 − qai | − qan ≥ 0
(2.65)
|qai−1 − qai | + qan ≥ 2qmax{ak ,aj }
(2.66)
|qai−1 − qai | − qan ≥ −2qmin{ak ,aj }
(2.67)
qa0 +
i=1
qa0 +
n i=1
−qa0 +
n i=1
The latter two inequalities hold for any k, j in {0, . . . , n}. In the case that the Kasner-like metric is diagonal, qa = pa . The metric in the frame {˜ ea } is 0 g˜ab = δab , g˜ab g˜ab
= δab + tα˜ b γ a b t|pa −pb | , a = δ ab + tα˜ b γ¯ ab t|pa −pb | . a
The extrinsic curvature satisfies t 0 k a b = −δ a b pb , t ka b t (k˜ab − 0 k˜ab )
a
= −δ a b pb + tα b κa b , = tα˜ b κa b , a
and t tr 0 k = −1, t tr k 2 2 0 2 t {(tr k) − (tr k) }
= −1 + tα0 trκ, = O(tα0 ).
(2.68) (2.69)
The following estimates will also be useful. −k a b k b a + 0 k a b 0 k b a
and
= −2t−2+α0 κa a pa − t−2+α = O(t−2+α0 ),
a
b b +α a
ea (tr k − tr 0 k) = ea (t−1+α0 tr κ) = O(t−1+α0 ).
κa b κb a (2.70)
(2.71)
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The structure functions of the frame {˜ ea } are c f˜ab
=
c tpc −pa −pb fab − ln t ea (pb ) t−pa δ c b + ln t eb (pa ) t−pb δ c a
tpc −pa −pb −δ .
˜ c , the connection coefficients (2.48) in It is convenient to have an estimate of Γ ab the frame {˜ ea }, term by term. Term A: g˜ch e˜a (˜ gbh ) t|pc −ph |−pa +|pb −ph |−δ , (2.72) h
g˜ e˜b (˜ gha ) ch
Term B:
t|pc −ph |−pb +|ph −pa |−δ ,
(2.73)
t|pc −ph |−ph +|pa −pb |−δ ,
(2.74)
h
gab ) g˜ch e˜h (˜
Term C:
h
h g˜ci g˜ha f˜bi
Term D:
t|pc −pi |+|ph −pa |+ph −pb −pi −δ , (2.75)
h,i=b ci
h g˜bh f˜ai
t|pc −pi |+|pb −ph |+ph −pa −pi −δ , (2.76)
Term E:
g˜
Term F:
c f˜ab tpc −pa −pb −δ .
h,i=a
(2.77)
The difference between the connection coefficients for the metric g˜ab and those for ˜ c . It is useful to have the estimates ˜c − 0Γ ˜c = Γ the Kasner-like metric is ∆Γ ab ab ab a ˜ aac = 1 g˜ab e˜c (˜ Γ gab ) + f˜ac t−pc −δ , 2
and ˜a = ∆Γ ac
1 ab g˜ e˜c (˜ gab ) t−pc +α0 −δ . 2
(2.78)
2.6.3 Momentum and Hamiltonian constraints First, we show (2.61) and (2.62). From equation (2.68), 1 + t tr k = O(tα0 ). Similarly, we can estimate ∇a C¯a , ∇a C¯a
=
˜¯ ˜ aC g˜ab ∇ b
=
˜ cab C¯c t−pc } g˜ab {t−pa ea (C¯b t−pb ) − Γ
The first term is g˜ab t−pa ea (C¯b t−pb ) t|pa −pb |−pa −pb −δ t−2pmin{a,b} −δ .
(2.79)
From (2.72)–(2.76) the second term is ˜ cab C¯c t−pc t−2p3 −δ . g˜ab Γ
(2.80)
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From (2.79) and (2.80), the desired estimate, ∇a C¯a = O(t−2+η1 −η2 ) is obtained. Thus, the system (2.59), (2.60) is Fuchsian. Next we turn to (2.63) and (2.64). A term that appears in the momentum constraint is ∇a k a b . The estimate is needed in the adapted frame, and the covariant derivative is calculated in the frame {˜ ea }. This adds a factor of tpb , ˜ aac k˜c b − Γ ˜ cab k˜ac } tpb ea (k˜ ab ) + Γ ∇a k a b = {˜ Furthermore, the quantity whose estimate will be required is the difference between this term and the corresponding term in the velocity-dominated constraint, ˜ a 0 k˜c + Γ ˜ a (k˜ c − 0 k˜c )}tpb ea (k˜ ab − 0 k˜ab ) + ∆Γ ∇a k a b − 0 ∇a 0 k a b = {˜ ac b ac b b ˜ cab k˜ac tpb + 0 Γ ˜ cab 0 k˜ac tpb −Γ
(2.81) (2.82)
The right-hand side of (2.81) is O(t−1+α0 −δ ). The terms in line (2.82) originating from Term E of the connection coefficients (see (2.76)) are cancelled in h and the symmetry of k˜ai and 0 k˜ai . For the sum, due to the antisymmetry of f˜ai estimating the rest of the terms in line (2.82), it is convenient to rewrite this line as, ˜ cab 0 k˜ac tpb = −∆Γ ˜ cab 0 k˜ac tpb − Γ ˜ cab (k˜ ac − 0 k˜ac ) tpb , ˜ cab k˜ac tpb + 0 Γ −Γ
(2.83)
with ˜ cab ∆Γ
=
b c a 1 {˜ ea (tα˜ c γ b c ) + e˜b (tα˜ a γ c a ) − e˜c (tα˜ b γ a b ) 2 c b h a + tα˜ h γ¯ c h [˜ ea (tα˜ h γ b h ) + e˜b (tα˜ a γ h a ) − e˜h (tα˜ b γ a b )]
h
−
i tα˜ a γ i a f˜bc − i
i
−t
α ˜b i b
i γ i f˜ac −
h
a tα˜ h γ¯ c h f˜bh − c
h
t
α ˜c h c
b γ¯ h f˜ah −
(2.84) (2.85)
i tα˜ h γ¯ c h tα˜ a γ i a f˜bh (2.86) c
i
hi
t
α ˜c h c
i γ¯ h tα˜ i γ b i f˜ah }. b
(2.87)
hi
The terms in line (2.87) need not be considered since they originate from Term E of the connection coefficients and as stated above the contribution from this term ˜ c 0 k˜ac tpb . So considering only lines (2.84)–(2.86), is cancelled by terms in Λ = Γ ab the first term on the right-hand side of (2.83) is ˜ a pa t−1+pb = O(t−1+α0 −δ ) + terms which are cancelled by Λ. ∆Γ ab
(2.88)
Since the terms in the sum come with different weights, pa , (2.78) cannot be used in (2.88). But the estimate is straightforward. For example, the term in (2.88) originating from the 3rd term in line (2.86) a,h,i t−1+|pa −ph |+|pi −pa |+pi −ph +2α0 −δ = O(t−1+α0 −δ ). Finally consider the rest of the right-hand side of (2.83), a 1 i i c tα˜ c κa c t−1+pb . − g˜ch {˜ ea (˜ gbh ) + e˜b (˜ gha ) − e˜h (˜ gab ) − g˜ia f˜bh − g˜bi f˜ah } + f˜ab 2 (2.89)
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For all terms except the 5th term in (2.89), the estimate, O(t−1+α0 −δ ) can be obtained from (2.72)–(2.77). The fifth term originates from Term E of the connection coefficients, and was already considered above. Therefore ∇a k a b − 0 ∇a 0 k a b = O(t−1+α0 −δ ).
(2.90)
Next the matter terms are estimated. For the Hamiltonian constraint, an estimate of ρ − 0 ρ is needed. (∂t φ)2 − (∂t 0 φ)2
g ab ea (φ)eb (φ) 1 gab E a E b e−λj φ g g ab g ch Fac Fbh eλj φ Therefore,
= =
{2 ∂t 0 φ + t−1+β (βψ + t∂t ψ)} t−1+β (βψ + t∂t ψ) {2A + tβ (βψ + t∂t ψ)} t−2+β (βψ + t∂t ψ)
=
o(t−2+η1 ), t−2p3 −δ− = o(t−2+η1 ),
=
O(t−2− +σ ) = o(t−2+η1 ),
=
O(t−2−2 +σ ) = o(t−2+η1 ). ρ − 0 ρ = o(t−2+η1 ).
(2.91)
The difference between the matter terms in the momentum constraint and in 0 Ca is −∂t φ ea (φ) + ∂t 0 φ ea ( 0 φ) = −∂t 0 φ ea (tβ ψ) − ∂t (tβ ψ)ea (φ) = O(t−1+β−δ ), 1 1 1 ( √ − ) 0 E b 0 Fab + √ (E b Fab − 0 E b 0 Fab ) = o(t−1+η2 ). (2.92) 0g g g Estimates related to the determinant which are relevant to (2.92) can be found immediately preceding Equation (2.55). From the estimates just obtained, ja − 0 ja = o(t−1+η2 ).
(2.93)
From R = O(t−2+α0 ) (shown in [15]) and from 0 C = 0, (2.70), (2.69), (2.91) and the relative magnitude of the various exponents, it follows that C = o(t−2+η1 ). From 0 Ca = 0, (2.90), (2.71), (2.93) and the relative magnitude of the various exponents, it follows that Ca = o(t−1+η2 ). Since (2.63)–(2.64) are satisfied, the Hamiltonian and momentum constraints are satisfied.
2.7
Counting the number of arbitrary functions
The number of degrees of freedom of the Einstein-Maxwell-dilaton system in 4 spacetime dimensions is 5 : 2 for the gravitational field, 2 for the electromagnetic field and 1 for the dilaton. Hence, a general solution to the equations of motion should contain 10 freely adjustable, physically relevant, functions of space (each degree of freedom needs two initial data, q and q). ˙ This is exactly the number that appears in the above Kasner-like solutions.
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• The metric carries four, physically relevant, distinct functions of space. This is the standard calculation [4]. • The scalar field carries two functions of space, A and B. • The electromagnetic field carries six functions of space, 0 E a and 0 Fab . These are physically relevant because they are gauge invariant, but they are subject to two constraints, leaving four independent functions. A different way to arrive at the same conclusions is to observe that the respective number of fields, dynamical equations and (first class) constraints are the same for the velocity-dominated system and the full system. Hence, a general solution of the velocity-dominated system (in the sense of function counting) will contain the same number of physically distinct, arbitrary functions as a general solution of the full system. This general argument applies to all systems considered below and hence will not be repeated. In [15] a different way of assessing the generality of the solutions constructed was used. This involved exhibiting a correspondence between solutions of the velocity-dominated constraints and solutions of the full constraints using the conformal method. That method starts with certain free data and shows the existence of a unique solution of the constraints corresponding to each set of free data. It is a standard method for exploring the solution space of the full Einstein constraints [42] and in [15] it was shown how to modify it to apply to the velocity-dominated constraints. While it is likely that the conformal method can be applied in some way to all the matter models considered in this paper, the details will only be worked out in two cases which suffice to illustrate the main aspects of the procedure. These are the Einstein-Maxwell-dilaton system with D = 4 (this section) and the Einstein vacuum equations with arbitrary D ≥ 4 (next section). Even in those cases no attempt will be made to give an exhaustive treatment of all issues arising. It will, however, be shown that the strategies presented for solving the velocity-dominated constraints are successful in some important situations. The procedure presented in the following is slightly different from that used in [15]. Even for the case of the Einstein-scalar field system with D = 4 it gives results which are in principle stronger than those in [15] since they are not confined to solutions which are close to isotropic ones. In the presence of exponential dilaton couplings a change of method seems unavoidable. One part of the conformal method concerns the construction of symmetric second rank tensors which are traceless and have prescribed divergence from the truly free data. In this step there is no difference between the full constraints and the velocity-dominated ones. An account of the methods applied to the full constraints in the case D = 4 can be found in [42]. (These arguments generalize in a straightforward way to other D. It is merely necessary to find the correct conformal rescalings. For D ≥ 4 and vacuum these are written down in the next section.) In view of this we say, with a slight abuse of terminology, that the free data consists of a collection g˜ab , k˜ab , H, φ, φ˜t , E a , Fab where g˜ab is a Riemannian metric, k˜ab is a symmetric tensor with
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vanishing trace and prescribed divergence with respect to g˜ab , H is a non-zero constant, φ and φ˜t are scalar functions and E a and Fab are objects of the same kind as elsewhere in this section. All these objects are defined on a three-dimensional manifold. Next we introduce a positive real-valued function ω which is used to construct solutions of the constraints from the free data. Define gab = ω 4 g˜ab , kab = ω −2 k˜ab + Hgab , φt = ω −6 φ˜t . The objects gab , kab , φ, φt , E a and Fab satisfy the constraints provided the divergence of k˜ab is prescribed as ω 6 ja and ω satisfies a nonlinear equation which in the case of the full Einstein equations is known as the Lichnerowicz equation. In the case of the velocity-dominated constraints it is an algebraic equation. The Lichnerowicz equation is of the form 3 1 3 ai ω αi − H 2 ω 5 = 0 ∆g˜ ω − Rg˜ ω + 8 4 i=1
(2.94)
Here α1 = 1, α2 = −3 and α3 = −7. The functions ai depend on the free data and their exact form is unimportant. All that is of interest are that each ai is non-negative and that at any point of space a1 = 0 iff ∇a φ = 0 , a2 = 0 iff the electromagnetic data vanish and a3 = 0 iff φt and k˜ab vanish. Next consider the velocity-dominated constraints for d = 3. The analogue of the elliptic Equation (2.94) is the algebraic equation 3 bω −7 − H 2 ω 5 = 0 4
(2.95)
Here b is a non-negative function which vanishes at a point of space iff φt and k˜ab vanish. This can be solved trivially for ω > 0 provided b does not vanish at any point since the mean curvature H is non-zero. For each choice of free data satisfying this non-vanishing condition there is a unique solution ω of (2.95). In order to compare the sets of solutions of the full and velocity-dominated constraints in these two cases it remains to investigate the solvability of the elliptic Equation (2.94) for ω. A discussion of this type of problem in any dimension can be found in [43]. We would like to show that for suitable metrics on a compact manifold the equation for ω always has a unique solution, i.e., the situation is exactly as in the case of the velocity-dominated equations. The problem can be simplified by the use of the Yamabe theorem, which says that any metric can be conformally transformed to a metric of constant scalar curvature −1, 0 or 1. In the following only the cases of negative and vanishing scalar curvature of the metric supplied by the Yamabe theorem will be considered. A key role in the existence and uniqueness theorems for Equation (2.94) is played by the positive zeros of 3 3 the algebraic expressions x + 8 i=1 ai xαi − 6H 2 x5 and 8 i=1 ai xαi − 6H 2 x5 . 3 Provided a=1 ai does not vanish anywhere it is possible to show that each of the algebraic expressions has a unique positive zero for each value of the parameters. The significance of the information which has been obtained concerning the zeros of certain algebraic expressions is that it guarantees the existence of a positive solution of the corresponding elliptic equations for any set of free data satisfying
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the inequalities already stated using the method of sub- and supersolutions (cf. [43]). In the case of Equation (2.94) uniqueness also holds. For in that case the equation has a form considered in [44] for which uniqueness is demonstrated in that paper. The advantage of the three-dimensional case is that there the problem reduces to the analysis of the roots of a cubic equation, a relatively simple task compared to the analysis of the zeros of the more complicated algebraic expressions occurring in higher dimensions. For the purpose of analyzing Kasner-like (monotone) singularities it is not enough to know about producing just any solutions of the constraints. What we have shown is that (i) if the Kasner constraints are satisfied at time t0 , then they are propagated at all times by the velocity-dominated evolution equations; and (ii) if the Kasner constraints are satisfied, the exact constraints are also satisfied. It is also necessary to verify, however, that one can satisfy simultaneously the Kasner constraints and the inequalities necessary for applying the Fuchsian arguments, i.e., we must make sure that we can produce a sufficiently large class of solutions which satisfy the inequalities necessary to make them consistent with Kasner behaviour. Because of the indirect nature of the way of solving the momentum constraint (which has not been explained here) it is not easy to control the generalized Kasner exponents of the resulting spacetime. There is however, one practical possibility. Choose a spatially homogeneous solution with Abelian isometry group (for d = 3 this means Bianchi type I) which satisfies the necessary inequalities. Take the free data from that solution and deform it slightly. Then the generalized Kasner exponents of the final solution of the velocity-dominated equations will also only be changed slightly. If the homogeneous solution is defined on the torus T 3 then it is known that any other metric of constant scalar curvature has non-positive scalar curvature. Therefore we are in the case for which existence and uniqueness is discussed above. We could also start with a negatively curved Friedmann model.
3 Vacuum solutions with D ≥ 11 The second class of solutions we consider is governed by the Action (1.2), with D ≥ 11. The d + 1 decomposition is as in Section 1.6, with the matter terms vanishing. The Kasner-like evolution equations are (2.7) and (2.8). The general analytic solution of these equations is given by (2.12) and (2.13). To obtain this form, we again adapt a global time coordinate such that the singularity is at t = 0. We label the eigenvalues of K, p1 , . . . , pd , such that pa ≤ pb if a < b. The d eigenvalues again satisfy i=1 pi = 1, coming from tr K = 1. As in the D = 4 case, in order to preserve analyticity even near the points where some of the eigenvalues coincide, while retaining control of the solution in terms of the eigenvalues, we introduce a special construction involving auxiliary exponents and an adapted frame. In higher dimensions, there are more possibilities to take care of, but the idea is the same as in the D = 4 case. Consider an arbitrary point x0 ∈ Σ. Let
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n be the number of distinct eigenvalues of K at x0 . Let mi be the multiplicity of pAi , i ∈ {1, . . . , n}, with pAi such that pb is strictly less than pAi if b < Ai . Thus pAi , . . . , pAi +mi −1 are equal at x0 . For each integer a ∈ {Ai , . . . , Ai + mi − 1}, define A +m −1 1 i i qa = pj mi j=Ai
on a neighborhood of x0 , U0 . Note that if mi = 1, then qAi = pAi . Shrinking U0 if necessary, choose > 0 such that for x ∈ U0 , for a ∈ {Ai , . . . , Ai + mi − 1} and for b ∈ {Aj , . . . , Aj + mj − 1}, if i = j, then |pa − pb | < /2, while if i = j, |pa − pb | > /2. The adapted frame {ea } is again required to be analytic and such that the related frame {˜ ea } is orthonormal with respect to the Kasner-like metric at some time t0 > 0, with e˜a = t−qa ea . In addition, it is required that eAi , . . . , eAi +mi −1 span the eigenspace of K corresponding to the eigenvalues pAi , . . . , pAi +mi −1 . Note qAi . that if mi = 1 then eAi is an eigenvector of K corresponding to the eigenvalue The auxiliary exponents, qa , are analytic, satisfy the Kasner relation ( qa = 1), are ordered (qa ≤ qb for a < b), and satisfy q1 ≥ p1 , qd ≤ pd and maxa |qa − pa | < /2. If qa = qb , then 0 gab , 0 g ab , 0 g˜ab and 0 g˜ab all vanish, and the same is true with g replaced by k. The velocity-dominated constraints corresponding to the Hamiltonian and momentum constraints are 0 C = 0 and 0 Ca = 0, with 0 C and 0 Ca as in Equations (2.18) and (2.19), with the matter terms vanishing. For the Solution (2.12)– (2.13) 2 the velocity-dominated Hamiltonian constraint equation is equivalent to pa = 1. Equations (2.21)–(2.22) are satisfied, so if the velocity-dominated constraints are satisfied at some t0 , then they are satisfied for all t > 0. For this class of solutions, the Inequality [13], 2p1 + p2 + · · · + pd−2 > 0, or equivalently, (3.1) 1 + p1 − pd − pd−1 > 0, defines the set V which was referred to in the introduction. As shown in [13], this inequality can be realized when the spacetime dimension D is equal to or greater than 11. As in our Maxwell archetypal example above, we expect that this inequality will be crucial to control the effect of the source terms (here linked to the spatial curvature) near the singularity. It is again convenient to introduce a number σ > 0 so that, for all x ∈ U0 , 4σ < 1 + p1 − pd − pd−1 . Reduce if necessary so that < σ/(2d + 1). If is reduced, it may be necessary to shrink U0 so that the conditions imposed above remain satisfied. It is assumed that and U0 are such that the conditions imposed above and the condition imposed in this paragraph are all satisfied. We again recast the evolution equations in the form (1.5) and show, for D ≥ 11, that (1.5) is Fuchsian and equivalent to the vacuum Einstein equation, with quantities u, A and f as follows. Let u = (γ a b , λc ef , κh i ) be related to the Einstein variables by (2.31)–(2.33). For general d define α0 = (d + 1) and define αa b
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in terms of α0 as in Section 2. Let A and f , be given by Equations (2.39)–(2.41), with M a b = 0. The argument that A in Equation (1.5) satisfies the appropriate positivity condition is analogous to the part of the argument concerning the submatrix of A corresponding to (γ, κ) in [15]. A transformation to a frame in which 0 gab is diagonal induces a similarity transformation of A. The eigenvalues of the submatrix are calculated in this representation in [15], and the generalization of the calculation to integer d ≥ 2 is straightforward. To obtain f = O(tδ ) requires the estimate t2−α b S Ra b = O(tδ ). The strategy used here is different from that used to estimate the curvature in [15]. The general problem is one of organization. There are many terms to be estimated, each of which on its own is not too difficult to handle. The difficulty is to maintain an overview of the different terms. The procedure in [15] made essential use of the fact that the indices only take three distinct values and in the case of higher dimensions, where this simplification is not available, an alternative approach had to be developed. a
First S Ra b is estimated by considering each of the five terms in the Expression (2.47). These five terms are expanded by considering each of the six terms in (2.48) if the indices on S Γcab are distinct, but carrying out the summation before estimating S Γaab . There are thus 55 terms to estimate. While many of these terms are actually identical up to numerical factors, the ease with which each term can be estimated, using the Inequalities (2.65)–(2.67), led to estimation of all 55 terms, i =0 rather than first combining terms. We do however, take into account that fjk if j = k for obtaining the estimates. Once an equation such as (1.5) is shown to be Fuchsian, then it follows that spatial derivatives of u of any order are o(1). At the stage of the argument we are at here, we cannot assume uxx = O(1). This means that t−ζ λa bc must be used for eb (γ a c ) in places where a spatial derivative of eb (γ a c ) occurs. This makes a slight difference, compared to Section 2.6, in what estimate of the terms in the connection coefficients is used for the first and second terms of (2.47) (t−δ is replaced by t−ζ ). There are additional differences from (2.72)–(2.77), because there it is assumed that the Kasner-like metric is diagonal. The estimates 0 g˜ab = O(t− ) and 0 g˜ab = O(t− ), obtained in Lemma 2 of [15], hold in the case we are considering, so that g˜(ab) t|qa −qb |− and (see [15]) S g˜ab t|qa −qb |− . This adds factors of t− to the estimate of terms in the connection coefficients. With these considerations, from (2.48), S ˜a Γac
=
1 S ab a g˜ e˜c (˜ g(ab) ) + f˜ac t−qc −2 −ζ . 2
(3.2)
Here we do not write out the estimates of all 55 terms, but instead give some examples, with a number designating which term of (2.47) is being considered (1–5), and a letter designating which term of (2.48) is being considered (A–F).
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Thus, for example, term 1C is g(cb) )) t−qa +qb g˜ac e˜h ( S g˜hi e˜i (˜ t−qa +qb +|qa −qc |−qh +|qh −qi |−qi +|qc −qb |−δ−3 −ζ c,h,i
t−2qa +2qmax{a,b} −2qd −δ−3 −ζ .
(3.3)
Term 3E is j S ˜h Γhi t−qa +qb g˜ac S g˜ik g˜(bj) f˜ck t−qa +qb +|qa −qc |+|qi −qk |+|qb −qj |+qj −qc −qk −qi −δ−5 c,i,j,k=c
t−2qmin{a,c} +2qb −2qmin{d,k} −δ−5 .
(3.4)
c,k=c
˜h S Γ ˜ i , the terms resulting from expanding S Γ ˜ h are desIn term 4, t−qa +qb g˜ac S Γ ib ch ib S ˜i ignated by small letters a–f, and those from Γch are designated by capital letters A–F. Term 4dA is g(hk) ) t−qa +qb g˜ac S g˜hl g˜(ji) f˜blj S g˜ik e˜c (˜ −qa +qb +|qa −qc |+|qh −ql |+|qj −qi |+qj −qb −ql +|qi −qk |−qc +|qh −qk |−δ−5 t c,h,i,j,k,l
t−2qa −δ−5 .
(3.5)
Term 4eD is j S il k g˜ g˜(kc) f˜hl t−qa +qb g˜ac S g˜hn g˜(bj) f˜in c,h,i,j,k,l=h,n=i
t−qa +qb +|qa −qc |+|qh −qn |+|qb −qj |+qj −qi −qn +|qi −ql |+|qk −qc |+qk −qh −ql −δ−5 t2qb −2qd −2qd−1 −δ−5
(3.6)
Term 5D is i S jk h t−qa +qb g˜ac f˜cj g˜ g˜(hi) f˜bk t−qa +qb +|qa −qc |+qi −qc −qj +|qj −qk |+|qh −qi |+qh −qb −qk −δ−3 c,h,i,j=c,k=b
t−2qmin{a,c} +2q1 −2qmin{j,k} −δ−3 .
(3.7)
c,j=c,k=b
The estimates of the remaining terms are obtained as these. The examples include one of the terms which limits the estimate for each possible choice of indices
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a and b. The result is, S
Ra b t2qb −2qd −2qd−1 −δ−5 +
t−2qmin{a,c} +2q1 −2qmin{j,k} −δ−5 .
(3.8)
c,j=c,k=b
And t2−α
a
b
S
Ra b
{t2qmin{a,b} −2qd −2qd−1 + t−2qmax{a,b} +2q1 −2qmin{j,k} }t2−α0 −δ−5 c≥a,j=c,k=b
t
2−2qd −2qd−1 +2q1 −(d+7)
t8σ−(d+7) = O(tδ ).
(3.9)
The estimate of the rest of the terms in f is obtained straightforwardly by checking that the exponent of t in each case is strictly positive. The other regularity conditions that f should satisfy are shown to hold by Equation (31) in [15] and the remarks following Equation (31). The symmetry of gab is shown for all d’s in Subsection 2.5.3. That the Hamiltonian and momentum constraints are satisfied is shown by the direct analogue of argument made in Section 2.6 and the estimate R = o(t−2+η1 ) obtained from Equation (3.8). The only change is that Equation (2.80) is replaced by ˜ c C¯c t−pc t−2pd −δ . g˜ab Γ ab
(3.10)
To conclude this section we discuss the solution of the velocity-dominated constraints for the vacuum equations and D ≥ 4. The case D = 3 could be discussed in a similar way but the analogue of the Lichnerowicz equation has a different form and so for brevity that case will be omitted. The discussion proceeds in a way which is parallel to that of the last section. As already indicated there, the essential task is the analysis of the Lichnerowicz equation. In the present case we start with free data g˜ab , k˜ab and H where k˜ab has zero divergence. The actual data are defined by gab = ω 4/(d−2) g˜ab and kab = ω −2 k˜ab + Hgab . The constraints will be satisfied is ω satisfies the following analogue of the Lichnerowicz equation: ∆g˜ ω +
3d−2 d+2 d−2 d(d − 2) 2 d−2 (−Rg˜ ω + k˜ ab k˜ab ω d−2 ) − H ω =0 4(d − 1) 4
(3.11)
The corresponding equation in the velocity-dominated case is 3d−2 d+2 d − 2 ˜ ab ˜ d(d − 2) 2 d−2 H ω k kab ω d−2 − =0 4(d − 1) 4
(3.12)
As in the case of (2.95) it is trivial to solve (3.12) provided k˜ab does not vanish at any point. To determine the solvability of Equation (3.11) it is necessary to study 3d−2 d+2 3d−2 the positive zeros of the algebraic expressions x + bx d−2 − ax d−2 and bx d−2 − d+2 ax d−2 where a > 0 and b > 0. The second expression is very close to what we
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had in the velocity-dominated case and clearly has a unique positive zero for any values of a and b satisfying the inequalities assumed. Looking for positive zeros of the first algebraic expression is equivalent to looking for positive solutions of d+2 x− d−2 + ax−2 − b = 0. Note that the function on the left-hand side of this equation is evidently decreasing for all positive x, tends to infinity as x → 0 and tends to −b as x → ∞. Hence as long as the constant b is non-zero this function has exactly one positive zero, as desired. This is what is needed to obtain an existence theorem. It would be desirable to also obtain a uniqueness theorem for the solution of (3.11). To obtain solutions of the velocity-dominated constraints of the right kind to be consistent with Kasner-like behaviour we can use the same approach as in the last section, starting with Kasner solutions with an appropriate set of Kasner exponents.
4
Massless scalar field, D ≥ 3
Consider Einstein’s equations, D ≥ 3, with a massless scalar field as source, the action given by SE [gαβ ] + Sφ [gαβ , φ], and d + 1 decomposition as in Section 1.6. The stress-energy tensor is Tµν =
(D)
1 ∇µ φ (D) ∇ν φ − gµν (D) ∇α φ (D) ∇α φ. 2
(4.1)
Thus ρ = 12 {(∂t φ)2 + g ab ea (φ)eb (φ)}, ja = −∂t φ ea (φ), and M a b = g ac eb (φ) ec (φ). A crucial step in the generalization to arbitrary D ≥ 3 is that the cancellation of terms involving ∂t φ in the expression for M a b is not particular to D = 4. The scalar field satisfies (D) ∇α (D) ∇α φ = 0, which has d + 1 decomposition ∂t2 φ − (trk)∂t φ = g ab ∇a ∇b φ.
(4.2)
Let the Kasner-like evolution equations be Equations (2.7)–(2.9), with Solutions (2.12)–(2.14) for time coordinate as in Section 3. Given a point x0 ∈ Σ, let the neighborhood U0 , the (local) adapted frame and the constant be as in Section 3. Define 0 ρ = 12 (∂t 0 φ)2 and 0 ja = −∂t 0 φ ea ( 0 φ). The velocity-dominated constraints corresponding to the Hamiltonian and momentum constraints are 0 C = 0 and 0 Ca = 0, with 0 C and 0 Ca given by Equations (2.18) and (2.19). For the Solution the velocity-dominated Hamiltonian constraint is equivalent (2.12)–(2.14) to pa 2 + A2 = 1. Equations (2.21) and (2.22) are satisfied so if the velocitydominated constraints are satisfied at some t0 , then they are satisfied for all t > 0. The restriction defining the set V is the Inequality (3.1). (If D < 11, then satis 2 pa + A2 = 1 requires A = 0. Note fying simultaneously (3.1), pa = 1 and that conversely, for D = 3, the restrictions defining V are simply equivalent to A = 0, since (3.1) is in this case a consequence of p1 + p2 = 1 and p21 + p22 < 1). The constant σ > 0 is chosen so that, for all x ∈ U0 , 4σ < 1 + p1 − pd − pd−1 from which it follows that σ < 2 − 2pd . (4.3)
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Now reduce if necessary so that < σ/(2d + 1). As before, this may in turn require shrinking U0 . The unknown u = (γ a b , λa bc , κa b , ψ, ωa , χ) is related to the Einstein-matter variables by (2.31)–(2.36). The quantities A and f appearing in Equation (1.5) are given by the evolution Equations (2.39)–(2.43) and t ∂t χ + βχ = tα0 −β (tr κ)(A + tβ χ) + t2−β S g ab S ∇a S ∇b 0 φ + t2−ζ S ∇a ωa . (4.4) The argument that the matrix A satisfies the appropriate positivity condition is analogous to the argument in [15]. Regarding the estimate f = O(tδ ), the estimate a a t2−α b S Ra b = O(tδ ) was obtained in Equation (3.9). The estimate t2−α b M a b = O(tδ ) follows from the Inequality (4.3) and from qd < pd . The only other terms in f whose estimates are not immediate from the estimates made in [15] are the last two terms on the right-hand side of Equation (4.4). The covariant derivative compatible with the symmetrized metric is used in Equation (4.4) for convenience. From the estimate S g˜ab t|qa −qb |− [15], Equations (2.53) and (2.54), S ab
g
t−2qmin{a,b} −
and
ec (g(ab) ) t2qmax{a,b} −δ− .
Therefore, S ab S
g
Γcab
=
S ab S ch
t−2qd −δ−3
g
g
1 a ea (g(bh) ) − eh (g(ab) ) − S g ch fah 2
and t2−β S g ab S ∇a S ∇b 0 φ = t2−ζ S ∇a ωa
=
t2−β S g ab {ea (eb ( 0 φ)) − S Γcab ec ( 0 φ)} t2−2qd −β−δ−3 = O(tδ ), t2−ζ S g ab {ea (wb ) − S Γcab wc } t2−2qd −ζ−δ−3 = O(tδ )
The other regularity conditions that f should satisfy are again shown to hold by Equation (31) in [15] and the remarks following Equation (31). That gab is symmetric (so that Equation (4.4) and Equation (4.2) are equivalent) is shown as in Subsection 2.5.3. That the Hamiltonian and momentum constraints are satisfied is shown by the analogue of the argument made in Section 2.6 and the estimate R = o(t−2+η1 ) obtained from Equation (3.8). Note that the case D = 3 of this result has an interesting connection to the Einstein vacuum equations in D = 4. As it follows from standard Kaluza-Klein lines, the solutions of the latter with polarized U (1) symmetry are equivalent to the Einstein-scalar field system in D = 3 (see e.g. [40] and [41], Section 5). Hence the result of this section implies that we have constructed the most general known class of singular solutions of the Einstein vacuum equations in four spacetime dimensions. These spacetimes have one spacelike Killing vector.
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5 Matter fields derived from n-form potentials 5.1
Equations of motion
We now turn to the general system (1.1), but without the interaction terms “more”. These are considered in Section 7 below. The action is the sum of (1.2), (1.3) and k additional terms, each of the form (1.4). The argument is based on that of Section 4. It is enough here to note the differences. Furthermore, since there is no coupling between additional matter fields, the differences from the argument made in Section 4 can be noted for each additional matter field independently of the others. Therefore consider the jth additional matter field, Fµ0 ···µnj = (nj + 1)∇[µ0 Aµ1 ···µnj ] , with A an nj -form. This matter field contributes the following additional terms to the stress-energy tensor, Equation (4.1), 1 1 Tµν = · · · + Fµα1 ···αnj Fν α1 ···αnj − gµν Fα0 ···αnj F α0 ···αnj eλj φ . nj ! 2(nj + 1)! √ Define E a1 ···anj = g F 0a1 ···anj eλj φ . If nj = 0, E is a spatial scalar density. Throughout this section and the next we use the following conventions. If nj = 0, then Pa1 ···anj is a scalar, ga1 b1 · · · ganj bnj = 1, etc. The d + 1 decomposition of the contribution of this matter field to the stress-energy tensor is ρ
=
ja
=
M ab
=
1 ga b · · · ganj bnj E a1 ···anj E b1 ···bnj e−λj φ 2 g nj ! 1 1 1 g a0 b0 · · · g anj bnj Fa0 ···anj Fb0 ···bnj eλj φ , + (5.1) 2(nj + 1)! 1 E b1 ···bnj Fab1 ···bnj , ···+ √ (5.2) g nj ! 1 nj gbh1 gc2 h2 · · · gcnj hnj E ac2 ···cnj E h1 ···hnj ···− g nj !
nj δ a b gc1 h1 · · · gcnj hnj E c1 ···cnj E h1 ···hnj e−λj φ − (d − 1)nj ! 1 g ac g h1 i1 · · · g hnj inj Fch1 ···hnj Fbi1 ···inj + nj !
nj − δ a b g c0 h0 · · · g cnj hnj Fc0 ···cnj Fh0 ···hnj eλj φ . (d − 1)(nj + 1)! ···+
The jth matter field satisfies (D)
∇µ (F µν1 ···νnj eλj φ ) = (D)
∇[µ Fν0 ···νnj ]
=
0,
(5.3)
0,
(5.4)
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with d + 1 decomposition into constraint equations, ea (E
ab2 ···bnj
)+
c fca E ab2 ···bnj
e[a (Fb0 ···bnj ] ) −
1 bi ab2 ···c···bnj + f E 2 i=2 ac
=
0,
(5.5)
(nj + 1) c f[ab0 F|c|b1 ···bnj ] 2
=
0,
(5.6)
nj
and evolution equations, √ h bc0 a1 c1 g g · · · g anj cnj ∂t E a1 ···anj = −eb ( gg bc0 g a1 c1 · · · g anj cnj Fc0 ···cnj eλj φ ) − {fhb 1 ai bc0 a1 c1 √ fbh g g · · · g hci · · · g anj cnj } gFc0 ···cnj eλj φ , 2 i=1 nj
+
1 ∂t Fa0 ···anj = −(nj + 1)e[a0 ( √ ga1 |b1 | · · · ganj ]bnj E b1 ···bnj e−λj φ ) g (nj + 1)nj c f[a0 a1 g|c||b1 | ga2 |b2 | · · · ganj ]bnj E b1 ···bnj e−λj φ . + √ 2 g
(5.7)
(5.8)
The jth matter field contributes the following terms to the evolution Equation (4.2) for φ. λj ga b · · · ganj bnj E a1 ···anj E b1 ···bnj e−λj φ 2 g nj ! 1 1 λj g a0 b0 · · · g anj bnj Fa0 ···anj Fb0 ···bnj eλj φ − 2(nj + 1)!
∂t2 φ − (trk)∂t φ = · · · +
5.2
(5.9)
Velocity-dominated system
The Kasner-like evolution equations corresponding to this matter field are ∂t 0 E a1 ···anj = 0 and ∂t 0 Fa0 ···anj = 0. The quantities 0 E a1 ···anj and 0 Fa0 ···anj are constant in time with analytic spatial dependence and both are totally antisymmetric. The velocity-dominated matter constraint equations are Equations (5.5) and (5.6) with 0 E and 0 F substituted for E and F . Since all quantities in the velocitydominated matter constraints are independent of time, if the matter constraints are satisfied at some time t0 > 0, then they are satisfied for all t > 0. This matter field does not contribute to 0 ρ. Its contribution to 0 ja is the term shown on the right-hand side of Equation (5.2) with 0 g, 0 E and 0 F substituted for g, E and F . The velocity-dominated constraints corresponding to the Hamiltonian and momentum constraints are 0 C = 0 and 0 Ca = 0, with 0 C and 0 Ca given by Equations (2.18) and (2.19). Equations (2.21) and (2.22) are satisfied, so as before, if the velocity-dominated constraints are satisfied at some t0 > 0, then they are satisfied for all t > 0.
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The presence of the matter field A(j) puts the following restrictions on the set V [1]. 2p1 + · · · + 2pnj − λj A > 0
and
2p1 + · · · + 2pd−nj −1 + λj A > 0. (5.10)
The restrictions generalize the Inequalities (2.24) found for a Maxwell field in 4 dimensions and, like them, guarantee that one can asymptotically neglect the pform A(j) in the Einstein-dilaton dynamical equations. (For nj = 0, the inequality on the left of (5.10) is −λj A > 0 while for nj = 1 it is 2p1 −λj A > 0. For nj = d−1, the inequality on the right is λj A > 0, while for nj = d − 2 it is 2p1 + λj A > 0.) The constant σ is reduced from its value in Section 4, if necessary, so that, for all x ∈ U0 , σ < 2p1 + · · · + 2pnj − λj A and σ < 2p1 + · · · + 2pd−nj −1 + λj A. If σ is reduced, it may be necessary to reduce , and in turn shrink U0 , so that the conditions imposed in Section 4 are still all satisfied.
5.3
Fuchsian property – estimates
The jth matter field contributes the following components to the unknown u in the Fuchsian Equation (1.5). E a1 ···anj Fa0 ···anj
= =
0 a1 ···anj
E + tβ ξ a1 ···anj , 0 Fa0 ···anj + tβ ϕa0 ···anj .
(5.11) (5.12)
a1 ···anj Here, β = /100 as above, is a totally antisymmetric spatial tensor den ξ d sity, so contributes nj independent components to u, and ϕa0 ···anj is a totally antisymmetric spatial tensor, so contributes njd+1 components to u. This is consistent with the form of the evolution equations. Note that E a1 ···anj = O(1) and Fa0 ···anj = O(1). This matter field contributes additional rows and columns to the matrix A such that the only non-vanishing new entries are on the diagonal and strictly positive. Therefore, the presence of this matter field does not alter that A satisfies the appropriate positivity condition. The terms in the source f which must be estimated on account of the jth matter field are the following. It contributes terms to the components of f corresponding to κ through its contribution to M a b .
1 gbh gc h · · · gcnj hnj E ac2 ···cnj E h1 ···hnj e−λj φ g 1 2 2 −2qmax{a,b} +2qa +2qmax{b,h } +···+2qmax{c ,h } −λj A−α0 −nj nj nj 1 t
t2−α
a
b
t2q1 +···+2qnj −λj A−α0 −nj = O(t−α0 −nj +σ ) = O(tδ ) t
2−αa b ac h1 i1
g g t
···g
hnj inj
Fch1 ···hnj Fbi1 ···inj e
(5.13)
λj φ
2−2qmax{a,b} +2qa −2qmin{a,c} −2qmin{h1 ,i1 } −···−2qmin{hn
j
,in } +λj A−α0 −(nj +1) j
t2q1 +···+2qd−nj −1 +λj A−α0 −(nj +1) = O(t−α0 −(nj +1) +σ ) = O(tδ )
(5.14)
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Here it is used that both E a1 ···anj and Fa0 ···anj are totally antisymmetric, so that the sums indicated by a summation symbol are not over all indices. Note that the Inequalities (5.10) have been crucially used in getting the Estimates (5.13) and (5.14). The desired estimates for the other two terms are obtained similarly. The terms contributed to the component of f corresponding to χ by the jth matter field are obtained by multiplying the right-hand side of Equation (5.9) by t2−β . 1 t2−β ga1 b1 · · · ganj bnj E a1 ···anj E b1 ···bnj e−λj φ = O(t−β−nj +σ ) = O(tδ ) g
(5.15)
t2−β g a0 b0 · · · g anj bnj Fa0 ···anj Fb0 ···bnj eλj φ = O(t−β−(nj +1) +σ ) = O(tδ ).
(5.16)
a1 ···anj
The terms in f corresponding to ξ for the jth matter field are obtained by multiplying the right-hand side of Equation (5.7) by t1−β . These terms are O(t−β−δ−(nj +1) +σ ) = O(tδ ). The terms in f corresponding to ϕa0 ···anj for the jth matter field are obtained by multiplying the right-hand side of Equation (5.8) by t1−β . These terms are O(t−β−δ−nj +σ ) = O(tδ ). Thus the terms which occur in f due to the jth matter field are O(tδ ). The time derivative of the matter constraint quantities for the jth field (the left-hand side of Equations (5.5) and (5.6)) vanishes. If the velocity-dominated matter constraints are satisfied, the matter constraint quantities are o(1). A quantity which is both constant in time and o(1) must vanish. Therefore the matter constraints for the jth field are satisfied. Next the matter terms due to the jth field in the Einstein constraints are estimated, in order to verify that they are consistent with Equations (2.63) and (2.64). The contribution to the Hamiltonian constraint is, from Equation (5.1), 1 ga b · · · ganj bnj E a1 ···anj E b1 ···bnj e−λj φ = O(t−2−nj +σ ) = o(t−2+η1 ), g 11 g a0 b0 · · · g anj bnj Fa0 ···anj Fb0 ···bnj eλj φ = O(t−2−(nj +1) +σ ) = o(t−2+η1 ).
(5.17) (5.18)
The contribution to the momentum constraint is 1 1 (5.19) ja −0 ja = · · · + ( √ − ) 0 E b1 ···bnj 0 Fab1 ···bnj 0g g 1 + √ (E b1 ···bnj Fab1 ···bnj − 0 E b1 ···bnj 0 Fab1 ···bnj ) = o(t−1+η2 ). g Estimates related to the determinant which are relevant to (5.19) are analogues of the estimates for d = 3 immediately preceding Equation (2.55). The form of √ these estimates for general d will now be presented. These are 1/ g − 1/ 0 g = O(t−1+α0 −d ), ea (˜ g) = O(tα0 −δ−d ), ea (g) = O(t2+α0 −δ−d ), and ea (g −1/2 ) = −
ea (g) = O(t−1+α0 −δ−d ). 2g 3/2
(5.20)
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6 Determination of subcritical domain The explicit determination of the subcritical range of the dilaton couplings for which the inequalities on the Kasner exponents are consistent so that V exists may be a complicated matter. We consider in this section a few cases and give some general rules. As in Subsection 2.4, we introduce the metric dpa 2 − ( dpa )2 + (dA)2 (6.1) dS 2 = Gµν dpµ dpν = in the D-dimensional space of the Kasner exponents (pa , A) ≡ (pµ ). This metric has again Minkowskian signature (−, +, +, · · · , +). The forward light cone is defined by pa > 0. (6.2) Gµν pµ pν = 0, The Kasner conditions met in the previous section are equivalent to the conditions that the Kasner exponents be on the forward light cone (since pa = 1 can always be achieved by positive rescalings). The wall chamber W is now defined by p1 ≤ p2 ≤ · · · ≤ pd 2p1 + p2 + · · · + pd−2 ≥ 0
(6.3) (6.4)
and, for each p-form, λj A≥0 2 λj p1 + p2 + · · · + pd−nj −1 + A ≥ 0. 2 p1 + p2 + · · · + pnj −
(6.5) (6.6)
These inequalities may not be all independent. The question is to determine the “allowed” values of the dilaton couplings for which the wall chamber contains in its interior future-directed lightlike vectors. It is clear that this set is non-empty since the inequalities can be all fulfilled when the couplings are zero (the pa ’s can be chosen to be positive in the presence of a dilaton).
6.1
Einstein-dilaton-Maxwell system in D dimensions
We consider first the case of a single 1-form in D ≥ 4 dimensions. This case is simple because the Inequalities (6.4) are then consequences of (6.5) and (6.6), which read λ λ p1 + p2 + · · · + pd−2 + A ≥ 0. (6.7) p1 − A ≥ 0, 2 2 Furthermore, the number of faces of the wall chamber (defined by these inequalities and (6.3)) is exactly D and the edge vectors form a basis. Thus, the analysis of Subsection 2.4 can be repeated.
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A basis of edge vectors can be taken to be (0,0,···,0,1,0) d−k −2 d−k −2 2(d − k − 2) − ,···,− ,1,···,1,− , k = 1,2,···,d − 2 k+1 k+1 λ(k + 1) 2 (1,1,···,1, ) λ 2(d − 2) ) (1,1,···,1,− λ
(6.8) (6.9) (6.10) (6.11)
In (6.9), the first k components are equal to − d−k−2 k+1 and the next d−k components are equal to 1. The first vector is lightlike. The kth vector in the group (6.9) has squared norm −
(d − 1)[k 2 − k(d − 3) + d] 4(d − k − 2)2 + , (k + 1)2 λ2 (k + 1)2
k = 1, 2, · · · , d − 2
(6.12)
while (6.10) and (6.11) have norm squared equal to −d(d − 1) +
4 λ2
(6.13)
and
4(d − 2)2 , (6.14) λ2 respectively. The subcritical values of λ must (by definition) be such that at least one of the Expressions (6.12), (6.13) or (6.14) is positive. To determine the boundaries ±λc of the subcritical interval, we first note that (6.13) is positive whenever 2/ d(d − 1). Similarly, (6.14) is positive whenever |λ| < Λ2 |λ| < Λ1 , with Λ1 = with Λ2 = 2(d − 2)/ d(d − 1). To analyze the sign of (6.12), we must consider two cases, according to whether k 2 − k(d − 3) + d is positive or negative. If d < 9, the factor k 2 − k(d − 3) + d is always positive (for any choice of k, k = 1, 2, · · · , d − 2) and the Expression (6.12) is positive provided |λ| < Πk , with −d(d − 1) +
2(d − k − 2) Πk = . (d − 1)[k 2 − k(d − 3) + d]
(6.15)
The critical value λc is equal to the largest number among Λ1 , Λ2 and Πk . This largest number is Λ2 for d = 3, 4, 5, 6, Π1 for d = 7 and Π2 for d = 8. We thus have the following list of critical couplings: 2 , d=3 λc = 3 2 λc = √ , d=4 3
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d=5 d=6 d=7 d = 8.
(6.16)
Note that the value of the dilaton coupling that comes from dimensional reduction of vacuum gravity in one dimension higher d λKK = 2 (6.17) d−1 is always strictly greater than the critical value, except for d = 8, where λKK = λc . (The corresponding values of the Kasner exponents are those of the point on the Kasner sphere exhibited in [13] for D = 10, where all gravitational inequalities are marginally fulfilled.) If d ≥ 9, the factor k 2 − k(d − 3) + d is non-positive for d − 3 + (d − 9)(d − 1) d − 3 − (d − 9)(d − 1) ≤k≤ (6.18) 2 2 (this always occurs for k = 3). Thus, the Expression (6.12) is positive for such k’s no matter what λ is. This implies that the critical value of λ is infinite, λc = ∞,
d ≥ 9.
(6.19)
The fact that D = 10 appears as a critical dimension for the Einstein-dilatonMaxwell system, above which the system is velocity-dominated no matter what the value of the dilaton coupling is in the line of the findings of [13], since the edges (6.9) differ from those of the pure gravity wall chambers only by an additional component along the spacelike dilaton direction.
6.2
Einstein-dilaton system with one p-form (p = 0, p = D − 2)
The same geometrical procedure for determining the critical values of the dilaton couplings can be followed when there is only one p-form in the system (p = 0, p = D − 2), because in that case the wall chamber has exactly D faces and the edge vectors form a basis. Indeed, the gravitational Inequalities (6.4) are always consequences of the symmetry Inequalities (6.3) and the form Inequalities (6.5) and (6.6) (for nj = 0 and nj = D − 2), 2p1 +p2 +· · ·+pd−2 = (p1 +· · ·+pnj −
λj λj A)+(p1 +pnj +1 +· · ·+pd−2 + A). (6.20) 2 2
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So, if there is only one p-form (with p = 0 and p = D − 2), the D − 2 symmetry Inequalities (6.3) together with the two form Inequalities (6.5) and (6.6) completely define the wall chamber, which has D faces. We shall not provide an explicit example of a calculation of λc for such a system, since it proceeds as for a 1-form. When there is more than one exterior form, one can still drop the gravitational inequalities (if there is at least one p-form with p = 0 and p = D − 2), but the situation is more involved because the inequalities corresponding to different forms are usually independent, so that the wall chamber has more than D faces (its intersection with the hyperplane pa = 1 is not a simplex). The calculation is then more laborious. The same feature arises for a 0-form, which we now examine.
6.3
0-form in 4 dimensions
We consider the case of a 0-form in 4 spacetime dimensions. As explained above, we impose the condition λ = 0 to the corresponding dilaton coupling7 . Without loss of generality (in view of the φ → −φ symmetry), we can assume λ > 0. The inequalities defining the subcritical domain relevant to the 0-form case can be brought to the form p1 > 0 A>0 λ A>0 2 p2 − p1 > 0
p1 + p2 −
p3 − p2 > 0
(6.21) (6.22) (6.23) (6.24) (6.25)
We denote by α, β, γ, δ and the corresponding border hyperplanes (i.e., α : p1 = 0, β : A = 0 etc). The Inequalities (6.21)–(6.25) guarantee that all potential walls are negligible asymptotically. They are independent. The five faces α, β, γ, δ and intersect along the 7 one-dimensional edges generated by the vectors: e1 = (0, 0, 1, 0) ∈ α ∩ β ∩ γ = α ∩ β ∩ δ = α ∩ γ ∩ δ = β ∩ γ ∩ δ (6.26) e2 = (0, 1, 1, 0) ∈ α ∩ β ∩ 2 e3 = (0, 1, 1, ) ∈ α ∩ γ ∩ λ e4 = (0, 0, 0, 1) ∈ α ∩ δ ∩ e5 = (−1, 1, 1, 0) ∈ β ∩ γ ∩ e6 = (1, 1, 1, 0) ∈ β ∩ δ ∩ 4 e7 = (1, 1, 1, ) ∈ γ ∩ δ ∩ λ
(6.27) (6.28) (6.29) (6.30) (6.31) (6.32)
7 The case λ = 0 is clearly in the subcritical region but must be treated separately because there are then two dilatons. The Kasner conditions read p1 +p2 +· · ·+pd = 1 and p21 +· · · p2d +A21 +A22 = 1, where the scalar fields behave as φ1 ∼ A1 ln t, φ2 ∼ A2 ln t. This allows positive pi ’s, which enables one to drop spatial derivatives as t → 0. The system is velocity-dominated.
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Among these vectors, neither e4 nor e5 bound the subcritical domain since e4 is such that p1 + p2 − (λ/2)A < 0 (changing its sign would make A < 0), while e5 is such that p1 < 0 (changing its sign would make p2 − p1 < 0). The edge-vectors {e1 , e2 , e3 , e6 , e7 } form a complete (but not linearly independent) set. Any vector can be expanded as v = v1 e1 + v2 e2 + v3 e3 + v6 e6 + v7 e7
(6.33)
The coefficients v1 , v2 , v3 , v6 , v7 are not independent but can be changed as v2 → v2 + 2k, v3 → v3 − 2k, v6 → v6 − k, v7 → v7 + k
(6.34)
For (6.33) to be interior to the wall chamber, the coefficients v1 , v2 , v3 , v6 and v7 must fulfill v1 > 0, v2 + v3 > 0, v2 + 2v6 > 0, v3 + 2v7 > 0, v6 + v7 > 0.
(6.35)
Using the above redefinitions, which leave the inequalities invariant, we can make vA ≥ 0, A = 1, 2, 3, 6, 7, with at most two vA ’s equal to zero. Indeed, let s = min(v2 , v3 , 2v6 , 2v7 ). Assume for definiteness that s = v2 (the other cases are treated in exactly the same way). One has then v2 ≤ 2v7 . Take 2k = −s in the redefinitions (6.34). This makes v2 equal to zero and makes v7 equal to v7 −(v2 /2) ≥ 0. Because of (6.35), the new v3 and v6 are strictly positive, as claimed. Thus, one sees that any vector in the wall chamber can be expanded as in (6.33) with nonnegative coefficients. But the vectors e1 , e2 , e3 , e6 and e7 are all future-pointing and timelike or null when λ ≥ 8/3. It follows that for such λ’s, there is no lightlike direction in the interior of the wall chamber. Conversely, if λ < 8/3, the vector e7 is spacelike and one can find an interior vector αe1 + βe2 + e7 (α, β > 0) that is lightlike. We can thus conclude: 8 for a 0-form in 4 dimensions, (6.36) λc = 3 i.e., the system is velocity-dominated for |λ| < 8/3. The action for the matter fields in the case of a 0-form A coupled to a dilaton φ is √ 1 (6.37) Sφ [gαβ , φ, A] = − (∂µ φ ∂ µ φ + eλφ ∂µ A ∂ µ A) −g d4 x 2 Note that this is the action for a wave map (also known as a nonlinear σ-model or hyperbolic harmonic map) with values in a two-dimensional Riemannian manifold of constant negative curvature. Its curvature is proportional to λ2 . Thus we obtain an interesting statement on velocity-dominated behaviour for the Einstein equations coupled to certain wave maps. Note for comparison that wave maps in flat space occurring naturally in the context of solutions of the vacuum Einstein equations with symmetry, for instance in Gowdy spacetimes (cf. [34]), are defined by a Lagrangian of the above type (using the flat metric) with λ = 2.
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Collection of 1-forms
We now turn to a system of several 1-forms. It is clear that if these have all the same dilaton coupling, as in the Yang-Mills Action (1.7), then, the critical value of λ is just that computed in (6.16) and (6.19) since each form brings in the same walls. The situation is more complicated if the dilaton couplings are different. One could naively think that the subcritical domain is then just the Cartesian product (j) (j) of the individual subcritical intervals [−λc , λc ], but this is not true because the intersection of the wall chambers associated with each 1-form may have no interior lightlike direction, even if each wall chamber has some. This is best seen on the example of two 1-forms in D spacetime dimensions with opposite dilaton couplings. The relevant inequalities, from which all others follow, are in this case λ λ A > 0, p1 + A > 0 2 2 p1 < p2 < · · · < pd
p1 −
(6.38) (6.39)
and can be easily analyzed because they determine, in this particular instance, a simplex in the hyperplane pa = 1. It follows from (6.38) that p1 > 0. The edgevectors can be taken to be (0, . . . , 0, 1, . . . , 1, 0) (k zeros, d−k ones, k = 1, . . . , d−1) and (1, 1, . . . , 1, ±2/λ). The first d − 1 edge-vectors are timelike or null, while the last two are spacelike provided −d(d − 1)λ2 + 4 > 0. This yields 2 λc = d(d − 1)
for two 1-forms with opposite dilaton couplings
(6.40)
Accordingly, λc is finite for any spacetime dimension (and in fact, tends to zero as d → ∞), even though λc = ∞ for a single 1-form whenever d > 8.
7 Coupling between the matter fields The actions for the bosonic sectors of the low-energy limits of superstring theories or M-theory contain coupling terms between the p-forms, indicated by “more” in (1.1). These coupling terms are of the Chern-Simons or the Chapline-Manton type. In this section, we show that these terms are consistent with the results obtained in Section 5, in that they are also asymptotically negligible in the dynamical equations of motion when the Kasner exponents are subject to the above Inequalities (6.3)– (6.6). More precisely, the form of the velocity-dominated evolution equations and solutions are in each case exactly as in Section 5. The velocity-dominated matter constraints have additional terms, but as before, the velocity-dominated matter variables (besides the dilaton) are constant in time, so if the constraints are satisfied at some t > 0 they are satisfied for all t > 0. The quantities 0 ρ and 0 ja are defined exactly as in Section 5. Since the velocity-dominated evolution equations
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are also the same, there is nothing additional to check concerning the velocitydominated Hamiltonian and momentum constraints. Turning now to the exact equations, the restrictions defining the set V are unchanged from Section 5. The form of the evolution equation for the dilaton is unchanged. The form of the stress-energy tensor is also unchanged, and so the form of the Einstein evolution equations and the Einstein constraints is unchanged. The additional matter field variables considered in Section 5 are still all O(1), so estimates of terms involving the matter fields do not need to be reconsidered, as long as their form has not changed, for instance, in the argument that the Einstein constraints are satisfied. That the matter constraints are satisfied follows as in the other cases, once it is verified that their time derivative vanishes and that they are o(1). Since so much of the argument is identical to that of Section 5, we only point out the few places where there are differences.
7.1
Chern-Simons terms
First we consider the coupling of i of the additional matter fields via a ChernSimons term in the action. These additional matter fields should be such that i−1+
i
nj = D.
(7.1)
j=1
The Chern-Simons term which is added to the action is (1) (i) SCS [Aγ1 ···γn1 , · · · , Aγ1 ···γni ] = A(1) ∧ dA(2) ∧ · · · ∧ dA(i) .
(7.2)
The variation of this term with respect to both the metric and the dilaton field, φ, vanishes. The matter Equation (5.4) is unchanged, since it is still the case that F (j) = dA(j) for all j. But Equation (5.3) for each of the i coupled matter fields acquires a non-vanishing right-hand side. (D)
√ ∇µ (F (j)µν1 ···νnj eλj φ ) −g (1)
(j−1)
= Cj ···ν1 ···νnj ··· F··· · · · F···
(j+1)
F···
(i)
· · · F···
(7.3)
Here 0...d = 1 and Cj is a numerical factor. Next, considering the d + 1 decomposition of Equation (7.3), the constraint Equation (5.5), for the jth coupled matter field, acquires the following term on its right-hand side, (1)
(j−1)
−Cj ···0b1 ··· F··· · · · F···
(j+1)
F···
(i)
· · · F···
(7.4)
Here all indices which are not explicit are spatial. So, only magnetic fields appear in (7.4). The following term is added to the right-hand side of the evolution
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Equation (5.7) for the jth coupled matter field. −Cj
j−1
(1)
(nm + 1)···0c1 ···cnm ···a1 ···anj ··· F··· · · ·
m=1
1
(j−1) (j+1) (i) × √ gc1 h1 · · · gcnm hnm E (m)h1 ···hnm e−λm φ · · · F··· F··· · · · F··· g i (1) (j−1) (j+1) −Cj (nm + 1)···a1 ···anj ···0c1 ···cm ··· F··· · · · F··· F··· ··· (7.5) m=j+1
1 (i) × √ gc1 h1 · · · gcnm hnm E (m)h1 ···hnm e−λm φ · · · F··· . g Again, all indices which are not explicit are spatial. There is in each term only one electric field. The velocity-dominated matter constraint equations for the jth coupled matter field can be obtained from the “full” matter constraint equations for the same field by substituting the velocity-dominated quantities for all variables. The only additional terms occurring in f are due to Equation (7.5). The form of the mth term on the right-hand side of Equation (7.5) is just like the form of the terms on the right-hand side of Equation (5.8) for the mth coupled field. The factors which differ, comparing the mth term of (7.5) to Equation (5.8) for the mth field, are 0(1). Since in both cases a factor of t1−β is added in order to obtain the terms appearing in f , the estimate that the additional terms in f due to the Chern-Simons coupling are O(tδ ) is obtained just as the corresponding previously obtained estimates.
7.2
Chapline-Manton couplings
Next we consider Chapline-Manton couplings. For definiteness, we treat two explicit examples, leaving to the reader the task of checking that the general case works in exactly the same way. The first coupling is between an n-form A and an (n + 1)-form B and is equivalent to making B massive. Let F = dA + B and H = dB. The gauge transformations are B → B + dη, for arbitrary n-form η, and A → A − η + dγ, for arbitrary (n − 1)-form γ. (If n = 0, then dγ is replaced by a constant scalar and we require that the corresponding constant, λA , in the coupling to the dilaton be nonzero.) The form of the action is the same as in Section 5, but since F now depends on B and not just on A, the variation of the action with respect to B acquires an additional term. Also, it is now the case that dF = H. The matter Equation (5.3) is unchanged for F and Equation (5.4) is unchanged for H. Equation (5.3) for H and Equation (5.4) for F are now as follows. (D)
∇µ (H µν0 ···νn eλB φ ) = (D)
∇[µ Fν0 ···νn ]
=
F ν0 ···νn eλA φ , 1 Hµν0 ···νn . (n + 2)
(7.6) (7.7)
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√ √ Define E a1 ···an = g F 0a1 ···an eλA φ and Da0 ···an = g H 0a0 ···an eλB φ . The matter constraint equations which are affected are 1 bi ab1 ···c···bn f D = −E b1 ···bn , 2 i=1 ac n
c Dab1 ···bn + ea (Dab1 ···bn ) + fca
e[a (Fb0 ···bn ] ) −
(n + 1) c 1 f[ab0 F|c|b1 ···bn ] = Hab0 ···bn , 2 n+2
(7.8) (7.9)
The additional term which appears on the right-hand side of Equation (5.7) for Da0 ···an is √ a0 b0 gg · · · g an bn Fb0 ···bn eλA φ . (7.10) The additional term which appears on the right-hand side of Equation (5.8) for Fa0 ···an is −1 (7.11) √ ga0 b0 · · · gan bn Db0 ···bn e−λB φ . g The velocity-dominated matter constraint equations which are affected can be obtained from Equation (7.8) and (7.9) by substituting the corresponding velocitydominated quantities for all variables. The only additional terms occurring in f are due to Equations (7.10) for D and (7.11) for F . The form of the additional terms in these equations is just like the form of the terms which appear in Equations (5.7) for E and in (5.8) for H. Therefore the estimate that the additional terms are O(tδ ) is obtained just as the corresponding previously obtained estimates. The second Chapline-Manton type coupling is between an n-form A and a (2n)-form B. Let F = dA and H = dB + A ∧ F . The gauge transformations are A → A + dγ, for arbitrary (n − 1)-form γ, and B → B + dη − γ ∧ F , for arbitrary (2n − 1)-form η. (If n = 0 the gauge transformations are A → A + C and B → B + D − CA for constant scalars C and D and we require both λA = 0 and also λB = 0.) The form of the action is again the same as in Section 5. √ √ Define E a1 ···an = g F 0a1 ···an eλA φ and Da1 ···a2n = g H 0a1 ···a2n eλB φ . The matter Equations (5.3) for F and (5.4) for H are affected, only if n is odd. The equation for F which is affected (if n is odd) and its d + 1 decomposition are (D)
∇µ (F µν1 ···νn eλA φ ) =
2 H µν1 ···νn σ1 ···σn Fµσ1 ···σn eλB φ , (n + 1)!
(7.12)
1 bi ab2 ···c···bn 2 Dab2 ···bn h1 ···hn Fah1 ···hn , f E = 2 i=2 ac (n + 1)! (7.13) n
c E ab2 ···bn + ea (E ab2 ···bn ) + fca
∂t E a1 ···an
2 ···− √ (7.14) Da1 ···an b1 ···bn gb1 c1 · · · gbn cn E c1 ···cn g n! 2 √ bh0 a1 h1 + gg g · · · g an hn g c1 hn+1 · · · g cn h2n Hh0 ···h2n Fbc1 ···cn eλB φ . (n + 1)! =
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The equation for H which is affected (if n is odd) and its d + 1 decomposition are (D)
∇[µ0 Hµ1 ···µ2n+1 ] =
(2n + 1)! F[µ ···µ Fµ ···µ ], (n + 1)!(n + 1)! 0 n n+1 2n+1
(7.15)
(2n + 1) c (2n + 1)! f[ab0 H|c|b1 ···b2n ] = F[ab0 ···bn−1 Fbn ···b2n ] , 2 (n + 1)!(n + 1)! (7.16) (2n + 2)! b1 ···bn −λA φ g[a |b | · · · gan−1 |bn | E e Fan ···a2n ] . ∂t Ha0 ···a2n = · · · + √ g (n + 1)!(n + 1)! 0 1 (7.17) The velocity-dominated matter constraint equations which are affected can be obtained from Equation (7.13) and (7.16) by substituting the corresponding velocitydominated quantities for all variables. The only additional terms occurring in f are due to Equations (7.14) for E and (7.17) for H. Here again, the estimate that the additional terms in f are O(tδ ), is just as the estimate of terms appearing already in Section 5, either in Equation (5.7) for D or in Equation (5.8) for F . e[a (Hb0 ···b2n ] ) −
8 Yang-Mills We complete our analysis by proving that Yang-Mills couplings also enjoy the property of not modifying the conclusions. The action is (1.7), with a Yang-Mills field as source in addition to the scalar field considered in Section 4 and with |λ| < λc . The argument is again based on that of Sections 2–5 and it is enough here to note differences. The main one is that one must work with the vector potential instead of the fields themselves, because bare A’s appear in the equations. We could, in fact, have developed the entire previous analysis in terms of the vector potentials, thereby reducing the number of matter constraint equations. We followed a manifestly gauge-invariant approach for easing the physical understanding, but this was not mandatory. The stress-energy tensor is 1 1 ∇µ φ (D) ∇ν φ − gµν (D) ∇α φ (D) ∇α φ + [Fµα · Fν α − gµν Fαβ · F αβ ]eλφ . 2 4 (8.1) We work in the temporal gauge, A0 = 0. The matter fields satisfy the following equations. λ (D) ∇α (D) ∇α φ − Fαβ · F αβ eλφ = 0 (8.2) 4 Tµν =
(D)
(D)
∇µ (F µν eλφ ) + [Aµ , F µν ]eλφ = 0,
Fµν = ∂µ Aν − ∂ν Aµ + [Aµ , Aν ].
(8.3) (8.4)
The Lie Bracket has no intrinsic time dependence. The d+1 decomposition of the stress-energy tensor is expressed in terms of the spatial tensor density
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√ 0a λφ g F e and the antisymmetric spatial tensor Fab . 1 1 1 {(∂t φ)2 + g ab ea (φ)eb (φ) + gab E a · E b e−λφ + g ab g ch Fac · Fbh eλφ }, 2 g 2 (8.5) 1 b = −∂t φ ea (φ) + √ E · Fab , (8.6) g 1 1 = g ac eb (φ) ec (φ) − {gbc E a · E c − δ a b gch E c · E h }e−λφ g 2 1 +{g ac g hi Fch · Fbi − δ a b g ch g ij Fci · Fhj }eλφ . (8.7) 4 =
The matter constraint equation is b ea (E a ) + fba E a + [Aa , E a ] = 0.
(8.8)
The matter evolution equations are λ λ gab E a · E b e−λφ − g ab g ch Fac · Fbh eλφ , (8.9) 2g 4 1 √ i ac a ic √ bh = eb ( gg ac g bh Fch eλφ ) + (fib g + fbi g ) gg Fch eλφ (8.10) 2 1 = − √ gab E b e−λφ . (8.11) g
∂t2 φ − (trk)∂t φ = g ab ∇a ∇b φ + ∂t E a ∂t Aa
Note that we use as basic matter variables Aa and E b (the quantity Fab being then defined in terms of Aa as Fab = ∂a Ab − ∂b Aa + [Aa , Ab ]). The Kasner-like evolution equations are Equations (2.7)–(2.10) and ∂t 0 Aa = 0. We consider analytic solutions of the Kasner-like evolution equations of the form (2.12)–(2.15) along with the quantity 0 Aa which is constant in time. Given a point x0 ∈ Σ, we use an adapted spatial frame on a neighborhood of x0 , U0 , as in Section 3. Thus, 0 gab (t0 ) and K a b are specialized as in that section. There is one velocity-dominated matter constraint equation, obtained from Equations elconstraintym) by replacing E a and Aa with 0 E a and 0 Aa . If the velocity-dominated matter constraint is satisfied at some time t0 > 0, then it is satisfied for all t > 0. Define 0 0
ρ
ja
=
1 (∂t 0 φ)2 , 2
(8.12)
1 = −∂t 0 φ ea ( 0 φ) +
0g
0 b
E · 0 Fab .
(8.13)
The velocity-dominated Einstein constraints are defined as in the other cases. Equations (2.21) and (2.22) are again satisfied, so if the velocity-dominated constraints are satisfied at some t0 , then they are satisfied for all t > 0. The restrictions
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defining the set V are as in Section 5, with nj = 1. The relation of the unknown, u, in Equation (1.5) to the Einstein-matter variables is given by Equations (2.31)– (2.37) and (8.14) Aa = 0 Aa + tβ ϕa . The quantities A and f in Equation (1.5) are given by Equations (2.39)–(2.43) and t ∂t χ + βχ = tα0 −β (tr κ)(A + tβ χ) + t2−β S g ab S ∇a S ∇b 0 φ + t2−ζ S ∇a ωa λ λ +t2−β { gab E a · E b e−λφ − g ab g ch Fac · Fbh eλφ }, (8.15) 2g 4 √ t ∂t ξ a + βξ a = t1−β {eb ( gg ac g bh Fch eλφ ) 1 a ic √ bh i ac +(fib g + fbi g ) gg Fch eλφ }, (8.16) 2 1 t ∂t ϕa + βϕa = −t1−β √ gab E b e−λφ . (8.17) g The estimate that f = O(tδ ) is obtained as before, using E a = O(1) and Fab = O(1). The matter constraint quantity, the left-hand side of Equation (8.8), is o(1) and its time derivative vanishes, so the matter constraint is satisfied. The estimate of the matter terms in the Einstein constraints is obtained as in Section 5 for nj = 1. To conclude: the whole analysis goes through even in the presence of the Yang-Mills coupling terms and the system is asymptotically Kasner-like provided |λ| < λc , where λc is the same as in the abelian case and explicitly given by (6.16) and (6.19).
9 Self-interacting scalar field Consider Einstein’s equations, D ≥ 3, with sources as in Sections 4, 5, 7 or 8, except that the massless scalar field, φ, is replaced by a self-interacting scalar field. That is, the Expression (1.8) is added to the action. Solutions with a monotone singularity can be constructed as in Sections 4–8, with assumptions regarding the function V (φ) which appears in (1.8) given below. There is no change in the velocity-dominated evolution equations and solutions, nor in the velocitydominated constraints. The only change to Equation (1.5) is that two new terms appear in f . There is a new term, t2−α0 δ a b 2 V (φ)/(D − 2), on the right-hand side of the evolution equation for κa b (through M a b ). There is also a new term, −t2−β V (φ), on the right-hand side of the evolution equation for χ. For Equation (1.5) to be Fuchsian, it must be the case that f = O(tδ ) and, in addition, that f satisfy other regularity conditions [15, 28]. Some examples were considered in [31]. A trivial example is obtained by taking V to be a constant. Then the equation for the scalar field is not changed by the potential while its effect on the Einstein equations is equivalent to the
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addition of a cosmological constant. Thus we see that the analysis of [15] generalizes directly to the case of the Einstein-scalar field system with non-zero cosmological constant. Of course the analogous statement applies to the other dimensions and matter fields considered in previous sections. To get another simple example take V (φ) = λφp for a constant λ and an integer p ≥ 2. Showing that the equation is Fuchsian involves examining the expression V (A ln t + B + tβ ψ) = λ(A ln t + B + tβ ψ)p
(9.1)
and corresponding expressions for the first and second derivatives of V . Of course in this particular case these are given by multiples of smaller powers of t. The aim is to estimate these quantities by suitable powers of t. In this case a Fuchsian system is always obtained. A linear massive scalar field is obtained by choosing p = 2. Another interesting possibility is to choose V (φ) = eλφ for a constant λ, in which case the derivatives of V are also exponentials. Then V (A ln t + B + tβ ψ) = eλB tλA exp(λtβ ψ)
(9.2)
Note that such an exponential potential can be (formally) generated by adding, as matter field, a d-form Aµ1 ···µd with dilaton coupling λd = −λ. Indeed, eliminating the field-strength F = dA (which satisfies eλd φ F = Cη, where C is a constant and η the volume form), leads to a term in the action proportional to e−λd φ C 2 . A Fuchsian system is obtained provided the general “electric” p-form condition (5.10) (with nj = d), 2p1 +· · ·+2pd −λd A > 0 is satisfied, i.e., (after using p1 +· · ·+pd = 1 and λd = −λ) provided λA > −2. This therefore yields a restriction on the data. More generally, it is enough to have a function V on the real line which has an analytic continuation to the whole complex plane and which satisfies estimates of the form ˜ = O(1), ˜ + tβ ψ) t2−c1 V˜ (A˜ ln t + B ˜ = O(1), ˜ + tβ ψ) t2−c2 V˜ (A˜ ln t + B 2−c3 ˜ ˜ β ˜ = O(1), ˜ + t ψ) V (A ln t + B t
(9.3)
˜ are the analytic continuafor some positive numbers c1 , c2 and c3 . Here A˜ and B tions of A(x) and B(x), to some (small, simply connected) complex neighborhood of the range of a coordinate chart. And ψ˜ lies in some region of the complex plane containing the origin. For f to be regular, it must be the case that c1 ≥ α0 and c2 ≥ β, which can be achieved by reducing , if necessary, and also possibly U0 , so that previous assumptions are satisfied. By taking suitable account of the domains of the functions involved it is also possible to obtain an analogue of this result when the functions V and V˜ are only defined on some open subsets of R and C. The only other change to the construction given in Sections 4–8 is that ρ → ρ + V (φ). It is still the case that (D) ∇µ T µν = 0, so Equations (2.59) and (2.60) are satisfied. Equation (2.63) is satisfied due to the assumptions concerning V (φ), so the Einstein constraints are satisfied.
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10 Conclusions Our paper establishes the Kasner-like behaviour for vacuum gravity in spacetime dimensions greater than or equal to 11, as well as the Kasner-like behaviour for the Einstein-dilaton-matter systems with subcritical dilaton couplings. Our results can be summarized as follows Theorem 10.1 Let Σ be a d-dimensional analytic manifold, d ≥ 10 and let ( 0 gab , 0 kab ) be a C ω solution of the Kasner-like vacuum Einstein equations on (0, ∞) × Σ such that t tr 0 k = −1 and such that the ordered eigenvalues of −t 0 kab satisfy 1 + p1 − pd − pd−1 > 0. Then there exists an open neighborhood U of {0} × Σ in [0, ∞) × Σ and a C ω solution (gab , kab ) of the Einstein vacuum field equations on U ∩ ((0, ∞) × Σ) such that for each compact subset K ⊂ Σ there are positive real numbers αab for which the following estimates hold uniformly on K: 1. 0 g ac gcb = δ ab + o(tα b ) a
2. k ab = 0 k ab + o(t−1+α b ) a
Theorem 10.2 Let Σ be a d-dimensional analytic manifold, d ≥ 2 and let (j)
X = ( 0 gab , 0 kab , 0 φ, 0 E (j)a1 ···anj , 0 Fa0 ···anj ), with j taking on values 1 through k for some non-negative integer k (possibly 0, in which case j takes on no values), 0 ≤ nj ≤ d−1. Let λj be constants in the subcritical range. Let X be a C ω solution of the Kasner-like Einstein-matter equations on (0, ∞) × Σ such that t tr 0 k = −1, and such that the ordered eigenvalues of −t 0 kab satisfy 1 + p1 − pd − pd−1 > 0 and, for each j, 2p1 + · · · + 2pnj − λj t ∂t 0 φ > 0 and 2p1 + · · · + 2pd−nj −1 + λj t ∂t 0 φ > 0. Then there exists an open neighborhood U of {0} × Σ in [0, ∞) × Σ and a C ω (j) solution (gab , kab , φ, E (j)a1 ···anj , Fa0 ···anj ) of the Einstein-matter field equations on U ∩ ((0, ∞) × Σ) such that for each compact subset K ⊂ Σ there are positive real numbers β, αab , with β < αab , for which the following estimates hold uniformly on K: 1. 0 g ac gcb = δ ab + o(tα b ) a
2. k ab = 0 k ab + o(t−1+α b ) a
3. φ = 0 φ + o(tβ ) 4. E (j)a1 ···anj = 0 E (j)a1 ···anj + o(tβ ) (j)
(j)
5. Fa0 ···anj = 0 Fa0 ···anj + o(tβ )
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Remarks 1. Corresponding estimates hold for certain first order derivatives of the basic unknowns in Theorems 10.1 and 10.2 (cf. Theorem 2.1 in [15]). These are the derivatives which arise in the definition of new unknowns when second order equations are reduced to first order so as to produce a first order Fuchsian system. 2. Our analysis shows that a solution of the full subcritical Einstein-matter equations satisfying the estimates given in the theorems and the corresponding estimates for first order derivatives just mentioned is uniquely determined by the solution of the velocity-dominated equations (the integration functions are included in the zeroth order, Kasner-like solutions; the deviation from them is uniquely determined). 3. The Einstein-matter field equations may include interaction terms of ChernSimons, Chapline-Manton and Yang-Mills type, and the scalar field may be self-interacting, with assumptions on V (φ) as stated in Section 9. If the (j) (j) (j) jth field is a Yang-Mills field, then Fab is obtained from Aa and 0 Fab is (j) obtained from 0 Aa through Equation (8.4). Note that the condition on tr 0 k which is assumed in both theorems can always be arranged by means of a time translation. 4. The spacetimes of the class whose existence is established by these theorems have the desirable property that it is possible to determine the detailed nature of their singularities by algebraic calculations. This allows them to be checked for consistency with the cosmic censorship hypothesis. What should be done from this point of view is to check that some invariantly defined physical quantity is unbounded as the singularity at t = 0 is approached. This shows that t = 0 is a genuine spacetime singularity beyond which no regular extension of the spacetime is possible. For this purpose it is common to examine curvature invariants but in fact it is just as good if an invariant of the matter fields can be found which is unbounded in the approach to t = 0. This is particularly convenient in the cases where a dilaton is present. Then ∇α φ∇α φ is equal in leading order to the corresponding velocity-dominated quantity and the latter is easily seen to diverge like t−4 for t → 0. The vacuum case is more difficult. It will be shown below that the approximation of the full solution by the velocity-dominated solution is sufficiently good that it is enough to do the calculation for the velocity-dominated metric. This means that it is enough to do the calculation for the Kasner metric in D dimensions. Note that the Kasner metric is invariant under reflection in each of the spatial coordinates. Hence curvature components of the form R0abc vanish, as do components of the form R0a0b with a = b. Hence the Kretschmann scalar Rαβγδ Rαβγδ is a sum of non-negative terms of the form Rabcd Rabcd and (Ra 0a0 )2 . In order to show that the Kretschmann scalar
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is unbounded it is enough to show that one of these terms is unbounded. A simple calculation shows that (Ra 0a0 )2 = p2a (1 − pa )2 t−4 in a Kasner spacetime. Thus the curvature invariant under consideration can only be bounded as t → 0 if all Kasner exponents are zero or one, which does not occur for the solutions we construct. To see that the approximation of the full solution by the velocity-dominated solution is valid for determining the asymptotics of the Kretschmann scalar it is enough to note that all terms appearing in the Kretschmann scalar which were not just considered are o(t−4 ). Only two estimates additional to those already obtained are needed ˜ h = O(t−2+ ) and ∇ ˜ a k˜b = O(t−2+ ) are – for these, the estimates R c abc sufficient. Both of these estimates are straightforward to obtain. The main ˜ c = O(t−1+4σ−2 −δ ) (i.e., the connection coefficients do not need input is Γ ab to be expanded). The expression for the Kretschmann scalar is 4((tr k)k a b − k a c κc b )((tr k)k b a − k b h k h a ) +(k a b k c h − k a h k c b )(k b a k h c − k h a k b c ) +4{(Ra b − M a b )(Rb a − M b a ) + 2(Ra b − M a b )((tr k)k b a − k b h k h a ) ˜ jR ˜ i ˜ a k˜b )(∇ ˜ h k˜c )˜ ˜ a k˜b )(∇ ˜ b k˜a )˜ −2(∇ g ah − 2(∇ g ch } − g˜ab g˜ch R c
b
c
h
aci
bhj
˜ abc h (k˜ ai k˜b h − k˜ah k˜b i )˜ +2R g ci . Apart from the Kasner terms (which can each be written as two factors, with each factor O(t−2 )), the remaining terms can each be written as two factors, with each factor O(t−2 ) and at least one of the two factors o(t−2 ). 5. We have constructed large classes of solutions of the Einstein-matter equations with velocity-dominated singularities for matter models defined by those field theories where the BKL picture predicts that solutions of this kind should exist. No symmetry assumptions were made. When symmetry assumptions are made there are more possibilities of finding specialized classes of spacetimes with velocity-dominated singularities. See for instance [45], where there are results for the Einstein-Maxwell-dilaton and other systems under symmetry assumptions. There are also results for the case where the Einstein equations are coupled to phenomenological matter models such as a perfect fluid and certain symmetry assumptions are made. For one of the most general results of this kind so far see [46]. 6. When solutions are constructed by Fuchsian methods as is done is this paper there is the possibility of algorithmically constructing an expansion of the solution about the singularity to all orders which is convergent when the input data are analytic, as in this paper. (If the input data are only C ∞ the expansion is asymptotic in a rigorous sense when Fuchsian techniques can be applied.) At the same time, there is the possibility of providing a rigorous confirmation of the reliability of existing expansions such as those of [24] and [25]. This is worked out for the case of [24] in [28].
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Acknowledgments We thank Mme Choquet-Bruhat for comments which led to clarifications in the exposition. The work of MH and MW is supported in part by the “Actions de Recherche Concert´ees” of the “Direction de la Recherche Scientifique – Communaut´e Fran¸caise de Belgique”, by a “Pˆ ole d’Attraction Interuniversitaire” (Belgium) and by IISN-Belgium (convention 4.4505.86). The research of MH is also supported by Proyectos FONDECYT 1970151 and 7960001 (Chile) and by the European Commission RTN programme HPRN-CT-00131, in which he is associated to K. U. Leuven. MW would also like to thank the organizers of the Mathematical Cosmology Program at the Erwin Schr¨ odinger Institute, Summer 2001, where a portion of this work was completed.
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[24] B. Grubiˇsi´c and V. Moncrief, Asymptotic behaviour of the T 3 × R Gowdy spacetimes, Phys. Rev. D 47 2371–2382 (1993). [25] A. Buonanno, T. Damour and G. Veneziano, Pre-big bang bubbles from the gravitational instability of generic string vacua, Nucl. Phys. B 543, 275 (1999) [arXiv:hep-th/9806230]. [26] B.K. Berger and V. Moncrief, Exact U(1) symmetric cosmologies with local Mixmaster dynamics, Phys. Rev. D 62 02359 (2000) [arXiv:gr-qc/0001083]. [27] H. Ringstr¨ om, The Bianchi IX attractor, Ann. H. Poincar´e 2, 405–500 (2001) [arXiv:gr-qc/0006035]. [28] S. Kichenassamy, and A.D. Rendall, Analytic description of singularities in Gowdy spacetimes, Class. Quantum Grav. 15 1339–1355 (1998). [29] A.D. Rendall, Fuchsian analysis of singularities in Gowdy spacetimes beyond analyticity, Class. Quantum Grav. 17 3305-3316 (2000) [arXiv:gr-qc/0004044]. [30] A.D. Rendall, Applications of the theory of evolution equations to general relativity, arXiv:gr-qc/0109028. [31] A.D. Rendall, Blow-up for solutions of hyperbolic PDE and spacetime singularities, in Proceedings of Journees EDP Atlantique, 2000 [arXiv:grqc/0006060]. [32] J. Demaret, J.L. Hanquin, M. Henneaux, P. Spindel and A. Taormina, The Fate Of The Mixmaster Behavior In Vacuum Inhomogeneous Kaluza-Klein Cosmological Models, Phys. Lett. B 175, 129 (1986). [33] B.K. Berger, Influence of scalar fields on the approach to a cosmological singularity, Phys. Rev. D 61, 023508–1-6 (1999) [arXiv:gr-qc/9907083]. [34] B.K. Berger and D. Garfinkle, Phenomenology of the Gowdy universe on T 3 × R, Phys. Rev. D 57, 4767–4777 (1998) [arXiv:gr-qc/9710102]. [35] A.D. Rendall and M. Weaver, Manufacture of Gowdy spacetimes with spikes, Class. Quantum Grav. 18, 2959–2975 (2001) [arXiv:gr-qc/0103102]. [36] D.M. Chitre, Ph. D. thesis, University of Maryland, 1972. [37] C.W. Misner, in: D. Hobill et al. (Eds), Deterministic chaos in general relativity, Plenum, 1994, pp. 317–328 [gr-qc/9405068]. [38] A.A. Kirillov and V.N. Melnikov, Dynamics of Inhomogeneities of Metric in The Vicinity of a Singularity in Multidimensional Cosmology Phys. Rev. D 52, 723 (1995) [gr-qc/9408004]; V.D. Ivashchuk and V.N. Melnikov, Billiard Representation for Multidimensional Cosmology with Multicomponent Perfect Fluid Near the Singularity, Class. Quantum Grav. 12, 809 (1995).
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[39] T. Damour, M. Henneaux, B. Julia and H. Nicolai, Hyperbolic Kac-Moody algebras and chaos in Kaluza-Klein models, Phys. Lett. B 509, 323 (2001) [arXiv:hepn-th/0103094]. [40] H. Nicolai, Two-dimensional gravities and supergravities as integrable system, DESY-91-038 Lectures presented at 30th Schladming Winter School, Schladming, Austria, Feb 27–Mar 5, 1991. [41] L. Andersson, The global existence problem in general relativity, arXiv:grqc/9911032. [42] Y. Choquet-Bruhat and J.W. York, The Cauchy problem, in A. Held (ed.), General Relativity Plenum, New York, 1980. [43] Y. Choquet-Bruhat, J. Isenberg and J.W. York, Einstein constraints on asymptotically Euclidean manifolds, Phys. Rev. D 61, 084034 (2000). ´ Murchadha and J.W. York, Existence and uniqueness of solutions of the [44] N.O. Hamiltonian constraint of general relativity on compact manifolds, J. Math. Phys. 14, 1551–1557, 1973. [45] M. Narita, T. Torii and K. Maeda, Asymptotic singular behaviour of Gowdy spacetimes in string theory, Class. Quantum Grav 17, 4597–4613 (2000) [arXiv:gr-qc/0003013]. [46] K. Anguige, A class of perfect-fluid cosmologies with polarized Gowdy symmetry and a Kasner-like singularity, arXiv:gr-qc/0005086.
Thibault Damour Institut des Hautes Etudes Scientifiques 35, Route de Chartres F-91440 Bures-sur-Yvette France email: [email protected] Marc Henneaux Physique Th´eorique et Math´ematique Universit´e Libre de Bruxelles C.P. 231 B-1050, Bruxelles Belgium and Centro de Estudios Cient´ıficos Casilla 1469 Valdivia
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Chile email: [email protected] Alan D. Rendall Max-Planck-Institut f¨ ur Gravitationsphysik Albert-Einstein-Institut Am M¨ uhlenberg 1 D-14476 Golm Germany email: [email protected] Marsha Weaver Physique Th´eorique et Math´ematique Universit´e Libre de Bruxelles C.P. 231 B-1050, Bruxelles Belgium email: [email protected] Communicated by Sergiu Klainerman submitted 19/02/02, accepted 15/07/02
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Ann. Henri Poincar´e 3 (2002) 1113 – 1181 c Birkh¨ auser Verlag, Basel, 2002 1424-0637/02/0601113-69
Annales Henri Poincar´ e
Adiabatic Vacuum States on General Spacetime Manifolds: Definition, Construction, and Physical Properties W. Junker and E. Schrohe Abstract. Adiabatic vacuum states are a well-known class of physical states for linear quantum fields on Robertson-Walker spacetimes. We extend the definition of adiabatic vacua to general spacetime manifolds by using the notion of the Sobolev wavefront set. This definition is also applicable to interacting field theories. Hadamard states form a special subclass of the adiabatic vacua. We analyze physical properties of adiabatic vacuum representations of the Klein-Gordon field on globally hyperbolic spacetime manifolds (factoriality, quasiequivalence, local definiteness, Haag duality) and construct them explicitly, if the manifold has a compact Cauchy surface.
1 Introduction It has always been one of the main problems of quantum field theory on curved spacetimes to single out a class of physical states among the huge set of positive linear functionals on the algebra of observables. One prominent choice for linear field theories is the class of Hadamard states. It has been much investigated in the past, but only recently gained a deeper understanding due to the work of Radzikowski [41]. He showed that the Hadamard states are characterized by the wavefront set of their two-point functions (see Definition 3.1). This characterization immediately allows for a generalization to interacting fields [8] and puts all the techniques of microlocal analysis at our disposal [27, 28, 29]. They have made possible the construction of the free field theory [31] and the perturbation theory [7] on general spacetime manifolds. On the other hand, there is another well-known class of states for linear field theories on Robertson-Walker spaces, the so-called adiabatic vacuum states. They were introduced by Parker [38] to describe the particle creation by the expansion of cosmological spacetime models. Much work has also been devoted to the investigation of the physical (for a review see [18]) and mathematical [35] properties of these states, but it has never been known how to extend their definition to field theories on general spacetime manifolds. Hollands [24] recently defined these states for Dirac fields on Robertson-Walker spaces and observed that they are in general not of the Hadamard form (correcting an erroneous claim in [31]). It has been the aim of the present work to find a microlocal definition of adiabatic vacuum states which makes sense on arbitrary spacetime manifolds and
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can be extended to interacting fields, in close analogy to the Hadamard states. It turned out that the notion of the Sobolev (or H s -) wavefront set is the appropriate mathematical tool for this purpose. In Appendix B we review this notion and the calculus related to it. After an introduction to the structure of the algebra of observables of the KleinGordon quantum field on a globally hyperbolic spacetime manifold (M, g) in Section 2 we present our definition of adiabatic states of order N (Definition 3.2) in Section 3. It contains the Hadamard states as a special case: They are adiabatic states “of infinite order”. To decide which order of adiabatic vacuum is physically admissible we investigate the algebraic structure of the corresponding GNS-representations. Haag, Narnhofer & Stein [23] suggested as a criterion for physical representations that they should locally generate von Neumann factors that have all the same set of normal states (in other words, the representations are locally primary and quasiequivalent). We show in Section 4.1 (Theorem 4.5 and Theorem 4.7) that this is generally the case if N > 5/2. For the case of pure states on a spacetime with compact Cauchy surface, which often occurs in applications, we improve the admissible order to N > 3/2. In addition, in Section 4.2 we show that adiabatic vacua of order N > 5/2 satisfy the properties of local definiteness (Corollary 4.13) and those of order N > 3/2 Haag duality (Theorem 4.15). These results extend corresponding statements for adiabatic vacuum states on Robertson-Walker spacetimes due to L¨ uders & Roberts [35], and for Hadamard states due to Verch [49]; for their discussion in the framework of algebraic quantum field theory we refer to [21]. In Section 5 we explicitly construct pure adiabatic vacuum states on an arbitrary spacetime manifold with compact Cauchy surface (Theorem 5.10). In Section 6 we show that our adiabatic states are indeed a generalization of the well-known adiabatic vacua on Robertson-Walker spaces: Theorem 6.3 states that the adiabatic vacua of order n (according to the definition of [35]) on a Robertson-Walker spacetime with compact spatial section are adiabatic vacua of order 2n in the sense of our microlocal Definition 3.2. We conclude in Section 7 by summarizing the physical interpretation of our mathematical analysis and calculating the response of an Unruh detector to an adiabatic vacuum state. It allows in principle to physically distinguish adiabatic states of different orders. Appendix A provides a survey of the Sobolev spaces which are used in this paper.
2 The Klein-Gordon field in globally hyperbolic spacetimes We assume that spacetime is modeled by a 4-dimensional paracompact C ∞ -manifold M without boundary endowed with a Lorentzian metric g of signature (+ − −−) such that (M, g) is globally hyperbolic. This means that there is a 3-dimensional smooth spacelike hypersurface Σ (without boundary) which is intersected by each inextendible causal (null or timelike) curve in M exactly once. As a consequence M is time-orientable, and we fix one orientation once and for all defining
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“future” and “past”. Σ is also assumed to be orientable. Our units are chosen such that = c = G = 1. In this work, we are concerned with the quantum theory of the linear KleinGordon field in globally hyperbolic spacetimes. We first present the properties of the classical scalar field in order to introduce the phase space that underlies the quantization procedure. Then we construct the Weyl algebra and define the set of quasifree states on it. The material in this section is based on the papers [36, 13, 32]. Here, all function spaces are considered to be spaces of real-valued functions. Let us start with the Klein-Gordon equation (2g + m2 )Φ
= =
(g µν ∇µ ∇ν + m2 )Φ √ 1 √ ∂µ (g µν g ∂ν Φ) + m2 Φ = 0 g
(1)
for a scalar field Φ : M → R on a globally hyperbolic spacetime (M, g) where g µν is the inverse matrix of g = (gµν ), g := | det(gµν )|, ∇µ the Levi-Civita connection associated to g and m > 0 the mass of the field. Since (1) is a hyperbolic differential equation, the Cauchy problem on a globally hyperbolic space is well posed. As a consequence (see e.g. [13]), there are two unique continuous linear operators E R,A : D(M) → C ∞ (M) with the properties (2g + m2 )E R,A f = E R,A (2g + m2 )f = f supp (E A f ) ⊂ J − (supp f ) supp (E R f ) ⊂ J + (supp f ) for f ∈ D(M) where J +/− (S) denotes the causal future/past of a set S ⊂ M, i.e., the set of all points x ∈ M that can be reached by future/past-directed causal (i.e., null or timelike) curves emanating from S. They are called the advanced (E A ) and retarded (E R ) fundamental solutions of the Klein-Gordon equation (1). E := E R − E A is called the fundamental solution or classical propagator of (1). It has the properties (2g + m2 )Ef = E(2g + m2 )f = 0 supp (Ef ) ⊂ J + (supp f ) ∪ J − (supp f )
(2)
for f ∈ D(M). E R , E A and E can be continuously extended to the adjoint operators E R , E A , E : E (M) → D (M) by E R = E A , E A = E R , E = −E. Let Σ be a given Cauchy surface of M with future-directed unit normal field nα .
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Then we denote by ρ0 : C ∞ (M) → C ∞ (Σ) u → u|Σ ρ1 : C (M) → C ∞ (Σ) ∞
(3)
→ ∂n u|Σ := (n ∇α u)|Σ α
u
the usual restriction operators, while ρ0 , ρ1 : E (Σ) → E (M) denote their adjoints. Dimock [13] proves the following existence and uniqueness result for the Cauchy problem: Proposition 2.1 (a) Eρ0 , Eρ1 restrict to continuous operators from D(Σ) (⊂ E (Σ)) to E(M) (⊂ D (M)), and the unique solution of the Cauchy problem (1) with initial data u0 , u1 ∈ D(Σ) is given by u = Eρ0 u1 − Eρ1 u0 .
(4)
(b) Furthermore, (4) also holds in the sense of distributions, i.e., given u0 , u1 ∈ D (Σ), there exists a unique distribution u ∈ D (M) which is a (weak) solution of (1) and has initial data u0 = ρ0 u, u1 = ρ1 u (the restrictions in the sense of Proposition B.7). It is given by u(f ) = −u1 (ρ0 Ef ) + u0 (ρ1 Ef ) for f ∈ D(M). (c) If u is a smooth solution of (1) with supp u0,1 contained in a bounded subset O ⊂ Σ then, for any open neighborhood U of O in M, there exists an f ∈ D(U) with u = Ef . Inserting u = Ef into both sides of Equation (4) we get the identity E = Eρ0 ρ1 E − Eρ1 ρ0 E
(5)
on D(M). Proposition 2.1 allows us to describe the phase space of the classical field theory and the local observable algebras of the quantum field theory in two different (but equivalent) ways. One uses test functions in D(M), the other the Cauchy data with compact support on Σ. The relation between them is then established with the help of the fundamental solution E and Proposition 2.1: ˜ σ ˜ := D(M)/ker E, Let (Γ, ˜ ) be the real linear symplectic space defined by Γ ˜ is independent of the choice of representatives f1 , f2 ∈ σ ˜ ([f1 ], [f2 ]) := f1 , Ef2 . σ ˜ For any open D(M) and defines a non-degenerate symplectic bilinear form on Γ. ˜ ˜ ˜ U ⊂ M there is a local symplectic subspace (Γ(U), σ ˜ ) of (Γ, σ ˜ ) defined by Γ(U) := ˜ D(U)/ker E. To a symplectic space (Γ, σ ˜ ) there is associated (uniquely up to ∗˜ σ isomorphism) a Weyl algebra A[Γ, ˜ ], which is a simple abstract C ∗ -algebra gen˜ that satisfy erated by the elements W ([f ]), [f ] ∈ Γ, W ([f ])∗ = W ([f ])−1 = W ([−f ]) (unitarity) i
W ([f1 ])W ([f2 ]) = e− 2 σ˜ ([f1 ],[f2 ]) W ([f1 + f2 ])
(Weyl relations)
(6)
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˜ (see e.g. [3]). The Weyl elements satisfy the “field equation” for all [f ], [f1 ], [f2 ] ∈ Γ 2 W ([(2g + m )f ]) = W (0) = 1. (In a regular representation we can think of the ˆ ˆ ]) is the usual field elements W ([f ]) as the unitary operators eiΦ([f ]) where Φ([f operator smeared with test functions f ∈ D(M) and satisfying the field equation 2 ˆ ]) = Φ([(2 ˆ (2g + m2 )Φ([f g + m )f ]) = 0. (6) then corresponds to the canonical commutation relations.) A local subalgebra A(U) (U an open bounded subset of ˜ M) is then given by A[Γ(U), σ ˜ ]. It is the C ∗ -algebra generated by the elements W ([f ]) with supp f ⊂ U and contains thequantum observables measurable in the ˜ σ spacetime region U. Then A[Γ, ˜ ] = C ∗ ( U A(U)). Dimock [13] has shown that U → A(U) is a net of local observable algebras in the sense of Haag and Kastler [22], i.e., it satisfies (i) (ii) (iii) (iv) (v)
U1 ⊂ U2 ⇒ A(U 1 ) ⊂ A(U 2 ) (isotony). U1 spacelike separated from U2 ⇒ [A(U 1 ), A(U 2 )] = {0} (locality). There is a faithful irreducible representation of A (primitivity). U1 ⊂ D(U2 ) ⇒ A(U 1 ) ⊂ A(U 2 ). For any isometry κ : (M, g) → (M, g) there is an isomorphism ακ : A → A such that ακ [A(U)] = A(κ(U)) and ακ1 ◦ ακ2 = ακ1 ◦κ2 (covariance).
In (iv), D(U) denotes the domain of dependence of U ⊂ M, i.e., the set of all points x ∈ M such that every inextendible causal curve through x passes through U. Since we are dealing with a linear field equation we can equivalently use the time zero algebras for the description of the quantum √ field theory. To this end we pick a Cauchy surface Σ with volume element d3 σ := h d3 x, where h := det(hij ) and hij is the Riemannian metric induced on Σ by g, and define a classical phase space (Γ, σ) of the Klein-Gordon field by the space Γ := D(Σ)⊕D(Σ) of real-valued initial data with compact support and the real symplectic bilinear form σ :Γ×Γ (F1 , F2 )
→ R → − [q1 p2 − q2 p1 ] d3 σ,
(7)
Σ
Fi := (qi , pi ) ∈ Γ, i = 1, 2. In this case, the local subspaces Γ(O) := D(O) ⊕ D(O) are associated to bounded open subsets O ⊂ Σ. The next proposition establishes the equivalence between the two formulations of the phase space: ˜ Proposition 2.2 The spaces (Γ(O), σ) and (Γ(D(O)), σ˜ ) are symplectically isomorphic. The isomorphism is given by ˜ ρΣ : Γ(D(O))
→ Γ(O)
[f ] → (ρ0 Ef, ρ1 Ef ). The proof of the proposition is a simple application of Proposition 2.1 and Equation (5). It shows in particular that the symplectic form σ, Equation (7), is independent of the choice of Cauchy surface Σ.
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Now, to (Γ, σ) we can associate the Weyl algebra A[Γ, σ] with its local subalgebras A(O) := A[Γ(O), σ]. By uniqueness, A(O) is isomorphic (as a C ∗ -algebra) to A(D(O)) which should justify our misuse of the same letter A. The ∗-isomorphism is explicitly given by α : A(D(O)) → A(O),
αW ([f ]) := W (ρΣ ([f ])).
In the rest of the paper we will only have to deal with the net O → A(O) of local time zero algebras, since they naturally occur when one discusses properties of a linear quantum field theory. Nevertheless, by the above isomorphism, one can translate all properties of this net easily into statements about the net U → A(U) and vice versa. Let us only mention here that locality of the time zero algebras means that [A(O1 ), A(O2 )] = {0} if O1 ∩ O2 = ∅. The states on an observable algebra A are the linear functionals ω : A → C satisfying ω(1) = 1 (normalization) and ω(A∗ A) ≥ 0 ∀A ∈ A (positivity). The set of states on our Weyl algebra A[Γ, σ] is by far too large to be tractable in a concrete way. Therefore, for linear systems, one usually restricts oneself to the quasifree states, all of whose truncated n-point functions vanish for n = 2: Definition 2.3 Let µ : Γ × Γ → R be a real scalar product satisfying 1 |σ(F1 , F2 )|2 ≤ µ(F1 , F1 )µ(F2 , F2 ) 4
(8)
for all F1 , F2 ∈ Γ. Then the quasifree state ωµ associated with µ is given by 1
ωµ (W (F )) = e− 2 µ(F,F ) . If ωµ is pure it is called a Fock state. The connection between this algebraic notion of a quasifree state and the usual notion of “vacuum state” in a Hilbert space is established by the following proposition which we cite from [32]: Proposition 2.4 Let ωµ be a quasifree state on A[Γ, σ]. (a) There exists a one-particle Hilbert space structure, i.e., a Hilbert space H and a real-linear map k : Γ → H such that (i) kΓ + ikΓ is dense in H, (ii) µ(F1 , F2 ) = RekF1 , kF2 H ∀F1 , F2 ∈ Γ, (iii) σ(F1 , F2 ) = 2ImkF1 , kF2 H ∀F1 , F2 ∈ Γ. The pair (k, H) is uniquely determined up to unitary equivalence. Moreover: ωµ is pure ⇔ k(Γ) is dense in H. (b) The GNS-triple (Hωµ , πωµ , Ωωµ ) of the state ωµ can be represented as (F s (H), ρµ , ΩF ), where (i) F s (H) is the symmetric Fock space over the one-particle Hilbert space H,
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(ii) ρµ [W (F )] = exp{−i[a∗ (kF ) + a(kF )]}, where a∗ and a are the standard creation and annihilation operators on F s (H) satisfying [a(u), a∗ (v)] = u, v H and a(u)ΩF = 0 for u, v ∈ H. (The bar over a∗ (kF ) + a(kF ) indicates that we take the closure of this operator initially defined on the space of vectors of finite particle number.) (iii) ΩF := 1 ⊕ 0 ⊕ 0 ⊕ . . . is the (cyclic) Fock vacuum. Moreover: ωµ is pure ⇔ ρµ is irreducible. Thus, ωµ can also be represented as ωµ (W (F )) = exp{− 21 ||kF ||2H } (by (a)) or ˆ ) := a∗ (kF ) + a(kF ) is the usual field ωµ (W (F )) = ΩF , ρµ (F )ΩF (by (b)). Φ(F operator on F s (H) and we can determine the (“symplectically smeared”) two-point function as λ(F1 , F2 ) = = =
ˆ 1 )Φ(F ˆ 2 )ΩF ΩF , Φ(F kF1 , kF2 H i µ(F1 , F2 ) + σ(F1 , F2 ) 2
for F1 , F2 ∈ Γ, resp. the Wightman two-point function Λ as ρ0 Ef1 ρ0 Ef2 Λ(f1 , f2 ) = λ , ρ1 Ef1 ρ1 Ef2
(9)
(10)
for f1 , f2 ∈ D(M). The fact that the antisymmetric (= imaginary) part of λ is the symplectic form σ implies for Λ: 1 [f1 E ρ0 ρ1 Ef2 − f1 E ρ1 ρ0 Ef2 ] d3 σ Im Λ(f1 , f2 ) = − 2 Σ 1 f1 , Ef2 = (11) 2 by Equation (5). All the other n-point functions can also be calculated, one finds that they vanish if n is odd and that the n-point functions for n even are sums of products of two-point functions. Once a (quasifree) state ω on the algebra A has been chosen the GNSrepresentation (Hω , πω , Ωω ) of Proposition 2.4 allows us to represent all the algebras A(O) as concrete algebras πω (A(O)) of bounded operators on Hω . The weak closure of πω (A(O)) in B(Hω ), which, by von Neumann’s double commutant theorem, is equal to πω (A(O)) (the prime denoting the commutant of a subalgebra of B(Hω )), is denoted by Rω (O). It is the net of von Neumann algebras O → Rω (O) which contains all the physical information of the theory and is therefore the main object of study in algebraic quantum field theory (see e.g. [21]). One
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of the most straightforward properties is the so-called additivity. It states that if an open bounded subset O ⊂ Σ is the union of open subsets O = i Oi then the von Neumann algebra Rω (O) is generated by the subalgebras Rω (Oi ), i.e., Rω (O) =
Rω (Oi )
.
(12)
i
Additivity expresses the fact that the physical information contained in Rω (O) is entirely encoded in the observables that are localized in arbitrarily small subsets of O. The following result is well known: Lemma 2.5 Let ω be a quasifree state of the Weyl algebra, O an open bounded subset of Σ. Then Rω (O) is additive. Proof. Let (k, H) be the one-particle Hilbert space structure of ω (Proposition 2.4). According to results of Araki [1, 34] Equation (12) holds iff kΓ(O) = span kΓ(Oi )
(13)
where the closure is taken w.r.t. the norm in H. With the help of a partition of unity {χi ; supp χi ⊂ Oi } it is clear that any u = k(F ) ∈ kΓ(O), F ∈ Γ(O), can be written as u = i k(χi F ) ∈ span kΓ(Oi ) (note that the sum is finite since F has compact support in O), and therefore kΓ(O) ⊂ span kΓ(Oi ). The converse inclusion is obvious, and therefore also (13) holds. (More generally, additivity even holds for arbitrary states since already the Weyl algebra A(O) has an analogous property, cf. [3].) Other, more specific, properties of the net of von Neumann algebras will not hold in such general circumstances, but will depend on a judicious selection of (a class of) physically relevant states ω. For the choice of states we make in Section 3 we will investigate the properties of the local von Neumann algebras Rω (O) in Section 4.
3 Definition of adiabatic states As we have seen in the last section, the algebra of observables can easily be defined on any globally hyperbolic spacetime manifold. This is essentially due to the fact that there is a well-defined global causal structure on such a manifold, which allows to solve the classical Cauchy problem and formulate the canonical commutation relations, Equations (6) and (11). Symmetries of the spacetime do not play any role. This changes when one asks for the physical states of the theory. For quantum field theory on Minkowski space the state space is built on the vacuum state which is defined to be the Poincar´e invariant state of lowest energy. A generic spacetime manifold however neither admits any symmetries nor the notion of energy, and it has always been the main problem of quantum field theory on curved spacetime to find a specification of the physical states of the theory in such a situation.
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Using Hadamard’s elementary solution of the wave equation DeWitt & Brehme [11] wrote down an asymptotic expansion of the singular kernel of a distribution which they called the Feynman propagator of a quantum field on a generic spacetime manifold. Since then quantum states whose two-point functions exhibit these prescribed local short-distance singularities have been called Hadamard states. Much work has been devoted to the investigation of the mathematical and physical properties of these states (for the literature see e.g. [32]), but only Kay & Wald [32] succeeded in giving a rigorous mathematical definition of them. Shortly later, in a seminal paper Radzikowski [41] found a characterization of the Hadamard states in terms of the wavefront set of their two-point functions. This result proved to be fundamental to all ensuing work on quantum field theory in gravitational background fields. Since we do not want to recall the old definition of Hadamard states (it does not play any role in this paper) we reformulate Radzikowski’s main theorem as a definition of Hadamard states: Definition 3.1 A quasifree state ωH on the Weyl algebra A[Γ, σ] of the KleinGordon field on (M, g) is called a Hadamard state if its two-point function is a distribution ΛH ∈ D (M × M) that satisfies the following wavefront set condition W F (ΛH ) = C + .
(14)
Here, C + is the positive frequency component of the bicharacteristic relation C = ˙ − that is associated to the principal symbol of the Klein-Gordon operator C + ∪C 2g + m2 (for this notion see [16]), more precisely C C
±
:=
{((x1 , ξ1 ; x2 , ξ2 ) ∈ T ∗ (M × M) \ 0; g µν (x1 )ξ1µ ξ1ν = 0,
:=
g µν (x2 )ξ2µ ξ2ν = 0, (x1 , ξ1 ) ∼ (x2 , ξ2 )} {(x1 , ξ1 ; x2 , ξ2 ) ∈ C; ξ10 ≷0, ξ20 ≷0}
(15) (16)
where (x1 , ξ1 ) ∼ (x2 , ξ2 ) means that there is a null geodesic γ : τ → x(τ ) such that x(τ1 ) = x1 , x(τ2 ) = x2 and ξ1ν = x˙ µ (τ1 )gµν (x1 ), ξ2ν = x˙ µ (τ2 )gµν (x2 ), i.e., ξ1 , ξ2 are cotangent to the null geodesic γ at x1 resp. x2 and parallel transports of each other along γ. The fact that only positive frequencies occur in (14) can be viewed as a remnant of the spectrum condition in flat spacetime, therefore (14) (and its generalization to higher n-point functions in [8]) is also called microlocal spectrum condition. However, condition (14) does not fix a unique state, but a class of states that generate locally quasiequivalent GNS-representations [48]. Now to which extent is condition (14) also necessary to characterize locally quasiequivalent states? In [31] one of us gave a construction of Hadamard states by a microlocal separation of positive and negative frequency solutions of the KleinGordon equation. From these solutions we observed that a truncation of the corresponding asymptotic expansions destroys the microlocal spectrum condition (14) but preserves local quasiequivalence, at least if the Sobolev order of the perturbation is sufficiently low (for Dirac fields an analogous observation was made by
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Hollands [24]). In other words, the positive frequency condition in (14) is not necessary to have local quasiequivalence, but can be perturbed by non-positive frequency or even non-local singularities of sufficiently low order. We formalize this observation by defining a new class of states with the help of the Sobolev (or H s -) wavefront set. For a definition and explanation of this notion see Appendix B. Definition 3.2 A quasifree state ωN on the Weyl algebra A[Γ, σ] of the KleinGordon field on (M, g) is called an adiabatic state of order N ∈ R if its twopoint function ΛN is a distribution that satisfies the following H s -wavefront set condition for all s < N + 32 W F s (ΛN ) ⊂ C + .
(17)
Note, that we did not specify W F s for s ≥ N + 32 in the definition. Hence every adiabatic state of order N is also one of order N ≤ N . In particular, every Hadamard state is also an adiabatic state (of any order). Now the task is to identify those adiabatic states that are physically admissible, i.e., generate the same local quasiequivalence class as the Hadamard states. In [31, Section 3.6] an example of an adiabatic state of order −1 was given that does not satisfy this condition. In Theorem 4.7 we will prove that for N > 5/2 (and in the special case of pure states on a spacetime with compact Cauchy surface already for N > 3/2) the condition is satisfied (and the gap in between will remain unexplored in this paper). For this purpose the following simple lemma will be fundamental: Lemma 3.3 Let ΛH and ΛN be the two-point functions of an arbitrary Hadamard state and an adiabatic state of order N , respectively, of the Klein-Gordon field on (M, g). Then (18) W F s (ΛH − ΛN ) = ∅ ∀s < N + 32 . Proof. From Lemma 5.2 it follows that ∅, s < − 12 s W F (ΛH ) = + C , − 21 ≤ s and therefore W F s (ΛH − ΛN ) ⊂ W F s (ΛH ) ∪ W F s (ΛN ) ⊂ C + , s < N + 32 .
(19)
On the other hand, since ΛH and ΛN have the same antisymmetric part σ ˜ , ΛH − ΛN must be a symmetric distribution, and thus also W F s (ΛH − ΛN ) must be a symmetric subset of T ∗ (M × M), i.e., W F s (ΛH − ΛN ) antisymmetric. However, the only antisymmetric subset of the right-hand side of (19) is the empty set and hence W F s (ΛH − ΛN ) = ∅ for s < N + 32 . In the next section we will use this lemma to prove the result mentioned above and some other algebraic properties of the Hilbert space representations generated by our new states. In Section 5 we will explicitly construct these states
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and in Section 6 we will show that the old and well-known class of adiabatic vacuum states on Robertson-Walker spacetimes satisfies our Definition 3.2 (the comparison with the order of these states led us to the normalization of s chosen in Definition 3.2). Contrary to an erroneous claim in [31], these states are in general no Hadamard states, but in fact “adiabatic states” in our sense. This justifies our naming of the new class of quantum states on curved spacetimes in Definition 3.2.
4 The algebraic structure of adiabatic vacuum representations 4.1
Primarity and local quasiequivalence of adiabatic and Hadamard states
Let A := A[Γ, σ] be the Weyl algebra associated to our phase space (Γ, σ) introduced in Section 2 and A(O) := A[Γ(O), σ] the subalgebra of observables localized in an open, relatively compact subset O ⊂ Σ. Let ωH denote some Hadamard state on A and ωN an adiabatic vacuum state of order N . It is the main aim of this section to show that ωH and ωN are locally quasiequivalent states for all sufficiently large N , i.e., the GNS-representations πωH and πωN are quasiequivalent when restricted to A(O), or, equivalently, there is an isomorphism τ between the von Neumann algebras πωH (A(O)) and πωN (A(O)) such that τ ◦ πωH = πωN on A(O) (see e.g. [6, Section 2.4]). To prove this statement we will proceed as follows: We first notice that πωH A(O) ˜ is quasiequivalent to πωN A(O) ˜ for is quasiequivalent to πωN A(O) if πωH A(O) ˜ ⊃ O. Since to any open, relatively compact set O we can find an open, some O ˜ containing O and having a smooth boundary we can relatively compact set O assume without loss of generality that O has a smooth boundary. Under this assumption we first show that πωN (A(O)) is a factor (for N > 3/2, Theorem 4.5). Now we note that the GNS-representation (πω˜ , Hω˜ , Ωω˜ ) of the partial state ω ˜ := ωN A(O) is a subrepresentation of (πωN A(O), HωN , ΩωN ). This is easy to see: K := {πωN (A)ΩωN ; A ∈ A(O)} is a closed subspace of HωN which is left invariant by πωN (A(O)). Since for all A ∈ A(O) (Ωω˜ , πω˜ (A)Ωω˜ ) = ω ˜ (A) = ωN (A) = (ΩωN , πωN (A)ΩωN ), the uniqueness of the GNS-representation implies that πω˜ and πωN A(O) coincide on K and (πω˜ , Hω˜ , Ωω˜ ) can be identified with (πωN A(O), K, ΩωN ) (up to unitary equivalence). We recall that a primary representation (which means that the corresponding von Neumann algebra is a factor) is quasiequivalent to all its (non-trivial) subrepresentations (see [14, Prop. 5.3.5]). Therefore, πωN A(O) is quasiequivalent to πω˜ = π(ωN A(O)) , and analogously πωH A(O) is quasiequivalent to π(ωH A(O)) . To prove that πωN A(O) and πωH A(O) are quasiequivalent it is therefore sufficient to prove the quasiequivalence of the GNS-representations π(ωN A(O)) and π(ωH A(O)) of the partial states. This will be done in Theorem 4.7 for N > 5/2.
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To get started we have to prove in a first step that the real scalar products µN and µH associated to the states ωN and ωH , respectively, induce the same topology on Γ(O) = C0∞ (O) ⊕ C0∞ (O). Let us denote by HµN (O) and HµH (O) the completion of Γ(O) w.r.t. µN and µH , respectively. R. Verch showed the following result [49, Prop. 3.5]: Proposition 4.1 For every open, relatively compact set O ⊂ Σ there exist positive constants C1 , C2 such that
q q 2 2 C1 qH 1/2 (O) + pH −1/2 (O) ≤ µH , p p
≤ C2 q2H 1/2 (O) + p2H −1/2 (O) for all pq ∈ Γ(O). Theorem 4.2 The topology of HµN (O) coincides with that of HµH (O) whenever ΛN satisfies (17) for N > 3/2. Proof. If (Σ, h) is not a complete Riemannian manifold we can find a function ˜ := f h) is complete [12, f ∈ C ∞ (Σ), f > 0, with f |O = const. such that (Σ, h Ch. XX.18, Problem 6]. Then the Laplace-Beltrami operator ∆h˜ associated with ˜ is essentially selfadjoint on C ∞ (Σ) [9]. The topology on Γ(O) will not be affected h 0 ˜ Without loss of generality we can therefore assume that by switching from h to h. ∆ is selfadjoint. Lemma 3.3 shows that 3 s (M × M) ∀ s < N + . ΛH − ΛN ∈ Hloc 2 In view of the fact that Σ is a hyperplane, Proposition B.7 implies that, for 1 < s < N + 3/2, (ΛH − ΛN )|Σ×Σ
∈
∂n1 (ΛH − ΛN ), ∂n2 (ΛH − ΛN )|Σ×Σ
∈
∂n1 ∂n2 (ΛH − ΛN )|Σ×Σ
∈
s−1 Hloc (Σ × Σ)
(20)
s−2 Hloc (Σ s−3 Hloc (Σ
× Σ)
(21)
× Σ).
(22)
Here, ∂n1 and ∂n2 denote the normal derivatives with respect to the first and second variable, respectively. We denote by λH and λN the scalar products on Γ induced via Equation (10) by ΛH and ΛN , respectively. Since ΛH and ΛN have the same antisymmetric parts we have q1 q1 q2 q2 (µH − µN ) , = (λH − λN ) , p1 p2 p1 p2 q1 q2 = (23) ,M p1 p2 L2 (Σ)⊕L2 (Σ)
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q1 q2 p1 , p2 ∈ Γ, where M is the integral operator with the kernel function ∂n1 ∂n2 (ΛH − ΛN )|Σ×Σ −∂n1 (ΛH − ΛN )|Σ×Σ M (x, y) = . (24) −∂n2 (ΛH − ΛN )|Σ×Σ (ΛH − ΛN )|Σ×Σ
˜ of O and a function Note that M (x, y) = M (y, x)∗ . We next fix a neighborhood O ∞ K = K(x, y) ∈ C0 (Σ × Σ) taking values in 2 × 2 real matrices such that K(x, y) = K(y, x)∗ , x, y ∈ Σ, and the entries Kij of K and Mij of M satisfy the relations K11 − M11 L2 (O× ˜ O) ˜ < K21 − M21 H 1/2 (O× ˜ O) ˜ <
K12 − M12 H 1/2 (O× ˜ O) ˜ < K22 − M22 H 1 (O× ˜ O) ˜ < ,
(25)
where > 0 is to be specified lateron. By K we denote the integral operator induced by K. We let µN
:=
λN
:=
µH + ·, (K − M)· = µN + ·, K· i µN + σ. 2
By ΛN we denote the associated bilinear form on C0∞ (M) × C0∞ (M) ρ0 ρ0 ΛN (f, g) := λN Ef, Eg ρ1 ρ1
(26)
(note that, in spite of our notation, ΛN is not the two-point function of a quasifree state in general). Recall from (3) that ρ0 , ρ1 are the usual restriction operators. The definition of ΛN makes sense, since both ρ0 Eg and ρ1 Eg have compact support in Σ so that λN can be applied. In view of the fact that K is an integral operator with a smooth kernel, also λN − λN = µN − µN is given by a smooth kernel. We claim that also ΛN − ΛN is smooth on M × M: In fact, ρ0 ρ0 (ΛN − ΛN )(f, g) = Ef, K Eg ρ1 ρ1 is given by the Schwartz kernel ∗ ρ0 ρ0 E K E. ρ1 ρ1 Since E is a Lagrangian distribution of order µ = −3/2 (for more details see Section 5 below), while K is a compactly supported smooth function, the calculus of Fourier integral operators [28, Theorems 25.2.2, 25.2.3] show that the composition is also smooth. It follows from an argument of Verch [48, Prop. 3.8] that there are functions φj , ψj ∈ C0∞ (M), j = 1, 2, . . . , such that ΛN (f, g) − ΛN (f, g) =
∞ j=1
σ(f, φj )σ(g, ψj )
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for all f, g ∈ C0∞ (D(O)), satisfying moreover ∞
ΛN (φj , φj )1/2 ΛN (ψj , ψj )1/2 < ∞.
j=1
(An inspection of the proof of [48, Prop. 3.8] shows that it is sufficient for the validity of these statements that ΛN is the two-point function of a quasifree state, ΛN need not be one.) It follows that |ΛN (f, f ) − ΛN (f, f )| ≤
|σ(f, φj )σ(f, ψj )|
j
≤
4ΛN (f, f )1/2 ΛN (φj , φj )1/2 ΛN (f, f )1/2 ΛN (ψj , ψj )1/2
j
= 4ΛN (f, f )
ΛN (φj , φj )1/2 ΛN (ψj , ψj )1/2
j
≤ CΛN (f, f ). Therefore
|ΛN (f, f )| ≤ (1 + C)ΛN (f, f ).
Given q, p ∈ C0∞ (O), we can find f ∈ C0∞ (D(O)) such that q = ρ0 Ef, p = ρ1 Ef (cf. Proposition 2.1). Hence q q q q µN , = λN , = ΛN (f, f ) ≤ (1 + C)ΛN (f, f ) p p p p q q q q = (1 + C)λN , = (1 + C)µN , . (27) p p p p We next claim that for all pq ∈ Γ(O)
q q 2 ≤ C3 q2 1/2 + p , M −1/2 H H (O) (O) p p and
q q 2 ≤ C q2 1/2 , (K − M) + p −1/2 H H (O) (O) , p p
(28)
(29)
where C3 and C are positive constants and C can be made arbitrarily small by taking small in (25). Indeed, in order to see this, we may first multiply the kernel functions M and K − M , respectively, by ϕ(x)ϕ(y) where ϕ is a smooth function ˜ of O and ϕ ≡ 1 on O. The above expressions supported in the neighborhood O (28) and (29) will not be affected by this change. We may then localize the kernel functions to R3 × R3 noting that the Sobolev regularity is preserved. Now we can apply Lemma 4.3 and Corollary 4.4 to derive (28) and (29).
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We finally obtain the statement of the theorem from the estimates
C1 qH 1/2 (O) + pH −1/2 (O) ≤ (C1 − C ) qH 1/2 (O) + pH −1/2 (O) 2 if is sufficiently small q q q q , + , (K − M) ≤ µH p p p p by Prop. 4.1 and (29) q q = µN , p p q q ≤ (1 + C)µN , by (27) p p q q q q , − ,M = (1 + C) µH p p p p
≤ (1 + C)(C2 + C3 ) qH 1/2 (O) + pH −1/2 (O) by Prop. 4.1 and (28). Lemma 4.3 Let k ∈ H 1/2 (Rn × Rn ). Then the integral operator K with kernel k induces an operator in B(H 1/2 (Rn ), H 1/2 (Rn )) and B(H −1/2 (Rn ), H −1/2 (Rn )). If we even have k ∈ H 1 (Rn × Rn ), then K induces an operator in B(H −1/2 (Rn ), H 1/2 (Rn )). In both cases, the operator norm of K can be estimated by the Sobolev norm of k. Proof. The boundedness of K : H ±1/2 → H ±1/2 is equivalent to the boundedness of L := D ±1/2 KD ∓1/2 on L2 (Rn ). (Here D ±1/2 := (1 − ∆)±1/4 , where ∆ is the Euclidean Laplacian.) This in turn will be true, if its integral kernel l(x, y) := Dx ±1/2 Dy ∓1/2 k(x, y) is in L2 (Rn × Rn ). In this case LB(L2 (Rn )) ≤ lL2(Rn ×Rn ) . We know that Dx ±1/2 Dy ∓1/2 are pseudodifferential operators on Rn × Rn with 1/2 on and Vaillancourt’s Theorem (cf. [33, symbols in S0,0 (R2n × R2n ). By Calder´ Theorem 7.1.6]), they yield bounded maps H 1/2 (Rn × Rn ) → L2 (Rn × Rn ). Hence l ∈ L2 (Rn ×Rn ) and we obtain the first assertion. For the second assertion we check that Dx 1/2 Dy 1/2 k(x, y) ∈ L2 (Rn × Rn ). Since the symbol of Dx 1/2 Dy 1/2 is 1 (R2n × R2n ), this holds whenever k ∈ H 1 . in S0,0 Corollary 4.4 If
k=
k11 k21
k12 k22
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with k11 ∈ L2 (Rn × Rn ),
k12 , k21 ∈ H 1/2 (Rn × Rn ),
k22 ∈ H 1 (Rn × Rn ),
then the integral operator K with kernel k induces a bounded map H 1/2 (Rn ) ⊕ H −1/2 (Rn ) → H −1/2 (Rn ) ⊕ H 1/2 (Rn ). Given (q, p) ∈ C0∞ (Rn ) ⊕ C0∞ (Rn ) we can estimate q q q K q ,K ≤ p p H −1/2 ⊕H 1/2 p H 1/2 ⊕H −1/2 p L2 ⊕L2 2 q ≤ K p 1/2 −1/2 . H ⊕H (Here we used the fact that, for u ∈ H s (Rn ) and v ∈ H −s (Rn ), u, v can be understood as the extension of the L2 bilinear form and |u, v | ≤ uH s vH −s .) We now apply Theorem 4.2 to show that adiabatic vacua (of order N > 3/2) generate primary representations. The proof is a modification of the corresponding argument for Hadamard states due to Verch [48]. Theorem 4.5 Let ωN be an adiabatic vacuum state of order N > 3/2 on the Weyl algebra A[Γ, σ] of the Klein-Gordon field on (M, g) and πωN its GNSrepresentation. Then, for any open, relatively compact subset O ⊂ Σ with smooth boundary, πωN (A(O)) is a factor. In the proof of the theorem we will need the following lemma. Recall that the ˜ introduced in the proof of Theorem 4.2 differs from h only by a conformal metric h factor which is constant on O. ˜ Lemma 4.6 C0∞ (O) + C0∞ (Σ \ O) is dense in C0∞ (Σ) w.r.t. the norm of H 1/2 (Σ, h) −1/2 ˜ (Σ, h)). (and hence also w.r.t. the norm of H Proof. Using a partition of unity we see that the problem is local. We can therefore confine ourselves to a single relatively compact coordinate neighborhood and work on Euclidean space. In view of the fact that ˜h is positive definite, the topology of the Sobolev spaces on Σ locally yields the usual Sobolev topology. The problem therefore reduces to showing that every function in C0∞ (Rn ), n ∈ N, can be approximated by functions in C0∞ (Rn+ ) + C0∞ (Rn− ) in the topology of H 1/2 (Rn ). Following essentially a standard argument [45, 2.9.3] we proceed as follows. We choose a function χ ∈ C ∞ (R) with χ(t) = 1 for |t| ≥ 2 and χ(t) = 0 for |t| ≤ 1, 0 ≤ χ ≤ 1. We define χ : Rn → R by χ (x) := χ(xn / ). Given f ∈ C0∞ (Rn ) we have √ f − χ f L2 (Rn ) ≤ C1 C2 f − χ f H 1 (Rn ) ≤ √ .
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Interpolation shows that {f − χ f }0<<1 is bounded in H 1/2 (Rn ) [45, Theorem 1.9.3]. Since H 1/2 is a reflexive space, there is a sequence j → 0 such that f −χj f converges weakly [51, Theorem V.2.1]. The limit necessarily is zero, since it is zero in L2 . According to Mazur’s Theorem [51, Theorem V.1.2] there is, for each k k δ > 0, a finite convex combination j=1 αj (f − χj f ) (with αj ≥ 0, j=1 αj = 1) such that k αj (f − χj f ) − 0H 1/2 < δ.
j=1
C0∞ (Rn+ )
+ C0∞ (Rn− ), the proof is complete. Since αj χj f ∈ Proof of Theorem 4.5. Let (kN , HN ) be the one-particle Hilbert space structure of ωN , let kN (Γ(O))∨ := {u ∈ HN ; Imu, v HN = 0 ∀v ∈ kN (Γ(O))}
(30)
denote the symplectic complement of kN (Γ(O)). It is a closed, real subspace of HN . According to results of Araki [1, 34] πωN (A(O)) is a factor iff kN (Γ(O)) ∩ kN (Γ(O))∨ = {0},
(31)
where the closure is taken w.r.t. the norm in HN . ˜ H) ˜ of In a first step we prove (31) for the one-particle Hilbert space structure (k, an auxiliary quasifree state on A[Γ, σ ˜ ], where σ ˜ is the symplectic form w.r.t. the metric ˜ h, ˜ =: H ˜ k˜ : Γ → L2 (Σ, h)
q 1 → √ iD 1/2 q + D −1/2 p p 2
(32)
(which, in general, induces neither a Hadamard state nor an admissible adiabatic vacuum state). As before, D := (1 − ∆h˜ )1/2 . Since ∆h˜ is essentially selfadjoint √ ˜ ˜ (since on C0∞ (Σ), 2 k(Γ) = iD 1/2 C0∞ (Σ) + D −1/2 C0∞ (Σ) is dense in L2 (Σ, h) ∞ 1/2 −1/2 ˜ C0 (Σ) is dense in H (Σ) as well as in H (Σ)), i.e., k describes a pure state. Note that locally, i.e., on Γ(O), the norm given by ˜ kF ˜ ˜ = 1 D 1/2 q2˜ + D −1/2 p2˜ , F := (q, p) ∈ Γ(O), µ ˜ (F, F ) := kF, H H H 2 (33) is independent of the choice of metric and hence equivalent to the norm of H 1/2 (O) ⊕ H −1/2 (O). Also, since the conformal factor f satisfies f = C > 0 on O, we have ˜ 1 , kF ˜ 2 ˜ = σ ˜ (F1 , F2 ) = 2ImkF d3 σh˜ (p1 q2 − q1 p2 ) H O 3/2 = C d3 σh (p1 q2 − q1 p2 ) = C 3/2 σ(F1 , F2 ) O
for all Fi = (qi , pi ) ∈ Γ(O), i = 1, 2.
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∨ ˜ ˜ Imu, v ˜ = 0 ∀v ∈ k(Γ(O))} ˜ Define now k(Γ(O)) := {u ∈ H; and let u ∈ H ∨ ˜ ˜ ˜ k(Γ(O)) ∩ k(Γ(O)) . Then Imu, k(F ) H˜ = 0 for all F ∈ Γ(O) (by the definition of ∨ ∨ ˜ ˜ ) ˜ = 0 for all F ∈ Γ(Σ\ O) (since k(F ˜ ) ∈ k(Γ(O)) ˜ k(Γ(O)) ) and also Imu, k(F H for F ∈ Γ(Σ\O)). This, together with the density statement of Lemma 4.6, implies ˜ ) ˜ = 0 for all F ∈ Γ, and, since k(Γ) ˜ ˜ it follows that that Imu, k(F is dense in H, H u = 0, i.e., (31) is proven for the auxiliary state given by k˜ on A[Γ, σ ˜ ]. Let us now show (31) for an adiabatic vacuum state ωN , N > 3/2, on A[Γ, σ]. Let u ∈ kN (Γ(O)) ∩ kN (Γ(O))∨ , then there is a sequence {Fn , n ∈ N} ⊂ Γ(O) with kN (Fn ) → u in HN . Of course, kN (Fn ) is in particular a Cauchy sequence in HN , i.e., µN (Fn − Fm , Fn − Fm ) = kN (Fn ) − kN (Fm )2HN → 0.
By Theorem 4.2, the norm given by µN , N > 3/2, on Γ(O) is equivalent to the norm given by µ ˜, namely that of H 1/2 (O) ⊕ H −1/2 (O). Therefore we also have ˜ n ) − k(F ˜ m )2˜ = µ k(F ˜(Fn − Fm , Fn − Fm ) → 0 H ˜ n ) → v in H ˜ ˜ for some v ∈ k(Γ(O)). and it follows that also k(F For all G ∈ Γ(O) we have the equalities 0 = = =
Imu, kN (G) HN = lim ImkN (Fn ), kN (G) HN n→∞
1 1 lim σ(Fn , G) = C −3/2 lim σ ˜ (Fn , G) n→∞ n→∞ 2 2 −3/2 ˜ n ), k(G) ˜ ˜ C −3/2 lim Imk(F Imv, k(G) ˜ = C ˜, H H n→∞
∨ ˜ n ) → 0 in ˜ ˜ = {0} and therefore k(F ∩ k(Γ(O)) which imply that v ∈ k(Γ(O)) ˜ ˜ H. Since the norms given by kN and k are equivalent on Γ(O) we also have u = limn→∞ kN (Fn ) = 0 in HN , which proves the theorem. Our main theorem is the following:
Theorem 4.7 Let ωN be an adiabatic vacuum state of order N and ωH a Hadamard state on the Weyl algebra A[Γ, σ] of the Klein-Gordon field in the globally hyperbolic spacetime (M, g), and let πωN and πωH be their associated GNSrepresentations. (i) If N > 5/2, then πωN A(O) and πωH A(O) are quasiequivalent for every open, relatively compact subset O ⊂ Σ. (ii) If ωN and ωH are pure states on a spacetime with compact Cauchy surface and N > 3/2, then πωN and πωH are unitarily equivalent. As explained at the beginning of this section it is sufficient to prove the quasiequivalence of the GNS-representations of the partial states ωN A(O) and ωH A(O) for part (i) of the theorem, for part (ii) we can take O = Σ. To this end we shall use a result of Araki & Yamagami [2]. To state it we first need some notation.
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Given a bilinear form µ on a real vector space K we shall denote by µC the extension of µ to the complexification K C of K (such that it is antilinear in the first argument): µC (F1 + iF2 , G1 + iG2 ) := µ(F1 , G1 ) + µ(F2 , G2 ) + iµ(F1 , G2 ) − iµ(F2 , G1 ). The theorem of Araki & Yamagami gives necessary and sufficient conditions for the quasiequivalence of two quasifree states ωµ1 and ωµ2 on the Weyl algebra A[K, σ] of a phase space (K, σ) in terms of the complexified data K C , σ C , and C C C µC i , i = 1, 2. Assuming that µ1 and µ2 induce the same topology on K , denote i i C C C C C C C C C ¯ the nletion. Then µ , µ , and λ := µ + σ , λ := µ + σ extend to by K 1 2 1 1 2 2 2 2 ¯ C by continuity (σ C extends due to (8)). We define bounded positive selfadjoint K ¯ C by operators S1 , S2 , and S2 on K λC j (F, G) λC 2 (F, G)
=
2µC j (F, Sj G),
=
2µC 1 (F, S2 G),
j = 1, 2, ¯ C. F, G ∈ K
(34)
Note that Sj is a projection operator if and only if ωµj is a Fock state. The theorem of Araki & Yamagami [2] then states that the corresponding GNS-representations πω1 and πω2 are quasiequivalent if and only if both of the following two conditions are satisfied: C C (AY1) µC 1 and µ2 induce the same topology on K , 1/2
(AY2) S1
1/2
− S2
¯ C , µC is a Hilbert-Schmidt operator on (K 1 ).
Proof of Theorem 4.7. (i) We choose K = Γ(O) = C0∞ (O) ⊕ C0∞ (O), σ our real symplectic form (7), µH and µN the real scalar products on K defining a Hadamard state and an adiabatic vacuum state of order N > 5/2, respectively, C and check (AY1) and (AY2) for the data K C , σ C , µC H and µN . From Theorem 4.2 we know that the topologies induced by µH and µN on Γ(O) coincide. In view of the fact that µC H (F1 + iF2 , F1 + iF2 ) = µH (F1 , F1 ) + µH (F2 , F2 ),
F1 , F2 ∈ Γ(O),
(and the corresponding relation for µN ), we see that the topologies coincide also on the complexification. Hence (AY1) holds. 1/2 1/2 In order to prove (AY2), we first note that the difference SH − SN for the operators SH and SN induced by µH and µN via (34) will be a Hilbert-Schmidt operator provided that SH −SN is of trace class, cf. [40, Lemma 4.1]. By definition, µC H (F, (SH − SN )G) =
1 C 1 C λH − λC µH − µC N (F, G) = N (F, G). 2 2
(35)
As in (23), (24), our assumption N > 5/2 and Lemma 3.3 imply that there is an integral kernel M = M (x, y) on O × O, given by (24) with entries satisfying (20)–(22), such that
1 C µH − µC N (F, G) = F, MG L2 (O)⊕L2 (O) , 2
F, G ∈ Γ(O),
(36)
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where M is the integral operator with kernel M . We may multiply M by ϕ(x)ϕ(y) ˜ where ϕ is a smooth function supported in a relatively compact neighborhood O of O with ϕ ≡ 1 on O. Equality (36) is not affected by this change. Moreover, as ˜ have smooth we saw in the beginning of this section we may suppose that O and O ˜ boundary. Using a partition of unity it is no loss of generality to assume that O 3 is contained in a single coordinate neighborhood. We then denote by O∗ ⊂ R the C image of O under the coordinate map. We shall use the notation µC H , µN , and M, M also for the push-forwards of these objects. We note that the closure of Γ(O∗ ) with 1/2 −1/2 respect to the topology of H 1/2 (R3 ) ⊕ H −1/2 (R3 ) is H0 (O∗ ) ⊕ H0 (O∗ ) =: H, cf. Appendix A for the notation. The dual space H w.r.t. the extension of ·, · L2 (O∗ )⊕L2 (O∗ ) , denoted by ·, · , is H −1/2 (O∗ ) ⊕ H 1/2 (O∗ ). The inner product C ˜ µC H extends to H. By Riesz’ theorem, µH induces an antilinear isometry θ : H → H C ˜ by θF, G = µH (F, G). Defining instead F, θG = µC H (F, G)
(37)
˜ of antilinear functionals we obtain a linear isometry θ from H to the space H ˜, on H. Complex conjugation provides a (real-linear) isometry between H and H −1/2 1/2 ˜ hence H = H (O∗ ) ⊕ H (O∗ ) as a normed space (and hence as a Hilbert space). We deduce from Lemma 4.8 and Corollary 4.9 below, in connection with the continuity of the extension operator H → H 1/2 (R3 ) ⊕ H −1/2 (R3 ) and the restriction operator H −1/2 (R3 ) ⊕ H 1/2 (R3 ) → H −1/2 (O∗ ) ⊕ H 1/2 (O∗ ), that M induces a mapping M : H → H −1/2 (O∗ ) ⊕ H 1/2 (O∗ ) ˜ by which is trace class. In particular, for G ∈ H, MG defines an element of H F → F, MG . Combining (35)–(37), we see that, for F, G ∈ H, F, MG = F, θ(SH − SN )G .
Hence )=M θ(SH − SN
˜ ), in B(H, H
so that SH − SN = θ−1 M in B(H).
˜ → H is an isometry while M : H → H ˜ As a consequence of the fact that θ−1 : H is trace class, this implies that SH − SN is trace class. (ii) To prove (ii) we apply the technique of Bogoljubov transformations (we follow be the [35] and [50, p. 68f.]). Assume that Σ is compact and let SH , SN , and SN operators induced by a pure Hadamard state ωH resp. a pure adiabatic state ωN of order N > 3/2 via (34). As remarked above, SH and SN are projection operators ¯ C , the closure of the complexification of K := Γ(Σ) w.r.t. µC or µC (since Σ on K H N C is compact, µC H and µN are equivalent on all of Γ(Σ), Theorem 4.2). We make a
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¯ C into direct sum decomposition of K + + HN HH K = ⊕ = ⊕ − − HN HH
¯C
(38)
+ − and 0 on HH/N , and the first such that SH/N has the eigenvalue 1 on HH/N C C decomposition is orthogonal w.r.t. µN , the second w.r.t. µH . We also denote the − + ¯ C onto H+ corresponding orthogonal projections of K H/N resp. HH/N by PH/N := − := 1 − SH/N . From Equations (9) and (34) we obtain for j ∈ SH/N resp. PH/N {H, N }
2µC j (F, Sj G) = ⇒ σ C (F, G)
=
i C C λC j (F, G) = µj (F, G) − σ (F, G) 2 C 2µC j (F, i(2Sj − 1)G) = 2µj (F, Jj G)
(39)
¯ C with the properties J 2 = where Jj := i(2Sj − 1) is a bounded operator on K j + ∗ C −1, Jj = −Jj (w.r.t. µj ). It has eigenvalue +i on Hj and −i on Hj− and is called the complex structure associated to µj . Because of (39) both decompositions in (38) are orthogonal w.r.t. σ C . We now define the Bogoljubov transformation
A B
C D
+ + HN HH : ⊕ → ⊕ − − HN HH
(40)
by the bounded operators + − + − A := PH |H+ , B := PH |H+ , C := PH |H− , D := PH |H− . N
N
N
N
Taking into account Equation (39) and the fact that the decomposition (38) is + orthogonal w.r.t. σ C we obtain for F, G ∈ HN µC N (F, G)
i C = µC N (F, −iJN G) = − σ (F, G) 2 i i + + − − = − σ C (PH F, PH G) − σ C (PH F, PH G) 2 2 i i = − σ C (AF, AG) − σ C (BF, BG) 2 2 C = −iµC H (AF, JH AG) − iµH (BF, JH BG) C = µC H (AF, AG) − µH (BF, BG),
− similarly for F, G ∈ HN C C µC N (F, G) = µH (DF, DG) − µH (CF, CG),
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+ − and for F ∈ HN , G ∈ HN
0 = µC N (F, G)
=
i C σ (F, G) 2 i C + i + − − σ (PH F, PH G) + σ C (PH F, PH G) 2 2 i C i σ (AF, CG) + σ C (BF, DG) 2 2 C iµC H (AF, JH CG) + iµH (BF, JH DG)
=
C −µC H (AF, CG) + µH (BF, DG),
= = =
µC N (F, iJN G) =
hence + + A∗ A − B ∗ B = 1 in B(HN , HN ) − − ∗ ∗ ) D D − C C = 1 in B(HN , HN ∗
∗
B D − A C = 0 in
(41)
− + B(HN , HN ).
In a completely analogous way we can define the inverse Bogoljubov transformation
by
A˜ ˜ B
C˜ ˜ D
+ + HH HN : ⊕ → ⊕ − − HH HN
(42)
˜ := P − | + , C˜ := P + | − , D ˜ := P − | − . A˜ := PN+ |H+ , B N H N H N H H
H
H
H
+ These operators satisfy relations analogous to (41). Moreover, for F ∈ HN ,G ∈ + HH
˜ µC N (F, AG) = = =
i C i C + ˜ ˜ µC N (F, −iJN AG) = − σ (F, AG) = − σ (F, PN G) 2 2 i i + − σ C (F, G) = − σ C (PH F, G) = −iµC H (AF, JH G) 2 2 µC H (AF, G),
+ + i.e. A˜ = A∗ : HH → HN ˜ = −C ∗ : H+ → H− and similarly B H N C˜ = −B ∗ : H− → H+ H
˜ D
N
− − = D∗ : HH → HN .
(43)
From (41) and (43) one easily finds that AA∗ − CC ∗ = 1 DD∗ − BB ∗ = 1 ∗
∗
AB − CD = 0
+ + in B(HH , HH ) − − in B(HH , HH )
in
− + B(HH , HH )
(44)
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and that (42) is the inverse of (40). Moreover, A is invertible with bounded inverse: + It follows from the first Equations in (41) and (44) that A∗ A ≥ 1 on HN and + ∗ ∗ ∗ AA ≥ 1 on HH , hence A and A are injective. Since {0} = Ker(A ) = Ran(A)⊥ , + . For F = AG ∈ Ran(A) we have A−1 F 2H+ = G2H+ ≤ A has dense range in HH N
N
G, A∗ AG H+ = AG2H+ = F 2H+ , i.e., A−1 is bounded and can be defined on N
H
H
+ . all of HH 1/2 1/2 We are now prepared to show that SH − SN is a Hilbert-Schmidt operator on ¯ C , µC ): We write F ∈ K ¯ C as a column vector w.r.t. the decomposition of K ¯C (K H C w.r.t. µH : + + + HH PH F F F = ∈ ⊕ . =: − F− PH F − HH
Then SH F =
+ PH F
=
1 0
0 0
F+ F−
.
(45)
¯C we get by the basis transformation (42) for all F, G ∈ K For SN
1 C λ (G, F ) = µC N (G, SN F ) 2 N + 1 0 PN F PN+ G C = µN , 0 0 PN− G PN− F + A˜ C˜ PH G 1 0 C = µN , − ˜ D ˜ 0 0 PH G B + ∗ ˜∗ A˜ B G 1 0 = µC , H ˜∗ G− 0 0 C˜ ∗ D
µC H (G, SN F ) =
and hence, utilizing (43), + F A −C 1 0 A∗ SN = − F −C ∗ −B D 0 0 F+ AA∗ −AB ∗ . = ∗ ∗ −BA BB F−
A˜ ˜ B
C˜ ˜ D
A˜ ˜ B
C˜ ˜ D
−B ∗ D∗
+ PH F − PH F + F , F−
F+ F−
(46)
+ − ⊕ HH From (45) and (46) we have now on HH 1/2 SH
−
1/2 SN
= =
1 0 0 0 1 0 0 0
1/2
−
−
AA∗ −BA∗
−AB ∗ BB ∗
AZ −1/2 A∗ −BZ −1/2 A∗
1/2
−AZ −1/2 B ∗ BZ −1/2 B ∗
,
(47)
where Z := A∗ A + B ∗ B = 1 + 2B ∗ B is a bounded selfadjoint positive operator + on HN , which has a bounded inverse due to the fact that Z ≥ 1. In Lemma 4.11
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+ − below we will show that (47) is a Hilbert-Schmidt operator on HH ⊕ HH if and only if the operator AA∗ −AB ∗ 1 0 Y := − 0 0 −BA∗ BB ∗
is Hilbert-Schmidt. From (46) and (34) we see that
1 C λ (G, F ) − λC N (G, F ) 2 H
1 C µ (G, F ) − µC N (G, F ) . 2 H
C µC H (G, Y F ) = µH (G, (SH − SN )F ) =
=
C Now we argue as in the proof of part (i): µC H − µN is given by an integral operator M with kernel M , where M has the form (24) with entries satisfying (20)–(22). Using a partition of unity we can transfer the problem to Rn with the Sobolev regularity of the entries preserved. For N > 3/2 the conditions in Remark 4.10 are satisfied. Hence M and also Y are Hilbert-Schmidt operators, i.e., ωH and ωN are quasiequivalent on A[Γ, σ], which, in turn, is equivalent to the unitary equivalence of the representations πωH and πωN , if ωH and ωN are pure states. s Lemma 4.8 Let M ∈ Hcomp (Rn × Rn ), s ≥ 0, and consider the integral operator M with kernel M , defined by (Mu)(x) = M (x, y)u(y) dy, u ∈ C0∞ (Rn ).
If s >
n−1 n n+1 2 , 2, 2 ,
and
n 2
+ 1, respectively, then M yields trace class operators in
B(H 1/2 (Rn ), H −1/2 (Rn )), B(H −1/2 (Rn )), B(H 1/2 (Rn )), B(H −1/2 (Rn ), H 1/2 (Rn )), respectively. Proof. For the first case, s > n−1 2 , write
1 1 1 1 x s+ 2 D s+ 2 M M = D −s− 2 x −s− 2 where x s is the operator of multiplication by x s and D s = op(ξ s ) w.r.t. the flat Euclidean metric of Rn . The first factor is known to be a Hilbert-Schmidt operator on H −1/2 (Rn ). Since M has compact support, it is sufficient to check 1 the Hilbert-Schmidt property of D s+ 2 M in B(H 1/2 (Rn ), H −1/2 (Rn )) or, equivs −1/2 2 n on L (R ). This operator, however, has the integral alently, of D MD kernel Dx s Dy −1/2 M (x, y). We may consider Dx s as the pseudodifferential s operator Dx s ⊗ I on Rn × Rn with a symbol in the class S0,0 (R2n × R2n ) and Dy −1/2 as the pseudodifferential operator I ⊗ Dy −1/2 on Rn × Rn with sym-
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0 bol in S0,0 (R2n × R2n ). By Calder´ on and Vaillancourt’s Theorem, Dx s Dy −1/2 s n n 2 n n maps H (R × R ) to L (R × R ), hence D s MD −1/2 is an integral operator with a square integrable kernel, hence Hilbert-Schmidt, and M is the composition of two Hilbert-Schmidt operators, hence trace class. The proofs of the other cases are similar.
Corollary 4.9 It is well known that the operator M=
M11 M21
M12 M22
H 1/2 (Rn ) H −1/2 (Rn ) ⊕ ⊕ : → H −1/2 (Rn ) H 1/2 (Rn )
is trace class if and only if each of the entries Mij of the matrix is a trace class operator between the respective spaces, cf. e.g., [42, Section 4.1.1.2, Lemma 2]. Denoting by Mij the integral kernel of Mij , M will be trace class if n−1 2 n M12 ∈ H s (Rn × Rn ), s > 2 n+1 s n n M21 ∈ H (R × R ), s > 2 n s n n M22 ∈ H (R × R ), s > + 1. 2 Remark 4.10 In the situation of Corollary 4.9, M will be a Hilbert-Schmidt operator if each of its entries has this property. Using the fact that an integral operator on L2 (Rn ) is Hilbert-Schmidt if its kernel is in L2 (Rn × Rn ), we easily see that it is sufficient for the Hilbert-Schmidt property of M that M11
∈ H s (Rn × Rn ),
M11 M12 , M21 M22
s>
∈ L2 (Rn × Rn ) ∈ H 1/2 (Rn × Rn ) ∈ H 1 (Rn × Rn ).
Lemma 4.11 In the notation of above, the following statements are equivalent: (i) The operator X :=
1 − AZ −1/2 A∗ BZ −1/2 A∗
AZ −1/2 B ∗ −BZ −1/2 B ∗
+ + HH HH : ⊕ → ⊕ − − HH HH
is Hilbert-Schmidt. (ii) The operator Y :=
1 − AA∗ BA∗
AB ∗ −BB ∗
+ + HH HH : ⊕ → ⊕ − − HH HH
is Hilbert-Schmidt. − − (iii) The operator BB ∗ : HH → HH is of trace class.
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Proof. Using again the fact that a 2 × 2-matrix of operators is trace class if and only if each of its entries is a trace class operator [42, Sec. 4.1.1.2, Lemma 2] it is sufficient to show the equivalence of the following statements: (i) Each of the entries of the operator
∗
X X=
1 − A(2Z −1/2 − 1)A∗ B(Z −1/2 − 1)A∗
A(Z −1/2 − 1)B ∗ BB ∗
+ + HH HH : ⊕ → ⊕ − − HH HH
is trace class. (ii) Each of the entries of the operator ∗
Y Y =
1 − A(2 − Z)A∗ B(1 − Z)A∗
A(1 − Z)B ∗ BZB ∗
+ + HH HH : ⊕ → ⊕ − − HH HH
is trace class. − − (iii) BB ∗ : HH → HH is trace class. Remember that a compact operator T : H1 → H2 acting between two (possibly different) Hilbert spaces H1 andH2 is said to be trace class, T ∈ B1 (H1 , H2 ), ∞ if it has finite trace norm T 1 := i=1 si < ∞, where si are the eigenvalues of |T | := (T ∗ T )1/2 on H1 . − − + − Note first, that BB ∗ ∈ B1 (HH , HH ) ⇔ B : HN → HH is Hilbert-Schmidt − + + + ⇔ B ∗ : HH → HN is Hilbert-Schmidt ⇔ B ∗ B ∈ B1 (HN , HN ). + + → HN is a bounded operator with bounded inverse we Since Z := 1 + 2B ∗ B : HN have + + + + + + , HN ) ⇔ ZB ∗ B ∈ B1 (HN , HN ) ⇔ BZB ∗ ∈ B1 (HN , HN ) B ∗ B ∈ B1 (HN
which proves the assertion for the 22-components of X ∗ X and Y ∗ Y . For the 12-components we note that −2B ∗ B = 1 − Z = (Z −1/2 − 1)(Z 1/2 + Z)
(48)
+ where Z 1/2 + Z is a bounded operator on HN with bounded inverse. As shown + + after Equation (44), also A : HN → HH is a bounded operator with bounded inverse, therefore + + + + , HN ) ⇔ A(1 − Z) = −2AB ∗ B ∈ B1 (HN , HH ) B ∗ B ∈ B1 (HN − + ∗ ⇒ A(1 − Z)B ∈ B1 (HH , HH ) − + ⇒ A(Z −1/2 − 1)B ∗ = A(1 − Z)(Z 1/2 + Z)−1 B ∗ ∈ B1 (HH , HH ).
The argument for the 21-component is analogous. As for the 11-component of Y ∗ Y we note, using the invertibility of A, the identity Z = 1 + 2B ∗ B, and (41), that + + + + , HH ) ⇔ A∗ A − A∗ A(2 − Z)A∗ A ∈ B1 (HN , HN ) 1 − A(2 − Z)A∗ ∈ B1 (HH + + + + ⇔ (1 + B ∗ B)B ∗ B(1 + 2B ∗ B) ∈ B1 (HN , HN ) ⇔ B ∗ B ∈ B1 (HN , HN ).
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Similarly, using A∗ A = 1 + B ∗ B = 12 (1 + Z), we rewrite the 11-component of X ∗ X in terms of Z and obtain + + 1 − A(2Z −1/2 − 1)A∗ ∈ B1 (HH , HH ) + + ⇔ A∗ A − A∗ A(2Z −1/2 − 1)A∗ A ∈ B1 (HN , HN ) + + , HN ). ⇔ (1 + Z)(1 − Z −1/2 )(2 − Z 1/2 + Z) ∈ B1 (HN
(49)
Taking into account (48) and the identity (2 − Z 1/2 + Z)(Z 1/2 + Z + 2) = 4 + 3Z + Z 2 , where both Z 1/2 + Z + 2 and 4 + 3Z + Z 2 are bounded operators with bounded inverse, we note that (49) is equivalent to + + , HN ) (1 + Z)(Z − 1)(4 + 3Z + Z 2 ) ∈ B1 (HN
+ + + + ⇔ Z − 1 ∈ B1 (HN , HN ) ⇔ B ∗ B ∈ B1 (HN , HN ).
This finishes the proof. Theorem 4.5 and Theorem 4.7 imply that, for N > 5/2, πωN (A(O)) and πωH (A(O)) , and, if Σ is compact, for N > 3/2, πωN (A[Γ, σ]) and πωH (A[Γ, σ]) are isomorphic von Neumann factors. Therefore it follows from the corresponding results for Hadamard representations due to Verch [49, Theorem 3.6] that πωN (A(O)) is isomorphic to the unique hyperfinite type III1 factor if Oc is nonempty, and is a type I∞ factor if Oc = ∅ (i.e., Σ = O is a compact Cauchy surface). Our Theorem 4.7 is the analogue of Theorem 3.3 of L¨ uders & Roberts [35] extended to our definition of adiabatic states on arbitrary curved spacetime manifolds. The loss of order 3/2 + in the compact case and 1/2 + in the non-compact case ( > 0 arbitrary) compared to their result is probably due to the fact that we use the regularity of ΛH − ΛN rather generously in the part of the proof of Theorem 4.2 between Equations (20) and (25).
4.2
Local definiteness and Haag duality
The next property of adiabatic vacua we check is that of local definiteness. It says that any two adiabatic vacua (of order > 5/2) get indistinguishable upon measurements in smaller and smaller spacetime regions. In a first step let us show that in the representation πωN generated by an adiabatic vacuum state ωN (of order N > 3/2) there are no nontrivial observables which are localized at a single point, more precisely: Theorem 4.12 Let x ∈ Σ. Then, for N > 3/2, πωN (A(O)) = C1, O x
where the intersection is taken over all open bounded subsets O ⊂ Σ.
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Before we prove the theorem let us recall how this, combined with Theorem 4.7, implies the property of local definiteness: Corollary 4.13 Let ωN be an adiabatic vacuum state of order N > 5/2 and ωH a Hadamard state. Let On , n ∈ N0 , be a sequence of open bounded subsets of Σ shrinking to a point x ∈ Σ, i.e., On+1 ⊂ On and n∈N0 On = {x}. Then (ωN − ωH )|A(On ) → 0
as n → ∞.
Proof. Let (πωN , HωN , ΩωN ) be the GNS-triple generated by ωN , and let RN (On ) := πωN (A(On )) be the corresponding von Neumann algebras associated to the regions On ⊂ Σ. Due to Theorem 4.7 and the remarks at the beginning of Section 4.1 π(ωH A(O0 )) is quasiequivalent to πωN A(O0 ). This implies [6, Theorem 2.4.21] that ωH A(O0 ) can be represented in HωN as a den ∈ H with ψ 2 = 1 such that sity matrix, i.e., there is a sequence ψ m ω m N m ωH (A) = m ψm , Aψm for all A ∈ A(O0 ). Let now An ∈ RN (On ) ⊂ RN (O0 ) be a sequence of observables with An = 1. From Theorem 4.12 it follows that An → c1 in the topology of RN (O0 ) for some c ∈ C. In particular, An → c1 in the weak topology, thus |ΩωN , (An − c1)ΩωN | → 0 as n → ∞, and An → c1 in the σ-weak topology, thus |ψm , (An − c1)ψm | → 0 as n → ∞. m
From this we can now conclude |(ωN − ωH )(An )|
=
|ΩωN , An ΩωN −
ψm , An ψm |
m
=
|ΩωN , (An − c1)ΩωN −
ψm , (An − c1)ψm |
m
≤
|ΩωN , (An − c1)ΩωN | +
|ψm , (An − c1)ψm |
m
→ 0
as n → ∞,
(50)
i.e., (ωN − ωH )(An ) converges to 0 pointwise for each sequence An . To show the uniform convergence we note that due to A(On+1 ) ⊂ A(On ) rn := sup{|(ωN − ωH )(A)|; A ∈ A(On ), A = 1} is a bounded monotonically decreasing sequence in n ∈ N0 with values in R+ 0. Hence rn → r for some r ∈ R+ . To show that r = 0 let
> 0. For all n ∈ N there 0 0 is an An ∈ A(On ) with An = 1 such that 0 ≤ rn − |(ωN − ωH )(An )| ≤ .
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Furthermore, due to (50) there is an no ∈ N0 such that for all n ≥ no |(ωN − ωH )(An )| ≤ . From these inequalities we obtain for n ≥ no 0 ≤ r ≤ rn ≤ + |(ωN − ωH )(An )| ≤ 2 and hence r = 0. This proves the assertion. To prove Theorem 4.12 we show an even stronger statement, namely πωN (A(O)) = C1
(51)
O⊃S
for any smooth 2-dimensional closed submanifold S of Σ. The statement of Theorem 4.12 then follows if we choose x ∈ S. In the proof we will need the following lemma:
Lemma 4.14
C0∞ (O) = {0},
O⊃S
where the closure is taken w.r.t. the norm of H −1/2 (Σ) (and hence it also holds w.r.t. the norm of H 1/2 (Σ)). Note that we can confine the intersection to all sets O contained in a suitable compact subset of Σ. Hence we can assume that (Σ, h) is a complete Riemannian manifold (otherwise we modify h as in the proof of Theorem 4.2), so that H ±1/2 (Σ) is well defined. Proof of Lemma 4.14. The problem is local, so it suffices to consider the case Σ = Rn , S = Rn−1 ×{0}. Suppose the above intersection contains some f ∈ H −1/2 (Rn ), say f H −1/2 = 1. Fix 0 < < 1/2. Since f H −1/2 = sup{|f (F )|; F ∈ H 1/2 , F H 1/2 = 1} we find some F ∈ H 1/2 (Rn ) such that F H 1/2 = 1 and f (F ) > 1− . According to Lemma 4.6 there exists an F0 ∈ C0∞ (Rn \ (Rn−1 × {0})) such that F −F0 H 1/2 < and therefore f (F0 ) = f (F ) + f (F0 − F ) > 1 − 2 . Clearly there is a δ > 0 such that |xn | > 2δ for each x = (x , xn ) ∈ supp F0 . H −1/2
On the other hand, f ∈ C0∞ (Rn−1 × (−δ, δ)) , hence supp f ⊂ Rn−1 × [−δ, δ] (in order to see this, use the fact that the closure of C0∞ (Rn+ ) in the topology of H s (Rn ) is equal to {u ∈ H s (Rn ); supp u ⊂ Rn+ } for s ∈ R, cf. [45, 2.10.3]). Denoting by χδ a smooth function, equal to 1 on Rn−1 × [−δ, δ] and vanishing outside Rn−1 × (−2δ, 2δ), we have f = χδ f and therefore 1 − 2 < f (F0 ) = (χδ f )(F0 ) = f (χδ F0 ) = f (0) = 0, a contradiction.
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Proof of Theorem 4.12. Let (kN , HN ) be the one-particle Hilbert space structure of ωN . According to results of Araki [1, 34] (51) holds iff kN (Γ(O)) = {0}, (52) O⊃S
where the closure is taken w.r.t. the norm in HN . As in the proof of Theorem 4.5, let us define a one-particle Hilbert space structure ˜ H) ˜ of an auxiliary pure quasifree state on A[Γ, σ] by (k, ˜ k˜ : Γ → L2 (Σ, h) =: H
q 1 → √ iD 1/2 q + D −1/2 p ; p 2 as before, we may change h near infinity to obtain completeness. Note that the norm given by ˜ kF ˜ ˜ = 1 D 1/2 q2 2 + D −1/2 p2 2 , F := (q, p) ∈ Γ, µ ˜(F, F ) := kF, L L H 2 is equivalent to the norm of H 1/2 (Σ) ⊕ H −1/2 (Σ). Let u ∈ kN (Γ(O)) for all O ⊃ S. Thus for every O there is a sequence {FnO , n ∈ N} ⊂ Γ(O) with kN (FnO ) → u in HN . By Theorem 4.2 the norm given by µN , N > 3/2, on Γ(O) is equivalent to the norm given by µ ˜, namely that of H 1/2 (O) ⊕ ˜ O ) → v O in H ˜ ˜ for some v O ∈ k(Γ(O)). H −1/2 (O). Therefore it follows that also k(F n Moreover, v O must be independent of O: To see this, suppose that O1 and O2 are ˜ ⊂ Σ, and let > 0. Then there is an contained in a common open, bounded set O n ∈ N such that ˜ O1 ) ˜ + k(F ˜ O1 ) − k(F ˜ O2 ) ˜ + k(F ˜ O2 ) − v O2 ˜ v O1 − v O2 H˜ ≤ v O1 − k(F n n n n H H H O1 O2 O1 O2 O1 O2 1/2 ˜ ≤ 2 + k(F − F ) ˜ = 2 + µ ˜(F − F , F − F ) n
≤ = ≤ ≤
n
H
n
n
n
n
˜ µN (FnO1 − FnO2 , FnO1 − FnO2 )1/2 2 + C(O) ˜ N (F O1 ) − kN (F O2 )HN 2 + C(O)k n n
˜ kN (FnO1 ) − uHN + u − kN (FnO2 )HN 2 + C(O) ˜ 2 (1 + C(O)),
˜ by v. hence v O1 = v O2 , and we denote this unique element of H ˜ H ˜ Since v ∈ O⊃S k(Γ(O)) it follows from Lemma 4.14 that v = 0 and therefore O ˜ ˜ k(Fn ) → 0 in H. Since the norms given by kN and k˜ are equivalent on Γ(O) we also have kN (FnO ) → 0 in HN and thus u = limn→∞ kN (FnO ) = 0, which proves the theorem. In the following theorem we show that the observable algebras RN (O) := πωN (A(O)) generated by adiabatic vacuum states (of order N > 3/2) satisfy a
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certain maximality property, called Haag duality. Due to the locality requirement it is clear that all observables localized in spacelike separated regions of spacetime commute. If O is some open, relatively compact subset of the Cauchy surface Σ with smooth boundary, this means that RN (Oc ) ⊂ RN (O) ,
(53)
where Oc := Σ \ O and RN (Oc ) :=
πωN (A(O1 ))
(54)
O1 ⊂O c
is the von Neumann algebra generated by all πωN (A(O1 )) with O1 bounded and O1 ⊂ Oc . One says that Haag duality holds for the net of von Neumann algebras generated by a pure state if (53) is even an equality. For mixed states (i.e., reducible GNS-representations) this can certainly not be true, because in this case, by Schur’s lemma [6, Prop. 2.3.8], there is a set S of non-trivial operators commuting with the representation πωN , i.e., S ⊂ RN (O) ∩ RN (Oc ) .
(55)
If equality held in (53) then the right-hand side of (55) would be equal to RN (O) ∩ RN (O), i.e., to the centre of RN (O), which, however, is trivial due to the local primarity (Theorem 4.5) of the representation πωN , hence S ⊂ C1, a contradiction. Therefore, in the reducible case one has to take the intersection with RN := πωN (A[Γ, σ]) on the right-hand side of (53) to get equality1 (in the irreducible case, again by Schur’s lemma, πωN (A) = C1 ⇒ πωN (A) = B(HωN ), hence the intersection with RN is redundant). Haag duality is an important assumption in the theory of superselection sectors [21] and has therefore been checked in many models of physical interest. For our situation at hand, Haag duality has been shown by L¨ uders & Roberts [35] to hold for the GNS-representations of adiabatic vacua on Robertson-Walker spacetimes and by Verch [47, 49] for those of Hadamard Fock states. He also noticed that it extends to all Fock states that are locally quasiequivalent to Hadamard states, hence, by our Theorem 4.7, to pure adiabatic states of order N > 5/2. Nevertheless, we present an independent proof of Haag duality for adiabatic states that does not rely on quasiequivalence but only on Theorem 4.2 and also holds for mixed states. Theorem 4.15 Let ωN be an adiabatic state of order N > 3/2. Then, for any open, relatively compact subset O ⊂ Σ with smooth boundary, RN (Oc ) = RN (O) ∩ RN , where Oc := Σ \ O and RN (Oc ) is defined by (54). 1 We are grateful to Fernando Lled´ o for pointing out to us this generalization of Haag duality and discussions about this topic.
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Proof. Denoting again by (kN , HN ) the one-particle Hilbert space structure of ωN , it follows from results of Araki [1, 34] that the assertion is equivalent to the statement kN (Γ(Oc )) = kN (Γ(O))∨ ∩ kN (Γ), where the closure has to be taken w.r.t. HN and kN (Γ(O))∨ was defined in Equation (30). Since kN (Γ(Oc )) ⊂ kN (Γ(O))∨ ∩ kN (Γ) (due to the locality of σ, compare (53) above), we only have to show that kN (Γ(Oc )) is dense in kN (Γ(O))∨ ∩ kN (Γ). This in turn is the case iff kN (Γ(O)) + kN (Γ(Oc )) is dense in kN (Γ)
(56)
(for the convenience of the reader, the argument will be given in Lemma 4.16 below). (56) will follow if we show that every element u = kN (F ) ∈ kN (Γ), F = (q, p) ∈ Γ, can be approximated by a sequence in kN (Γ(O)) + kN (Γ(Oc )). To this end we fix a bounded open set O0 ⊂ Σ with smooth boundary such that supp p and supp q ⊂ O0 . According to Lemma 4.6 we find sequences {qn }, {pn } ⊂ C0∞ (O), {qnc }, {pcn} ⊂ C0∞ (Oc ) such that q − (qn + qnc ) p − (pn + pcn )
→ 0 in H 1/2 (O0 ) → 0 in H −1/2 (O0 ).
(57) (58)
Note that it is no restriction to ask that the supports of all functions are contained ˜ H) ˜ the one-particle Hilbert space structure introduced in O0 . Let us denote by (k, in (32) with the real scalar product µ ˜ given by (33). The relations (57) and (58) imply that Γ(O0 ) Fn := (q − (qn + qnc ), p − (pn + pcn )) → 0 with respect to the norm induced by µ ˜. According to Theorem 4.2 it also tends to zero with respect to the norm induced by µN , in other words kN (Fn ) → 0 in HN . This completes the argument.
Lemma 4.16 kN (Γ(O)) + kN (Γ(Oc )) is dense in kN (Γ) ⇔ kN (Γ(Oc )) is dense in kN (Γ(O))∨ ∩ kN (Γ). Proof. ⇒. Let u ∈ kN (Γ(O))∨ ∩ kN (Γ), and choose vn ∈ kN (Γ(O)), wn ∈ kN (Γ(Oc )) such that vn + wn → u in HN . (59) In view of the fact that kN (Γ(Oc )) ⊂ kN (Γ(O))∨ we have kN (Γ(O)) ∩ kN (Γ(Oc )) ⊂ kN (Γ(O)) ∩ kN (Γ(O))∨ = {0}.
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Indeed, the last equality is a consequence of Theorem 4.5, cf. (31). We can therefore define a continuous map π : kN (Γ(O)) ⊕ kN (Γ(Oc )) → HN v ⊕ w → v. Now (59) implies that {vn + wn } is a Cauchy sequence in HN , hence so are {vn } = {π(vn + wn )} and {wn }. Let v0 := lim vn ∈ kN (Γ(O)), w0 := lim wn ∈ kN (Γ(Oc )). By (59), u − w0 = v0 ∈ kN (Γ(O))∨ ∩ kN (Γ(O)) = {0}. Therefore u = w0 ∈ kN (Γ(Oc )). ⇐. Denoting by ⊥ the orthogonal complement in kN (Γ), we clearly have from the ∨ ∩ kN (Γ). Since kN (Γ(O)) + definition (30) of ∨ that kN (Γ(O))⊥ ⊂ kN (Γ(O))
kN (Γ(O))⊥ = kN (Γ) it follows that kN (Γ(O)) + kN (Γ(O))∨ ∩ kN (Γ) is dense in
kN (Γ). From the assumption that kN (Γ(Oc )) is dense in kN (Γ(O))∨ ∩ kN (Γ) the assertion follows.
5 Construction of adiabatic vacuum states The existence of Hadamard states on arbitrary globally hyperbolic spacetimes has been proven by Fulling, Narcowich & Wald [19] using an elegant deformation argument. Presumably, the existence of adiabatic vacuum states could be shown in a similar way employing the propagation of the Sobolev wavefront set, Proposition B.4. Instead of a mere existence argument, however, we prefer to explicitly construct a large class of adiabatic vacuum states as it is indispensable for the extraction of concrete information in physically relevant situations to have available a detailed construction of the solutions of the theory. In Section 5.1 we first present a parametrization of quasifree states in terms of two operators R and J acting on the L2 -Hilbert space w.r.t. a Cauchy surface Σ, Theorem 5.1. The main technical result is Theorem 5.3 which gives a sufficient condition on R and J such that the associated quasifree states are adiabatic of a certain order. In Section 5.2 we construct operators R and J satisfying the above assumptions, Theorem 5.10.
5.1
Criteria for initial data of adiabatic vacuum states
We recall the following theorem from [31, Theorem 3.11]: Theorem 5.1 Let (M, g) be a globally hyperbolic spacetime with Cauchy surface Σ. Let J, R be operators on L2 (Σ, d3 σ) satisfying the following conditions: (i) (ii) (iii) (iv)
C0∞ (Σ) ⊂ dom(J), J and R map C0∞ (Σ, R) to L2R (Σ, d3 σ), J is selfadjoint and positive with bounded inverse, R is bounded and selfadjoint.
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Then k:Γ
→ H := k(Γ) ⊂ L2 (Σ, d3 σ)
(q, p)
→ (2J)−1/2 [(R − iJ)q − p]
(60)
is the one-particle Hilbert space structure of a pure quasifree state. Note that we can define the inverse square root by 1 ∞ −1/2 λ (λ + 2J)−1 dλ. (2J)−1/2 = π 0
(61)
The integral converges since λ + 2J ≥ λ and hence (λ + 2J)−1 ≤ λ−1 for λ ≥ 0. Therefore (2J)−1/2 is a bounded operator on L2 (Σ, d3 σ). Moreover, (2J)−1/2 maps L2R (Σ, d3 σ) to itself since λ + 2J and therefore (λ + 2J)−1 commutes with complex conjugation (λ ≥ 0). Proof. A short computation shows that for Fj = (qj , pj ) ∈ Γ, j = 1, 2, we have σ(F1 , F2 )
= −q1 , p2 + p1 , q2 = 2Im kF1 , kF2 .
Here, ·, · is the scalar product of L2 (Σ, d3 σ). We then let µ(F1 , F2 ) := Re kF1 , kF2 . We note that |σ(F1 , F2 )|2
≤ =
4|kF1 , kF2 |2 ≤ 4kF1 , kF1 kF2 , kF2 4µ(F1 , F1 )µ(F2 , F2 );
hence k defines the one-particle Hilbert space structure of a quasifree state with real scalar product µ (Definition 2.3) and one-particle Hilbert space H = kΓ + ikΓ (Proposition 2.4). Let us next show that the state is pure, i.e., kΓ is dense in H (see Proposition 2.4). We apply a criterion by Araki & Yamagami [2] and check that the operator S : Γ → L2 (Σ, d3 σ) ⊕ L2 (Σ, d3 σ) defined by kF1 , kF2 = 2µ(F1 , SF2 ) is a projection (cf. Equation (34)). Indeed, this relation implies that 1 −iJ −1 iJ −1 R + 1 S= . iRJ −1 R + iJ −iRJ −1 + 1 2 Therefore S 2 = S, and the proof is complete. The distinguished parametrices of the Klein-Gordon operator In the following we shall use the calculus of Fourier integral operators of Duistermaat & H¨ ormander [16] in order to analyze the wavefront set of certain bilinear forms related to fundamental solutions of the Klein-Gordon operator P = 2g +m2 . We recall from [16, Theorem 6.5.3] that P on a globally hyperbolic spacetime
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(which is known to be pseudo-convex w.r.t. P , see [41]) has 22 = 4 orientations ˙ ν2 of the bicharacteristic relation C, Equation (15). Here, C \ diag (C) = Cν1 ∪C ν is one of the four sets {∅, N+ , N− , N+ ∪ N− } of components of the lightcone 1(2) N := char P , with N± := N ∩ {ξ0 ≷0}. Cν are the subsets of the bicharacteristic +(−) in [16]. Associated to these orientations there relation which are denoted by Cν 1 2 are four pairs Eν , Eν of distinguished parametrices with W F (Eν1 ) = ∆∗ ∪ Cν1 ,
W F (Eν2 ) = ∆∗ ∪ Cν2
where ∆∗ is the diagonal in (T ∗ X \ 0) × (T ∗ X \ 0). Moreover, Duistermaat & H¨ormander show that every parametrix E with W F (E) contained in ∆∗ ∪ Cν1 resp. ∆∗ ∪ Cν2 must be equal to Eν1 resp. Eν2 modulo C ∞ . In addition, Eν1 − Eν2 ∈ I 1/2−2 (M × M, C ) and Eν1 − Eν2 is noncharacteristic at every point of C . Here, I µ (X, Λ) denotes the space of Lagrangian distributions of order µ over the manifold X associated to the Lagrangian submanifold Λ ⊂ T ∗ X \ 0, cf. [28, Def. 25.1.1]. We shall need three particular parametrices: For the forward light cone N+ we 1 obtain EN = E R (mod C ∞ ), the retarded Green’s function, for the backward + 1 light cone N− we have the advanced Green’s function EN = E A (mod C ∞ ) while − 1 F EN+ ∪N− is the so-called Feynman parametrix E (mod C ∞ ). We deduce that 1 2 = EN (mod C ∞ ), in particular EN + − E = E R − E A ∈ I −3/2 (M × M, C ). We next write E = E + + E − with E + := E F − E A , E − := E R − E F . We deduce from [16, Theorem 6.5.7] that E−
1 1 = E R − E F = EN − EN ∈ I −3/2 (M × M, (C − ) ) + + ∪N−
(62)
E+
1 1 = E F − E A = EN − EN ∈ I −3/2 (M × M, (C + ) ) + ∪N− −
(63)
where C + = C ∩ (N+ × N+ ), C − = C ∩ (N− × N− ) as in Equation (16). It follows from [41, Theorem 5.1] that the two-point function ΛH of every Hadamard state coincides with iE + (mod C ∞ ). (We define the physical Feynman propagator by F (x, y) := −iT Φ(x)Φ(y) , i.e., −i × the expectation value of the time ordered product of two field operators. From this choice it follows that iF = ΛH +iE A and hence F = E F (mod C ∞ ) and ΛH = iE + (mod C ∞ ).) Lemma 5.2 For every Hadamard state ΛH we have ∅, s < − 12 s s + . W F (ΛH ) = W F (E ) = C + , s ≥ − 12
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Proof. The statement for s < −1/2 follows from Equation (63) and Proposition B.10. For s ≥ −1/2 we rely on [16, Section 6]. According to [16, Equation (6.6.1)] 1 1 1 EN + E∅1 = EN + EN mod C ∞ , + ∪N− + −
so that, in the notation of [16, Equation (6.6.3)], 1 E + = EN − E∅1 = SN+ . +
The symbol of SN+ is computed in [16, Theorem 6.6.1]. It is non-zero on the diagonal ∆N in N × N . Moreover, it satisfies a homogeneous first order ODE along the bicharacteristics of P in each pair of variables, so that it is non-zero everywhere on C + . Hence E + is non-characteristic at every point of C + . Now Proposition B.10 gives the assertion. We fix a normal coordinate t which allows us to identify a neighborhood of Σ in M with (−T, T ) × Σ =: MT . We assume that Rl = {Rl (t); −T < t < T } and Jl = {Jl (t); −T < t < T }, l = 1, 2, are smooth families of properly supported pseudodifferential operators on Σ with local symbols rl = rl (t) ∈ C ∞ ((−T, T ), S 0 (Σ × R3 )) and jl = jl (t) ∈ C ∞ ((−T, T ), S 1(Σ × R3 )). Moreover, let H = {H(t); −T < t < T } be a smooth family of properly supported pseudodifferential operators of order −1 on Σ. We can then also view Rl , Jl , and H as operators on, say, C0∞ ((−T, T ) × Σ). Theorem 5.3 Let Rl , Jl , and H be as above, and let Ql be a properly supported first order pseudodifferential operator on (−T, T ) × Σ such that (N )
Ql (Rl − iJl − ∂t )E − = Sl (N )
E−,
l = 1, 2,
(64)
(N )
with Sl = Sl (t) ∈ C ∞ ((−T, T ), L−N (Σ)) a smooth family of properly supported pseudodifferential operators on Σ of order −N . Moreover, we assume that Ql has a real-valued principal symbol such that char Ql ∩ N− = ∅. Then the distribution DN ∈ D (M × M), defined by DN (f1 , f2 ) = [(R1 − iJ1 )ρ0 − ρ1 ] Ef1 , H [(R2 − iJ2 )ρ0 − ρ1 ] Ef2 satisfies the relation s
W F (DN ) ⊂
∅, s < −1/2 C + , −1/2 ≤ s < N + 3/2.
(65)
Note that DN will in general not be a two-point function unless R1 = R2 and J1 = J2 = H −1 are selfadjoint and J is positive (compare Theorem 5.1).
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Proof. Since ρ0 commutes with Rl , Jl and H we have DN (f1 , f2 ) = ρ0 [R1 − iJ1 − ∂t ] Ef1 , ρ0 H [R2 − iJ2 − ∂t ] Ef2 .
(66)
Denoting by K1 and K2 the distributional kernels of (R1 − iJ1 − ∂t )E and H(R2 − iJ2 − ∂t )E, respectively, we see that DN = (ρ0 K1 )∗ (ρ0 K2 ). We shall apply the calculus of Fourier integral operators in order to analyze the composition (ρ0 K1 )∗ (ρ0 K2 ). The following lemma is similar in spirit to [28, Theorem 25.2.4]. Lemma 5.4 Let X ⊂ Rn1 , Y ⊂ Rn2 be open sets and A ∈ Lk (X) be a properly supported pseudodifferential operator with symbol a(x, ξ). Assume that C is a homogeneous canonical relation from T ∗ Y \ 0 to T ∗ X \ 0 and that a(x, ξ) vanishes on a conic neighborhood of the projection of C in T ∗ X \ 0. If B ∈ I m (X × Y, C ) then AB ∈ I −∞ (X × Y, C ). Proof. The problem is microlocal, so we may assume that B has the form Bu(x) = eiφ(x,y,ξ) b(x, y, ξ)u(y) dy dN ξ, where φ is a non-degenerate phase function on X × Y × (RN \ {0}) and b ∈ S m+(n1 +n2 −2N )/4 (X × Y × RN ) an amplitude. We know that C = Tφ (Cφ ), where Cφ := {(x, y, ξ) ∈ X × Y × (RN \ {0}); dξ φ(x, y, ξ) = 0}, and Tφ is the map Tφ : X × Y × (RN \ {0}) → T ∗ (X × Y ) \ 0 (x, y, ξ)
→ (x, dx φ; y, dy φ).
We recall that ess supp b is the smallest closed conic subset of X × Y × (RN \ {0}) outside of which b is of class S −∞ and that the wavefront set of the kernel of B is contained in the set Tφ (Cφ ∩ ess supp b), cf. [15, Theorem 2.2.2]. Hence we may assume that b vanishes outside a conic neighborhood N of Cφ in X × Y × RN . In fact we can choose this neighborhood so small that a(x, ξ ) = 0 whenever (x, ξ ) lies in the projection of Tφ (N ) ⊂ T ∗ X × T ∗ Y onto the first component (we call this projection π1 ). Then ABu(x) = eiφ(x,y,ξ) c(x, y, ξ)u(y) dy dN ξ
1150
where
W. Junker and E. Schrohe
Ann. Henri Poincar´e
c(x, y, ξ) = e−iφ(x,y,ξ) A(b(·, y, ξ)eiφ(·,y,ξ) ).
According to [44, Ch. VIII, Equation (7.8)], c has the asymptotic expansion c(x, y, ξ) ∼ Dξα a(x, dx φ(x, y, ξ))Dxβ b(x, y, ξ)ψαβ (x, y, ξ) (67) α≥0 β≤α
where ψαβ is a polynomial in ξ of degree ≤ |α − β|/2. Now from our assumptions on a and b it follows that in Equation (67) b(x, y, ξ) = 0 if a(x, ξ ) = 0 if
(x, y, ξ) ∈ /N (x, ξ ) ∈ π1 Tφ (N ) ⇒ a(x, dx φ(x, y, ξ)) = 0 if (x, y, ξ) ∈ N ,
and hence c ∼ 0. This proves that AB ∈ I −∞ (X × Y, C ). ∞
Lemma 5.5 Let A ∈ C ((−T, T ), L (Σ)) be properly supported and B ∈ I (M × M, (C ± ) ). Then AB ∈ I m+k (MT × M, (C ± ) ). k
m
Proof. Choosing local coordinates and a partition of unity we may assume that M = R4 , Σ = R3 ∼ = R3 × {0} ⊂ R4 and that A is supported in a compact set. We let X = op χ where χ = χ(τ, ξ) ∈ C ∞ (R4 ) vanishes near (τ, ξ) = 0 and is homogeneous of degree 0 for |(τ, ξ)| ≥ 1 with χ(τ, ξ) = 1 for (τ, ξ) in a conic neighborhood of {ξ = 0}, and χ(τ, ξ) = 0 for (τ, ξ) outside a larger conic neighborhood of {ξ = 0}, such that, in particular, χ(τ, ξ) = 0 in a neighborhood of π1 (C ± ) (by π1 we denote the projection onto the first component in T ∗ M × T ∗ M, i.e., π1 (x, ξ; y, η) := (x, ξ)). We have AB = AXB + A(1 − X)B. Denoting by a(t, x, ξ) the local symbol of A, the operator A(1 − X) has the symbol a(t, x, ξ)(1 − χ(τ, ξ)) ∈ S k (R4 × R4 ). (Here we have used the fact that (1−χ(τ, ξ)) is non-zero only in the area where τ can be estimated by ξ .) Hence A(1−X) is a properly supported pseudodifferential operator of order k on MT . We may apply [28, Theorem 25.2.3] with excess equal to zero and obtain that A(1 − X)B ∈ I m+k (MT × M, (C ± ) ). On the other hand, X is a pseudodifferential operator with symbol vanishing in a neighborhood of π1 (C ± ). According to Lemma 5.4, XB ∈ I −∞ (M × M, (C ± ) ). Hence XB is an integral operator with a smooth kernel on M × M, and so is AXB, since A is continuous on C ∞ (M). Lemma 5.6 (i) (Rl − iJl − ∂t )E + ∈ I −1/2 (MT × M, (C + ) ), l = 1, 2; (ii) H(R2 − iJ2 − ∂t )E + ∈ I −3/2 (MT × M, (C + ) ); (iii) (Rl − iJl − ∂t )E − ∈ I −N −5/2 (MT × M, (C − ) ), l = 1, 2; (iv) H(R2 − iJ2 − ∂t )E − ∈ I −N −7/2 (MT × M, (C − ) ).
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Proof. (i) It follows from (63) and Lemma 5.5 that (Rl − iJl )E + ∈ I −1/2 (MT × M, (C + ) ). Since ∂t is a differential operator, the assumptions of the composition theorem for Fourier integral operators [28, Theorem 25.2.3] are met with excess equal to zero, and we conclude from (63) that also ∂t E + ∈ I −1/2 (MT ×M, (C + ) ). Since, by assumption, H ∈ C ∞ ((−T, T ), L−1 (Σ)) is properly supported we also obtain (ii). (iii) We know from (64) that (N )
Ql (Rl − iJl − ∂t )E − = Sl
E−.
(68)
Applying Lemma 5.5 and (62), the right-hand side is an element of I −3/2−N (MT × (N ) M, (C − ) ) (note that Sl is properly supported). We next observe that the question is local, so that we can focus on a small neighborhood U of a point x0 ∈ M. (1) (2) Here, we write Ql = Ql +Ql as a sum of two pseudodifferential operators, where (1) (2) Ql is elliptic, and the essential support of Ql is contained in the complement of N− . To this end choose a real-valued function χ ∈ C ∞ (T ∗ U ) with the following properties: (α) (β) (γ) (δ)
χ(x, ξ) = 0 for small |ξ|, χ is homogeneous of degree 1 for |ξ| ≥ 1, χ(x, ξ) ≡ 0 on a conic neighborhood of N− , χ(x, ξ) ≡ |ξ| on a neighborhood of char Ql ∩ {|ξ| ≥ 1}.
We denote the local symbol of Ql by ql and let (1)
Ql
:= op (ql (x, ξ) + iχ(x, ξ)),
(2)
Ql
:= op (−iχ(x, ξ)).
By the Lemmata 5.4 and 5.5 we have (2)
Ql (Rl − iJl − ∂t )E − ∈ I −∞ (MT × M, (C − ) ). (1)
Moreover, Ql is elliptic of order 1, since ql is real-valued and χ(x, ξ) = |ξ| on char Ql . We conclude from (68) that (1)
Ql (Rl − iJl − ∂t )E − ∈ I −3/2−N (MT × M, (C − ) ). (1)
Multiplication by a parametrix to Ql
shows that
(Rl − iJl − ∂t )E − ∈ I −5/2−N (MT × M, (C − ) ). (iv) follows from (iii) and Lemma 5.5. We next analyze the effect of the restriction operator ρ0 . We recall from [15, p. 113] that ρ0 ∈ I 1/4 (Σ × M, R0 ) (69) where R0 := {(xo , ξo ; x, ξ) ∈ (T ∗ Σ × T ∗ M) \ 0; xo = x, ξo = ξ|Txo Σ }.
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Lemma 5.7 ρ0 H(R2 − iJ2 − ∂t )E − ρ0 (R1 − iJ1 − ∂t )E −
∈ ∈
I −N −13/4 (Σ × M, (R0 ◦ C − ) ) I −N −9/4 (Σ × M, (R0 ◦ C − ) )
ρ0 (R1 − iJ1 − ∂t )E + ρ0 H(R2 − iJ2 − ∂t )E +
∈ ∈
I −1/4 (Σ × M, (R0 ◦ C + ) ) I −5/4 (Σ × M, (R0 ◦ C + ) ).
Proof. All these statements follow from (69), Lemma 5.6 and the composition formula for Fourier integral operators [28, Theorem 25.2.3], provided that the compositions R0 ◦ C − and R0 ◦ C + of the canonical relations are clean, proper and connected with excess zero (cf. [27, C.3] and [28, p. 18] for notation). We note that (R0 × C + ) ∩ (T ∗ Σ × diag (T ∗ M) × T ∗ M) = {(xo , ξo ; x, ξ; x, ξ; y, η); x = xo , ξo = ξ|Txo Σ , (x, ξ; y, η) ∈ C + }.
(70)
Given (xo , ξo ) ∈ T ∗ Σ \ 0 there is precisely one (x, ξ) ∈ N+ such that x = xo and ξ|Txo Σ = ξo ; given (x, ξ) ∈ N+ there is a 1-parameter family of (y, η) such that (x, ξ; y, η) ∈ C + . We deduce that codim (R0 × C + ) + codim (T ∗ Σ × diag (T ∗ M) × T ∗ M) = 6 dim M − 1 = codim (R0 × C + ) ∩ (T ∗ Σ × diag (T ∗ M) × T ∗ M); here the codimension is taken in T ∗ Σ × (T ∗M)3 . Hence the excess of the intersection, i.e., the difference of the left- and the right-hand side, is zero. In particular, the intersection is transversal, hence clean. Moreover, the fact that in (70) the (x, ξ) is uniquely determined as soon as (xo , ξo ) and (y, η) are given shows that the associated map (xo , ξo ; x, ξ; x, ξ; y, η) → (xo , ξo ; y, η) is proper (i.e., the pre-image of a compact set is compact). Indeed, the pre-image of a closed and bounded set is trivially closed; it is bounded, because |ξ| ≤ C|ξo | for some constant C. Finally, the pre-image of a single point (xo , ξo ; y, η) is again a single point, in particular a connected set. An analogous argument applies to R0 ◦ C − . Lemma 5.8 (i) (ρ0 (R1 − iJ1 − ∂t )E − )∗ (ρ0 H(R2 − iJ2 − ∂t )E − ) ∈ I −2N −11/2 (M × M, (C − ) ), (ii) (ρ0 (R1 − iJ1 − ∂t )E + )∗ (ρ0 H(R2 − iJ2 − ∂t )E + ∈ I −3/2 (M × M, (C + ) ). Denoting by D± the relation (R0 ◦ C ∓ )−1 ◦ (R0 ◦ C ± ) we have (iii) (ρ0 (R1 − iJ1 − ∂t )E + )∗ (ρ0 H(R2 − iJ2 − ∂t )E − ) ∈ I −N −7/2 (M × M, (D− ) ), (iv) (ρ0 (R1 − iJ1 − ∂t )E − )∗ (ρ0 H(R2 − iJ2 − ∂t )E + ) ∈ I −N −7/2 (M × M, (D+ ) ).
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Proof. (i) According to [28, Theorem 25.2.2] and Lemma 5.7 (ρ0 (R1 − iJ1 − ∂t )E − )∗ ∈ I −N −9/4 (M × Σ, ((R0 ◦ C − )−1 ) ). We first note that the composition (R0 ◦ C − )−1 ◦ (R0 ◦ C − ) equals C − : In fact, (R0 ◦ C − )−1 is the set of all (y, η; xo , ξo ), where (xo , ξo ) ∈ T ∗ Σ, y is joined to xo by a null geodesic γ, η ∈ N− is cotangent to γ at y and the projection Pγ (η)|Txo Σ of the parallel transport of η along γ coincides with ξo . The codimension of (R0 ◦ C − )−1 × (R0 ◦ C − ) in T ∗ M × T ∗ Σ × T ∗ Σ × T ∗ M therefore equals 4 dim Σ + 2, and we have codim ((R0 ◦ C − )−1 × (R0 ◦ C − )) + codim (T ∗ M × diag (T ∗ Σ) × T ∗ M) = 6 dim Σ + 2 = codim ((R0 ◦ C − )−1 × (R0 ◦ C − )) ∩ (T ∗ M × diag (T ∗ Σ) × T ∗ M). In particular, the intersection of (R0 ◦ C − )−1 × (R0 ◦ C − ) and T ∗ M × diag (T ∗ Σ) × T ∗ M is transversal in T ∗ M × T ∗ Σ × T ∗ Σ × T ∗ M, hence clean with excess 0. Given (y, η; xo , ξo ; xo , ξo ; y˜, η˜) in the intersection, the element (xo , ξo ) is uniquely determined by (y, η) and (˜ y , η˜). The mapping (y, η; xo , ξo ; xo , ξo ; y˜, η˜) → (y, η; y˜, η˜) therefore is proper. The pre-image of each element is a single point, hence a connected set. We can apply the composition theorem [28, Theorem 25.2.3] and obtain the assertion. The proof of (ii), (iii) and (iv) is analogous. We can now finish the proof of Theorem 5.3. According to (66) and the following remarks we have to find the wavefront set of (ρ0 (R1 − iJ1 − ∂t )(E + + E − ))∗ (ρ0 H(R2 − iJ2 − ∂t )(E + + E − )). By Proposition B.10 we have, for an arbitrary canonical relation Λ, I µ (M × M, Λ) ⊂ H s (M × M) if µ + 12 dim M + s < 0; moreover, the wavefront set of elements of I µ (M × M, Λ) is a subset of Λ. Lemma 5.8 therefore immediately implies (65).
5.2
Construction on a compact Cauchy surface
Following the idea in [31] we shall now show that one can construct adiabatic vacuum states on any globally hyperbolic spacetime M with compact Cauchy surface Σ. In Gaußian normal coordinates w.r.t. Σ the metric reads 1 gµν = −hij (t, x)
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and the Klein-Gordon operator reduces to 1 2g + m2 = √ ∂t ( h∂t ·) − ∆Σ + m2 , h where hij is the induced Riemannian metric on Σ, h its determinant and ∆Σ the Laplace-Beltrami operator w.r.t. hij acting on Σ. Following [31, Equation (130)ff.] (N ) (N ) there exist operators P1 , P2 , N = 0, 1, 2, . . . , of the form (N )
P1
(N )
P2
1 = −a(N ) (t, x, Dx ) − √ ∂t h h = a(N ) (t, x, Dx ) − ∂t
with a(N ) = a(N ) (t, x, Dx ) ∈ C ∞ ([−T, T ], L1(Σ)) such that (N )
P1
(N )
◦ P2
− (2g + m2 ) = sN (t, x, Dx )
(71)
with sN ∈ C ∞ ([−T, T ], L−N (Σ)). In fact one gets a(N ) (t, x, Dx ) = −iA1/2 +
N +1
b(ν) (t, x, Dx );
ν=1
here A is the selfadjoint extension of −∆Σ + m2 on L2 (Σ), so that A1/2 is an elliptic pseudodifferential operator of order 1. The b(ν) are elements of C ∞ ([−T, T ], L1−ν (Σ)) defined recursively so that (71) holds. One then sets similarly as in [31, Equation (134)] j (N ) (t, x, ξ) r
(N )
:=
(t, x, ξ)
:=
J(t)
:=
R(t) :=
−
N +1 1 (ν) b (t, x, ξ) − b(ν) (t, x, −ξ) ∈ S 0 2i ν=1
N +1 1 (ν) b (t, x, ξ) + b(ν) (t, x, −ξ) ∈ S 0 2 ν=1 1 (N ) j (t, x, Dx ) + j (N ) (t, x, Dx )∗ ∈ L1 A1/2 + 2 1 (N ) r (t, x, Dx ) + r(N ) (t, x, Dx )∗ ∈ L0 . 2
(72)
(73)
Lemma 5.9 We can change the operator J defined above by a family of regularizing operators such that the assumptions of Theorem 5.1 are met. Proof. It is easily checked that a pseudodifferential operator on Rn with symbol a(x, ξ) maps C0∞ (Rn , R) to L2R (Rn ) (i.e., commutes with complex conjugation) iff a(x, ξ) = a(x, −ξ). The symbols j (N ) and r(N ) have this property
by construction. The operator family R(t) = 12 r(N ) (t, x, Dx ) + r(N ) (t, x, Dx )∗ ∈ L0 is bounded
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and symmetric, hence selfadjoint. Moreover, it commutes with complex conjugation: If v ∈ L2R (Σ, d3 σ), then R(t)v is real-valued, since for every u ∈ L2R (Σ, d3 σ) 2u, R(t)v = u, (r(N ) + r(N )∗ )v = u, r(N ) v + r(N ) u, v ∈ R. The operator A1/2 maps C0∞ (Σ, R) to L2R (Σ, d3 σ) by (61); it is selfadjoint on D(A1/2 ) = H 1 (Σ). Hence J defines a selfadjoint family of pseudodifferential operators of order 1; it is invariant under complex conjugation. Moreover, its principal symbol is hij ξi ξj > 0. According to [44, Ch. II, Lemma 6.2] there exists a family of regularizing operators J∞ = J∞ (t) such that J + J∞ is strictly positive. Replacing J∞ by 12 (J∞ + CJ∞ C), where C here denotes the operator of complex conjugation, we obtain an operator which is both strictly positive and invariant under complex conjugation. It differs from J by a regularizing family. Theorem 5.10 For N = 0, 1, 2, . . . we let ΛN (f1 , f2 ) =
1 [(R − iJ)ρ0 − ρ1 ] Ef1 , J −1 [(R − iJ)ρ0 − ρ1 ] Ef2 2
with J modified as in Lemma 5.9. Then ΛN is the two-point function of a (pure) adiabatic vacuum state of order N . Proof. By Theorem 5.1 and Lemma 5.9, ΛN defines the two-point function of a (pure) quasifree state. We write R(t) = J(t) =
1 1 (N ) r (t, x, Dx ) + r(N ) (t, x, Dx )∗ 2 2 1
1 1/2 (N ) A + j (t, x, Dx ) + A1/2 + j (N ) (t, x, Dx )∗ + j∞ (t, x, Dx ) 2 2
with r(N ) , j (N ) as in (72), (73) and j∞ the regularizing modification of Lemma 5.9. We shall use Theorem 5.3 to analyze the wavefront set of ΛN . We decompose ΛN (f1 , f2 ) = 1 (N ) r (t, x, Dx ) − i(A1/2 + j (N ) (t, x, Dx ) + 2j∞ (t, x, Dx )) ρ0 − ρ1 Ef1 8
r(N ) (t, x, Dx )∗ − i(A1/2 + j (N ) (t, x, Dx )∗ ) ρ0 − ρ1 Ef1 ,
J(t)−1 r(N ) (t, x, Dx ) − i(A1/2 + j (N ) (t, x, Dx ) + 2j∞ (t, x, Dx )) ρ0 − ρ1 Ef2
(74) +J(t)−1 r(N ) (t, x, Dx )∗ − i(A1/2 + j (N ) (t, x, Dx )∗ ) ρ0 − ρ1 Ef2 .
+
Now we let ˜ 1 (t) := Q =
i A1/2 + i r(N ) (t, x, Dx ) − ij (N ) (t, x, Dx ) + √ ∂t h h 1 (N ) i a(N ) (t, x, Dx ) + √ ∂t h = −iP1 h
(75)
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and
˜ 2 (t) := A1/2 + i r(N ) (t, x, Dx )∗ − ij (N ) (t, x, Dx )∗ + √i ∂t h. Q h
(76)
Equation (71) implies that
˜ 1 (t) r(N ) (t, x, Dx ) − i(A1/2 + j (N ) (t, x, Dx )) − 2ij∞ (t, x, Dx ) − ∂t iQ
(N ) (N ) P2 − 2ij∞ (t, x, Dx ) = P1 = 2g + m2 + s˜N (t, x, Dx )
(77)
where s˜N differs from sN in (71) by an element in C ∞ ([−T, T ], L−∞(Σ)). Next we note that (71) is equivalent to the identity 1 (N ) r − i(A1/2 + j (N ) ) − ∂t −r(N ) + i(A1/2 + j (N ) ) − √ ∂t h h = 2g + m2 + sN which in turn is equivalent to
−r(N ) + i(A1/2 + j (N ) ) r(N ) − i(A1/2 + j (N ) )
1 h r(N ) − i(A1/2 + j (N ) ) − √ ∂t h 2 = −∆Σ + m + sN or – taking adjoints and conjugating with the operator C of complex conjugation –
−C r(N )∗ + i(A1/2 + j (N )∗ ) CC r(N )∗ + i(A1/2 + j (N )∗ ) C
1 hC r(N )∗ + i(A1/2 + j (N )∗ ) C − √ ∂t h (78) = C(−∆Σ + m2 + s∗N )C = −∆Σ + m2 + Cs∗N C. Here we have used the fact that
∗ ∗ ∂t h r(N ) − i(A1/2 + j (N ) ) = ∂t h r(N ) − i(A1/2 + j (N ) ) . Using that r(N )∗ , j (N )∗ and A1/2 commute with C, (78) reads
− r(N )∗ − i(A1/2 + j (N )∗ ) r(N )∗ − i(A1/2 + j (N )∗ )
1 − √ ∂t h r(N )∗ − i(A1/2 + j (N )∗ ) h 2 ∗ = −∆Σ + m + CsN C.
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Adding the time derivatives, it follows that
˜ 2 (t) r(N )∗ − i(A1/2 + j (N )∗ ) − ∂t iQ 1 (N )∗ (N )∗ 1/2 (N )∗ = −r + i(A + j ) − √ ∂t h r − i(A1/2 + j (N )∗ ) − ∂t h = 2g + m2 + Cs∗N C. (79) ˜ 1 and Q ˜ 2 , defined by Equations (75) and (76), are not Note that the operators Q yet pseudodifferential operators since their symbols will not decay in the covariable of t, say τ , if we take derivatives w.r.t. the covariables of x, say ξ. To make them pseudodifferential operators we choose a finite number of coordinate neighborhoods {Uj } for Σ, which yields finitely many coordinate neighborhoods for (−T, T ) × Σ. As (t, x) varies over (−T, T ) × Uj , the negative light cone will not intersect a fixed conic neighborhood N of {ξ = 0} in T ∗ ((−T, T ) × Uj ). We choose a realvalued function χ which is smooth on T ∗ ((−T, T ) × Uj ), zero for |(τ, ξ)| ≤ 1/2, homogeneous of degree zero for |(τ, ξ)| ≥ 1 such that χ(t, x, τ, ξ) χ(t, x, τ, ξ)
= 0 = 1
on a conic neighborhood of {ξ = 0} outside N .
(80)
We now let X := op χ. Then ˜1 Q1 := X Q
˜2 and Q2 := X Q
are pseudodifferential operators due to (80). Their principal symbols are ((hij ξi ξj )1/2 − τ )χ(t, x, τ, ξ), so that their characteristic set does not intersect N− . Equations (77), (79) and the fact that (2g + m2 )E − = 0 show that the assumption of Theorem 5.3 is satisfied for each of the four terms arising from the decomposition of ΛN in (74). This yields the assertion. Lemma 5.8 explicitly shows that the non-Hadamard like singularities of the two-point function ΛN in Theorem 5.10 (i.e., those not contained in the canonical relation C + ) are either pairs of purely negative frequency singularities lying on a common bicharacteristic (C − ) or pairs of mixed positive/negative frequency singularities (D± ) which lie on bicharacteristics that are “reflected” by the Cauchy surface. They may have spacelike separation. For the states constructed in Theorem 5.10 one can explicitly find the Bogoljubov B-operator, which was introduced in the proof of Theorem 4.7(ii), in terms of the operators R and J. Applying the criterion of Lemma 4.11(iii) one can check the unitary equivalence of the GNS-representations generated by these states. A straightforward (but tedious) calculation shows that unitary equivalence already holds for N ≥ 0, thus improving the statement of Theorem 4.7(ii) for these particular examples.
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6 Adiabatic vacua on Robertson-Walker spaces By introducing adiabatic vacua on Robertson-Walker spaces Parker [38] was among the first to construct a quantum field theory in a non-trivial background spacetime. A mathematically precise version of his construction and an analysis of the corresponding Hilbert space representations along the same lines as in our Section 4 were given by L¨ uders & Roberts [35]. Relying on their work we want to show in this section that these adiabatic vacua on Robertson-Walker spaces are indeed adiabatic vacua in the sense of our Definition 3.2. This justifies our naming and gives a mathematically intrinsic meaning to the “order” of an adiabatic vacuum. In [31] one of us had claimed to have shown that all adiabatic vacua on RobertsonWalker spaces are Hadamard states. This turned out to be wrong in general, when the same question was investigated for Dirac fields [24]2 . So the present section also serves to correct this mistake. Our presentation follows [31]. In order to be able to apply our Theorem 5.3 without technical complications we restrict our attention to Robertson-Walker spaces with compact spatial sections. These are the 4-dimensional Lorentz manifolds M = R × Σ where Σ is regarded as being embedded in R4 as Σ = {x ∈ R4 ; (x0 )2 +
3
(xi )2 = 1} ∼ = S3,
i=1
and M is endowed with the homogeneous and isotropic metric " ! dr2 2 2 2 2 2 2 2 + r (dθ + sin θ dϕ ) ; ds = dt − a(t) 1 − r2
(81)
here ϕ ∈ [0, 2π], θ ∈ [0, π], r ∈ [0, 1) are polar coordinates for the unit ball in R3 , and a is a strictly positive smooth function. In [31] it was shown that an adiabatic vacuum state of order n (as defined in [35]) is a pure quasifree state on the Weyl algebra of the Klein-Gordon field on the spacetime (81) given by a one-particle Hilbert space structure w.r.t. a fixed Cauchy surface Σt = {t = const.} = Σ × {t} (equipped with the induced metric from (81)) kn : Γ → Hn := kn (Γ) ⊂ L2 (Σt ) (q, p) → (2Jn )−1/2 [(Rn − iJn )q − p] of the form (60) of Theorem 5.1, where the operator families Rn (t), Jn (t) acting on L2 (Σ, d3 σ) are defined in the following way: ˙ (n) (t) Ω 1 a(t) ˙ k + f˜(t, k)φ k (x) (Rn f )(t, x) := − dµ(k) 3 2 a(t) Ω(n) (t) k (n) (Jn f )(t, x) := dµ(k) Ωk (t)f˜(t, k)φ k (x), (82) 2 We
want to thank S. Hollands for discussions about this point.
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with t in some fixed finite interval I ⊂ R, say. Here, {φ k , k := (k, l, m), k = 0, 1, 2, . . . ; l = 0, 1, . . . , k; m = −l, . . . , l} are the t-independent eigenfunctions of the Laplace-Beltrami operator ∆Σ w.r.t. the Riemannian metric 1 sij =
1−r 2
r2
2
2
(83)
r sin θ on the hypersurface Σ: # 2 2 − 3r2 ∂ 1 2 ∂ + ∆(θ, ϕ) φ k = −k(k + 2)φ k , ∆Σ φ k ≡ (1 − r ) 2 + ∂r r ∂r r2 2
2
∂ ∂ 1 ∂ 2 where ∆(θ, ϕ) := ∂θ 2 + cot θ ∂θ + sin2 θ ∂ϕ2 is the Laplace operator on S . They form an orthonormal basis of L2 (Σ, d3 σ) with d3 σ := r2 (1 − r2 )−1/2 dr sin θdθ dϕ. ˜denotes the generalized Fourier transform
˜: L2 (Σ, d3 σ) f
˜ dµ(k)) → L2 (Σ, ˜ → f (k) := d3 σ φ k (x)f (x),
(84)
Σ
˜ dµ(k)) where Σ ˜ is the space of which is a unitary map from L2 (Σ, d3 σ) to L2 (Σ, $ ∞ values of k = (k, l, m) equipped with the measure dµ(k) := k=0 kl=0 lm=−l [35]. (Note that (84) is defined w.r.t. Σ with the metric sij , Equation (83), and not w.r.t. Σt with the metric a2 (t)sij .) The inverse is given by f (x) = dµ(k) f˜(k)φ k (x). Using duality and interpolation of the Sobolev spaces one deduces Lemma 6.1
H s (Σ) = {f =
dµ(k) f˜(k)φ k ;
dµ(k) (1 + k 2 )s |f˜(k)|2 < ∞}.
The Klein-Gordon operator associated to the metric (81) is given by 2g + m2 =
1 ∂2 a˙ ∂ − ∆Σ + m2 . +3 ∂t2 a ∂t a2
(85)
(n)
The functions Ωk (t), n ∈ N0 , in (82) were introduced by L¨ uders & Roberts. (n) Ωk (t) is strictly positive and plays the role of a generalized frequency which is determined by a WKB approximation to Equation (85). It is iteratively defined by the following recursion relations (0)
(Ωk )2 (n+1) 2
(Ωk
)
k(k + 2) + m2 a2 2 2 ˙ (n) ¨ (n) Ω a ˙ a ¨ 3 3 3 1Ω k k + ωk2 − − − . 4 a 2 a 4 Ω(n) 2 Ω(n) k k
:= ωk2 ≡ =
(86)
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In the following we shall study the analytic properties of these functions. We shall (n) see that, using (86), we may express Ωk as a function of t and ωk . As a function of these two variables, it turns out to behave like a classical pseudodifferential symbol. In Lemma 6.2 below we shall derive the corresponding estimates and expansions to consider a ‘covariable’ in R+ ; in later for (t, ωk ) ∈ I × R+ . It is a little unusual applications, however, we will have ωk = k(k + 2)/a2 + m2 bounded away from zero, so that the behaviour of ωk near zero is irrelevant. (n+1) 2 We first observe that (Ωk ) can be determined by an iteration involving only (n) (Ωk )2 and its time derivatives: Since, for an arbitrary F , we have ∂t (F 2 )/F 2 = 2F˙ /F , we obtain (n) 2 d 2 (Ωk )2 a ˙ a ¨ 1 3 3 (n+1) 2 dt + ) = ωk2 − − (Ωk (n) 2 4 a 2 a 16 Ωk (n) d 2 1 d dt (Ωk ) − . (87) (n) 4 dt (Ω )2 k
(n)
An induction argument shows that (Ωk )2 − ωk2 is a rational function in ωk of degree ≤ 0 with coefficients in C ∞ (I). Indeed this is trivially true for n = 0. Suppose it is proven for some fixed n. We write (n)
(Ωk )2 − ωk2 = r(t, ωk ) =
p(t, ωk ) q(t, ωk )
(88)
with polynomials p and q in ωk such that deg (p) ≤ deg (q) and the leading coefficient of q is 1. Then
(n) 2 d ) (Ω k dt 2ωk ω˙ k + r˙ . (89) = (n) 2 ωk2 + r (Ωk ) In view of the fact that a˙ m2 ω˙ k = − ωk − a ωk
and r˙ =
q∂t p − p∂t q q∂ωk p − p∂ωk q ω˙ k + , q2 q2
(89) is again rational of degree 0 and the leading coefficient of the polynomial in the denominator again equals 1. The same is true for the time derivatives of (89). The recursion formula (87) then shows the assertion for n + 1. (n) 2 dl 2 (Ω is a rational function of ωk with Next we observe that also dt ) − ω l k k
coefficients in C ∞ (I) of degree ≤ 0. Moreover, this shows that, for sufficiently (n) large ωk (equivalently for sufficiently large k), (Ωk )2 is a strictly positive function (n) (uniformly in t ∈ I). We may redefine (Ωk )2 for small values of ωk so that it is strictly positive and bounded away from zero on I × R+ . It makes sense to take its (n) square root, and in the following we shall denote this function by Ωk . We note:
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(n)
Lemma 6.2 Ωk , considered as a function of (t, ω) ∈ I × R+ , is an element of 1 Scl (I × R+ ), i.e., (n) ∂tl ∂ωm Ωk (t, ω) = O(ω 1−m ) (90) ∞ (n) (n) (n) and, in addition, Ωk has an asymptotic expansion Ωk ∼ j=0 (Ωk )j into (n)
symbols (Ωk )j ∈ S 1−j which are positively homogeneous for large ω. Its principal symbol is ω. With the same understanding ˙ (n) Ω k (n)
Ωk (n+1) 2
(n)
) − (Ωk )2
(Ωk
0 ∈ Scl (I × R+ ),
(91)
−2n ∈ Scl (I × R+ ).
(92)
Proof. By induction, (90) is immediate from (88) together with the formulae 1 2ω + ∂ω r(t, ω) . ω 2 + r(t, ω) = 2 ω 2 + r(t, ω)
˙ (n) /Ω(n) = d (Ω(n) )2 Relation (91) is immediate from (89), noting that 2Ω k k k dt ∂t
1 ∂t r(t, ω) ω 2 + r(t, ω) = 2 ω 2 + r(t, ω)
and ∂ω
(n)
/(Ωk )2 . In both cases the existence of the asymptotic expansion follows from [44, Ch. II, Theorem 3.2] and the expansion % j ∞ 1/2 r(t, ω) r(t, ω) ω 2 + r(t, ω) = ω 1 + = ω j ω2 ω2 j=0 (n)
(n)
valid for large ω. The principal symbol of Ωk is ω since (Ωk )2 = ω 2 modulo rational functions of degree ≤ 0, as shown above (cf. Equation (88)). In order to show (92) we write, following L¨ uders & Roberts, (n+1) 2
(Ωk
(n)
) = (Ωk )2 (1 + n+1 );
this yields [35, Equation (3.9)]
n+1
=
1 ω˙ ˙n 1
˙n 1 1 ˙1 + + ··· 2 ω (1 + 1 ) · · · (1 + n ) 4 ω 1 + n 8 1 + 1 1 + n 1 ˙n−1
˙n 5 ˙2n 1 ¨n + + − . 8 1 + n−1 1 + n 16 1 + n 4 1 + n (1)
(0)
(1)
We know already that (Ωk )2 − (Ωk )2 = (Ωk )2 − ω 2 is rational in ω of degree ≤ 0, hence 1 is rational of degree −2. Noting that ˙n = (∂ω n )ω˙ + ∂t n , we deduce from the above recursion that n is rational of degree −2n. This completes the argument. The operators Rn and Jn , Equation (82), are unitarily equivalent to mul˜ dµ(k)). From the fact that Ω ˙ (n) /Ω(n) is bounded tiplication operators on L2 (Σ, k k
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(n)
and Ωk is strictly positive with principal symbol ωk (Lemma 6.2) we can immediately deduce that the assumptions of Theorem 5.1 are satisfied if we let dom J(t) = H 1 (Σ) for t ∈ I. We are now ready to state the theorem that connects the adiabatic vacua of L¨ uders & Roberts [35] to our more general Definition 3.2: Theorem 6.3 For fixed t let Λn (f, g) := (Rn − iJn − ∂t )Ef, Jn−1 (Rn − iJn − ∂t )Eg L2 (Σt ) be the two-point function of a pure quasifree state of the Klein-Gordon field on the Robertson-Walker spacetime (81) with Rn , Jn given by Equations (82) and (86). Then ∅, s < − 21 s , W F (Λn ) ⊂ C + , − 12 ≤ s < 2n + 32 i.e., Λn describes an adiabatic vacuum state of order 2n in the sense of our Definition 3.2. To prove the theorem we shall need the following observations: Lemma 6.4 Let m ∈ R. Let M be a compact manifold and A : D(M ) → D (M ) a linear operator. Suppose that, for each k ∈ N, we can write A = Pk + Rk
(93)
where Pk is a pseudodifferential operator of order m and Rk is an integral operator with a kernel function in C k (M × M ). Then A is a pseudodifferential operator of order m. Proof. Generalizing a result by R. Beals [4], Coifman & Meyer showed the following: A linear operator T : D(M ) → D (M ) is a pseudodifferential operator of order 0 if and only if T as well as its iterated commutators with smooth vector fields are bounded on L2 (M ) [10, Theorem III.15]. As a corollary, T is a pseudodifferential operator of order m if and only if T and its iterated commutators induce bounded maps L2 (M ) → H −m (M ). Given the iterated commutator of A with, say, l vector fields V1 , . . . , Vl , we write A = Pk + Rk with k ≥ l + |m|. The iterated commutator [V1 , [. . . [Vl , Pk ] . . .]] is a pseudodifferential operator of order m and hence induces a bounded map L2 (M ) → H −m (M ). The analogous commutator with Rk has an integral kernel in C k−l (M × M ). As k − l ≥ |m|, it furnishes even a bounded operator L2 (M ) → H |m| (M ). µ Lemma 6.5 Let µ ∈ Z and b = b(t, τ ) ∈ Scl (I × R) with principal symbol b−µ (t)τ µ . 1/2
2 Replacing τ by ωk (t) = k(k+2) , b defines a family {B(t); t ∈ I} of a2 (t) + m
operators B(t) : D(Σ) → D (Σ) by
)(t, k) := b(t, ωk (t))f˜(k). (B(t)f
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We claim that this is a smooth family of pseudodifferential operators of order µ with principal symbol σ (µ) (B(t)) = b−µ (t)(|ξ|Σ /a(t))µ ,
(94)
where the length |ξ|Σ of a covector ξ is taken w.r.t. the (inverse of the) metric (83). Note that for the definition of B(t) we only need to know b(t, τ ) for τ ≥ m. We may therefore also apply this result to the symbols that appear in Lemma 6.2. Proof. The fact that b is a classical symbol allows us to write, for each N , b(t, τ ) =
N
bj (t)τ −j + b(N ) (t, τ ),
(95)
j=−µ
where bj ∈ C ∞ (I) and |∂tj ∂τl b(N ) (t, τ )| ≤ Cjl (1 + |τ |)−N for all t ∈ I and τ ≥ ,
> 0 fixed. (Note that we will not obtain the estimates for all τ , since we have a fully homogeneous expansion in (95), but as we shall substitute τ by ωk and ωk is bounded away from 0, this will not be important.) Equation (95) induces an analogous decomposition of B: B(t) =
N
Bj (t) + B (N ) (t),
j=−µ
where Bj (t) is given by −j ˜ (B j (t)f )(t, k) = bj (t)ωk (t) f (k)
and B (N ) (t) by (N ) (t)f )(t, k) = b(N ) (t, ωk (t))f˜(k). (B
In view of the fact that ∆Σ φ k = −k(k + 2)φ k , we have
−j/2 . Bj (t) = bj (t) m2 − ∆Σ /a2 (t) According to Seeley [43], Bj is a smooth family of pseudodifferential operators of order −j. Next, we observe that, by (90), ∂tl ωk = O(ωk ) and hence, for each l ∈ N,
|∂tl b(N ) (t, ωk (t)) | ≤ C(1 + ωk (t))−N ≤ C (1 + k)−N for all t ∈ I. Lemma 6.1 therefore shows that, for each s ∈ R ∂tl B (N ) (t) : H s (Σ) → H s+N (Σ)
(96)
is bounded, uniformly in t ∈ I. On the other hand, it is well known that a linear operator T which maps H −s−k (Σ) to H s+k (Σ) for some s > 3/2 (dim Σ = 3) has
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an integral kernel of class C k on Σ× Σ. It is given by K(x, y) = T δy , δx . Choosing N > 3 + 2k, the family B (N ) will therefore have integral kernels of class C k . Now we apply Lemma 6.4 to conclude that B is a smooth family of pseudodifferential operators of order µ. Since Bj is of order −j the principal symbol is that of
−µ/2 . This yields (94). B−µ = b−µ (t) m2 − ∆Σ /a2 (t) Now let us define the family of operators An (t) acting on L2 (Σ, d3 σ) by (An f )(t, x) := dµ(k) a(n) (t, k)f˜(t, k)φ k (x), with a(n) given by the function a
(n)
˙ (n) 1Ω 3 a˙ (n) k − (t, k) := − iΩk . 2 a 2 Ω(n) k
Moreover let Qn := iX(∂t + An (t)), where X := op χ and χ = χ(t, x, τ, ξ) is as in (80). Lemma 6.6 An ∈ C ∞ (I, L1cl (Σ)) with principal symbol σ (1) (An (t)) = i|ξ|Σ /a(t). Qn is a pseudodifferential operator of order 1 on I × Σ with real-valued principal symbol whose characteristic does not intersect N− . Proof. We apply Lemma 6.5 in connection with Lemma 6.2 to see that An ∈ C ∞ (I, L1cl (Σ)). The operator Qn clearly is an element of L1cl (I × Σ). Outside a small neighborhood of {ξ = 0} its characteristic set is {(t, x, τ, ξ) ∈ T ∗ (I × Σ); −τ + |ξ|Σ /a(t) = 0}. Since N− = {τ = −|ξ|Σ /a(t)}, the intersection is empty.
Proof of Theorem 6.3. In view of Theorem 5.3 we only have to check that Qn (Rn − iJn − ∂t )E − = S (2n) E −
(97)
for a pseudodifferential operator S (2n) of order −2n. A straightforward computation shows that (∂t + An )(Rn − iJn − ∂t ) is the operator defined by a˙ (n) 2 (n+1) 2 2 2 − ∂t + 3 ∂t + ωk + (Ωk ) − (Ωk ) . a Now ∂t2 + 3 aa˙ ∂t + ωk2 induces 2g + m2 , Equation (85), while, by Lemma 6.2 com(n) (n+1) 2 bined with Lemma 6.5, (Ωk )2 − (Ωk ) induces an element of C ∞ (I, L−2n (Σ)). Composing with the operator X from the left and noting that (2g + m2 )E − = 0, we obtain (97).
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7 Physical interpretation Using the notion of the Sobolev wavefront set (Definition B.1) we have generalized in this paper the previously known positive frequency conditions to define a large new class of quantum states for the Klein-Gordon field on arbitrary globally hyperbolic spacetime manifolds (Definition 3.2). Employing the techniques of pseudodifferential and Fourier integral operators we have explicitly constructed examples of them on spacetimes with a compact Cauchy surface (Theorem 5.10). We call these states adiabatic vacua because on Robertson-Walker spacetimes they include a class of quantum states which is already well known under this name (Theorem 6.3). We order the adiabatic vacua by a real number N which describes the Sobolev order beyond which the positive frequency condition may be perturbed by singularities of a weaker nature. Our examples show that these additional singularities may be of negative frequency or even non-local type (Lemma 5.8). Hadamard states are naturally included in our definition as the adiabatic states of infinite order. To decide which orders of adiabatic vacua are physically admissible we have investigated their corresponding GNS-representations: Adiabatic vacua of order N > 5/2 generate a quasiequivalence class of local factor representations (in other words, a unique local primary folium). For pure states on a spacetime with compact Cauchy surface – a case which often occurs in applications – this holds true already for N > 3/2 (Theorems 4.5 and 4.7). Physically, locally quasiequivalent states can be thought of as having a finite energy density relative to each other. Primarity means that there are no classical observables contained in the local algebras. Hence there are no local superselection rules, i.e., the local states can be coherently superimposed without restriction. For N > 3/2 the local von Neumann algebras generated by these representations contain no observables which are localized at a single point (Theorem 4.12). Together with quasiequivalence this implies that all the states become indistinguishable upon measurements in smaller and smaller spacetime regions (Corollary 4.13). This complies well with the fact that the correlation functions have the same leading short-distance singularities, whence the states should have the same high energy behaviour. Finally, the algebras are maximal in the sense of Haag duality (Theorem 4.15) and additive (Lemma 2.5). For a more thorough discussion of all these properties in the framework of algebraic quantum field theory we refer to [21]. Taken together, all these results suggest that adiabatic vacua of order N > 5/2 are physically meaningful states. Furthermore we expect that the energy momentum tensor of the Klein-Gordon field can be defined in these states by an appropriate regularisation generalizing the corresponding results for Hadamard states [8, 50] and adiabatic vacua on Robertson-Walker spaces [39]. However, all the mentioned physical properties of the GNS-representations are of a rather universal nature and therefore cannot serve to distinguish between different types of states. How can we physically discern an adiabatic state of order
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N from one of order N , say, or from a Hadamard state? To answer this question we investigate the response of a quantum mechanical model detector (a so-called Unruh detector [5, 46]) to the coupling with the Klein-Gordon field in an N -th order adiabatic vacuum state. So let us assume we are given an adiabatic state ˆ on the spacetime M and its ωN of order N of the Klein-Gordon quantum field Φ associated GNS-triple (HωN , πωN , ΩωN ) as in Proposition 2.4(b). We consider a detector that moves on a wordline γ : R → M, τ → x(τ ), in M and is described as a quantum mechanical system by a Hilbert space HD and a free time evolution w.r.t. proper time τ . It shall be determined by a free Hamiltonian H0 with a discrete energy spectrum E0 < E1 < E2 < · · · , E0 being the groundstate energy of H0 (e.g. a harmonic oscillator). We assume that the detector has negligible ˆ via the interaction Hamiltonian extension and is coupled to the quantum field Φ ˆ ))χ(τ ) HI := λM (τ )Φ(x(τ
(98)
acting on HD ⊗HωN , where λ ∈ R is a small coupling constant, M (τ ) = eiH0 τ M (0) e−iH0 τ the monopole moment operator characterizing the detector and χ ∈ C0∞ (R) a cutoff function that describes the adiabatic switching on and off of the interaction. To calculate transition amplitudes between states of HD ⊗ HωN under the interaction (98) one uses most conveniently the interaction picture, in which the ˆ and the operator M evolve with the free time evolution (but the full coufield Φ pling to the gravitational background) whereas the time evolution of the states is determined by the interaction HI . In this formulation the perturbative S-matrix is given by [5, 7] S
=
=
1+ 1+
∞ (−i)j j=1 ∞ j=1
j! (−iλ)j j!
dτ1 . . . dτj T [HI (τ1 ) . . . HI (τj )]
dτ1 χ(τ1 ) . . . dτj χ(τj ) T [M (τ1 ) . . . M (τj )]
ˆ ˆ T [Φ(x(τ 1 )) . . . Φ(x(τj ))],
(99)
where T denotes the operation of time ordering. Let us assume that the detector is prepared in its ground state |E0 prior to switching on the interaction, and calculate in first order perturbation theory (j = 1 in (99)) the transition amplitude between the incoming state ψin := |E0 ⊗ ΩωN ∈ HD ⊗ HωN and some outgoing state ψout := |En ⊗ψ, n = 0, where |En ∈ HD is the eigenstate of H0 corresponding to the energy En and ψ some one-particle state in the Fock space HωN (the ˆ scalar products of Φ(x)Ω ωN with other states vanish in a quasifree representation): ˆ ψout , Sψin = −iλEn |M (0)|E0 dτ χ(τ )ei(En −E0 )τ ψ|Φ(x(τ ))ΩωN . From this we obtain the probability P (En ) that a transition to the state |En occurs in the detector by summing over a complete set of (unobserved) one-particle
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states in HωN : λ2 |En |M (0)|E0 |2
P (En ) =
dτ1
dτ2 e−i(En −E0 )(τ1 −τ2 )
χ(τ1 )χ(τ2 )ΛN (x(τ1 ), x(τ2 )) λ |En |M (0)|E0 | F (En − E0 ). 2
=
2
Here, |En |M (0)|E0 |2 describes the model dependent sensitivity of the detector, whereas ∞ ∞ F (E) := dτ1 dτ2 e−iE(τ1 −τ2 ) χ(τ1 )χ(τ2 )ΛN (x(τ1 ), x(τ2 )) −∞
−∞
is the well-known expression for the detector response function depending on the two-point function ΛN of the adiabatic state ωN . Inspection of the formula shows that it is in fact obtained from ΛN ∈ D (M × M) by restricting ΛN to γ × γ ⊂ M × M, multiplying this restricted distribution pointwise by χ ⊗ χ and taking the Fourier transform at (−E, E): ∧
F (E) = 2π ((ΛN |γ×γ ) · (χ ⊗ χ)) (−E, E). It follows from the very definition of ΛN (Definition 3.2) and Proposition B.7 that ΛN |γ×γ is a well-defined distribution on R × R if N > 3/2, since N ∗ (γ) consists only of space-like covectors. It holds W F s (ΛN |γ×γ ) ⊂ ϕ∗ (C + ) for s < N − 3/2, where ϕ∗ is the pullback of the embedding ϕ : γ × γ → M × M. We now observe that (100) {(τ1 , −E; τ2 , E) ∈ R4 ; E ≥ 0} ∩ ϕ∗ (C + ) = ∅ (this observation was already made by Fewster [17] in the investigation of energy mean values of Hadamard states). Hence there is an open cone Γ in R2 \ {0} containing (−E, E), E > 0, such that W F s (ΛN |γ×γ )∩Γ = ∅. By (102) we can write s (R2 ) for s < N −3/2 and W F (u2 )∩Γ = (ΛN |γ×γ )·(χ⊗χ) = u1 +u2 with u1 ∈ Hloc ∅. Since (ΛN |γ×γ ) · (χ ⊗ χ) has compact support we can assume without loss of generality that also u1 and u2 have compact supports. From W F (u2 ) ∩ Γ = ∅ it follows then that u ˆ2 (ξ) = O(ξ −k ) ∀k ∈ N ∀ξ ∈ Γ, s (R2 ) implies that whereas u1 ∈ Hcomp
Dα u1 ∈ L2comp (R2 ) ∧
for |α| ≤ s < N − 3/2, cf. Prop. B.3
⇒ (D u1 ) (ξ) = ξ u ˆ1 (ξ) is bounded α
α
⇒ u ˆ1 (ξ) = O(ξ −|α| ).
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Taken together, we find that ((ΛN |γ×γ ) · (χ ⊗ χ))∧ (ξ) = O(ξ −[N −3/2] ) for ξ ∈ Γ, where [N − 3/2] := max{n ∈ N0 ; n < N − 3/2}. Since (−E, E) ∈ Γ, E > 0, we can now conclude that F (E) = O(E −[N −3/2] ) for an adiabatic vacuum state of order N > 3/2. (Note that this estimate could be improved for the states constructed in Section 5 by taking into account that for them the singularities of lower order are explicitly known, cf. Lemma 5.8, and the sub-leading singularities also satisfy relation (100).) This means that the probability of a detector, moving in an adiabatic vacuum of order N , to get excited to the energy E decreases like E −[N −3/2] for large E, in a Hadamard state it decreases faster than any inverse power of E. We can therefore interpret adiabatic states of lower order as higher excited states of the quantum field. One should however keep in mind that all the states usually considered in elementary particle physics (on a static spacetime, say) are of the Hadamard type: ground states and thermodynamic equilibrium states are Hadamard states [31], particle states satisfy the microlocal spectrum condition (the generalization of the Hadamard condition to higher n-point functions) [8]. We do not know by which physical operation an adiabatic state of finite order could be prepared. Although all results in this paper are concerned with the free Klein-Gordon field, it is clear that our Definition 3.2 is capable of a generalization to higher npoint functions of an interacting quantum field theory in analogy to the microlocal spectrum condition of Brunetti et al. [8]. In order to treat the pointwise product (ΛN )2 we write (ΛN )2 = (ΛH )2 + (ΛN − ΛH )2 + 2(ΛN − ΛH )ΛH , where ΛH is the two-point function of any Hadamard state. It follows from Proposition B.6, Lemma 3.3, and Lemma 5.2 that these pointwise products are well defined if N > −1. It is well known that W F ((ΛH )2 ) ⊂ C + ⊕ C + , where C + ⊕ C + := {(x1 , ξ1 + η1 ; x2 , ξ2 + η2 ) ∈ T ∗ (M × M) \ 0; (x1 , ξ1 ; x2 , ξ2 ), (x1 , η1 ; x2 , η2 ) ∈ C + } [29, Theorem 8.2.10]. The regularity of the other two terms can be estimated by Theorems 8.3.1 and 10.2.10 in [30] such that finally W F s ((ΛN )2 ) ⊂ C + ⊕ C +
if s < N − 3.
Higher powers of ΛN can be treated similarly. Thus, for sufficiently large adiabatic order N , finite Wick powers (with finitely many derivatives) can be defined. This should be sufficient for the perturbative construction of a quantum field theory with an interaction Lagrangian involving a fixed number of derivatives and powers of the fields. Obviously one has to require more and more regularity of the states if one wants to define higher and higher Wick powers. This complies with a recent result of Hollands & Ruan [25].
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It is also clear that the notion of adiabatic vacua can be extended to other field theory models than merely the scalar field. A first step in this direction has been taken by Hollands [24] for Dirac fields. Finally we want to point out that, although the whole analysis in this paper has been based on a given C ∞ -manifold M with smooth Lorentz metric g, the notion of adiabatic vacua should be particularly relevant for manifolds with C k -metric. Typical examples that occur in general relativity are star models: here the metric outside the star satisfies Einstein’s vacuum field equations and is matched on the boundary C 1 to the metric inside the star where it satisfies Einstein’s equations with an energy momentum tensor of a suitable matter model as a source term. In such a situation Hadamard states cannot even be defined on a part of the spacetime that contains the boundary of the star, whereas adiabatic states up to a certain order should still be meaningful. This remark could e.g. be relevant for the derivation of the Hawking radiation from a realistic stellar collapse to a black hole.
A
Sobolev spaces
H s (Rn ), s ∈ R, is the set of all tempered distributions u on Rn whose Fourier transforms u ˆ are regular distributions satisfying 2 u(ξ)|2 dn ξ < ∞. uH s (Rn ) := ξ 2s |ˆ For a domain U ⊂ Rn we let H s (U) := {rU u; u ∈ H s (Rn )} be the space of all restrictions to U of H s -distributions on Rn , equipped with the quotient topology uH s (U ) := inf{U H s (Rn ) ; U ∈ H s (Rn ), rU U = u}. Moreover, we denote by H0s (U ) the space of all elements in H s (Rn ) whose support is contained in U. If U is bounded with smooth boundary, then it follows from [28, Theorem B.2.1] that C0∞ (U) is dense in H0s (U ) for every s and that H0s (U ) is the dual space of H s (U) with respect to the extension of the sesquilinear form u ¯v dn x, u ∈ C0∞ (U), v ∈ C ∞ (U). If Σ is a compact manifold without boundary we choose a covering by coordinate neighborhoods with associated coordinate maps, say {Uj , κj }j=1,...,J with a subordinate partition of unity {ϕj }j=1,...,J . Given a distribution u on Σ, we shall say that u ∈ H s (Σ) if, for each j, the push-forward of ϕj u under κj is an element of
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H s (Rn ). It is easy to see that this definition is independent of the choices made for Uj , κj and ϕj . The space H s (Σ) is a Hilbert space with the norm uH s (Σ)
1/2 J := (κj )∗ (ϕj u)2H s (Rn ) . j=1
We denote by ∆ the Laplace-Beltrami operator with respect to an arbitrary metric on Σ. Then we have H 2k (Σ) = {u ∈ L2 (Σ); (1 − ∆)k u ∈ L2 (Σ)} for k = 0, 1, 2, . . .: Clearly, the left-hand side is a subset of the right-hand side. Conversely, we may assume that u has support in a single coordinate neighborhood, so that we can look at the push-forward u∗ under the coordinate map. The fact that both u∗ and ((1 − ∆)k u)∗ belong to L2 (Rn ) implies that u∗ ∈ H 2k (Rn ), hence u ∈ H 2k (Σ). Moreover, this consideration shows that the two topologies are equivalent (and in particular independent of the choice of metric on Σ). We may identify H −s (Σ) with the dual of H s (Σ) with respect to the L2 -inner product in Σ. Now let Σ be a (possibly) non-compact Riemannian manifold which is geodesically complete. The Laplace-Beltrami operator ∆ : C0∞ (Σ) → C0∞ (Σ) is essentially selfadjoint by Chernoff’s theorem [9]. We can therefore define the powers (1−∆)s/2 for all s ∈ R. By H s (Σ) we denote the completion of C0∞ (Σ) with respect to the norm uH s (Σ) := (1 − ∆)s/2 uL2 (Σ) . For s ∈ 2N0 , this shows that H 2k (Σ) is the set of all u ∈ L2 (Σ) for which (1 − ∆)k u ∈ L2 (Σ). We deduce that this definition coincides with the previous one if Σ is compact and s = 2k; using complex interpolation, cf. [44, Ch. I, Theorem 4.2], equality holds for all s ≥ 0. Moreover, we can define a sesquilinear form ., . : H −s (Σ) × H s (Σ) → C by letting
u, v := (1 − ∆)−s/2 u, (1 − ∆)s/2 v
L2 (Σ)
.
This allows us to identify H −s (Σ) with the dual of H s (Σ), as in the compact case. In particular, the definition of the Sobolev spaces on compact manifolds coincides also for negative s. Now suppose that O is a relatively compact subset of Σ. We let H s (O) := rO H (Σ), the restriction to O of H s -distributions on Σ, endowed with the quotient topology uH s (O) := inf{U H s (Σ) ; U ∈ H s (Σ), rO U = u}. s
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This definition is local: If O is another relatively compact subset with smooth boundary containing O, then we can find a function f ∈ C0∞ (O ) with f ≡ 1 on O. Hence, whenever there exists a U ∈ H s (Σ) with rO U = u, then there is a U1 ∈ H s (Σ) with supp U1 ⊂ O and rO U1 = u, namely U1 = f U . We therefore obtain the same space and the same topology, if we replace the right-hand side by inf{U H s (Σ) ; U ∈ H s (Σ), supp U ⊂ O , rO U = u}. Indeed, both definitions yield the same space, which also is a Banach space with respect to both norms. As the first norm can be estimated by the second, the open mapping theorem shows that both are equivalent. Note that H s (O) is independent of the particular choice of O . On C0∞ (O) the topology of H s (Σ) is independent of the choice of the Riemannian metric; moreover it coincides with that induced from H s (Rn ) via the coordinate maps: This follows from the fact that, for s = 0, 2, 4, . . . , the spaces H s (Σ) are the domains of powers of the Laplacian, together with interpolation and duality. As a consequence, H s (O) does not depend on the choice of the metric, and its topology is that induced by the Euclidean H s -topology. Finally we define the local Sobolev spaces s 2 n Hloc (Σ) := {u ∈ D (Σ); ξ 2s |κ ∗ (ϕu)(ξ)| d ξ < ∞ for all coordinate maps s (Σ) Hcomp
κ : U → Rn , U ⊂ Σ, and all ϕ ∈ C0∞ (U)} s := {u ∈ Hloc (Σ); supp u compact}.
We have the following inclusions of sets s s s s (O) ⊂ Hcomp (Σ) ⊂ Hloc (Σ) ⊂ H s (O) ⊂ Hloc (O) Hcomp
for any relatively compact subset O of Σ.
B Microlocal analysis with finite Sobolev regularity The C ∞ -wavefront set W F of a distribution u characterizes the directions in Fourier space which cause the appearance of singularities of u. It does however not specify the strength with which the different directions contribute to the singularities. To give a precise quantitative measure of the strength of singular directions of u the notion of the H s -wavefront set W F s was introduced by Duistermaat & H¨ ormander [16]. It is the mathematical tool which we use in the main part of the paper to characterize the adiabatic vacua of a quantum field on a curved spacetime manifold. To make the paper reasonably self-contained we present the definition of W F s and collect some results of the calculus related to it which are otherwise spread over the literature. They are mainly taken from [16, 20, 27, 30, 44]. All other notions from microlocal analysis which we use can also be found there or, in a short synopsis, in [31]. In the following let X denote an open subset of Rn .
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Definition B.1 Let u ∈ D (X), x0 ∈ X, ξ0 ∈ Rn \ {0}, s ∈ R. We say that u is H s (microlocally) in (x0 , ξ0 ) or that (x0 , ξ0 ) is not in the H s -wavefront set of u ((x0 , ξ0 ) ∈ / W F s (u)) if there is a test function ϕ ∈ C0∞ (X) with ϕ(x0 ) = 0 and an open conic neighborhood Γ of ξ0 in Rn \ {0} such that 2 n ξ 2s |ϕu(ξ)| & d ξ < ∞, (101) Γ
2 1/2
where ξ := (1 + |ξ| )
.
Note that, since ϕu ∈ E (X), there is for all (x, ξ) ∈ X × Rn \ 0 a sufficiently small s ∈ R such that (x, ξ) ∈ / W F s (u). From the definition the following properties of s W F are immediate: (i) W F s (u) is a local property of u, depending only on an infinitesimal neighborhood of a point x0 , in the following sense: If u ∈ D (X), ϕ ∈ C0∞ (X) with ϕ(x0 ) = 0 then (x0 , ξ0 ) ∈ W F s (u) ⇔ (x0 , ξ0 ) ∈ W F s (ϕu) (ii) W F s (u) is a closed cone in X × (Rn \ {0}), i.e., in particular (x, ξ) ∈ W F s (u) ⇒ (x, λξ) ∈ W F s (u)
∀λ > 0.
(iii) s (X) W F s (u) = ∅ ⇔ u ∈ Hloc
(iv) s (X) : (x, ξ) ∈ W F (u − v) (x, ξ) ∈ W F s (u) ⇔ ∀v ∈ Hloc
(102)
(v) W F s1 (u) ⊂ W F s2 (u) ⊂ W F (u) ∀ s1 ≤ s2 (vi) W F s (u1 + u2 ) ⊂ W F s (u1 ) ∪ W F s (u2 ) (vii) W F (u) =
W F s (u)
s∈R
As an example consider the δ-distribution in D (Rn ). One easily calculates from the criterion of the definition ∅, s < −n/2 s W F (δ) = (103) {(0, ξ); ξ ∈ Rn \ {0}}, s ≥ −n/2. The following proposition gives an important characterization of the H s -wavefront m (X × Rn ) is the set in terms of pseudodifferential operators. Remember that Sρ,δ space of symbols of order m and type ρ, δ (m ∈ R, 0 ≤ δ, ρ ≤ 1), and Lm ρ,δ (X) the corresponding space of pseudodifferential operators on X.
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Proposition B.2 Let u ∈ D (X). Then W F s (u) = char A = A ∈ L01,0 s Au ∈ Hloc (X)
char A,
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(104)
A ∈ Ls1,0 Au ∈ L2loc (X)
where the intersection is taken over all properly supported classical pseudodifferential operators A (having principal symbol a(x, ξ)) and char A := a−1 (0) \ 0 is the characteristic set of A. Also the pseudolocal property of pseudodifferential operators can be stated in a refined way taking into account the finite Sobolev regularity: Proposition B.3 If A ∈ Lm ρ,δ (X) is properly supported, with 0 ≤ δ < ρ ≤ 1, and u ∈ D (X), then W F s−m (Au) ⊂ W F s (u) for all s ∈ R, in particular s−m s A : Hloc (X) → Hloc (X).
From Propositions B.2 and B.3 we can draw the following important conclusions: (i) Since the principal symbol of a pseudodifferential operator is an invariant function on the cotangent bundle T ∗ X we see from (104) that W F s (u) is well defined as a subset of T ∗ X \0, i.e., does not depend on a particular choice of coordinates. By a partition of unity one can therefore define W F s (u) for any paracompact smooth manifold M and u ∈ D (M) as a subset of T ∗ M\0 and all results in this appendix remain valid when replacing X by M. s (ii) If Au ∈ Hloc (X) for some properly supported A ∈ Lm 1,0 (X) then
W F s+m (u) ⊂ char A.
(105)
This follows from Proposition B.2 because, choosing some elliptic B ∈ s+m 0 L−m 1,0 (X), we have BA ∈ L1,0 (X) and, by Proposition B.3, BAu ∈ Hloc (X), s+m and therefore, by (104), W F (u) ⊂ char(BA) = char(A). (iii) If A ∈ L−∞ (X), then W F (Au) = ∅ and hence W F s (Au) = ∅ for all s ∈ R. (iv) If A ∈ Lm ρ,δ (X), 0 ≤ δ < ρ ≤ 1, is a properly supported elliptic pseudodifferential operator, u ∈ D (X), then W F s−m (Au) = W F s (u) for all s ∈ R. This is a consequence of the fact that an elliptic pseudodifferential operator has a parametrix, i.e., there is a properly supported Q ∈ L−m ρ,δ (X) with
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QAu = u + Ru and AQu = u + R u for some R, R ∈ L−∞ (X). Therefore, by Proposition B.3, W F s (u) = W F s (QAu) ⊂ W F s−m (Au) ⊂ W F s (u). The behaviour of W F s (u) for hyperbolic operators (like the Klein-Gordon operator, which plays an important role in this work) is determined by the theorem of propagation of singularities due to Duistermaat & H¨ ormander [16, Thes (X) for A ∈ orem 6.1.1’]. It states in particular that, if u satisfies Au ∈ Hloc m L1,0 (X) with real principal symbol a(x, ξ) which is homogeneous of degree m, then W F s+m−1 (u) consists of complete bicharacteristics of A, i.e., complete integral curves in a−1 (0) ⊂ T ∗ X of the Hamiltonian vector field " n ! ∂a(x, ξ) ∂ ∂a(x, ξ) ∂ − Ha (x, ξ) := . ∂xi ∂ξi ∂ξi ∂xi i=1 The precise statement is as follows: Proposition B.4 Let A ∈ Lm 1,0 (X) be a properly supported pseudodifferential operator with real principal symbol a(x, ξ) which is homogeneous of degree m. If u ∈ D (X) and Au = f it follows for any s ∈ R that W F s+m−1 (u) \ W F s (f ) ⊂ a−1 (0) \ 0 and W F s+m−1 (u) \ W F s (f ) is invariant under the Hamiltonian vector field Ha . It is well known that the wavefront set gives sufficient criteria when two distributions can be pointwise multiplied, composed or restricted to submanifolds. We reconsider these operations from the point of view of finite Sobolev regularity and obtain weaker conditions in terms of W F s . We start with the regularity of the tensor product of two distributions: Proposition B.5 Let X ⊂ Rn , Y ⊂ Rm be open sets and u ∈ D (X), v ∈ D (Y ). Then the tensor product w := u ⊗ v ∈ D (X × Y ) satisfies W F r (w) ⊂ W F s (u) × W F (v) ∪ W F (u) × W F t (v) (supp u × {0}) × W F (v) ∪ W F (u) × (supp v × {0}), r = s + t ∪ (supp u × {0}) × W F t (v) ∪ W F s (u) × (supp v × {0}), r = min{s, t, s + t}. The proof of this proposition can be adapted from the proof of Lemma 11.6.3 in [30]. The pointwise product of two distributions u1 , u2 ∈ D (X) – if it exists – is defined by convolution of Fourier transforms as the distribution v ∈ D (X) such that ∀x ∈ X ∃f ∈ D(X) with f = 1 near x such that for all ξ ∈ Rn 1 2 v(ξ) = u2 (ξ − η) dn η f' u1 (η)f' f' (2π)n/2 Rn
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with absolutely convergent integral. It is clear that for the integral to be absolutely convergent it is sufficient that f' u1 (η) and f' u2 (ξ − η) decay sufficiently fast in the opposite directions η resp. −η, i.e., that u1 and u2 are in Sobolev spaces of sufficiently high order at (x, η) resp. (x, −η). The precise condition is the following: Proposition B.6 Let u1 , u2 ∈ D (X). Suppose that ∀(x, ξ) ∈ T ∗ X \ 0 ∃s1 , s2 ∈ R with s1 + s2 ≥ 0 such that (x, ξ) ∈ / W F s1 (u1 ) and (x, −ξ) ∈ / W F s2 (u2 ). Then the pointwise product u1 u2 exists. For a proof see [37]. Next we consider the restriction of distributions to submanifolds. Let Σ be an (n − 1)-dimensional hypersurface of X (i.e., there exists a C ∞ -embedding ϕ : Σ → X) with conormal bundle N ∗ Σ := {(ϕ(y), ξ) ∈ T ∗ X; y ∈ Σ, ϕ∗ (ξ) = 0}. We can define the restriction uΣ ∈ D (Σ) of u ∈ D (X) to Σ – if it exists – as the mapping f →$ u · (f δΣ ), 1 , where f δΣ : C ∞ (X) → C is the distribution given by (f δΣ )(g) := Σ f g, f ∈ D(Σ). If Σ is locally given by x0 = 0 then f δΣ is locally given by f (x)δ(x0 ), where δ(x0 ) is the delta-function in the x0 -variable. By a consideration analogous to (103) we see that ∅, s < −1/2 W F s (f δΣ ) ⊂ . (106) N ∗ Σ, s ≥ −1/2 We obtain Proposition B.7 Let u ∈ D (X) with W F s (u) ∩ N ∗ Σ = ∅ for some s > 1/2. Then the restriction uΣ of u is a well-defined distribution in D (Σ), and W F r−1/2 (uΣ ) ⊂ ϕ∗ W F r (u) := {(y, ϕ∗ (ξ)) ∈ T ∗ Σ; (ϕ(y), ξ) ∈ W F r (u)} for all r > 1/2. Proof. Let s > 1/2 and W F s (u) ∩ N ∗ Σ = ∅. It follows from (106) and Proposition B.6 that the product u·f δΣ is defined. Suppose that (y, η) ∈ W F r−1/2 (uΣ ) for some r−1/2 r > 1/2. By (102) we have (y, η) ∈ W F (uΣ − w) for each w ∈ Hloc (Σ). Since r−1/2 r the restriction operator Hloc (X) → Hloc (Σ) is onto [44, Ch. I, Theorem 3.5], r there exists a v ∈ Hloc (X) for each w such that w = vΣ . Hence we have for every r (X) v ∈ Hloc (y, η) ∈ W F (uΣ − vΣ ) = W F ((u − v)Σ ) ⊂ ϕ∗ W F (u − v) where we have used the standard result on the wavefront set of a restricted distribution [26, Theorem 2.5.11’]. Applying (102) again we obtain the assertion.
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The proposition can easily be generalized to submanifolds of higher codimension by repeated projection. From Proposition B.5 and B.7 one can get an estimate for the H s -wavefront set of the pointwise product in Proposition B.6 when noticing that u1 u2 is the pull-back of u1 ⊗ u2 under the map ϕ : X → X × X, x → (x, x) and that ϕ∗ (ξ1 , ξ2 ) = ξ1 + ξ2 . This estimate, however, is rather poor and we will not present it here, better information on the regularity of products can be gained e.g. from [30, Theorem 8.3.1 and Theorem 10.2.10]. Proposition B.8 Let X ⊂ Rn , Y ⊂ Rm be open sets, u ∈ C0∞ (Y ) and let K ∈ D (X × Y ) be the kernel of the continuous map K : C0∞ (Y ) → D (X). Then we have for all s ∈ R s W F s (Ku) ⊂ W FX (K) := {(x, ξ) ∈ T ∗ X \ 0; (x, ξ; y, 0) ∈ W F s (K) for some y ∈ Y }. Proof. Assume that (x, ξ; y, 0) ∈ / W F s (K) for some (x, ξ) ∈ T ∗ X \ 0, y ∈ Y . s (X × Y ) and (x, ξ; y, 0) ∈ / By (102) we can write K = K1 + K2 with K1 ∈ Hloc W F (K2 ). Since Ku = K1 u + K2 u and W F (K2 u) ⊂ W FX (K2 ) it follows that s (X), because then it (x, ξ) ∈ / W F (K2 u). It remains to be shown that K1 u ∈ Hloc s s s follows from (102) that (x, ξ) ∈ / W F (Ku), i.e., W F (Ku) ⊂ W FX (K). ∞ To this end we localize K1 with test functions ϕ ∈ C0 (X) and ψ ∈ C0∞ (Y ) such $ that ψ = 1 on supp u and estimate for ϕ(K1 u) = ϕ(K1 ψu) = K1 (x, y)u(y) dm y ∈ E (X), where K1 (x, y) := ϕ(x)K1 (x, y)ψ(y):
2 m ˆ = d ξ (1 + |ξ| ) d η K1 (ξ, −η)ˆ u(η) ˆ 1 (ξ, −η)|2 ≤ dn ξ (1 + |ξ|2 )s dm η (1 + |η|2 )t |K u(θ)|2 dm θ (1 + |θ|2 )−t |ˆ ˆ (ξ, −η)|2 = C dn ξ dm η (1 + |ξ|2 )s (1 + |η|2 )t |K 1 ˆ 1 (ξ, −η)|2 ≤ C dn ξ dm η (1 + |ξ|2 + |η|2 )s |K
2 d ξ (1 + |ξ| ) |ϕ(K 1 u)(ξ)| n
2 s
n
2 s
s which is finite since K1 ∈ Hcomp (X ×Y ). The last estimate was obtained by putting t := 0 if s ≥ 0, and t := s if s < 0.
In the next proposition we generalize this result to the case where u is a distribution in E (Y ). Then Ku – if it exists – is defined as the distribution in D (X) such that, for ϕ ∈ C0∞ (X), Ku, ϕ = K(1 ⊗ u), ϕ ⊗ 1 .
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Proposition B.9 Let X ⊂ Rn , Y ⊂ Rm be open sets, K ∈ D (X × Y ) be the kernel of the continuous map K : C0∞ (Y ) → D (X), u ∈ E (Y ) and denote W FYs (K) := {(y, η) ∈ T ∗ Y \ 0; (x, 0; y, −η) ∈ W F s (K) for some x ∈ X}. If ∀(y, η) ∈ T ∗ Y \ 0 ∃s1 , s2 ∈ R with s1 + s2 ≥ 0 such that (y, η) ∈ / W FYs1 (K) ∩ W F s2 (u), s then Ku exists. If, in addition, W FY (K) = ∅ and K(Hcomp (Y )) ⊂
(107) s−µ Hloc (X),
then
W F s−µ (Ku) ⊂ W F (K) ◦ W F s (u) ∪ W FX (K), where W F (K) := {(x, ξ; y, −η) ∈ T ∗ X × T ∗ Y ; (x, ξ; y, η) ∈ W F (K)} is to be regarded as a relation mapping elements of T ∗ Y to elements in T ∗ X. Proof. For the first part of the statement we only have to check that the product K(1 ⊗ u) exists. Indeed, by Proposition B.5 we have W F s2 (1 ⊗ u) ⊂ (X × {0}) × W F s2 (u) and, because of (107), for no point (y, η) ∈ T ∗ Y \ 0 is (x, 0; y, −η) in W F s1 (K) and at the same time (x, 0; y, η) in W F s2 (1 ⊗ u). Therefore, according to Proposition B.6, the pointwise product K(1 ⊗ u) exists. Given an open conic neighborhood Γ of W F s (u) in T ∗ Y , we can write u = u1 + u2 s with u1 ∈ Hloc (Y ) and W F (u2 ) ⊂ Γ. This is immediate from (102) with the help s−µ (Y ), and of a microlocal partition of unity. By assumption we have Ku1 ∈ Hloc hence, by [29, Theorem 8.2.13], W F s−µ (Ku) ⊂ ⊂
W F (Ku2 ) ⊂ W F (K) ◦ W F (u2 ) ∪ W FX (K) W F (K) ◦ Γ ∪ W FX (K).
Since Γ was arbitrary, we obtain W F s−µ (Ku) ⊂ W F (K) ◦ W F s (u) ∪ W FX (K). The assumptions in the last proposition are tailored for application to the case that K is the kernel of a Fourier integral operator. Indeed, if K ∈ Iρµ (X × Y, C ), 1/2 < ρ ≤ 1, where C is locally the graph of a canonical transformation from T ∗ Y \ 0 to T ∗ X \ 0, then W F (K) ⊂ C [26, Theorem 3.2.6] and s−µ s K(Hcomp (Y )) ⊂ Hloc (X) [28, Cor. 25.3.2] and the proposition applies. For pseudodifferential operators we have C = id and hence we get back the result of Proposition B.3. In the next proposition we give information about the smoothness of the kernel K itself: Proposition B.10 Let K ∈ Iρµ (X × Y, Λ), 1/2 < ρ ≤ 1, where Λ is a closed Lagrangian submanifold of T ∗ (X × Y ) \ 0, and K ∈ D (X × Y ) its kernel. Then W F s (K) ⊂ W F (K) ⊂ Λ, more precisely n+m , W F s (K) = ∅ if s < −µ − 4 n+m and λ ∈ Λ is a non-characteristic point of K. λ ∈ W F s (K) if s ≥ −µ − 4
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K ∈ Iρµ (X × Y, Λ) is said to be non-characteristic at a point λ ∈ Λ if the principal symbol has an inverse (as a symbol) in a conic neighborhood of λ. A proof of the proposition can be found in [16, Theorem 5.4.1].
Acknowledgments We want to thank Stefan Hollands, Fernando Lled´ o, J¨ org Seiler and Ingo Witt for helpful discussions. W.J. is grateful to the DFG for financial support, to Bernd Schmidt for moral support, and to Prof. B.-W. Schulze for the hospitable reception in his “Arbeitsgruppe Partielle Differentialgleichungen und komplexe Analysis” at Potsdam University.
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[20] L. Garding, Singularities in Linear Wave Propagation. Lecture Notes in Mathematics 1241. Springer Verlag, Berlin, 1987. [21] R. Haag, Local Quantum Physics: Fields, Particles, Algebras. Springer Verlag, Berlin, 2nd edition, 1996. [22] R. Haag and D. Kastler, An algebraic approach to quantum field theory, J. Math. Phys. 5, 848–861 (1964). [23] R. Haag, H. Narnhofer, and U. Stein, On quantum field theory in gravitational background, Comm. Math. Phys. 94, 219–238 (1984). [24] S. Hollands, The Hadamard condition for Dirac fields and adiabatic states on Robertson-Walker spacetimes, Comm. Math. Phys. 216, 635–661 (2001). [25] S. Hollands and W. Ruan, The state space of perturbative quantum field theory in curved spacetimes. Preprint gr-qc/0108032, 2001. [26] L. H¨ormander, Fourier integral operators I. Acta Math. 127, 79–183 (1971). [27] L. H¨ormander, The Analysis of Linear Partial Differential Operators III: Pseudo-Differential Operators. Grundlehren der mathematischen Wissenschaften 274. Springer Verlag, Berlin, 1985.
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[28] L. H¨ormander, The Analysis of Linear Partial Differential Operators IV: Fourier Integral Operators. Grundlehren der mathematischen Wissenschaften 275. Springer Verlag, Berlin, 1985. [29] L. H¨ormander, The Analysis of Linear Partial Differential Operators I: Distribution Theory and Fourier Analysis. Grundlehren der mathematischen Wissenschaften 256. Springer Verlag, Berlin, 2nd edition, 1990. [30] L. H¨ormander, Lectures on Nonlinear Hyperbolic Differential Equations. Math´ematiques & Applications 26. Springer Verlag, Berlin, 1997. [31] W. Junker, Hadamard states, adiabatic vacua and the construction of physical states for scalar quantum fields on curved spacetime. Rev. Math. Phys 8, 1091–1159 (1996) and 14, 511–517 (2002). [32] B.S. Kay and R.M. Wald, Theorems on the uniqueness and thermal properties of stationary, nonsingular, quasifree states on spacetimes with a bifurcate Killing horizon. Phys. Rep. 207, 49–136 (1991). [33] H. Kumano-go, Pseudo-Differential Operators, The MIT Press, Cambridge (Mass.), 1981. [34] P. Leyland, J. Roberts, and D. Testard, Duality for quantum free fields, Preprint, Marseille, 1978. [35] C. L¨ uders and J.E. Roberts, Local quasiequivalence and adiabatic vacuum states, Comm. Math. Phys. 134, 29–63 (1990). [36] J. Manuceau and A. Verbeure, Quasi-free states of the C.C.R.-algebra and Bogoliubov transformations, Comm. Math. Phys. 9, 293–302 (1968). [37] M. Oberguggenberger, Products of distributions, J. reine angew. Math. 365, 1–11 (1986). [38] L. Parker, Quantized fields and particle creation in expanding universes. I, Phys. Rev. 183, 1057–1068 (1969). [39] L. Parker and S.A. Fulling, Adiabatic regularization of the energy-momentum tensor of a quantized field in homogeneous spaces, Phys. Rev. D 9, 341–354 (1974). [40] R.T. Powers and E. Størmer, Free states of the canonical anticommutation relations, Comm. Math. Phys. 16, 1–33 (1970). [41] M.J. Radzikowski, Micro-local approach to the Hadamard condition in quantum field theory on curved space-time, Comm. Math. Phys. 179, 529–553 (1996). [42] S. Rempel and B.-W. Schulze, Index Theory of Elliptic Boundary Problems, Akademie-Verlag, Berlin, 1982.
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[43] R.T. Seeley, Complex powers of an elliptic operator, Amer. Math. Soc. Proc. Symp. Pure Math. 10, 288–307 (1968). [44] M.E. Taylor, Pseudodifferential Operators, Princeton, New Jersey, 1981.
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[45] H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, North-Holland Publ. Company, Amsterdam, 1978. [46] W.G. Unruh, Notes on black hole evaporation, Phys. Rev. D. 14, 870–892 (1976). [47] R. Verch, Nuclearity, split property, and duality for the Klein-Gordon field in curved spacetime, Lett. Math. Phys. 29, 297–310 (1993). [48] R. Verch, Local definiteness, primarity and quasiequivalence of quasifree Hadamard quantum states in curved spacetime, Comm. Math. Phys 160, 507–536 (1994). [49] R. Verch, Continuity of symplectically adjoint maps and the algebraic structure of Hadamard vacuum representations for quantum fields on curved spacetime, Rev. Math. Phys. 9, 635–674 (1997). [50] R.M. Wald, Quantum Field Theory in Curved Spacetime and Black Hole Thermodynamics, The University of Chicago Press, Chicago, 1994. [51] K. Yosida, Functional Analysis, Grundlehren der mathematischen Wissenschaften 123. Springer Verlag, Berlin, 6th edition, 1980. Wolfgang Junker Max-Planck-Institut f¨ ur Gravitationsphysik Albert-Einstein-Institut Am M¨ uhlenberg 1 D-14476 Golm Germany email: [email protected] Elmar Schrohe Universit¨ at Potsdam Institut f¨ ur Mathematik Am Neuen Palais 10 D-14415 Potsdam Germany email: [email protected] Communicated by Klaus Fredenhagen submitted 25/03/02, accepted 15/04/02
Ann. Henri Poincar´e 3 (2002) 1183 – 1213 c Birkh¨ auser Verlag, Basel, 2002 1424-0637/02/0601183-31
Annales Henri Poincar´ e
Marginal Fermi Liquid Behaviour in the d = 2 Hubbard Model with Cut-Off V. Mastropietro Abstract. We consider the half-filled Hubbard model with a cut-off forbidding momenta close to the angles of the square shaped Fermi surface. By renormalization group methods we find a convergent expansion for the Schwinger function up to exponentially small temperatures. We prove that the system is not a Fermi liquid, but on the contrary it behaves like a Marginal Fermi liquid, a behaviour observed in the normal phase of high Tc superconductors.
1 Main results 1.1
Motivations
The notion of Fermi liquids, introduced by Landau, refers to a wide class of interacting fermionic systems whose thermodynamic properties (like the specific heat or the resistivity) are qualitatively the same of a gas of non-interacting fermions. While there is an enormous number of metals having Fermi liquid behaviour, in recent times new materials has been found whose properties are qualitatively different. In particular the high-temperature superconducting materials (so anisotropic to be considered essentially bi-dimensional) in their normal phase have a non Fermi liquid behaviour, in striking contrast with previously known superconductors, which are Fermi liquids above the critical temperature. While in Fermi liquids the wave function renormalization Z is Z = 1 + O(λ2 ), where λ is the strength of the interaction, in such metals it was found Z 1+O(λ2 log T ) for temperatures T above the critical temperature, see [VLSAR] (see also [VNS] for a review); metals behaving in this way were called Marginal Fermi liquids. Such results stimulated an intense theoretical research. It was found by a perturbative analysis, see for instance [AGD] or [Sh], that in a system of weakly interacting fermions in d = 2 Z is essentially temperature independent, at least for circular or “almost” circular Fermi surfaces. Despite doubts appeared about the reliability of results obtained by perturbative expansions [A], such results were indeed confirmed recently by rigorous renormalization group methods. It was proved in [FMRT] and [DR] that indeed a weakly interacting Fermi system with a circular Fermi surface is a Fermi liquid, up to exponentially small temperatures. Such result was extended in [BGM] to all possible weakly interacting d = 2 fermionic systems with symmetric, smooth and convex Fermi surfaces, up to exponentially small temperatures. These results cannot be obtained by dimensional power counting arguments as such arguments
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give a bound |Z − 1| ≤ Cλ2 | log T | from which one cannot distinguish Fermi or non Fermi liquid behaviour; for obtaining Z = 1 + O(λ2 ) one has instead to use delicate volume improvements in the integrals expressing Z, based on the geometrical constraints to which the momenta close to the Fermi surface (assumed convex, regular and symmetric) are subjected. As Fermi liquid behaviour is found in systems with symmetric, smooth and convex Fermi surfaces, in order to find non Fermi liquid behaviour one has to relax some of such conditions. It was pointed out, see for instance [VR] and [ZYD], that the presence in the Fermi surface of flat regions in opposite sides could produce a non Fermi liquid behaviour; flat regions are indeed present in the Fermi surfaces of high Tc superconductors [S]. The simplest model exhibiting a Fermi surface with flat pieces is the half-filled Hubbard model, describing a system of spinning d = 2 fermions with local interaction and dispersion relation given by ε(kx , ky ) = cos kx + cos ky . The Fermi surface is the set of momenta such that ε(kx , ky ) = 0 and it is a square with corners (±π, 0) and (0, ±π). However this model has the complicating feature of vanishing Fermi velocity at the points (±π, 0) and (0, ±π), i.e., at the corners of the Fermi surface; this originates to the so-called Van Hove singularities in the density of states. In order to investigate the possible non Fermi liquid behaviour of interacting fermions with a Fermi surface with flat pieces, independently from the presence of Van Hove singularities, one can introduce in the half-filled Hubbard model a cut-off forbidding momenta near the corners of the Fermi surface. The half-filled Hubbard model with cut-off (or the essentially equivalent, but slightly simpler, problem of fermions with the linearized dispersion relation ε(kx , ky ) = |kx | + |ky | − π) has been extensively studied in literature, see for instance [M], [L], [ZYD], [VR], [FSW], [DAD], [FSL]. The cut-off is somewhat artificially introduced but the idea is that the model, at least for same values of the parameters, belongs to the same university class of models with “almost” squared and smooth Fermi surface, like the anisotropic Hubbard models [Sh], the Hubbard model with nearest and next to nearest neighbor interaction [Me], or the half-filled Hubbard model close to half-filling. Aim of this paper is to compute in a rigorous way the asymptotic behaviour of the Schwinger functions of the half-filled Hubbard model with cut-off up to exponentially small temperatures. We will show that such a system is indeed a Marginal Fermi liquid, and our result furnishes indeed the first example rigorously established of such behaviour in d = 2. k +k For our convenience, we will consider new variables k+ = x 2 y and k− = kx −ky so that the dispersion relation of the half-filled Hubbard model is given by 2 ε(k+ , k− ) = 2 cos k+ cos k− and the Fermi surface is the set k+ = ± π2 or k− = ± π2 .
(1.1)
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The model
Given a square [0, L]2 ∈ R2 , the inverse temperature β and the (large) integer M , we introduce in Λ = [0, L]2 × [0, β] a lattice ΛM , whose sites are given by the space-time points x = (x0 , x+ , x− ) with (x+ , x− ) ∈ Z2 and x0 = n0 β/M , n0 = 0, 1, . . . , M − 1. We also consider the set D of space-time momenta k = 2 ± (k0 , k+ , k− ) ≡ (k0 , k), with k± , = 2πn L , (n+ , n− ) ∈ Z , [−L/2] ≤ n± ≤ [L − 1/2]; 1 k0 = 2π β (n0 + 2 ), n0 = 0, 1, . . . , M − 1. With each k ∈ D we associate four Grassε , ε, s ∈ {+, −}; s is the spin. The lattice ΛM is introduced manian variables ψˆk,s only for technical reasons so that the number of Grassmann variables is finite, and eventually the (essentially trivial) limit M → ∞ is taken. We introduce also a linear functional P (dψ) on the Grassmanian algebra generated by the variables ε ψˆk,σ , such that P (dψ)ψˆk−1 ,s1 ψˆk+2 ,s2 = L2 βδs1 ,s2 δk1 ,k2 gˆ(k1 ) , (1.2) where g(k) is defined by gˆ(k) =
χ(k) , −ik0 + 2 cos k+ cos k−
(1.3)
where χ(k) is a cut-off function χ(k) = H(a20 sin2 k+ )C0−1 (k) + H(a20 sin2 k− )C0−1 (k) C0−1 (k) = H( k02 + 4 cos2 (k+ ) cos2 (k− )) √ and, if γ > 1 and a0 ≥ 2 1 if |t| < γ −1 H(t) = , 0 if |t| > 1
(1.4)
where
(1.5)
(1.6)
The function C0−1 (k) acts as an ultraviolet cut-off forcing the momenta k to be not too far from the Fermi surface, and k0 not too large; the cut-off on k0 is imposed only for technical convenience and it could be easily removed. The functions H(a20 sin2 k± ) forbid momenta near the corners of the Fermi surface, i.e., the points (±π/2, ±π/2). The Grassmanian field ψxε is defined by 1 ˆ± ±ik·x ± ψx,s = 2 . (1.7) ψk,s e L β k∈D
The “Gaussian measure” P (dψ) has a simple representation in terms of the “Lebesgue Grassmanian measure” Dψ =
∗ k∈D,s=±
+ − dψˆk,s dψˆk,s ,
(1.8)
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defined as the linear functional on the Grassmanian algebra, such that, given a − + monomial Q(ψˆ− , ψˆ+ ) in the variables ψˆk,s , ψˆk,s , its value is 0, except in the case ∗ − ˆ+ − ˆ+ ˆ ˆ Q(ψ , ψ ) = k,s ψk,s ψk,s , up to a permutation of the variables, in which case ∗ its value is 1. Finally k∈D,s=± means a product over the k such that χ(k) > 0. We define P (dψ) = N −1 Dψ · exp[−
1 L2 β
∗ k∈D,σ=±
+ ˆ− χ−1 (k)(−ik0 + 2 cos k+ cos k− )ψˆk,s ψk,s ] ,
(1.9) ∗ with N is a renormalization constant and again k means a sum over k such that χ(k) > 0. The two point Schwinger function is defined by the following Grassman functional integral − + P (dψ)e−V(ψ) ψx,s ψy,s S(x − y) = lim lim , (1.10) L→∞ M→∞ P (dψ)e−V(ψ) where, if we use
dx as a shorthand for V(ψ) = λ
β M
x∈ΛM ,
+ − + − ψx,s ψx,−s ψx,−s . dxψx,s
(1.11)
s
ˆ We call S(k) the Fourier transform of S(x − y).
1.3
Main theorem
Our main results are summarized by the following Theorem, which will be proved in the following sections. Theorem. Given a0 large enough, there exist two positive constants ε and c¯ such π ≤ that, for all |λ| ≤ ε and T ≥ exp{−(¯ c|λ|)−1 }, for all k ∈ D such that 2β 2 3π 2 2 2 2 k0 + 4 cos k+ cos k− ≤ 2β and H(a0 sin k− ) = 1 then (k 2 + 4 cos2 k+ cos2 k− )η(k− ) ˆ S(k) = 0 (1 + λ2 AI (k)) , (1.12) −ik0 + 2 cos k+ cos k− 2 π 2 ≤ k02 + 4 cos2 k+ cos2 k− ) ≤ 3π and for k ∈ D such that 2β 2β and H(a0 sin k+ ) = 1 then (k 2 + 4 cos2 k+ cos2 k− )η(k+ ) ˆ S(k) = 0 (1 + λ2 AII (k)) , (1.13) −ik0 + 2 cos k+ cos k− where |Ai (k)| ≤ c, where c > 0 is a constant, and η(k± ) = a(k± )λ2 + O(λ3 ) is a critical index expressed by a convergent series with a(k± ) ≥ 0 a not identically vanishing smooth function.
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Remarks
The above theorem describes the behaviour of the two point Schwinger function up to exponentially small temperatures, i.e., T ≥ exp{−(¯ c|λ|)−1 }; the constant c¯ is essentially given by the second order terms of the perturbative expansion. A straightforward consequence of (1.12), (1.13) is that the wave function renormalization is Z = 1 + O(λ2 log β), which means that the half-filled Hubbard model with cut-off is a marginal Fermi liquid up to exponentially small temperatures. From (1.12), (1.13) we see that the behaviour of the Schwinger function close to the Fermi surface is anomalous and described by critical indices which are functions of the projection of the momentum on the Fermi surface. Critical indices which are momentum dependent were found for the same model also in [FSL] by heuristic bosonization methods. The presence of the critical indices makes the Schwinger function quite similar to the one for d = 1 interacting spinless fermionic systems, characterized by Luttinger liquid behaviour (see for instance [A]). However an important difference is that the critical exponent η in a Luttinger liquid is a number, while here is a function of the momenta. Another crucial difference ˆ is that in a Luttinger liquid S(k) gˆ(k)|k|η , with η = aλ2 + O(λ3 ) up to T = 0; hence a Luttinger liquid is a Marginal Fermi liquid for high enough temperatures but not all the marginal Fermi liquids are Luttinger liquids. The paper is organized in the following way. In §2 we implement renormalization group ideas by writing the Grassman integration in (1.10) as the product of many integrations at different scales. The integration of a single scale leads to new effective interactions, and the renormalization consists in subtracting from the kernels of the effective interaction (which are not dimensionally irrelevant) their value computed at the Fermi surface. One obtains an expansion for the Schwinger functions as power series of a set of running couplings functions (depending from the momentum on the Fermi surface and the scale). In §3 we prove that this series is convergent if the running coupling functions are small enough; the convergence radius is finite and temperature independent, and this means that the theory is renormalizable. In the proof of convergence one uses the Gram-Hadamard inequality. In §4 we show that the running coupling functions obey to a recursive set of integral equation, called beta function, and we show that the running coupling functions remain small up to exponentially small temperatures T ≥ exp{−(¯ c|λ|)−1 }. Moreover we show that the wave function renormalization has an anomalous flow, with a non-vanishing exponent (contrary to what happens for instance in the case of circular Fermi surfaces), and this essentially concludes the proof of the Theorem. It would be possible to use our beta function to detect (at least numerically) the main instabilities of the system at very low temperatures. At the moment, this kind of numerical analysis was done for this model only in [ZYD] in the parquet approximations, with no control on higher orders which are simply neglected. Finally in §5 we compare the Marginal Fermi liquid behaviour we find in this model with the Luttinger liquid behaviour, and we discuss briefly what happens in the Hubbard model with cut-off close to half-filling.
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It is very likely that the half-filled Hubbard model with cut-off can work as a paradigm for a large class of systems, in which the Fermi surface is flat or almost flat but there are no Van Hove singularities. Marginal Fermi liquid behaviour can be surely found in the Hubbard model with cut-off and close to half-filling, up to temperatures above the inverse of the radius of curvature of the Fermi surface. Another model in which one could possibly find Marginal Fermi liquid behaviour is the anisotropic Hubbard model introduced in [Sh] with dispersion relation cos k1 + t cos k2 , with t = 1 + ε. Such model has a Fermi surface with no van Hove singularities and four “almost” flat and parallel pieces, and one can c|λ|)}]. Another expect Z = 1 + O(λ2 | log(|ε|)| log β) for β ≤ O(min[ε−1 , exp{(¯ interesting question is the possibility of Marginal Fermi liquid behaviour in the Hubbard model close to half-filling (with no cut-off). At half-filling it is believed Z 1 + O(λ2 log2 β), so a different behaviour with respect to Marginal Fermi liquid behaviour. A renormalization group analysis for this problem was begun in [R], and it was proved the convergence of the series not containing subgraphs with −1 two external lines for T ≥ exp{−(c0 |λ|) 2 }.
2 Renormalization group analysis 2.1
The scale decomposition
As the spin index will play no role in the following analysis (on the contrary it is expected to have an important role at lower temperatures) we simply omit it. The cut-off function χ(k) defined in (1.4) has a support in the k space which is given by four disconnected regions, each one containing only one flat side of the Fermi surface. It is natural then to write each Grassman variable as a sum of four independent Grassman variables, with momentum k having value in one of the four disconnected regions; each field will be labeled by a couple of indices, σ = I, II and ω = ±1, so that each field has spatial momenta with values in the region containing (ωpF , 0) if σ = I or (0, ωpF ) if σ = II. We write the Grassman integration as
P (dψ)F (ψ) =
Pσ,ω (dψ)F (
σ=I,II ω=±1
ψσ,ω ) ,
(2.1)
σ=I,II ω=±1
where F is any monomial, ω = ±1 and
− ˆ+ PI,ω (dψ)ψˆI,ω,k +ω F,I pF,I ψI,ω ,k +ω p 1
= δω,ω δk1 ,k H(a20 sin2 k− )
, k− ) Cω−1 (k0 , k+ cos k −ik0 + 2ω sin k+ −
(2.2)
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− ˆ+ PII,ω (dψ)ψˆII,ω,k +ω F,II pF,II ψII,ω ,k +ω p 1
= δk1 ,k δω,ω H(a20 sin2 k+ )
, k+ ) Cω−1 (k0 , k− , (2.3) cos k −ik0 + 2ω sin k− +
cos k ) , k− ) = θ(ωk+ + pF )H( k02 + 4 sin2 k+ Cω−1 (k0 , k+ − cos k ) , k+ ) = θ(ωk− + pF )H( k02 + 4 sin2 k− Cω−1 (k0 , k− +
where
(2.4) (2.5)
and pF,σ is defined such that pF,I = ( π2 , 0) and pF,II = (0, π2 ); moreover pF = π2 and k = k + ω pF,σ ( k is the momentum measured from the Fermi surface). It is convenient, for clarify reasons, to start by studying the “free energy” of the model, defined as 1 (2.6) − 2 log P (dψ)e−V , L β where, calling with a slight abuse of notation ψˆσ,ω,k +ωpF,σ ≡ ψˆσ,ω,k , V is equal to
λ
ω1 ,dots,ω4 σ1 ,dots,σ4 =I,II
4 εi (ki + ωi pF,σi ))ψˆσ+1 ,ω1 ,k ψˆσ+2 ,ω2 ,k ψˆσ−3 ,ω3 ,k ψˆσ−4 ,ω4 ,k , (2.7) dk1 . . . dk4 δ( i=1
1
2
3
4
where dk = L12 β k and δ(k − k ) = L2 βδk,k . We will evaluate the Grassman integral (2.6) by a multiscale analysis based on (Wilsonian) renormalization group ideas. The starting point is the following decomposition of the cut-off functions (2.4), (2.5) H( k02 + 4 cos2 kˆσ sin2 k σ ) 0 0 = fk (k0 , k σ , kˆσ ) , (2.8) f¯k ( k02 + 4 cos2 kˆσ sin2 k σ ) ≡ k=−∞
k=−∞
with f¯k (t) = H(γ −k t) − H(γ −k+1 t) is a smooth compact support function, with support γ k−1 ≤ |t| ≤ γ k+1 ; moreover: a) kˆσ = k− if σ = I and kˆσ = k+ if σ = II; kˆσ is the projection of k in the direction parallel to the Fermi surface. b) k σ = k+ if σ = I and k σ = k− if σ = II; k σ + ωpF,σ is the projection of k in the direction normal to the Fermi surface. For each σ, the function fk (k0 , kσ , kˆσ ) has a support in two regions of thickness O(γ k ) around each flat side of the Fermi surface, at a distance O(γ k ) from it. We
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will assume L = ∞ for simplicity and it follows that there is a hβ = O(log β) such that fk = 0 for k < hβ , while fk is not identically vanishing for k ≥ hβ . The integration of (2.6) will be done iteratively integrating out the fields with momenta closer and closer to the Fermi surface. We will prove by induction that ) and a sequence of effective it is possible to define a sequence of functions Zh (k¯σ,ω (h) potentials V such that
PI (dψ)PII (dψ)e−V
√ ( Zh ψ (≤h) )
, (2.9)
)ψ ˆ(≤h) , Zh (k¯ )ψˆ(≤h) , ( Zh (k¯I,1 I,1,k I,−1,k I,−1 (≤h) (≤h) Zh (k¯II,1 )ψˆII,1,k , Zh (k¯II,−1 )ψˆII,−1,k )
(2.10)
= e−L
2
PZh ,I (dψ (≤h) )PZh ,II (dψ (≤h) )e−V
βEh
where Eh is a constant and
(h)
√ Zh ψˆ(≤h) equal to
and PZh ,σ (dψ (≤h) ) is the fermionic integration with propagator (≤h) gσ,ω (k )
−1 (k0 , kσ , kˆσ ) H(a20 sin2 kˆσ )Ch,ω 1 = θ(ωk σ + ωpF ) ) Zh (k¯σ,ω −ik0 + 2ω cos kˆσ sin k σ
with −1 Ch,ω (k0 , k σ , kˆσ ) =
h
fk (k0 , k σ , kˆσ ).
(2.11)
(2.12)
k=−∞
The θ-function in (2.11) can be omitted by the definition of the variables k σ . We define k¯σ,ω = ( πβ , ωpF , k− ) if σ = I and k¯σ,ω = ( πβ , k+ , ωpF ) if σ = II; moreover k¯σ,ω = ( πβ , 0, k− ) if σ = I and k¯σ,ω = ( πβ , k+ , 0) if σ = II; moreover we π ¯ ¯ call kσ,ω = (− β , 0, k− ) if σ = I and kσ,ω = (− πβ , k+ , 0) if σ = II. If ε = ± ∞ V (h) (ψ ≤h ) = n=1 ω1 ,...,ω2n σ1 ,...,σ2n ε1 ,...,εn
dk1
. . . dk2n δ(
i
εi (ki
+ ωi pF,σi ))
2n
i=1
(≤h)ε ψˆσi ,ωi ,ki i
ˆ (h) (k , . . . , k W 1 2n−1 ) , (2.13) 2n
where h h ˆ 2n ˆ 2n W (k1 . . . k2n−1 ) = W (k1 + ω1 pF,σ1 . . . k2n−1 + ω2n−1 pF,σ2n−1 ) h ˆ 2n (k1 . . . k2n−1 ) . (2.14) =W
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The renormalization procedure
Let us show that (2.9) is true for h − 1, assuming that it is true for h. We define an L operator acting linearly on the kernels of the effective potential (2.13): ˆ (h) = 0 if n ≥ 2 1) LW 2n 2) If n = 1 1 h ¯ ˆ h (k¯ ) + sin k¯ ∂σ W ˆ h (k¯ ) , ˆ h (k¯ )] + k0 ∂k0 W [W (k ) + W 2 σ,ω 2 σ,ω σ 2 σ,ω 2 2 σ,ω (2.15) where ∂k0 means the discrete derivative and ∂σ = ∂k+ is σ = I and ∂σ = ∂k− ˆ h (k¯ ) + W ˆ h (k¯ )] = 0. is σ = II. We will prove in §4 that [W 2 σ,ω 2 σ,ω 3) If n = 2 ˆ 4h (k1 , k2 , k3 ) = W ˆ 4h (k¯σ ,ω , k¯σ ,ω , k¯σ ,ω ) . LW (2.16) 1 1 2 2 3 3 ˆ h (k ) = LW 2
ˆ 2h (k¯ω,σ ˆ 2h (k¯ω,σ Calling ∂0 W ) = −iah (k¯ω,σ ), ∂σ W ) = 2ω cos kˆσ zh (k¯ω,σ ) and
ˆ 4h (k¯σ ,ω , k¯σ ,ω , k¯σ ,ω ) , lh (k¯σ 1 ,ω1 , k¯σ 2 ,ω2 , k¯σ 3 ,ω3 ) = W 1 1 2 2 3 3
(2.17)
we can write +(≤h) −(≤h) h )2ω cos kˆσ sin k σ − ik0 ah (k¯ω,σ )]ψˆk ,σ,ω ψˆk ,σ,ω + dk [zh (k¯ω,σ LV = {ω},{σ}
σ=I,II
+(≤h) +(≤h) −(≤h) −(≤h) dk1 . . . dk4 lh (k¯ω 1 ,σ1 , k¯ω 2 ,σ2 , k¯ω 3 ,σ3 )ψˆk ,σ1 ,ω1 ψˆk ,σ2 ,ω2 ψˆk ,σ3 ,ω3 ψˆk ,σ4 ,ω4 1 2 3 4 δ( εi (ki + pF,σi )) . (2.18) i
We write the right-hand side of (2.9) as √ √ (h) (≤h) )−RV (h) ( Zh ψ (≤h) ) PI,Zh (dψ (≤h) ) PII,Zh (dψ (≤h) )e−LV ( Zh ψ
(2.19)
with R = 1 − L.
2.3
Remark 1.
The non-trivial action of R on the kernel with n = 2 can be written as ˆ h (k , k , k ) = [W ˆ h (k , k , k ) − W ˆ h (k¯ RW 4 1 2 3 4 1 2 3 4 σ1 ,ω1 , k2 , k3 )] ˆ h (k¯ ˆ h (k¯ , k , k ) − W , k¯ , k )] + [W 4
σ1 ,ω1
2
3
4
σ1 ,ω1
σ2 ,ω2
3
ˆ 4h (k¯σ ,ω , k¯σ ,ω , k¯σ ,ω )] . (2.20) ˆ 4h (k¯σ ,ω , k¯σ ,ω , k3 ) − W + [W 1 1 2 2 1 1 2 2 3 3
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The first addend can be written as, if σ1 = I (say), in the limit L → ∞ π (k0,1 − ) β
1
ˆ h ( π + t(k0,1 − π ), k , k−,1 ; k , k ) dt∂k0,1 W 4 +,1 2 3 β β 0 1 ˆ h (0, tk , k−,1 ; k , k ) . W dt∂k+,1 +k+,1 4 +,1 2 3
(2.21)
0
The factors k0,1 − π/β and k+,1 are O(γ h ), for the compact support properties +(≤h)
of the propagator associated to ψI,ω1 ,k , with h ≤ h, while the derivatives are 1
dimensionally O(γ −h−1 ); hence the effect of R is to produce a factor γ h −h−1 < 1. Similar considerations can be done for the other addenda and for the action of R on the n = 1 terms.
Remark 2. From (2.16) we see that the effect of the L operation is to replace in W2h (k) the momentum k with its projection on the closest flat side of the Fermi surface. Hence the fact that the propagator is singular over an extended region (the Fermi surface) and not simply in a point has the effect that the renormalization point cannot be fixed but it must be left moving on the Fermi surface.
2.4
The anomalous integration
In order to integrate the field ψ (h) we can write
√ (h) √ (≤h) )−RV (h) ( Zh ψ (≤h) ) PI,Zh (dψ (≤h) ) PII,Zh (dψ (≤h) )e−LV ( Zh ψ √ √ (≤h) ˜h )−RV (h) ( Zh ψ (≤h) ) = PI,Zh−1 (dψ (≤h) ) PII,Zh−1 (dψ (≤h) )e−LV ( Zh ψ (2.22)
where Pσ,Zh−1 (dψ (≤h) ) is the fermionic integration with propagator H(a20 sin2 kˆσ )Ch−1 (k0 , k σ , kˆσ ) Zh−1 (k ) −ik0 + 2ω cos kˆσ sin k 1
(2.23)
σ
and Zh−1 (k ) = Zh (k¯σ,ω )[1 + H(a20 sin2 kˆσ )Ch−1 (k0 , k σ , kˆσ )ah (k¯σ,ω )] .
(2.24)
Moreover LV˜ h = LV h −
σ=I,II
+(≤h)
−(≤h)
)[2ω cos kˆσ sin k σ − ik0 ]ψˆk ,σ,ω ψˆk ,σ,ω . (2.25) dk zh (k¯σ,ω
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We rescale the fields by rewriting the right-hand side of (2.22) as √ √ (≤h) ˆh (≤h) )−RV (h) ( Zh−1 ψ (≤h) ) ) PII,Zh−1 (dψ (≤h) )e−LV ( Zh−1 ψ , PI,Zh−1 (dψ (2.26)
where ˆh LV = dk [δh,ω (k¯σ,ω )2ω cos kˆσ sin kσ )]ψˆk+ ,σ,ω ψˆk− ,σ,ω + σ=I,II
dk1 . . . dk4
σ1 ,...,σ4 =I,II
λh (k¯σ 1 ,ω1 , k¯σ 2 ,ω2 , k¯σ 3 ,ω3 )ψˆk+ ,σ1 ,ω1 ψˆk+ ,σ2 ,ω2 ψˆk− ,σ3 ,ω3 ψˆk− ,σ4 ,ω4 δ( εi (ki + pF,σi ) 1
2
3
4
i
(2.27) and
Zh (k¯σ,ω ) )) (2.28) (z (k¯ ) − ah (k¯σ,ω ) h σ,ω Zh−1 (k¯σ,ω 4 Zh (k¯σ i ,ωi ) λh (k¯σ1 ,ω1 , k¯σ2 ,ω2 , k¯σ3 ,ω3 ) = [ ]lh (k¯σ 1 ,ω1 , k¯σ 2 ,ω2 , k¯σ 3 ,ω3 ) . ¯ Z ( k ) h−1 σi ,ωi i=1 )= δh (k¯ω,σ
We will call δh and λh running coupling functions; the above procedure allow to write a recursive equation for them, see §5. Then we write (≤h−1) (≤h−1) (h) ) PII,Zh−1 (dψ ) PI,Zh−1 (dψ ) PII,Zh−1 (dψ (h) ) PI,Zh−1 (dψ ˆ (h) (
e−LV
√
Zh−1 ψ (≤h) )−RV (h) (
√
Zh−1 ψ (≤h) )
(2.29)
and the propagator of Pσ,Zh−1 (dψ) is (h) (k ) = H(a20 sin2 kˆσ ) gˆω,σ
1 f˜h (k0 , k σ , kˆσ ) ¯ Zh−1 (kω,σ ) −ik0 + 2ω cos kˆσ sin k σ
(2.30)
and −1 C −1 (k0 , k σ , kˆσ ) Ch−1 (k0 , k σ , kˆσ ) − f˜h (k0 , kσ , kˆσ ) = Zh−1 (k¯ω,σ )[ h ] ) Zh−1 (k ) Zh−1 (k¯σ,ω
(2.31)
with H(a20 sin2 kˆσ )f˜h (k0 , k σ , kˆσ ) having the same support that H(a20 sin2 kˆσ ) fh (k0 , k σ , kˆσ ). We integrate then the field ψ (h) and we get e−L
2
βEh−1
PI,Zh−1 (dψ (≤h−1) )
PII,Zh−1 (dψ (≤h−1) )e−V
(h−1)
(
√
Zh−1 ψ (≤h−1) )
(2.32) and the procedure can be iterated.
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We will see in the following section that if the running coupling functions are small Zk−1 (k¯σ,ω ) )| ≤ 2|λ| sup sup ≤ e2|λ| sup sup |δk (k¯σ,ω ¯ Z ( k ) ¯ ¯ k≥h k k≥h k k σ,ω σ,ω σ,ω sup
¯ k≥h k σ
sup
¯ ¯ ,k σ2 ,ω2 ,kσ3 ,ω3 1 ,ω1
|λk (k¯σ 1 ,ω1 , k¯σ 2 ,ω2 , k¯σ 3 ,ω3 )| ≤ 2|λ| ,
(2.33)
then the effective potential is given by a convergent series. In §4 we will show that up to exponentially small temperatures this is indeed true.
3 Analyticity of the effective potential 3.1
Coordinate representation
ε It is convenient to perform bounds to introduce the variables ψx,ω,σ . We define ε iεω pF,σ x ˜ε ψx,ω,σ , or more explicitly the fields ψx,ω,σ = e ε ε ψx,ω,I = eiεωpF x1 ψ˜x,ω,I
ε ε ψx,ω,II = eiεωpF x2 ψ˜x,ω,II
(3.1)
and the propagators of such fields is (h) g˜ω,σ (x − y) =
dk
ˆ 2ˆ 2 ˜ 1 −ik (x−y) H(a0 sin kσ )fh (k0 , k σ , kσ ) e ) Zh−1 (k¯ω,σ −ik0 + ω2 sin k σ cos kˆσ
(3.2)
It is easy to prove, by integration by parts, that for any integer N , for L → ∞ (h)
|∂xn00 ∂xn++ ∂xn−− g˜I,ω (x − y)| ≤
(h)
|∂xn00 ∂xn++ ∂xn−− g˜II,ω (x − y)| ≤ where d(x0 ) =
β π
1+
[γ h |d(x0
Cn0 ,n+ ,n− ,N γ h(1+n0 +n+ ) − y0 )| + γ h |x+ − y+ | + |x− − y− |]N (3.3)
Cn0 ,n+ ,n− ,N γ h(1+n0 +n− ) 1 + [γ h |d(x0 − y0 )| + |x+ − y+ | + γ h |x− − y− |]N (3.4)
sin xβ0 π .
Proof. The above formula can be derived by integration by parts; note that, if for instance σ = I ∂k−
1 1 = cos k cos k )2 2ω sin k+ sin k− −ik0 + 2ω sin k+ (−ik0 + 2ω sin k+ − −
(3.5)
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which is O(γ −h ); in the same way the n-th derivative with respect to k− is still n O(γ −h ). On the other hand ∂kn00 ∂k++ is bounded by γ −h−n0 h−n+ h ; finally the integration gives a volume factor γ 2h . We define h W2n (x1 , . . . , x2n ) = 2n 1 −i εr kr xr ˆ h r=1 W2n (k1 , . . . k2n−1 )δ( e εi (ki + ωi pF,σi )) . (3.6) 2 2n (L β) i k1 ,...,k2n
Hence (2.13) can be written as V (h) (ψ ≤h ) =
∞
dx1 . . . dx2n
n=1 ω1 ,...,ω2n σ1 ,...,σ2n ε1 ,...,ε2n
2n
i ψ˜σ(≤h)ε i ,xi ,ωi
(h)
W2n (x1 , . . . , x2n ) .
i=1
(3.7) We now discuss the action of the operator L and R = 1 − L on the effective potential in the x-space representation. Noting that from (3.6), if ε1 = ε2 = −ε3 = −ε4 = + W4h (x1 , x2 , x3 , x4 ) ˜ 4h (x1 − x4 , x2 − x4 , x3 − x4 ) , (3.8) = eix4 (ω1 pF,σ1 +ω2 pF,σ2 −ω3 pF,σ3 −ω4 pF,σ4 ) W we can write the action of R (2.16) as R
4
dxi
i=1
4
˜ 4h ψ˜xεii ,σi ,ωi eix4 (ω1 pF,σ1 +ω2 pF,σ2 −ω3 pF,σ3 −ω4 pF,σ4 ) W
i=1
=
4 i=1
dxi
4
ψ˜xεii ,σi ,ωi eix4 (ω1 pF,σ1 +ω2 pF,σ2 −ω3 pF,σ3 −ω4 pF,σ4 )
i=1
˜ 4h (x1 −x4 , x2 −x4 , x2 −x4 )−δ(x0,1 −x0,4 )δ(x0,2 −x0,4 )δ(x0,3 −x0,4 )δ(x [W σ1 ,1 −xσ1 ,4 ) ˜ (h) (t1 , t2 , t3 )] , δ(xσ2 ,2 −xσ2 ,4 )δ(xσ3 ,3 −xσ3 ,4 ) dt0,1 dt0,2 dt0,3 dt1,σ1 dt2,σ2 dt3,σ3 W 4 (3.9) β ˆ where dx = M x∈Λ and ti = (t0,i , tσ,i , tσ,i ) where tI,i = t+,i ; tII,i = t−,i and tˆI,i = t−,i ; tˆII,i = t+,i . On the other hand we can equivalently write the R operation as acting on the fields, and such two representations of the R operation will be used in the
1196
V. Mastropietro
Ann. Henri Poincar´e
following. It holds that, by simply integrating the deltas in (3.9) R
4 1=1
=
dxi W4h ({x})ψ˜x+1 ,σ1 ,ω1 ψ˜x+2 ,σ2 ,ω2 ψ˜x−3 ,σ3 ,ω3 ψ˜x−4 ,σ4 ,ω4
4 1=1
dxi W4h ({x}) Dx+1 ,¯x4,σ1 ,σ1 ,ω1 ψ˜x+2 ,σ2 ,ω2 ψ˜x−3 ,σ3 ,ω3 ψ˜x−4 ,σ4 ,ω4 + ˜− ˜− + ψ˜x+ ¯ 4 ,σ1 ,ω1 Dx2 ,¯ x4,σ2 ,σ2 ,ω2 ψx3 ,σ3 ,ω3 ψx4 ,σ4 ,ω4 +
− ˜+ ˜− ψ˜x+ ¯ 4 ,σ1 ,ω1 ψx ¯ 4 ,σ2 ,ω2 Dx3 ,¯ x4,σ3 ,σ3 ,ω3 ψx4 ,σ4 ,ω4 , (3.10)
¯ 4,σi = (x0,4 , x+,i , x−,4 ) if σi = II; ¯ 4,σi = (x0,4 , x+,4 , x−,i ) if σi = I and x where x moreover Dxε i ,¯x4,σi ,σi ,ωi = ψ˜xε i ,σi ,ωi − ψ˜xε¯ 4,σi ,σi ,ωi . (3.11) This means that the action of the renormalization operator R can be seen as the ε(≤h) replacement of a ψ ε(≤h) field with a Dxi ,¯x4,σi ,σi ,ωi field and some of the other ε(≤h) ψ˜(≤h) fields are “translated” in the localization point. The field Dx ,¯x ,σ ,ω is i
4,σi
i
i
¯ 4 . We can write Dε(≤h) as sum antiperiodic in the time components of xi , and x of two terms (if σi = I for instance): ε(≤h)
Dxi ,¯x4,σi ,σi ,ωi = [ψ˜xε i ,I − ψ˜xε 0,4 ,x+,i ,x−,i ,I ]+[ψ˜xε 0,4 ,x+,i ,x−,i ,I − ψ˜xε 0,4 ,x+,4 ,x−,i ,I ] (3.12) and the second addend can be written as, for L → ∞ ψ˜xε 0,4 ,x+,i ,x−,i ,I − ψ˜xε 0,4 ,x+,4 ,x−,i ,I = (x+,i − x+,4 )
1 0
ε dt∂x+ ψ˜I,ω,x 0,4 ,x+,i −t(x+,i −x+,4 ),x−,i
(3.13)
and x+,i − t(x+,i − x+,4 ) ≡ x+,i,4 (t) is called interpolated point. This means that it is dimensionally equivalent to the product of the “zero” (x+,i − x+,4 ) and the derivative of the field, so that the bound of its contraction with another field variable on a scale h < h will produce a “gain” γ −(h−h ) , see (≤h)σ (3.3), (3.4), with respect to the contraction of ψ˜x,ω . Similar considerations can be repeated for the first addend of (3.3); some care has to be done as β is finite, and we refer §3.5 of [BM]. If there are two external lines. R
dx1 dx2 W2h (x1 , x2 )ψ˜x+1 ,σ1 ,ω1 ψ˜x−2 ,σ2 ,ω2 = dx1 dx2 W2h (x1 , x2 )ψ˜x+1 ,σ1 ,ω1 Tx−2 ,¯x1,σ2 ,σ2 ,ω2
(3.14)
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where Txε2 ,¯x1,σ2 ,σ2 ,ω = ψ˜xε 2 ,σ2 ,ω − ψ˜xε¯ 1,σ2 ,ω,σ2 − (x0,2 − x0,1 )∂0 ψ˜xε¯ 1,σ2 ,ω,σ2 − (xσ,2 − x1,σ )∂σ ψ˜x− ¯ 1,σ2 ,ω,σ2
(3.15)
and ∂σ = ∂x+ if σ = I and ∂σ = ∂x− if σ = II. In this case the “gain” produced by the R operation is γ −2(h−h ) . We can write the local part of the effective potential (2.18) in the following way + xσ,2 [ωδh,ω ((ˆ x1 − x ˆ2 )σ1 )eix2 (ω1 pF,σ1 −ω2 pF,σ2 ) ψ˜ω,σ ∂ ψ˜− x1,σ2 dx1 d˜ LV h = 1 ;x1 σ ω,σ2 ;¯ σ1 =σ2
+
xσ1 ,1 d˜ xσ2 ,2 d˜ xσ3 ,3 λh;ω1 ,...ω4 ((ˆ x1 − x ˆ4 )σ1 ,(ˆ x2 − x ˆ4 )σ1 ,(ˆ x3 − x ˆ4 )σ3 ) dx4 d˜
σ1 ,...,σ4 =I,II
˜+ ˜− ˜− eix4 (ω1 pF,σ1 +ω2 pF,σ2 −ω3 pF,σ3 −ω4 pF,σ4 ) ψ˜x+ ¯ 4,σ1 ,σ1 ,ω1 ψx ¯ 4,σ2 ,σ2 ,ω2 ψx ¯ 4,σ3 ,σ3 ,ω3 ψx4 ,σ4 ,ω4 (3.16) where δh (x) is the Fourier transform of δh (kˆσ )2 cos kˆσ with respect to kˆσ and xi − x ˆj )σ = x+,i − x+,j if σ = II; moreover (ˆ xi − xˆj )σ = x−,i − x−,j if σ = I and (ˆ x ˜σi = x− if σ = I and x ˜σi = x+ if σ = II.
3.2
Tree expansion
By using iteratively the “single scale expansion” we can write the effective potential V (h) (ψ (≤h) ), for h ≤ 0, in terms of a tree expansion. For a tutorial introduction to the tree formalism we will refer to the review [GM].
v r
v0
h
h+1
hv
We need some definitions and notations.
−1 0
+1
1198
V. Mastropietro
Ann. Henri Poincar´e
1) Let us consider the family of trees which can be constructed by joining a point r, the root, with an ordered set of n ≥ 1 points, the endpoints of the unlabeled tree (see Fig. 1), so that r is not a branching point. n will be called the order of the unlabeled tree and the branching points will be called the non-trivial vertices. The unlabeled trees are partially ordered from the root to the endpoints in the natural way; we shall use the symbol < to denote the partial order. Two unlabeled trees are identified if they can be superposed by a suitable continuous deformation, so that the endpoints with the same index coincide. It is then easy to see that the number of unlabeled trees with n endpoints is bounded by 4n . We shall consider also the labeled trees (to be called simply trees in the following); they are defined by associating some labels with the unlabeled trees, as explained in the following items. 2) We associate a label h ≤ −1 with the root and we denote Th,n the corresponding set of labeled trees with n endpoints. Moreover, we introduce a family of vertical lines, labeled by an integer taking values in [h, 1], and we represent any tree τ ∈ Th,n so that, if v is an endpoint or a non-trivial vertex, it is contained in a vertical line with index hv > h, to be called the scale of v, while the root is on the line with index h. There is the constraint that, if v is an endpoint, hv > h + 1. The tree will intersect in general the vertical lines in set of points different from the root, the endpoints and the non-trivial vertices; these points will be called trivial vertices. The set of the vertices of τ will be the union of the endpoints, the trivial vertices and the non-trivial vertices. Note that, if v1 and v2 are two vertices and v1 < v2 , then hv1 < hv2 . We will call sv the number of subtrees coming out from v. Moreover, there is only one vertex immediately following the root, which will be denoted v0 and cannot be an endpoint; its scale is h + 1. 3) To each endpoint of scale +1 we associate V (1.11). With each endpoint v of scale hv ≤ 0 we associate one of the two terms appearing in (3.16), with coupling λhv −1 or δhv −1 . Moreover, we impose the constraint that, if v is an endpoint and hv ≤ 0, hv = hv + 1, if v is the non-trivial vertex immediately preceding v. 4) We introduce a field label f to distinguish the field variables appearing in the terms associated with the endpoints as in item 3); the set of field labels associated with the endpoint v will be called Iv . Analogously, if v is not an endpoint, we shall call Iv the set of field labels associated with the endpoints following the vertex v; x(f ), ε(f ) and ω(f ) will denote the space-time point, the ε index and the ω index, respectively, of the field variable with label f . If hv ≤ 0, one of the field variables belonging to Iv carries also a derivative ∂σ if the corresponding local term is of type δ, see (3.16). Hence we can associate with each field label f an integer m(f ) ∈ {0, 1}, denoting the order of the derivative.
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Marginal Fermi Liquid Behaviour in Hubbard Model with Cut-off
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If h ≤ −1, the effective potential can be written in the following way: V (h) (ψ (≤h) ) + LβEh+1 =
∞
V (h) (τ, ψ (≤h) ) ,
(3.17)
n=1 τ ∈Th,n
where, if v0 is the first vertex of τ and τ1 , . . . , τs (s = sv0 ) are the subtrees of τ with root v0 , V (h) (τ, ψ (≤h) ) is defined inductively by the relation V (h) (τ, ψ (≤h) ) =
(−1)s+1 T ¯ (h+1) Eh+1 [V (τ1 , ψ (≤h+1) ); . . . ; V¯ (h+1) (τs , ψ (≤h+1) )] , (3.18) s!
and V¯ (h+1) (τi , ψ (≤h+1) ) a) is equal to RV (h+1) (τi , ψ (≤h+1) ) if the subtree τi is not trivial, with R defined as acting on kernels according to (3.9) and its analogous for n = 1; b) if τi is trivial and h < −1, it is equal to LV (h+1) (ψ (≤h+1) ) (3.16) or, if h = −1, to V. T Eh+1 denotes the truncated expectation with respect to the measure Pσ,Zh (dψ (h+1) ), that is
T (X1 ; . . . ; Xp ) ≡ Eh+1
p
∂ log ∂λ1 . . . ∂λp
Pσ,Zh (dψ (h+1) )eλ1 X1 +···λp Xp σ=I,II
σ=I,II
. λi =0
(3.19)
We write (3.18) in a more explicit way. If h = −1, the right-hand side of (3.18) can be written in the following way. Given τ ∈ T−1,n , there are n endpoints of scale 1 and only another one vertex, v0 , of scale 0; let us call v1 , . . . , vn the endpoints. We choose, in any set Ivi , a subset Qvi and we define Pv0 = ∪i Qvi . We have V (−1) (τ, ψ (≤−1) ) = V (−1) (τ, Pv0 ) , (3.20) V (−1) (τ, Pv0 ) = (0)
Kτ,Pv (xv0 ) = 0
Pv0 (0) dxv0 ψ˜≤−1 (Pv0 )Kτ,Pv (xv0 ) , 0
n 1 T ˜(0) E0 [ψ (Pv1 \Qv1 ), . . . , ψ˜(0) (Pvn \Qvn )] Kv(1) (xvi ) , i n! i=1
(3.21) (3.22)
where we use the definitions ψ˜(≤h) (Pv ) =
f ∈Pv
m(f ) (≤h)ε(f ) ∂ˆσ(f ) ψ˜x(f )
,
h ≤ −1 ,
(3.23)
1200
V. Mastropietro
ψ˜(0) (Pv ) =
f ∈Pv
Kv(1) (xvi ) i
=e
i
f ∈Iv
i
Ann. Henri Poincar´e
(0)σ(f ) ψ˜x(f ) ,
εf x(f )ω(f ) pF,σ(f )
λ
(3.24)
xvi = x
(3.25)
It is not hard to see that, by iterating the previous procedure, one gets for V (h) (τ, ψ (≤h) ), for any τ ∈ Th,n , the representation described below. We associate with any vertex v of the tree a subset Pv of Iv , the external fields of v. These subsets must satisfy various constraints. First of all, if v is not an endpoint and v1 , . . . , vsv are the vertices immediately following it, then Pv ⊂ ∪i Pvi ; if v is an endpoint, Pv = Iv . We shall denote Qvi the intersection of Pv and Pvi ; this definition implies that Pv = ∪i Qvi . The subsets Pvi \Qvi , whose union Iv will be made, by definition, of the internal fields of v, have to be non-empty, if sv > 1. Moreover, we associate with any f ∈ Iv a scale label h(f ) = hv . Given τ ∈ Th,n , there are many possible choices of the subsets Pv , v ∈ τ , compatible with all the constraints; we shall denote Pτ the family of all these choices and P the elements of Pτ . Then we can write V (h) (τ, ψ (≤h) ) =
V (h) (τ, P) .
(3.26)
P∈Pτ
V (h) (τ, P) can be represented as V (h) (τ, P) =
(h+1) dxv0 ψ˜(≤h) (Pv0 )Kτ,P (xv0 ) ,
(3.27)
(h+1)
with Kτ,P (xv0 ) defined inductively (recall that hv0 = h + 1) by the equation, valid for any v ∈ τ which is not an endpoint, (h ) Kτ,Pv (xv )
sv 1 = [K (hv +1) (xvi )] EhTv [ψ˜(hv ) (Pv1 \Qv1 ), . . . , ψ˜(hv ) (Pvsv \Qvsv )] , sv ! i=1 vi
(3.28) where ψ˜(hv ) (Pv ) is defined as in (3.23), with (hv ) in place of (≤ hv ), if hv ≤ −1, while, if hv = 0, it is defined as in (3.24). (1)
Moreover, if v is an endpoint and hv = 0, Kv (xv ) is given by (3.25), otherwise, see (3.16) Kv(hv ) (xv ) =
lhv −1 (x1 , x2 , x3 , x4 ) if v is of type λ, d
x1 , x2 ) hv −1 (
if v is of type z,
(3.29)
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Marginal Fermi Liquid Behaviour in Hubbard Model with Cut-off
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where lhv −1 (x1 , x2 , x3 , x4 ) = eix4 (ε1 ω1 pF,σ1 +ε2 ω2 pF,σ2 −ω3 pF,σ3 −ε4 ω4 pF,σ4 ) λhv −1,ω1 ,...,ω4 ((ˆ x1 − x ˆ4 )σ1 , (ˆ x2 − x ˆ4 )σ2 , (ˆ x3 − xˆ4 )σ3 ) dhv −1,ω (x1 , x2 ) = eix2 (ω1 pF,σ1 −ε2 ω2 pF,σ2 ) δhv −1 ((ˆ x1 − x ˆ2 )σ1 ) If v is not an endpoint, (h +1)
v Kv(hi v +1) (xvi ) = RKτi ,P (i) ,Ω(i) (xvi ) ,
(3.30)
where τi is the subtree of τ starting from v and passing through vi (hence with root the vertex immediately preceding v), P(i) and is the restrictions to τi of P. The action of R is defined using the representation (3.9) of the R operation. (3.26) is not the final form of our expansion, since we further decompose V (h) (τ, P), by using the following representation of the truncated expectation in the right-hand side of (3.28). Let us put s = sv , Pi ≡ Pvi \Qvi ; moreover we order in an arbitrary way the sets Pi± ≡ {f ∈ Pi , σ(f ) = ±}, we call fij± their elements and − + we define x(i) = ∪f ∈P − x(f ), y(i) = ∪f ∈P + x(f ), xij = x(fi,j ), yij = x(fi,j ). Note i i s s − + that i=1 |Pi | = i=1 |Pi | ≡ n, otherwise the truncated expectation vanishes. A couple l ≡ (fij− , fi+ j ) ≡ (fl− , fl+ ) will be called a line joining the fields with labels fij− , fi+ j and sector indices ωl− = ω(fl− ), ωl+ = ω(fl+ ) and connecting the points xl ≡ xi,j and yl ≡ yi j , the endpoints of l. Moreover, we shall put ml = m(fl− ) + m(fl+ ) and, if ωl− = ωl+ , ωl ≡ ωl− = ωl+ . A similar definition is repeated for σ. Then, it is well known (see [Le], [BM], [GM] for example) that, up to a sign, if s > 1, EhT (ψ˜(h) (P1 ), . . . , ψ˜(h) (Ps )) (h) = g˜ω− ,σ− (xl − yl )δω− ,ω+ δσ− ,σ+ dPT (t) det Gh,T (t) , (3.31) T
l∈T
l
l
l
l
l
l
where T is a set of lines forming an anchored tree graph between the clusters of points x(i) ∪ y(i) , that is T is a set of lines, which becomes a tree graph if one identifies all the points in the same cluster. Moreover t = {ti,i ∈ [0, 1], 1 ≤ i, i ≤ s}, dPT (t) is a probability measure with support on a set of t such that ti,i = ui · ui for some family of vectors ui ∈ Rs of unit norm. Finally Gh,T (t) is a (n − s + 1) × (n − s + 1) matrix, whose elements are given by Gh,T ij,i j = −
+
m(f ) m(f ) (h) ti,i ∂ˆσ(f −ij) ∂ˆσ(f +ij) g˜ωl (xij − yi j )δω− ,ω+ δσ− ,σ+ with (fij− , fi+ j ) not belonging to T . ij
ij
l
l
l
l
In the following we shall use (3.31) even for s = 1, when T is empty, by interpreting the right-hand side as equal to 1, if |P1 | = 0, otherwise as equal to det Gh = EhT (ψ˜(h) (P1 )).
1202
V. Mastropietro
Ann. Henri Poincar´e
If we apply the expansion (3.31) in each non-trivial vertex of τ , we get an expression of the form (h) V (h) (τ, P, T ) , (3.32) dxv0 ψ˜(≤h) (Pv0 )Wτ,P,T (xv0 ) ≡ V (h) (τ, P) = T ∈T
T ∈T
where T is a special family of graphs on the set of points xv0 , obtained by putting together an anchored tree graph Tv for each non-trivial vertex v. Note that any graph T ∈ T becomes a tree graph on xv0 , if one identifies all the points in the sets xv , for any vertex v which is also an endpoint. We are writing the R operation as acting on the kernels, according to (3.9) and its analogous for n = 1. Such representation for the R operation is however not suitable to “gain” the convergence factor γ −(h−h ) , or γ −2(h−h ) , for which is much more convenient representation of R in (3.10), (3.14). However if we write simply all the R operations as in (3.10), (3.14) one gets possibly factors (xi −xj )αn with αn = O(n), which when integrated give O(n!) terms. One has to proceed in a more subtle way starting from the vertices of τ closest to the root from which the R operation is non-trivial, and writing R as in (3.10), (3.14) leaving all the other R operations as in (3.9). One distributes the “zero” along a path connecting ˜ 4h = (xi − xj )W ˜ 4h , if xi , xj are a family of end points, and from (3.9) (xi − xj )RW h h ˜ is the term in square brackets in the left-hand ˜ and RW two coordinates of W 4 4 ˜ 2h . There are same technical side of (3.9); an analogous property holds for RW complications in implementing this idea, which are discussed in [BM] (see also [BoM]), §3.2, §3.3 for a different model, but the adapting of such argument to the present case is straightforward. We obtain, in the L → ∞ limit V (h) (τ, P) =
dxv0
|Pv0 | Zh Wτ,P,T,α (xv0 )
T ∈T α∈AT
·
f ∈Pv0
q (f ) (≤h)σ(f ) [∂ˆjαα(f ) ψ]xα (f ),ω(f ) , (3.33)
where
·
n i=1
Wτ,P,T,α (xv0 ) =
Zhv /Zhv −1
v not e.p.
|Pv |/2
b (v ∗ ) djαα(vi∗ ) (xi , yi )Kvh∗i (xvi∗ ) i i
v not e.p.
1 sv !
dPTv (tv )
qα (fl− ) ¯qα (fl+ ) bα (l) ˆml g (h−v ) − + + (xl − yl )] , · det Ghαv ,Tv (tv ) ∂¯j (f − ∂ + [dj (l) (xl , yl )∂ α ) j (f ) σ ,ω ;σ ,ω l∈Tv
α
l
α
l
l
l
l
l
(3.34) where:
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Marginal Fermi Liquid Behaviour in Hubbard Model with Cut-off
1203
1) P is the set of {Pv }; 2) T is the set of the tree graphs on xv0 , obtained by putting together an anchored tree graph Tv for each non-trivial vertex v; 3) AT is a set of indices which allows to distinguish the different terms produced by the non-trivial R operations and the iterative decomposition of the zeros; v1∗ , . . . , vn∗ are the endpoints of τ , fl− and fl+ are the labels of the two fields forming the line l, “e.p.” is an abbreviation of “endpoint”. 4) Ghαv ,Tv (tv ) is obtained from the matrix Ghv ,Tv (tv ), associated with the vertex v and Tv , by substituting + m(f − ) m(f ) (h )
v ,Tv ˆ ij− ∂ˆx i+j g −v − + + (xij − yi j ) Ghij,i j = tv,i,i ∂x ) σ ,ω ;σ ,ω σ(f ) σ(f ij
i j
l
l
l
l
with −
+
−
ij
ij
m(f + )
) q (f ) m(fij ) (h ) ij ¯ α ij ˆ v ,Tv ¯qα (f− ∂ ∂xσ(f − ) ∂ˆxσ(fi+j ) gσ−v,ω− ;σ+ ,ω+ (xij − yi j ) . Ghα,ij,i j = tv,i,i ∂ j (f ) j (f + ) α
ij
α
i j
l
l
l
l
(3.35) 5) ∂¯jq , q = 0, 1, 2, are discrete derivatives or operators dimensionally equivalent to derivatives, due to the presence of the lattice and the fact that β is finite, see [BM] §3. Moreover ∂¯j0 denotes the identity and j = 0, +, −. According to q (3.13), (3.15) if σ(f ) = I then in ∂¯j(f ) one has j(f ) = 0, + and if σ(f ) = II then j(f ) = 0, −. 6) d0 (xl − yl ) = πβ sin πβ (x0,l − y0,l ) and di (xl − yl ) = (xi,l − yi,l ), i = ± are the “zeros” produced by the R operation, see (3.13), (3.15). Finally by construction ba (l) ≤ 2. are functions of the coordinates, and such dependence is 7) The factors ZZh−1 h not explicitly written. Of course the coefficients bα and qα are not independent, and, by the definition of R (see the discussion after (3.13)) it holds for any α ∈ AT , the following inequality γ hα (f )qα (f ) γ −hα (l)bα (l) ≤ γ −z(Pv ) , (3.36) f ∈Iv0
l∈T
v not e.p.
where hα (f ) = hv0 − 1 if f ∈ Pv0 , otherwise it is the scale of the vertex where the field with label f is contracted; hα (l) = hv , if l ∈ Tv and 1 z(Pv ) = 2 0
if |Pv | = 4, if |Pv | = 2; otherwise.
(3.37)
1204
V. Mastropietro
Ann. Henri Poincar´e
It holds
sv | det Ghαv ,Tv (tv )| ≤ C i=1 |Pvi |−|Pv |−2(sv −1) sv sv hv · γ 2 ( i=1 |Pvi |−|Pv |−2(sv −1)) γ hv i=1 [qα (Pvi \Qvi )+m(Pvi \Qvi )] −h [q (f + )+qα (fl− )+m(fl+ )+m(fl− )] . (3.38) · γ v l∈Tv α l
This follows from the well-known Gram-Hadamard inequality, see also [Le], [BM], [GM], stating that, if M is a square matrix with elements Mij of the form Mij =< Ai , Bj >, where Ai , Bj are vectors in a Hilbert space with scalar product < ·, · >, then | det M | ≤ ||Ai || · ||Bi || . (3.39) i
where || · || is the norm induced by the scalar product. In our case it can be shown that −
+
ij ) ¯qα (fij ) ˆm(fij ) ˆm(fi j ) (hv ) v ,Tv ¯qα (f− Ghα,ij,i gωl ,σl (xij − yi j ) ∂ ∂ j = ti,i ∂ + ∂ jα (fij ) jα (fij ) (hv ) (hv ) = ui ⊗ Ax(f − ),ω ,σ , ui ⊗ Bx(f + ),ω ,σ , (3.40) +
−
ij
l
l
l
i j
l
(h )
where ui ∈ Rs , i = 1, . . . , s, are the vectors such that ti,i = ui ·ui , and Ax(fv − ),ω ,σ , (h ) Bx(fv + ),ω ,σ l l i j
l
ij
l
are such that (in the case q = m = 0 for simplicity):
v) gω(hl ,σ (xij − yi j ) = l
(h ) (h ) Ax(fv − ),ω ,σ , Bx(fv + ),ω ,σ l l l l ij i j (h ) (hv ) ≡ dyAx(fv − )−y,ω ,σ By−x(f + ij
l
l
),ωl ,σl i j
(3.41)
For instance A and B can be chosen as: (h ) Ax,ωv l = −i dk e−ik x H(a20 sin2 kˆσl )f˜h (k0 , k σl , kˆσl ) k2 +(2 cos kˆ1 sin k )2 (3.42) σl 0 σl (h ) Bx,ωv l ,σl = dk e−ik x H(a20 sin2 kˆσl )f˜h (k0 , kσl , kˆσl ) ik0 + 2ω cos kˆσl sin k σl and from (3.39) we easily get (3.38). By using (3.34) and (3.38) we get, assuming (2.33) dxv0 |Wτ,P,T,α (xv0 )| ≤ C n Jτ,P,r,T,α ·C
sv
· γ hv
i=1
|Pvi |−|Pv |−2(sv −1)
sv
i=1
γ
hv 2
(
sv i=1
v not e.p. |Pvi |−|Pv |−2(sv −1))
[qα (Pvi \Qvi )+m(Pvi \Qvi )] γ −hv
(3.43) ] ,
qα (fl+ )+qα (fl− )+m(fl+ )+m(fl− ) l∈Tv
[
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Marginal Fermi Liquid Behaviour in Hubbard Model with Cut-off
1205
where Jτ,P,T,α = ·
n b (v ∗ ) dxv0 djαα (vi∗ ) (xi , yi )Kvh∗i (xvi∗ )
v not e.p.
i
i
i=1
1 ¯qα (fl− ) ¯qα (fl+ ) bα (l) (h ) ∂j (f − ) ∂j (f + ) [djα (l) (xl , yl )∂ˆml gω−v,σ− ;ω+ ,σ+ (xl − yl )] . α l α l sv ! l l l l l∈Tv
(3.44) In [BM], [BoM] it is proved that x d(xv0 ) = d¯
drl ,
(3.45)
l∈T
where rl = xl (tl ) − yl (sl ) and xl (tl ), yl (sl ) are interpolated points, see (3.13), ¯ is an arbitrary point of xv0 . By using (3.12), (3.4) we bound dimen(3.15), and x sionally each propagator, each derivative and each zero and we find Jτ,P,T,α ≤ C n
1 −h b (l)+˜ bα (l) C 2(sv −1) γ v l∈Tv α sv ! v not e.p. h [q (f + )+qα (fl− )+m(fl+ )+m(fl− )] . (3.46) · γ −hv (sv −1) γ v l∈Tv α l
We find then dxv0 |Wτ,P,T,α (xv0 )|
≤ C n L2 β|λ|n γ −hD(Pv0 )
v not e.p.
1 sv |Pvi |−|Pv | −[−2+ |Pv | +z(Pv )] 2 C i=1 γ sv !
(3.47)
where D(Pv0 ) = −2 + m. The sum over t, P, T, α is standard and we refer to [BM], §3.15; at the end the following theorem is proved. Theorem. Let h > hβ ≥ 0. If (2.33) holds then there exists a constant c0 such that
P τ ∈Th,n |Pv0 |=2m
dxv0 |Wτ,P,T,α (xv0 )| ≤ L2 βγ −hD(Pv0 ) (c0 λ)n , (3.48)
T ∈T α∈AT
where D(Pv0 ) = −2 + m .
(3.49)
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V. Mastropietro
Ann. Henri Poincar´e
4 The flow of running coupling functions 4.1
Lemma
It holds that
η=±
ˆ 2h (η π , ± π , k− ) = W β 2
η=±
ˆ 2h (η π , k+ , ± π ) = 0 . W β 2
ˆ 2h (k) also by a “single scale” integration; in fact W ˆ 2h (k) Proof. We can compute W +(≤h) −(≤h) ,h is the kernel of the term ψk ψk in V defined by ,h [h,0] (
g [h,0] (k0 , k+ , k− ) =
(4.2)
where χh,0 (k) = [H(a20 sin2 k+ ) + H(a20 sin2 k− )]Ch−1 ( k02 + 4 cos2 k+ cos2 k− )
(4.3)
0 with Ch−1 = k=h fk . We can write W2h ( πβ , π2 , k− ) as sum over Feynman graphs (see for instance [GM]) and each Feynman diagram can be written as 1 1 [h,0] ... 2 g (k1 ) . . . g [h,0] (km ) 2 L β L β k1 kn m m [h,0] 1 1 [h,0] n n σi ki + σ k . . . g σi ki + σ k g i=1
(4.4)
i=1
where n + m is an odd number, σ j , σij = 0, 1, −1, σ j + i σij is an odd integer and π π k = ( β , 2 , k2 ). In order to write (4.4) we consider a spanning tree T formed by propagators connecting all the vertices of the graphs. We will call the propagators not belonging to T loop lines and we write the momenta of the propagators of T as a linear combination of the momenta of the loop propagators and of the external + π2 , and the summation domain is momentum. We perform the shift k+,i → k+,i not changed by periodicity. The loop propagators become g¯[h,0] (k ) =
χ ¯h,0 (k ) cos k −ik0 + 2 sin k+ −
(4.5)
with
χ ¯h,0 (k ) =
[H(a20
2
cos
k+ )
+
H(a20
2
sin
cos2 k ) k02 + 4 sin2 k+ −
k− )]Ch−1 (
(4.6)
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1207
Of course g¯[h,0] (k) is odd in the exchange k0 , k+ , k− → −k0 , −k+ , k− . On the other hand the momenta of the propagators belonging to T becomes m
m m m m π j π j σij ki + σ j k = ( σij k0,i + σ j , σi k+,i + ( σij + σ j ) , σi k−,i + σ j k− ) β 2 i=1 i=1 i=1 i=1 i=1 (4.7) m with ( i=1 σij + σ j ) an odd integer; hence the propagators belonging to T have the form
χ ¯h,0 (k ) π )][cos( k −i[ i σi k0,i + σ β ] + 2[(−1) sin( i σi ki,+ i σi k−,i + σk− )]
(4.8)
and
χ ¯h,0 (k ) = H [Ch−1
a20
cos
σi k0,i
2
2
σi k+,i
i 2
+ 4 sin
i
+H
σi ki,+
i
a20
2
sin
cos2
σi k−,i + k−
i
σi k−,i + σk−
. (4.9)
i
, k− → −k0 , −k+ , k− we find Hence by performing the change of variables k0 , k+
ˆ 2h W
4.2
π π , , k− β 2
!
ˆ 2h − π , π , k− = −W β 2
! .
(4.10)
Finite temperature flow
The multiscale analysis defined above has the effect that the running coupling ), Zh (k¯σ,ω ) and λh (k¯σ 1 ,ω1 , k¯σ 2,ω2 , k¯σ 3 ,ω3 ) verify a recursive relation functions δ¯h (kσ,ω of the form ) = δh (k¯σ,ω ) + βδh (k¯σ,ω ) δh−1 (k¯σ,ω ) Zh−1 (k¯σ,ω ) = 1 + βξh (k¯σ,ω Zh (k¯σ,ω )
(4.11)
λh−1 (k¯σ 1 ,ω1 , k¯σ 2 ,ω2 , k¯σ 3 ,ω3 ) = λh (k¯σ 1 ,ω1 , k¯σ 2 ,ω2 , k¯σ 3 ,ω3 ) + βλh (k¯σ 1 ,ω1 , k¯σ 2 ,ω2 , k¯σ 3 ,ω3 ) . It is quite easy to prove that, at temperature not too low, indeed (2.33) hold. The proof is done by induction assuming that (2.33) holds for h and proving that it holds also for h − 1, if h − 1 ≥ hβ and β ≤ exp c¯|λ|−1 , where c¯ is a suitable constant. In fact iterating for instance the last of (4.11) we find λh−1 (k¯σ 1 ,ω1 , k¯σ 2 ,ω2 , k¯σ 3 ,ω3 ) = λ +
0 k=h+1
βλh (k¯σ 1 ,ω1 , k¯σ 2 ,ω2 , k¯σ 3 ,ω3 )
(4.12)
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V. Mastropietro
and from (2.33) and (3.48) we find, for |λ| ≤
Ann. Henri Poincar´e
cλ 2 2c30
| sup βkλ (k¯σ 1 ,ω1 , k¯σ 2 ,ω2 , k¯σ 3 ,ω3 )| ≤ 2cλ2 λ2 , {k },{σ}
if cλ2 > 0 is a bound for the norm of the second order contribution to λh . Hence sup |λh−1 (k¯σ1 ,ω1 , k¯σ2 ,ω2 , k¯σ3 ,ω3 , k¯σ4 ,ω4 )| ≤ [|λ| + |h|2cλ2 λ2 ] .
(4.13)
{k},{σ}
1 λ
Then sup |λh−1 | ≤ 2|λ| if β < e 2c2 |λ| , as |h| ≤ |hβ |. The same argument can be Z repeated for δh and Zh−1 , and (2.33) holds. h
4.3
Flow of the wave function renormalization
To complete the proof of the main Theorem one has to check that indeed the critical indices η(k+ ) or η(k− ) are non identically vanishing, and this is equivalent to show that there exists a non-vanishing function a(k¯σ,ω ) > 0 such that e−
¯ a(k σ,ω ) 2
λ2 h
¯
2
≤ Zh (k¯σ,ω ) ≤ e−2a(kσ,ω )λ
h
.
(4.14)
∞ h(n) h(n) ) = n=2 βξ (k¯σ,ω ) with |βξ (k¯σ,ω )| ≤ cn0 |λ|n , as a From the fact that βξh (k¯σ,ω consequence of (3.49) and (2.33), it is sufficient to find an upper and lower bound h(2) for βξ . From an explicit computation one finds ∂ h(2) = 24 [ dk1 dk2 dk3 gσ(≤h) (k1 ) (4.15) 2ω cos kˆσ βξ 1 ,ω1 ∂kσ ω ,ω ,ω σ ,σ ,σ 1
2
3
1
gσ(h) (k2 )gσ(≤h) (k3 )δ(k 2 ,ω2 3 ,ω3
2
3
¯σ , k ¯ 3,σ , k ¯ 1,σ ) + k3 − k1 − k2 )λh (k 3 1 ¯ 1,σ , k ¯ 2,σ , k ¯ σ )]| ¯ . λh (k 1 2 k=kσ,ω
h (≤h) (k) where gσ,ω = k=hβ gσ,ω . As the dependence from the momenta of λh is quite complex, it is convenient to replace in the above integral λh with λ; if the integral −1 so obtained is non-vanishing, the correction will be surely smaller for T ≥ e−(¯c|λ|) for a suitable c¯, as λh = λ + O(λ2 | log β|) from (4.13). We can choose σ = I for definiteness (the analysis for σ = II is identical), and we can distinguish two kind of contributions in the sum over σ1 , σ2 , σ3 ; one in which all the propagators are gI , and the other such that there is at least a propagator gII . The estimate of this second contribution is O(λ2 γ h ), as it can be immediately checked by dimensional considerations and applying the derivative in (4.15) over the gII propagators (one can always do that). We can further simplify the expression we have to compute noting that H(a20 sin2 k− )f˜h (k0 , k+ ) (h) (h) + g¯I,ω (x − y) , (4.16) gI,ω (x − y) = dke−ikx −ik0 + 2ω sin k+
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1209
with (h)
|¯ gI,ω (x − y)| ≤ a−2 0
CN γ h , 1 + [γ h |x0 − y0 | + γ h |x+ − y+ | + |x− − y− |]N
(4.17)
i.e., similar to (3.3) with an extra a−2 0 . We can replace in (4.15) the propagators (h) gI,ω with the first addend in the right-hand side of (4.16); if such term will be given by a non-vanishing constant, the correction will be surely smaller at least for a0 large enough. Hence the dominant contribution to (4.15) is given by dk−,1 dk−,3 H(a20 sin2 k−,1 )H(a20 sin2 k−,3 )H(a20 sin2 (−k−,1 +k−,3 +k− ))A, ω1 ,ω2 ,ω3
(4.18) with A=
dk0,1 dk0,3
ω1 ,ω2 ,ω3
dk+,1 dk+,3
f ≤h (k0,1 , k+,1 ) f ≤h (k0,3 , k+,3 ) ∂k −ik0,1 + ω1 2k+,1 −ik0,3 + ω3 2k+,3 +
f h (−k0,1 + k0,3 + k0 , −k+,1 + k+,3 + k+ ) −i(−k0,1 + k0,3 + k0 ) + 2ω2 (−k+,1 + k+,3 + k+ ) k+ −ωpF =k0 =0
(4.19)
with ω = ω1 + ω2 − ω3 ; it is easy to check that this term is indeed non-vanishing. Note also that A is the first non-trivial contribution to the critical index η of the Schwinger function of a d = 1 systems of interacting fermions.
4.4
Schwinger functions
We will not repeat here the analysis of the Schwinger functions at the temperature scale, as one can proceed as in the d = 1 to obtain an expansion for the Schwinger function once that the expansion for the effective potential is understood; see for instance [GM]. We only remark that the AI and AII in (1.12) and (1.13) are indeed O(λ2 ) as a consequence of h dk− gIh (k0 , k+ + ωpF , k− ) = 0 dk0 dk+ dk− gI (k0 , k+ , k− + ωpF ) = 0 . dk0 dk+ (4.20)
5 Conclusions 5.1
Marginal Fermi liquids and Luttinger liquids
We can compare the behaviour of the half-filled Hubbard model with cut-off with other models. We have found that the wave function renormalization has Z an anomalous flow up to exponentially small temperatures, Zh−1 = 1 + O(λ2 ), see h (4.14); in the case of circular Fermi surfaces one finds instead, see [DR], for |λ| ≤ ε h Zh−1 = 1 + O(ε2 γ 2 ) , Zh
(5.1)
1210
V. Mastropietro
Ann. Henri Poincar´e
which means that Zh = 1 + O(λ2 ), up to exponentially small temperatures; the factor γ h/2 in the right-hand side of (5.1) is an improvement with respect to a power counting bound and is found by using a volume improvement based on the geometrical constraints to which the momenta close to the Fermi surface are subjected. An equation similar to (5.1) holds also for any symmetric smooth Fermi surfaces with non-vanishing curvature; a proof can be obtained by combining the results of [BGM] with Appendix 2 of [DR]. The similarity of the equation for Zh with its analogous for one-dimensional systems may suggest that the behaviour of the half-filled Hubbard model with cutoff up to zero temperature is similar to the one of a system of spinless interacting fermions in d = 1 (the so-called Luttinger liquid behaviour). However this is false; in a Luttinger liquid in fact one has that h
λh−1 = λh + O(ε2 γ 2 ) ,
(5.2)
a property known as vanishing of beta function. One can easily check that this cancellation is not present in the half-filled Hubbard model with cut-off; in fact the dominant second order contribution to λh (k¯1,I , k¯−1,I , k¯1,I ) containing only σ = I internal lines is 1 2 2 dk f h (k0 , k+ )f ≥h (k0 , k+ ) 2 2 H(a0 sin k− ) k0 + k+ [H(a20 sin2 (k1,− + k2,− − k− )λh λh − H(a20 sin2 (k3,− − k2,− + k− )λh λh ] ,
(5.3)
where the dependence from k of the λh has not been explicitated. It is clear then that even at the second order the flow of λh is quite complex, and we plan to analyze it in a future work, in order to understand the leading instabilities. Replacing H with 1 and having λh not momentum dependent one recovers the d = 1 situation in which the beta function is vanishing. The theory resembles the theory of d = 1 Fermi systems in which each particle has an extra degree of freedom, the component of the momentum parallel to the flat Fermi surface, playing the role of a “continuous” spin index; and it is known that in d = 1 even a spin- 12 index can destroy the Luttinger liquid behaviour [BoM].
5.2
Marginal Fermi liquid behaviour close to half-filling
A similar analysis can be performed in the case of the Hubbard model with cut-off close to half-filling (µ = −ε with ε small and positive); in such a case the Fermi surface is convex and with finite radius of curvature but still resembles a square with non-flat sides and rounded corners. The propagator has the form χ(k) −ik0 + 2 cos k+ cos k− − ε and it is easy to verify that, if β < C min[ ρ1 ] where ρ is the radius of curvature h of the Fermi surface, the bounds (3.3), (3.4) for the single scale propagator gσ,ω
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still holds; the reason is that, up to temperatures greater than the inverse of the curvature radius, the bounds are insensitive to the fact that the sides of the Fermi surface are not perfectly flat. One can repeat all the analysis of the preceding sections and it is found that the Schwinger functions behave like (1.12), (1.13) for −1 λ small enough and β < C min[min[ ρ1 ], e(¯c|λ|) ]; in other words marginal Fermi liquid behaviour is still found close to half-filling, up to such temperatures. −1 On the other hand at lower temperatures, for min[ ρ1 ] ≤ β ≤ e(¯c|λ|) (of −1
course assuming min[ ρ1 ] ≤ e(¯c|λ|) ) one can apply the results of [BGM] (valid for any convex symmetric and regular Fermi surface) so finding Z = 1+Cρ [λ2 +O(λ3 )] where Cρ is a constant which is very large for small ε (and diverging at half-filling ε = 0). Hence, depending on the values of the parameters, one can have, in the low temperature region and before the critical temperature, two possibilities: the first is to have only marginal Fermi liquid behaviour Z = 1 + O(λ2 log β), and the second is to have marginal Fermi liquid behaviour up to temperatures O(min[ρ−1 ]) and then Fermi liquid behaviour up to the critical temperature.
Acknowledgments This work was partly written during a visit at the Ecole Polytechnique in Paris. I thank J. Magnen and V. Rivasseau for their warm hospitality and useful discussions. I thank also G. Benfatto and G. Gallavotti for important remarks.
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P.W. Anderson, The theory of superconductivity in high Tc cuprates, Princeton University Press, Princeton (1997).
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A.A. Abrikosov, L.P. Gorkov, I. Dzialoshinsky, Methods of quantum field theory in statistical physics, Prentice hall (1963).
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G. Benfatto, A. Giuliani, V. Mastropietro, to appear in Annales Henri Poincar´e.
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G. Benfatto, V. Mastropietro, Renormalization group, hidden symmetries and approximate Ward identities in the XY Z model, Rev. Math. Phys. 13, 1323–143 (2001), no. 11.
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F. Bonetto, V. Mastropietro, Beta function and anomaly of the Fermi surface for a d = 1 system of interacting fermions in a periodic potential, Comm. Math. Phys. 172, 57–93 (1995), no. 1.
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S. Dusuel, F. De Abreu, B. Doucot, Renormalization Group for 2D fermions with a flat Fermi surface, cond-mat 0107548.
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M. Disertori, V. Rivasseau, Interacting Fermi liquid in two dimensions, Comm. Math. Phys. 215, 251–290 and 291–341 (2000).
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[FMRT] J. Feldman, J. Magnen, V. Rivasseau, E. Trubowitz, An infinite volume expansion for Many Fermions Green functions, Helvetica Physica Acta 65, 679–721 (1992). [FSW]
A. Ferraz, T. Saikawa, Z.Y. Weng, marginal fermi liquid with a twodimensional Patched Fermi surface, cond-mat9908111.
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J.O. Fjaerested, A. Sudbo, A. Luther, Correlation function for a twodimensional electron system with bosonic interactions and a square Fermi surface, Phys. Rev. B. 60, 19, 13361–13370 (1999).
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G. Gentile, V. Mastropietro, Renormalization group for one-dimensional fermions. A review on mathematical results, Phys. Rep. 352, 273–43 (2001), no. 4–6.
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A. Lesniewski, Effective action for the Yukawa 2 quantum field Theory, Comm. Math. Phys. 108, 437–467 (1987).
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A. Luther, Interacting electrons on a square Fermi surface, PRB vol.30, n.16 (1994).
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D.C. Mattis, Implications of infrared instability in a two-dimensional electron gas, PRB, vol.36, n.1 (1987).
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W. Metzner, Renormalization Group analysis of a two-dimensional interacting electron system, Int. Jour. mod. Phys. 16, 11, 1889–1898 (2001).
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V. Rivasseau, The two-dimensional Hubbard model at half-filling. I. Convergent contributions, J. Statist. Phys. 106, no. 3-4, 693–72 (2002).
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Z-X. Shen et al, Science 267, 343 (1995).
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R. Shankar, Renormalization Group approach to interacting fermions, Rev. Mod. Phys. 66 (1), 129–192 (1994).
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C.M. Varma, Z. Nussinov, W. van Saarloos, Singular Fermi liquids, cond-mat0103393.
[VLSAR] C.M. Varma, P.B. Littlewood, S. Schmitt-Rink, E. Abrahams, A.E. Ruckestein, Phenomenology of the normal state of the Cu-O high Tc superconductors, Phys. Rev. Lett. 63, 1996 (1989). [VR]
A. Viroszteck, J. Ruvalds, Nested fermi liquid theory, Phys. Rev. B 42, 4064 (1990).
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Vieri Mastropietro Dipartimento di Matematica Universit` a di Roma “Tor Vergata” Via della Ricerca Scientifica I-00133, Roma Italy email: [email protected] Communicated by Vincent Rivasseau submitted 17/05/02, accepted 14/10/02
To access this journal online: http://www.birkhauser.ch
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Propagation Properties for Schr¨ odinger Operators Affiliated with Certain C*-Algebras W.O. Amrein, M. M˘ antoiu and R. Purice Abstract. We consider anisotropic Schr¨ odinger operators H = −∆ + V in L2 (Ên ). To certain asymptotic regions F we assign asymptotic Hamiltonians HF such that (a) σ(HF ) ⊂ σess (H), (b) states with energies not belonging to σ(HF ) do not propagate into a neighbourhood of F under the evolution group defined by H. The proof relies on C ∗ -algebra techniques. We can treat in particular potentials that tend asymptotically to different periodic functions in different cones, potentials with oscillation that decays at infinity, as well as some examples considered before by Davies and Simon in [4].
1 Introduction This paper is concerned with propagation properties of scattering states of selfadjoint n-dimensional Schr¨ odinger operators H = −∆+V with potentials V having different asymptotics in different directions. We recall that the scattering states of H are defined by the property that, as the time t tends to ±∞, they propagate away from each bounded region of the configuration space Rn (at least in some time average [2]). In many situations, in particular if V is a bounded function, they can be identified with the states in the continuous spectral subspace of H. If the potential V tends to zero (or to some other constant) sufficiently rapidly at infinity, standard scattering theory provides a description of the behaviour of e−itH f for a scattering state f at large times t. In more complicated situations, in particular if the asymptotic behaviour of V is highly anisotropic, little is known about the propagation of the scattering states. One may expect that certain asymptotic regions of configuration space should be inaccessible to states of certain energies, as illustrated by the following two examples. (1) In one dimension (n = 1), assume that V (x) → V± as x → ±∞, with V+ = V− . If for example V+ > V− , a state in the continuous spectral subspace of H with spectral support in the interval (V− , V+ ) will not propagate to the right. (2) In higher dimension (n ≥ 2) consider a potential V approaching a periodic function V0 as the argument x tends to infinity inside some cone C ⊂ Rn . In addition to the Hamiltonian H = −∆ + V one may introduce the periodic Schr¨ odinger operator H0 = −∆ + V0 . Bearing in mind some hypothetical scattering theory (e−itH0 should furnish a suitable comparison dynamics for the propagation inside C), one could expect that scattering states of H with energy disjoint from the continuous spectrum of H0 will not be able to propagate into C (such states may exist: the
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spectrum of H0 has a band structure, whereas the continuous spectrum of H will depend also on the behaviour of V outside the cone C and thus could intersect some gap in the band spectrum of H0 ). Detailed results about the propagation and non-propagation of states for onedimensional Schr¨ odinger operators with different spatial asymptotics at ±∞ (in particular for the first example above) and for multi-dimensional operators periodic in all but one dimension have been obtained by Davies and Simon [4]. The investigation of these authors includes a careful spectral analysis of the Hamiltonians under study. We propose here a different method for obtaining non-propagation properties, based on a relatively recent approach to spectral theory in the framework of C ∗ -algebras and without invoking any scattering theory. We shall in particular obtain the non-propagation result stated in the second example above. Below we give a brief non-technical description of this method. Typically the potential V to be considered is an element of a C ∗ -algebra A of bounded, continuous functions on Rn . The functions in A are characterized by a specific asymptotic behaviour (for example asymptotic periodicity in certain cones). Then, by invoking the Neumann series for (H − z)−1 (which is convergent for z large enough), one finds that the resolvent of the operator H = −∆ + V belongs to a C ∗ -algebra CA generated by products of elements of A (viewed as multiplication operators in L2 (Rn )) and suitable functions of momentum. We shall say that H is affiliated with CA . A central concept is that of the spectrum of H relative to an ideal K of CA of the form K = CK , where K is an ideal of A and CK is defined similarly to CA (just replace A by K in the definition of CA ). Let us denote this spectrum by σ K (H) and call it the essential spectrum associated with the ideal K (a precise definition is given in Section 2). For K = {0}, σ K (H) is the usual spectrum of H; for K = C0 (Rn ) (the space of continuous functions converging to zero at infinity), K will be the ideal of compact operators and σ K (H) the essential spectrum σess (H) of H. For an ideal K of A such that C0 (Rn ) ⊂ K, σ K (H) is a subset of σess (H). A typical non-propagation result will assert that scattering states of H with spectral support disjoint from σ K (H) will essentially never be localized in certain spatial domains W determined by K. Using the essential spectrum associated with such ideals to characterize some geometric properties for quantum Hamiltonians seems to be new, although in the literature ideals have been used in connection with spectral theory (e.g. in [3], [5]). In these considerations the ideal K will be given in terms of the asymptotic behaviour of the elements of A in some neighbourhood of infinity, and the theory will apply if the spatial domains W associated with K cover or intersect this neighbourhood of infinity. In the second example mentioned at the beginning K could be the set of functions in A that are asymptotically periodic in the cone C and tend to zero in directions not belonging to C, W could be the intersection of C with the complement of a compact set and σ K (H) would be the spectrum of H0 . Our treatment consists in the introduction of a compactification X of Rn related to the algebra A (in fact X will be the character space of A). The ideal K is determined by some closed subset F of the frontier X \ Rn of Rn in X and
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W is a subset of Rn which is sufficiently close to F . In the cited example one may think, intuitively, of F as the part of the compactification attached at infinity to the cone C. In this context the behaviour of various objects under translations is important. The algebra A is assumed to be invariant under translations; then the natural action of the translation group in Rn has a continuous extension to the compactification X of Rn and F must be invariant under this extension in order to allow the application of a result from the theory of crossed products of C ∗ -algebras to determine the spectrum σ K (H). In Section 2 we present some of the C ∗ -algebraic concepts that are useful in the spectral theory of self-adjoint operators. In Section 3 we discuss some algebras A of continuous functions on Rn , the associated compactifications of Rn and the continuous extension of the translation group to these compactifications. Section 4 contains a few remarks on crossed product algebras, and in Section 5 we give the proof of an abstract theorem on non-propagation. An application of this theorem to Schr¨ odinger Hamiltonians that are asymptotically periodic in several cones is studied in detail in Section 6, and in Section 7 we mention other classes of Hamiltonians that can be treated in a similar way. We use the terminology and results of [12] for the theory of C ∗ -algebras. We shall not try to discuss the various applications of C ∗ -algebraic methods in the study of quantum Hamiltonians, in the literature there exist several excellent reviews on these problems. Nevertheless we refer to Chapter 8 of [1] for a presentation of the algebraic approach to spectral theory that we shall use. The algebras CA and CK mentioned above have the structure of a crossed product; Reference [6] contains a description of such algebras that is well adapted to our applications in spectral theory. Finally we point out that various generalizations of our results are possible with almost no extra effort (cf. also [1], [6], [10] and [11]). Local singularities of the potential can easily be taken into account. The kinetic energy −∆ can be replaced by h(P ), an arbitrary continuous function of momentum satisfying |h(p)| → ∞ when |p| → ∞. Instead of the configuration space Rn , one can work with any abelian locally compact group X. The case X = Zn leads to finite difference operators.
2 C*-algebras and generalized essential spectra If H is a self-adjoint operator in a Hilbert space H, the spectral theorem allows one to associate an operator η(H) to a large class of functions η : R → C. We shall be here concerned with the set C0 (R) consisting of all continuous functions η : R → C that vanish at infinity (i.e., satisfying limx→±∞ η(x) = 0). Some parts of the spectrum of H can easily be characterized in terms of these functions: (i) a number λ ∈ R belongs to the spectrum σ(H) of H if η(H) = 0 whenever η ∈ C0 (R)
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and η(λ) = 0, (ii) λ belongs to the essential spectrum σess (H) of H if η(H) is a non-compact operator whenever η ∈ C0 (R) and η(λ) = 0. If C is a C ∗ -algebra of bounded operators in H such that η(H) ∈ C for each η ∈ C0 (R), then H is said to be affiliated with C. A sufficient condition for H to be affiliated with C is the requirement that (H − z)−1 ∈ C for some complex number z∈ / σ(H). The preceding situation can be viewed as a special case of the following abstract definition: Definition 1. (a) An observable affiliated with a C ∗ -algebra C is a ∗- homomorphism from the C ∗ -algebra C0 (R) to C (i.e., a linear mapping Φ : C0 (R) → C satisfying Φ(ξη) = Φ(ξ)Φ(η) and Φ(η)∗ = Φ(η) if ξ, η ∈ C0 (R)). (b) The spectrum σ(Φ) of the observable Φ is defined as the set of real numbers λ such that Φ(η) = 0 whenever η(λ) = 0. σ(Φ) is a closed subset of R. ˆ≡ Now let K be a (closed, self-adjoint, bilateral) ideal in C. We denote by C ∗ C/K the associated quotient C -algebra and by Π the canonical ∗-homomorphism ˆ If Φ is an observable affiliated with C, then clearly Π ◦ Φ determines of C onto C. ˆ an observable affiliated with C. ˆ is called Definition 2. The spectrum σ(Π◦Φ) of the observable Π◦Φ (relative to C) the K-essential spectrum of Φ and will be denoted by σK (Φ): σK (Φ) ≡ σ(Π ◦ Φ). Equivalently, a real number λ belongs to σK (Φ) if and only if Φ(η) ∈ / K whenever η ∈ C0 (R) is such that η(λ) = 0. To motivate the present terminology, let us consider the situation introduced at the beginning, where C is a C ∗ -subalgebra of B(H) and ΦH is the observable determined by a self-adjoint operator H affiliated with C (so ΦH (η) = η(H)). Assume that C contains the ideal K(H) of all compact operators in H. Then σK(H) (ΦH ) is just the essential spectrum σess (H) of the self-adjoint operator H. Now let us observe that, if K1 and K2 are two ideals in C satisfying K1 ⊂ K2 , then σK2 (Φ) ⊂ σK1 (Φ) ⊂ σ(Φ). In particular, if H is a self-adjoint operator affiliated with a C ∗ -subalgebra C of B(H) and if K is an ideal in C with K(H) ⊂ K, then σK (ΦH ) ⊂ σess (H). One of the interesting aspects of the preceding framework in the study of self-adjoint operators in a Hilbert space H is as follows. Let C be a C ∗ -subalgebra of B(H) and consider a class Θ of self-adjoint operators H affiliated with C such that, for some ideal K of C, the ∗-homomorphisms Π ◦ ΦH do not depend on H. So all members H of Θ have the same K-essential spectrum σK . In some situations ˆ will not be it is rather easy to determine σK : although the quotient C ∗ -algebra C identifiable with a subalgebra of B(H), it may be possible to specify a faithful ˆ in a Hilbert space H ˆ (an injective ∗-homomorphism π : C ˆ → representation of C ˆ ˆ ˆ ˆ B(H)) and a simple self-adjoint operator H in H affiliated with π(C) such that
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ˆ for all η ∈ C0 (R) and all H ∈ Θ. Then σK is just the spectrum π [Π(η(H))] = η(H) ˆ Examples will be considered in Sections 6 of the (presumably simple) operator H. and 7. The following result will be used in Section 5: Lemma 1. Let K be an ideal in a C ∗ -algebra C and Φ an observable affiliated with C. If η ∈ C0 (R) is such that η(µ) = 0 for all µ ∈ σK (Φ), then Φ(η) ∈ K. Proof. (i) Let λ ∈ R \ σK (Φ). There are a number ε > 0 and a function θ ∈ C0 (R) such that |θ(µ)| > ε for all µ ∈ (λ − ε, λ + ε) and Φ(θ) ∈ K. Now let ξ ∈ C0 (R) be such that supp ξ ⊂ (λ−ε, λ+ε). Since ξ/θ ∈ C0 (R), we have Φ(ξ) = Φ(θ)Φ(ξ/θ) ∈ K. In conclusion: each λ in R \ σK (Φ) has an open neighbourhood Vλ with the property that Φ(ξ) ∈ K for each ξ ∈ C0 (R) having support in Vλ . (ii) Since K is norm-closed, it is enough to establish the conclusion of the lemma under the additional assumption that η has compact support in R \ σK (Φ). Choose a finite collection of numbers λ1 , . . . , λM ∈ supp η such that supp η ⊂ ∪k Vλk and a corresponding partition of unity on supp η, i.e., a collection of funcM tions ξk in C0 (R) such that supp ξk ⊂ Vλk and k=1 ξk (λ) = 1 for all λ ∈ supp η. M Since Φ(ξk ) ∈ K by (i), we get Φ(η) = k=1 Φ(η)Φ(ξk ) ∈ K.
3 Some abelian C*-algebra If Y is a locally compact, Hausdorff space, we denote by Cb (Y ) the abelian C ∗ algebra of all bounded, continuous complex functions defined on Y . If G is a closed subset of Y , we set C G (Y ) = {ϕ ∈ Cb (Y ) | ϕ(y) = 0, ∀y ∈ G}. Certain C ∗ -subalgebras of Cb (Y ) will be important further on, in particular the algebras Cbu (Y ) and C0 (Y ) consisting respectively of all bounded, uniformly continuous functions and of all continuous functions vanishing at infinity. In fact C0 (Y ) is an ideal of Cb (Y ). Throughout this paper we set X = Rn . Let Y be as above and assume that X acts on Y as a group of homeomorphisms: so if αx denotes the homeomorphism in Y associated with the element x ∈ X, we have αx ◦ αx = αx+x . The mapping X × Y (x, y) → αx (y) ∈ Y is assumed continuous. Then α induces a representation of the group X by ∗-automorphisms of Cb (Y ) as well as of various C ∗ -subalgebras of Cb (Y ): for ϕ ∈ Cb (Y ) and x ∈ X, define ax (ϕ) ∈ Cb (Y ) by [ax (ϕ)](y) = ϕ(αx (y)) (y ∈ Y ). We observe that a C ∗ -subalgebra form C G (Y ) is invariant under G of the G this automorphism group (i.e., ax C (Y ) ⊂ C (Y )) if and only if the closed set G is invariant under each αx . Let A be a unital C ∗ -subalgebra of Cb (X) containing C0 (X). We denote its character space Ω(A) by X and we recall that X is a compactification of X, i.e., X is a compact topological space and there is a homeomorphism i from X to a dense subset of X (see e.g. §8.1 of [8]). For x ∈ X, the character i(x) is given by the formula [i(x)](ϕ) = ϕ(x), for ϕ ∈ A. We write Z = X \ i(X) and call it the frontier of X in X . By the Gelfand theorem, A is isomorphic to the C ∗ -algebra C(X ) of
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continuous functions on Ω(A). We shall use the notation G : C[Ω(A)] → A for the inverse of the Gelfand isomorphism. The C ∗ -subalgebra C Z (X ) (consisting of continuous functions on X that vanish on the frontier Z of X ) can be naturally identified with C0 (X), more precisely C0 (X) = GC Z [Ω(A)]. There is a one-to-one correspondence between (self-adjoint, closed) ideals K of A and closed subsets G of X , given by K = GC G (X ) (Theorem 3.4.1. of [9]). In particular each closed subset F of the frontier Z determines an ideal KF in A, viz. KF = GC F (X ). It is clear that such an ideal contains C0 (X). Suppose now that the C ∗ -algebra A considered above is contained in Cbu (X) and invariant under translations, i.e., such that ax A ⊂ A for all x ∈ X, with [ax (ϕ)](y) = ϕ(x + y). Since A ⊂ Cbu (X), the mapping x → ax (ϕ) is norm continuous for each ϕ ∈ A. Furthermore the action of X on itself (given as αx (y) = x+y) induces a continuous representation ρ of X by homeomorphisms of the character space X = Ω(A): for τ ∈ X the character ρx τ is defined as [ρx τ ](ϕ) = τ [ax (ϕ)]. For y ∈ X, set τy = i(y); then ρx τy = τx+y (x ∈ X). We end this section with a result which will be useful in the examples presented further on. Let τ ∈ X be a character of A. A neighbourhood base of τ in X is given by the collection {VF ,ε (τ )}, where ε varies over (0, ∞) and F over all finite families {ϕ1 , . . . , ϕm } of elements of A and where VF ,ε (τ ) = {τ ∈ X | |τ (ϕi ) − τ (ϕi )| < ε for each ϕi ∈ F }. Lemma 2. Let A be a unital C ∗ -subalgebra of Cb (X). Let F be a closed subset of Ω(A) and W a neighbourhood of F . Then there exist ε > 0 and a finite family F = {ϕ1 , . . . , ϕm } of elements of A such that F ⊂ ∪τ ∈F VF ,ε (τ ) ⊂ W. Proof. Let τ ∈ F . Then W is a neighbourhood of τ , hence there are a finite family F (τ ) of elements of A and a number ε(τ ) > 0 such that VF (τ ),ε(τ )(τ ) ⊂ W. Since F is compact, there are a finite number of points τ1 , . . . , τM in F such that 1 M F ⊂ ∪M j=1 VF (τj ),ε(τj )/2 (τj ). Let F = ∪j=1 F (τj ) and ε = 2 min{ε(τ1 ), . . . , ε(τM )}. The result of the lemma is true if we can show that, for each τ ∈ F , there is j ∈ {1, . . . , M } such that VF ,ε (τ ) ⊂ VF (τj ),ε(τj ) (τj ). Since clearly VF ,ε (τ ) ⊂ VF (τj ),ε (τ ) ⊂ VF (τj ),ε(τj )/2 (τ ) for each j, it is enough to show that for some j ∈ {1, . . . , M } one has VF (τj ),ε(τj )/2 (τ ) ⊂ VF (τj ),ε(τj ) (τj ). To prove this last inclusion, observe that τ belongs to VF (τj ),ε(τj )/2 (τj ) for at least one value of j. Choose one of these values of j and let τ ∈ VF (τj ),ε(τj )/2 (τ ). By the triangle inequality one has for each ϕ ∈ A: |τj (ϕ) − τ (ϕ)| ≤ |τj (ϕ) − τ (ϕ)| + |τ (ϕ) − τ (ϕ)|. For every ϕ ∈ F(τj ) each term on the right-hand side is less than ε(τj )/2. Hence τ belongs to VF (τj ),ε(τj ) (τj ).
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4 Some crossed product C*-algebras We consider some C ∗ -subalgebras of the space B(H) of all bounded, linear operators in the Hilbert space H = L2 (X). If ϕ : X → C is a bounded, measurable function, we denote by ϕ(Q) the operator of multiplication by ϕ in H and by ϕ(P ) the operator F∗ ϕ(Q)F (the operator of multiplication by ϕ in momentum space), where F is the Fourier transformation. A C ∗ -subalgebra A of Cbu (X) will be identified with the subalgebra of B(H) consisting of all multiplication operators ϕ(Q) with ϕ ∈ A. If A is a C ∗ -subalgebra of Cub (X), we write CA for the norm closure in B(H) of the set of finite sums of the form ϕ1 (Q)ψ1 (P ) + · · · + ϕN (Q)ψN (P ) with ϕk ∈ A and ψk ∈ C0 (X). We mention the fact that, if A = C0 (X), then CA is the ideal of all compact operators in L2 (X). If A is invariant under translations, then CA is a C ∗ -algebra isomorphic to the crossed product algebra A X defined in terms of the action ax of X on A. In the proof of Lemma 6 we shall use the following result from the theory of crossed products: If K is an ideal in A that is invariant under translations, then the quotient C ∗ -algebra CA /CK is isomorphic to [A/K] X. The point is that the general theory allows us to define the crossed product [A/K] X only by using the continuous action of X by ∗-automorphisms of A/K (the quotient action); the fact that A/K is not a C ∗ -subalgebra of B(H) does not matter. Remark. If V ∈ A, where A is a unital C ∗ -subalgebra of Cbu (X), then the selfadjoint operator H = −∆ + V is affiliated with CA . This is easily seen from the k ∞ fact that the Neumann series [H − z]−1 = k=0 (P 2 − z)−1 −V (Q)[P 2 − z]−1 converges in the norm of B(H) if z is sufficiently large.
5 A non-propagation theorem For our principal theorem and its corollary we consider the following Framework. A is a unital C ∗ -subalgebra of Cbu (X), invariant under translations and such that C0 (X) ⊂ A, and CA is the associated C ∗ -subalgebra of B(H) introduced in §4 (with H = L2 (X)). X = Ω(A) is the character space of A, F a translation invariant, closed subset of Z = X \ i(X) and C F (X ) the ideal in C(X ) determined by F . We set KF ≡ GC F (X ), which is a translation invariant ideal in A. Then KF ≡ CKF is an ideal in CA that contains all the compact operators in H. We shall work with families W of subsets of X such that their images through i in X are close to F . W will have the structure of a filter base, i.e., a non-void collection of non-void subsets of X such that for any W1 , W2 ∈ W there is a W ∈ W with W ⊂ W1 ∩ W2 . If W is a filter base in X, then the family {i(W ) | W ∈ W } is a filter base in X and we say that W is adjacent to F if all cluster points in X of this family {i(W ) | W ∈ W } belong to F , i.e., if ∩W ∈W W i(W ) ⊂ F , where the closures are taken in X . We observe that the set of these cluster points is
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non-empty since X is a compact space. In the majority of situations considered further on it will suffice to take for W the family {W = i−1 [W ∩ i(X) | W ∈ W]}, where W is a neighbourhood base of F in X (since i(X) is dense in X , each of these sets W is non-void). If W is a filter base adjacent to F and ϕ ∈ C(X ) is such that ϕ|F = 0, then given any ε > 0, there is some W ∈ W such that | ϕ(τ ) |< ε for all τ ∈ i(W ). In the sequel we shall denote by χW the characteristic function of W . Theorem. Let A and F be as in the framework and let W a filter base in X that is adjacent to F . Let H be a self-adjoint operator in H affiliated with CA . Let ε > 0 and η ∈ C0 (R) with supp η ∩ σKF (ΦH ) = ∅. Then there is a W ∈ W such that χW (Q)η(H) ≤ ε.
(1)
Proof. (i) We use the notation τ for characters in Ω(A) and observe that KF = G{ϕ ∈ C(X ) | ϕ|F = 0} = {ϕ ∈ A | τ (ϕ) = 0 ∀τ ∈ F }. So if ϕ belongs to KF , then for each δ > 0 there exists W ∈ W such that |τ (ϕ)| ≤ δ ∀τ ∈ i(W ). Thus, if ϕ ∈ KF , we have | ϕ(x) |≤ δ for all x ∈ W . (ii) By the hypothesis on the support of η we have η(H) ∈ KF (see Lemma 1). So there are a finite number of functions ϕ1 , . . . , ϕN ∈ KF and ψ1 , . . . , ψN ∈ C0 (X) such that N η(H) − ϕk (Q)ψk (P ) ≤ ε/2. k=1
We also have χW (Q)η(H) ≤
N
ϕk L∞ (W ) ψk L∞ (X) +
k=1
+ η(H) −
N
ϕk (Q)ψk (P ) .
(2)
k=1
The first term in the right-hand side of (2) can be made less than ε/2 by using −1 the result of (i) with δ = N · supk=1,...,N ψk L∞ (X) · ε/2, so the proof is finished. Corollary. Let A, F , W and H be as in the theorem. Then for each ε > 0 and each η ∈ C0 (R) with supp η ⊂ R \ σKF (ΦH ), there exists W ∈ W such that χW (Q)e−itH η(H)f ≤ ε f for all t ∈ R and all f ∈ L2 (X).
(3)
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(3) is a straightforward consequence of (1). Note the obvious fact that one may replace in the corollary {e−itH } by any bounded family of bounded operators commuting with H. Remark. The corollary gives the precise meaning, in our framework, of the notion of non-propagation described in the introduction. To be more specific, let us denote by suppH (f ) the spectral support with respect to H of the vector f ∈ H, defined as follows in terms of the spectral measure EH of H: λ∈ / suppH (f ) ⇔ ∃ε > 0 such that EH (λ − ε, λ + ε)f = 0. suppH (f ) is the smallest closed set J ⊂ R such that EH (J)f = f . Then it follows easily that, under the hypotheses of the corollary, for each ε > 0 and each closed subset L of R \ σKF (ΦH ) there exists an element W of W such that χW (Q)e−itH f ≤ ε f for all t ∈ R and all f ∈ L2 (X) with suppH (f ) ⊂ L. In the situation just described, let us take for H a self-adjoint Schr¨ odinger operator affiliated with CA . Then, if f is a unit vector with suppH (f ) ⊂ L, one has χW (Q)e−itH f ≤ ε for all t ∈ R. In physical terms: the probability of finding f localized in W is less than ε2 at all times. If K is a compact subset of X and if the preceding vector f belongs to the absolutely continuous subspace of H, then there is t0 ∈ R such that χK (Q)e−itH f ≤ ε for all t > t0 [2]. It follows that χK c ∩W c (Q)e−itH f 2 ≥ 1−2ε2 for all t > t0 , which (for small ε) essentially means that f describes a state that will propagate into the complement (K ∪ W )c of the set K ∪ W . If f belongs to the singularly continuous subspace of H, a similar conclusion is true, except that some averaging over time may be necessary [2]: t there are 0 < t0 < t1 such that (t − t0 )−1 t0 χK c ∩W c (Q)e−iτ H f 2 dτ ≥ 1 − 3ε2 for all t > t1 .
6 Example: Non-propagation in multicrystalline systems As an application we present in some detail the situation where the potential V of a Schr¨ odinger Hamiltonian becomes asymptotically periodic, with different periodic limit functions in different cones (the more general case in which the limit functions are only almost periodic can be treated analogously). More precisely V will belong to the C ∗ -algebra A introduced below, so that H = −∆ + V will be affiliated with CA . Let S be the unit sphere in X = Rn . For j = 1, . . . , N let Γj be a periodic lattice in X and Σj a non-empty open subset of S, with Σj ∩ Σk = ∅ if j = k. We denote by Cj (X) the C ∗ -algebra Cj (X) = {ϕ ∈ Cbu (X) | ϕ(x + γ) = ϕ(x) ∀x ∈ X, ∀γ ∈ Γj } and we define A as the set of bounded, uniformly continuous complex functions ϕ on X such that for each j ∈ {1, . . . , N } there exists ϕj ∈ Cj (X) such that limr→∞ |ϕ(rω) − ϕj (rω)| = 0 for all ω ∈ Σj , uniformly in ω on each
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compact subset of Σj . The (uniquely determined) collection {ϕj | j = 1, . . . , N } corresponding to ϕ ∈ A will be called the asymptotic functions of ϕ. If Σ is some subset of Σj and R > 0, let WjR (Σ) be the subset {rω | r > R, ω ∈ Σ} of X. The application of the results of Section 5 leads to the following non-propagation property into the cone subtended by Σj : Proposition 1. Let V ∈ A be real and denote its asymptotic functions by {Vj }. Set H = −∆ + V . Fix a number j ∈ {1, . . . , N } and choose η ∈ C0 (R) with supp η disjoint from the spectrum of the periodic Schr¨ odinger operator −∆ + Vj . Then, given a compact subset K of Σj and ε > 0, there is R ∈ (0, ∞) such that for each f ∈ L2 (X) we have sup χW R (K) (Q)e−itH η(H)f ≤ ε f . t∈R
(4)
j
The proof will be given in a series of lemmas. The validity of (4) is obtained by combining the last two lemmas (Lemma 5 and Lemma 6) with the corollary given in Section 5 and with the remark at the end of Section 4. The estimate (4) gives a precise meaning to the statement made in the second example of the introduction that states with spectral support away from certain subsets of R will not propagate into the asymptotic part of the cone Cj subtended by Σj . We shall use the following notations: WjR ≡ WjR (Σj ) = {rω | r > R > 0, ω ∈ Σj } ⊂ X, and Tj = X/Γj (the class of z ∈ X in Tj , denoted by ζ, is given as ζ = {x ∈ X | x = z + γ, γ ∈ Γj }). We observe that Cj (X) is isomorphic to C(Tj ) and that the correspondence ϕ → ϕj defines a ∗-homomorphism Ψj from A to Cj (X). Lemma 3. (a) A is a unital C ∗ -algebra containing C0 (X) and invariant under translations. (b) The ∗-homomorphism Ψj : A → Cj (X) is surjective. Proof. (a) It is clear that A contains C0 (X) and the constants. To see that A is closed, let {ϕ(k) } be a Cauchy sequence in A, denote by ϕ ∈ Cbu (X) its limit (k) and by ϕj (j = 1, . . . , N ) the asymptotic functions of ϕ(k) . Let us show that, for (k)
fixed j, {ϕj (k)
| k ∈ N} is Cauchy in the norm of Cbu (X). We have for any γ ∈ Γj :
(l)
|ϕj (x) − ϕj (x)|
(k)
(l)
(k)
= |ϕj (x + γ) − ϕj (x + γ)| ≤ |ϕj (x + γ) − ϕ(k) (x + γ)| (l)
+ |ϕ(k) (x + γ) − ϕ(l) (x + γ)| + |ϕ(l) (x + γ) − ϕj (x + γ)|. Fix ε > 0. The second term on the right-hand side is less than ε/3 for all x and all γ if k, l > L for some L ∈ N. For fixed k and l the first and the third term are less than ε/3 in the sup norm (with respect to x) since for each fixed x ∈ X and any R > 0 one may find γ ∈ Γj such that x + γ ∈ WjR (K) if K is a compact subset of Σj with non-empty interior.
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(k)
Now define ϕj = limk→∞ ϕj and observe that ϕj ∈ Cj (X). By another ε/3 type argument one then finds that these functions ϕj are asymptotic functions of ϕ. Let us show that A is invariant under translations. If x ∈ X and ϕ ∈ A with asymptotic functions {ϕj }, then the collection {ax (ϕj )} are asymptotic functions of ax (ϕ), hence ax (ϕ) ∈ A:
x x |ax (ϕ)(rω) − ax (ϕj )(rω)| = ϕ r ω + − ϕj r ω + → 0 as r → ∞, r r uniformly in ω belonging to compact subsets of Σj . (b) Let ψ ∈ Cj (X). Let ϕ = θψ, where θ is a function in Cbu (X) that is homogeneous of degree zero outside the unit ball of X and satisfies θ(x) = 1 on Wj1 and θ(x) = 0 on Wk1 if k = j. Then ϕ ∈ A, with asymptotic functions ϕj = ψ, ϕk = 0 for k = j. We next show that there is a canonical identification of Tj with a closed subset Tj of Z = X \ i(X), where X = Ω(A) as before. This is a direct consequence of the fact that there is a surjective ∗-homomorphism Φj : C(Z) → C(Tj ), deduced from Ψj : A → Cj (X), which is a surjective ∗-homomorphism with kernel including C0 (X), and from the natural isomorphisms A/C0 (X) ∼ = C(Z) and Cj (X) ∼ = C(Tj ). Below we shall make the construction as explicit as possible. For this we introduce a mapping ij : Tj → X that associates to ζ ∈ Tj (j) (j) the character ij (ζ) ≡ τζ ∈ X given as τζ (ϕ) = ϕj (z), where ϕj is the j-th asymptotic function of ϕ and z is any representative of the class ζ. In other terms (j) τζ = τz ◦ Ψj , where τz is interpreted as a character of C(Tj ). We set Tj = ij (Tj ). Lemma 4. (a) Tj is contained in X \ i(X). (j)
(b) The correspondence ζ → τζ
is injective and continuous.
(c) The set Tj is closed. (d) The set Tj is invariant under all translations. (j)
Proof. (a) If ζ ∈ Tj , then τζ
does not belong to i(X): if x ∈ X, choose a (j)
function ϕ ∈ C0 (X) such that ϕ(x) = 0: then ϕj = 0, so that τζ (ϕ) = 0 = (j)
ϕ(x) = [i(x)](ϕ). Thus τζ = i(x) for each x ∈ X. / Γj . Choose ϕj ∈ Cj (X) such that ϕj (z1 ) = ϕj (z2 ) (b) Assume that z1 − z2 ∈ (j) and let ϕ ∈ A be such that ϕj = Ψj (ϕ) (Lemma 3.(b)). Then τζ1 (ϕ) = ϕj (z1 ) = (j)
(j)
(j)
ϕj (z2 ) = τζ2 (ϕ), so τζ1 = τζ2 . Thus the mapping ij is injective. Its continuity is easy to establish. (c) Tj is the continuous image of the compact space Tj , hence it is a compact subset of Z.
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(d) Let z ∈ X be a representative of ζ ∈ Tj . For x ∈ X, denote the class of z + x in Tj by ζ + ξ. Then (j) (j) (j) ρx (τζ ) (ϕ) = τζ [ax (ϕ)] = [ax (ϕ)]j (z) = [ax (ϕj )](z) = ϕj (z + x) = τζ+ξ (ϕ). (j)
(j)
Hence ρx (τζ ) = τζ+ξ , the character associated with the class of z + x in Tj . We conclude that Tj is a single orbit under the representation ρ of X in X . We need to find subsets of i−1 [W ∩ i(X)] easy to express in terms of the geometry of X, for arbitrary neighbourhoods W of Tj . Lemma 5. Let W be a neighbourhood of Tj in X and set W = i−1 [W ∩ i(X)]. Given any compact subset K of Σj there is R ∈ (0, ∞) such that WjR (K) ⊂ W . Proof. By Lemma 2 applied to F = Tj , there are ε > 0 and a finite family F = {ϕ1 , . . . , ϕm } of elements of A such that ∪τ ∈Tj VF ,ε (τ ) ⊂ W, in other terms such (j) that ∩i {τ ∈ X | |τ (ϕi ) − τζ (ϕi )| < ε} ⊂ W for each ζ ∈ Tj . Upon restricting to characters τ belonging to i(X) and denoting the j-th asymptotic function of ϕi by ϕi,j , we get immediately that ∩i {x ∈ X | |ϕi (x) − ϕi,j (x)| < ε} ⊂ W . Now for each i there is Ri ∈ (0, ∞) such that |ϕi (x) − ϕi,j (x)| < ε for all x ∈ WjRi (K). Then clearly the assertion of the Lemma holds for R = max{R1 , . . . , Rm }. We finally specify the KTj -essential spectrum of H. Lemma 6. The set σKTj (ΦH ) coincides with the (band) spectrum of the periodic Schr¨ odinger operator Hj = −∆ + Vj . Proof. Let us denote by Πj : CA → CA /KTj the canonical ∗-homomorphism. By definition, σKTj (ΦH ) is the spectrum of the observable Πj ◦ ΦH affiliated with CA /KTj . It is enough to show that CA /KTj is isomorphic to CCj (X) and that the image of Πj ◦ΦH under this isomorphism is ΦHj ; this will conclude the proof, since isomorphisms of C ∗ -algebras leave the spectra of observables invariant. For any M ∈ N, ϕ1 , . . . , ϕM ∈ A and ψ1 , . . . , ψM ∈ C0 (X) we set
M M Θj ϕi (Q)ψi (P ) = ϕi,j (Q)ψi (P ), i=1
i=1
where ϕi,j ∈ Cj (X) is the j-th asymptotic function of ϕi . By the discussion in Section 4, Θj extends to a surjective ∗-homomorphism CA → CCj (X) with kernel KTj = CKTj . A simple argument in terms of the Neumann series shows that Θj [(H− z)−1 ] = (Hj − z)−1 , so that Θj (ΦH ) = ΦHj . Remark. Since K(H) = CC0 (X) = KZ ⊂ KTj , we have ∪N j=1 σ(Hj ) ⊂ σess (H). The behaviour of the bounded, uniformly continuous function V outside the cones
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{Cj }j=1,...,N is submitted to no constraint and the asymptotic functions Vj are not related. So it is possible to have a large set σess (H) \ ∪N j=1 σ(Hj ) on which the result of the proposition (non-propagation into the asymptotic part of ∪N j=1 Cj ) is relevant and non-trivial. Of course the simplest situation is that where N = 1, and the general case (N > 1) can be reduced to it since Proposition 1 involves only one value of j.
7 Other examples, comments Some other examples will be discussed briefly in the present section. Most of the proofs consist in suitable adaptations of arguments already used above and we shall just sketch them.
7.1
Example 1. Potentials that are asymptotically periodic in a half-space
This is the example that is the most close to that treated in [4]. It is also related to Section 6. Let us write X = R × X , with X = Rn−1 . For a periodic lattice Γ+ of X, we denote by C+ (X) the C ∗ -algebra of all complex continuous functions on X that are Γ+ -periodic. We shall consider the unital C ∗ -algebra A+ of all bounded, uniformly continuous functions ϕ : X → C such that there exist a (necessarily unique) element ϕ+ ∈ C+ (X) such that | ϕ(x1 , x ) − ϕ+ (x1 , x ) |→ 0 when x1 → +∞, uniformly in x ∈ X . As before we call ϕ+ the asymptotic function of ϕ. It is easy to see that A+ is invariant under translations. Since we imposed no conditions on the behaviour of ϕ outside a remote half-space, we cannot determine the character space X of A+ precisely. But it is straightforward to show that the torus T+ = X/Γ+ is a closed invariant subset of its frontier and that any neighbourhood of T+ in X contains {(x1 , x ) ∈ X | x1 > R} for some R > 0 large enough (use Lemma 2). It is also easy to show that the quotient C ∗ -algebra CA+ /KT+ is isomorphic to CC+ (X) in such a way that ϕ(Q)ψ(P ) corresponds to ϕ+ (Q)ψ(P ); here ϕ ∈ A+ , ϕ+ ∈ C+ (X) is its asymptotic function and ψ ∈ C0 (X). By applying the results of Section 5 and the discussion above one gets Proposition 2. Let V ∈ A+ be real and let V+ ∈ C+ (X) be its asymptotic function. Let H = −∆ + V and H+ = −∆ + V+ be the associated self-adjoint operators in H = L2 (X). Let η ∈ C0 (R) with supp η ∩ σ(H+ ) = ∅. Then for each ε > 0 there exist R > 0 such that χ(Q1 ≥ R)e−itH η(H)f ≤ ε f for all f ∈ H and all t ∈ R. Of course, one can also introduce the C ∗ -algebra A− , consisting of all bounded, uniformly continuous functions that become Γ− -periodic (for some other periodic lattice Γ− ) at x1 = −∞, uniformly in the orthogonal variable x . The elements
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of A = A− ∩ A+ are bounded, uniformly continuous functions which have (different) periodic limits at x1 = ±∞; their behaviour in a vertical strip is submitted to no constraint.
7.2
Example 2. Potentials with asymptotic vanishing oscillation
We say that a function ϕ ∈ Cb (X) has asymptotic vanishing oscillation, and we write ϕ ∈ V O(X), if the function x → sup|y|≤1 | ϕ(x+y)−ϕ(x) | is of class C0 (X). We remark that V O(X) contains the C ∗ -algebra C rad (X) of all complex continuous functions on X that have radial limits at infinity, uniformly in all directions. Sums of the form ϕ0 +ϕ1 with ϕ0 ∈ C0 (X) and ϕ1 continuous and homogeneous of degree zero outside a ball are the most general elements of C rad (X). Note that V O(X) is considerably larger than C rad (X). A C 1 -function ϕ with ∂j ϕ ∈ C0 (X) for all j is in V O(X). This includes ϕ(x) = φ [(1 + |x|)p ] for p < 1 and φ, φ continuous and bounded. We point out that Proposition 3 will be particularly simple to interpret for potentials in C rad (X). Let X be the character space of V O(X). By identifying X with its homeomorphic image in X , we can express this character space as the disjoint union X = X Z. The nice feature is that V O(X) is the largest unital, translation invariant C ∗ -subalgebra of Cbu (X) such that all the elements of Z are fixed points under the extension of the action of the group X. This was used in [11] to show that, for V ∈ V O(X), the essential spectrum of the Schr¨odinger operator H = −∆ + V is [min V (X)asy , ∞), where the asymptotic range of V is defined as V (X)asy = ∩K V (X \ K) with K varying over all compact subsets of X. This result is specific to the class V O(X). The frontier Z is not easy to understand, so we shall consider only closed sets F ⊂ Z which are suitably related to a given potential V . Let Vˆ be the continuous extension of V to X and G a closed subset of R such that its interior Go meets V (X)asy . We set F = Vˆ −1 (G) ∩ Z; it is a closed, non-void subset of Z, and it is automatically invariant under translations (this is the point which makes our analysis possible without extra information on Z). To apply the theorem, one has to find in X a filter base adjacent to F and to calculate the KF -essential spectrum of H. For the latter problem we proceed as in [11], where more details can be found. The set σKF (H) is the spectrum in CV O(X) /KF of the image of the observable ΦH through the canonical ∗-homomorphism CV O(X) → CV O(X) /KF . The quotient C ∗ -algebra CV O(X) /KF is isomorphic to C(F ) X (crossed product constructed in terms of the trivial action of X on C(F )). The latter can be embedded in the direct sum ⊕τ ∈F C X ∼ = ⊕τ ∈F C0P (X), where C0P (X) is ∗ the C -subalgebra of B(H) of all the operators of the form ψ(P ), with ψ ∈ C0 (X). This leads to a ∗-homomorphism ΠF : CV O(X) → ⊕τ ∈F C0P (X) with kerˆ )ψ(P ))τ ∈F , where ϕˆ is nel KF . This ∗-homomorphism maps ϕ(Q)ψ(P ) to (ϕ(τ the continuous extension of ϕ to X . With the Neumann series for the resolvent, it follows easily that the observable ΦH is mapped to ΦHτ τ ∈F , where Hτ =
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ˆ −∆ operator in H). It follows that σKF (H) = ∪τ ∈F σ(Hτ ) = + V (τ ) (self-adjoint ˆ min V (F ), ∞ . By taking into account the definition of the closed set F and the fact that Vˆ (Z) = V (X)asy one easily shows that Vˆ (F ) = G ∩ V (X)asy , thus σKF (H) = [min{G ∩ V (X)asy }, ∞). We next indicate a suitable filter base in X adjacent to F . For any compact subset K of X we set WK = V −1 (G) ∩ K c . The assumption Go ∩ V (X)asy = ∅ implies that WK = ∅. Since WK ∩ WK = WK∪K , W = {WK }K is a filter base in X. Then (all closures are taken in X ): ∩K WK = ∩K [V −1 (G) ∩ K c ] ⊂ ∩K V −1 (G) ∩ K c ⊂ Vˆ −1 (G) ∩ ∩K K c = Vˆ −1 (G) ∩ Z = F. Thus W is adjacent to F . By applying the corollary in Section 5 we obtain Proposition 3. Let V ∈ V O(X) and consider the self-adjoint operator H = −∆ + V in H = L2 (X), which defines an observable affiliated with CV O(X) . Let G ⊂ R be a closed set such that Go ∩V (X)asy = ∅, and let ε > 0. Then for each η ∈ C0 (R) with supp η ⊂ (−∞, min{G ∩ V (X)asy }) there is a compact subset K of X such that χV −1 (G)\K (Q)e−itH η(H)f ≤ ε f for all f ∈ H and all t ∈ R. To illustrate this result, let us take G = [λ, ∞) with min [V (X)asy ] < λ < max [V (X)asy ]. Then, roughly, scattering states at energies situated below λ will not propagate into the asymptotic part of the set {x ∈ X | V (x) ≥ λ}. For a one-dimensional system with a slowly oscillating potential, this corresponds to tunneling through an infinite sequence of more and more widely separated barriers of increasing length. The effective parts of these barriers, for states with energy less than λ, occupy the intervals {x ∈ R \ K | V (x) ≥ λ} for some compact set K ⊂ R (as an example one may consider a potential that is asymptotically of the form cos(|x|β ) with 0 < β < 1; the essential spectrum of the associated Sturm-Liouville operator will often be continuous, cf. Theorem 4 in [7], in particular vectors with spectral support in the interval (−1, λ) will propagate away from each compact set K and thus undergo tunneling of the indicated type). For multi-dimensional systems there are various possibilities: for a spherically symmetric slowly oscillating potential there will be an infinite sequence of spherically symmetric barriers arranged (as a function of the radial variable r) in analogy with the one-dimensional case; for a potential having radial limits (V ∈ C rad (X)) there will be no propagation into the asymptotic part of the cone subtended by {ω ∈ S | limr→∞ V (rω) ≥ λ}; for certain slowly oscillating non-spherically symmetric potentials there may be an infinite collection of inaccessible regions of increasing size towards infinity.
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The above type of behaviour is specific to the class V O(X) and is related to the fact that the action of translations on the frontier Z is trivial. If V would tend at infinity to a periodic function for instance, the connection between the localization in energy and the domains of non-propagation has a different nature, as seen in Section 6.
7.3
Example 3. Potentials with cartesian anisotropy
We shall work here with X = R2 ≡ R1 × R2 . The generalization to arbitrary dimension is straightforward. For j = 1, 2 let us denote by Aj the C ∗ -algebra of all continuous functions ϕ : Rj → C such that the limits c± j = limxj →±∞ ϕ(xj ) exist. Then A = A1 ⊗ A2 is a unital C ∗ -subalgebra of Cbu (X) that is invariant under translations by elements of X. j j = Rj ∪{−∞j , +∞j } be the two-point compactification of Rj . Then R Let R 1 × R 2 is the spectrum of A. The closed invariant is the spectrum of Aj and X = R subsets of the frontier Z = X \ X are as follows: the four corners {(±∞1 , ±∞2 )}, 1 ×{±∞2 } and {±∞1 }× R 2 and all their unions. We shall illustrate the four edges R 1 × {+∞2 }. our theorem for F = {(+∞1 , +∞2 )} and for F = R So let us consider the self-adjoint operator H = −∆ + V in H = L2 (X), for V ∈ A a real function. It is easy to show that V has the following property (which in fact characterizes the elements of A): for j, k ∈ {1, 2} and k = j, the limits Vj± (xj ) = limxk →±∞k V (x) exist uniformly in xj ∈ Rj and define elements of Aj . The values taken by the continuous extension of V to X on the four edges coincide respectively with Vj± , j = 1, 2. Its values at the four corners will be denoted by c++ , c+− , c−+ , c−− . Note that for example c++ = limx1 →+∞1 V1+ (x1 ) = limx2 →+∞2 V2+ (x2 ). Let us consider the operator H1+ = −∆+V1+ = −∆1 + V1+ ⊗12 +11 ⊗(−∆2 ) + in the representation L2 (R1 ) ⊗ L2 (R2 ). Its spectrum equals [a+ 1 , ∞), where a1 + 1,+ is the infimum of the spectrum of the operator H = −∆1 + V1 acting in of this kind are available and, with obvious H1 = L2 (R1 ). Three other operators − + − notations, we have σess (H) = min{a+ , a , a , a }, ∞ . This follows quite easily 1 1 2 2 from our arguments and was proved in a greater generality in §3 of [11]. Remark also that the spectrum of H ±± = −∆ + c±± is [c±± , ∞) and that inequalities such −+ ++ as a+ , c } are true. 1 ≤ min{c A neighbourhood base of the point {(+∞1 , +∞2 )} is composed of all the rectangles {(y1 , +∞1 ] × (y2 , +∞2 ] | y1 ∈ R1 , y2 ∈ R2 } and a neighbourhood base 1 × (y2 , +∞2 ] | y2 ∈ R2 }. We get: 1 × {+∞2 } consists of the rectangles {R of R Proposition 4. Let ε > 0. (a) For any η ∈ C0 (R) with supp η ⊂ (−∞, c++ ) there exist y1 ∈ R1 , y2 ∈ R2 such that for all f ∈ H χ(Q1 > y1 )χ(Q2 > y2 )e−itH η(H)f ≤ ε f .
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(b) For any η ∈ C0 (R) with supp η ⊂ (−∞, a+ 1 ) there exists y2 ∈ R2 such that for all f ∈ H χ(Q2 > y2 )e−itH η(H)f ≤ ε f . So, at energies below a+ 1 there is no propagation towards x2 = +∞2 . At ++ this becomes possible, but then the obenergies comprised between a+ 1 and c servable Q1 cannot diverge through positive values. Results of this type are by no means trivial in the sense that, in certain situations, propagation away from any compact subset of X does occur at energies as above (under suitable assumptions on V there will be intervals of purely absolutely continuous spectrum of H in the considered energy range, and associated states must propagate to infinity by the results of [2]).
Acknowledgments We are grateful to G¨ unter Stolz for correspondence and for pointing out to us the paper [7] and to the referee for drawing our attention to Refs. [3] and [5].
References [1] W.O. Amrein, A. Boutet de Monvel and V. Georgescu, C0 -Groups, Commutator Methods and Spectral Theory of N-Body Hamiltonians, Birkh¨ auserVerlag, Basel-Boston-Berlin, 1996. [2] W.O. Amrein and V. Georgescu, On the Characterization of Bound States and Scattering States in Quantum Mechanics, Helv. Phys. Acta 46, 635–658 (1973). [3] J. Bellissard, K-Theory of C ∗ -algebras in Solid State Physics, Lecture Notes in Physics 257, 99–156 (1986). [4] E.B. Davies and B. Simon, Scattering Theory for Systems with Different Spatial Asymptotics on the Left and Right, Commun. Math. Phys. 63, 277– 301 (1978). [5] G.A. Elliott, Gaps in the Spectrum of an Almost Periodic Schr¨ odinger Operator, C.R. Math. Rep. Acad. Sci. Canada 4, 255–259 (1982). [6] V. Georgescu and A. Iftimovici, C ∗ -Algebras of Energy Observables: I. General Theory and Bumps Algebras, preprint 00–520 at http://www.ma.utexas.edu/mp arc/. [7] A.Ya. Gordon, Pure Point Spectrum under 1-Parameter Perturbations and Instability of Anderson Localizaton, Commun. Math. Phys. 164, 489–505 (1994).
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[8] A. Guichardet, Special Topics in Topological Algebras, Gordon and Breach, New York (1968). [9] R.V. Kadison and J.R. Ringrose, Fundamentals of the Theory of Operator Algebras, Academic Press, New York (1983). [10] M. M˘ antoiu, On a Class of Anisotropic Schr¨ odinger Operators, preprint 01– 201 at http://www. ma.utexas.edu/mp arc/. [11] M. M˘ antoiu, C ∗ -Algebras, Dynamical Systems at Infinity and the Essential Spectrum of Generalized Schr¨ odinger Operators, preprint 01–298 at http://www.ma.utexas.edu/mp arc/ and to appear in J. Reine Angew. Math. [12] G.J. Murphy, C ∗ -Algebras and Operator Theory, Academic Press, Boston (1990).
W.O. Amrein D´epartement de Physique Th´eorique Universit´e de Gen`eve 1211 Gen`eve 4 Switzerland email: [email protected] M. M˘ antoiu, R. Purice Institute of Mathematics of the Romanian Academy P.O.Box 1-764 Bucharest, RO-70700 Romania email: [email protected] email: [email protected] Communicated by Jean Bellissard submitted 15/05/02, accepted 22/06/02
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