Communications In Mathematical Physics - Volume 281

Commun. Math. Phys. 281, 1–21 (2008) Digital Object Identifier (DOI) 10.1007/s00220-008-0457-x

Communications in

Mathematical Physics

How Smooth is Your Wavelet? Wavelet Regularity via Thermodynamic Formalism Mark Pollicott1 , Howard Weiss2,
1 Mathematics Institute, University of Warwick, Coventry CV4 7AL, United Kingdom 2 School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30334, USA.

E-mail: [email protected] Received: 26 April 2006 / Accepted: 3 October 2007 Published online: 6 May 2008 – © Springer-Verlag 2008

Abstract: A popular wavelet reference [W] states that “in theoretical and practical studies, the notion of (wavelet) regularity has been increasing in importance.” Not surprisingly, the study of wavelet regularity is currently a major topic of investigation. Smoother wavelets provide sharper frequency resolution of functions. Also, the iterative algorithms to construct wavelets converge faster for smoother wavelets. The main goals of this paper are to extend, refine, and unify the thermodynamic approach to the regularity of wavelets and to devise a faster algorithm for estimating regularity. The thermodynamic approach works equally well for compactly supported and non-compactly supported wavelets, and also applies to non-analytic wavelet filters. We present an algorithm for computing the Sobolev regularity of wavelets and prove that it converges with super-exponential speed. As an application we construct new examples of wavelets that are smoother than the Daubechies wavelets and have the same support. We establish smooth dependence of the regularity for wavelet families, and we derive a variational formula for the regularity. We also show a general relation between the asymptotic regularity of wavelet families and maximal measures for the doubling map. Finally, we describe how these results generalize to higher dimensional wavelets. 0. Introduction While the Fourier transform is useful for analyzing stationary functions, it is much less useful for analyzing non-stationary cases, where the frequency content evolves over time. In many applications one needs to estimate the frequency content of a nonstationary function locally in time, for example, to determine when a transient event occurred. This might arise from a sudden computer fan failure or from a pop on a music compact disk. The usual Fourier transform does not provide simultaneous time and frequency localization of a function.
The work of the second author was partially supported by a National Science Foundation grant DMS0355180.

2

M. Pollicott, H. Weiss

The windowed or short-time Fourier transform does provide simultaneous time and frequency localization. However, since it uses a fixed time window width, the same window width is used over the entire frequency domain. In applications, a fixed window width is frequently unnecessarily large for a signal having strong high frequency components and unnecessarily small for a signal having strong low frequency components. In contrast, the wavelet transform provides a decomposition of a function into components from different scales whose degree of localization is connected to the size of the scale window. This is achieved by integer translations and dyadic dilations of a single function: the wavelet. The special class of orthogonal wavelets are those for which the translations and dilations of a fixed function, say, ψ form an orthonormal basis of L 2 (R). Examples include the Haar wavelet, the Shannon wavelet, Meyer wavelet, Battle-Lemarié wavelets, Daubechies wavelets, and Coiffman wavelets. A standard way to construct an orthogonal wavelet is to solve the dilation equation ∞ √
φ(x) = 2 ck φ(2x − k),

(0.1)

k=−∞

 with the normalization k ck = 1. Very few solutions of the dilation equation for wavelets are known to have closed form expressions. This is one reason why it is difficult to determine the regularity of wavelets. 1 Provided the solution φ satisfies some additional conditions, the function ψ defined by ψ(x) =

∞


(−1)k c1−k φ(2x − k),

n=−∞

is an orthogonal wavelet, i.e., the set {2 j/2 ψ(2 j x − n) : j, n ∈ Z} is an orthonormal basis for L 2 (R). Henceforth, the term wavelet will mean orthogonal wavelet. Daubechies [Da1] made the breakthrough of constructing smoother compactly supported wavelets. For these wavelets, the wavelet coefficients {ck } satisfy simple recursion relations, and thus can be quickly computed. However, the Daubechies wavelets necessarily possess only finite smoothness.2 Smoother wavelets provide sharper frequency resolution of functions. Also, the regularity of the wavelet determines the speed of convergence of the cascade algorithm [Da1], which is another standard method to construct the wavelet from Eq. (0.1). Thus knowing the smoothness or regularity of wavelets has important practical and theoretical consequences. The main goals of this paper are to extend, refine, and unify the thermodynamic approach to the regularity of wavelets and to devise a faster algorithm for estimating regularity. In Sect. 1 we recall the construction of wavelets using multiresolution analysis. We then discuss how the L p -Sobolev regularity of wavelets is related to the thermodynamic pressure of the doubling map E 2 : [0, 2π ) → [0, 2π ). The latter is the crucial link between wavelets and dynamical systems. Previous authors have noted that for compactly supported wavelets, the transfer operator preserves a finite dimensional subspace and is thus represented by a d × d matrix. Providing the matrix is small, its maximal eigenvalue, which is related to the pressure, 1 We learned in [Da2] (see references) that this dilation equation arises in other areas, including subdivision schemes for computer aided design, where the goal is the fast generation of smooth curves and surface. 2 Frequently, increasing the support of the function leads to increased smoothness.

Wavelet Regularity via Thermodynamic Formalism

3

can be explicitly computed, and thus the L p -Sobolev regularity can be determined by matrix algebra. Cohen and Daubechies identified the L p -Sobolev regularity in terms of the smallest zero of the Fredholm-Ruelle determinant for the transfer operator acting on a certain Hilbert space of analytic functions [CD1,Da2]. Their analysis leads to an algorithm for the Sobolev regularity of wavelets having analytic filters that converges with exponential speed. In Sect. 2, we study the transfer operator acting on the Bergman space of analytic functions and effect a more refined analysis. This leads to a more efficient algorithm which converges with super-exponential speed to the L p -Sobolev regularity of the wavelet. The algorithm involves computing certain orbital averages over the periodic points of the doubling map E 2 . In Sect. 3 we illustrate, in a number of examples, the advantage of this method over the earlier Cohen-Daubechies method. Running on a fast PC, this algorithm provides a highly accurate approximation for the Sobolev regularity in less than two minutes. A feature of this thermodynamic approach is that it works equally well for compactly supported and non-compactly supported wavelets (e.g., Butterworth filters). In practice, for wavelets with large support, the distinction may become immaterial, and the use of our algorithm which applies to arbitrary wavelets may prove quite useful. In Sect. 4, we use this improved algorithm to study the parameter dependence of the L p -Sobolev regularity in several smooth families of wavelets. In particular, we construct new examples of wavelets having the same support as Daubechies wavelets, but with higher regularity. In Sect. 5, we show that the L p -Sobolev regularity depends smoothly on the wavelet, and we provide, via thermodynamic formalism, a variational formula. This problem has been considered experimentally by Daubechies [Da1] and Ojanen [Oj1] for different parameterized families. Our thermodynamic approach, based on properties of pressure, gives a sound theoretical foundation for this analysis. In Sect. 6, we study the asymptotic behavior of the L p -Sobolev regularity of wavelet families. We establish a general relation between this asymptotic regularity and maximal measures for the doubling map, extending work of Cohen and Conze [CC1]. The study of maximal measures is currently an active research area in dynamical systems. In Sect. 7, we describe the natural generalization of these results to arbitrary dimensions. Finally, in Sect. 8, we briefly describe the natural generalization of these results to wavelets whose filters are not analytic. 1. Wavelets: Construction and Regularity Constructing wavelets. We begin by recalling the construction of wavelets via multiresolution analysis. A comprehensive reference for wavelets is [Da1], which also contains an extensive bibliography. The Fourier transform of the scaling equation (0.1) satisfies 1
(ξ ) = m(ξ/2)φ (ξ/2), where m(ξ ) = √ φ ck e−ikξ . (1.1) 2 k The function m is sometimes called the filter defined by the scaling equation. Iterating this expression, and assuming that m(0) = 1, one obtains that (ξ ) = φ

∞  j=1

m(2− j ξ ).

(1.2)

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M. Pollicott, H. Weiss

Conditions (1)–(3) below allow one to invert this Fourier transform and obtain the scaling function φ. The wavelet ψ is then explicitly given by ψ(t) =

√

2

∞


dk φ(2t − k), where

dk = (−1)k−1 c−1−k ,

(1.3)

k=−∞

and thus the family

  ψ j,k = 2 j/2 ψ(2 j t − k)

j,k∈Z

L 2 (R).3

forms an orthonormal basis of A major result in the theory is the following. Theorem 1.1. [Da1,p.186] Consider an analytic 2π -periodic function m with Fourier expansion m(ξ ) = n cn einξ satisfying the following conditions: (1) m(0) = 1; (2) |m(ξ )|2 + |m(ξ + π )|2 = 1; and (3) |m(ξ )|2 has no zeros in the interval [− π3 , π3 ].  defined in (1.2) is the Fourier transform of a function φ ∈ L 2 (R) Then the function φ and the function ψ defined by (1.3) defines a wavelet. (0) = 0, which is necessary to ensure the completeCondition (1) is equivalent to φ ness in the multiresolution defined by φ. Condition (1) is also necessary for the infinite product (1.2) to converge. Condition (2) is necessary for the wavelet to be orthogonal, and condition (3), called the “Cohen condition”, is a sufficient condition for the wavelet to be orthogonal [Da1, Cor. 6.3.2]. If one wishes to construct compactly supported wavelets, then one needs the additional assumption: (4) There exists N ≥ 1 with cn = 0 for |n| ≥ N . If ck = 0 for k ∈ [N1 , N2 ], then the support of φ is also contained in [N1 , N2 ], and the support of ψ is contained in [(N1 − N2 + 1)/2, (N2 − N1 + 1)/2]. It is frequently useful to work with the function q(ξ ) = |m(ξ )|2 . This function is positive, analytic, and even. Condition (1) is equivalent to q(0) = 1 and (2)  is equivalent to q(ξ ) +q(ξ + π ) = 1. A simple exercise shows that (1) is equivalent to k∈Z ck = 1/2 and (2) is equivalent to q(ξ ) = 1/2 + k∈Z c2k+1 cos((2k + 1)ξ ). Thus the set K of positive even analytic functions satisfying both (1) and (2) is a convex subset of the positive analytic functions. Example (Daubechies wavelet family). This family of continuous and compactly supported wavelets φ Daub {N ∈ N}, is obtained by choosing the filter m such that N |m(ξ )|2 = cos2N (ξ/2)PN (sin2 (ξ/2)), where PN (y) =

N −1


 N −1+k  k

yk .

k= 0

The wavelets in this family have no known closed form expression. Note that |m(ξ )|2 vanishes to order 2N at ξ = π . More generally, a trigonometric ξ 2 2 2N polynomial m with |m(ξ )| = cos 2 u(ξ ) satisfies (2) only if u(ξ ) = Q(sin (ξ/2)), where Q(y) = PN (y) + y N R(1/2 − y), and R is an odd polynomial, chosen such that Q(y) ≥ 0 for 0 ≤ y ≤ 1 [Da1, 171]. 3 Engineers often call the family {c , d } a quadratic mirror filter (QMF). k k

Wavelet Regularity via Thermodynamic Formalism

5

Wavelet regularity. Since the Fourier transform is such a useful tool for constructing wavelets, and since wavelets (and other solutions of the dilation equation) rarely have closed form expressions, it seems useful to measure the regularity of wavelets using the Fourier transform. For p ≥ 1, the L p -Sobolov regularity of φ is defined by   f (ξ )| p ∈ L 1 (R) . s p ( f ) = sup s : (1 + |ξ | p )s |  The most commonly studied case is p = 2, where if s2 ( f ) > 1/2, then f ∈ C s2 −1/2− . One can also easily see that s p ( f ) − sq ( f ) ≤

1 1 − , for 0 < p < q. p q

Thermodynamic approach to wavelet regularity. We now discuss a thermodynamic formalism approach to wavelet regularity. We assume that |m(ξ )|2 has a maximal zero of order M at ±π and write |m(ξ )|2 = cos2M (ξ/2)r (ξ ),

(1.4)

where r ∈ C ω ([0, 2π )) is an analytic function with no zeros at ±π . We further assume that r > 0, and thus log r ∈ C ω ([0, 2π )). Given a function g : [0, 2π ) → R and the doubling map E 2 : [0, 2π ) → [0, 2π ) defined by E 2 (x) = 2x(mod 2π ), the pressure of g is defined by  P(g) = sup h(µ) +

2π

g dµ : µ a E 2 -invariant probability measure ,

0

where h(µ) is the measure theoretic entropy of µ. If g is Hölder continuous, then this supremum is realized by a unique E 2 −invariant measure µg , called the equilibrium state for g. The following theorem expresses the relationship between the pressure and the Sobolev regularity s p (φ), and provides the crucial link between wavelets and dynamical systems. Theorem 1.2. The L p -Sobolov regularity satisfies the expression s p (φ) = M −

P( p log r ) . p log 2

Proof. The proof for p = 2 is due to Cohen and Daubechies [CD1] and the general case is due to Eirola [Ei1] and Villemoes [Vi1]. See also [He1,He2,Sun]. In the literature this result is formulated in terms of the spectra radius of the transfer operator on continuous functions; but this is precisely the pressure (see Proposition 2.1).  See Appendix I for a proof of the lower estimate in arbitrary dimensions.

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M. Pollicott, H. Weiss

2. Determinants and Transfer Operators Given h : [0, 2π ) → R we define the semi-norm |h|β = sup x= y

|h(x) − h(y)| , |x − y|β

and the Hölder norm ||h||β = |h|∞ + |h|β , where 0 < β < 1. Given g ∈ C β [0, 2π ), the transfer operator L g : C β ([0, 2π ) → C β ([0, 2π ) is the positive linear operator defined by x

x

x

x

k + exp g +π k +π . L g k(x) = exp g 2 2 2 2 The following properties of pressure are well known [Ru1,PP1]. Proposition 2.1. (1) The spectral radius of L g is exp(P(g)); (2) The pressure P(g) has an analytic dependence on g ∈ C β ([0, 2π )); and  2π (3) The derivative Dg P( f ) = 0 f dµg . The transfer operator L g C β ([0, 2π )) → C β ([0, 2π )) is not trace class on this class of functions. However, following Jenkinson and Pollicott [JP1,JP2], we consider the transfer operator acting on the Bergman space A2 ( ) of square-integrable analytic functions on an open unit disk ⊂ C containing [0, 2π ). In Appendix I we show that for analytic functions g, the transfer operator L g . A2 ( ) → A2 ( ) is trace class and L g has a welldefined Ruelle-Fredholm determinant. For small z this can be defined by det(I − z L g ) =  n k exp(tr(log(I − z L g ))) [Si1]. We can write tr(log(I − z L g )) = ∞ k=1 (z /n)tr(L g ), and k each operator L g is the sum of weighted composition operators having sharp-trace (in the sense of Atiyah and Bott) tr(L kg ) =


exp(Sk g(x)) , 1 − 2−k k

E 2 x=x

Sk (x) = g(x) + g(E 2 x) + · · · + g(E 2k−1 x). The Ruelle-Fredholm determinant for L g can be expressed as ⎛ ⎞ ∞ n

z exp(S g(x)) n ⎠, det(I − z L g ) = exp ⎝− n n 1 − 2−n n=1

E 2 x=x

(2.1)

 and has the “usual determinant property” det(I − z L g ) = ∞ n=1 (1 − zλn ), where λn denotes the n th eigenvalue of L g . Note that this determinant is formally quite similar to the Ruelle zeta function, and has an additional factor of the form 1/(1 − 2−n ) in the inner sum. More generally, L g need not be trace class for expression (2.1) to be well defined for small z (see Sect. 8). For our applications to wavelet regularity, we consider g = p log r (see (1.1)). To compute the determinant, one needs to compute the orbital averages of the function g over all the periodic points of E 2 . Every root of unity of order 2n − 1 is a periodic point

Wavelet Regularity via Thermodynamic Formalism

7

for E 2 , and there are exactly 2n − 1 such roots of unity, hence there are exactly 2n − 1 periodic points of period n for E 2 . If one expands the exponential function in a Taylor series and applies Cauchy’s convolution formula for a power of a Taylor series, one obtains the following formula: d p (z) := det(I − z L plogr ) = 1 +

∞


bn z n ,

n=1

where bn =

n
(−1)k k!

k= 0

and  p

an =

1 1 − 2−n




p

n 1 +···+n k = n

p

an 1 · · · an k , n1 · · · nk

⎡ ⎤p n −1 n−1   2
jk   2 ⎣ ⎦ . r 2n − 1 k= 0

j=0

The next theorem summarizes some important properties of this determinant d p (z). Theorem 2.2. Assume log r ∈ C ω [0, 2π ). Then (1) The function d p (z) can be extended to an entire function of z and a real-analytic function of p; (2) The reciprocal of the exponential of pressure z p := exp(−P( p log r ))) is the smallest zero for d p (z); (3) The Taylor coefficients bn of d p (z) decay to zero at a super-exponential rate, i.e., 2 there exists 0 < A <1 such that bn = O(An ); N (4) Let d p,N (z) = 1 + n=1 bn z n be the truncation of the Taylor series to N terms and consider smallest zero d p,N (z Np ) = 0. Then z Np → z p at a super-exponential 2

rate, i.e., there exists 0 < B < 1 such that |z p − z Np | = O(B N ). In fact, in Theorem 2.2 (3) and (4) we can choose A and B arbitrarily close to 1/2. Parts (1) and (2) follow from [Ru2] and (4) follows from (3) via the argument principle. The essential idea in proving (3) is that one can also express the determinant d p (z) = 1 +

∞
n=1

bn z n =

∞ 

(1 − zλn ) ,

n=1

where λn denotes the n th eigenvalue of L p log r [Ru2]. It follows that the coefficient bn = i1 <···
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M. Pollicott, H. Weiss

For our applications to wavelets, we shall take either g = log |m|2 or g = log r . In fact, both functions lead to equivalent results. The following proposition relates the zeros of det(1 − z L log r ) and det(1 − z L |m|2 ). 4 Proposition 2.3. The determinant det(1 − z L log |m|2 ) has zeros at (1) 1, 2, 4, . . . , 2 M−1 ; and (2) 2 M z, where z is a zero for det(1 − z L log r ). This was observed by Cohen and Daubechies [CD1]. 3. Examples: Computation of Sobolev Regularity Using the Determinant In this section we shall apply Theorem 2.2 to estimate the Sobolev exponent s p (φ) of certain interesting examples. 3.1. Daubechies wavelet family. Example 1: (N = 3 and p = 2). In this case d2 (z) is a polynomial of degree 5 with rational coefficients: d2 (z) = 1 − 2z − 57.1875z 2 − 74.53125z 3 + 210.09375z 4 − 91.125z 5 , which has its first zero at z 0 = 0.111 . . .. By Theorem 1.2 we deduce the precise value s2 (ψ3Daub ) = 3 + log(0.1111 . . .)/2 log 2 = 1.415037 . . ..5 The following table illustrates the remarkable speed of approximation of this algorithm by computing the first few approximations to s2 (ψ3Daub ). The level k approximation to the determinant is obtained by summing over all periodic points with period less than or equal to k. Level 6 approximations take less than a minute on a desktop computer. N

Smallest root of d2

Approx to s2 (ψ3Daub )

1 2 3 4 5 6

0.50000000000000000 0.11590073269851241 0.10935186260699924 0.11120565746767627 0.11111111111111109 0.11111111111111111

2.5000000000000000 1.4454807959406297 1.4035248496765980 1.4156510452962938 1.4150374992788437 1.4150374992788437

Example 2: (N = 2 and p = 4). In this case d4 (z) is a polynomial of degree 5 with rational cofficients: d4 (z) = 1 − 2z − 50.75z 2 − 76.25z 3 + 115z 4 − 32z 5 , which has its first zero at z 0 = 0.115146 . . .. By Theorem 2.1 we deduce that s4 (ψ2Daub ) = 3 + log(0.115146 . . .)/2 log 2 = 1.22038 . . ..6 4 In our method it is desirable to factor out the zeros and work with r because this reduces the amount of computation and may increase the efficiency of estimating the determinant by decreasing C > 0. 5 This compares with the numerical estimate of Cohen-Daubechies of 1.414947, which is accurate to two decimal places. 6 This compares with the estimate of Cohen-Daubechies of 1.220150 . . ., which is accurate to three decimal places.

Wavelet Regularity via Thermodynamic Formalism

9

Example 3: (N = 4 and p = 2). In this case d2 (z) is a polynomial of degree 7 with rational coefficients d2 (z) = 1 − 2z − 433z 2 − 996z 3 + 23572z 4 − 32288z 5 − 68627z 6 − 24414z 7 , which has its first zero at z 0 = 0.045788 . . .. By Theorem 2.1 we deduce that s2 (ψ4Daub ) = 4 + log(0.045788 . . .)/2 log 2 = 1.77557 . . ..7 Based on the same approximation scheme, the following table contains our approximations to s2 (ψ Ndaub ) for N = 1, 2, . . . , 10, and compares them with the values obtained by Cohen and Daubechies in [CD1]. N

New approx to s2 (ψ4Daub )

Previous approx to s2 (ψ4Daub )

1 2 3 4 5 6 7 8 9 10

0.500000 1.000000 1.415037 1.775565 2.096786 2.388374 2.658660 2.914722 3.161667 3.402723

0.338856 0.999820 1.414947 1.775305 2.096541 2.388060 2.658569 2.914556 3.161380 3.402546

Remark. In Examples 1–3 the determinant is a rational function. This is an immediate consequence of the fact that the transfer operator preserves the finite dimensional subspace of functions spanned by {1, cos(ξ ), cos(2ξ ), · · · , cos(N ξ )}. This fact has been observed by several authors, cf., [Oj1]. 3.2. Butterworth filter family. The Butterworth filter is the filter type with flattest pass band and which allows a moderate group delay [OS1]. These filters form an important family of non-compactly supported wavelets and are defined by |m(ξ )|2 =

cos2N (ξ/2) , N ∈ N. sin2N (ξ/2) + cos2N (ξ/2)

We suppress the index n for notational convenience. It is easy to check that |m(ξ )|2 satisfies conditions (1) − (3) of Theorem 1.1. Example 4: (Butterworth filter with N=2, p=2). For this case the determinant d2 (z) is a power series of the form d2 (z) = 1 − 2. z − 2.08 z 2 − 0.479053 z 3 − 0.0124521 z 4 − 0.000086246 z 5 −2.06875 10−9 z 6 − 3.2482 10−13 z 7 − 7.38964 10−13 z 8 + 4.24431 10−12 z 9 −2.94676 10−11 z 10 + · · · , which has its first zero at z 0 = 0.356702 . . ., and thus by Theorem 1.2 we deduce that s2 (ψ2Butt ) = 2 + log(0.356702 . . .)/2 log 2 = 1.25497 . . .. 7 This compares with the estimate of Cohen-Daubechies of 1.775305, which is accurate to three decimal places.

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M. Pollicott, H. Weiss

The following table contains the first few approximations of s2 (ψ2Butt ): N

Root

Approx to s2 (ψ2Butt )

1 2 3 4 5 6 7

0.500000000000000 1.256072268804003 0.356757200613385 0.356702225093496 0.356702089349239 0.356702089348077 0.356702089348077

1.0000000000000000 1.2689757055091953 1.2565072268804003 1.2563960602082473 1.2563957856969186 1.2563957856945698 1.2563957856945698

We can similarly compute s2 (ψ NButt ) for N = 1, 2, . . . , 10 and compare these with the values in [CD1]. N

New approx to s2 (ψ NButt )

Previous approx to s2 (ψ NButt )

1 2 3 4 5 6 7 8 9 10

0.500000 1.256211 2.044109 2.843768 3.648646 4.456118 5.264533 6.072947 6.881125 7.688598

0.338856 1.256395 2.044117 2.843771 3.648817 4.456252 5.264586 6.073034 6.881173 7.688776

4. Compactly Supported Wavelets Smoother than the Daubechies Family We now consider smooth finite dimensional families of wavelets and attempt to find the maximally regular wavelets in the family. Several authors have used this approach to find compactly supported wavelets that are more regular than the Daubechies wavelets with the same support. The most straightforward approach to the problem of maximizing the regularity of compactly supported wavelets in parameterized families reduces to the problem of maximizing the maximal eigenvalue of an N × N matrix. However, except for very small N , there is no algebraic expression for the eigenvalue. In this section, we illustrate the advantages of our fast converging algorithm in seeking more regular wavelets. Example 1. Ojanen [Oj1] experimentally studied the regularity of wavelets corresponding to filters |m(ξ )|2 of width N and which vanish to order M at π . In the case that M ≥ N − 1 he found that the Daubechies polynomials (where M = N ) are apparently the smoothest. For N = 2 and M = 1, say, one can define the family of filters |m(ξ )|2 =

1 (1 + cos ξ ) (1 + 8a cos2 ξ − 8a cos ξ ) .    2

(4.1)

:=r (ξ )

The transfer operator preserves the subspace spanned by {1, cos(ξ ), cos(2ξ )}√and thus can be represented by a 3×3 matrix whose maximal eigenvalue is exp P = 1+ 1 + 16a. In the special case when a = −1/16 one obtains the filter corresponding to ψ2Daub , for which r (π ) = 0. One can easily show that this wavelet is the smoothest in this family. However, for the broader class M ≤ N − 2, this is no longer the case.

Wavelet Regularity via Thermodynamic Formalism

11

Example 2. We consider the two-parameter family of putative wavelets corresponding to filters   1 1 2 − a − b cos(5x). (4.2) |m(x)| = + a cos(x) + b cos(3x) + 2 2 We say putative, because all the conditions in Theorem 1.1 need to be verified for all parameter values. The Daubechies wavelet ψ3Daub corresponds to parameters a = 75/128 and b = −25/256. Here |m(x)|2 has a double zero at x = π and thus one can always take out (at least) a factor of (1 + cos(ξ )) /2. However, it is always necessary to check “by hand” that for each particular choice of a and b expression (4.2) is positive and the Cohen condition holds. It is useful to write (4.2) in the form r (ξ )(1 + cos(ξ ))/2, where r (ξ ) = 1 − 4 (−1 + 2 a + 4 b) cos(ξ ) + 4 (−1 + 2 a + 4 b) cos2 (ξ ) +16 (−1 + 2 a + 2 b) cos3 (ξ ) − 16 (−1 + 2 a + 2 b) cos4 (ξ ). Notice this trigonometric polynomial has the form r (ξ ) = 1 − α cos(ξ ) + α cos2 (ξ ) + β cos3 (ξ ) − β cos4 (ξ ). (A) The transfer matrix approach. We consider the finite dimensional space spanned by {1, cos(ξ ), cos2 (ξ ), cos3 (ξ ), cos4 (ξ )} and write the action of the transfer operator associated to weights of the general form r (ξ ) = 1 − α cos(ξ ) + α cos2 (ξ ) + β cos3 (ξ ) − β cos4 (ξ ). A routine calculation yields that the transfer operator on this space has the following matrix representation: ⎛ ⎞ 2 + α − β2 α−β − β2 0 0 ⎜ ⎟ β ⎜ −α + β −α + β 0 0 ⎟ 2 2 ⎜ ⎟ ⎜ ⎟ β 3β 3β β α α ⎜ L 2 log r = L 2 log r (α, β) = ⎜ 1 + 2 − 4 1 + α − 4 −4 0 ⎟ ⎟. 2 − 4 ⎜ ⎟ 3β β α ⎜ −α + β ⎟ −α + 3β − + 0 ⎝ 2 4 ⎠ 4 2 4 4 1 2

+

α 4

−

β 8

The matrix corresponding to ψ3Daub is

L 2 log r (−7/8, −3/8) =

1+ ⎛ 21 16 ⎜ 11 ⎜ ⎜ 16 ⎜ 21 ⎜ ⎜ 32 ⎜ ⎜ 11 ⎜ 32 ⎝ 21 64

3α 4

−

− 21 1 2 13 32 19 32 17 32

β 1 2 2

3 16 3 − 16 5 − 32 5 32 1 8

+

3α 4

0 0

−

0

3β α 4 4

−

β 2

− β8

⎞

⎟ 0⎟ ⎟ ⎟ 0⎟ ⎟, ⎟ 0⎟ ⎟ ⎠ 3

3 32 3 − 32 1 − 32 64

with eigenvalues 1, 9/16, ±1/4 and 3/64. Then det(I − z L 2 log r ) has a zero at z = 16/9. By a slightly modified version of Theorem 1.2 accounting for the fact that we have not

12

M. Pollicott, H. Weiss

factored out the maximal zero at ±π , we again recover the Sobolev regularity of ψ3Daub : s2 (ψ3Daub ) = 1 − (log(9/16)/(2 log 2) = 1.41504 . . . . By varying the parameters α and β, and always checking that expression (4.2) is positive and satisfies the Cohen condition, one can search for wavelets having maximal L p −Sobolev regularity, and, in particular, having greater regularity than the Daubechies wavelet. The two required conditions make this a difficult constrained optimization problem, which is even difficult numerically. In contrast with Example 1, there appears to be no simple algebraic expression for the eigenvalues for the matrix corresponding to L log r ; one must resort to empirical searches using different choices of α and β. However, it is more illuminating to implement this computation using determinants. (B) The determinant approach. If one computes the determinant det(1 − z L log |m|2 associated with the Daubechies wavelet obtains the power series expansion det(1 − z L log |m|2

Daub

ψ3Daub

Daub

)

using our determinant algorithm, one

) = 1 − 2 z + 1.31904 z 2 − 0.35310 z 3 + 0.03319 z 4 + 0.00120 z 5 − 0.000351 z 6 + · · · .

We observe that: (1) There are “trivial” zeros for det(1 − z L log |m|2 ) at the values z = 1, 2, 4, 8. Daub (2) There is a zero at z = 7.07337 . . . ., which by Proposition 2.3 corresponds to a zero of det(1 − z L log rDaub ) at 7.07337/26 = 0.111 . . . . This agrees with our previous calculation in Sect. 3. By Theorem 1.2, if we change the filter |m|2 to decrease the pressure, then the Sobolev regularity of the associated wavelet increases. One can easily check that if one replaces a = 75/128 by 75/128 − 0.00068, say, and b = −25/256 by −25/256 + 0.00002, say, then the associated filter |m(ξ )2 | remains positive, and by Theorems 1.2 and 2.1, the zero of the determinant increases from 7.1111 to 8.51989. This reflects an increase in the L 2 -Sobolev regularity from 1.415 to 1.545. Thus, this two-parameter family contains wavelets having the same support as ψ3Daub with greater L 2 -Sobolev regularity. The same idea can be applied for other N . Remarks. Other approaches. (a) Daubechies [Da1, p.242] considered the one-parameter family of filters |m(ξ )|2 = cos2(N −1) (ξ/2)(PN −1 (sin2 (ξ/2)) + a cos2N (ξ/2)) cos(ξ ), where a ∈ R. For definiteness, we discuss the case N = 3. For different values of the parameter a we calculate the first zero of the determinant, and thus the Sobolev regularity of the associated wavelet using Theorem 2.2. The following table shows that this zero is monotone increasing as a increases to 3. The zero of the determinant is 0.40344 . . . when a = 3, which corresponds to L 2 Sobolev regularity 1.34443 . . .. It is not necessary to study the case a > 3 since the expansion for |m|2 is no longer positive. To see this, we can expand in terms of ξ − π as |m(ξ )|2 = cos4 (ξ/2)(P2 (sin2 (ξ/2)) + a cos4 (ξ/2)) cos(ξ ) = cos4 (ξ/2 − π/2)(1 + 2 sin2 (ξ/2 − π/2) + a sin6 (ξ/2 − π/2) cos(ξ − π )) = sin4 (ξ/2)(1 + 2 cos2 (ξ/2) − a cos6 (ξ/2) cos(ξ ))

Wavelet Regularity via Thermodynamic Formalism

13

= 1 + 2(1 − (ξ/2)2 + · · · )2 − a(1 − (ξ/2)2 + · · · )6 (1 − ξ 2 + · · · ) = (3 − a) − (1 + 5a/2)ξ 2 + · · · . It is clear this expression is negative values for a > 3. We see that the parameter value a = 3 corresponding to the maximally regular wavelet lies on the boundary of an allowable parameter interval, and thus could not be detected using variational methods. a -1.0 -0.5 0.0 0.5 1.0 1.5 2.0 2.5 2.7 2.8 2.9 3.0

Zero of determinant 0.230 0.239 0.250 0.262 0.276 0.294 0.317 0.350 0.367 0.377 0.389 0.403

Sobolev regularity 0.939 0.967 1.000 1.030 1.071 1.116 1.710 1.242 1.276 1.296 1.318 1.344

(b) Daubechies [Da3] also considered the one-parameter family of functions  m(ξ ) =

1 + eiξ 2

2 Q(ξ ),

where |Q(ξ )|2 = P(cos ξ ) and P(x) = 2 − x + a4 (1 − x)2 , with the parameter a chosen such that P(x) > 0 for 0 ≤ x < 1. The value a = 3 corresponds to ψ2Daub , and the associated polynomial has a zero at x = −1. She reparameterizes this wavelet family using the zero at x = −1+δ, and obtains, numerically, a wavelet having the same support as ψ2Daub but with higher regularity. 5. Perturbation Theory and Variational Formulae In this section we observe that the analyticity of pressure implies that the L p Sobolev regularity depends smoothly on the wavelet. It thus seems natural to search for critical points (maximally smooth wavelets) in parametrized families of wavelets by explicitly calculating the derivative of the Sobolev regularity functional. We caution that the example in the previous section shows that maximally regular wavelets can lie on the boundary of allowable parameter intervals. We now restrict |m(ξ )|2 to the subset of analytic functions B satisfying conditions (1) − (3) and let r0 be a tangent vector to this space. It follows from remarks in Sect. 1 that r0 is represented by an analytic function of the form ∞

1
r0 = + d2n+1 cos((2n + 1)ξ ), 2 n= 0

We have the following result.

such that

∞
n= 0

dn =

1 . 2

14

M. Pollicott, H. Weiss

Proposition 5.1. (1) For every p ≥ 1, the regularity mapping K → R defined by |m(ξ )|2 → s p (φ) is analytic; (2) Let |m λ (ξ )|2 = cos2M (ξ/2) (r0 (ξ ) + λ r0 (ξ ) + · · · ) be a smooth parametrized family of filters. Then the change in the Sobolev regularity is given by   1 r0 λ 0 s p (φ ) = s p (φ ) + dµ λ + · · · , p log 2 r0 where µ is the unique equilibrium state for p log r0 . Proof. The analyticity of the Sobolev regularity follows from Proposition 2.1(2), the λ 0 analyticity of the pressure, and the implicit function  theorem. Since log r = log(r + r0 0 r0 + · · · ) = log r + r0 λ + · · · and Dg P(·) = ·dµg , this follows from the chain rule.  If we restrict to finite dimensional spaces we can consider formulae for the second (and higher) derivatives of the pressure, which translate into formulae for the next term in the expansion of s p (φλ ). Also observe that we can estimate the derivatives of the Sobolev exponent of a family s p (φλ ) using the series for d p (φλ , z). More precisely, by the implicit function theorem ds p (φ λ ) d(det(φ λ , z)) = dλ dλ

d(det(φ λ , z)) , dz

which is possible to estimate for periodic orbit data. We can similarly compute the higher derivatives ! d k s p (φ λ ) !! , for k ≥ 1 ! dλk ! λ = λ0

using Mathematica, say. We can then expand the power series for the Sobolev function: ! ! ds p (φ λ ) !! (λ − λ0 )2 d 2 s p (φ λ ) !! λ λ0 s p (φ ) = s p (φ ) + (λ − λ0 ) + + ··· . dλ !λ = λ0 2 dλ2 !λ = λ0 6. Asymptotic Sobolev Regularity The use of the variational principle to characterize pressure (and thus the Sobolev regularity) leads one naturally to the idea of maximizing measures. More precisely, one can associate to g:  2π

Q(g) = sup g dµ, µ an E 2 -invariant Borel probability measure . 0

Unlike pressure, which has analytic dependence on the function, the quantity Q(g) is less regular, but we have the following trivial bound.

Wavelet Regularity via Thermodynamic Formalism

15

Lemma 6.1. For any continuous function g : [0, 2π ) → R, Q(g) ≤ P(g) ≤ Q(g) + log 2. Proof. Since 0 ≤ h(µ) ≤ log 2, this follows from the definitions.



Proposition 6.2. Let φn , n ∈ N be a family of wavelets with filters |m(ξ )|2 = cos2Mn (ξ/2)rn (ξ ), where Mn ≥ 0. Then the asymptotic L p -Sobolev regularity of the family is given by s p (φn ) Mn Q(log rn ) = lim − lim . n→∞ n→∞ n n→∞ n log 2 n lim

Proof. Using Theorem 1.2, Lemma 6.1, and (1.1), one can easily show Q( p log rn ) ≤ p log 2(Mn − s p (φn )) ≤ Q( p log rn ) + log 2. The proposition immediately follows after dividing by n and taking the limit as n → ∞.  We now apply Proposition 6.2 to compute the L p -Sobolev regularity of the Daubechies family of wavelets for arbitrary p ≥ 2. Cohen and Conze [CC1] first proved this result for p = 2. Proposition 6.3. Let p ≥ 2. The asymptotic L p -Sobolev regularity of the Daubechies family is given by s p (ψ NDaub ) log 3 =1− . N →∞ N 2 log 2 lim

Proof. If one applies Proposition 6.2 to the Daubechies filter coefficients (see Sect. I), one immediately obtains " # Q(log PN (sin2 ( 2y ))) s p (φ Daub ) N lim = 1 − lim . N →∞ N →∞ N N log 2 Cohen and Conze [CC1, p. 362] show that y log PN (3/4) , Q log2 PN (sin2 ( )) = 2 2 log 2 and the sup is attained for the period two periodic orbit {1/3, 2/3}. Thus s p (φ Daub ) 1 log PN (3/4) N =1− lim . N →∞ N 2 log 2 N →∞ N  −1 3 N −1 The exponentially dominant term in PN (3/4) is easily seen to be 2N , and N −1 ( 4 ) a routine application of Stirling’s formula yields lim

log PN (3/4) = log 3. N →∞ N lim

The proposition immediately follows.



16

M. Pollicott, H. Weiss

Remark. As briefly discussed in Sect. 3.2, the Butterworth filter family ψ NButt , N ∈ N is the family of non-compactly supported wavelets defined by the filter |m(ξ )|2 =

cos2N (ξ/2) sin2N (ξ/2) + cos2N (ξ/2)

.

Fan and Sun [FS1] computed the asymptotic regularity for this wavelet family; they showed that s p (ψ NButt ) log 3 = . N →+∞ N log 2 lim

This asymptotic linear growth of regularity is similar to that for the Daubechies wavelet family. 7. Regularity of Wavelets in Higher Dimensions The dilation equation (0.1) has a natural generalization to d-dimensions of the form:
φ(x) = det(D) ck φ(Dx − k), (7.1) k∈Zd

where D ∈ G L(d, Z) is a d × d matrix with all eigenvalues having modulus strictly greater than 1. We are interested in solutions φ ∈ L 2 (Rd ). We consider the expanding map T : [0, 2π )d → [0, 2π )d by T (x) = Dx (mod 1). By taking d-dimensional Fourier transformations, (7.1) gives rise to the equation
(ξ ) = m(D −1 ξ )φ (D −1 ξ ), where m(ξ ) = φ ck e−i k,ξ . k∈Zd

 −n (ξ ) = ∞ Thus by iterating this identity we can write φ n=1 m(D ξ ). β d For g ∈ C ([0, 2π ) ), one can define a transfer operator L g : C β ([0, 2π )d ) → β C ([0, 2π )d ) by


L g k(x) = exp g(y) k(y), T y=x

where the sum is over the det(D) preimages of x. We consider the Banach space of periodic analytic functions A in a uniform neighbourhood of the torus [0, π )d and the associated Bergman space of functions. We have that for g ∈ A the operator L g : A → A and its iterates L kg are trace class operators for each k ≥ 1. We can compute tr(L kg ) =


|det(D)|n exp(Sn g(x)) , |det(D n − I )| n

T x =x

where Sn g(x) = g(x) + g(T x) + · · · + g(T n−1 x).

Wavelet Regularity via Thermodynamic Formalism

17

M a(Dx) Assume that we can write m(x) = |det(D)|a(x) r (x), where r (x) > 0.8 The Ruelle-Fredholm determinant of L p log r takes the form: ⎛ ⎞ ∞ n
n exp( pS log r (x))
z |det(D)| n ⎠. det(I − L p log r ) = exp ⎝− n n |det(D n − I )| n=1

T x =x

The statement of Theorem 1.2 has a partial generalization to d-dimensions. We define the L p -Sobolev regularity of p to be (ξ )| p (1 + ||ξ || p )s ∈ Ł1 (Rd )}. s p (φ) := sup{s > 0 : |φ Proposition 7.1. The L p -Sobolev regularity satisfies the inequality s p (φ) ≥

P( p log r ) Mdet(D) − , log λ p log λ

where λ > 1 is smallest modulus of an eigenvalue of D. n (ξ )| ≤ C|det(D)| Mn i=0 Proof. We can write |φ r (D −i ξ ), for some C > 0 and all   n n d ξ ∈ D ([0, 2π ) ). However, we can identify T n D i=0 r (D −i ξ ) p dξ = D L np log r 1(ξ )dξ . Moreover, since L p log r has maximal eigenvalue e P( p log r ) , we can bound this by C  en P( p log r ) , for some C  > 0. In particular, ∞
p p s psn  (ξ )| p dξ |φ (ξ )| (1 + ||ξ || ) dξ ≤ λ |φ Rd

n=1

≤ CC 

T n D−T n−1 D

∞


λ psn |det(D)| M pn en P( p log r ) .

n=1

In particular, the right-hand side converges if λ ps |det(D)| M p e P( p log r ) < 1, from which the result follows.  In certain cases, for example if D has eigenvalues of the same modulus and some other technical assumptions, then this inequality becomes an equality [CD1]. As in Sect. 2, a routine calculation gives that d p (z) := det(I − z L p log r ) = 1 + bn =

n
(−1)k k=0

k!




n=1 p p an 1 · · · an k

n 1 +···+n k =n

n1 · · · nk

and p an

=


|det(D)|n T n x=x

∞




n

T n x=x |det(D n −

bn z n , ,

i=0 r (x)

I )|

where

p .

The simplest case is that of diagonal matrices, but this reduces easily to tensor products (x1 , . . . , xd ) = φ1 (x1 ) · · · φd (xd ) of the one dimensional case. The next easiest example illustrates the “non-separable” case: 8 In particular, if T n x = x we have that n−1 |m(T i x)|2 = 2−n n−1 g(T i x). i =0 i =0

18

M. Pollicott, H. Weiss

Example. For d = 2 we let D =

1

1 1 −1

√  , which has eigenvalues λ = ± 2.

In the d-dimensional setting we have the following version of Theorem 2.2: Theorem 7.2. Assume that log r ∈ A. Then (1) The function d p (z) can be extended to an entire function of z and a real analytic function of p; (2) The exponential of pressure z p := exp(−P( p log r )) is the smallest zero for d p (z); (3) The Taylor coefficients bn of d p (z) decay to zero at a super-exponential rate, i.e., 1+1/d there exists 0 < A <1 such that bn = O(An ); N (4) Let d p,N (z) = 1 + n=1 bn z n be the truncation of the Taylor series to N terms and consider smallest zeros d p,N (z Np ) = 0. Then z Np → z p at a super-exponential rate, i.e., there exists 0 < B < 1 such that |z p − z Np | = O(B N

1+1/d

).

Furthermore, we can choose A and B arbitrarily close to λ−1 . The main difference for d ≥ 2 is that the exponent N 2 is replaced by N 1+1/d in parts 3 and 4. 8. Generalizations to Non Real Analytic Filters m We now consider the more general setting where the filter m is not√ analytic,  and we necessarily obtain weaker results. Recall that the filter m(ξ ) = (1/ 2) k ck e−ikξ , and assume there exists β > 0 such that |cn | = O(|n|−β ). In this case we can only expect that the determinant algorithm for the Sobolevregularity converges with exponential n speed. Recall d p (z) = det(I − z L plogr ) = 1 + ∞ n=1 bn z . Theorem 8.1.

(1) Let 0 < β < 1. If |cn | = O |n|1β , then bn = O(2−nβ(1−) ) for any  > 0;

(2) Let k ≥ 1. If |cn | = O |n|1 k , then bn = O(2−nk(1−) ) for any  > 0. This follows from a standard analysis of transfer operators, and in particular, studying their essential spectral radii. The radius of convergence of d p (z) is the reciprocal of the essential spectral radius of the operators. In the case that 0 < β < 1, this theorem follows from the Hölder theory of transfer operators [PP1]. But for k > 1 this requires a slightly different analysis [Ta1]. The necessary results are summerized in the following proposition. Proposition 8.2. (1) Assume cn = O(|n|−β ). The transfer operator L f has essential spectral radius (1/2)α e P( f ) (and the function d(z) is analytic in a disk |z| < 2β ). (2) Assume cn = O(|n|−k ). The transfer operator L f has essential spectral radius (1/2)k e P( f ) (and the function d(z) is analytic in a disk |z| < 2k ). The first part is proved in [Po1,PP2]. The second part is proved in [Ta1,Ru2]9 . As in Sect. 2, the speed of decay of the coefficients bn immediately translate into the corresponding speed of convergence for the main algorithm. 9 At least when β ∈ N.

Wavelet Regularity via Thermodynamic Formalism

19

Appendix I: Proof of Theorem 2.1 Given a bounded linear operator L : H → H on a Hilbert space H , its i th approximation number (or singular value) 10 si (L) is defined as si (L) = inf{||L − K || : rank(K ) ≤ i − 1}, where K is a bounded linear operator on H . Let r ⊂ C denote the open disk of radius r centered at the origin in the complex plane. The phase space [0, 2π ) for T = E 2 is contained in 2π + for any  > 0, and the two inverse branches T1 (x) = 21 x and T2 (x) = 21 x + 21 have analytic extensions to 2π + satisfying T1 ( 2π + ) ∪ T2 ( 2π + ) ⊂ 2π + 2 . Thus T1 and T2 are strict contractions of 2π + onto 2π + 2 with contraction ratio θ = (2π + )/(2π + 2) < 1. We can choose  > 0 arbitrarily large (and thus θ arbitrarily close to 1/2). Let A2 ( r ) denote  the Bergman Hilbert space of analytic functions on r with inner product f, g := f (z) g(z) d x d y [Ha1]. Lemma A.1. The approximation numbers of the transfer operator L p log r : A2 ( 2π ) → A2 ( 2π ) satisfy s j (L p log r ) ≤

||L p log r || A2 ( 2π ) j θ, 1−θ

for all j ≥ 1. This implies that the operator L p log r is of trace class.  k Proof. Let g ∈ A2 ( 2π + ) and write L p log r g = ∞ k (z) = z . k=−∞ lk (g) pk , where p$ $ π π We can easily check that || pk || A2 ( 2π ) = k+1 (2π )k+1 and || pk || A2 ( 2π + ) = k+1 (2π + )k+1 . The functions { pk }k∈Z form a complete orthogonal family for A2 ( 2π + ), and so L p log r g, pk A2 ( 2π + ) = lk (g)|| pk ||2A2 ( 2π + ) . The Cauchy-Schwarz inequality implies that |lk (g)| ≤ ||L p log r g|| A2 ( 2π + ) || pk ||−1 A2 ( 2π + ) .  j−1

( j)

We define the rank- j projection operator by L p log r (g) = g ∈ A2 ( θ ) we can estimate

k = 0 lk (g) pk .

∞


( j) || L p log r − L p log r (g)|| A2 ( 2π ) ≤ ||L p log r g|| A2 ( 2π ) θ k+1 . k= j

It follows that ( j)

||L p log r − L p log r || A2 ( 2π ) ≤

||L p log r || A2 ( 2π ) j+1 θ , 1−θ

and so s j+1 (L p log r ) ≤ and the result follows.

||L p log r || A2 ( 2π ) j+1 θ , 1−θ



10 We use these terms interchangeably because one can show that s (L) = k

% λk (L ∗ L).

For any

20

M. Pollicott, H. Weiss

We now show that the coefficients of the power series of the determinant decay to zero with super-exponential speed. Lemma A.2. If one writes det(I − z L p log r ) = 1 +  |cm | ≤ B where B =

∞

m=1 (1 − θ

m )−1

||L p log r || A2 ( 2π ) 1−θ

∞

m=1 cm z

m,

then

m θ m(m+1)/2 ,

< ∞.

Proof. By [Si1, Lemma 3.3], the coefficients cn in the power series expansion of the  determinant have the form cm = i 1 <···
|cm | = ≤ = ≤

where B =

∞

! ! !
!
! ! ! ! λ · · · λ si1 (L p log r ) · · · sim (L p log r ) i i m! ≤ 1 ! !i1 <···
m=1 (1 − θ

m )−1 .



Theorem 2.1 follows from Lemma 2.2. The coefficients of det(I − zLs ) = 1 + ∞ n n=1 bn z are given by Cauchy’s Theorem: |bn | ≤

1 sup |det(I − zLs )|, r n |z|= r

for any r > 0.

Using Weyl’s inequality, we can deduce that if |z| = r , then |det(I − zLs )| ≤

∞ 

(1 + |z|λ j ) ≤

"

j=1

≤ 1+ B

∞


∞ 

(1 + |z|s j )

j=1

(r α) θ m

m(m+1) 2

# ,

m=1

where α = ||Ls ||A2 ( 1 ) . If we choose r = r (n) appropriately we obtain the bounds given in the theorem. 

Wavelet Regularity via Thermodynamic Formalism

21

References [CC1] [CD1] [Da1] [Da2] [Da3] [Ei1] [FS1] [GGK1] [Ha1] [He1] [He2] [JP1] [JP2] [Oj1] [OS1] [PP1] [Po1] [Ru1] [Ru2] [Si1] [Sun] [Ta1] [Vi1] [W]

Cohen, A., Conze, J.-P.: Régularité des bases d’ondelettes et mesures ergodiques. Rev. Mat. Iberoamericana 8, 351–365 (1992) Cohen, A., Daubechies, I.: A new technique to estimate the regularity of refinable functions. Rev. Mat. Iberoamericana 12, 527–591 (1996) Daubechies, I.: Ten lectures on wavelets. CBMS-NSF Regional Conference Series in Applied Mathematics, Vol. 61, Philadelphia, PA: Society for Industrial and Applied Mathematics (SIAM), 1992 Daubechies, I.: Using Fredholm determinants to estimate the smoothness of refinable functions. Approximation theory VIII, Vol. 2 (College Station, TX, 1995), Ser. Approx. Decompos. 6 River Edge, NJ: World Sci. Publishing, 1995, pp. 89–112 Daubechies, I.: Orthonormal bases of compactly supported wavelets ii. SIAM J. Math. Anal. 24, 499–519 (1993) Eirola, T.: Sobolev characterization of solutions of dilation equations. SIAM J. Math. Anal. 23, 1015–103 (1992) Fan, A., Sun, Q.: Regularity of butterworth refinable functions. Asian J. Math. 5(3), 433– 440 (2001) Gohberg, I., Goldberg, S., Kaashoek, M.A.: Classes of linear operators. Vol 1, Basel: Birthauser Verlag, 1990 Halmos, P.: Introduction to Hilbert space and the theory of spectral multiplicity. Reprint of the Second (1957) edition, Providence, RI: AMS/Chelsea Publishing, 1998 Hervé, L.: Comportement asymptotique dans l’algorithme de transformée en ondelettes. Rev. Mat. Iberoamericana 11, 431–451 (1985) Hervé, L.: Construction et régularité des fonctions d’´chelle. SIAM J. Math. Anal. 26, 1361–1385 (1995) Jenkinson, O., Pollicott, M.: Computing invariant densities and metric entropy. Commun. Math. Phys. 211, 687–703 (2000) Jenkinson, O., Pollicott, M.: Orthonormal expansions of invariant densities for expanding maps. Adv. in Math. 192, 1–34 (2005) Ojanen, H.: Orthonormal compactly supported wavelets with optimal sobolev regularity. Applied and Computational Harmonic Analysis 10, 93–98 (2001) Oppenheim, A., Shaffer, R.: Digital Signal Processing. Upper Saddie River, NJ: Pearson Higher Education, 1986 Parry, W., Pollicott, M.: Zeta Functions and the Periodic Orbit Structures of Hyperbolic Dynamics. Astérisque 187–188 (1990) Pollicott, M.: Meromorphic extensions for generalized zeta functions. Invent. Math 85, 147– 164 (1986) Ruelle, D.: Thermodynamic Formalism. Reading, MA: Addison-Wesley, 1978 Ruelle, D.: An extension of the theory of fredholm determinants. Inst. Hautes Etudes Sci. Publ. Math. 72, 175–193 (1990) Simon, B.: Trace ideals and their applications. LMS Lecture Note Series, Vol 35, Cambridge: Cambridge University Press, 1979 Sun, Q.: Sobolev exponent estimate and asymptotic regularity of M band Daubechies scaling function. Constructive Approximation 15, 441–465 (1999) Tangerman, F.: Meromorphic continuation of Ruelle zeta functions. Ph.D. Thesis, Boston University, 1988 Villemoes, L.: Energy moments in time and frequency for two-scale difference equation solutions and wavelets. SIAM J. Math. Anal. 23, 1519–1543 (1992) Matlab, Wavelet Toolbox Documentation. www.mathworks.com/access/helpdesk/help/tool-box/ wavelet/

Communicated by J.L. Lebowitz

Commun. Math. Phys. 281, 23–127 (2008) Digital Object Identifier (DOI) 10.1007/s00220-008-0472-y

Communications in

Mathematical Physics

Noncommutative Finite Dimensional Manifolds II: Moduli Space and Structure of Noncommutative 3-Spheres Alain Connes1,2,3 , Michel Dubois-Violette4 1 2 3 4

Collège de France, 3, rue d’Ulm, Paris, F-75005 France. E-mail: [email protected] I.H.E.S., 35, route deChartres, F-91440 Bures-sur-Yuette, France Mathematics Department, Vanderbitt University, Nashvitte, TN 37235, USA Laboratoire de Physique Théorique, UMR 8627, Université Paris XI, Bâtiment 210, F-91 405 Orsay Cedex, France. E-mail: [email protected]

Received: 31 July 2006 / Accepted: 20 November 2007 Published online: 6 May 2008 – © Springer-Verlag 2008

Abstract: This paper contains detailed proofs of our results on the moduli space and the structure of noncommutative 3-spheres. We develop the notion of central quadratic form for quadratic algebras, and a general theory which creates a bridge between noncommutative differential geometry and its purely algebraic counterpart. It allows to construct a morphism from an involutive quadratic algebra to a C*-algebra constructed from the characteristic variety and the hermitian line bundle associated to the central quadratic form. We apply the general theory in the case of noncommutative 3-spheres and show that the above morphism corresponds to a natural ramified covering by a noncommutative 3-dimensional nilmanifold. We then compute the Jacobian of the ramified covering and obtain the answer as the product of a period (of an elliptic integral) by a rational function. We describe the real and complex moduli spaces of noncommutative 3-spheres, relate the real one to root systems and the complex one to the orbits of a birational cubic automorphism of three dimensional projective space. We classify the algebras and establish duality relations between them. Contents 1. 2.

3.

Introduction . . . . . . . . . . . . . . . . . . . . The Noncommutative 3-Spheres S 3 (
) ⊂ R4 (
) 2.1 Unitary “up to scale” . . . . . . . . . . . . . 2.2 Equation ch1/2 (U ) = 0 . . . . . . . . . . . 2.3 Noncommutative 3-spheres and 4-planes . . The Real Moduli Space M . . . . . . . . . . . . 3.1 M in terms of A3 . . . . . . . . . . . . . . 3.2 Trigonometric parameters ϕ of Sϕ3 . . . . . . 3.3 M in terms of D3 . . . . . . . . . . . . . . 3.4 Roots  = R(G, T) . . . . . . . . . . . . . 3.5 Singular hyperplanes Hα,n . . . . . . . . . .

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. . . . . . . . . . .

25 27 27 27 29 31 32 34 35 37 37

24

A. Connes, M. Dubois-Violette

3.6 Kernel of the exponential map: (T) (nodal group of T) . . . . . . . . 37 3.7 Group of nodal vectors N (G, T) ⊂ (T) . . . . . . . . . . . . . . . . 37 3.8 Affine Weyl group Wa . . . . . . . . . . . . . . . . . . . . . . . . . . 38 3.9 Affine Weyl group Wa . . . . . . . . . . . . . . . . . . . . . . . . . . 38 3.10 Alcoves and fundamental domain . . . . . . . . . . . . . . . . . . . . 38 4. The Flow F . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 4.1 Compatibility of F with the triangulation by alcoves . . . . . . . . . . 41 4.2 Invariance of R4ϕ under the flow F . . . . . . . . . . . . . . . . . . . . 42 5. The Geometric Data of R4ϕ . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 5.1 The definition and explicit matrices . . . . . . . . . . . . . . . . . . . 47 5.2 The table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 5.3 Generic case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 5.4 Face α = n and n even. F1 = {(ϕ
1 , ϕ1 , ϕ3 )} . . . . . . . . . . . . . . 51 5.5 Face α = n and n odd. F2 = { π2 , ϕ2 , ϕ3 } . . . . . . . . . . . . . . . 51
 5.6 Edge α ⊥ β and (n 1 , n 2 ) = (even, odd). L = { π2 , ϕ, ϕ } . . . . . . . 52
 5.7 Edge α − β and (n 1 , n 2 ) = (odd, odd) or (even, odd). L  = { π2 , π2 , ϕ } 53 5.8 Edge α − β and (n 1 , n 2 ) = (even, even). L  = {(ϕ, ϕ, ϕ)} . . . . . . . 53 5.9 Edge α ⊥ β and (n 1 , n 2 ) = (even, even). C+ =
{(ϕ, ϕ, 0)} . . . . . . . 53 5.10 Edge α ⊥ β and (n 1 , n 2 ) = (odd, odd). C− = { π2 + ϕ, π2 , ϕ } . . . . 54 5.11 Vertex P = ( π2 , π2 , π2 ) . . . . . . . . . . . . . . . . . . . . . . . . . . 54 5.12 Vertex P  = ( π2 , π2 , 0) . . . . . . . . . . . . . . . . . . . . . . . . . . 55 5.13 Vertex O = (0, 0, 0) . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 6. Isomorphism Classes of R4ϕ and Orbits of the Flow F . . . . . . . . . . . . 55 6.1 Proof in the non-generic case . . . . . . . . . . . . . . . . . . . . . . 55 6.2 Basic notations for elliptic curves . . . . . . . . . . . . . . . . . . . . 57 6.3 The generic case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 7. Dualities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 7.1 Semi-cross product. . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 7.2 Application to R4ϕ and Sϕ3 . . . . . . . . . . . . . . . . . . . . . . . . 65 8. The Algebras Calg (R4ϕ ) in the Nongeneric Cases . . . . . . . . . . . . . . . 67 8.1 Uq (sl(2)), Uq (su(2)) and their homogenized versions . . . . . . . . . 67 8.2 The algebras in the nongeneric cases . . . . . . . . . . . . . . . . . . 69 9. The Complex Moduli Space and its Net of Elliptic Curves . . . . . . . . . . 71 9.1 Complex moduli space MC . . . . . . . . . . . . . . . . . . . . . . . 73 9.2 Notations for ϑ-functions . . . . . . . . . . . . . . . . . . . . . . . . 75 9.3 Fiber = characteristic variety, and the birational automorphism σ of P3 (C) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 10. The Map from T2η × [0, τ ] to Sϕ3 and the Pairing . . . . . . . . . . . . . . . 82 10.1 Central elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 10.2 The Hochschild cycle ch3/2 (U ) . . . . . . . . . . . . . . . . . . . . . 84 10.3 Elliptic parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 10.4 The sphere Sϕ3 and the noncommutative torus T2η . . . . . . . . . . . . 88 10.5 Pairing with [T2η ] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 10.6 Simplifying the ∗-homomorphism ρ˜ . . . . . . . . . . . . . . . . . . . 93 11. Algebraic Geometry and C ∗ -Algebras . . . . . . . . . . . . . . . . . . . . . 96 11.1 Central quadratic forms and generalised cross-products . . . . . . . . . 96 11.2 Positive central quadratic forms on quadratic ∗-algebras . . . . . . . . 99 12. The Jacobian of the Covering of Sϕ3 . . . . . . . . . . . . . . . . . . . . . . 107

Noncommutative 3-Spheres

13. Calculus and Cyclic Cohomology . . . . 14. Appendix 1: The List of Minors . . . . . 15. Appendix 2: The Sixteen Theta Relations References . . . . . . . . . . . . . . . . . . .

25

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113 122 124 126

1. Introduction This paper contains detailed proofs of our results on the moduli space and the structure of noncommutative 3-spheres announced in [17]. Through the analysis of a specific class of noncommutative manifolds which arose as solutions of a simple equation of K -theoretic origin we discovered rather general structures which lie at the intersection of two fundamental aspects of noncommutative geometry, namely • Differential Geometry • Algebraic Geometry. This class of noncommutative manifolds, called noncommutative 3-spheres, has a very rich structure both at the level of the objects themselves as well as at the level of the moduli space which parameterizes these geometric objects. There are two aspects in the geometry of the moduli space: • The real moduli space and its scaling foliation, its link with the alcove structure of the root system D3 and the Morse theory of the character of the signature representation. • The complex moduli space and its net of elliptic curves, its link with the iteration of a cubic transformation of P3 (C). At the level of the structure of the noncommutative 3-spheres our main result is to relate them to very well understood noncommutative nilmanifolds which fall under the framework of the early theory developed in [10] and were analysed in great detail in [1,2]. The core of the paper is to construct the corresponding map of noncommutative spaces and compute its Jacobian. The essence of the work is to extract from very complicated computations the general concepts that allow not only to understand what is going on but also to extend the construction in full generality. Thus at center stage lies the computation of the Jacobian and the gradual simplification of the result which at first was expressed in terms of elliptic functions and the 9th power of the Dedekind η function. We shall reach at the end of the paper (Theorem 13.8) a result of utmost simplicity while the starting point was a computation performed even in the trigonometric case with the help of a computer.1 While we reach a reasonable level of conceptual understanding of the general construction of the map, we believe that a lot remains to be discovered for the abstract construction of the calculus as well as for the general computation of the Jacobian, let alone in the case of higher dimensional spheres which we do not address here. After recalling the basic definitions and properties of the noncommutative 3-spheres in Sect. 2 we analyse the real moduli space in Sect. 3 and exhibit a fundamental domain in terms of alcoves of the root system D3 . In Sect. 4 we define the scaling foliation and show its compatibility with the alcove structure of the real moduli space. We also show that the isomorphism class of the 1 We are grateful to Michael Trott for his help.

26

A. Connes, M. Dubois-Violette

4-spaces R4 (
) remains constant on the leaves of the foliation. In order to prove the converse, i.e. that isomorphism of the 4-spaces R4 (
) implies equality of the leaves we compute in detail in Sect. 5 the geometric data of the quadratic algebra of R4 (
). This allows us to finish the proof of the converse in Sect. 6. We exhibit in Sect. 7 more subtle relations between the 4-spaces R4 (
) given by dualities. At the level of the algebras these are obtained from the general notion of semicross product of graded algebras, a notion used and analysed in [5] where it is referred to as twisting. At the level of the moduli space these dualities shrink the fundamental domain further and that amounts essentially to the transition from the root system D3 to the larger one C3 . We then use these to describe the 4-spaces R4 (
) for degenerate values of the parameter in Sect. 8. In Sect. 9 we analyse the complex moduli space which appears naturally as a net of elliptic curves having eight points in common in P3 (C). We show that in the generic case these elliptic curves are the characteristic varieties of the algebras that their points label. Moreover the canonical correspondence σ is simply the restriction of a globally defined cubic map of P3 (C). This gives in particular a very natural choice of generators for the algebra. As a preparation for the next section we give the natural parameterization of the net of elliptic curves in terms of ϑ-functions. Section 10 is the most technical one and contains the root of the concepts developed in full generality in Sect. 11. In essence what we do first is, starting from unitary representations of the Sklyanin algebra constructed by Sklyanin in his original paper, we derive a one parameter family of ∗-homorphisms from the algebras of R4 (
) in the generic case, to the algebras of noncommutative tori. We then use a suitable restriction to the 3-spheres S 3 (
). After a lot of work on this construction we find that we can eliminate all occurrences of ϑ-functions from the formulas and obtain a purely algebraic formulation of the construction as a morphism to a twisted cross product C ∗ -algebra obtained from the geometric data. Section 11 describes the abstract general construction in the framework of involutive quadratic algebras. The key notions are those of central quadratic form and of positivity for such forms. It is in that section that the interaction between the two above aspects of noncommutative geometry is manifest. In fact we construct a bridge between the purely algebraic notions such as the geometric data of a quadratic algebra and the world of noncommutative geometry including the topological (C ∗ -algebraic) and differential geometric aspects (in the sense of [10,11]). At one end of the bridge one starts with the given involutive quadratic algebra. At the other one has the C ∗ -algebra obtained as the twisted cross product of the charateristic variety by the canonical correspondence. The twisting is effected by an hermitian line bundle and the construction is a special case of a general one due to M. Pimsner. The bridge provides a ∗-algebra morphism. This algebra morphism has a “trivial part” which does not make use of the central quadratic form and lands in a “triangular” subalgebra of the C ∗ -algebra. This part was well-known for quite some time to noncommutative algebraic geometors. The non-triviality of our construction lies in the involved relations coming from the cross terms mixing generators with their adjoints. We compute in Sect. 12 the Jacobian of the above map. We first define what we mean by the jacobian in the sense of noncommutative geometry where Hochschild homology replaces differential forms. The algebraic form of the result then suggests the existence of a calculus of purely algebraic nature allowing to express the cyclic cohomology fundamental class in terms of the algebraic geometry of the characteristic variety and a hermitian structure on the canonical line bundle. This is achieved in Sect. 13 and allows

Noncommutative 3-Spheres

27

us to finally obtain the purest form of the computation of the jacobian in the already mentioned Theorem 13.8. Needless to say this is a paper of highly “computational” nature and we tried to ease the reading by supplying in an appendix the basic factorisations of the minors and the sixteen theta relations which are often used in the text. 2. The Noncommutative 3-Spheres S3 (
) ⊂ R4 (
) We shall recall in this section the basic properties of the noncommutative spheres S 3 (
) and the corresponding 4-spaces R4 (
). 2.1. Unitary “up to scale”. We let A be a unital involutive algebra and first start with a unitary “up to scale” in Mq (A), i.e. U ∈ Mq (A), UU ∗ = U ∗ U ∈ A ⊗ ? ⊂ A ⊗ Mq (C).

(2.1)

Lemma 2.1. Let U ∈ Mq (A) satisfy (2.1) with UU ∗ = U ∗ U = C ⊗ ? ∈ A ⊗ Mq (C), then C is in the center of the ∗-algebra generated by the matrix elements of U . Proof. One has (C ⊗ ?)U = (UU ∗ )U = U (U ∗ U ) = U (C ⊗ ?) and (C ⊗ ?)U ∗ = (U ∗ U )U ∗ = U ∗ (UU ∗ ) = U ∗ (C ⊗ ?) by associativity in Mq (A), which implies the result.   Let

  τµ , µ ∈ 0, . . . , q 2 − 1

(2.2)

be an orthonormal basis of Mq (C) for the scalar product A|B = q1 Trace(A∗ B). Then one has U = τµ z µ , z λ ∈ A, (2.3) where we used the Einstein summation convention on “up-down indices”. The ∗-algebra generated by the matrix elements of U is the ∗-subalgebra of A generated by the z λ , z µ∗ (λ, µ ∈ {0, . . . , q 2 − 1}) and if U is as in the above lemma then one has the equalities   C= z µ z µ∗ = z µ∗ z µ (2.4) µ

µ

for the central element C of the ∗-algebra generated by the z µ . 2.2. Equation ch1/2 (U ) = 0. We now turn to the relation ch1/2 (U ) = 0, in the unreduced complex i.e. using the convention of summation on repeated indices, Uii10 ⊗ (U ∗ )ii10 − (U ∗ )ii01 ⊗ Uii01 = 0,

(2.5)

where the left-hand side belongs to the tensor square A⊗2 = A ⊗ A. Both terms in the left-hand side give sums of the q 2 terms of the form (z µ ⊗ z ν∗ ) (τµ )ii01 (τν∗ )ii10 and similarly for the other. The sum on i 0 , i 1 is thus simply Trace (τµ τν∗ ), i.e. the Hilbert Schmidt inner product (τµ , τν ) = q τν |τµ . This is 0 unless µ = ν and is q if µ = ν. Thus, up to an overall factor of q the equality (2.5) means:  (z µ ⊗ z µ∗ − z µ∗ ⊗ z µ ) = 0. (2.6)

28

A. Connes, M. Dubois-Violette

Lemma 2.2. Equation (2.6) holds iff there exists a unitary symmetric matrix
∈ Mq 2 (C) such that: ν z µ∗ =
µ ν z .

(2.7)

Proof. Let us first assume that the z µ are linearly independent elements of A. Let then ϕµ be linear forms on A with ϕµ (z ν ) = δµν . Applying 1 ⊗ ϕµ to (2.6) we get z µ∗ =



z ν ϕµ (z ν∗ ) =



ν
µ ν z ,

where the matrix
is uniquely prescribed by this relation. Then since the z µ ⊗ z ν are linearly independent in A ⊗ A, the relation (2.6) means, looking at the coefficient of z µ ⊗ z ν on both sides, ν
µ ν =
µ ,

(2.8)

so that the matrix
is symmetric. µ Taking the adjoint of both sides in z µ∗ =
ν z ν one gets ν∗ ν ρ ∗ µ ρ ¯µ ¯µ zµ =
=
ν z ν
ρ z = (

)ρ z

and the linear independence of the z ρ thus shows that:
∗
= 1.

(2.9)

For the general case2 note that Eq. (2.6) is invariant by the transformation z˜ µ = Wνµ z ν , where W ∈ U (q 2 ) is a unitary matrix. Moreover (2.7) implies a similar equation for (˜z µ ) with the matrix ˜ = W¯
W¯ t ,
which is still symmetric and unitary. This allows to assume that the kernel of the linear 2 map from Cq to A determined by eν → z ν is the linear span of a subset I of the basis ν vectors e . In other words the non-zero z ν are linearly independent and the above proof ensures the existence of a matrix fulfilling (2.7) which is symmetric and unitary once extended by the identity on the eν , ν ∈ I .   As pointed out in [15] and as will be explained in Part III, for the study of (2n + 1)dimensional spherical manifolds one should take q = 2n to be coherent in particular with the suspension functor. In the following we shall concentrate on the 3-dimensional case (and the corresponding noncommutative R4 ) which is the lowest dimensional nontrivial case from the noncommutative side and for which ch1/2 (U ) = 0 is the only K -homological condition. Accordingly we take q = 2 in the following. 2 The simplification of the argument of [15] given here is due to G. Skandalis.

Noncommutative 3-Spheres

29

2.3. Noncommutative 3-spheres and 4-planes. We now turn to the case q = 2 and we choose as orthonormal basis of M2 (C) the basis τ0 = ? and τk = iσk , k ∈ {1, 2, 3}, where the σk are the Pauli matrices, i.e.    0 1 0 , σ2 = σ1 = 1 0 i

  −i 1 , σ3 = 0 0

(2.10)

0 −1

 (2.11)

which satisfy σ j∗ = σ j , σ j2 = 1 and σk σ = i εkm σm for any permutation (k, , m) of (1, 2, 3), where ε123 = 1 and ε is totally antisymmetric. This allows to write U = z 0 + i σ1 z 1 + i σ2 z 2 + i σ3 z 3 ,

z µ ∈ A,

(2.12)

and one has U ∗ = z 0∗ − i σ1 z 1∗ − i σ2 z 2∗ − i σ3 z 3∗ . Thus UU ∗ = z 0 z 0∗ + z 1 z 1∗ + z 2 z 2∗ + z 3 z 3∗ + i σ1 (z 1 z 0∗ − z 0 z 1∗ + z 2 z 3∗ − z 3 z 2∗ ) + i σ2 (z 2 z 0∗ − z 0 z 2∗ + z 3 z 1∗ − z 1 z 3∗ ) + i σ3 (z 3 z 0∗ − z 0 z 3∗ + z 1 z 2∗ − z 2 z 1∗ ). Similarly we get, U ∗ U = z 0∗ z 0 + z 1∗ z 1 + z 2∗ z 2 + z 3∗ z 3 + i σ1 (z 0∗ z 1 − z 1∗ z 0 + z 2∗ z 3 − z 3∗ z 2 ) + i σ2 (z 0∗ z 2 − z 2∗ z 0 + z 3∗ z 1 − z 1∗ z 3 ) + i σ3 (z 0∗ z 3 − z 3∗ z 0 + z 1∗ z 2 − z 2∗ z 1 ). Thus Eq. (2.1) is equivalent to 7 relations which are,   z µ∗ z µ , z µ z µ∗ =  εkm z  z m∗ = 0, z k z 0∗ − z 0 z k∗ +  εkm z ∗ z m = 0. z 0∗ z k − z k∗ z 0 +

(2.13) (2.14) (2.15)

We then let S be the space of unitary symmetric matrices, S = {
∈ M4 (C) ;

∗ =
∗
= 1,
t =
}.

(2.16)

We define for
∈ S the algebra Calg (R4 (
)) of coordinates on R4 (
) as the algebra generated by the z µ , z µ∗ with the relations (2.14), (2.15) together with, ν z µ∗ =
µ ν z .

(2.17)

Note that (2.13) follows automatically from (2.17). For S 3 (
) we add the inhomogeneous relation,  z µ∗ z µ = 1. (2.18)  µ∗ µ By Lemma 2.1 the element C = z z is in the center of the involutive algebra Calg (R4 (
)).

30

A. Connes, M. Dubois-Violette

Theorem 2.3. Let A be a unital involutive algebra and U ∈ M2 (A) a unitary such that ch1/2 (U ) = 0. Then there exists
∈ S and a homomorphism ϕ : Calg (S 3 (
)) → A such that: U = ϕ(τµ z µ ). Conversely for any
the unitary U = τµ z µ ∈ M2 (Calg (S 3 (
))) fulfills ch1/2 (U ) = 0. By construction we thus obtain involutive algebras parametrized by
∈ S. They are endowed with a canonical element of H3 (A, A) (in fact of Z 3 ) given by

S 3 (
) = ch3/2 (U ) = Uii10 ⊗Ui∗i2 1 ⊗Uii32 ⊗Ui∗i0 3 −Ui∗i1 0 ⊗Uii21 ⊗Ui∗i3 2 ⊗Uii03. (2.19) The operations U → λ U , U → V1 U V2 , U → U ∗ , for λ ∈ U (1) and V j ∈ SU (2) together with the universality described in Theorem 2.3 give natural isomorphisms between the S 3 (
) as follows, Proposition 2.4. The following define isomorphisms S 3 (
) → S 3 (
 ) (resp. R4 (
) → R4 (
 )) preserving [S 3 ] (resp. [R4 ]) in the first two cases and changing its sign in the last case: 1) For λ ∈ U (1),
 = λ2
, one lets µ

µ

ϕ(z
 ) = λ−1 z
. 2) For V ∈ S O(4),
 = V
V t , one lets µ

ν . ϕ(z
 ) = Vνµ z


3) For
 = ε
−1 ε, one lets µ

µ∗

ϕ(z
 ) = εµ z
, ε0 = 1, εk = −1, εµν = 0 µ = ν. Proof. 1) Let U = U
be the unitary in M2 (Calg (S 3 (
)). Then λ U still fulfills ch1/2 (λ U ) = 0 and ch3/2 (λ U ) = ch3/2 (U ). With z˜ µ = λ z µ one has (˜z µ )∗ = λ¯ z µ∗ = µ µ λ¯
ν z ν = λ¯ 2
ν z˜ ν . This shows that ϕ is a homomorphism ϕ : Calg (S 3 (
 )) → Calg (S 3 (
)) and that ϕ([S 3 (
 )]) = [S 3 (
)]. 2) One has Spin(4) = SU (2) × SU (2) and the covering map π : Spin(4) = SU (2) × SU (2) → S O(4) is given for any (u, v) ∈ SU (2) × SU (2) and ξ ∈ R4 by π(u, v)ξ = η with τµ ηµ = u (τµ ξ µ ) v ∗ .

(2.20) µ

This equality continues to hold for the natural complex linear extension Vν z ν of V = π(u, v) to C4 and it follows that with the notations of assertion 2) one has ν = u U v∗ τµ Vνµ z


with U = U
as above. The unitary u U v ∗ still fulfills ch1/2 (u U v ∗ ) = 0 and ch3/2 (u U v ∗ ) = ch3/2 (U ) since in M2 (C) it is the ordinary product and trace which are

Noncommutative 3-Spheres

31 µ

involved in the formulas for chk/2 . Thus, with z˜ µ = Vν z ν we just have to check that µ

z˜ µ∗ =
ν z˜ ν . One has
 z˜ = V
V t V z = V
z = V z ∗ , and since V has real coefficients this is (V z)∗ = (˜z )∗ . 3) Let U = U
as above, then U ∗ is still unitary and fulfills ch1/2 (U ∗ ) = 0, ch3/2 (U ∗ ) = − ch3/2 (U ). It corresponds to z˜ 0 = z 0∗ , z˜ k = −z k∗ . Then z˜ µ∗ = εµ z µ = µ µ εµ (
−1 )ν z ν∗ = εµ (
−1 )ν εν z˜ ν which gives the value of
 .   Corollary 2.5. For every
∈ S there exists ϕ j ∈ R/ π Z and isomorphisms S 3 (
) → Sϕ3 (resp. R4 (
) → R4ϕ ) where Sϕ3 corresponds to the diagonal matrix   1 0 .
(ϕ) = 0 e−2iϕk Proof. We just need to recall why the matrix
can be diagonalized by a W ∈ S O(4). In fact
is unitary and fulfills
t =
i.e.
∗ = J
J −1 , where J gives the real structure of C4 . An eigenspace E λ = {ξ ∈ C4 ;
ξ = λ ξ } is stable by J since
J −1 ξ = J −1
∗ ξ = J −1 λ¯ ξ = λ J −1 ξ (and J −1 = J ). Thus we can find an orthonormal basis of real vectors: J ξi = ξi which are eigenvectors for
. With ei the standard basis of C4 the map ei → ξi gives an element of O(4) and we can take it in S O(4). Thus
= W D W t with W ∈ S O(4).   The presentation of the algebra of R4ϕ is given by (2.14) (2.15) and the relation (2.7) with
=
(ϕ). This gives z 0∗ = z 0 and z k∗ = e−2iϕk z k so that with x 0 = z 0 , x k = e−iϕk z k we get x µ∗ = x µ , ∀µ ∈ {0, 1, 2, 3}, (2.21) and the six other relations give eiϕk x k x 0 − e−iϕk x 0 x k + eiϕk x 0 x k − e−iϕk xk x 0 +

 

εkm ei(ϕ −ϕm ) x  x m = 0,

(2.22)

εkm e−i(ϕ −ϕm ) x  x m = 0.

(2.23)

This gives, by combining (2.22) and (2.23) the relations sin(ϕk ) [x 0 , x k ]+ = i cos(ϕ − ϕm ) [x  , x m ], cos(ϕk ) [x 0 , x k ] = i sin(ϕ − ϕm ) [x  , x m ]+ ,

(2.24) (2.25)

where we let [a, b]+ = a b + b a be the anticommutator. We shall now study in much greater detail the corresponding moduli space. 3. The Real Moduli Space M We let M be the moduli space of noncommutative 3-spheres. It is obtained from Proposition 2.4 1) and 2) as the quotient M = (U (1) × S O(4))\S,

(3.1)

of S by the action of U (1) × S O(4). This action of U (1) × S O(4) on S is the restriction of the following action of U (4) on S:
∈ S → W
W t , ∀ W ∈ U (4),

(3.2)

32

A. Connes, M. Dubois-Violette

which allows to identify S with the homogeneous space U (4)/O(4)  S.

(3.3)

(Note that any
∈ S can be written as
= V V t for some V ∈ U (4) since it can be diagonalized by an orthogonal matrix.) The presence of U (1) in (3.1) allows to reduce to SU (4) and one obtains this way a first convenient description of M. 3.1. M in terms of A3 . The description of M in terms of the compact Lie group SU (4) is given by: Proposition 3.1. 1) Let N be the normalizer of S O(4) in SU (4). Then one has a canonical isomorphism M  S O(4)\SU (4)/N . (3.4) 2) Let T ⊂ SU (4) be the maximal torus of diagonal matrices, and W ⊂ Aut(T) the corresponding Weyl group. Let D = T ∩ N . Then the above restricts to an isomorphism M  W \T/D. (3.5) 3) The map u → u 2 from T to T induces an isomorphism M  W \T/D  Space of Conjugacy Classes in P SU (4).

(3.6)

Proof. 1) Let Z be the center of SU (4), it is generated by i which has order 4 and contains −1 ∈ S O(4). The normalizer N is generated by S O(4) and the element ⎡

−1

⎢ v= w⎣

0 1 1

0

⎤ ⎥ ⎦,

w = e2πi/8 ,

(3.7)

1

which implements the outer automorphism of S O(4) and whose square v 2 = i generates Z. Let X = U (1)\S considered as an homogeneous space on SU (4) using the action (3.2), i.e.
∈ X → W
W t . Given
∈ S we can find λ ∈ U (1) so that
= λ
1 with Det
1 = 1, thus with S1 = {
∈ S, Det
= 1} one has, X = U (1)\S  w Z \S1  SU (4)/N .

(3.8)

Indeed the first equality follows since the action (3.2) of w is multiplication by i and the second follows by computing the isotropy group K of 1 ∈ wZ \S1 . One has S O(4) ⊂ K and v ∈ K since v t v = v 2 = w 2 . Thus N ⊂ K . Conversely given V ∈ K one has V V t = i N for some N ∈ Z and thus for a suitable power of v one has v k V V t v k = 1, thus v k V ∈ S O(4). This shows that K = N and we get the first statement of the proposition since M  S O(4)\X by construction. Note the standard description of N which is given as follows. One lets θ ∈ Aut(SU (4)) be given by complex conjugation, θ (u) = u¯ = J u J −1

∀ u ∈ SU (4).

(3.9)

Noncommutative 3-Spheres

33

One has θ 2 = 1 and the fixed points of θ give S O(4) = SU (4)θ . The normalizer N of S O(4) is characterized by, u −1 θ (u) ∈ Z , (3.10) where Z is the center of SU (4). 2) Let σ be the map σ (u) = u u t from SU (4) to the space PS1 of classes of elements of S1 modulo the action of Z by multiplication. It follows from 1) that σ is an isomorphism of X = SU (4)/N with PS1 . Given u ∈ SU (4) we can find V ∈ S O(4) such that u ut = V D V t , (3.11) where D is a diagonal matrix. Then σ (u) = σ (V D 1/2 ), where D 1/2 is a diagonal square root of D with determinant equal to 1. Thus every element of the coset space S O(4)\X can be represented by a diagonal matrix, and the natural map given by inclusion W \T/D → S O(4)\X

(3.12)

is surjective. 3) When restricted to T the map σ is simply the squaring u → u 2 . Moreover the equality σ (u 1 ) = σ (u 2 ) in PS1 for u j ∈ T just means that the u 2j define the same element of the maximal torus T/Z of P SU (4). Thus the result follows. Let us check that the group D/Z is (Z/2)3 . The elements of D are the v k u with v as in (3.7) and u in T ∩ S O(4). One has v 2 ∈ Z and modulo Z ∩ S O(4) = ±1 the elements of T ∩ S O(4) form the Klein group H = (Z/2Z)2 . Thus one gets D/Z  (Z/2Z)3 .   We identify the Lie algebra of SU (4) with the Lie algebra of antihermitian matrices with trace 0, Lie (SU (4)) = {T ∈ M4 (C) ; T ∗ = −T, Trace T = 0},

(3.13)

where θ is still acting by complex conjugation. The diagonal matrices D ∈ D form a maximal abelian Lie subalgebra of the eigenspace, (3.14) Lie (SU (4))− = {T, θ (T ) = −T }. The roots α ∈  are given by, αµ,ν (δ) = δµ − δν .

(3.15)

Proposition 3.2. For δ ∈ D one has, eδ ∈ N ⇔ e2δ ∈ Z ⇔ αµ,ν (δ) ∈ i π Z, ∀µ, ν.

(3.16)

Proof. The equivalence between the last two conditions is a general fact for compact Lie groups (cf. [9]). The equivalence between the first two conditions follows from the third statement of Proposition 3.1.   We let  be the lattice  ⊂ D determined by the equivalent conditions (3.16), and T A = D/  be the quotient 3-dimensional torus.

(3.17)

34

A. Connes, M. Dubois-Violette

We let the group W of permutations of 4 elements act on T A by permutations of the δµ . In fact we view it as the Weyl group of the pair (SU (4), N ), i.e. as the quotient, W = N /C

(3.18)

of the normalizer of D in S O(4) by the centralizer of D. Note that v being diagonal is in the centralizer of D so that W does not change in replacing S O(4) by N since v k u normalizes D iff u does. Corollary 3.3. The map σ (δ) = e2δ defines an isomorphism of the quotient of T A by the action of W with the moduli space M = (U (1) × S O(4))\S. Proof. This is just another way to write the second statement of Proposition 3.1.

 

3.2. Trigonometric parameters ϕ of Sϕ3 . We shall now describe a convenient parametrization of the torus T A which gives the corresponding algebras in the form of Corollary 2.5. It is given by the map ϕ = (ϕ1 , ϕ2 , ϕ3 ) ∈ (R/π Z)3 → d(ϕ) = (α0 , α0 − i ϕ1 , α0 − i ϕ2 , α0 − i ϕ3 ), i  ϕj. (3.19) α0 = 4 One has α0,k (d(ϕ)) = i ϕk and the definition of  shows that d is an isomorphism. Also 

e

2d(ϕ)

1  0

0



e−2iϕk

up to multiplication by a scalar. In terms of the parameters ϕ j the twelve roots α ∈  are the following linear forms, (ϕ1 , ϕ2 , ϕ3 ) → i {±ϕ j , ϕk − ϕl }.

(3.20)

The action of the Weyl group W gives the following linear transformations of the ϕ j . Arbitrary permutations of the ϕ j ’s correspond to permutations of the last three αk ’s. The transposition of α0 with α1 corresponds to : T01 (ϕ1 , ϕ2 , ϕ3 ) = (−ϕ1 , ϕ2 − ϕ1 , ϕ3 − ϕ1 ).

(3.21)

The 3-spheres Sϕ3 are parametrized by ϕ = (ϕ1 , ϕ2 , ϕ3 ) ∈ (R/π Z)3

(3.22)

modulo the action of the Weyl group W  S4 generated by the permutation group S3 of the ϕ j ’s and (ϕ j ) → (− ϕ1 , ϕ3 − ϕ1 , ϕ2 − ϕ1 ) = (ϕ j ).

(3.23)

Noncommutative 3-Spheres

35

3.3. M in terms of D3 . To obtain another very convenient parametrization of the torus T A we use the isomorphism A3 ∼ D3 , i.e. SU (4)  Spin (6),

(3.24)

which simply comes from the spin representation of Spin (6). The Clifford algebra Cliff C (R6 ) has dimension 26 and is a matrix algebra Mn (C) with n = 23 = 8. We let γ µ be the corresponding γ -matrices, with γµ∗ = γµ , γµ γν + γν γµ = 2 δµ,ν . We then let σµν =

1 (γµ γν − γν γµ ) 2

which span a real subspace of the Clifford algebra stable under bracket and isomorphic (up to a factor of 2) to the Lie algebra of S O(6) of real antisymmetric 6 by 6 matrices. Proposition 3.4. 1) The following gives a parametrization τ of a maximal torus in Spin(6),  τ θ = (θ j ) ∈ (R/ 2π Z)3 −→ Exp θ j σ2 j−1,2 j ∈ Spin(6) ⊂ Cliff C (R6 ). (3.25) 2) The half Spin representation gives an isomorphism π : Spin(6) → SU (4). 3) One has π(τ (θ )) = eδθ with δθ diagonal given by δθ = i (θ1 + θ2 + θ3 , θ1 − θ2 − θ3 , − θ1 + θ2 − θ3 , − θ1 − θ2 + θ3 ).

(3.26)

Proof. This is a straightforward check, since in the half Spin representation π one has in a suitable basis ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ i 0 i 0 i 0 i −i −i ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ σ12 → ⎣ ⎦, σ34 → ⎣ ⎦, σ56 → ⎣ ⎦, −i i −i 0 −i 0 −i 0 i (3.27) thus π(τ (θ )) = eδθ with δθ given by (3.26).   Thus the trigonometric parameters ϕk are given in terms of the θ ’s by, ϕ1 = 2 (θ2 + θ3 ), ϕ2 = 2 (θ1 + θ3 ), ϕ3 = 2 (θ1 + θ2 ).

(3.28)

The natural parameters for the maximal torus T D of S O(6) are the ψ j = 2 θ j and are defined modulo 2 π Z, i.e. correspond to the Lie algebra element (ψ) = ψ1 β12 + ψ2 β34 + ψ3 β56 ,

(3.29)

where the βi j form the canonical basis of real antisymmetric matrices. The kernel of the covering Spin (6) → S O(6) corresponds to θ j = π so that for S O(6) the torus T D is parametrized by the ψ j ’s defined modulo 2π by ψ ∈ (R/2 π Z)3 → e(ψ) .

(3.30)

36

A. Connes, M. Dubois-Violette

The transition from ψ to ϕ’s is given then by, ϕ j = ψk + ψ , 2 ψ j = ϕk + ϕ − ϕ j ,

(3.31)

as well as ϕ1 − ϕ2 = ψ2 − ψ1 , ϕ2 − ϕ3 = ψ3 − ψ2 , ϕ3 − ϕ1 = ψ1 − ψ3 .

(3.32)

We shall now spellout in great detail the basic Lie group data for D3 and get a description of the moduli space M in these terms. Proposition 3.1 3) gives a natural isomorphism M  Space of Conjugacy Classes in P SU (4) of the moduli space M with the space of conjugacy classes of elements of P SU (4)  P S O(6). The general theory of compact Lie groups provides a natural triangulation of such a space of conjugacy classes in terms of alcoves. The latter are obtained as the connected components of the complement of the union of the singular hyperplanes. Our aim in this section is to describe such a triangulation in our specific case and to exhibit the role of the singular hyperplanes. We shall see in the next section their natural compatibility with the scaling foliation. In terms of the parameters ϕ j of the 3-spheres Sϕ3 the relations which specify the non-generic situations, are all of the form  π ϕ, α(ϕ) = n = G α,n , (3.33) 2 where n is an integer and α is one of the twelve “roots” i.e. α ∈  = ±{ϕ1 , ϕ2 , ϕ3 , ϕ1 − ϕ2 , ϕ2 − ϕ3 , ϕ3 − ϕ1 }. Moreover the periodicity lattice of ϕ is (π Z)3 which is specified by {ϕ, α(ϕ) ∈ π Z, ∀ α ∈ } = ϕ .

(3.34)

We now want to relate more precisely the above situation with canonical objects (root systems, alcoves, chambers, affine Weyl group, nodal vectors . . .) associated to the following data (G, T): G = P S O(6), T = Maximal torus T D / ± 1.

(3.35)

We use the natural parametrization of the Lie algebra Lie (T), (ξ ) = ξ1 β12 + ξ2 β34 + ξ3 β56 .

(3.36)

Since we used the “squaring” u → u 2 in the isomorphism of Proposition 3.1 3), the parameter ψ that appears in the transition ϕ → ψ of Eq. (3.31) is related to ξ by ξ = 2 ψ.

(3.37)

In other words the natural relation between the parameters ϕ and ξ is 2 ϕ j = ξk + ξ , ξ j = ϕk + ϕ − ϕ j .

(3.38)

This accounts for a factor of 21 in (3.33) but does not yet relate it to the equation of singular hyperplanes. To understand this relation more precisely we shall now review briefly the standard ingredients of the theory of alcoves for the specific data (G, T).

Noncommutative 3-Spheres

37

3.4. Roots  = R(G, T). They are by definition the linear forms α on Lie (T) given by eigenvalues X α ∈ Lie G C which fulfill: [ξ, X α ] = α(ξ ) X α

∀ ξ ∈ Lie T ⊂ Lie G.

(3.39)

They are complex valued as defined. In our case they are given by (up to multiplication by i) (± eµ ± eν ) ξ = ± ξµ ± ξν . (3.40) 3.5. Singular hyperplanes Hα,n . They are given by a root α ∈  and n ∈ Z, with Hα,n = {ξ, α(ξ ) = 2πin}.

(3.41)

As we shall explain below while these hyperplanes suffice to obtain a triangulation of the space of conjugacy classes of the simply connected covering of G, we shall need the additional ones with n ∈ 21 Z to describe the space of conjugacy classes in G itself. 3.6. Kernel of the exponential map: (T) (nodal group of T). In our case we are dealing with G = P S O(6) and thus for ξ ∈ Lie T, eξ is 1 in G iff Ad(eξ ) = 1 since the center of G is C(G) = {1}. This means exactly that all eigenvalues of ad(ξ ) belong to Ker (exp) = 2πiZ, thus (T) = {ξ, α(ξ ) ∈ 2πiZ ∀ α ∈ }.

(3.42)

Thus, after applying the change of variables (3.38) ϕ  (T)

(3.43)

which shows that the periodicity lattice we have is the nodal group of T. 3.7. Group of nodal vectors N (G, T) ⊂ (T). This group can be defined in terms of vectors K α which are associated to the roots α ∈  but in our case it is simpler to use the definition as follows: N (G, T) = Kernel of exp : Lie T → G˜ = Universal cover of G.

(3.44)

In our case G˜ = Spin 6 and in terms of γ -matrices the exponential map takes the form,    1 1 1 1 (ξ ) → cos ξ1 + sin ξ1 γ1 γ2 cos ξ2 + sin ξ2 γ3 γ4 2 2 2 2   1 1 (3.45) × cos ξ3 + sin ξ3 γ5 γ6 2 2 whose kernel is given by N (G, T) = {ξ ; ξ j = 2π n j , n j ∈ Z, n j even}.

(3.46)

One has N (G, T) ⊂ (T) and the quotient, of order 4, is generated by ξ = (π, π, π ) ∈ (T).

38

A. Connes, M. Dubois-Violette

3.8. Affine Weyl group Wa . The affine Weyl group Wa is the group generated by the reflexions associated to the hyperplane Hα,n for α ∈  and n ∈ Z. One has [8] (Chapter VI, Prop. 1, p. 173) Wa = N (G, T)
W,

(3.47)

where the Weyl group is generated by the reflexions associated to singular hyperplanes Hα,0 . In our case W = S4 and all its elements are of the form  W = ε σ, σ ∈ S3 , ε = (εi ), εi ∈ ±1, εi = 1, (3.48) where the action on ξ is by permutation of the ξ j for σ (careful that (σ ξ )(i) = ξ(σ −1 (i)) to get a covariant action) and by multiplication by εi for ε. For N (G, T) we can check that the αˇ = K α are simply given by the vectors ± eµ ± eν which correspond to (± 2π, ± 2π, 0) = K ±e1 ± e2 , K α , α = 2.

(3.49)

3.9. Affine Weyl group Wa . It is by definition the semi-direct product, Wa = (T)
W.

(3.50)

What matters is that it still acts on the set of singular hyperplanes. For γ ∈ (T) one has α(h + γ ) = α(h) + α(γ ) and α(γ ) ∈ 2πiZ thus one is just shifting the n in Hα,n . From [9] Prop. 2 (Chapter 9, p. 45) Wa is a normal subgroup of Wa . 3.10. Alcoves and fundamental domain. The alcoves are the connected components of (U Hα,n )c ⊂ Lie T. The chambers are the components of (U Hα,0 )c . By construction the alcoves are intersections of half spaces and are thus convex polyhedra. By [9] the affine Weyl group Wa acts simply transitively on  = the set of alcoves. Since Wa is still acting on  we can identify  with the homogeneous space  = Wa /H X ,

(3.51)

where H X is the finite isotropy group of an alcove X . In our case, we take the following alcove : X = {ξ, ξ1 + ξ2
0, ξ2 − ξ1
0, ξ3 − ξ2
0, ξ2 + ξ3  2π }.

(3.52)

It is a tetrahedron with all 4-faces congruent but 2 long edges and 4 short edges. 

Lemma 3.5. The isotropy subgroup H X ⊂ Wa of X is generated by w1 = 21 , 21 , 21 ,  ε12 σ13 . Proof. For convenience we rescale the ξ j by 2π . The coordinates of the vertices of X are 

then 0, p = 21 , 21 , 21 , q = − 21 , 21 , 21 , p  = (0, 0, 1). One has w1 (0) = p, w1 ( p) = 
p+ε12 σ13 ( p) = p+ − 21 , − 21 , 21 = p  , w1 ( p  ) = p+ε12 σ13 ( p  ) = p+(−1, 0, 0) = q, 
1 1 1
1  w1 (q) = p + ε12 σ13 (q) = 2 , 2 , 2 + − 2 , − 21 , − 21 = 0. 

Noncommutative 3-Spheres

39

Fig. 1. Tiling of the alcove X by 21 -alcoves

P’ Z

Q P

0 Fig. 2. Fundamental domain

We let σ be the orthogonal reflexion around the face ( pqp  ) of X . One has σ ∈ Wa and it is given explicitly by (3.53) σ = ((0, 1, 1), ε23 σ23 ). Indeed, since the face is given by ξ2 +ξ3 = 2π the corresponding nodal vector is (0, 1, 1). One checks that σ 2 = 1 and that σ fixes p, q, p  , (σ ( p  ) = (0, 1, 1) + (0, −1, 0) = (0, 0, 1) = p  ). We let Y = σ (X ) be the reflexion of X along that face which yields the convex pentahedra X ∪ Y . Proposition 3.6. Let X be the alcove given by (3.52) and Y = σ (X ) its reflexion along the face ( p, q, p  ). Then 21 (X ∪ Y ) is a fundamental domain for the action of (T)
W in Lie T. Proof. By the above lemma the action of Wa on the set of alcoves is the same as the action of Wa on Wa /H X . Let Wa = 2
W . The left coset space Wa /Wa is identified with the 8 elements set, /2, where the corresponding map is given by: (γ , w) ∈ Wa → Class of w −1 (γ ) in /2.

(3.54)

40

A. Connes, M. Dubois-Violette

Indeed (0, w −1 )(γ , w) = (w −1 (γ ), 1). We can thus display the action (on the right) of H X as γ → Class of σ13 ε12 ( p + γ ). Let us write this transformation in terms of the ϕ-coordinates. One obtains    1 1 1 , w(ϕ) = (−ϕ3 , ϕ1 − ϕ3 , ϕ2 − ϕ3 ). ϕ→w ϕ+ (3.55) , , 2 2 2 Thus in fact we look at the following transformation of (Z/2)3 ,   1 (3.56) S(a1 , a2 , a3 ) = − a3 , a1 − a3 , a2 − a3 . 2

 



One has S(0) = 21 , 0, 0 , S 21 , 0, 0 = 21 , 21 , 0 , S 21 , 21 , 0 = 21 , 21 , 21 , S 21 , 21 , 21 


1
1  = 0. The other orbit is S 0, , 0 = 2 , 0, 21 , S 21 , 0, 21 = 0, 0, 21 , S(0, 0, 21 ) =
1 1
1 1
1 2  0, 2 , 2 , S 0, 2 , 2 = 0, 2 , 0 . This shows that the double coset space Wa \Wa /H X has cardinality 2 and thus that Wa just has 2 orbits in its action on the set of alcoves. What remains is to show that Y ∈ / Orbit of X . But one has Y = σ (X ) with σ given by (3.53). Thus we just need to determine the double coset of σ , i.e. by (3.54) the class of σ23 ε23 (0, 1, 1) in /2. One just checks that it is in the other orbit.   We thus have a fairly simple fundamental domain for the parameter space of noncommutative 3-spheres. In terms of the ϕ’s a simple translation of the above result gives, Proposition 3.7. 1) The union A ∪ B of the following simplices:     π π π
ϕ1
ϕ2
ϕ3
0 , B = ϕ ; ϕ3 +
ϕ1

ϕ2
ϕ3 , A = ϕ; 2 2 2 (3.57) gives a fundamental domain for the action of W on R3 / ϕ = (R/π Z)3 . 2) The real moduli space M is obtained by glueing the face3 (P  Q Z ) to the face (Z P P  ) by the transformation γ ∈ ϕ
W γ (ϕ1 , ϕ2 , ϕ3 ) = (π − ϕ1 + ϕ2 , π − ϕ1 + ϕ3 , π − ϕ1 ) and crossing each wall, i.e. each face whose supporting hyperplane contains 0, by the corresponding reflexion (in W ). The two simplices A and B have a common face (P Q P  ) supported by the hyperplane ϕ1 = π2 . In order to describe the moduli space one needs to give the identifcations of the boundary components. All faces of A other than (P Q P  ) are walls of chambers and thus crossing them leads to a simple reflexion on that wall. 4. The Flow F While the space M is the natural moduli space for noncommuative 3-spheres, the moduli space of the corresponding 4-spaces R4ϕ is obtained as a space of flow lines for a natural flow F on M. We define this flow F as the gradient flow for the Killing metric on the Lie algebra of S O(6) of the character of the virtual representation given by the “signature”. We first show that F is nicely compatible with the triangulation by alcoves and then check that the isomorphism class of the 4-spaces R4ϕ is constant along the flow 3 We use capital letters P, Q, Q  for the vertices : P = ( π , π , π ), P  = ( π , π , 0), Q = ( π , 0, 0), 2 2 2 2 2 2 Z = P + Q.

Noncommutative 3-Spheres

41

lines. We shall need for the converse to have a complete knowledge of the geometric datas of these quadratic algebras, and this will be obtained in Sect. 5. Thus the converse will be proved later on in Sect. 6. 4.1. Compatibility of F with the triangulation by alcoves. The basic compatibility of the flow F with the structure of the real moduli space M is given by : Proposition 4.1. a) The character of the signature representation is  χ (ξ ) = Trace (∗ π((ξ )) = − 8 sin ξ j .

(4.1)

b) The flow F = ∇ χ (2ψ) is invariant under the action of the Weyl group W and leaves each of the singular hyperplanes Hα,n , n ∈ 21 Z globally invariant. c) In terms of the variables ϕ j one has F=



sin(2ϕ j ) sin(ϕk + ϕ − ϕ j )

∂ . ∂ϕ j

(4.2)

Proof. a) We parametrize the maximal torus T D of S O(6) by the ξ j as in (3.29), i.e. (ξ ) = ξ1 β12 + ξ2 β34 + ξ3 β56 .

 The Killing metric is simply given there (up to scale and sign) by dξ 2j . We identify ∧3 C6 with the linear span of the ei jk = γi γ j γk , card {i, j, k} = 3 in Cliff C (R6 ). We then use γ7 = γ1 γ2 γ3 γ4 γ5 γ6 , which fulfills γ72 = −1 to define the ∗ operation by: ∗ ei jk = γ7 ei jk .

(4.3)

One checks that the matrix of ∗ π((ξ )) restricted to the 12 dim subspace spanned by the ei jk , where two indices belong to one of the 3 subsets {1, 2}, {3, 4}, {5, 6} is off diagonal and hence has vanishing trace. Indeed for instance ∗ π((ξ )) e123 is a linear combination of e356 and e456 which are orthogonal to e123 . On the 8 dimensional subspace of ei jk , i ∈ {1, 2}, j ∈ {3, 4}, k ∈ {5, 6} one gets the product of what happens in the 2-dimensional case with a basis e1 , e2 of ∧1 C2 . One has ∗ e1 = e2 , ∗ e2 = −e1 and the representation is given by e1 → cosξ e1 + sinξ e2 , e2 → cosξ e2 − sinξ e1 so that the trace of ∗ π((ξ )) is −2sinξ . Thus we get, Trace (∗ π((ξ ))) = − 8 sin ξ1 sin ξ2 sin ξ3 .

(4.4)

b) In terms of S O(6) the Weyl group W maps to the permutations of (ψ1 , ψ2 , ψ3 ) and the kernel of this map is the Klein subgroup which is given by the transformation, ψj → εj ψj

3 

εj = 1

(ε j ∈ {±1}).

(4.5)

1

By construction the function χ (ξ ) being a (virtual) character is invariant by these transformations and so is the flow X.  ∂χ ∂ c) Let us rewrite X = ∂ψ j ∂ ψ j in terms of the coordinates ϕ j . One has

  ∂h   ∂h ∂h d ψj, d ϕ j = d ψk + d ψ , d ϕj = + ∂ϕ j ∂ϕk ∂ϕ

∂h ∂h ∂h = + ∂ψ j ∂ϕk ∂ϕ

42

A. Connes, M. Dubois-Violette

and X is given by    ∂χ  ∂ ∂ ∂χ X= + =2 (cos2 ψk sin2 ψ + cos2 ψ sin2 ψk ) sin2 ψ j ∂ψk ∂ψ ∂ϕ j ∂ϕ j  ∂ =2 sin(2 ϕ j ) sin(ϕk + ϕ − ϕ j ) . ∂ϕ j   4.2. Invariance of R4ϕ under the flow F. We let C+ = {(ϕ, ϕ, 0)}, C− =

π 2

, ϕ, ϕ +

π . 2

(4.6)

The critical set of the flow X is described as follows in terms of the action of the Weyl group W : Lemma 4.2. The critical set C of X is given by C = W (C+ ) ∪ W (C− ) ∪ W (P),

P=

π π π  , , . 2 2 2

One checks this directly in the ψ variables. In order to perform a change of variables we use the function,  δ(ϕ) = sinϕ j cos(ϕk − ϕ ) (4.7) and we let

D = {ϕ, δ(ϕ) = 0}.

(4.8)

By construction D is invariant under the group S3 of permutations of the ϕ j ’s. Lemma 4.3. Let ϕ ∈ / C, then there exists g ∈ W such that gϕ ∈ / D. Proof. Let us first show that the conclusion holds if one of the ϕ j vanishes (we always work modulo π and all equalities below have this meaning). Thus assume that ϕ3 = 0, i.e. that ϕ = (ϕ1 , ϕ2 , 0). Then if ϕ1 = ϕ2 one is in C thus we can assume that ϕ1 = ϕ2 . By the transformation (3.23) we get ϕ  = (− ϕ1 , − ϕ1 , ϕ2 − ϕ1 ). If ϕ1 = 0 we treat (0, ϕ2 , 0) ∼ (ϕ2 , 0, 0) by applying (3.23) which gives (−ϕ2 , −ϕ2 , −ϕ2 ) for which δ(ϕ) is 0 only if ϕ2 = 0 in which case we are dealing with the point O =  (0, 0, 0) which is in C. Thus we can assume ϕ1 = 0, ϕ2 − ϕ1 = 0 and we get sinϕ j = 0.  One has ϕk −ϕ ∈ {0, ±ϕ2 } thus the product cos(ϕk −ϕ ) vanishes only if ϕ2 = π2 . 

We are thus dealing with ϕ1 , π2 , 0 (with ϕ1 = 0). We apply (3.23) to π2 , ϕ1 , 0 which gives  π π π . ϕ  = − , − , ϕ1 − 2 2 2  If ϕ1 = π2 then one is in C otherwise sinϕ j = 0. One has ϕk − ϕ ∈ {0, ±ϕ1 } and  since ϕ1 = π2 we get cos(ϕk − ϕ ) = 0. We have thus shown that if ϕ j = 0 for some

Noncommutative 3-Spheres

43

j and  ϕ∈ / C we can find / D. Thus we can now assume that all ϕ j = 0.  g ∈ W with gϕ ∈ Thus sinϕ j = 0. If cos(ϕk − ϕ ) = 0 we can assume ϕ1 − ϕ2 = π2 . We then apply (3.23) and get  π ϕ  = − ϕ1 , ϕ3 − ϕ1 , − . 2 If one of the components of ϕ  is 0 we are back to the previous case, thus we can assume  sinϕ j = 0. The ϕk − ϕ give (up to sign) ϕ3 , ϕ3 − ϕ1 + π2 , ϕ1 − π2 and since ϕ1 = 0  and ϕ2 = ϕ3 − ϕ
1 = 0, cos(ϕk − ϕ ) = 0 can occur only if cos ϕ3 = 0. In that case the original ϕ is ϕ1 , ϕ1 − π2 , π2 which is in C.   Note then that one can always choose the g ∈ W in the Klein subgroup H ⊂ W . Indeed H is a normal subgroup and S3 ⊂ W acts as the group of permutation of the ϕ’s and preserves D. Thus for g = σ k1 , σ ∈ S3 , σ k1 ϕ ∈ / D ⇒ k1 ϕ ∈ / D. Let us now use this lemma to simplify the presentation of the algebra. Lemma 4.4. If δ(ϕ) = 0, there exists 4 non zero scalars ∈ i N R∗ , such that one has and



(sinϕk ) λ λm + cos(ϕ − ϕm ) λ0 λk = 0 λµ = −δ(ϕ).

Proof. Let us choose the square roots 

λ0 = in such a way that



sinϕ j

1/2

⎛

, λk = ⎝sinϕk



⎞1/2 cos(ϕk − ϕ )⎠

 =k

λ j = −δ(ϕ). Note indeed that the product of the squares gives   (sinϕ j )2 (cos(ϕk − ϕ ))2 = δ(ϕ)2 , k<

and thus one can fix the λk and then choose the sign of λ0 so that it fits. Then to prove the 3 equalities one can multiply by λ0 λk which gives (sinϕk )(−δ(ϕ))+cos(ϕ −ϕm ) λ20 λ2k . Thus one needs to check   cos(ϕk − ϕ1 )sinϕk , sinϕk cos(ϕ1 − ϕm 1 ) = cos(ϕ − ϕm ) 1 =k

which is an identity.

 

One then lets

Sµ = λµ x µ , (no summation on µ). Multiplying (2.24) by λ0 λ1 λ2 λ3 one gets [S , Sm ] = i [S0 , Sk ]+

(4.9) (4.10)

while (2.25) gives: cosϕk [S0 , Sk ] = i Note that

λ0 λk sin(ϕ − ϕm ) [S , Sm ]+ . λ λm

 λµ λ0 λk −δ(ϕ) = 2 2 = 2 2 . λ λm λ λm λ λm

(4.11)

44

A. Connes, M. Dubois-Violette

Proposition 4.5. If δ(ϕ) = 0 the quadratic algebra of R4ϕ admits the presentation given by (4.10) and (4.12) [S0 , Sk ] = i Jm [S , Sm ]+ , where the Jm are given by Jm = −tan ϕk tan(ϕ − ϕm ). If ϕk = π2 and ϕ = ϕm the corresponding relation gives 0 = 0. If ϕk = it gives [S , Sm ]+ = 0.

(4.13) π 2

and ϕ = ϕm

Proof. One needs to show that −δ(ϕ) sin(ϕ − ϕm ) = −tan ϕk tan(ϕ − ϕm ). cosϕk λ2 λ2m This amounts to the equality sinϕ sinϕm cos2 (ϕ − ϕm ) cos(ϕ − ϕk )cos(ϕm − ϕk ) = λ2 λ2m , which follows from the definition of the λk ’s.

 

We now find a better way to write the Jm . We first pass in a faithful manner from the ϕ j to π t j = tan ϕ j ∈ R (recall ϕ j = ). (4.14) 2 Note also that t j = 0 since we are on δ(ϕ) = 0. We then introduce the following new parameters, (4.15) sk (ϕ) = 1 + t tm . σ

Lemma 4.6. a) The map ϕ −→ s = (sk (ϕ)) on  = {ϕ | cos(ϕk ) = 0 ∀ k, δ(ϕ) = 0} is a double cover of the open subset of R3 ,   sk = 0 and (sk − 1) > 0. (4.16) b) The map σ is a diffeomorphism of the interior A◦ of A with σ (A◦ ) = {s | 1 < s1 < s2 < s3 }. c) The map σ is a diffeomorphism of B ◦ with σ (B ◦ ) = {s | s3 < s2 < 0, 1 < s1 }. Proof. a) The condition δ(ϕ) = 0 shows that sk = 0, ∀ k. Indeed ϕm = −1  tan ϕ tan means cos(ϕ − ϕm ) = 0. By construction sk = 1 and (sk − 1) = t2 > 0.  2  one gets ( tk ) = (sk − 1) and choosing the sign of the square Knowing the sk ’s root gives p = tk and then tk = p(sk − 1)−1 . Thus one gets a double cover and the range is characterized by the conditions (4.16). The deck transformation is simply ϕ → −ϕ.  b) On A◦ = {ϕ | π2 > ϕ1 > ϕ2 > ϕ3 > 0} one has tk > 0 and thus tk > 0 so that the above map is one to one. One checks that the range σ (A◦ ) is given by σ (A◦ ) = {s | 1 < s1 < s2 < s3 }. c)  On B ◦ = {ϕ | π2 + ϕ3 ϕ1 > π2 > ϕ2 > ϕ3 } one has t1 < 0, t2 > 0, t3 > 0 and thus tk < 0 so that the above map is one to one. One checks that the range σ (B ◦ ) is given by σ (B ◦ ) = {s | s3 < s2 < 0, 1 < s1 }.  

Noncommutative 3-Spheres

45

We define the transformation ρ by s − sm . sk   Lemma 4.7. a) Let sk = 0, then one has (1 + ρ(s)k ) = (1 − ρ(s)k ).   b) If ρ(s) = ρ(s ) there exists λ = 0 with s = λ s. c) The transformation s → s˜ , ρ(s)k =

s˜k =

−sk + s + sm s sm

is involutive and ρ(˜s ) = −ρ(s).

 Proof. a) Both products give, up to the denominator sk , the product of (sk + s − sm ). b) Consider the 3 linear equations (for fixed ρk ) given by s − sm − ρk sk = 0. This corresponds to the 3 × 3 matrix given by ⎡ ρ1 −1 ρ2 M(ρ) = ⎣ 1 −1 1

⎤ 1 −1⎦ . ρ3

(4.17)

(4.18)

The condition a) means exactly that Det (M(ρ)) = 0.  we claim that the rank  Moreover 1 ρ2 which gives 1 + ρ2 and of M(ρ) is 2. Indeed ρ2 appears in the two minors −1 1   −1 1 which gives 1 − ρ2 and one of them is = 0. This shows that the kernel is ρ2 −1 1-dimensional and hence ρ(s) = ρ(s  ) implies s  = λ s for some λ. c) The relation between s and s˜ can be written as s s˜m + sm s˜ = 2.

(4.19)

The determinant of the system is 2 s1 s2 s3 so that (4.19) determines s˜k uniquely. The relation is clearly symmetric. Finally ρ(˜s )1 =

s 3 − s2 s˜2 − s˜3 (− s2 + s1 + s3 ) s2 − (− s3 + s1 + s2 ) s3 = = = −ρ(s)1 . s˜1 s1 (− s1 + s2 + s3 ) s1

  We now get ρ ◦ s (ϕ)k = Jm



ϕk =

 π and δ(ϕ) = 0 . 2

(4.20)

Indeed s − sm tk tm − tk t = = tan ϕk tan(ϕm − ϕ ) = Jm . sk 1 + t tm Lemma 4.8. 1) The flow X fulfills X sk (ϕ) = 4



sin ϕ j sk (ϕ).

(4.21)

46

A. Connes, M. Dubois-Violette

2) Let ϕ, ϕ  ∈ A with ϕ3 > 0, ϕ3 > 0. The following conditions are equivalent: a) Jm (ϕ  ) = Jm (ϕ), ∀k. b) ϕ  belongs to the orbit of ϕ by the flow X . 3) The same statement holds for ϕ, ϕ  ∈ B with ϕ3 + π2 > ϕ1 , ϕ3 + π2 ϕ1 . Proof. 1) One has X (tk ) = sin(2 ϕk )sin(− ϕk + ϕ + ϕn )

∂ tan ϕk = 2 tan ϕk sin(− ϕk + ϕ + ϕm ), ∂ϕk

X (sk ) = X (t ) tm + t X (tm ) = 2 t tm (sin(− ϕ + ϕm + ϕk ) + sin(− ϕm + ϕ + ϕk )), and using sin(a + b) + sin(a − b) = 2sina cosb, one gets X (sk ) = 4 t tm (sinϕk cos(ϕm − ϕ ))   cos(ϕ − ϕ )   m  =4 =4 sinϕ j sinϕ j (1 + tm t ). cosϕm cosϕ 2) In fact 1) shows that the flow X is up to a non-zero change of speed the scaling flow in σ (A). By construction Jm has homogeneity degree zero in sk , thus it is preserved by X and b) ⇒ a). Let us show that a) ⇒ b). We assume first that π2 > ϕ1 . The same then holds for ϕ  uisng a). By Lemma 4.7 the equality a) implies that σ (ϕ  ) = λ σ (ϕ) for some non-zero scalar λ. By Lemma 4.6 the image σ (A◦ ) is convex (as well as its closure) and thus the segment [σ (ϕ), σ (ϕ  )] is contained in σ (A) and its preimage under σ is a segment in a flow line. On the face Y determined by ϕ1 = π2 one has, assuming ϕ2 < π2 , the equalities J12 = − tan ϕ3 tan

π 2

 − ϕ2 = − t3 /t2 ,

 π = t2 /t3 . J31 = − tan ϕ2 tan ϕ3 − 2

Moreover one has X (t2 ) = cos(ϕ2 − ϕ3 ) 2 t2 , X (t3 ) = cos(ϕ2 − ϕ3 ) 2 t3 so that the flow X restricts as the scaling flow (up to a non-zero change of speed) in the parameters t j . Since J23 = ∞ holds iff ϕ1 = π2 for ϕ ∈ A we see that a) then implies ϕ1 = π2 and the proportionality t j = λt j . Since the allowed t j are simply constrained by the inequalities t2 ≥ t1 > 0 the same convexity argument applies. Finally let ϕ with ϕ1 = ϕ2 = π2 , ϕ3 = π2 . Then the only remaining parameter is t3 and one has X (t3 ) = 2sinϕ3 t3 . Thus one is dealing with a single flow line. The proof of 3) is similar.   5. The Geometric Data of R4ϕ In this section we compute the geometric data of the quadratic algebras of functions on R4ϕ for all values of the parameter ϕ. These fall in eleven different classes in each of which one gets further invariants.

Noncommutative 3-Spheres

47

5.1. The definition and explicit matrices. We give a list of the characteristic varieties and correspondences. There are 11 different cases. They are described in terms of the ϕ-coordinates but the result will then be translated in invariant terms using the roots. Let us recall the definition of the geometric data {E, σ, L} for quadratic algebras. Let A = A(V, R) = T (V )/(R) be a quadratic algebra, where V is a finite-dimensional complex vector space and where (R) is the two-sided ideal of the tensor algebra T (V ) of V generated by the subspace R of V ⊗ V . Consider the subset of V ∗ × V ∗ of pairs (α, β) such that ω, α ⊗ β = 0, α = 0, β = 0 (5.1) for any ω ∈ R. Since R is homogeneous, (5.1) defines a subset  ⊂ P(V ∗ ) × P(V ∗ ), where P(V ∗ ) is the complex projective space of one-dimensional complex subspaces of V ∗ . Let E 1 and E 2 be the first and the second projection of  in P(V ∗ ). It is usually assumed that they coincide, i.e. that one has E 1 = E 2 = E ⊂ P(V ∗ ),

(5.2)

and that the correspondence σ with graph  is an automorphism of E, L being the pullback on E of the dual of the tautological line bundle of P(V ∗ ). The algebraic variety E is referred to as the characteristic variety. The algebras we consider have 4 generators and six relations, thus their characteristic variety is obtained as the locus of points where a 4 × 6 matrix has rank less than 4. The various matrices depend upon the choice of the parameters and are listed below. We first give the matrix corresponding to the original quadratic algebra of R4ϕ , ⎡

−cos(ϕ1 ) x1

⎢−cos(ϕ2 ) x2 ⎢ ⎢ ⎢−cos(ϕ3 ) x3 ⎢ ⎢ i sin(ϕ ) x 3 3 ⎢ ⎢ ⎣ i sin(ϕ1 ) x1 i sin(ϕ2 ) x2

cos(ϕ1 ) x0

−i sin(ϕ2 − ϕ3 ) x3

i sin(ϕ1 − ϕ3 ) x3

cos(ϕ2 ) x0

−i sin(ϕ1 − ϕ2 ) x2

−i sin(ϕ1 − ϕ2 ) x1

−cos(ϕ1 − ϕ2 ) x2

cos(ϕ1 − ϕ2 ) x1

i sin(ϕ1 ) x0

−cos(ϕ2 − ϕ3 ) x3

cos(ϕ1 − ϕ3 ) x3

i sin(ϕ2 ) x0

−i sin(ϕ2 − ϕ3 ) x2

⎤

i sin(ϕ1 − ϕ3 ) x1 ⎥ ⎥ ⎥ ⎥ cos(ϕ3 ) x0 ⎥. ⎥ i sin(ϕ3 ) x0 ⎥ ⎥ cos(ϕ2 − ϕ3 ) x2 ⎦

−cos(ϕ1 − ϕ3 ) x1

(5.3) When we pass to the Sklyanin algebra and eliminate the factors i in replacing Z 0 = i S0 , Z k = Sk , we get the following matrix, with α = −J23 , β = −J31 , γ = −J12 , ⎡

z1

⎢z 2 ⎢ ⎢ ⎢z 3 ⎢ ⎢z ⎢ 3 ⎢ ⎣z 1 z2

−z 0

α z3

β z3

−z 0

γ z2

γ z1

z2

−z 1

z0

z3

−z 3

z0

α z2

⎤

β z1⎥ ⎥ ⎥ −z 0 ⎥ ⎥. z0 ⎥ ⎥ ⎥ −z 2 ⎦

(5.4)

z1

The fifteen minors of the matrix (5.3) are listed in factorized form in Appendix 1 and those of the matrix (5.4) in Subsect. 5.3 below (see [29]).

48

A. Connes, M. Dubois-Violette

5.2. The table. We give below the list of the characteristic varieties and correspondences in all cases. Given z ∈ C we let σ (z) be the conjugacy class of semi-simple automorphisms of a curve of genus 0 with eigenvalues {z, z −1 }. By construction σ (z) = σ (z −1 ). The identification of the corresponding algebras Calg (R4ϕ ) in the nongeneric cases (Cases 2 to 11 in the table below) are described in Sect. 8 and may be summarized as follows. In Case 2, Calg (R4ϕ ) is isomorphic to a homogenized version (quadratic) Uq (su(2))hom of Uq (su(2)) with either q ∈ C with |q| = 1 or q ∈ R with 0 < q < 1 or −1 < q < 0; it is important to notice that this is a ∗-isomorphism, (that is the su(2) does really matter in this notation). Case 3 is obtained by duality from Case 2 as explained in Sect. 7. In Case 4, there is a missing relation so Calg (R4ϕ ), which corresponds formally to a version q = 0 of Uq (su(2))hom , is of exponential growth. Case 5 is obtained by α3 -duality (Sect. 7) from Case 6 which is isomorphic to a homogenized version U (su(2))hom of the universal enveloping algebra of su(2) (i.e. q = 1 in Uq (su(2))hom ). Case 7 is the θ -deformation studied in Part I [15] and in [19] while Case 8 (anti θ -deformation) is obtained by α1 -duality (cf. Sect. 7) from Case 7. Case 9 is very singular: 3 relations are missing. Case 10 is obtained by α3 -duality (cf. Sect. 7) from Case 11 which is the ordinary algebra of polynomials with 4 indeterminates C[x 0 , x 1 , x 2 , x 3 ](classical case). Note that the reality conditions above correspond to the hermiticity of the x µ and not of the Sklyanin generators Sµ . It is important to describe the stratification in terms of the roots. The stratas of codimension k correspond to intersections of k singular hyperplanes (up to a factor 21 ) of the form F((α1 , . . . , αk ), (n 1 , . . . , n k )) = ∩

1 H(α j ,n j ) , 2

α j ∈ , n j ∈ Z.

The two dimensional stratas are defined using a single root α ∈  (i.e. k = 1) and since the Weyl group W acts transitively on  only the parity of n matters which gives the two kinds of faces F1 corresponding to n even and F2 to n odd. The one dimensional stratas are defined using two roots α, β ∈  (i.e. k = 2). The roots only matter up to sign and their relative position is described by their angle which (up to the sign) can be π2 in which case we write α ⊥ β, or 2π 3 in which case we write α − β. We thus have the following one dimensional stratas according to the parity of the n j. • α ⊥ β and (n 1 , n 2 ) = (even, odd) gives the line L of Case 4 below. • α − β and (n 1 , n 2 ) = (even, odd) or (n 1 , n 2 ) = (odd, odd) gives the line L  of Case 5 below. • α − β and (n 1 , n 2 ) = (even, even) gives the line L  of Case 6 below. • α ⊥ β and (n 1 , n 2 ) = (even, even) gives the line C+ of Case 7 below. • α ⊥ β and (n 1 , n 2 ) = (odd, odd) gives the line C− of Case 8 below. To double check that the list is complete4 one can use Lemma 4.2 to control the critical set C and then assume by Lemma 4.3 that δ(ϕ) = 0. Then if ϕ ∈ H(α,n) and n is even (resp. odd) the root α is one of the differences ϕk − ϕl (resp. ϕk ). Thus up to permutations of the ϕk one obtains one of the Cases 1)-6). The complete table giving the geometric datas is the following: 4 We are grateful to Marc Bellon for pointing out the subtelty of case 9) which was incomplete in an earlier version.

Noncommutative 3-Spheres

49

Point in M

Case

π 2

Z

1 Generic

α(ϕ) ∈ /

2 Even

ϕ1 = ϕ2 , ϕ2 − ϕ3 ∈ /

3

Odd Face

π 2

ϕj ∈ /

Face

ϕ1 =

π 2

ϕk ∈ /

Z

Correspondence

4 points ∪ Elliptic curve

(id, id, id, id, translation)

2 points, 1 line, 2 conics

  1/2   1/2  i+α i+α id, id, id, σ i−α , σ i−α 1/2 1/2

Z,

α = −J23

, ϕ2 − ϕ3 ∈ / π 2

π 2

Characteristic Variety

π 2

Z

 2 points, 1 line, 2 conics

id, id, −id, exchange with square  1/2 2  i+β σ i−β , β = −J31 1/2

six lines

(id, −id, cyclic permutation of 4 lines (iso, coarse, iso, coarse))

point ∪ P2 (C)

(id, Symmetry of determinant −1)

Z, k = 2, 3

4 α⊥β (0,1)

L=


π

, ϕ, ϕ

5 α−β (0,1)

L =


π

,

6 α−β (0,0)

L  = {(ϕ, ϕ, ϕ)}

point ∪ P2 (C)

(id, id)

7 α⊥β (0,0)

C+ = {(ϕ, ϕ, 0)}

six lines

(id, id, σ (e±2iϕ ))

six lines

(−id, −id, exchanges with square σ (e±4iϕ ))

P3 (C)

Symmetry of determinant − 1 and point → line on a quadric

P3 (C)

Symmetry of determinant 1

P3 (C)

id

8 α⊥β (1,1)

C− =

2

2




9

P=

10

P =

ϕ+


π 2

2

,ϕ

π 2

,

π 2

π 2

,

π 2

,


π

π 2



,

π 2



,ϕ

,0







(in C+ ∩ C− )

11

0 = (0, 0, 0) (in C+ )

The Geometric Data

(5.5)

5.3. Generic case. We now give the detailed description of the geometric data starting with the generic case. This case is defined by  π π  G = ϕ ; ϕj ∈ Z, ϕk − ϕ ∈ Z . (5.6) / / 2 2

50

A. Connes, M. Dubois-Violette

Then the Jk are well defined and are = 0. Let us show that we cannot have J12 = 1, J23 = −1. Indeed tan ϕ3 tan(ϕ1 − ϕ2 ) = −1 means π/2 − ϕ3 = ϕ2 − ϕ1 , while tan ϕ1 tan(ϕ2 −ϕ3 ) = 1 means π2 −ϕ1 = ϕ2 −ϕ3 . This gives π −ϕ1 −ϕ3 = 2 ϕ2 −ϕ1 −ϕ3 and 2 ϕ2 = π which is not allowed by (5.6). We can thus apply the result of Smith-Stafford [29] and get: Proposition 5.1. For ϕ ∈ G the geometric data of R4ϕ is given by 4 points and a nondegenerate elliptic curve E, and σ is identity on the 4 points and a translation of E given explicitly in terms of the parameters α1 = α, α2 = β, α3 = γ with αk = − Jm

(5.7)

by  E = z;

3  0

z 2j

 1−γ 2 1+γ 2 2 z + z + z3 = 0 , = 0, 1+α 1 1−β 2

(5.8)

and ⎡ ⎤ − 2 α β γ z 1 z 2 z 3 − z 0 (− z 02 + β γ z 12 + α γ z 22 + α β z 32 ) ⎡ ⎤ ⎢ ⎥ z0 ⎢ ⎥ 2 α z 0 z 2 z 3 + z 1 (z 02 − β γ z 12 + α γ z 22 + α β z 32 ) ⎥ ⎢z 1 ⎥ σ ⎢ ⎥. ⎣z ⎦ −→ ⎢ 2 2 2 2 ⎢ ⎥ 2 2 β z 0 z 1 z 3 + z 2 (z 0 + β γ z 1 − α γ z 2 + α β z 3 ) ⎣ ⎦ z3 2 γ z 0 z 1 z 2 + z 3 (z 02 + β γ z 12 + α γ z 22 − α β z 32 )

(5.9)

Proof. We rely on Smith-Stafford [29]. The first point is to rewrite the algebra of Proposition (4.5) in the form, [T0 , Tk ]+ = [T , Tm ], [T0 , Tk ] = αk [T , Tm ]+ .

(5.10)

One simply lets T0 = i S0 , Tk = Sk . Then (4.10) means [T , Tm ] = [T0 , Tk ]+ and (4.12) means [T0 , Tk ] = − Jm [T1 , Tm ]+ . Note the crucial − sign in (5.7). By hypothesis on ϕ ∈ G one has α β γ = 0. Moreover we have seen above that for ϕ ∈ G one cannot have J12 = 1, J23 = −1, i.e. α = −1, β = 1 (or any cyclic transformed of that). Now of the α’s, say α in case one  belongs to ±1 the equality α + β + γ + α β γ = 0, i.e. (1 + α) = (1 − α) shows that both +1 and −1 occur and since (−1, 1, x) is excluded the only remaining case is (1, −1, γ ) with γ ∈ / {−1, 0, 1}. Thus the hypotheses of Smith-Stafford are fulfilled and one gets from [29] that besides the 4 points (1, 0, 0, 0) . . . the characteristic variety is the curve in P3 (C) with Eqs. (5.8), the translation σ being given explicity by (5.9).   It will be useful for the computations in the degenerate case to display the list of the 15 minors in the case of the Sklyanin algebra, in the above parameters (α, β, γ ). Their list is given below ([29]).

Noncommutative 3-Spheres

51

{−2 (γ z 1 z 2 − z 0 z 3 ) (z 02 + z 12 + z 22 + z 32 ), 2 (z 0 z 2 − β z 1 z 3 ) (z 02 + z 12 + z 22 + z 32 ), −2 (z 0 z 2 + z 1 z 3 ) (z 02 − γ (β z 12 + z 22 ) + β z 32 ), 2 (z 02 (−(1 + γ ) z 22 + (−1 + β) z 32 ) + z 12 ((−1 + β) γ z 22 + β (1 + γ ) z 32 )), 2 (−z 1 z 2 + z 0 z 3 ) (z 02 − γ (β z 12 + z 22 ) + β z 32 ), 2 (z 0 z 1 − α z 2 z 3 ) (z 02 + z 12 + z 22 + z 32 ), −2 (z 0 z 1 − z 2 z 3 ) (z 02 + γ z 12 − α (γ z 22 + z 32 )), −2 (z 1 z 2 + z 0 z 3 ) (z 02 + γ z 12 − α (γ z 22 + z 32 )),

(5.11)

2 (z 02 ((−1 + γ ) z 12 − (1 + α) z 32 ) + z 22 ((1 + α) γ z 12 + α (−1 + γ ) z 32 )), 2 (z 02 ((1 + β) z 12 − (−1 + α) z 22 ) − ((−1 + α) β z 12 + α (1 + β) z 22 ) z 32 ), −2 (z 0 z 2 − z 1 z 3 ) (z 02 + α z 22 − β (z 12 + α z 32 )), −2 (z 0 z 1 + z 2 z 3 ) (z 02 + α z 22 − β (z 12 + α z 32 )), 2 (z 0 z 2 + β z 1 z 3 ) (z 02 + γ z 12 − α (γ z 22 + z 32 )), 2 (z 0 z 1 + α z 2 z 3 ) (z 02 − γ (β z 12 + z 22 ) + β z 32 ), 2 (γ z 1 z 2 + z 0 z 3 ) (z 02 + α z 22 − β (z 12 + α z 32 ))}. 5.4. Face α = n and n even. F1 = {(ϕ1 , ϕ1 , ϕ3 )}. In that case we have a Sklyanin algebra and with the above notations the parameters are γ = 0 while β = −α = tanϕ1 tan(ϕ3 − ϕ1 ). The list of minors then simplifies as follows: {z 0 z 3 (z 02 + z 12 + z 22 + z 32 ), (z 0 z 2 + α z 1 z 3 ) (z 02 + z 12 + z 22 + z 32 ), −(z 0 z 2 + z 1 z 3 ) (z 02 − α z 32 ), −α z 12 z 32 − z 02 (z 22 + (1 + α) z 32 ), (−z 1 z 2 + z 0 z 3 ) (z 02 − α z 32 ), (z 0 z 1 − α z 2 z 3 ) (z 02 + z 12 + z 22 + z 32 ), −(z 0 z 1 − z 2 z 3 ) (z 02 − α z 32 ), −(z 1 z 2 + z 0 z 3 ) (z 02 − α z 32 ), −α z 22 z 32 − z 02 (z 12 + (1 + α) z 32 ), (−1 + α) (z 12 + z 22 ) (−z 02 + α z 32 ), −(z 0 z 2 − z 1 z 3 ) (z 02 + α (z 12 + z 22 + α z 32 )), −(z 0 z 1 + z 2 z 3 ) (z 02 + α (z 12 + z 22 + α z 32 )), (z 0 z 2 − α z 1 z 3 ) (z 02 − α z 32 ), (z 0 z 1 + α z 2 z 3 ) (z 02 − α z 32 ), z 0 z 3 (z 02 + α (z 12 + z 22 + α z 32 ))}.

(5.12) The detailed analysis shows that the characteristic variety is the union of the two points with coordinates (z 0 , z 1 , z 2 , z 3 ) = (1, 0, 0, 0), (z 0 , z 1 , z 2 , z 3 ) = (0, 0, 0, 1), of the line {(0, z 1 , z 2 , 0)} and of the two conics obtained by intersecting the quadric (z 02 + z 12 + z 22 + z 32 ) = 0 with the two hyperplanes (z 02 − α z 32 ) = 0. The correspondence σ is the identity on the two points, and on the line. It restricts to the two conics and is a rational automorphism of each. It admits two fixed points on each of them and its derivative at the fixed points is given by the following complex numbers: √ √ i− α i+ α √ , √ . i+ α i− α 
5.5. Face α = n and n odd. F2 = { π2 , ϕ2 , ϕ3 }. In that case we have a limiting case of the Sklyanin algebra where the parameter α = ∞ while γ = − β1 . The list of minors

52

A. Connes, M. Dubois-Violette

can be computed directly and gives the following, up to non-zero scalar factors, {(z 1 z 2 + β z 0 z 3 ) (z 02 + z 12 + z 22 + z 32 ), (z 0 z 2 − β z 1 z 3 ) (z 02 + z 12 + z 22 + z 32 ), (−z 0 z 2 − z 1 z 3 ) (β (z 02 + z 12 ) + z 22 + β 2 z 32 ), (−1 + β) (z 02 + z 12 ) (−z 22 + β z 32 ), (−z 1 z 2 + z 0 z 3 ) (β (z 02 + z 12 ) + z 22 + β 2 z 32 ), −z 2 z 3 (z 02 + z 12 + z 22 + z 32 ), (z 0 z 1 − z 2 z 3 ) (−z 22 + β z 32 ), (z 1 z 2 + z 0 z 3 ) (−z 22 + β z 32 ), −β z 02 z 32 − z 22 (z 12 + (1 + β) z 32 ), −β z 12 z 32 − z 22 (z 02 + (1 + β) z 32 ), (−z 0 z 2 + z 1 z 3 ) (z 22 − β z 32 ), (−z 0 z 1 − z 2 z 3 ) (z 22 − β z 32 ), (z 0 z 2 + β z 1 z 3 ) (z 22 − β z 32 ), z 2 z 3 (β (z 02 + z 12 ) + z 22 + β 2 z 32 ), (−z 1 z 2 + β z 0 z 3 ) (z 22 − β z 32 )}.

(5.13)

The detailed analysis shows that the characteristic variety is the union of the two points with coordinates (z 0 , z 1 , z 2 , z 3 ) = (0, 0, 1, 0), (z 0 , z 1 , z 2 , z 3 ) = (0, 0, 0, 1), of the line {(z 0 , z 1 , 0, 0)} and of the two conics obtained by intersecting the quadric (z 02 + z 12 + z 22 + z 32 ) = 0 with the two hyperplanes (z 22 − β z 32 ) = 0. The correspondence σ is the identity on the two points, and is given on the line by σ (z 0 , z 1 , 0, 0) = (z 0 , −z 1 , 0, 0). It exchanges the two conics and is a rational isomorphism of one with the other, moreover the square of σ admits two fixed points on each of them and its derivative at the fixed points is given by the squares of the following complex numbers: √ √ i+ β i− β √ , √ . i− β i+ β
 5.6. Edge α ⊥ β and (n 1 , n 2 ) = (even, odd). L = { π2 , ϕ, ϕ }. From now on we no longer use the change of variables to the Sklyanin algebras but we rely directly on the explicit form of the minors of the original matrix as computed in Appendix 14. Note in particular that the parameters x j are no longer the same as the above z j , but this is irrelevant since we compute intrinsic invariants of the quadratic algebra. In the case at hand the list of minors simplifies (with non-zero scale factors removed) to the following: {(x1 x2 + i x0 x3 ) (−x02 + x12 + x22 + x32 ), (x0 x2 + i x1 x3 ) (x02 − x12 − x22 − x32 ), (x0 x2 + i x1 x3 ) (x02 − x12 + x22 + x32 ), (x02 − x12 ) (x22 + x32 ), (−i x1 x2 +

x0 x3 ) (−x02

+ − − 0, 0, 0}. x12

x22

(5.14)

x32 ), 0, 0, 0, 0, 0, 0, 0,

Thus the characteristic variety consists in the six lines  j given in terms of free parameters x j by {(0, 0, x2 , x3 )}, {(x0 , x1 , 0, 0)}, {(x0 , x0 , x2 , −i x2 )}, {(x0 , x0 , x2 , i x2 )}, {(x0 , −x0 , x2 , i x2 )}, {(x0 , −x0 , x2 , −i x2 )}.

(5.15)

The correspondence σ is the identity on the first line, −1 on the second and permutes cyclically the four others  j . Passing from 4 to 5 or from 6 to 3 one gets the coarse correspondence, while the other maps are rational isomorphisms. The following three lines meet and their point of intersection is mapped by the coarse correspondence to the indicated line: 1 ∩ 4 ∩ 5 → 5 , 1 ∩ 3 ∩ 6 → 3 , 2 ∩ 3 ∩ 4 → 5 , 2 ∩ 5 ∩ 6 → 3 . This is coherent as the restriction of the relevant coarse correspondence.

Noncommutative 3-Spheres

53


 5.7. Edge α − β and (n 1 , n 2 ) = (odd, odd) or (even, odd). L  = { π2 , π2 , ϕ } . In that case the list of minors simplifies (with non-zero scale factors removed) to the following: { x0 x3 (−x02 + x12 + x22 + sin(ϕ)2 x32 ), x1 x3 (−x02 + x12 + x22 + sin(ϕ)2 x32 ), x32 (x0 x2 + i sin(ϕ) x1 x3 ), (x02 − x12 ) x32 , x32 (−i x1 x2 + sin(ϕ) x0 x3 ), x2 x3 (−x02 + x12 + x22 + sin(ϕ)2 x32 ), x32 (x0 x1 − i sin(ϕ) x2 x3 ),

(5.16)

x32 (i x1 x2 + sin(ϕ) x0 x3 ), (x02 − x22 ) x32 , (x12 + x22 ) x32 , x32 (−x0 x2 + i sin(ϕ) x1 x3 ), x32 (x0 x1 + i sin(ϕ) x2 x3 ), x1 x33 , x2 x33 , x0 x33 }. Thus the characteristic variety contains the hyperplane x3 = 0. For x3 = 0 the last three minors show that all other coordinates vanish and this gives an additional point, not in the above hyperplane. The correspondence σ is the symmetry σ (x0 , x1 , x2 , 0) = (−x0 , x1 , x2 , 0). 5.8. Edge α − β and (n 1 , n 2 ) = (even, even). L  = {(ϕ, ϕ, ϕ)}. In that case the list of minors simplifies (with non-zero scale factors removed) to the following: { x0 x3 (sin(ϕ)2 x02 − x12 − x22 − x32 ), x0 x2 (sin(ϕ)2 x02 − x12 − x22 − x32 ), x02 (sin(ϕ) x0 x2 + i x1 x3 ), x02 (x22 + x32 ), x02 (−i x1 x2 + sin(ϕ) x0 x3 ), x0 x1 (sin(ϕ)2 x02 − x12 − x22 − x32 ), x02 (sin(ϕ) x0 x1 − i x2 x3 ), x02 (i x1 x2 + sin(ϕ) x0 x3 ),

(5.17)

x02 (x12 + x32 ), x02 (x12 + x22 ), x02 (sin(ϕ) x0 x2 − i x1 x3 ), x02 (sin(ϕ) x0 x1 + i x2 x3 ), x03 x2 , x03 x1 , x03 x3 }. Thus the characteristic variety contains the hyperplane x0 = 0. For x0 = 0 the last three minors show that all other coordinates vanish and this gives an additional point, not in the above hyperplane. The correspondence σ is the identity. 5.9. Edge α ⊥ β and (n 1 , n 2 ) = (even, even). C+ = {(ϕ, ϕ, 0)}. In that case the list of minors simplifies (with non-zero scale factors removed) to the following: { x0 (x12 + x22 ) x3 , (x12 + x22 ) (−i cos(ϕ) x0 x2 + sin(ϕ) x1 x3 ), (sin(ϕ) x0 x2 + i cos(ϕ) x1 x3 ) (x02 + x32 ), (x02 x22 − x12 x32 ), x1 x2 (x02 + x32 ), (x12 + x22 ) (cos(ϕ) x0 x1 − i sin(ϕ) x2 x3 ), (sin(ϕ) x0 x1 − i cos(ϕ) x2 x3 ) (x02 + x32 ), x1 x2 (x02 + x32 ), (x02 x12 − x22 x32 ), sin(4 ϕ) (x12 + x22 ) (x02 + x32 ), + x22 ) (i sin(ϕ) x0 x2 + cos(ϕ) x1 x3 ), + x22 ) (sin(ϕ) x0 x1 + i cos(ϕ) x2 x3 ), (cos(ϕ) x0 x2 − i sin(ϕ) x1 x3 ) (x02 + x32 ), cos(ϕ) x0 x1 + sin(ϕ) x2 x3 ) (x02 + x32 ), x0 (x12 + x22 ) x3 }. (x12 (x12

(−i

(5.18)

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A. Connes, M. Dubois-Violette

Thus the characteristic variety consists in the six lines  j given in terms of free parameters x j by {(0, x1 , x2 , 0)}, {(x0 , 0, 0, x3 )}, {(x0 , x1 , i x1 , i x0 )}, {(x0 , x1 , −i x1 , i x0 )}, {(x0 , x1 , i x1 , −i x0 )}, {(x0 , x1 , −i x1 , −i x0 )}.

(5.19)

The correspondence σ is the identity on the first two lines. It preserves globally the other  j and induces on each of them the rational automorphism given by multiplication by e±2iϕ .
 5.10. Edge α ⊥ β and (n 1 , n 2 ) = (odd, odd). C− = { π2 + ϕ, π2 , ϕ }. In that case the list of minors simplifies (with non-zero scale factors removed) to the following: {(sin(ϕ)x1 x2 + icos(ϕ)x0 x3 )(x02 − x22 ), x1 x3 (x02 − x22 ), x0 x2 (x12 − x32 ), x02 x32 − x12 x22 , (icos(ϕ)x1 x2 − sin(ϕ)x0 x3 )(x12 − x32 ), (−isin(ϕ)x0 x1 + cos(ϕ)x2 x3 )(x02 − x22 ), (cos(ϕ)x0 x1 − isin(ϕ)x2 x3 )(x02 − x22 ), (cos(ϕ)x1 x2 − isin(ϕ)x0 x3 )(x02 − x22 ), sin(4ϕ)(x02 − x22 )(x12 − x32 ), (x02 x12 − x22 x32 ), x0 x2 (x12 − x32 ), (cos(ϕ)x0 x1 + isin(ϕ)x2 x3 )(x12 − x32 ), x1 x3 (x02 (isin(ϕ)x0 x1 + cos(ϕ)x2 x3 )(x12 − x32 ), (sin(ϕ)x1 x2 − icos(ϕ)x0 x3 )(x12 − x32 )}.

(5.20)

− x22 ),

Thus the characteristic variety consists in the six lines  j given in terms of free parameters x j by {(0, x1 , 0, x3 )}, {(x0 , 0, x2 , 0)}, {(x0 , x1 , x0 , x1 )}, {(x0 , x1 , −x0 , −x1 )}, {(x0 , x1 , x0 , −x1 )}, {(x0 , x1 , −x0 , x1 )}.

(5.21)

The correspondence σ is −1 on the first two lines. It permutes 3 with 4 and its square is the rational automorphism multiplying the ratio x1 /x0 by e4iϕ . It permutes 5 with 6 and its square is the rational automorphism multiplying the ratio x1 /x0 by e−4iϕ . 5.11. Vertex P = ( π2 , π2 , π2 ). In that case all minors vanish identically. Thus the characteristic variety is all projective space. The correspondence σ is given by σ ((x0 , x1 , x2 , x3 )) = (−x0 , x1 , x2 , x3 ), but it degenerates on the quadric Q=



x | x02 −



 xk2 = 0 ,

to a correspondence which assigns to any point p ∈ Q a line ( p) ⊂ Q containing the point σ ( p) and belonging to one of the two families of lines that rule the surface Q.

Noncommutative 3-Spheres

55

5.12. Vertex P  = ( π2 , π2 , 0). In that case all minors vanish identically. Thus the characteristic variety is all projective space. The correspondence σ is given by σ ((x0 , x1 , x2 , x3 )) = (−x0 , x1 , −x2 , x3 ). 5.13. Vertex O = (0, 0, 0). In that case all minors vanish identically and the correspondence σ is the identity. 6. Isomorphism Classes of R4ϕ and Orbits of the Flow F We let as above M be the moduli space of oriented non-commutative 3-spheres and PM its quotient by the symmetry given by Proposition 2.4, 3). For ϕ ∈ M we view the algebra A = Calg (R4ϕ ) as a graded algebra, i.e. we endow it with the one parameter group of automorphisms which rescale the generators xν , θλ ∈ AutA, θλ (xν ) = λ xν , ∀λ ∈ R∗ .

(6.1)

We let PM be the quotient of the real moduli space M by the symmetry of Proposition 2.4, 3). This section will be devoted to prove the following result: Theorem 6.1. Let ϕ j ∈ M; the following conditions are equivalent: a) The graded algebras Calg (R4ϕ j ) are isomorphic. b) ϕ2 ∈ Flow line of ϕ1 in PM. The proof of b) ⇒ a) was given above in Sect. 4.2. The converse is based on the information given by the geometric data which is by construction an invariant of the graded algebra. The proof will be broken up in the non-generic and generic cases.

6.1. Proof in the non-generic case. To analyze the information given by the geometric data we can restrict the parameters ϕ to the fundamental domain A∪ B of Proposition 3.7. The symmetry given by Proposition 2.4, 3) is given explicitly by the transformation ρ(ϕ1 , ϕ2 , ϕ3 ) = (ϕ1 , ϕ1 − ϕ3 , ϕ1 − ϕ2 ).

(6.2)

The identification γ of the face (P  Q Z ) with the face (Z P P  ) (Proposition 3.7) followed by the symmetry ρ (6.2) gives the following symmetry σ = ρ ◦ γ of the face (P  Q Z ), (6.3) σ (ϕ1 , ϕ2 , ϕ3 ) = (π − ϕ1 + ϕ2 , ϕ2 , ϕ2 − ϕ3 ). For vertices we have three elements 0, P, P  modulo the action of 
W . The geometric data allows to separate them. One has a priori nine edges. They fall in five different classes 4) 5) 6) 7) 8). Using γ and ρ one sees that the following four edges are equivalent: [Z P] ∼ [P P  ] ∼ [Q P  ] ∼ [Z Q], and are all in Case 5).

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Similarly, using ρ one sees that the following two edges are equivalent: [O P] ∼ [O Q] and are both in Case 6). The table of geometric datas shows that the geometric data of R4ϕ determines in which of the five Cases 4)-8) one is. Thus when the flow is transitive in the corresponding edge there is nothing to prove. This covers Cases 4), 5), 6). The two Cases 7), 8) correspond to fixed points of the flow. For C+ one gets the edge in [O P  ] ⊂ A¯ given by {ϕ, ϕ, 0} with 0 < ϕ < π2 . The geometric data gives back the set e±2iϕ and this allows to recover ϕ. Thus distinct ϕ give non-isomorphic quadratic algebras. For C− one gets the edge [P  Z ] ⊂ B¯ i.e. {( π2 + ϕ, π2 , ϕ)} whose interior corresponds to 0 < ϕ < π2 . The geometric data gives back the set e±4iϕ . Thus there is an ambiguity ϕ → π2 − ϕ in ϕ knowing the geometric data. To understand it let us note that in fact one checks that σ restricts to the edge [P  Z ] as ϕ → π2 − ϕ. This then accounts for the above ambiguity. We now have to deal with the faces. We start with those which are odd (i.e. Hα,n with n odd). The identification γ of the face (P  Q Z ) with the face (Z P P  ) (Proposition 3.7) shows that we just need to deal with (Z P P  ) and with the face (Q P P  ) which is common to A and B. For the face (Z P P  ) the equation of the supporting hyperplane is ϕ2 = π2 and one is in Case 3) with generic elements of the form (ϕ1 , π2 , ϕ3 ), where ϕ3 + π2 > ϕ1 > π2 > ϕ3 . The geometric data determines the square 2  i + β 1/2 σ , i − β 1/2 1/2

i+β where β = −tanϕ3 /tanϕ1 > 0. Then i−β 1/2 is of modulus one and the geometric data determines β up to the ambiguity given by β → 1/β. But the face (Z P P  ) admits the symmetry given by

γ ◦ ρ(ϕ1 , ϕ2 , ϕ3 ) = (π − ϕ3 , π − ϕ2 , π − ϕ1 ),

(6.4)

whose effect is precisely the transformation β → 1/β. Note that the segment joining P to the middle of [P  Z ] is globally invariant under the flow X . For the face (Q P P  ) the equation of the supporting hyperplane is ϕ1 = π2 and one is in Case 3) with generic elements of the form ( π2 , ϕ2 , ϕ3 ), where π2 > ϕ2 > ϕ3 > 0. The geometric data determines the square 2  i + β 1/2 , σ i − β 1/2 1/2

i+β where β = −tanϕ2 /tanϕ3 < 0. Then i−β 1/2 is real and the geometric data determines β up to the ambiguity given by β → 1/β. But the inequality tanϕ2 > tanϕ3 > 0 shows that in fact β ∈] − ∞, −1[ so that the geometric data determines β uniquely. To summarize we have up to symmetry only two odd faces, the geometric data allows to decide (by |q| = 1 or q ∈ R) in which case one is, and gives back the flow line up to the remaining symmetries. Let us now consider the even faces (i.e. Hα,n with n even). Using ρ we get the equivalence (O P P  ) ∼ (O Q P  ). To be able to use Lemma 4.8 we concentrate on (O P P  )

Noncommutative 3-Spheres

57

on which ϕ3 > 0. The equation of the supporting hyperplane is ϕ1 = ϕ2 and one is in Case 2) with generic elements of the form (ϕ1 , ϕ1 , ϕ3 ), where π2 > ϕ1 > ϕ3 > 0. The geometric data determines   i + α 1/2 σ , i − α 1/2 1/2

i+α where α = tan ϕ1 tan(ϕ1 − ϕ3 ) > 0. Then q = i−α 1/2 is of modulus one. Thus the 1/2 geometric data determines α (since it determines α up to sign). There are however two other even faces namely (O P Q) and (Z P Q). The equation of the supporting hyperplane is the same in both cases and is ϕ2 = ϕ3 and one is in Case 2) with generic elements of the form (ϕ1 , ϕ2 , ϕ2 ), where for (O P Q) one has π π π 2 > ϕ1 > ϕ2 > 0, while for (Z P Q) one gets 2 + ϕ2 > ϕ1 > 2 > ϕ2 . The geometric data determines   i + α 1/2 , σ i − α 1/2 1/2

i+α where α = tan ϕ2 tan(ϕ2 − ϕ1 ) < 0. Then q = i−α 1/2 is real. This first allows to distinguish these faces from the other even faces treated above. Moreover one checks that

α ∈] − 1, 0[, ∀ϕ ∈ (O P Q), α ∈] − ∞, −1[, ∀ϕ ∈ (Z P Q). Thus q > 0 on (O P Q) and q < 0 on (Z P Q) which allows to distinguish these two faces from each other. Finally on each of these faces the geometric data determines α and hence the flow line of ϕ using Lemma 4.8. Thus we get the proof in all cases except the generic case which we shall now treat in detail separately.

6.2. Basic notations for elliptic curves. We recall that given an elliptic curve E viewed as a 1-dimensional complex manifold and choosing a base point p0 ∈ E one gets an isomorphism of the universal cover E˜ of E with base point p0 I E˜  C,

given by the integral  I ( p) =

p

ω,

p0

where ω is a holomorphic (1, 0)-form. This isomorphism is unique up to multiplication by λ ∈ C∗ . Let L ⊂ C be the lattice of periods, then I −1 (L) is the kernel of the covering map π : E˜ → E and one has an isomorphism E ∼ C/L. To eliminate the choice of the base point we let T (E) be the group of translations of E and note that the universal cover T˜ (E) identifies with the additive group C, T˜ (E)  C, T (E)  C/L .

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One can moreover take L of the form L = Z+Zτ , where τ ∈ H/ [1] and H is the upper half plane H = {z ∈ C | Imz > 0}, while in general [n] is the congruence subgroup of level n in S L(2, Z). With H∗ obtained from H by adjoining the rational points of the boundary one has a canonical isomorphism H∗ / [1] → P1 (C) given by Jacobi’s j function. In terms of the elliptic curve E defined by the equation y 2 = 4x 3 − g2 x − g3 , in P2 (C) one has j (E) = 1728

g23 , 

where the discriminant is  = g23 − 27 g32 . One obtains a finer invariant λ(E) if one has the additional structure given by an isomorphism of abelian groups φ:

(Z/2Z)2 →

1 L/L = T2 (E), 2

where T2 (E) is the group of two torsion elements of T (E). This allows to choose the module τ in a finer manner as τ ∈ H/ [2] and one has a canonical isomorphism H∗ / [2] → P1 (C) given by Jacobi’s λ function. In terms of the elliptic curve E defined by the equation  y2 = (x − e j ) in P2 (C) one has λ(E) = Cross Ratio(e1 , e2 ; e3 , e4 ). In more intrinsic terms the labelling of the two torsion T2 (E), ω j ∈ T (E), ω1 = φ(1, 0), ω3 = φ(0, 1), ω2 = −ω1 − ω3

(6.5)

allows to define the following function on the group T (E) of translations of E: Fφ (u) =

℘3 (u) , ℘3 (ω1 )

where ℘3 is defined using a fixed isomorphism T˜ (E)  C, as the sum ⎛ ⎞ ⎛ ⎞   ℘3 (u) = ⎝ y −2 ⎠ − ⎝ y −2 ⎠, π(y)=u

(6.6)

(6.7)

π(y)=ω3

where one defines the sums by restricting y to |y| < R on both sides and then taking the limit. In standard notation with the Weierstrass ℘-function given by   1 1  1 , (6.8) ℘ (z) = 2 + − z (z + )2 2 ∗ ∈L

one gets

℘3 (u) = ℘ (z) − ℘ (w3 ), π(z) = u.

(6.9)

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59

Note that the ratios involved in (6.6) eliminate the scale factor λ in the isomorphism T˜ (E)  C. With these notations one has ℘ (ω2 ) − ℘ (ω3 ) . ℘ (ω1 ) − ℘ (ω3 )

λ(E, φ) = F(w2 ) =

(6.10)

Finally the covering H∗ / [2] → H∗ / [1] is simply given by the algebraic map λ

→

256

(1 − λ + λ2 )3 , λ2 (1 − λ)2

while the group  of deck transformations is the dihedral group generated by the two symmetries u(z) = 1/z v(z) = 1 − z. One has a unique anti-isomorphism w : P S L(2, Z/2Z) →  such that λ(E, φ ◦ α) = w(α)(λ(E, φ)).

(6.11)

Moreover w(t) = v, where t is the transposition of (a, b) → (b, a). Finally one gets, Fφ◦t (u) = 1 − Fφ (u).

(6.12)

6.3. The generic case. We now deal with the generic case. Let s j ∈ R be three real numbers and (α, β, γ ) be given by α=

s3 − s 2 s1 − s3 s2 − s1 , β= , γ = . s1 s2 s3

(6.13)

Let (E, σ ) be the pair of an elliptic curve and a translation associated to (α, β, γ ) by Proposition 5.1. Lemma 6.2. There exists an isomorphism φ : (Z/2Z)2 → T2 (E) such that with e j = s −1 j one has (6.14) λ(E, φ) = Cross Ratio (e2 , e1 ; e3 , ∞), and with Fφ defined by (6.6), Fφ (σ ) =

s1 . s1 − s3

(6.15)

Proof. The proof is straightforward using θ -functions to parametrize the elliptic curve (5.1) but we prefer to give an elementary direct proof. In order to prove (6.14) and (6.15) we start from Proposition 5.1 and replace (α, β, γ ) by their value. The equation of E simplifies to 3 3   xµ2 = 0, sk xk2 = 0. (6.16) 0

1

We then rescale x1 = a X 1 , x2 = b X 2 , x3 = X 3 , where a2 = −

s3 s3 , b2 = − . s1 s2

(6.17)

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A. Connes, M. Dubois-Violette

After this rescaling the second equation of (6.16) gives X 32 = X 12 + X 22 , and one uses the standard rational parametrization of the conic to parametrize the solutions of this equation as x1 = 2 a t, x2 = b(1 − t 2 ), x3 = (1 + t 2 ). (6.18) One then writes the first equation in (5.1) as x02 + 4 a 2 t 2 + b2 (1 − t 2 )2 + (1 + t 2 )2 = 0, and since 1 + b2 = equation

(6.19)

(s2 − s3 )

= 0 this reduces to the elliptic curve defined by the s2 y 2 = t 4 − 2 r t 2 + 1,

(6.20)

where r =−

2 a 2 − b2 + 1 . b2 + 1

Using (6.17) one gets

s1 (s2 + s3 ) − 2 s2 s3 . (6.21) s1 (s3 − s2 ) One checks that r = ±1 since the s j are pairwise distinct. Thus the roots of x 4 − 2 r x 2 + 1 = 0 are all distinct and we write them as ± u, ± v, where uv = 1, u 2 + v 2 = 2r. (6.22) r=

The cross ratio of (−v, u; v, −u) is independent of the above choices and is given by −v − v u − v 4 uv 2 1 + b2 −4 s1 s2 − s3 : =− = − = = = . 2 2 2 2 −v + u u + u (u − v) u +v −2 r −1 1+a s2 s1 − s3 (6.23) one has In other words with e j = s −1 j Cross Ratio(−v, u; v, −u) = Cross Ratio(e2 , e1 ; e3 , ∞).

(6.24)

which transforms (e2 , e1 , e3 , ∞) to (−v, u, v, −u). Let then γ ∈ SL(2, C), γ (X ) = Then for a suitable choice of λ = 0 the transformation, a X +b cX +d

t = γ (X ),

y=

λY , (cX + d)2

gives a birational isomorphism γ˜ of the elliptic curve defined by the equation Y2 =

3 

(X − ei )

(6.25)

1

with the curve (6.20). Choosing the origin of (6.25) as the point at infinity, one gets the point P with coordinates t = −u, y = 0 as the origin of (6.20). One has P = ( p0 , p1 , p2 , p3 ) and p0 = 0 since y = 0. Thus the other coordinates fulfill (6.16) in the simplified form p12 + p22 + p32 = 0, s1 p12 + s2 p22 + s3 p32 = 0, and in homogeneous expressions one can replace pk2 by s − sm .

(6.26)

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61

Let us now determine the translation σ . We just compute the t parameter of σ (P) = (x0 , x1 , x2 , x3 ). The parameter t of (6.18) is recovered in homogeneous coordinates as, t=

b x1 . a x2 + a b x3

(6.27)

One first starts by simplifying (5.9) when applied to P. One has p0 = 0 and up to an overall scale (−α β γ ) one gets using (6.26), i.e. the replacement pk2 → s − sm , −β γ p12 + α γ p22 + α β p32 → − s1 + s2 + s3 , β γ p12 − α γ p22 + α β p32 → s1 − s2 + s3 , β γ p12 + α γ p22 − α β p32 → s1 + s2 − s3 . Thus the coordinates of σ (P) are up to an overall scale x1 = p1 (−s1 + s2 + s3 ), x2 = p2 (s1 − s2 + s3 ), x3 = p3 (s1 + s2 − s3 ),

(6.28)

where the p j are the coordinates of P. These are given by (6.18) taking t = −u, with u, v as above, thus, p1 = −2 a u, p2 = b(1 − u 2 ), p3 = 1 + u 2 .

(6.29)

We then need to compute: t (σ ) =

b(−2 a u)(−s1 + s2 + s3 ) b x1 . (6.30) = a x2 + a b x3 a b(1 − u 2 )(s1 − s2 + s3 ) + a b(1 + u 2 )(s1 + s2 − s3 )

We see that a and b drop out and we remain with t (σ ) =

− u(−s1 + s2 + s3 ) . s1 + (s2 − s3 ) u 2

(6.31)

We just need to compute Cross Ratio(−v, u, v, t (σ )) which is the same as Cross Ratio (e2 , e1 ; e3 , τ ), where τ is ℘ (σ ), up to an affine transformation which transforms the e j to ℘ (ω j ). One has −v − v u−v u − t (σ ) 2v : = −v − t (σ ) u − t (σ ) u − v v + t (σ ) 2 u 2 − u t (σ ) = u 2 − 1 1 + u t (σ )

Cross Ratio(−v, u, v, t (σ )) =

using u v = 1. The u t (σ ) only involves u 2 and one can simplify by its denominator to get (s2 − s3 ) u 4 + (s2 + s3 ) u 2 Cross Ratio(−v, u, v, t (σ )) = 2 . (6.32) (s1 − 2 s3 ) u 4 + 2 s3 u 2 − s1 We then claim that one has: Cross Ratio(−v, u, v, t (σ )) =

s2 − s3 = Cross Ratio(e2 , e1 ; e3 , 0), s1 − s3

(6.33)

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A. Connes, M. Dubois-Violette

i.e. that τ = 0. To see this we replace u 4 by 2 r u 2 − 1 in the expression (6.32) for the cross ratio, which gives Cross Ratio(−v, u, v, t (σ )) =

(2(s2 − s3 ) r + s2 + s3 ) u 2 − (s2 − s3 ) . ((s1 − 2 s3 ) r + s3 ) u 2 − (s1 − s3 )

The computation using (6.21) shows that this fraction is independent of u 2 and equal s2 − s3 . This ends the proof of the lemma since we have shown that, up to an affine to s1 − s3 tranformation, ℘ (ω j ) = e j , ℘ (σ ) = 0, so that Fφ (σ ) =

0 − e3 s1 = . e1 − e3 s1 − s3

  We shall now give the proof of Theorem 6.1 in the generic case. We start with the alcove A. Given an elliptic curve E and a translation σ of E we claim that if there is a labelling φ of T2 (E) such that λ(E, φ) ∈]0, 1[,

Fφ (σ ) < 0,

(6.34)

then this labelling is unique. Indeed the only element of P S L(2, Z/2Z) which preserves the first condition is the transposition t and this does not preserve the second by (6.12). On σ (A) = {s | 1 < s1 < s2 < s3 } one has (Lemma 4.6) Cross Ratio (s2 , s1 ; s3 , 0) ∈]0, 1[,

s1 < 0, s1 − s3

thus by Lemma 6.2 there exists a unique labelling φ of T2 (E) such that (6.34) holds. This s1 s3 gives back both Cross Ratio (s2 , s1 ; s3 , 0) and . The latter gives the ratio =a s1 − s3 s1 s2 − as1 s2 s2 − as1 s1 = , which gives the ratio = b. Thus and the former then gives s1 − as1 s2 (1 − a)s2 s1 we recover the flow line using the convexity of σ (A). Let us now look at the alcove B. One has σ (B) = {s | s3 < s2 < 0, 1 < s1 }. Thus s2−1 < s3−1 < s1−1 . Exactly as above, given an elliptic curve E and a translation σ of E we claim that if there is at most one labelling φ of T2 (E) such that 0 < Fφ (σ ) < λ(E, φ) < 1,

(6.35)

the algebra associated to the s j is unchanged, up to isomorphism, by cyclic permutations of the s j , and the same holds for the associated geometric data. Thus Lemma 6.2 gives a labelling φ such that λ(E, φ) = Cross Ratio (s3 , s2 ; s1 , 0),

Fφ (σ ) =

s2 , s2 − s1

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One checks that it then fulfills (6.35) since with e j = s −1 j one has Cross Ratio(s3 , s2 ; s1 , 0) =

e1 − e3 , e1 − e2

s2 e1 = , s2 − s1 e1 − e2

and e1 − e3 e1 > > 0. e1 − e2 e1 − e2 s1 s3 Thus as above we recover the ratios and and the flow line of ϕ using the convexity s2 s2 of σ (B). Finally note that the conditions (6.34) and (6.35) are exclusive and thus allow to decide using the geometric data whether ϕ ∈ σ (A) or ϕ ∈ σ (B). 1>

7. Dualities We show in this section that there are unexpected dualities between the noncommutative spaces R4ϕ in the following cases of Table 5.5: A ↔ B, 2 ↔ 3, 5 ↔ 6, 7 ↔ 8, 10 ↔ 11. Modulo these dualities the fundamental domain gets reduced from an alcove of the root system D3 to the smaller alcove of the root system C3 . 7.1. Semi-cross product.. Let A be a graded algebra and let α ∈ Aut(A) be a symmetry commuting with the grading (i.e. homogeneous of degree 0). Definition 7.1. The semi-cross product A(α) of A by α is the graded vector space A equipped with the bilinear product .α defined by a .α b = a α n (b), ∀a ∈ An , b ∈ A. It is easily verified that this product is associative and that Am .α An ⊂ Am+n so that A(α) is a graded algebra and that if A is unital then A(α) is also unital with the same unit. As pointed out in the introduction, this notion has been analysed in [5] and used there to reduce and complete the classification of regular algebras of dimension 3 (of [3]) ; the graded algebra A(α) is denoted there by Aα and called the twist of A by α. Some basic properties of the semi-cross product are given by the following proposition. Proposition 7.2. Let A be a graded algebra and let α be an automorphism of degree 0 of A. (i) If β is an automorphism of degree 0 of A which commutes with α, then β is also canonically an automorphism of degree 0 of A(α) and one has A(α)(β) = A(α ◦ β). In particular one has A(α)(α −1 ) = A.

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(ii) If A is a quadratic algebra, then A(α) is also a quadratic algebra and its geometric datas (E  , σ  , L ) are deduced from those (E, σ, L) of A as follows: E  = E, σ  = α t ◦ σ, L = L, where α t is induced by the transposed of (α  A1 ). (iii) If A is an involutive algebra with involution x → x ∗ homogeneous of degree 0 (in short, if A is a graded ∗-algebra) and if α commutes with the involution, then one defines an antilinear antimultiplicative mapping x → x ∗α of A(α) onto A(α −1 ) by setting a ∗α = α −n (a ∗ ) for a ∈ An . In particular if α 2 = 1, then A(α) equipped with the involution x → x ∗α is a graded ∗-algebra. Proof. (i) One has for a ∈ An and b ∈ A, β(a .α b) = β(aα n (b)) = β(a)β(α n (b)) = β(a)α n (β(b)) = β(a) .α β(b), which shows that β is an automorphism of A(α). One has also a .α β n (b) = aα n (β n (b)) = a(α ◦ β)n (b) = a .α◦β b, which implies A(α)(β) = A(α ◦ β). (ii) Assume that A is a quadratic algebra, i.e. A = A(V, R) = T (V )/(R), where V is finite-dimensional and where (R) is the two-sided ideal of the tensor algebra T (V ) of V generated by the subspace R of V ⊗ V . Let m denote the product of A and m denote the product of A(α). Since V = A1 = A(α)1 , one has m = m ◦ (Id ⊗ α) on V ⊗ V and thus m(R) = 0 is equivalent to m ((Id ⊗ α −1 )(R)) = 0 from which it follows easily that A(α) = A(V, R  ) = T (V )/(R  ) with R  = (Id ⊗ α −1 )R so A(α) is quadratic. By definition the graph of σ  is the subset   ⊂ P(V ∗ ) × P(V ∗ ) obtained from the subset of V ∗ × V ∗ of pairs (ω, π ), ω = 0, π = 0 such that ω ⊗ π, r = 0, ∀r ∈ R  . Since R  = (Id ⊗ α −1 )R we thus get σ  = α t ◦ σ . (iii) One has for a ∈ An and b ∈ Am , α −(n+m) ((a .α b)∗ ) = α −(n+m) (α n (b∗ )a ∗ ) = α −m (b∗ )α −m (α −n (a ∗ )), and thus (a .α b)∗α = b∗α .α −1 a ∗α , which implies (iii).

 

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More generally if A is finitely generated in degree 1 and finitely presented, i.e. if A = T (V )/(R) with V finite-dimensional and (R) is the two-sided ideal of T (V ) genN R (R ⊂ V ⊗n ) of T (V ), one has A(α) = erated by the graded subspace R = ⊕n=2 n n N R (α), T (V )/(R(α)) with R(α) = ⊕n=2 n Rn (α) = (Id ⊗ α −1 ⊗ · · · ⊗ α −(n−1) )Rn . In particular A(α) is a N -homogeneous algebra whenever A is N -homogeneous. Let us N recall that an algebra A as above is said to be N -homogeneous iff R = R N ⊂ V ⊗ [6,7]. For these algebras, which generalize the quadratic algebras (N = 2), one has a direct extension of the Koszul duality of quadratic algebras as well as a natural generalization of the notion of Koszulity [6,7]. The stability of the corresponding homological notions with respect to the semi-cross product construction will be studied elsewhere. The terminology semi-cross product of A by α adopted here for A(α) comes from the fact that it can be identified with a subalgebra of the crossed product A
α Z, namely the subalgebra generated by the elements x W, x ∈ A1 , where W denotes the new invertible generator of the crossed product defined by W aW −1 = α(a) for a ∈ A. Indeed one has for a ∈ Am b ∈ A p , aW n bW p = aα n (b) W n+ p . If A is a graded ∗-algebra with α a ∗-homomorphism of degree 0, one endows the crossed product of a structure of ∗-algebra by setting W ∗ = W −1 . The involution of the crossed product induces then, by restriction, the antilinear antimultiplicative mapping ∗α : A(α) → A(α −1 ) of (iii) in the above proposition. For the following application, we shall have α 2 = 1, so A(α) can then be identified with the corresponding subalgebra of the crossed product A
α Z/2Z, and if A is a graded ∗-algebra with α a ∗-homomorphism of degree 0, A
α Z/2Z becomes a ∗-algebra by setting W = W ∗ (= W −1 ) and A(α) is a ∗-subalgebra. 7.2. Application to R4ϕ and Sϕ3 . The above construction allows to give a duality between the following cases of Table 5.5: A ↔ B, 2 ↔ 3, 5 ↔ 6, 7 ↔ 8, 10 ↔ 11, where A and B are the two simplices that together complete Case 1). The explicit transformation on the ϕ-parameters is   π π f 1 (ϕ) = π − ϕ1 , − ϕ1 + ϕ2 , − ϕ1 + ϕ3 , (7.1) 2 2 but there are two others which give relevant additional dualities,  π π f 2 (ϕ) = + ϕ2 − ϕ3 , ϕ2 , − ϕ1 + ϕ2 , 2 2  π π f 3 (ϕ) = − ϕ2 + ϕ3 , − ϕ1 + ϕ3 , ϕ3 . 2 2

(7.2) (7.3)

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P’

Z

Q P

0 Fig. 3. The hyperplanes f j (ϕ) = ϕ

They are all involutions f j2 = 1 and on the fundamental domain A ∪ B one has f 1 (A) = B,

f 1 (B) = A,

f 2 (B) = B,

f 3 (A) = A,

with f 1 (0P Q P  Z ) = (Z P Q P  0), f 2 (P Q P  Z ) = (P Q Z P  ) and f 3 (O P Q P  ) = (P  P Q O). Thus the duality f 2 (resp. f 3 ) operates only in B (resp. A). Moreover the transformations f 2 and f 3 are conjugate under f 1 i.e. f 2 = f 1 ◦ f 3 ◦ f 1 . By Proposition 2.4, 2) any element v of the centralizer C ⊂ SO(4) of the diagonal matrices in SU(4) defines an automorphism of R4ϕ acting by v on the generators. We thus let α j ∈ AutA correspond to the diagonal matrices with respective diagonals given by α1 : (1, 1, −1, −1), α2 : (1, −1, 1, −1), α3 : (1, −1, −1, 1). (7.4) Proposition 7.3. For j ∈ {1, 2, 3} one has canonical ∗-isomorphisms     ρ j : Calg R4f j (ϕ) → Calg R4ϕ (α j ) which preserve the central element S 3f j (ϕ)  Sϕ3 (α j ).



xµ2 and induce corresponding isomorphisms

Proof. One writes explicitly the isomorphisms as follows on the canonical (self-adjoint) µ generators x j of Calg (R4f j (ϕ) ) (which are the canonical noncommutative coordinates of

R4f j (ϕ) ):

ρ1 (x10 ) = x 1 W1 , ρ1 (x11 ) = x 0 W1 , ρ1 (x12 ) = −i x 2 W1 , ρ1 (x13 ) = −i x 3 W1 , ρ2 (x20 ) = x 2 W2 , ρ2 (x21 ) = −i x 3 W2 , ρ2 (x22 ) = x 0 W2 , ρ2 (x23 ) = i x 1 W2 , ρ3 (x30 ) = x 3 W3 , ρ3 (x31 ) = −i x 2 W3 , ρ3 (x32 ) = i x 1 W3 , ρ3 (x33 ) = x 0 W3 . Using the signs in (7.4) one checks that one gets ∗-isomorphisms Calg (R4f j (ϕ) )    µ  Calg (R4ϕ )(α j ) and that one has ρ j ( µ (x j )2 ) = µ (x µ )2 . 

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Corollary 7.4. (1) Let ϕ be generic and (E, σ, L) be the geometric data of Calg (R4ϕ ). Then the geometric data of Calg (R4f j (ϕ) ) is (E, σ + τ j , L), where τ j ∈ T2 (E). (2) The hyperplane f j (ϕ) = ϕ is globally invariant under the flow Z . (3) Let ϕ ∈ A ∪ B be generic, then it belongs to the hyperplane f j (ϕ) = ϕ (with j = 2 on B and j = 3 on A) iff the translation σ fulfills σ ∈ T4 (E). Proof. 1) Using Proposition 7.2 the required equality follows if one shows that the action of α j on E is indeed given by a translation τ j ∈ T2 (E). This can be checked directly using ϑ-functions, i.e. Proposition 9.3.   These new symmetries suggest that one extends the group (T)
W of Sect. 3.3 to include the T j . This amounts to adding the following new roots to the root system  of Sect. 3.3. To the ±ψµ ± ψν we adjoin the ±2ψµ . This is in fact the same as the replacement S O(6) → Sp(3), of the compact group S O(6) by the symplectic group Sp(3) (i.e. of D3 by C3 ). The corresponding Weyl group is now O(3, Z). The description of Sp(3) is obtained as follows using the natural representation ρ : H →   M2 (C) of the field of quaternions H as two by two matrices of the form α β with α, β ∈ C. By definition Sp(3) is the group of three by three matrices −β¯ α¯ Q ∈ M3 (H) whose image ρ(Q) ∈ M6 (C) is unitary. Thus it contains as a subgroup {Q ∈ M3 (H) | ρ(Q) ∈ U (6) ∩ M6 (R)}, which is isomorphic to U (3) and is contained in S O(6) but has only 9 parameters. 8. The Algebras Calg (R4ϕ ) in the Nongeneric Cases In this section we shall describe the algebras Calg (R4ϕ ) in the nongeneric cases using the above dualities. 8.1. Uq (sl(2)), Uq (su(2)) and their homogenized versions. In this section for q ∈ C∗ , Uq (sl(2)) is considered as an associative algebra while for |q| = 1 or q ∈ R∗ , Uq (su(2)) is considered as a ∗-algebra which is a real form of Uq (sl(2)). The coalgebra aspect plays no role in the following and we refer to [23] for a very complete discussion of these topics. It is convenient to start by the homogenized version. For q ∈ C\{−1, 0, 1} we define Uq (sl(2))hom to be the quadratic algebra generated by 4 elements X + , X − , K + , K − with relations K+K− K +X+ K +X− K −X+ K −X−

= K − K +, = q X + K +, = q −1 X − K + , = q −1 X + K − , = q X − K −, (K + )2 − (K − )2 [X + , X − ] = . q − q −1

(8.1) (8.2) (8.3) (8.4) (8.5) (8.6)

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The “classical limit” U (sl(2))hom = U1 (sl(2))hom for q = 1 is obtained by setting q = 1 + ε and K ± = X 0 ± 2ε X 3 . Letting ε → 0, the relations read then [X 0 , X 3 ] = 0,

(8.7)

[X 0 , X + ] = 0,

(8.8)

−

[X , X ] = 0, 0

(8.9)

[X , X ] = 2X X , 3

+

0

−

+

−

[X , X ] = −2X X , 3

−

0

[X , X ] = X X . +

0

3

(8.10) (8.11) (8.12)

To obtain Uq (sl(2)), one notices that relations (8.1) …(8.5) imply that K + K − is central so that one may add the inhomogeneous relation K + K − = ?,

(8.13)

which together with relations (8.1) …(8.6) defines Uq (sl(2)), i.e. Uq (sl(2)) is (as associative algebra) the quotient of Uq (sl(2))hom by the two-sided ideal generated by K + K −−?. Similarily one notices that relations (8.7)…(8.11) imply that X 0 is central and the universal enveloping algebra U (sl(2)) is obtained by adding X 0 = 1 to the relations (8.7) …(8.12). One has of course limq→1 Uq (sl(2)) = U (sl(2)). For q ∈ C\R with |q| = 1 the real version Uq (su(2))hom of Uq (sl(2))hom is obtained by endowing the algebra with the unique antilinear antimultiplicative involution such that (X ± )∗ = X ∓ , ± ∗

∓

(K ) = K ,

(8.14) (8.15)

while for q ∈ R\{−1, 0, 1}, Uq (su(2))hom corresponds to the unique antilinear antimultiplicative involution such that (X ± )∗ = X ∓ , (K ± )∗ = K ± ,

(8.16)

which gives (X ± )∗ = X ∓ , (X 0 )∗ = X 0 ,

(8.17)

(X 3 )∗ = X 3 ,

(8.18)

for the limiting case q = 1, i.e. for U (su(2))hom . These involutions pass to the quotient to define the ∗-algebras Uq (su(2)) for |q| = 1 or q ∈ R∗ . Notice that the involution of Uq (su(2)) is obtained from the one of U (su(2)) 1 3 by setting K ± = q ± 2 X (the relations in terms of X ± and X 3 differ of course).

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8.2. The algebras in the nongeneric cases. We now identify the algebra Calg (R4ϕ ) in the Cases 2 to 11 of Table 5.5. 2. Even face : ϕ1 = ϕ2 = ϕ, ϕ − ϕ3 ∈ π2 Z, ϕk ∈ π2 Z. By setting X ± = x 1 ± i x 2,

(8.19)

the relations (2.24), (2.25) of the algebra read then [x 0 , x 3 ] = 0, cos(ϕ)[x 0 , X + ] = sin(ϕ − ϕ3 )(x 3 X + + X + x 3 ), cos(ϕ − ϕ3 )[x 3 , X + ] = −sin(ϕ)(x 0 X + + X + x 3 ), cos(ϕ)[x 0 , X − ] = −sin(ϕ − ϕ3 )(x 3 X − + X − x 3 ), cos(ϕ − ϕ3 )[x 3 , X − ] = sin(ϕ)(x 0 X − + X − x 0 ), [X + , X − ] = −2sin(ϕ3 )(x 0 x 3 + x 3 x 0 ),

(8.20) (8.21) (8.22) (8.23) (8.24) (8.25)

and since (8.19) implies (X + )∗ = X − , one sees that relations (8.21) and (8.22) are the adjoints of relations (8.23) and (8.24) respectively. We now distinguish the following 2 regions R and R  : R : π2 > ϕ > ϕ3 ≥ 0. By setting K ± = (2tan(ϕ3 ))1/2 ((sin(2ϕ))1/2 x 0 ± i(sin(2(ϕ − ϕ3 )))1/2 x 3 )

(8.26)

the relations (8.20) to (8.25) become (8.1) to (8.6) with q=

1 − i(tan(ϕ − ϕ3 )tan(ϕ))1/2 , 1 + i(tan(ϕ − ϕ3 )tan(ϕ))1/2

(8.27)

so Calg (R4ϕ ) coincides then with Uq (su(2))hom for |q| = 1, q = ±1 as ∗-algebra (one has (K + )∗ = K − ). R  : π2 > ϕ > 0 and π2 + ϕ > ϕ3 > ϕ. By setting K ± = (2tan(ϕ3 ))1/2 ((sin(2ϕ))1/2 x 0 ± (sin(2(ϕ3 − ϕ)))1/2 x 3 ),

(8.28)

the relations (8.20) to (8.25) become (8.1) to (8.6) with q=

1 − (tan(ϕ3 − ϕ)tan(ϕ))1/2 , 1 + (tan(ϕ3 − ϕ)tan(ϕ))1/2

(8.29)

so Calg (R4ϕ ) coincides then with Uq (su(2))hom for q ∈]−1, 0[∪]0, 1[ (one has (K ± )∗ = K ± here). In general for Case 2, it is easy to see that Calg (R4ϕ ) is isomorphic as ∗-algebra either to an algebra of R or of R  . Notice that q > 1, for instance, is the same as q ∈]0, 1[ by exchanging q and q −1 and K + and K − . Thus for Case 2 the Calg (R4ϕ ) are the Uq (su(2))hom . 3. Odd face : ϕ1 = π2 , ϕ2 − ϕ3 ∈ π2 Z, ϕ2 ∈ π2 Z, ϕ3 ∈ π2 Z. Using the analysis of Sect. 7, one sees that the cases corresponding to the plane ϕ1 = π2 are in α3 -duality with the cases corresponding to the plane ϕ2 = ϕ3 . On the other hand, the cases corresponding to the plane ϕ2 = ϕ3 are the same as the cases corresponding to the plane ϕ1 = ϕ2 . So finally (taking into account the forbidden values π2 Z) one sees that Case 3

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(odd face) is obtained by duality (in the sense of Sect. 9) from Case 2 (even face) i.e. from the Uq (su(2))hom . 4. α ⊥ β (0, 1) : ϕ1 = π2 , ϕ2 = ϕ3 = ϕ ∈ π2 Z. This case is singular in the sense that one of the 6 relations is missing, namely: cos(ϕ1 )[x 0 , x 1 ] = isin(ϕ2 − ϕ3 )[x 2 , x 3 ]+ , which gives “0 = 0”. This implies exponential growth. This is a singular limiting case of Uq (su(2))hom for q = 0 which separates the regions 1 < q < 0 and 0 < q < 1 of Case 2. 5. α − β (0, 1) : ϕ1 = π2 , ϕ2 = π2 , ϕ3 ∈ π2 Z. This case is obtained by α3 -duality (Sect. 7) from ϕ1 = ϕ2 = ϕ3 ∈ π2 Z which is Case 6 below. It corresponds to U−1 (su(2))hom . 6. α − β (0, 0) : ϕ1 = ϕ2 = ϕ3 ∈ π2 Z. The relations (2.24), (2.25) reduce in this case to the relations (8.7) to (8.12) by setting X ± = x 1 ± i x 2 , X 3 = 2x 3 and X 0 = −2sin(ϕ1 )x 0 . Thus in this case the ∗-algebra is isomorphic to U (su(2))hom = U1 (su(2))hom . 7. α ⊥ β (0, 0) : ϕ1 = ϕ2 = − 21 θ ∈ π2 Z, ϕ3 = 0. This is the “θ -deformation” studied in [19] and [15]. By setting z 1 = x 0 + i x 3 , z 2 = x 1 + i x 2 , z¯ 1 = (z 1 )∗ , z¯ 2 = (z 2 )∗

(8.30)

the relations (2.24), (2.25) read ⎧ 1 2 z z ⎪ ⎪ ⎨ 1 2 z¯ z z 1 z¯ 2 ⎪ ⎪ ⎩ 1 2 z¯ z¯

= eiθ z 2 z 1 = e−iθ z 2 z¯ 1 = e−iθ z¯ 2 z 1 = eiθ z¯ 2 z¯ 1

(8.31)

and we refer to Part I [15] for more details and generalizations of this algebra. 8. α ⊥ β (1, 1) : ϕ1 = π2 + ϕ, ϕ2 = π2 , ϕ3 = ϕ ∈ π2 Z. This case is obtained from the preceding Case 7 (θ -deformation) by α1 -duality as explained in Sect. 7. 9. ϕ1 = ϕ2 = ϕ3 = π2 . This is the most singular case, 3 relations are missing (i.e. reduce to “0 = 0”) among the 6 relations (2.24), (2.25). The algebra has exponential growth. 10. ϕ1 = ϕ2 = π2 , ϕ3 = 0. This case which is at the intersection of the lines carrying Case 7 and Case 8 is obtained by α3 -duality (Sect. 7) from the next Case 11. 11. ϕ1 = ϕ2 = ϕ3 = 0. This is the “classical case”, the relations (2.24), (2.25) reduce to x µ x ν = x ν x µ for µ, ν ∈ {0, 1, 2, 3} so the algebra reduces to the algebra of polynomials C[x 0 , x 1 , x 2 , x 3 ]. Remark. One can go much further and describe the C ∗ -algebras corresponding to the noncommutative spheres Sϕ3 in all these degenerate cases. It is important in that respect to classify the discrete series besides the obvious continuous series of representations.

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9. The Complex Moduli Space and its Net of Elliptic Curves We first explain in this section a striking coincidence in the generic case between the geometric data of R4ϕ and the fiber of the double cover 4.6. This takes place in the real moduli space and leads us to introduce the complex moduli space in which the equality between the two elliptic curves makes full sense. We then describe the geometric structure of the complex moduli space as a net of elliptic curves in three dimensional projective space and exhibit a presentation of the algebras making the equality “fiber = characteristic” manifest. We showed above in the proof of Lemma 6.2 that the geometric data of R4ϕ can be interpreted as the elliptic curve   3  2 E 2 = (X, Y ) | Y = (X − ei ) (9.1) 1

with e j = s −1 j and endowed with a translation σ sending the point at ∞ to a point of E 2 whose X coordinate vanishes.5 One can write E 2 in the equivalent form,   3  2 E 3 (ϕ) = (X, Y ) | Y = (X s j − 1) . (9.2) 1

In this form this equation is the same as the one involved in the double cover 4.6. More precisely one gets Proposition 9.1. Let ϕ be generic, and Fiber (ϕ) = {ϕ  generic | Jm (ϕ  ) = Jm (ϕ), ∀k}. a) There is a natural isomorphism ϕ : Fiber (ϕ)  E 3 (ϕ) ∩ (R × R∗ ) determined by the equalities  tan(ϕ j ). (9.3) X (ϕ  ) = s j (ϕ  )/s j , ∀ j, Y (ϕ  ) = b) The closure Fiber (ϕ) is the union Fiber (ϕ)∪W (P) and ϕ extends to an isomorphism ϕ : Fiber (ϕ) → E 3 (ϕ) ∩ P2 (R). Proof. a) Let ϕ  ∈ Fiber (ϕ) then by Lemma 4.7 the ratio X = s j (ϕ  )/s j is independent of j. Moreover one has 3 

(X s j − 1) =



tan(ϕ j )

2

,

1

thus the map is well defined. Conversely given (X, Y ) ∈ E 3 (ϕ) ∩ (R × R∗ ) one lets sk = X sk and tk = Y (sk −1)−1 . This determines uniquely the ϕ j such that tan(ϕ j ) = t j . b) Note that Fiber (ϕ) is not connected but falls in four connected components each a flow line of the scaling flow X . These correspond to the four components of E 3 (ϕ) ∩ (R × R∗ ), i.e. of the set of real points of E 3 (ϕ) which are not of two torsion. The inverse map −1 ϕ extends to the four points {∞, e1 , e2 , e3 } of E 3 (ϕ) and one thus obtains a 5 The two choices give isomorphic pairs (E , σ ). 2

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O

S

R

Q

R

S

O

Q

Fig. 4. The elliptic curve E 3 and the fiber for ϕ ∈ A and ϕ ∈ B

compactification of Fiber (ϕ) by adding the four points {P, Q, R, S} in the orbit W (P). The above map ϕ extends continuously to the compactification and the four points {P, Q, R, S} map to the four two torsion points {∞, e1 , e2 , e3 } of E 3 (ϕ) and in that order for ϕ ∈ A as in Fig. 4. The situation is similar on B but the order of e2 and e3 is reversed as well as that of R and S. Moreover the point 0 is now between Q and S instead of being on the left of S as in case ϕ ∈ A (see Fig. 4).   In fact the action of the discrete symmetry given by the Klein group H ⊂ W preserves globally Fiber (ϕ) since it does not alter the Jm (ϕ). In fact this discrete symmetry admits a simple interpretation in terms of translations of order two τ ∈ T2 (E 3 ) as follows, where we let k j ∈ H be given by k1 (ϕ) = (−ϕ1 , ϕ3 − ϕ1 , ϕ2 − ϕ1 ). Proposition 9.2. 1) Given ϕ  ∈ Fiber (ϕ) the map (X, Y ) → (λ X, Y ) for λ = s j (ϕ  )/s j , ∀ j is an isomorphism E 3 (ϕ  )  E 3 (ϕ) compatible with the isomorphisms ϕ and ϕ  of Lemma 9.1. 2) Fiber (ϕ) is globally invariant under k j and under the above isomorphism Fiber (ϕ)  R2 ∩ E 3 (ϕ) the action of k j is given by the translation by the two torsion element (e j , 0) ∈ E 3 (ϕ). Proof. 1) By construction the s j (ϕ  ) are proportional to the s j , so the conclusion follows looking at Definition 9.3 of the identification Fiber (ϕ)  R2 ∩ E 3 (ϕ). 2) Using 1) it is enough to show that the point of E 3 (ϕ) associated to k1 (ϕ) is obtained from (1, t1 t2 t3 ) ∈ E 3 (ϕ) (with t j = tan(ϕ j )) by the translation of order two associated to (e1 , 0) ∈ E 3 (ϕ). One checks that the effect of k1 on the s j (ϕ) is to multiply  tan(ϕ j ) is to replace it by all of them by X  = (1 + t12 )/(s2 s3 ). Its effect on Y = Y  = − t1 (t3 − t1 ) (t2 − t1 )/(s2 s3 ). One then checks by direct computation that the line joining the points (1, t1 t2 t3 ) ∈ E 3 (ϕ) and (X  , −Y  ) ∈ E 3 (ϕ) intersects E 3 (ϕ) in the other point (e1 , 0). The

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73

P

S R Q

Q R S

P Fig. 5. The flow lines

result follows since colinearity of three points A, B, C on the elliptic curve E 3 means A + B + C = 0 in that abelian group, while the opposite of A = (X, Y ) is −A = (X, −Y ).   The above proposition shows that one should give a direct definition of E 3 (ϕ) depending only upon the Jm (ϕ) rather than the s j (ϕ). There is also a strong reason to define in a natural manner the complex points of E 3 (ϕ). Indeed the translation σ corresponds to points where the coordinate X = 0 and the corresponding equation for Y is Y 2 = −1, which does not admit real solutions. 9.1. Complex moduli space MC . We shall show now that there is a natural way of extending the moduli space from the real to the complex domain. The E 3 (ϕ) then appear as a net of degree 4 elliptic curves in P3 (C) having 8 points in common. These elliptic curves will turn out to play a fundamental role and to be closely related to the elliptic curves of the geometric data of the quadratic algebras which their elements label. To extend the moduli space to the complex domain we start with the relations defining the involutive algebra Calg (S 3 (
)) and take for
the diagonal matrix with −1
µ µ := u µ ,

(9.4) µ

where (u 0 , u 1 , u 2 , u 3 ) are the coordinates of u ∈ P3 (C). Using yµ :=
ν z ν one obtains the homogeneous defining relations in the form, u k yk y0 − u 0 y0 yk + u  y ym − u m ym y = 0, u k y0 yk − u 0 yk y0 + u m y ym − u  ym y = 0

(9.5)

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for any cyclic permutation (k, , m) of (1,2,3). The inhomogeneous relation becomes,  (9.6) u µ yµ2 = 1, 3 (u)) only depends upon the class of u ∈ P (C) and the corresponding algebra Calg (SC 3 (see the remark at the end of 9.2). We let Calg (C4 (u)) be the quadratic algebra defined by the six relations (9.5). Taking u µ = e2i ϕµ , ϕ0 = 0, for all µ and x µ := ei ϕµ yµ , we obtain the defining relations of Calg (R4ϕ ) (except for x µ∗ = x µ which allows to pass from C4 (u) to R4ϕ ). Thus the torus T A sits naturally in P3 (C) as

T A = {u ∈ P3 (C) | |u µ | = |u ν |, ∀µ, ν}.

(9.7)

In terms of homogeneous parameters the functions Jm (ϕ) read as Jm (ϕ) = tan(ϕ0 − ϕk )tan(ϕ − ϕm )

(9.8)

for any cyclic permutation (k, , m) of (1,2,3), and extend to the complex domain u ∈ P3 (C) as, Jm (u) =

(u 0 + u  )(u m + u k ) − (u 0 + u m )(u k + u  ) . (u 0 + u k )(u  + u m )

(9.9)

It follows easily from the argument of Proposition 4.5 that for generic values of u ∈ P3 (C) the quadratic algebra Calg (C4 (u)) only depends upon Jk (u). We thus define the complex fiber as F(u) := {v ∈ P3 (C) | Jk (v) = Jk (u)}.

(9.10)

Let then (u) = (a, b, c) = {(u 0 + u 1 )(u 2 + u 3 ), (u 0 + u 2 )(u 3 + u 1 ), (u 0 + u 3 )(u 1 + u 2 )} (9.11)  be the three roots of the Lagrange resolvent of the 4th degree equation (x − u j ) = 0. We view  as a map  : P3 (C)\S → P2 (C), (9.12) where S is the following set of 8 points p0 = (1, 0, 0, 0), p1 = (0, 1, 0, 0), p2 = (0, 0, 1, 0), p3 = (0, 0, 0, 1), (9.13) q0 = (−1, 1, 1, 1), q1 = (1, −1, 1, 1), q2 = (1, 1, −1, 1), q3 = (1, 1, 1, −1). The points q j belong to the torus T A of (9.7), they correspond to the orbit W (P) = (P Q RS). We extend the generic definition (9.10) to arbitrary u ∈ P3 (C)\S and define Fu in general as the union of S with the fiber F(u) of  through u. It can be understood geometrically as follows: Let N be the net of quadrics in P3 (C) which contain S. Given u ∈ P3 (C)\S the elements of N which contain u form a pencil of quadrics with base locus ∩ {Q | Q ∈ N , u ∈ Q} = Yu , (9.14) which is an elliptic curve of degree 4 containing S and u. One has Yu = Fu .

(9.15)

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75

q1 p3

p0 q2

p2

q3

q0 p1

Fig. 6. The Elliptic Curve Fu ∩ P3 (R)

With (u) = (s1 , s2 , s3 ), an explicit isomorphism of the elliptic curve Fu with the elliptic curve  E 3 = {(X, Y ) | Y 2 = (X s j − 1)} of (9.2) is given by (X, Y ) → u, (u k − u 0 )(X sk − 1) − i (u 0 + u k ) Y = 0, ∀k.

(9.16)

Under this isomorphism the point at infinity in E 3 maps to q0 ∈ Fu . 9.2. Notations for ϑ-functions. Let us fix our notations for elliptic ϑ-functions. We fix τ ∈ H a complex number of strictly positive imaginary part and let q = eπiτ , so that |q| < 1. The basic ϑ-function is  2 ϑ3 (z) = q n e2πinz . (9.17) Z

It is periodic in z with period 1 and is (up to scale) the only holomorphic section on E = C/L, L = Z + τ Z of the line bundle associated to the periodicity conditions ξ(z + 1) = ξ(z), ξ(z + τ ) = q −1 e−2πi z ξ(z).

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In particular it is equal (up to scale) to the infinite product ∞ 

(1 + q 2n−1 e2πi z )(1 + q 2n−1 e−2πi z ),

1

and only vanishes at ω2 = 21 + τ2 modulo L. The three other ϑ-functions are deduced from ϑ3 (z) by the translations of order two of E, more precisely one lets,     1 1 1 , ϑ4 (z) = ϑ3 z + . (9.18) iϑ1 (z) = q 4 eπi z ϑ3 (z+ω2 ), ϑ2 (z) = ϑ1 z + 2 2 They all are holomorphic sections on E = C/L, L = Z + τ Z of the line bundles associated to the periodicity conditions of the form ξ(z + 1) = ± ξ(z), ξ(z + τ ) = ± q −1 e−2πi z ξ(z), and it follows that the linear span of their squares ϑ 2j (z) is a complex vector space of dimension two since all are holomorpic sections of a line bundle of degree two. It follows in particular that given any two theta functions the square of any other is a linear combination of the squares of the first two. The relevant coefficients are easy to compute from the values ϑ 2j (0) and if one lets k=

ϑ22 (0) ϑ32 (0)

, k =

ϑ42 (0) ϑ32 (0)

,

one gets ϑ42 (z) = kϑ12 (z)+k  ϑ32 (z), ϑ22 (z) = −k  ϑ12 (z)+k ϑ32 (z), ϑ12 (z) = k ϑ42 (z)−k  ϑ22 (z). (9.19) The λ-function of Jacobi is given by λ(τ ) = k 2 and the ℘-function of Weierstrass by ϑ 2 (z) ℘ (z) = α 42 + β, (9.20) ϑ1 (z) up to irrelevant normalization constants α and β. The main identities we shall use for ϑ-functions are the sixteen theta relations recalled in Appendix 2. 9.3. Fiber = characteristic variety, and the birational automorphism σ of P3 (C). We shall now give, for generic values of (a, b, c) a parametrization of Fu by ϑ-functions. We start with the equations for Fu , (u 0 + u 1 )(u 2 + u 3 ) (u 0 + u 2 )(u 3 + u 1 ) (u 0 + u 3 )(u 1 + u 2 ) = = , a b c

(9.21)

and we diagonalize the above quadratic forms as follows: (u 0 + u 1 ) (u 2 + u 3 ) = Z 02 − Z 12 , (u 0 + u 2 ) (u 3 + u 1 ) = Z 02 − Z 22 , (u 0 + u 3 ) (u 1 + u 2 ) = Z 02 − Z 32 ,

(9.22)

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77

where (Z 0 , Z 1 , Z 2 , Z 3 ) = M.u,

(9.23)

where M is the involution, ⎡

1 1 ⎢1 M := ⎣ 2 1 1

1 1 −1 −1

1 −1 1 −1

⎤ 1 −1 ⎥ . −1 ⎦ 1

(9.24)

In these terms the equations for Fu read Z 2 − Z 22 Z 2 − Z 32 Z 02 − Z 12 = 0 = 0 . a b c

(9.25)

Let now τ ∈ C, Im τ > 0 and η ∈ C be such that one has, modulo projective equivalence,   ϑ2 (0)2 ϑ3 (0)2 ϑ4 (0)2 (a, b, c) ∼ , (9.26) , , ϑ2 (η)2 ϑ3 (η)2 ϑ4 (η)2 where ϑ1 , ϑ2 , ϑ3 , ϑ4 are the theta functions associated as above to τ . More precisely let (a, b, c) be distinct non-zero complex numbers and λ=

a c−b , b c−a

p=

a , a−c

(9.27)

and let τ ∈ C, Im τ > 0 such that λ(τ ) = λ, where λ is Jacobi’s λ function (cf. Subsect. 6.2). Let then η ∈ C be such that with ω1 = 21 , ω3 = τ2 , one has ℘ (η) − ℘ (ω3 ) = p, ℘ (ω1 ) − ℘ (ω3 ) where ℘ is the Weierstrass ℘-function for the lattice L = Z + Zτ ⊂ C. This last equality does determine η only up to sign and modulo the lattice L but this ambiguity does not affect the validity of (9.26) which one checks using the basic properties of ϑ-functions recalled in Subsect. 9.2. Indeed in these terms one has λ(τ ) = k 2 which gives the first equality in (9.26) with (a, b, c) replaced by the right hand side of (9.26). Using (9.20) one gets the second equality since for any z one has ϑ4 (z)2 ℘ (z) − ℘ (ω3 ) = k . ℘ (ω1 ) − ℘ (ω3 ) ϑ1 (z)2 Let then τ and η be fixed by the above conditions; one gets: Proposition 9.3. The following define isomorphisms of C/L with Fu ,   ϑ1 (2z) ϑ2 (2z) ϑ3 (2z) ϑ4 (2z) , , , = (Z 0 , Z 1 , Z 2 , Z 3 ) ϕ(z) = ϑ1 (η) ϑ2 (η) ϑ3 (η) ϑ4 (η) and ψ(z) = ϕ(z − η/2).

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Proof. Up to an affine transformation, ϕ (and ψ are) is the classical projective embedding of C/L in P3 (C). Thus we only need to check that the biquadratic curve Im ϕ = Im ψ is given by (9.25). It is thus enough to check (9.25) on ϕ(z). This follows from the basic relations (9.19) written as ϑ32 (z)ϑ22 (0) = ϑ22 (z)ϑ32 (0) + ϑ12 (z)ϑ42 (0)

(9.28)

ϑ42 (z)ϑ32 (0) = ϑ12 (z)ϑ22 (0) + ϑ32 (z)ϑ42 (0),

(9.29)

and Z 2 −Z 2

Z 2 −Z 2

Z 2 −Z 2

Z 2 −Z 2

which one uses to check 0 a 1 = 0 b 2 and 0 b 2 = 0 c 3 respectively. Let us check the first one using (9.28) to replace all occurrences of ϑ32 (2z) and ϑ32 (η) by the value given by (9.28). One gets Z 02 − Z 22 ϑ 2 (2z) ϑ32 (η) ϑ 2 (2z) ϑ 2 (2z) ϑ22 (η) ϑ12 (2z)ϑ42 (0) = 12 − 32 = 12 + 2 2 b ϑ1 (η) ϑ3 (0) ϑ3 (0) ϑ1 (η) ϑ22 (0) ϑ2 (0)ϑ32 (0) $ % ϑ22 (2z) ϑ12 (2z)ϑ42 (0) − + 2 ϑ22 (0) ϑ2 (0)ϑ32 (0) =

ϑ12 (2z) ϑ22 (η) ϑ12 (η) ϑ22 (0)

−

ϑ22 (2z) ϑ22 (0)

=

Z 02 − Z 12 . a

Let us check the second one using (9.29) to replace all occurrences of ϑ42 (2z) and ϑ42 (η) by the value given by (9.29). One gets ϑ 2 (2z) ϑ42 (η) Z 02 − Z 32 ϑ42 (2z) ϑ12 (2z) ϑ32 (η) ϑ12 (2z)ϑ22 (0) = 12 − = + 2 c ϑ1 (η) ϑ42 (0) ϑ42 (0) ϑ12 (η) ϑ32 (0) ϑ3 (0)ϑ42 (0) $ % ϑ32 (2z) ϑ12 (2z)ϑ22 (0) − + 2 ϑ32 (0) ϑ3 (0)ϑ42 (0) ×

ϑ12 (2z) ϑ32 (η) ϑ12 (η)

ϑ32 (0)

−

ϑ32 (2z) ϑ32 (0)

=

Z 02 − Z 22 . b

  The elements of S are obtained from the following values of z:      1 1 τ τ ψ(η) = p0 , ψ η + = p1 , ψ η + + = p2 , ψ η + = p3 2 2 2 2 and ψ(0) = q0 , ψ

    τ  1 1 τ = q1 , ψ + = q2 , ψ = q3 . 2 2 2 2

(9.30)

(9.31)

(We used M −1 .ψ to go back to the coordinates u µ , but note that M −1 = M and M(q j ) = q j .) Let H ∼ Z2 × Z2 be the Klein subgroup of the symmetric group S4 acting on P3 (C) by permutation of the coordinates (u 0 , u 1 , u 2 , u 3 ).

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79

For ρ in H one has  ◦ ρ = , so that ρ defines for each u an automorphism of Fu . For ρ in H the matrix Mρ M −1 is diagonal with ±1 on the diagonal and the quasiperiodicity of the ϑ-functions allows to check that these automorphisms are translations on Fu by the following 2-torsion elements of C/L,  ρ=  ρ=

0 1

1 0

2 3

3 2

0 2

1 3

2 0

3 1

 is translation by ω1 = 21 , (9.32)

 is translation by ω2 =

1 2

+ τ2 .

Let O ⊂ P3 (C) be the complement of the 4 hyperplanes {u µ = 0} with µ ∈ {0, 1, 2, 3}. −1 −1 −1 Then (u 0 , u 1 , u 2 , u 3 ) → (u −1 0 , u 1 , u 2 , u 3 ) defines an involutive automorphism I of O and since one has −1 −1 −1 −1 (u −1 0 + u k )(u  + u m ) = (u 0 u 1 u 2 u 3 ) (u 0 + u k )(u  + u m ),

(9.33)

it follows that  ◦ I = , so that I defines for each u ∈ O\{q0 , q1 , q2 , q3 } an involutive automorphism of Fu ∩ O which extends canonically to Fu . Note in fact that, as a birational map I continues to make sense on the complement of the 6 lines µ,ν = {u | u µ = 0, u ν = 0} for µ, ν ∈ {0, 1, 2, 3} using (u 0 , u 1 , u 2 , u 3 ) → (u 1 u 2 u 3 , u 0 u 2 u 3 , u 0 u 1 u 3 , u 0 u 1 u 2 ). Proposition 9.4. The restriction of I to Fu is the symmetry ψ(z) → ψ(−z) around any of the points qµ ∈ Fu in the elliptic curve Fu . This symmetry, as well as the above translations by two torsion elements does not refer to a choice of origin in the curve Fu . The proof follows from identities on theta functions but it can be seen directly using the isomorphism E 3  Fu of (9.16). Indeed the symmetry around q0 ∈ Fu corresponds to the transformation (X, Y ) → (X, −Y ) on E 3 and −1 −1 −1 the isomorphism (9.16) carries this back to (u 0 , u 1 , u 2 , u 3 ) → (u −1 0 , u 1 , u 2 , u 3 ) as one checks directly dividing each of Eqs. (9.16) (u k −u 0 )(X sk −1)− i (u 0 +u k ) Y = 0 by u 0 u k to get the same equation but with −Y instead of Y for the u −1 j . The torus T A of (9.7) gives a covering of the real moduli space M. For u ∈ T A , the point (u) is real with projective coordinates (u) = (s1 , s2 , s3 ), sk := 1 + t tm , tk := tan(ϕk − ϕ0 ).

(9.34)

The corresponding fiber Fu is stable under complex conjugation v → v and the intersection of Fu with the real moduli space is given by, FT (u) = Fu ∩ T A = {v ∈ Fu |I (v) = v}.

(9.35)

The curve Fu is defined over R and (9.35) determines its purely imaginary points. Note that FT (u) (9.35) is invariant under the Klein group H and thus has two connected components; we let FT (u)0 be the component containing q0 . The real points, {v ∈ Fu |v = v} = Fu ∩ P3 (R) of Fu do play a complementary role in the characteristic variety as we shall see below.

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Our aim now is to show that for u ∈ P3 (C) generic, there is an astute choice of generators of the quadratic algebra Au = Calg (C4 (u)) for which the characteristic variety E u actually coincides with the fiber variety Fu and to identify the corresponding automorphism σ . Since this coincidence no longer holds for non-generic values it is a quite remarkable fact which we first noticed by comparing the j-invariants of these two elliptic curves. Let u ∈ P3 (C) be generic; we perform the following change of generators: y0 = y1 = y2 = y3 =

√ √ √ √

u1 − u2 u0 + u2 u0 + u3 u0 + u1

√ √ √ √

u2 − u3

u2 − u3

√ √

u 3 − u 1 Y0 ,

u 0 + u 3 Y1 ,

u3 − u1

√ u 0 + u 1 Y2 ,

u1 − u2

√ u 0 + u 2 Y3 .

(9.36)

We let Jm be as before, given by (9.9), J12 =

a−b b−c c−a , J23 = , J31 = , c a b

(9.37)

with a, b, c given by (9.11). Finally let eν be the 4 points of P3 (C) whose homogeneous coordinates (Z µ ) all vanish but one. Theorem 9.5. 1) In terms of the Yµ , the relations of Au take the form [Y0 , Yk ]− = [Y , Ym ]+ , [Y , Ym ]− = −Jm [Y0 , Yk ]+

(9.38) (9.39)

for any k ∈ {1, 2, 3}, (k, , m) being a cyclic permutation of (1,2,3) 2) The characteristic variety E u is the union of Fu with the 4 points eν . 3) The automorphism σ of the characteristic variety E u is given by ψ(z) → ψ(z − η)

(9.40)

on Fu and σ = Id on the 4 points eν . 4) The automorphism σ is the restriction to Fu of a birational automorphism of P3 (C) independent of u and defined over Q. The similarity between the above presentation and the Sklyanin one (cf. (4.10), (4.12)) is misleading, indeed for the latter all the characteristic varieties are contained in the same quadric (cf. [29] § 2.4) 

xµ2 = 0

and can’t of course form a net of essentially disjoint curves.

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81

Proof. By construction E u = {Z | Rank N (Z ) < 4}, where ⎛

Z1

⎜ ⎜ Z2 ⎜ ⎜ ⎜ ⎜ Z3 ⎜ N (Z ) = ⎜ ⎜ ⎜ (b − c)Z 1 ⎜ ⎜ ⎜ ⎜ (c − a)Z 2 ⎝ (a − b)Z 3

−Z 0

Z3

Z2

Z3

−Z 0

Z1

Z2

Z1

−Z 0

(b − c)Z 0

−a Z 3

a Z2

bZ 3

(c − a)Z 0

−bZ 1

−cZ 2

cZ 1

(a − b)Z 0

⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟. ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

(9.41)

One checks that it is the union of the fiber Fu (in the generic case) with the above 4 points. In fact in terms of the original presentation (9.5), i.e. in terms of the y j the characteristic variety in the generic case is the intersection of the two quadrics,   y 2j = 0, u 2j y 2j = 0, (9.42) and after the change of variables (9.36) it just becomes Z 02 − Z 12 Z 2 − Z 22 Z 2 − Z 32 = 0 = 0 , a b c

(9.43)

as can easily be checked since only the squares Z 2j are involved (and linearly). Note that we already knew that the fiber Fu is abstractly isomorphic to the elliptic curve of the characteristic variety using Lemma 6.2, Proposition 9.2 and the isomorphism (9.16). But here we have shown that their respective embeddings in P3 (C) are the same (i.e. the corresponding line bundles are the same). The automorphism σ of the characteristic variety E u is given by definition by the equation, N (Z ) σ (Z ) = 0, (9.44) where σ (Z ) is the column vector σ (Z µ ) := M · σ (u) (in the variables Z λ ). One checks that σ (Z ) is already determined by the equations in (9.44) corresponding to the first three lines in N (Z ) which are independent of a, b, c (see below). Thus σ is in fact an automorphism of P3 (C) which is the identity on the above four points and which restricts as automorphism of Fu for each u generic. One checks that σ is the product of two involutions which both restrict to E u (for u generic) σ = I ◦ I0 ,

(9.45)

where I is the involution of Proposition 9.4 corresponding to u µ → u −1 µ and where I0 is given by (9.46) I0 (Z 0 ) = −Z 0 , I0 (Z k ) = Z k for k ∈ {1, 2, 3}, and which restricts obviously to E u in view of (9.22). Both I and I0 are the identity on the above four points and since I0 induces the symmetry ϕ(z) → ϕ(−z) around ϕ(0) = ψ(η/2) (Proposition 9.3) one gets the result using Proposition 9.4.

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The fact that σ does not depend on the parameters a, b, c plays an important role. Explicitly we get from the first 3 equations (9.44), ⎛ ⎞   Z ν2 − 2 Zλ⎠ (9.47) σ (Z )µ = ηµµ ⎝ Z µ3 − Z µ ν =µ

λ =µ

for µ ∈ {0, 1, 2, 3}, where η00 = 1 and ηnn = −1 for n ∈ {1, 2, 3}.

 

Remark. Notice that, since the u µ are homogeneous coordinates on P3 (C), the generators yµ as well as the generators Yµ are only defined modulo a non-zero multiplicative scalar. Indeed under a change u µ → ν 2 u µ of homogeneous coordinates (ν ∈ C\{0}), the yµ transform as yµ → ν −1 yµ while the Yµ transform as Yµ → ν −4 Yµ which leaves invariant (9.6) and (9.36) as well as the x µ ((x µ )2 = u µ (yµ )2 ). Later on, it will be convenient to choose a normalization for the generators, e.g. in (11.32). 10. The Map from T2η × [0, τ ] to Sϕ3 and the Pairing This section contains the main technical result of the paper, i.e. both the construction of the one-parameter family of ∗-homomorphisms from the algebra of Sϕ3 (with ϕ generic) to the algebra of the noncommutative torus T2η and the computation of the pairing of the image of the Hochschild 3-cycle with the natural “fundamental class” for the product T2η × [0, τ ]. 10.1. Central elements. We already saw in Sect. 2 (Lemma 2.1) that the algebra Calg (R4ϕ ) of R4ϕ contains in its center the following element:  (x µ )2 . (10.1) Q1 = C = To get another one, one first looks at the Sklyanin algebra defined by (4.10) and (4.12) whose center contains two natural “Casimir” elements C j ,   C1 = Sµ2 , C2 = jk Sk2 , (10.2) where the jk fulfill the relations −

j − jm = Jm . jk

(10.3)

Let us check that C1 commutes with S0 and S . The idea is to only use the relation (4.10) so that the Jm do not interfere with the computation. This means that one treats the commutators as follows: [S0 , Sk2 ] = (S0 Sk + Sk S0 ) Sk − Sk (S0 Sk + Sk S0 ) = i[Sk [S Sm ]]. Thus the sum over k gives 0 in view of the Jacobi identity: [S1 , S02 ] = (S1 S0 + S0 S1 ) S0 − S0 (S1 S0 + S0 S1 ) = − i(S2 S3 − S3 S2 ) S0 +i S0 (S2 S3 − S3 S2 ), [S1 , S22 ] = [S1 , S2 ] S2 + S2 [S1 , S2 ] = i(S0 S3 + S3 S0 ) S2 + i S2 (S0 S3 + S3 S0 ), [S1 , S32 ] = [S1 , S3 ] S3 + S3 [S1 , S3 ] = − i(S0 S2 + S2 S0 ) S3 − i S3 (S0 S2 + S2 S0 ).

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83

One checks that the sum of these terms gives 0. Using cyclic permutations the commutation with Sk easily follows. Remark. It is worth noticing here that the relation [C1 , Sν ] = 0 can be written in the form [18]  [Sµ , [Sµ , Sν ]+ ] = 0, µ

where it becomes apparent that it is a super Lie algebra version of the relation defining the Yang-Mills algebra studied in [16]. As pointed out in [18] the relation (4.10) is the corresponding super analog of the self-duality relation and the fact that it implies [C1 , Sν ] = 0 is the content of Lemma 1 in [18]. Using (4.9) we can then assert that in the generic case the following element is in the center, 

3   cos(ϕk − ϕ )cos(ϕk − ϕm )sinϕk (x k )2 . sin ϕk (x 0 )2 + 1

Subtracting (10.1) multiplied by



(10.4)

sinϕk and using

cos(ϕk − ϕ ) cos(ϕk − ϕm ) − sinϕ sinϕm = cosϕk cos(− ϕk + ϕ + ϕm ), we then get: Q2 =

3 1  sin 2ϕk cos(− ϕk + ϕ + ϕm ) (x k )2 2

(10.5)

1

as a central element. Note that to get that in (10.2), C1 is central, we did not use relation (4.12) and thus it holds irrespective of the finiteness of the Jm . Thus the case δ(ϕ) = 0 is entirely covered to show that (10.5) is central. Proposition 10.1. 1) Both Q j are in the center of Calg (R4ϕ ) for all values of ϕ. 2) Let Sµ = λµ x µ as in (4.9) and λ such that λ jk = (−sk + s + sm )/s sm , then C1 − λ C2 , Q 2 = λC2 . Q1 =  sin ϕk

(10.6)

Proof. 1) We just have to check for Q and this will be done replacing it by (10.4) and using instead of (4.10) the three relations (2.24), sinϕk [x 0 , x k ]+ = icos(ϕ − ϕm ) [x  , x m ].

(10.7)

We just have to repeat the same proof as for the Sµ ’s making sure that any [x 0 , x k ]+ has a sinϕk as coefficient and every [x  , x m ] a cos(ϕ − ϕm ). For the commutator with x 0 this follows from the term sinϕk (x k )2 in (10.4). For the commutator with x 1 this follows from the terms sinϕ1 (x 0 )2 and cos(ϕ1 − ϕk ) (x k )2 , k = 1. 2) Note that the existence of λ follows from Lemma 4.7, i.e. with s˜k = (−sk + s + sm )/s sm the equality −

s˜ − s˜m = Jm . s˜k

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 We have already shown that C1 = sin ϕk Q 1 + Q 2 . It remains to check that Q 2 = λ C2 . Since λ jk = s˜k this amounts to λ2k s˜k = 21 sin 2ϕk cos(− ϕk + ϕ + ϕm ), i.e. s˜k =

cos ϕk cos(− ϕk + ϕ + ϕm ) , cos(ϕk − ϕ ) cos(ϕk − ϕm )

which follows from the definition of s˜k = (−sk + s + sm )/s sm .

 

In terms of the presentation of Theorem 9.5 one gets: Proposition 10.2. Let Au = Calg (C4 (u)) at generic u, then the following three linearly dependent quadratic elements belong to the center of Au : Q m = Jk (Y02 + Ym2 ) + Yk2 − Y2 .

(10.8)

10.2. The Hochschild cycle ch3/2 (U ). With z k = eiϕk x k we have, with ϕ0 = 0,  τ0 = 1, τk = i σk . (10.9) U= τµ z µ , We need to compute ch3/2 (U ) = Ui0 i1 ⊗ Ui∗1 i2 ⊗ Ui2 i3 ⊗ Ui∗3 i3 − Ui∗0 i1 ⊗ Ui1 i2 ⊗ Ui∗2 i3 ⊗ Ui3 i0 .

(10.10)

The trace computation is given by: Lemma 10.3. One has 1 Trace (τα τβ∗ τγ τδ∗ ) = δαβ δγ δ + δβγ δδα − δαγ δβδ + εαβγ δ . 2 Proof. If the set {α, β, γ , δ} is {0, 1, 2, 3} we can assume α = 0 or β = 0 by symmetry in (α, β, γ , δ) → (γ , δ, α, β). For α = 0 we get 21 Trace (τβ τγ τδ ) since the two − signs from τ ∗ cancel. This is cyclic and antisymmetric and gives for (1, 2, 3) using σ1 σ2 = i σ3 the result i 4 × 21 × 2 = 1. For β = 0 we get 1 1 Trace (τα τγ τδ∗ ) = − Trace (τα τγ τδ ) = − ε0αγ δ = εαβγ δ . 2 2 If two of the elements α, β, γ , δ are equal and the two others are different we get 0. Thus there are 3 cases α = β, α = γ , α = δ. One has τα τα∗ = 1, thus if α = β we get 1. For α = γ we get 21 Trace (τα τβ∗ τα τβ∗ ). The two − signs cancel and give 1 1 2 2 Trace (τα τβ τα τβ ). We can assume α = β. If 0 ∈ {α, β} we get 2 Trace (τk ) = −1. If 0∈ / {α, β}, τβ τα = − τα τβ and we get again −1. The case α = δ, β = γ is as α = β, γ = δ. Finally if all indices are equal we get 1.   Proposition 10.4. The Hochschild cycle ch3/2 (U ) ∈ H Z 3 (C ∞ (Sϕ3 )) is given (using ϕ0 = 0 and up to a scalar factor) by  ch3/2 (U ) = εαβγ δ cos(ϕα − ϕβ + ϕγ − ϕδ ) x α ⊗ x β ⊗ x γ ⊗ x δ  −i sin2(ϕµ − ϕν ) x µ ⊗ x ν ⊗ x µ ⊗ x ν . µ,ν

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Proof. The coefficient of x α ⊗ x β ⊗ x γ ⊗ x δ in 21 ch3/2 (U ) is given by × 21 1 1 Trace (τα τβ∗ τγ τδ∗ ) ei(ϕα −ϕβ +ϕγ −ϕδ ) − Trace (τα∗ τβ τγ∗ τδ ) ei(−ϕα +ϕβ −ϕγ +ϕδ ) . 2 2 (10.11) It is non-zero only in the two cases (with cardinality denoted as #), # {α, β, γ , δ} = 4, # {α, β, γ , δ} ≤ 2. In the first case we get as coefficient of x α ⊗ x β ⊗ x γ ⊗ x δ the term εαβγ δ ei(ϕα −ϕβ +ϕγ −ϕδ ) − εβγ δα e−i(ϕα −ϕβ +ϕγ −ϕδ ) = 2εαβγ δ cos(ϕα − ϕβ + ϕγ − ϕδ ), since the cyclic permutation has signature −1. In the second case the terms δαβ δγ δ ei(ϕα −ϕβ +ϕγ −ϕδ ) and δβγ δδα ei(ϕα −ϕβ +ϕγ −ϕδ ) are just δαβ δγ δ and δβγ δδα and they cancel with the terms coming from the second part of 10.10, −δβγ δδα e−i(ϕα −ϕβ +ϕγ −ϕδ ) = −δβγ δδα and −δγ δ δαβ e−i(ϕα −ϕβ +ϕγ −ϕδ ) = −δγ δ δαβ . Thus the following terms remain: − δαγ δβδ ei(ϕα −ϕβ +ϕγ −ϕδ ) − (− δβδ δγ α e−i(ϕα −ϕβ +ϕγ −ϕδ ) ) = −2 i sin(2(ϕα − ϕβ )), which yield the second term in the proposition.

 

We now use the rescaling (4.9) in the case δ(ϕ) = 0 and rewrite ch3/2 in terms of the generators Sµ . We let
=

3 

(tan (ϕ j ) cos(ϕk − ϕ )),

(10.12)

1

s0 = 0, s j = 1 + tan ϕk tan ϕl , ∀ j ∈ {1, 2, 3},

(10.13)

n 0 = 0, n k = 1, ∀k ∈ {1, 2, 3}.

(10.14)

and

Corollary 10.5. One has 
ch3/2 = − εαβγ δ (n α − n β + n γ − n δ )(sα − sβ + sγ − sδ ) Sα ⊗ Sβ ⊗ Sγ ⊗ Sδ α,β,γ ,δ

+ 2i

 (−1)n µ −n ν (sµ − sν ) Sµ ⊗ Sν ⊗ Sµ ⊗ Sν . µ,ν

Proof. We write the above formula as
ch3/2 = −A + 2i B. We let the λµ be as in Lemma 4.4 so that 3  0

and λ20 =

3  1

λµ = −δ(ϕ) = −

3 

(sinϕk cos(ϕ − ϕm ))

(10.15)

1

sin ϕ j , λ2k = sin ϕk cos(ϕk − ϕ ) cos(ϕk − ϕm ).

(10.16)

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One has x µ =

Sµ λµ

and thus in
ch3/2 the first terms of Proposition 10.4 give


εαβγ δ  cos(ϕα − ϕβ + ϕγ − ϕδ )Sα ⊗ Sβ ⊗ Sγ ⊗ Sδ λµ

(ϕ0 = 0).

(10.17)

cos(ϕα − ϕβ + ϕγ − ϕδ )  = (n α − n β + n γ − n δ )(sα − sβ + sγ − sδ ) cosϕk

(10.18)

One has


1  = − . λµ cosϕk

Thus the presence of the term −A follows from the equality

which we now check. We let t j = tan ϕ j . One gets cos(ϕ1 − ϕ2 − ϕ3 )  = 1 − t2 t 3 + t 1 t 3 + t 1 t 2 , cosϕk and more generally for any permutation σ of 1, 2, 3 one has cos(ϕσ (1) − ϕσ (2) − ϕσ (3) )  = −sσ (1) + sσ (2) + sσ (3) . cosϕk

(10.19)

Let us prove (10.18). Since (α, β, γ , δ) is a permutation of (0, 1, 2, 3) one of the indices is 0. For α = 0 we get ϕα = 0 and cos(ϕα − ϕβ + ϕγ − ϕδ )  = −(s0 − sβ + sγ − sδ ), cosϕk since s0 = 0. But (n α − n β + n γ − n δ ) = −1 so that (10.18) holds. For β = 0 we get ϕβ = 0 and cos(ϕα − ϕβ + ϕγ − ϕδ )  = (sα − s0 + sγ − sδ ), cosϕk since s0 = 0. But (n α − n β + n γ − n δ ) = 1 so that (10.18) holds. For γ = 0 we get ϕγ = 0 and cos(ϕα − ϕβ + ϕγ − ϕδ )  = −(sα − sβ + s0 − sδ ), cosϕk since s0 = 0. But (n α − n β + n γ − n δ ) = −1 so that (10.18) holds. Finally for δ = 0, ϕδ = 0 and cos(ϕα − ϕβ + ϕγ − ϕδ )  = (sα − sβ + sγ − s0 ), cosϕk since s0 = 0. But (n α − n β + n γ − n δ ) = 1 so that (10.18) holds and we checked it in all cases. Let us now compute the contribution of the second terms of Proposition 10.4. For i, j ∈ {1, 2, 3} one has −


sin 2(ϕi − ϕ j ) = si − s j . 2λi2 λ2j

(10.20)

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87

Indeed say with i = 2, j = 3, one gets
sinϕ1 = . cosϕ1 cosϕ2 cosϕ3 cos(ϕ2 − ϕ3 ) λ22 λ23 Thus −


sinϕ1 sin(ϕ2 − ϕ3 ) sin 2(ϕ2 − ϕ3 ) = − cosϕ1 cosϕ2 cosϕ3 2λ22 λ23 = −t1 (t2 − t3 ) = (1 + t1 t3 ) − (1 + t1 t2 ) = s2 − s3 .

Next let us check that


sin 2ϕk = sk . (10.21) 2λ20 λ2k  Say with k = 1 one has λ20 = sinϕ j , λ21 = sinϕ1 cos(ϕ1 − ϕ2 )cos(ϕ1 − ϕ3 ) and
cos(ϕ2 − ϕ3 ) sin 2ϕ1 = = s1 . 2 2 cosϕ2 cosϕ3 2λ0 λ1 We thus get in general,
sin2(ϕµ − ϕν ) = −(−1)n µ −n ν 2(sµ − sν ). λ2µ λ2ν

(10.22)

Indeed, for µ, ν ∈ {1, 2, 3} this is (10.20). If both µ, ν = 0, both sides are 0. Now both sides are antisymmetric in µ, ν, thus one can take ν = 0, µ ∈ {1, 2, 3}. Then n µ − n ν = 1 and the result follows from (10.21).   10.3. Elliptic parameters. Let ϕ ∈ T A and using (9.26) let τ ∈ C, Im τ > 0 and η ∈ C and λ ∈ C such that   ϑ2 (0)2 ϑ3 (0)2 ϑ4 (0)2 . (10.23) λ (s1 , s2 , s3 ) = , , ϑ2 (η)2 ϑ3 (η)2 ϑ4 (η)2 We shall call the triplet (τ, η, λ) elliptic parameters for ϕ. They are not unique given ϕ but they determine uniquely the s j , and hence ϕ up to an overall sign. Lemma 10.6. With the above notations (9.27) is equivalent to   ϑ2 (0) ϑ2 (2η) ϑ3 (0) ϑ3 (2η) ϑ4 (0) ϑ4 (2η) (˜s1 , s˜2 , s˜3 ) = λ . , , ϑ2 (η)2 ϑ3 (η)2 ϑ4 (η)2 Proof. By Lemma 4.7 we just need to check that with   ϑ2 (0) ϑ2 (2η) ϑ3 (0) ϑ3 (2η) ϑ4 (0) ϑ4 (2η) ( j1 , j2 , j3 ) = , , ϑ2 (η)2 ϑ3 (η)2 ϑ4 (η)2 one has ( j˜1 , j˜2 , j˜3 ) =



 ϑ2 (0)2 ϑ3 (0)2 ϑ4 (0)2 . , , ϑ2 (η)2 ϑ3 (η)2 ϑ4 (η)2

(10.24)

(10.25)

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The three equalities jk j˜ + j˜k j = 2 follow from the following identities on ϑ-functions: ϑ3 (0)2 ϑ2 (0) ϑ2 (2η) = ϑ2 (η)2 ϑ3 (η)2 − ϑ1 (η)2 ϑ4 (η)2 , ϑ2 (0)2 ϑ3 (0) ϑ3 (2η) = ϑ2 (η)2 ϑ3 (η)2 + ϑ1 (η)2 ϑ4 (η)2 , and similarly ϑ4 (0)2 ϑ3 (0) ϑ3 (2η) = ϑ3 (η)2 ϑ4 (η)2 − ϑ1 (η)2 ϑ2 (η)2 , ϑ3 (0)2 ϑ4 (0) ϑ4 (2η) = ϑ3 (η)2 ϑ4 (η)2 + ϑ1 (η)2 ϑ2 (η)2 , and ϑ2 (0)2 ϑ4 (0) ϑ4 (2η) = ϑ2 (η)2 ϑ4 (η)2 + ϑ1 (η)2 ϑ3 (η)2 , ϑ4 (0)2 ϑ2 (0) ϑ2 (2η) = ϑ2 (η)2 ϑ4 (η)2 − ϑ1 (η)2 ϑ3 (η)2 .   This lemma allows to relate the above parameters with those used by Sklyanin and one has with the above notations ([28]), J12 =

ϑ1 (η)2 ϑ4 (η)2 , ϑ2 (η)2 ϑ3 (η)2

J23 =

ϑ1 (η)2 ϑ2 (η)2 , ϑ3 (η)2 ϑ4 (η)2

J31 = −

ϑ1 (η)2 ϑ3 (η)2 , ϑ2 (η)2 ϑ4 (η)2

(10.26)

which follows from the definition (9.27) of the elliptic parameters together with (9.19). 10.4. The sphere Sϕ3 and the noncommutative torus T2η . Let ϕ ∈ A◦ so that ϕ2 > ϕ3 > 0. We can then choose the elliptic parameters τ and η such that τ ∈ i R+ , η ∈ [0, 1].

π 2

> ϕ1 > (10.27)

Then the module q = eiτ ∈ ]0, 1[ and the ϑ functions ϑ j (z) are all real functions, i.e. fulfill ϑ j (¯z ) = ϑ j (z), ∀z ∈ C. (10.28) In particular the last elliptic parameter λ determined by (9.27) fulfills λ > 0. We shall explain in this section how to use the representations constructed by Sklyanin [28] to obtain ∗-homomorphisms from Calg (Sϕ3 ) to the algebra C ∞ (T2η ) = C ∞ (R/Z)
η Z,

(10.29)

obtained as the crossed product of the algebra C ∞ (R/Z) of smooth periodic functions by the translation η. Recall that a generic element of C ∞ (T2η ) is of the form f =



fn V n

Z

while the basic algebraic rule is given by V f V −1 (u) = f (u + η), ∀u ∈ R/Z, ∀ f ∈ C ∞ (R/Z).

(10.30)

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89

Moreover C ∞ (T2η ) is an involutive algebra with involution turning V into a unitary operator. Starting from the representations constructed in [28] and conjugating by the operator M(ξ )(u) = e−2πiu v/η ξ(u + τ/4) one performs a shift in the indices of the ϑ-functions based on   1 1 τ τ ϑ1 z + = q − 4 e−πi z iϑ4 (z), ϑ2 z + = q − 4 e−πi z ϑ3 (z), 2 2   1 1 τ τ ϑ3 z + = q − 4 e−πi z ϑ2 (z), ϑ4 z + = q − 4 e−πi z iϑ1 (z), 2 2 which allows to replace the singular denominator ϑ1 (2u) by ϑ4 (2u) which no longer vanishes for u ∈ R/Z. One obtains in this way a homomorphism from the Sklyanin algebra to C ∞ (T2η ), but it is not yet unitary and to make it so one needs to conjugate again by a multiplication operator of the form, ¯ ξ(u), N (ξ )(u) = d(u) where the function d ∈ C ∞ (R/Z) fulfills the following conditions, ¯ d(u)d(u) = ϑ4 (2u), ∀u ∈ R/Z. We use the identity ϑ3 (0)2 ϑ4 (0) ϑ4 (2u) = ϑ3 (u)2 ϑ4 (u)2 + ϑ1 (u)2 ϑ2 (u)2 , and thus take c d(u) = ϑ3 (u) ϑ4 (u) + i ϑ1 (u) ϑ2 (u), c2 = ϑ3 (0)2 ϑ4 (0).

(10.31)

Note that one has ¯ d(u) = d(−u), ∀u ∈ R/Z. The effect of the conjugacy N . N −1 on simple monomials is the following: f (u) f (u) V → V, ϑ4 (2 u) d(u) d(−u − η)

f (u) f (u) V∗ → V ∗. ϑ4 (2 u) d(u) d(−u + η)

The formulas which define the images ρ(Sα ) then become, with m ∈ [0, τ ], ϑ3 (2u + η + im) ϑ3 (2u − η − im) ∗ V + ϑ1 (η) V , (10.32) d(u) d(−u − η) d(u) d(−u + η) ϑ4 (2u + η + im) ϑ4 (2u − η − im) ∗ V + i ϑ2 (η) V , (10.33) ρ(S1 ) = −i ϑ2 (η) d(u) d(−u − η) d(u) d(−u + η) ϑ1 (2u + η + im) ϑ1 (2u − η − im) ∗ V + ϑ3 (η) V , (10.34) ρ(S2 ) = ϑ3 (η) d(u) d(−u − η) d(u) d(−u + η) ϑ2 (2u + η + im) ϑ2 (2u − η − im) ∗ V − ϑ4 (η) V , (10.35) ρ(S3 ) = −ϑ4 (η) d(u) d(−u − η) d(u) d(−u + η)

ρ(S0 ) = ϑ1 (η)

and one has

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Theorem 10.7. The formulas (10.32)...(10.35) define a ∗-homomorphism from the (C ∞ ([0, τ ]). Sklyanin algebra to C ∞ (T2η )⊗ Proof. Since by construction ρ is conjugate to an homomorphism, it is an homomorphism and we just need to check that the images ρ(Sµ ) of the generators are self-adjoint elements of C ∞ (T2η ) for each value of m ∈ [0, τ ]. One has ∗  f (u) f¯(u) = V∗ V , d(u) d(−u − η) d(−u) d(u + η) ¯ since η ∈ R and d(x) = d(−x) for x ∈ R. Thus using (10.30) one gets  ∗ f (u) f¯(u − η) V V ∗. = d(u) d(−u − η) d(−u + η) d(u) Since m is real one has ϑ¯ j (x + im) = ϑ j (x − im), ∀x ∈ R, ∀ j using (10.28). Thus one checks directly the required self-adjointness of the ρ(Sµ ).

 

To obtain a ∗-homomorphism from Calg (Sϕ3 ) to C ∞ (T2η ) ⊗ C ∞ ([0, τ ]) we need to normalize the above formulas so that the element Q 1 in the center of Calg (R4ϕ ) gets mapped to 1. By Proposition 10.1 this amounts to introduce an overall scaling factor given by 1/2  σ (m) = (C1 − λ C2 )−1/2 . (10.36) sin ϕ j where the explicit values of the Casimirs C j are given from [28] by C1 = 4 ϑ22 (im), C2 = 4 ϑ2 (η + im) ϑ2 (η − im).

(10.37)

We can now normalize the above homomorphism ρ as ρ(S ˜ j ) = σ (m) ρ(S j ).

(10.38)

we then get using the change of variables Sµ = λµ x µ . Corollary 10.8. The map ρ˜ defines a ∗-homomorphism (C ∞ ([0, τ ]). Calg (Sϕ3 ) → C ∞ (T2η )⊗ Proof. Since ϕ ∈ A one has λµ ∈ R and the above change of variables is a ∗-isomorphism of Calg (R4ϕ ) with the Sklyanin algebra. Thus we only need to check that with the above normalization the ∗-homomorphism ρ˜ maps the central element Q 1 which determines the sphere to the element (C ∞ ([0, τ ]). 1 ∈ C ∞ (T2η )⊗ This follows from Eq. (10.6).

 

In the following we shall use the notation C ∞ (T2η × [0, τ ]) to denote the completed (C ∞ ([0, τ ]). tensor product C ∞ (T2η )⊗

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91

10.5. Pairing with [T2η ]. In order to test the non-triviality of the ∗-homomorphism ρ˜ we shall compute what will be later interpreted as its Jacobian. In order to do this we shall pair the image ρ˜∗ (ch3/2 ) ∈ H Z 3 (C ∞ (T2η × [0, τ ])) (10.39) with the natural Hochschild three cocycle obtained using the fundamental class [T2η ] introduced in [10]. Since the variable m ∈ [0, τ ] labels the center we shall view the above pairing as defining a function of m. The basic Hochschild three cocycle on C ∞ (T2η × [0, τ ])) is given by  i jk τ0 (a0 δi (a1 ) δ j (a2 ) δk (a3 )), (10.40) τ (a0 , . . . , a3 ) = where τ0 is the trace obtained as the tensor product of the canonical trace χ on C ∞ (T2η ) by the trace on C ∞ ([0, τ ]) given by integration,  τ  1 τ0 (a)) = χ (a(m))dm, χ ( f ) = f (u)du, χ ( f V n ) = 0, ∀n = 0. 0

0

(10.41)

The three basic derivations δ j are given by δ1 = ∂/∂m, δ2 = ∂/∂u, δ3 = 2πi V ∂/∂ V,

(10.42)

where in the last term the differentiation V ∂/∂ V has the effect of multiplying by n any monomial f V n . As the product of τ by any function h(m) viewed as an element of the center of C ∞ (T2η × [0, τ ])) is still a Hochschild three cocycle, we obtain a differential one form on [0, τ ] as the pairing ω = ch3/2 , τ .

(10.43)

The basic lemma is then the following using the notation   ϑ2 (0)2 ϑ3 (0)2 ϑ4 (0)2 . (σ0 , σ1 , σ2 , σ3 ) = 0, , , ϑ2 (η)2 ϑ3 (η)2 ϑ4 (η)2

(10.44)

Lemma 10.9. With the above notations one has )  τ, εαβγ δ (n α −n β +n γ −n δ )(σα −σβ +σγ −σδ )ρ(Sα ) ⊗ ρ(Sβ ) ⊗ ρ(Sγ ) ⊗ ρ(Sδ ) α,β,γ ,δ

 −2 i (−1)n µ −n ν (σµ − σν ) ρ(Sµ ) ⊗ ρ(Sν ) ⊗ ρ(Sµ ) ⊗ ρ(Sν )

*

µ,ν

= 24 (2πi)3

ϑ1 (0)3 ϑ1 (η) ϑ1 (2im) . π 3 ϑ2 (η)ϑ3 (η)ϑ4 (η)

The precise meaning of the equality is that for any h ∈ C ∞ ([0, τ ])) the evaluation of the Hochschild cocycle τ on the product of the Hochschild cycle of the right-hand side by h gives the integral  τ h(m) g(m) dm, 0

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where6 g(m) = 24 (2πi)3

ϑ1 (0)3 ϑ1 (η) ϑ1 (2im) . π 3 ϑ2 (η)ϑ3 (η)ϑ4 (η)

(10.45)

The proof of this is a long computation based on the “a priori” properties of the pairing which allow to show that the dependence in the parameters η and m is of the expected form, while the dependence in the module q is that of a modular form. It then follows from the explicit knowledge of enough terms in the q-expansion that the above formula is valid. So far we have not been able to eliminate completely the use of the computer to check this validity and the lemma should thus be given the status of a conjecture rather than that of a mathematical result, due to the non-zero probability of a computational mistake. However what we shall see is that starting from this formula we shall, through the gradual simplifications below, come to a beautifully simple final result. We shall now show that the dependance in m of the normalization factor σ (m) in the definition (10.38) of the homomorphism ρ˜ can be ignored when one computes the pairing (10.43). Lemma 10.10. Let δ, δ  be derivations of the unital algebra A preserving a trace τ0 on A. Let φ j be the multilinear forms on A given by φ1 (a0 , a1 , a2 , a3 ) = τ0 (a0 a1 δ(a2 ) δ  (a3 )), φ2 (a0 , a1 , a2 , a3 ) = τ0 (a0 δ(a1 ) a2 δ  (a3 )). Then for any invertible U ∈ A one has φ j (U, U −1 , U, U −1 ) − φ j (U −1 , U, U −1 , U ) = 0. Proof. One has φ1 (U, U −1 , U, U −1 ) = τ0 (U U −1 δ(U ) δ  (U −1 )) = −τ0 (δ(U ) U −1 δ  (U ) U −1 ), φ1 (U −1 , U, U −1 , U ) = τ0 (U −1 U δ(U −1 ) δ  (U )) = −τ0 (U −1 δ(U ) U −1 δ  (U )), thus the cyclicity of the trace proves the statement for j = 1. Similarly one has φ2 (U, U −1 , U, U −1 ) = τ0 (U δ(U −1 ) U δ  (U −1 )) = τ0 (δ(U ) U −1 δ  (U ) U −1 ) φ2 (U −1 , U, U −1 , U ) = τ0 (U −1 δ(U ) U −1 δ  (U )), and the cyclicity of the trace proves the statement for j = 2.

 

We thus get the following result Corollary 10.11. The pairing of ρ˜∗ (ch3/2 ) with τ is given by the differential form ω= −

σ (m)4 g(m) dm, λ


with
given in (10.12), λ by (9.27), σ (m) by (10.36) and g(m) by (10.45).  6 The q-expansion of the fraction ϑ1 (0) has rational coefficients. π

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93

Proof. Using Lemma 10.9 and Corollary 10.5 one just needs to show that the terms of the form dσ (m) ˜ j )) − σ (m) δ1 (ρ(S j )) = δ1 (ρ(S ρ(S j ) dm do not contribute. But their total contribution is a sum of six terms each of which is of the form φ j (U, U −1 , U, U −1 ) − φ j (U −1 , U, U −1 , U ), where U ∈ M2 (Calg (Sϕ3 )) is the basic unitary element while φ j is as in Lemma 10.10  with δ, δ  ∈ {δ2 , δ3 }. Thus each of these terms vanishes by Lemma 10.10.  10.6. Simplifying the ∗-homomorphism ρ. ˜ We shall make several simplifications in the formulas involved in the construction of the ∗-homomorphism ρ˜ of Corollary 10.8 in order to gradually eliminate all ϑ-functions and express the result in purely algebraic terms. The denominators involved in the construction of the ∗-homomorphism ρ˜ are of the form d(u) d(−u ± η), (10.46) where by (10.31), c d(u) = ϑ3 (u) ϑ4 (u) + i ϑ1 (u) ϑ2 (u), c2 = ϑ3 (0)2 ϑ4 (0). Our first task will be to rewrite (10.46) as a linear form in terms of the projective coordinates ψ(u) of Proposition 9.3, i.e.   ϑ1 (2u − η) ϑ2 (2u − η) ϑ3 (2u − η) ϑ4 (2u − η) , , , = (Z 0 , Z 1 , Z 2 , Z 3 ). ψ(u) = ϑ1 (η) ϑ2 (η) ϑ3 (η) ϑ4 (η) Lemma 10.12. With the above notations one has ϑ3 (0) d(u) d(−u + η) = i ϑ1 (η)ϑ2 (η) Z 1 + ϑ3 (η)ϑ4 (η) Z 3 .

(10.47)

Proof. One has c2 d(u) d(−u + η) = (ϑ3 (u) ϑ4 (u) + i ϑ1 (u) ϑ2 (u))(ϑ3 (u − η) ϑ4 (u − η) − i ϑ1 (u − η) ϑ2 (u − η)) = ϑ3 (u) ϑ4 (u) ϑ3 (u − η) ϑ4 (u − η) + ϑ1 (u) ϑ2 (u) ϑ1 (u − η) ϑ2 (u − η) + i ϑ1 (u) ϑ2 (u) ϑ3 (u − η) ϑ4 (u − η) − i ϑ3 (u) ϑ4 (u) ϑ1 (u − η) ϑ2 (u − η). Thus using the basic addition formulas (obtained from (15.6) and (15.15)) ϑ3 (x)ϑ4 (x)ϑ3 (y)ϑ4 (y) − ϑ1 (x)ϑ2 (x)ϑ1 (y)ϑ2 (y)) = ϑ3 (0)ϑ4 (0)ϑ3 (x + y)ϑ4 (x − y), ϑ1 (x)ϑ2 (x)ϑ3 (y)ϑ4 (y) + ϑ3 (x)ϑ4 (x)ϑ1 (y)ϑ2 (y)) = ϑ3 (0)ϑ4 (0)ϑ1 (x + y)ϑ2 (x − y) for x = u, y = η − u, we get c2 d(u) d(−u + η) = ϑ3 (0) ϑ4 (0) (ϑ3 (η) ϑ4 (2u − η) + i ϑ1 (η) ϑ2 (2u − η)), which gives the required equality.

 

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To simplify the numerators involved in the construction of the ∗-homomorphism ρ˜ we pass from generators Sµ of the Sklyanin algebra to the generators Yµ of Theorem 9.5 by the following transformation: S0 = d Y2 , S1 = i Y3 , S2 = d Y0 , S3 = −Y1 ,

(10.48)

ϑ1 (η)ϑ3 (η) . One checks that the Yµ fulfill the presentation of Theorem 9.5 ϑ2 (η)ϑ4 (η) using the equality (10.26) d 2 = − J31 . We can then reformulate the construction of the homomorphism ρ in the following terms:

where d =

Lemma 10.13. With the above notations one has, up to an overall scalar factor γ , ρ(Yµ ) =

ψµ (u − i m/2) ∗ ψµ (u + i m/2) V + µ V , ¯ L(u) L(u)

(10.49)

where  = (1, 1, 1, −1) and ¯ = L(u). L(u) = i ϑ1 (η)ϑ2 (η) ψ1 (u) + ϑ3 (η)ϑ4 (η) ψ3 (u), L(u) Proof. One just needs to perform the transformation (10.48) on Eqs. (10.32)....(10.35). One gets an overall scalar factor γ = ϑ2 (η) ϑ4 (η) ϑ3 (0), multiplying the right-hand side of (10.49) (or equivalently dividing L(u)).

 

In order to understand (10.49) we let ¯ −1 . Z  =  ψ(u + i m/2), W = L(u)−1 V ∗ , W  = V L(u) (10.50) Note that one has Z  =  Z¯ and W  = W ∗ but we shall ignore that for a while and treat for instance Z and Z  as independent variables. The multiplicative terms such as L(u)−1 do not alter the cross product rules (10.30) but they alter the simplification rule V V ∗ = V ∗ V = 1. Our next task will thus be to give a simple expression for W W  in terms of (Z , Z  ). One has by construction Z = ψ(u − i m/2),

−1 ¯ , W W  = (L(u) L(u))

(10.51)

and we need to express the denominator in terms of Z and Z  . Note that we have the freedom to multiply by an arbitrary function of m since this only alters the normalization of ρ which is needed in any case to pass to ρ. ˜ Lemma 10.14. With the above notations one has, ¯ ν(m) L(u) L(u) = J23 (Z 0 Z 0 + Z 1 Z 1 ) + Z 2 Z 2 − Z 3 Z 3 , where ν(m) =

2 ϑ32 (im) ϑ32 (0)ϑ32 (η)ϑ42 (η)

.

(10.52)

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95

Proof. one has ¯ L(u) L(u) = ϑ12 (η)ϑ22 (2u − η) + ϑ32 (η)ϑ42 (2u − η), thus with a = 2u − η + im, b = 2u − η − im,

a+b = 2u − η, 2

a−b = im 2

we get     a−b a+b ¯ ϑ22 ϑ32 (im) L(u) L(u) = ϑ12 (η) ϑ32 2 2     a−b a+b ϑ42 . +ϑ32 (η) ϑ32 2 2

(10.53)

We now use the addition formulas     a−b a+b 2ϑ32 ϑ22 = ϑ22 (0)ϑ3 (a)ϑ3 (b) + ϑ32 (0)ϑ2 (a)ϑ2 (b) 2 2 −ϑ42 (0)ϑ1 (a)ϑ1 (b) (adding (15.9) and (15.10)) and     2 a−b 2 a+b 2 ϑ3 ϑ4 = ϑ22 (0) ϑ1 (a) ϑ1 (b) + ϑ32 (0) ϑ4 (a) ϑ4 (b) 2 2 + ϑ42 (0) ϑ3 (a) ϑ3 (b) (adding (15.5) and (15.6)) which allow to write (10.53) as a symmetric bilinear form in (Z , Z  ). One then uses (9.19) and (10.26) J23 = to obtain the required equality.

ϑ12 (η)ϑ22 (η) ϑ32 (η)ϑ42 (η)

,

 

Proposition 10.15. With the above notations one has, up to an overall scalar factor δ(m), (10.54) ρ(Yµ ) = Z µ W + W  Z µ with algebraic rules given by Z i W W  Z j =

Z i Z j Q(Z , Z  )

, W f (Z , Z  ) = f (σ (Z ), σ −1 (Z  )) W,

where σ is the translation by −η as in Theorem 9.5 and Q(Z , Z  ) = J23 (Z 0 Z 0 + Z 1 Z 1 ) + Z 2 Z 2 − Z 3 Z 3 .

(10.55)

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Proof. The first equality follows from Lemma 10.13 and Definition 10.50 of Z , Z  , W, W  . The first algebraic rule follows from (10.51) and Lemma 10.14. To obtain the second we need to understand the transformation  ψ(u + im/2) →  ψ(u − η + im/2), and to compare it with σ −1 , where σ is the translation by −η as in Theorem 9.5. By construction σ is the product (9.45) of two involutions σ = I ◦ I0 , where I0 just alters the sign of Z 0 (cf. (9.46)). Thus σ −1 = I0 ◦ I = I0 ◦ σ ◦ I0 and to show that the above tranformation is σ −1 it is enough to show that σ commutes with I0 ◦ I3 where I3 (Z ) =  Z . This follows from the commutation of translations on the elliptic curve and can be checked directly using (9.47).   11. Algebraic Geometry and C ∗-Algebras In this section we shall develop the basic relation between noncommutative differential geometry in the sense of [10] and noncommutative algebraic geometry. This will be obtained by abstracting the results of Proposition 10.15 of Subsect. 10.6 and giving a general construction, independent of ϑ-functions, of a homomorphism from a quadratic algebra to a crossed product algebra constructed from the geometric data. 11.1. Central quadratic forms and generalised cross-products. Let A = A(V, R) = T (V )/(R) be a quadratic algebra. Its geometric data {E, σ, L} is defined in such a way that A maps homomorphically to a cross-product algebra obtained from sections of powers of the line bundle L on powers of the correspondence σ ([4]). We shall begin by a purely algebraic result which considerably refines the above homomorphism and lands in a richer cross-product. We use the notations of Sect. 5 for general quadratic algebras. Definition 11.1. Let Q ∈ S 2 (V ) be a symmetric bilinear form on V ∗ and C a component of E × E. We shall say that Q is central on C iff for all (Z , Z  ) in C and ω ∈ R one has, ω(Z , Z  ) Q(σ (Z  ), σ −1 (Z )) + Q(Z , Z  ) ω(σ (Z  ), σ −1 (Z )) = 0.

(11.1)

By construction the space of symmetric bilinear form on V ∗ which are central on C is a linear subspace of S 2 (V ). Let C be a component of E × E globally invariant under the map σ˜ (Z , Z  ) := (σ (Z ), σ −1 (Z  )). (11.2) Given a quadratic form Q central and not identically zero on the component C, we define as follows an algebra C Q as a generalised cross-product of the ring R of meromorphic functions on C by the transformation σ˜ . Let L, L  ∈ V be such that L(Z ) L  (Z  ) does not vanish identically on C. We adjoin two generators W L and W L  which besides the usual cross-product rules, W L f = ( f ◦ σ˜ ) W L , W L  f = ( f ◦ σ˜ −1 ) W L  , ∀ f ∈ R

(11.3)

fulfill the following relations: W L W L  := π(Z , Z  ),

W L  W L := π(σ −1 (Z ), σ (Z  )),

(11.4)

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where the function π(Z , Z  ) is given by the ratio, π(Z , Z  ) :=

L(Z ) L  (Z  ) . Q(Z , Z  )

(11.5)

The a priori dependence on L, L  is eliminated by the rules, W L 2 :=

L 2 (Z ) WL 1 L 1 (Z )

W L  := W L  2

1

L 2 (Z  ) L 1 (Z  )

(11.6)

which allow to adjoin all W L and W L  for L and L  not identically zero on the projections of C, without changing the algebra and provides an intrinsic definition of C Q . Our first result is Lemma 11.2. Let Q be central and not identically zero on the component C. (i) The following equality defines a homomorphism ρ, A → C Q , √

2 ρ(Y ) :=

Y (Z  ) Y (Z ) W L + W L    , L(Z ) L (Z )

∀Y ∈ V.

(11.7)

(ii) If σ 4 = ?, then ρ(Q) = 1, where Q is viewed as an element of T (V )/(R). √ Proof. (i) Formula (11.7) is independent of L, L  using (11.6) and reduces (up to 2) to WY + WY when Y is non-trivial on the two projections of C. It is enough to check that the ρ(Y ) ∈ C Q fulfill the quadratic relations ω ∈ R. Let ω ∈ R,  ω(Z , Z  ) = ωi j Yi (Z ) Y j (Z  ), viewed as a bilinear form on V ∗ . One has    2 ωi j ρ(Yi )ρ(Y j ) = ωi j (WYi + WY i )(WY j + WY j ) = ωi j Yi (Z )Y j (σ (Z )W 2 +



ωi j

Yi (Z )Y j (Z  )  Yi (σ (Z  ))Y j (σ −1 (Z ))   + ω + ωi j W 2 Yi (σ −1 (Z  ))Y j (Z  ), i j  −1  Q(Z , Z ) Q(σ (Z ), σ (Z ))

where W2 =

1 1   W L2 , W 2 = W L2  −1    . L(Z )L(σ (Z )) L (σ (Z ))L (Z )

The vanishing of the terms in W 2 and in W correspondence σ , i.e. the equality

2

is automatic by construction of the

ω(Z , σ (Z )) = 0, ∀Z ∈ E. The sum of the middle terms is just ω(Z , Z  ) ω(σ (Z  ), σ −1 (Z )) + = 0,  Q(Z , Z ) Q(σ −1 (Z ), σ (Z  )) as follows from Definition 11.1 and the symmetry of Q. (ii) The above computation shows that ρ(Q) = 1, provided one can show that Q(Z , σ (Z )) = 0 and Q(σ −1 (Z  ), Z  ) = 0 for all Z , Z  in the projections E, E  of C. We

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assume that σ 4 (Z ) is not identically equal to Z on each connected component of E (resp. E  ) and use (11.1) with Z  = σ (Z ). The first term vanishes and we get ω(σ 2 (Z ), σ −1 (Z )) Q(Z , σ (Z )) = 0, ∀ω ∈ R. Thus if Q(Z , σ (Z )) does not vanish identically on a given connected component E 1 of E one gets that ω(σ 2 (Z ), σ −1 (Z )) = 0, ∀Z ∈ E 1 , ω ∈ R, so that σ −1 (Z ) = σ 3 (Z ) for all Z ∈ E 1 which contradicts the hypothesis.

 

Let Au = Calg (C4 (u)) at generic u, then by Proposition 10.2 the center of Au contains the three linearly dependent quadratic elements Q m := Jk (Y02 + Ym2 ) + Yk2 − Y2 .

(11.8)

Proposition 11.3. Let Au = Calg (C4 (u)) at generic u, then each Q m is central on Fu × Fu (⊂ E u × E u ).  

Proof. One uses (9.45) to check the algebraic identity.

Together with Lemma 11.2 this yields non-trivial homomorphisms of Au whose unitarity will be analysed in the next section. Note that for a general quadratic algebra A = A(V, R) = T (V )/(R) and a quadratic form Q ∈ S 2 (V ), such that Q ∈ Center(A), it does not automatically follow that Q is central on E × E. For instance Proposition 11.3 no longer holds on Fu × {eν }, where eν is any of the four points of E u not in Fu . In fact let us describe in some detail what happens in the case of the θ -deformations, i.e. C+ = {(ϕ, ϕ, 0)} (Case 7). We take the notations of Subsect. 5.9 to write the characteristic variety as the union of six lines  j . Proposition 11.4. Let Calg (R4ϕ ) for ϕ ∈ C+ , and Q k be defined by (10.1) and (10.5). (1) Each Q k is central on i × j provided i and j belong to the same subsets I = {1, 2} and J = {3, 4, 5, 6}. (2) The bilinear form Q 1 does not vanish identically on i ×  j iff i = j for i, j ∈ I and iff i = j for i, j ∈ J . (3) The bilinear form Q 2 vanishes identically on 2 ×  j and  j × 2 for all j. This is proved by direct computations. Note that since Q 1 fails to be central on 1 ×3 for instance, it was crucial to “localize” the notion of central quadratic form to components of the square E × E of the characteristic variety E. It is of course also crucial to check the non-vanishing of Q when applying Lemma 11.2, and the component 2 does not work for Q 2 in that respect. The precise table for the vanishing of the form Q 2 is the following, where = at (i, j) means that Q 2 does not vanish identically on i ×  j ;

= 0

=

=

=

=

0 0 0 0 0 0

= 0 0

=

= 0

= 0

= 0 0

=

= 0

= 0 0

=

= 0 0

=

= 0

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11.2. Positive central quadratic forms on quadratic ∗-algebras. The algebra Au , u ∈ T is by construction a quadratic ∗-algebra, i.e. a quadratic complex algebra A = A(V, R) which is also a ∗-algebra with involution x → x ∗ preserving the subspace V of generators. Equivalently one can take the generators of A (spanning V ) to be hermitian elements of A. In such a case the complex finite-dimensional vector space V has a real structure given by the antilinear involution v → j (v) obtained by restriction of x → x ∗ . Since one has (x y)∗ = y ∗ x ∗ for x, y ∈ A, it follows that the set R of relations satisfies ( j ⊗ j)(R) = t (R)

(11.9)

in V ⊗ V , where t : V ⊗ V → V ⊗ V is the transposition v ⊗ w → t (v ⊗ w) = w ⊗ v. This implies Lemma 11.5. The characteristic variety is stable under the involution Z → j (Z ) and one has σ ( j (Z )) = j (σ −1 (Z )). We let C be as above an invariant component of E × E; we say that C is j-real when it is globally invariant under the involution ˜ , Z  ) := ( j (Z  ), j (Z )). j(Z

(11.10)

By Lemma 11.5 this involution commutes with the automorphism σ˜ (11.2) and one has Proposition 11.6. Let C be a j-real invariant component of E × E and Q central on C be such that ˜ , Z  )) = Q(Z , Z  ), ∀(Z , Z  ) ∈ C. Q( j(Z (11.11) (1) The following turns the cross-product C Q into a ∗-algebra, ˜ , Z  )), f ∗ (Z , Z  ) := f ( j(Z

(W L )∗ = W j (L) ,

(W L  )∗ = W j (L  ) . (11.12)

(2) The homomorphism ρ of Lemma 11.2 is a ∗-homomorphism. Proof. We used the transpose of j to define j (L) in (11.12) by j (L)(Z ) = L( j (Z )),

∀Z ∈ V ∗ .

(11.13)

The compatibility of the involution with (11.3) follows from the commutation of j˜ with σ˜ . Its compatibility with (11.4) follows from π ∗ (Z , Z  ) :=



L( j (Z  )) L  ( j (Z )) ˜ , Z  )) Q( j(Z

−

=

j (L  )(Z ) j (L)(Z  ) . Q(Z , Z  )

To check 2) one writes for Y ∈ V , ρ(Y )∗ = (WY + WY )∗ = W j (Y ) + W j (Y ) = ρ( j (Y ).  

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We have treated so far Z and Z  as independent variables. We shall now restrict the above construction to the graph of j, i.e. to {(Z , Z  ) ∈ C | Z  = j (Z )}. Composing ρ with the restriction to the subset K = {Z | (Z , j (Z )) ∈ C}, one obtains in fact a ∗homomorphism θ of A = A(V, R) to a twisted cross-product C ∗ -algebra, C(K )×σ, L Z which involves the full geometric data (E, σ, L) and encodes the central quadratic form Q as a Hermitian metric on L provided Q fulfills the following positivity. Definition 11.7. Let C be a j-real invariant component of E × E and Q central on C. Then Q is positive on C iff it fullfills (11.11) and Q(Z , j (Z )) > 0, ∀Z ∈ K . One can then endow the line bundle L dual of the tautological line bundle on P(V ∗ ) with the Hermitian metric defined by f L , g L  Q (Z ) = f (Z ) g(Z )

L(Z ) L  (Z ) Q(Z , j (Z ))

L , L  ∈ V,

Z ∈ K , ∀ f, g ∈ C(K ).

(11.14) We view f L and g L  as sections of L and the right-hand side of the formula as a function on K which expresses their inner product f L , g L  . This defines a Hermitian metric on the restriction of L to K . Before we proceed we need to describe the general notion due to Pimsner [25] of twisted cross product. Given a compact space K , an homeomorphism σ of K and a hermitian line bundle L on K we define the C ∗ -algebra C(K ) ×σ, L Z as the twisted cross-product of C(K ) by the Hilbert C ∗ -bimodule associated to L and σ ([2,25]). We n let for each n ≥ 0, Lσ be the hermitian line bundle pullback of L by σ n and (cf. [4,29]) Ln := L ⊗ Lσ ⊗ · · · ⊗ Lσ

n−1

.

(11.15)

We first define a ∗-algebra as the linear span of the monomials ξ W n , W ∗n η∗ , ξ, η ∈ C(K , Ln )

(11.16)

with product given as in ([4,29]) for (ξ1 W n 1 ) (ξ2 W n 2 ) so that (ξ1 W n 1 ) (ξ2 W n 2 ) := (ξ1 ⊗ (ξ2 ◦ σ n 1 )) W n 1 +n 2 .

(11.17)

We use the hermitian structure of Ln to give meaning to the products η∗ ξ and ξ η∗ for ξ, η ∈ C(K , Ln ). The product then extends uniquely to an associative product of ∗-algebra fulfilling the following additional rules: (W ∗k η∗ ) (ξ W k ) := (η∗ ξ ) ◦ σ −k ,

(ξ W k ) (W ∗k η∗ ) := ξ η∗ .

(11.18)

The C ∗ -norm of C(K ) ×σ, L Z is defined as for ordinary cross-products and due to the amenability of the group Z there is no distinction between the reduced and maximal norms. The latter is obtained as the supremum of the norms in involutive representations in Hilbert space. The natural positive conditional expectation on the subalgebra C(K ) shows that the C ∗ -norm restricts to the usual sup norm on C(K ). To lighten notations in the next statement we abbreviate j (Z ) as Z¯ , but one should take care that in general the expression for j (Z ) can differ from Z¯ for instance with the notations of Subsect. 10.6 one gets j (Z ) =  Z¯ .

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Theorem 11.8. Let K ⊂ E be a compact σ -invariant subset and Q be central and strictly positive on {(Z , Z¯ ); Z ∈ K }. Let L be the restriction to K of the dual of the tautological line bundle on P(V ∗ ) endowed with the hermitian metric , Q . √ (i) The equality 2 θ (Y ) := Y W + W ∗ Y¯ ∗ yields a ∗-homomorphism θ : A = A(V, R) → C(K ) ×σ, L Z. (ii) For any Y ∈ V the C ∗ -norm of θ (Y ) fulfills √ Sup K !Y ! ≤ 2! θ (Y )! ≤ 2 Sup K !Y !. (iii) If σ 4 = ?, then θ (Q) = 1 where Q is viewed as an element of T (V )/(R). (i) The subset K˜ = {(Z , j (Z )); Z ∈ K } ⊂ C is globally invariant under σ˜ by Lemma 11.5. Moreover j˜ defined in (11.10) is the identity on K˜ . Each L ∈ V defines a section of L and hence an element L W ∈ C(K ) ×σ, L Z. The Definition (11.14) of the hermitian structure of L then shows that the elements L W and W ∗ j (L  )∗ of C(K ) ×σ, L Z fulfill the same algebraic rules (11.3), (11.4) as the W L and W L  while the involution of C(K ) ×σ, L Z is the restriction of the involution of Proposition 11.11. Thus the conclusion follows from Lemma 11.2. (ii) Since (Y W )(Y W )∗ = Y ∗ Y the C ∗ -norm of Y W is Sup K !Y !. It follows that √ 2! θ (Y )! ≤ 2 Sup K !Y !. For any complex number u of modulus one the map ξ W n → u n ξ W n extends to a ∗-automorphism of C(K ) ×σ, L Z. It √ follows taking u = i that !Y W − W ∗ Y¯ ∗ ! = !Y W + W ∗ Y¯ ∗ ! and Sup K !Y ! ≤ 2! θ (Y )!. (iii) follows from Lemma 11.2.  

Proof.

We shall now apply this general result to the algebras Calg (R4ϕ ). We take the quadratic form  X µ X µ (11.19) Q(X, X  ) := in the x-coordinates, so that Q is the canonical central element defining the sphere Sϕ3 by the equation Q = 1. Proposition 11.3 shows that in the generic case, i.e. for ϕ ∈ A ∪ B, the quadratic form Q is central on Fϕ × Fϕ with obvious notations. The positivity of Q is automatic since in the x-coordinates the involution jϕ coming from the involution of the quadratic ∗-algebra Calg (R4ϕ ) is simply complex conjugation jϕ (Z ) = Z¯ , so that Q(X, jϕ (X )) > 0 for X = 0. We thus get, Corollary 11.9. Let K ⊂ Fϕ be a compact σ -invariant subset. The homomorphism θ of Theorem 11.8 is a unital ∗-homomorphism from Calg (Sϕ3 ) to the cross-product C ∞ (K ) ×σ, L Z. This applies in particular to K = Fϕ . It follows that one obtains a non-trivial C ∗ -algebra C ∗ (Sϕ3 ) as the completion of Calg (Sϕ3 ) for the semi-norm, !P! := Sup! π(P)!,

(11.20)

where π varies through all unitary representations of Calg (Sϕ3 ). It was clear from the start that (11.20) defines a finite C ∗ -semi-norm on Calg (Sϕ3 ) since the equation of the sphere

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p3

q1

q2 p0 q0 p2 p1 q3 Fig. 7. The Elliptic Curve Fϕ ∩ P3 (R) (odd case)

 µ 2 (x ) = 1 together with the self-adjointness x µ = x µ∗ show that in any unitary representation one has ! π(x µ )! ≤ 1, ∀µ. What the above corollary gives is a lower bound for the C ∗ -norm such as that given by statement (ii) of Theorem 11.8 on the linear subspace V of generators. To analyse the compact σ -invariant subsets of Fϕ for generic ϕ, we distinguish the even case which corresponds to all sk having the same sign (cf. Fig. 4) (and holds for instance for ϕ ∈ A) from the odd case when all sk don’t have the same sign. First note that in all cases the real curve Fϕ ∩ P3 (R) is non-empty (it contains p0 ), and has two connected components since it is invariant under the Klein group H (9.32). In the even case σ preserves each of the two connected components of the real curve Fϕ ∩ P3 (R). In the odd case it permutes them (cf. Fig. 7). Proposition 11.10. Let ϕ be generic and even. (i) Each connected component of Fϕ ∩ P3 (R) is a minimal compact σ -invariant subset. (ii) Let K ⊂ Fϕ be a compact σ -invariant subset, then K is the sum in the elliptic curve Fϕ with origin p0 of K T = K ∩ FT (ϕ)0 (cf. 9.35) with the component Cϕ of Fϕ ∩ P3 (R) containing p0 . (iii) The cross-product C(Fϕ ) ×σ, L Z is isomorphic to the mapping torus of the automorphism β of the noncommutative torus T2η = Cϕ ×σ Z acting on the generators   1 4 by the matrix . 0 1 (i) This holds if we assume that ϕ is “generic” so that the elliptic parameter η fullfills η ∈ / Q. The diophantine approximations of η will play an important role later on. (ii) By construction the abelian compact group Fϕ is the product T1 × T2 of the one-dimensional tori T1 = Cϕ and T2 = FT (ϕ)0 , i.e. the component of FT (ϕ)

Proof.

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containing q0 (cf. 9.35). The translation σ is η × Id and the action of η is minimal on T1 = Cϕ . (iii) The isomorphism class of the cross-product C(Fϕ ) ×σ, L Z depends of L only through its class as a hermitian line bundle on the two torus Fϕ . This class is entirely specified by the first Chern class c1 (L). By construction one gets c1 (L) = 4 since the space of holomorphic sections of L is the 4-dimensional space V . Let U j be the generators of the algebra C(T2η ) where the presentation of the algebra is U1 U2 = e2πiη U2 U1 . For any integer k let βk be the automorphism of C(T2η ) acting on the generators U j by βk (U1 ) := U1 ,

βk (U2 ) := U1k U2 .

(11.21)

By construction the mapping torus T (βk ) of the automorphism βk is given by the algebra C(T (βk )) of continuous maps s ∈ R → x(s) ∈ C(T2η ) such that x(s + 1) = βk (x(s)), ∀s ∈ R. We just need to show that C(T (β4 )) is isomorphic to C(Fϕ ) ×σ, L Z and this follows from the general isomorphism C(T (βk ))  C(T1 × T2 ) ×η×Id, L Z,

(11.22)

(with T j = R/Z) for any hermitian line bundle L on T1 × T2 with c1 (L) = k. To check this one chooses L so that its continuous sections C(T1 × T2 , L) are scalar functions f (u, m) with u, m ∈ R such that f (u + 1, m) = f (u, m),

f (u, m + 1) = e2πik u f (u, m), ∀u, m ∈ R,

while its hermitian metric is given by f, g (u, m) = f (u, m) g(u, m), ∀u, m ∈ R. One defines a map α :

C(T (βk )) → C(T1 × T2 ) ×η×Id, L Z,

by writing for x ∈ C(T (βk )), x = (x(s)), x(s) ∈ C(T2η ) the Fourier expansion  x(s) = x(s, n) U2n , x(s, n) ∈ C(T1 ). Then the x(s, n) ∈ C(T1 ) define sections xn ∈ C(T1 × T2 , Ln ), and one just lets α(x) =



xn W n ∈ C(T1 × T2 ) ×η×Id, L Z, ∀x ∈ C(T (βk )).

One then checks that this gives the required isomorphism (11.22).

 

Corollary 11.11. Let ϕ be generic and even, then Fϕ ×σ, L Z is a noncommutative 3-manifold with an elliptic action of the three dimensional Heisenberg Lie algebra h3 and an invariant trace τ .

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Proof. This follows 7 from Proposition 11.10 (iii). One can construct directly the action of h3 on C ∞ (Fϕ ) ×σ, L Z by choosing a constant (translation invariant) curvature connection ∇, compatible with the metric, on the hermitian line bundle L on Fϕ (viewed in the C ∞ -category not in the holomorphic one). The two covariant differentials ∇ j corresponding to the two vector fields X j on Fϕ generating the translations of the elliptic curve, give a natural extension of X j as the unique derivations δ j of C ∞ (Fϕ ) ×σ, L Z fulfilling the rules, δ j ( f ) = X j ( f ), ∀ f ∈ C ∞ (Fϕ ), δ j (ξ W ) = ∇ j (ξ ) W, ∀ξ ∈ C ∞ (Fϕ , L).

(11.23)

We let δ be the unique derivation of C ∞ (Fϕ ) ×σ, L Z corresponding to the grading by powers of W . It vanishes on C ∞ (Fϕ ) and fulfills δ(ξ W k ) = i k ξ W k ,

δ(W ∗k η∗ ) = −i k W ∗k η∗ .

(11.24)

Let i κ be the constant curvature of the connection ∇; one gets [δ1 , δ2 ] = κ δ, [δ, δ j ] = 0, which provides the required action of the Lie algebra h3 on C ∞ (Fϕ ) ×σ, L Z.

(11.25)  

It follows that one is exactly in the framework developed in [10]. We refer to [26] and [1] where these noncommutative manifolds were analysed in terms of crossed products by Hilbert C ∗ -bimodules. Integration on the translation invariant volume form dv of Fϕ gives the h3 -invariant trace τ ,  τ( f ) = f dv, ∀ f ∈ C ∞ (Fϕ ), τ (ξ W k ) = τ (W ∗k η∗ ) = 0, ∀k = 0.

(11.26)

It follows in particular that the results of [10] apply to obtain the calculus. In particular the following gives the “fundamental class” as a 3-cyclic cocycle:  τ3 (a0 , a1 , a2 , a3 ) = i jk τ (a0 δi (a1 ) δ j (a2 ) δk (a3 )), (11.27) where the δ j are the above derivations with δ3 := δ. We shall in fact describe the same calculus in greater generality in the last section which will be devoted to the computation of the Jacobian of the homomorphism θ of Corollary 11.9. Similar results hold in the odd case. Then Fϕ ∩ P3 (R) is a minimal compact σ -invariant subset, any compact σ -invariant subset K ⊂ Fϕ is the sum in the elliptic curve Fϕ with origin p0 of Fϕ ∩ P3 (R) with K T = K ∩ FT (ϕ)0 but the latter is automatically invariant under the subgroup H0 ⊂ H of order 2 of the Klein group H (9.32), H0 := {h ∈ H | h(FT (ϕ)0 ) = FT (ϕ)0 }.

(11.28)

The group law in Fϕ is described geometrically as follows. It involves the point q0 . The sum z = x + y of two points x and y of Fϕ is z = I0 (w), where w is the 4th point of 7 It justifies the terminology “nilmanifold”.

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intersection of Fϕ with the plane determined by the three points {q0 , x, y}. It commutes by construction with complex conjugation so that x + y = x + y¯ , ∀x, y ∈ Fϕ . By Lemma 11.5 the translation σ is imaginary for the canonical involution jϕ . In terms of the coordinates Z µ this involution is described as follows, using (9.36) (multiplied by ei(π/4−ϕ1 −ϕ2 −ϕ3 ) 2−3/2 ) to change variables. Among the 3 real numbers λk = cosϕ cosϕm sin(ϕ − ϕm ),

k ∈ {1, 2, 3},

two have the same sign  and one, λk , k ∈ {1, 2, 3}, the opposite sign. Then jϕ =  Ik ◦ c,

(11.29)

where c is complex conjugation on the real elliptic curve Fϕ (Sect. 3) and Iµ the involution Iµ (Z µ ) = −Z µ , Iµ (Z ν ) = Z ν , ν = µ. (11.30) The index k and the sign  remain constant when ϕ varies in each of the four components of the complement of the four points qµ in FT (ϕ). The sign  matters for the action of jϕ on linear forms as in (11.13), but is irrelevant for the action on Fϕ . Each involution Iµ is a symmetry z → p − z in the elliptic curve Fϕ and the products Iµ ◦ Iν form the Klein subgroup H (9.32) acting by translations of order two on Fϕ . The quadratic form Q of (11.19) is given in the new coordinates by,   tk s k Q k (11.31) Q= cos2 ϕ with sk := 1 + t tm , tk := tan ϕk and Q k defined by (10.8). Let us assume that 0 < ϕ1 < ϕ2 < ϕ3 < π/2 for instance, then the relation between the x µ and the Yµ is given with the appropriate normalization of the Yµ by x µ = ρµ Yµ , where ρ02 = −sin(ϕ1 − ϕ2 ) sin(ϕ1 − ϕ3 ) sin(ϕ2 − ϕ3 ), ρ12 = −cosϕ2 cosϕ3 sin(ϕ2 − ϕ3 ), (11.32) 2 2 ρ2 = cosϕ1 cosϕ3 sin(ϕ1 − ϕ3 ), ρ3 = −cosϕ1 cosϕ2 sin(ϕ1 − ϕ2 )). All the ρµ are real except for ρ2 which is purely imaginary and the involution j is I2 ◦ c. One checks directly that    ρµ2 Yµ2 = tk s k Q k . cos2 ϕ We can now compare the ∗-homomorphism ρ˜ of Sect. 10.6 with the ∗-homomorphism obtained from a positive central quadratic form, one gets with the constants s and b given by,   s = −s2 cosϕ j cos(ϕk − ϕ ). sinϕ j , b2 = Proposition 11.12. Let 0 < ϕ1 < ϕ2 < ϕ3 < π/2. (i) The ∗-homomorphism ρ˜ is the ∗-homomorphism associated to the central quadratic form Q  , s Q  = Q 1 + Q 3 + s2 Q 2 , which is positive on E for the involution I3 ◦ c.

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(ii) Let β(Y0 ) = iY2 , β(Y1 ) =

+

+ + + J12 Y3 , β(Y2 ) = − J12 J23 Y0 , β(Y3 ) = i J23 Y1 ,

the map b β gives a ∗-isomorphism sending the form Q  into Q and the involution I3 ◦ c into I2 ◦ c.  Proof. (i) By Proposition 10.15 it is enough to show that Q  corresponds to (x µ )2 µ under the composition of the transformations Sµ = λµ x of Lemma 4.4 and (10.48), S0 = d Y2 , S1 = i Y3 , S2 = d Y0 , S3 = −Y1 , where d 2 = −J31 . (ii) The map b β is obtained as the composition of the isomorphisms (10.48), Sµ = λµ x µ of Lemma 4.4 and x µ = ρµ Yµ with ρµ given in (11.32). Thus the answer follows since each of these is a ∗-isomorphism and the image of Q  in the  maps µ µ 2 x variables is simply (x ) .   Let ϕ be generic and even and v ∈ FT (ϕ)0 . Let K (v) = v + Cϕ be the minimal compact σ -invariant subset containing v (Proposition 11.10 (ii)). By Corollary 11.9 we get a homomorphism, θv : Calg (Sϕ3 ) → C ∞ (T2η ), (11.33) whose non-triviality will be proved below in Corollary 12.8. We shall first show (Theorem 11.13) that it transits through the cross-product of the field K q of meromorphic functions on the elliptic curve by the subgroup of its Galois group AutC (K q ) generated by the translation σ . For Z = v + z , z ∈ Cϕ , one has using (11.29) and (9.35), jϕ (Z ) = Iµ (Z − v) − I (v),

(11.34)

which is a rational function r (v, Z ). Fixing ϕ, v and substituting Z and Z  = r (v, Z ) in the formulas (11.4) and (11.5) of Lemma 11.2 with L real such that 0 ∈ / L(K (v)), L  =  L ◦ Iµ and Q given by (11.31), we obtain rational formulas for a homomorphism θ˜v of Calg (Sϕ3 ) to the generalised cross-product of the field K q of meromorphic functions f (Z ) on the elliptic curve Fϕ by σ . The generalised cross-product rule (11.4) is given by W W  := γ (Z ), where γ is a rational function. Similarly W  W := γ (σ −1 (Z )). Using integration on the cycle K (v) to obtain a trace, together with Corollary 11.9, we get, Theorem 11.13. The homomorphism θv : Calg (Sϕ3 ) → C ∞ (T2η ) factorises with a homomorphism θ˜v : Calg (Sϕ3 ) → K q ×σ Z to the generalised cross-product of the field K q of meromorphic functions on the elliptic curve Fϕ by the subgroup of the Galois group AutC (K q ) generated by σ . Its image generates the hyperfinite factor of type II1 after weak closure relative to the trace given by integration on the cycle K (v). Elements of K q with poles on K (v) are unbounded and give elements of the regular ring of affiliated operators, but all elements of θv (Calg (Sϕ3 )) are regular on K (v). The above generalisation of the cross-product rules (11.4) with the rational formula for W W  := γ (Z ) is similar to the introduction of 2-cocycles in the standard Brauer theory of central simple algebras.

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12. The Jacobian of the Covering of Sϕ3 In this section we shall analyse the morphism of ∗-algebras θ : Calg (Sϕ3 ) → C ∞ (Fϕ ×σ, L Z)

(12.1)

of Corollary 11.9, by computing its Jacobian in the sense of noncommutative differential geometry ([11]). We postpone the analysis at the C ∗ -level, in particular the role of the discrete series, to another forthcoming publication. The usual Jacobian of a smooth map ϕ : M → N of manifolds is obtained as the ratio ϕ ∗ ( ω N )/ω M of the pullback of the volume form ω N of the target manifold N with the volume form ω M of the source manifold M. In noncommutative geometry, differential forms ω of degree d become Hochschild classes ω˜ ∈ H Hd (A), A = C ∞ (M). Moreover since one works with the dual formulation in terms of algebras, the pullback ϕ ∗ (ω N ) is replaced by the pushforward ϕ∗t (ω˜ N ) under the corresponding transposed morphism of algebras ϕ t ( f ) := f ◦ ϕ, ∀ f ∈ C ∞ (N ). The noncommutative sphere Sϕ3 admits a canonical “volume form” given by the Hochschild 3-cycle ch 3 (U ). Our goal is to compute the push-forward, 2

  θ∗ ch 3 (U ) ∈ H H3 (C ∞ (Fϕ ×σ, L Z)). 2

(12.2)

Let ϕ be generic and even. The noncommutative manifold Fϕ ×σ, L Z is, by Corollary 11.11, a noncommutative 3-dimensional nilmanifold isomorphic to the mapping torus of an automorphism of the noncommutative 2-torus Tη2 . Its Hochschild homology is easily computed using the corresponding result for the noncommutative torus ([11]). It admits in particular a canonical volume form V ∈ H H3 (C ∞ (Fϕ ×σ, L Z)) which corresponds to the natural class in H H2 (C ∞ (Tη2 )) ([11]). The volume form V is obtained in the cross-product Fϕ ×σ, L Z from the translation invariant 2-form dv on Fϕ . To compare θ∗ (ch 3 (U )) with V we shall pair it with the 3-dimensional Hochs2

child cocycle τh ∈ H H 3 (C ∞ (Fϕ ×σ, L Z)) given, for any element h of the center of C ∞ (Fϕ ×σ, L Z), by τh (a0 , a1 , a2 , a3 ) = τ3 (h a0 , a1 , a2 , a3 ),

(12.3)

H C 3 (C ∞ (Fϕ

where τ3 ∈ ×σ, L Z)) is the fundamental class in cyclic cohomology defined by (11.27). By Proposition 10.4 the component ch 3 (U ) of the Chern character 2 is given by,  ch 3 (U ) = αβγ δ cos(ϕα − ϕβ + ϕγ − ϕδ ) x α d x β d x γ d x δ 2  −i sin2(ϕµ − ϕν ) x µ d x ν d x µ d x ν , (12.4) where ϕ0 := 0. In terms of the Yµ one gets,  ch 3 (U ) = κ δαβγ δ (sα − sβ + sγ − sδ ) Yα dYβ dYγ dYδ 2  +κ αβγ δ (sα − sβ ) Yγ dYδ dYγ dYδ ,

(12.5)

where s0 := 0, sk := 1 + t tm , tk := tan ϕk and δαβγ δ = αβγ δ (n α − n β + n γ − n δ )

(12.6)

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with n 0 = 0 and n k = 1. The normalization factor is  κ= i cos2 (ϕk ) sin(ϕ − ϕm ).

(12.7)

Formula (12.5) shows that, up to normalization, ch 3 (U ) only depends on the fiber Fϕ 2 of ϕ. Let ϕ be generic and even; there is a similar formula in the odd case. In our case the involutions I and I0 are conjugate by a real translation κ of the elliptic curve Fϕ and we let Fϕ (0) be one of the two connected components of {Z ∈ Fϕ | I0 (Z ) = Z¯ }.

(12.8)

By Proposition 11.10 we can identify the center of C ∞ (Fϕ ×σ, L Z) with C ∞ (Fϕ (0)). We assume for simplicity that ϕ j ∈ [0, π2 ] are in cyclic order ϕk < ϕl < ϕm for some k ∈ {1, 2, 3}. Theorem 12.1. Let h ∈ Center (C ∞ (Fϕ ×σ, L Z)) ∼ C ∞ (Fϕ (0)). Then  h(Z ) d R(Z ), ch 3 (U ), τh = 6 π  Fϕ (0)

2

where  is the period given by the elliptic integral of the first kind,  Zk d Z0 − Z0d Zk , = Z Zm Cϕ and R the rational fraction, R(Z ) = tk

2 Zm 2 + c Z2 Zm k l

with ck = tg(ϕl ) cot(ϕk − ϕ ). We assume that 0 < ϕ1 < ϕ2 < ϕ3 < π/2, i.e. that k = 1. We start from Corollary 10.11 and express the result ω= −

σ (m)4 g(m) dm, λ


(12.9)

in terms of the trigonometric parameters ϕ j and the coordinates Z µ of Z . One has (cf. (10.36)) 2  σ (m)4 = (C1 − λ C2 )−2 , (12.10) sin ϕ j and we begin by giving a better formula for C1 − λ C2 . Lemma 12.2. One has C1 − λ C2 = b1 ϑ12 (im) + b2 ϑ22 (im), where b1 = 4

ϑ12 (η) s1 ϑ22 (η)

, b2 = 4

s1 − 1 . s1

(12.11)

(12.12)

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109

Proof. One uses (10.37), λ =

ϑ22 (0) s1 ϑ22 (η)

and the identity

ϑ22 (0) ϑ2 (η + im) ϑ2 (η − im) = ϑ22 (im) ϑ22 (η) − ϑ12 (im) ϑ12 (η).   The m-dependent terms are understood from the following lemma Lemma 12.3. Let b j be arbitrary constants, then ϑ32 (0)ϑ42 (0)

ϑ12 (u) ϑ1 (0)3 ϑ1 (2u) d . = b 2 2 2 2 2 du b1 ϑ1 (u) + b2 ϑ2 (u) π (b1 ϑ1 (u) + b2 ϑ22 (u))2 (12.13)

Proof. This follows from the classical identity d sn(u) = cn(u) dn(u), du

(12.14)

which implies ϑ  (0)3 d ϑ12 (u) ϑ1 (u)ϑ2 (u)ϑ3 (u)ϑ4 (u) ϑ1 (2u) = 2 π ϑ22 (0) = 2 21 , 2 4 du ϑ2 (u) ϑ2 (u) π ϑ3 (0)ϑ42 (0) ϑ24 (u) using the duplication formula ϑ2 (0)ϑ3 (0)ϑ4 (0)ϑ1 (2 u) = 2 ϑ1 (u)ϑ2 (u)ϑ3 (u)ϑ4 (u) and the Jacobi derivative formula ϑ1 (0) = ϑ2 (0)ϑ3 (0)ϑ4 (0). π   Taking u = im in (12.13) this allows to write ω as the differential of R(m) =

c ϑ12 (im) b1 ϑ12 (im) + b2 ϑ22 (im)

(12.15)

using (10.45)

ϑ1 (0)3 ϑ1 (η) ϑ1 (2im) . (12.16) π 3 ϑ2 (η)ϑ3 (η)ϑ4 (η) The constant c is uniquely determined and will be simplified later; we get so far,  sin ϕ j )2 ϑ1 (η) ϑ32 (0)ϑ42 (0) 3 ( . (12.17) c = 24 (2π ) π b2 λ
ϑ2 (η)ϑ3 (η)ϑ4 (η) g(m) = 24 (2πi)3

Lemma 12.4.

(i) The differential form χ :=

Zk d Z0 − Z0d Zk sk Z  Z m

(12.18)

is independent of k and is, up to scale, the only holomorphic form of type (1, 0) on Fϕ .

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(ii) One has 2π Proof.

ϑ42 (0)

ϑ1 (η) ϑ4 (η) = ϑ2 (η) ϑ3 (η)

 Cϕ

Z3d Z0 − Z0d Z3 . Z1 Z2

(12.19)

(i) Recall that the equations defining Fϕ are Z 02 − Z 12 Z 2 − Z 22 Z 2 − Z 32 = 0 = 0 . s1 s2 s3

(12.20)

One gets the required independence by differentiation. ϑ1 (η) ϑ4 (η) (ii) Let us check (12.19). The factor allows to replace the Z j by ϑ j+1 (2z) ϑ2 (η) ϑ3 (η) so that the right hand side gives, using (12.14), 

1

0

ϑ4 (2z)dϑ1 (2z) − ϑ1 (2z)dϑ4 (2z) = 2 π ϑ42 (0). ϑ2 (2z)ϑ3 (2z)

  In terms of Z = (Z j ) one has ϑ12 (η)

R(m) = c s1 where a1 =

Z 02

4 ϑ22 (η)

ϑ14 (η) ϑ24 (η)

a1 Z 02

+ a2 Z 12

,

(12.21)

, a2 = s1 − 1.

(12.22)

One can express the coefficient a1 in trigonometric terms using Lemma 12.5. a1 =

ϑ14 (η) ϑ24 (η)

=

(s1 − s2 )(s1 − s3 ) . s2 s3

(12.23)

Proof. By homogeneity of the right hand side one can replace the s j by the σ j of (10.44). One then gets (s1 − s2 )(s1 − s3 ) = − J12 J31 s2 s3 and the result follows from (10.26). We now simplify the product c s1 using (cf. (10.12))
=

3  1

  ϑ12 (η) 4 ϑ22 (η)

replacing ϑ42 (0) in (12.17) by (12.19) and

(tan (ϕ j ) cos(ϕk − ϕ )), λ =

ϑ32 (0) s2 ϑ32 (η)

.

(12.24)

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111

By elementary computations and using once more (12.23) one gets  ϑ 2 (η) s1 Z3d Z0 − Z0d Z3 c s1 1 2 . = 6 π t1 a 1 s3 C ϕ Z1 Z2 4 ϑ2 (η)

(12.25)

We thus obtain so far using the elementary equality a2 = cot(ϕ1 − ϕ2 ) cot(ϕ1 − ϕ3 ), a1 and Lemma 12.4 which allows to eliminate the term  the following formula for the rational fraction, R(Z ) = t1

s1 , using the definition of the period s3

Z 02 Z 02 + a Z 12

(12.26)

with a = cot(ϕ1 − ϕ2 ) cot(ϕ1 − ϕ3 ). What we have computed so far is the image of ch3/2 under the ∗-homomorphism ρ. ˜ By Proposition 11.12 (i) this amounts to the image of ch3/2 under the ∗-homomorphism associated to the central quadratic form Q  . By Proposition 11.12 (ii) we get the result for Q using the isomorphism β and this gives the following formula for R(Z ), R(Z ) = t1

Z 32 Z 32 + c1 Z 22

(12.27)

with c1 = tg(ϕ2 ) cot(ϕ1 − ϕ2 ). We shall explain below in Sect. 13 how to perform the transition from (12.26) to (12.27) in a conceptual manner. In fact the conceptual understanding of the simplicity of the final result of Theorem 12.1 is at the origin of many of the notions developed in the present paper and in particular of the “rational” formulation of the calculus which will be obtained in the last section. The geometric meaning of Theorem 12.1 is the computation of the Jacobian in the sense of noncommutative geometry of the morphism θ as explained above. The integral  is (up to a trivial normalization factor) a standard elliptic integral, it is given by an hypergeometric function in the variable m :=

sk (sl − sm ) sl (sk − sm )

(12.28)

or a modular form in terms of q. Lemma 12.6. The differential of R is given on Fϕ by d R = J (Z ) χ , where J (Z ) = 2 (sl − sm ) ck tk

Z0 Z1 Z2 Z3 2 + c Z 2 )2 (Z m k l

(12.29)

Proof. This can easily be checked using (12.14) but it is worthwhile to give a simple direct argument. One has indeed by definition of χ , d

Z 2j Z 02

= −2 s j

Z0 Z1 Z2 Z3 χ, Z 04

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which gives using 2 sl Z m − sm Z l2 = (sl − sm ) Z 02

the equality d and the required result.

Z l2 Z0 Z1 Z2 Z3 = 2 (sm − sl ) χ, 2 4 Zm Zm  

The period  does not vanish and J (Z ), Z ∈ Fϕ (0), only vanishes on the 4 “ramification points” necessarily present due to the symmetries. Corollary 12.7. The Jacobian of the map θ t is given by the equality θ∗ (ch 3 (U )) = 3  J V, 2

where J is the element of the center C ∞ (Fϕ (0)) of C ∞ (Fϕ ×σ, L Z) given by formula (12.29). This statement assumes that ϕ is generic in the measure theoretic sense so that η admits good diophantine approximation ([11]). It justifies in particular the terminology of “ramified covering” applied to θ t . The function J has only 4 zeros on Fϕ (0) which correspond to the ramification. As shown by Theorem 9.5 the algebra Aϕ is defined over R, i.e. admits a natural antilinear automorphism of period two, γ uniquely defined by γ (Yµ ) := Yµ , ∀µ.

(12.30)

Theorem 9.5 also shows that σ is defined over R and hence commutes with complex conjugation c(Z ) = Z¯ . This gives a natural real structure γ on the algebra C Q with C = Fϕ × Fϕ and Q as above,  γ ( f (Z , Z  )) := f (c(Z ), c(Z  )), γ (W L ) := Wc(L) , γ (W L  ) := Wc(L ).

One checks that the morphism ρ of Lemma 11.2 is “real”, i.e. that, γ ◦ ρ = ρ ◦ γ.

(12.31)

Since c(Z ) = Z¯ reverses the orientation of Fϕ , while γ preserves the orientation of Sϕ3 , it follows that J ( Z¯ ) = −J (Z ) and J necessarily vanishes on Fϕ (0) ∩ P3 (R). Note also that for general h one has ch 3 (U ), τh = 0 which shows that both 2

ch 3 (U ) ∈ H H3 and τh ∈ H H 3 are non trivial Hochschild classes. These results hold 2

in the smooth algebra C ∞ (Sϕ3 ) containing the closure of Calg (Sϕ3 ) under holomorphic functional calculus in the C ∗ algebra C ∗ (Sϕ3 ). We can also use Theorem 12.1 to show the non-triviality of the morphism θv : Calg (Sϕ3 ) → C ∞ (T2η ) of (11.33). Corollary 12.8. The pullback of the fundamental class [T2η ] of the noncommutative torus by the homomorphism θv : Calg (Sϕ3 ) → C ∞ (T2η ) of (11.33) is non zero, θv∗ ([T2η ]) = 0 ∈ H H 2 provided v is not a ramification point.

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We have shown above the non-triviality of the Hochschild homology and cohomology groups H H3 (C ∞ (Sϕ3 )) and H H 3 (C ∞ (Sϕ3 )) by exhibiting specific elements with nonzero pairing. Combining the ramified cover π = θ t with the natural spectral geometry (spectral triple) on the noncommutative 3-dimensional nilmanifold Fϕ ×σ, L Z yields a natural spectral triple on Sϕ3 in the generic case. 13. Calculus and Cyclic Cohomology Theorem 12.1 suggests the existence of a “rational” form of the calculus explaining the appearance of the elliptic period  and the rationality of R. We shall show in this last section that this is indeed the case. This will allow us to get a very simple conceptual form of the above computation of the Jacobian in Theorem 13.8 below. Let us first go back to the general framework of twisted cross products of the form A = C ∞ (M) ×σ, L Z,

(13.1)

where σ is a diffeomorphism of the manifold M. We shall follow [12] to construct cyclic cohomology classes from cocycles in the bicomplex of group cohomology (with group  = Z) with coefficients in de Rham currents on M. The twist by the line bundle L introduces a non-trivial interesting nuance. Let (M) be the algebra of smooth differential forms on M, endowed with the action of Z, α1,k (ω) := σ ∗k ω, k ∈ Z. (13.2) ˜ As in [14] p. 219 we let (M) be the graded algebra obtained as the (graded) tensor product of (M) by the exterior algebra ∧(C[Z] ) on the augmentation ideal C[Z] in the group ring C[Z]. With [n], n ∈ Z the canonical basis of C[Z], the augmentation  : C[Z] → C fulfills ([n]) = 1, ∀n , and δn := [n] − [0],

n ∈ Z, n = 0

(13.3)

is a linear basis of C[Z] . The left regular representation of Z on C[Z] restricts to C[Z] and is given on the above basis by α2,k (δn ) := δn+k − δk ,

k ∈ Z.

(13.4)

It extends to an action α2 of Z by automorphisms of ∧C[Z] . We let α = α1 ⊗ α2 be the tensor product action of Z on the graded tensor product ˜ (M) = (M) ⊗ ∧C[Z] .

(13.5)

We now use the hermitian line bundle L to form the twisted cross-product ˜ C := (M) ×α, L Z.

(13.6)

We let Ln be as in (11.15) for n > 0 and extend its definition for n < 0 so that L−n is the pullback by σ n of the dual Lˆ n of Ln for all n. The hermitian structure gives an antilinear isomorphism ∗ : Ln → Lˆ n . One has for all n, m ∈ Z a canonical isomorphism Ln ⊗ σ ∗m Lm  Ln+m , which is by construction compatible with the hermitian structures.

(13.7)

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The algebra C is the linear span of monomials ξ W n , where ˜ ξ ∈ C ∞ (M, Ln ) ⊗C ∞ (M) (M)

(13.8)

with the product rules (11.17), (11.18). Let ∇ be a hermitian connection on L. We shall turn C into a differential graded algebra. By functoriality ∇ gives a hermitian connection on the Lk and hence a graded derivation ∇n : C ∞ (M, Ln ) ⊗C ∞ (M) (M) → C ∞ (M, Ln ) ⊗C ∞ (M) (M)

(13.9)

whose square ∇n2 is multiplication by the curvature κn ∈ 2 (M) of Ln , κn+m = κn + σ ∗n (κm ), ∀n, m ∈ Z

(13.10)

2 (M)

the curvature of L. One has dκn = 0 and we extend the differwith κ1 = κ ∈ ˜ ential d to a graded derivation on (M) by dδn = κn .

(13.11)

We can then extend ∇n uniquely to the induced module ˜ En = C ∞ (M, Ln ) ⊗C ∞ (M) (M).

(13.12)

˜ ∇˜ n (ξ ω) = ∇n (ξ ) ω + (−1)deg(ξ ) ξ dω, ∀ω ∈ (M)

(13.13)

by the equality ˜ Proposition 13.1. a) The pair ((M), d) is a graded differential algebra. ˜ b) Let α be the tensor product action of Z, then α(k) ∈ Aut((M), d), ∀k ∈ Z. ˜ c) The following equality defines a flat connection on the induced module En on (M), ∇n (ξ ) = ∇˜ n (ξ ) − (−1)deg(ξ ) ξ δn .

(13.14)

˜ extends uniquely to a graded derivation of C such d) The graded derivation d of (M) that, d(ξ W n ) = ∇n (ξ ) W n , (13.15) which turns the pair (C, d) into a graded differential algebra. Proof. a) By construction d is the unique extension of the differential d of (M) to a graded derivation of the graded tensor product (13.5) such that (13.11) holds. One just needs to check that d 2 = 0 on simple tensors ω ⊗ δn , one gets d(ω ⊗ δn ) = dω ⊗ δn + (−1)deg(ω) ω κn ⊗ 1, d 2 (ω ⊗ δn ) = d 2 ω ⊗ δn + (−1)deg(ω)+1 dω κn ⊗ 1 + (−1)deg(ω) dω κn ⊗ 1 = 0. b) Let us check that α(k) commutes with the differentiation d on simple tensors ω ⊗δn . One has d(α(k)(ω ⊗ δn )) = d(σ ∗k (ω) ⊗ (δn+k − δk )) = σ ∗k (dω) ⊗ (δn+k − δk ) +(−1)deg(ω) σ ∗k (ω) (κn+k − κk ) ⊗ 1, ×α(k)(d(ω ⊗ δn )) = σ ∗k (dω) ⊗ (δn+k − δk ) +(−1)deg(ω) σ ∗k (ω κn ) ⊗ 1, and the equality follows from (13.10).

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c) Since δn2 = 0 one gets (∇n )2 (ξ ) = (∇˜ n )2 (ξ ) − ξ dδn = 0, since dδn = κn is the curvature. d) Since the algebra C is the linear span of monomials ξ W n the linear map d is well ˜ defined and coincides with the differential d on (M) since δ0 = 0. Let us show that the two terms of (13.14) separately define derivations of C. By construction the connections ∇n are compatible with the canonical isomorphisms (13.7) and the same holds for their extensions ∇˜ n which is enough to show that the first term of (13.14) separately defines a derivation of C. The proof for the second term 

d (ξ W n ) = (−1)deg(ξ ) ξ δn W n ,

(13.16)

follows from (13.4) and is identical to the proof of Lemma 12, Chapter III of [14]. The flatness (c) of the connections ∇n ensures that d 2 = 0 so that (C, d) is a graded differential algebra. To construct closed graded traces on this differential graded algebra we follow ([12]) and consider the double complex of group cochains (with group  = Z) with coefficients in de Rham currents on M. The cochains γ ∈ C n,m are totally antisymmetric maps from Zn+1 to the space −m (M) of de Rham currents of dimension −m, which fulfill γ (k0 + k, k1 + k, k2 + k, · · · , kn + k) = σ∗−k γ (k0 , k1 , k2 , · · · , kn ), ∀k, k j ∈ Z. (13.17) Besides the coboundary d1 of group cohomology, given by (d1 γ )(k0 , k1 , · · · , kn+1 ) =

n+1 

(−1) j+m γ (k0 , k1 , · · · , kˆj , · · · , kn+1 )

(13.18)

0

and the coboundary d2 of de Rham homology, (d2 γ )(k0 , k1 , · · · , kn ) = b(γ (k0 , k1 , · · · , kn )), the curvatures κn generate the further coboundary d3 defined on Ker d1 by, (d3 γ )(k0 , · · · , kn+1 ) =

n+1 

(−1) j+m+1 κ k j γ (k0 , · · · , kˆj , · · · , kn+1 ),

(13.19)

0

which maps Ker d1 ∩ C n,m to C n+1,m+2 . Translation invariance follows from (13.10) and ϕ∗ (ωC) = ϕ ∗−1 (ω)ϕ∗ (C) for C ∈ −m (M), ω ∈ ∗ (M). To each γ ∈ C n,m one associates the functional γ˜ on C given by, ˜ γ˜ ( ξ W n ) = 0, ∀ n = 0, ξ ∈ (M), γ˜ ( ω ⊗ δk1 · · · δkn ) = ω, γ (0, k1 · · · , kn ) , ∀k j ∈ Z. One then has

(13.20)

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Lemma 13.2. Let γ ∈ C n,m , then ˜ (i) One has for all ρ ∈ (M), γ˜ (ρ − α(−k)ρ) = −(d1˜γ )(δk ρ).

(13.21)

(ii) One has for all a, b ∈ C with d  defined in (13.16), γ˜ (a b − (−1)deg(a)deg(b) b a) = (−1)deg(a) (d1˜γ )(a d  b).

(13.22)

˜ (iii) One has for all ρ ∈ (M), γ˜ (dρ) = (d2˜γ )(ρ) + (d3˜γ )(ρ).

(13.23)

Proof. (i) We can assume that ρ is of the form ρ = ω ⊗ δk1 · · · δkn . The left side of (13.21) is by construction ω, γ (0, k1 · · · , kn ) − γ˜ (σ ∗−k (ω) ⊗ (δk1 −k − δ−k ) · · · (δkn −k − δ−k )). 2 = 0, When one expands the product one gets using δ−k

(δk1 −k − δ−k ) · · · (δkn −k − δ−k ) =



δk j −k + δ−k



(−1)i



δk j −k ,

j =i

and one uses the translation invariance (13.17) to write  γ˜ (σ ∗−k (ω) ⊗ δk j −k ) = ω, γ (k, k1 · · · , kn )

and



γ˜ (σ ∗−k (ω) ⊗ (−1)i δ−k

δk j −k ) = −(−1)i ω, γ (0, k, k1 · · · , kˆi , · · · , kn ) .

j =i

One thus obtains the same terms as in −(d1˜γ )(δk ρ) = − ω, (d1 γ )(0, k, k1 · · · , kn ) , using (13.18) and the graded commutation of δk with ω which yields a (−1)m overall sign. ˜ Eq. (11.13) holds (ii) It is enough to show that for any k ∈ Z and a  , b ∈ (M),  k  −k ˜ for a = a W and b = b W . The graded commutativity of (M) allows to write the ˜ graded commutator in (11.13) as ρ − α(−k)ρ, where ρ = a  α(k)(b ) ∈ (M). One has ρ = a b and (i) thus shows that γ˜ (a b − (−1)deg(a)deg(b) b a) = − (d1˜γ )(δk a b) = −(−1)deg(a)+deg(b) (d1˜γ )(a b δk ). One has 

a d (b) = a  α(k)((−1)deg(b) b δ−k ),

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thus the result follows from the equality α(k)(δ−k ) = − δk . (iii) We can assume that ρ is of the form ρ = ω ⊗ δk1 · · · δkn . One has dρ = dω ⊗ δk1 · · · δkn − (−1)deg(ω) which gives (13.23).



(−1) j ω κk j ⊗ δk1 · · · δkˆ j · · · δkn

 

To each γ ∈ C n,m one associates the (n − m + 1) linear form on A = C ∞ (M) ×σ, L Z given by, (γ )(a0 , a1 , · · · , an−m ) n−m  (−1) j (n−m− j) γ˜ (da j+1 · · · dan−m a0 da1 · · · da j−1 da j ), = λn,m

(13.24)

0

where λn,m :=

n! (n−m+1)! .

Lemma 13.3. (i) The Hochschild coboundary b(γ ) is equal to (d1 γ ). (ii) Let γ ∈ C n,m ∩ Ker d1 . Then (γ ) is a Hochschild cocycle and 1 B(γ ) = (d2 γ ) + n+1 (d3 γ ) Proof. (i) The proof is identical to that of Theorem 14 a) Chapter III of [14]. (ii) By (i) and the hypothesis d1 (γ ) = 0, (γ ) is a Hochschild cocycle. In fact by Lemma 13.2 the functional γ˜ is a graded trace and thus the formula for (γ ) simplifies to (γ )(a0 , a1 , · · · , an−m ) = (n − m + 1) λn,m γ˜ (a0 da1 · · · dan−m ).

(13.25)

It follows that B0 ((γ ))(a0 , a1 , · · · , an−m−1 ) = (n − m + 1) λn,m γ˜ (dρ), ρ = a0 da1 · · · dan−m−1 , and B0 ((γ )) is already cyclic so that B((γ )) = (n − m)B0 ((γ )). Since the coboundary d2 anticommutes with d1 , one has d2 γ ∈ Ker d1 and (d2 γ ) is also a Hochschild cocycle and is given by (13.25) for d2 γ . Let us check that d3 γ ∈ Ker d1 . One has up to an overall sign,  d1 d3 (γ ) = (−1)i d3 (γ )(k0 , · · · , kˆi · · · , kn+2 )  (i, j) κ k j γ (k0 , · · · , kˆi  , · · · , kˆj  , · · · , kn+1 ), = i, j

where (i  , j  ) is the permutation of (i, j) such that i  < j  and up to an overall sign (i, j) is the product of the signature of this permutation by (−1)i+ j . For each j the coefficient of κ k j is up to an overall sign given by d1 (γ )(k0 , · · · , kˆj · · · , kn+2 ) = 0, thus d1 d3 (γ ) = 0. The result thus follows from Lemma 13.2 (iii).  

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We shall now show how the above general framework allows to reformulate the calculus involved in Theorem 12.1 in rational terms. We let M be the elliptic curve Fϕ , where ϕ is generic and even. Let then ∇ be an  arbitrary hermitian connection on L and  κ its curvature. We first display a cocycle γ = γn,m ∈ C n,m which reproduces the cyclic cocycle τ3 . Lemma 13.4. There exists a two form α on M = Fϕ and a multiple λ dv of the translation invariant two form dv such that : (i) κn = n λ dv + (σ ∗n α − α), ∀n ∈ Z. (ii) d2 (γ j ) = 0, d1 (γ3 ) = 0, d1 (γ 1 ) + 21 d3 (γ3 ) = 0, B(γ1 ) = 0, where γ 1 ∈ C 1,0 and γ3 ∈ C 1,−2 are given by γ1 (k0 , k1 ) := 21 (k1 − k0 )(σ ∗k0 α + σ ∗k1 α), γ3 (k0 , k1 ) := k1 − k0 , ∀k j ∈ Z. (iii) The class of the cyclic cocycle (γ1 ) + (γ3 ) is equal to τ3 . We use the generic hypothesis in the measure theoretic sense to solve the “small denominator” problem in (i). In (ii) we identify differential forms ω ∈  d of degree d with the dual currents of dimension 2 − d. It is a general principle explained in [11] that a cyclic cocycle τ generates a calculus whose differential graded algebra is obtained as the quotient of the universal one by the radical of τ . We shall now explicitly describe the reduced calculus obtained from the cocycle of Lemma 13.4 (iii). We use as above the hermitian line bundle L to form the twisted cross-product B := (M) ×α, L Z (13.26) of the algebra (M) of differential forms on M by the diffeomorphism σ . Instead of having to adjoin the infinite number of odd elements δn we just adjoin two χ and X as follows. We let δ be the derivation of B such that δ(ξ W n ) := i n ξ W n , ∀ξ ∈ C ∞ (M, Ln ) ⊗C ∞ (M) (M).

(13.27)

We adjoin χ to B by tensoring B with the exterior algebra ∧{χ } generated by an element χ of degree 1, and extend the connection ∇ (13.9) to the unique graded derivation d  of  = B ⊗ ∧{χ } such that, d  ω = ∇ω + χ δ(ω), ∀ω ∈ B, d  χ = − λ dv

(13.28)

with λ dv as in Lemma 13.4. By construction, every element of  is of the form y = b0 + b1 χ , b j ∈ B. One does not yet have a graded differential algebra since d in Lemma 13.4 one has

(13.29) 2



= 0. However, with α as

d 2 (x) = [ x, α], ∀x ∈  = B ⊗ ∧{χ }

(13.30)

and one can apply Lemma 9, p. 229 of [14] to get a differential graded algebra by adjoining the degree 1 element X := “d1 fulfilling the rules X 2 = −α, x X y = 0, ∀x, y ∈  ,

(13.31)

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119

and defining the differential d by d x = d  x + [ X, x], ∀x ∈  , d X = 0,

(13.32)

where [ X, x] is the graded commutator. It follows from Lemma 9, p.229 of [14] that we obtain a differential graded algebra ∗ , generated by B, ξ and X . In fact using (13.31) every element of ∗ is of the form x = x1,1 + x1,2 X + X x2,1 + X x2,2 X, xi, j ∈  , , and we define the functional on ∗ by extending the ordinary integral,   ω := ω, ∀ω ∈ (M)

(13.33)

(13.34)

M

first to B := (M) ×α, L Z by

 ξ W n := 0, ∀n = 0

then to  by

(13.35)



 (b0 + b1 χ ) :=

b1 , ∀b j ∈ B

and finally to ∗ as in Lemma 9, p.229 of [14],    (x1,1 + x1,2 X + X x2,1 + X x2,2 X ) := x1,1 + (−1)deg( x2,2 ) x2,2 α.

(13.36)

(13.37)

∗ is a differential Theorem 13.5. Let M = Fϕ , ∇, α be as in Lemma 13.4. The algebra , ∞ graded algebra containing C (M) ×α, L Z. The functional is a closed graded trace on ∗ . The character of the corresponding cycle on C ∞ (M) ×α, L Z,  τ (a0 , · · · , a3 ) := a0 da1 · · · da3 , ∀a j ∈ C ∞ (M) ×α, L Z is cohomologous to the cyclic cocycle τ3 . It is worth noticing that the above calculus fits with [10,21 and 20]. Now in our case the line bundle L is holomorphic and we can apply Theorem 13.5 to its canonical hermitian connection ∇. We take the notations of Sect. 11, with C = Fϕ × Fϕ , and Q given by (11.31). This gives a particular “rational” form of the calculus which explains the rationality of the answer in Theorem 12.1. We first extend as follows the construction of C Q . We let (C, Q) be the generalised cross-product of the algebra (C) of meromorphic differential forms (in d Z and d Z  ) on C by the transformation σ˜ . The generators W L and W L  fulfill the cross-product rules, W L ω = σ˜ ∗ (ω) W L ,

W L  ω = (σ˜ −1 )∗ (ω) W L 

(13.38)

while (11.4) is unchanged. The connection ∇ is the restriction to the subspace {Z  = Z¯ } of the unique graded derivation ∇ on (C, Q) which induces the usual differential on (C) and satisfies, ∇W L = ( d Z log L(Z ) − d Z log Q(Z , Z  )) W L , ∇W L  = W L  ( d Z  log L  (Z  ) − d Z  log Q(Z , Z  )),

(13.39)

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where d Z and d Z  are the (partial) differentials relative to the variables Z and Z  . Note that one needs to check that the involved differential forms such as d Z log L(Z ) − d Z log Q(Z , Z  ) are not only invariant under the scaling transformations Z → λZ but are also basic, i.e. have zero restriction to the fibers of the map C4 → P3 (C), in both variables Z and Z  . By definition the derivation δκ = ∇ 2 of (C, Q) vanishes on (C) and fulfills δκ (W L ) = κ W L , δκ (W L  ) = − W L  κ, (13.40) where

κ = d Z d Z  log Q(Z , Z  )

(13.41) ¯ is a basic form which when restricted to the subspace = Z } is the curvature. We let as above δ be the derivation of (C, Q) which vanishes on (C) and is such that δW L = i W L and δW L  = −i W L  . We proceed exactly as above and get the graded algebras  = (C, Q) ⊗ ∧{χ } obtained by adjoining,χ and ∗ by adjoining X . We define d  , d as in (13.28) and (13.32) and the integral by integration (13.34) on the subspace {Z  = Z¯ } followed as above by steps (13.35), (13.36), (13.37). {Z 

Corollary 13.6. Let ρ: Calg (Sϕ3 ) → C Q be the morphism of Lemma 11.2. The equality    τalg (a0 , · · · , a3 ) := ρ(a0 ) d ρ(a1 ) · · · d ρ(a3 ) defines a 3-dimensional Hochschild cocycle τalg on Calg (Sϕ3 ). Let h ∈ Center (C ∞ (Fϕ ×σ, L Z)) ∼ C ∞ (Fϕ (0)). Then ch 3 (U ), τh = h ch 3 (U ), τalg . 2

2

The computation of d  only involves rational fractions in the variables Z , Z  (13.39), and the formula (12.5) for ch 3 (U ) is polynomial in the W L , W L  . 2 We are now ready for a better understanding and formulation of the result of Theorem 12.1. Indeed what the above shows is that the denominator that appears in the rational fraction R(Z ) of Theorem 12.1 should have to do with the central quadratic form Q(Z , Z  ) evaluated on the pairs (Z , Z¯ ). In fact the two dimensional space of central quadratic forms provides a natural space of functions of the form R(Z ) =

P(Z , Z¯ ) , Q(Z , Z¯ )

(13.42)

and this space is one dimensional when one mods out the constant functions. The following lemma shows that these functions are in fact invariant under the correspondence σ : Lemma 13.7. Let Q be central and not identically zero on the component C and P be central, then the function R(Z , Z  ) =

P(Z , Z  ) Q(Z , Z  )

is invariant under σ˜ . It is thus natural now to compare the differential form d R with the form that appears in Theorem 12.1.

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Theorem 13.8. Let ϕ be generic and even and let Q be the central quadratic form on R4ϕ defining the three sphere Sϕ3 . Let ρ Q be the associated ∗-homomorphism C ∞ (Sϕ3 ) → C ∞ (E ×σ, L Z). Then for any central quadratic form P not proportional to Q there exists a scalar µ such that  ch 3 (U ), τh = µ h(Z ) d R(Z ), ∀h ∈ Center C ∞ (E ×σ, L Z), 2

where R(Z ) =

P(Z , Z¯ ) . Q(Z , Z¯ )

Proof. Let us show that one can interpret (12.26) in the above terms. Thus with a = cot(ϕ1 − ϕ2 ) cot(ϕ1 − ϕ3 ) we need to show that R(Z ) = t1

Z 02 Z 02 + a Z 12

is in fact the restriction to the subset Fϕ (0) ⊂ E of elements Z with c(Z ) = I0 (Z ) of a ratio of the form P(Z , j (Z )) . Q(Z , j (Z )) One has j (Z ) = I3 (c(Z ) and thus j (Z ) = I3 ◦ I0 (Z ), ∀Z ∈ Fϕ (0).

(13.43)

We let Q  be the central quadratic form of Proposition 11.12, namely s Q  = Q 1 + Q 3 + s2 Q 2 , and we let P  be given by

s P  = Q1 + Q3.

(13.44)

A simple computation using (13.43) then shows that b

Z 02 P  (Z , j (Z )) 1 = − , ∀Z ∈ Fϕ (0), 2 2 Q  (Z , j (Z )) s Z0 + a Z1 1

(13.45)

where b=

sinϕ2 sinϕ3 . cos(ϕ2 − ϕ3 )

This gives the required result for the central quadratic form P  and since the space of central quadratic forms is two dimensional its quotient by multiples of Q  is one dimensional so that the result holds for all non-zero elements of this quotient.  

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With the above “invariant” formulation of the formulas of Theorem 12.1 we can now perform the change of variables required in the last part of its proof, i.e. explain how to pass from (12.26) to (12.27). We let P and Q be the central quadratic forms obtained from P  and Q  by the isomorphism β of Proposition 11.12. The form Q is given by (11.31), i.e. by    tk sk Q k = sin(ϕ − ϕm ) Z 02 (13.46) Q = ( cos2 ϕ )  − cosϕ cosϕm sin(ϕ − ϕm ) Z k2 . The form P is given by P =

3 

sinϕk sin(ϕ − ϕm ) cos(ϕk − ϕ − ϕm ) Z k2 .

(13.47)

1

The involution j2 is now given by j2 (Z ) = I2 (c(Z ) and one has j2 (Z ) = I2 ◦ I0 (Z ), ∀Z ∈ Fϕ (0).

(13.48)

A simple computation using (13.48) then shows that b

Z2 P(Z , j2 (Z )) 1 , ∀Z ∈ Fϕ (0), = 2 3 2− Q(Z , j2 (Z )) s Z 3 + c1 Z 2 1

(13.49)

with c1 = tg(ϕ2 ) cot(ϕ1 − ϕ2 ) and we thus obtain the formula required by Theorem 12.1. 14. Appendix 1: The List of Minors We give for convenience the list of the 15 minors of the matrix (5.3), which labels the missing lines, and in factorized form. By setting ⎧  ⎨ A = x02 + 3k=1 cos(2ϕk )xk2 (14.1)  ⎩ B = 3k=1 sin(2ϕk )xk2 one sees that these minors are combinations of the form Mi j = Pi j A + Q i j B: M12 = 2 (sin(ϕ1 − ϕ2 ) x1 x2 + i cos(ϕ3 ) x0 x3 ) (−cos(ϕ1 − ϕ2 ) (cos(ϕ1 − ϕ3 ) sin(ϕ1 ) x12 + cos(ϕ2 − ϕ3 ) sin(ϕ2 ) x22 ) +sin(ϕ3 ) (sin(ϕ1 ) sin(ϕ2 ) x02 − cos(ϕ1 − ϕ3 ) cos(ϕ2 − ϕ3 ) x32 )) = (sin(ϕ1 − ϕ2 ) x1 x2 + i cos(ϕ3 ) x0 x3 ) (2sin(ϕ1 ) sin(ϕ2 ) sin(ϕ3 ) A − (cos(ϕ1 − ϕ2 ) cos(ϕ3 ) + sin(ϕ1 + ϕ2 ) sin(ϕ3 ))B), (14.2) M13 = 2 i (cos(ϕ2 ) x0 x2 + i sin(ϕ1 − ϕ3 ) x1 x3 ) (−cos(ϕ1 − ϕ2 ) (cos(ϕ1 − ϕ3 ) sin(ϕ1 ) x12 + cos(ϕ2 − ϕ3 ) sin(ϕ2 ) x22 ) +sin(ϕ3 ) (sin(ϕ1 ) sin(ϕ2 ) x02 − cos(ϕ1 − ϕ3 ) cos(ϕ2 − ϕ3 ) x32 )) = i (cos(ϕ2 ) x0 x2 + i sin(ϕ1 − ϕ3 ) x1 x3 ) (2sin(ϕ1 ) sin(ϕ2 ) sin(ϕ3 ) A − (cos(ϕ1 − ϕ2 ) cos(ϕ3 ) + sin(ϕ1 + ϕ2 ) sin(ϕ3 ))B), (14.3)

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M14 = 2 (sin(ϕ2 ) x0 x2 + i cos(ϕ1 − ϕ3 ) x1 x3 ) (cos(ϕ2 ) (cos(ϕ3 ) sin(ϕ1 ) x02 + cos(ϕ2 − ϕ3 ) sin(ϕ1 − ϕ2 ) x22 ) +sin(ϕ1 − ϕ3 ) (−sin(ϕ1 ) sin(ϕ1 − ϕ2 ) x12 + cos(ϕ2 − ϕ3 ) cos(ϕ3 ) x32 )) = (sin(ϕ2 ) x0 x2 + i cos(ϕ1 − ϕ3 ) x1 x3 ) (2sin(ϕ1 ) cos(ϕ2 ) cos(ϕ3 ) A − (cos(ϕ1 ) cos(ϕ2 − ϕ3 ) − sin(ϕ1 ) sin(ϕ2 + ϕ3 )) B), (14.4) M15 = −i cos(ϕ1 − ϕ2 − ϕ3 ) (sin(2 (ϕ1 − ϕ2 )) x12 x22

+ sin(2 (ϕ1 − ϕ3 )) x12 x32 − x02 (sin(2 ϕ2 ) x22 + sin(2 ϕ3 ) x32 )) = i cos(ϕ1 − ϕ2 − ϕ3 ) ((sin(2ϕ2 ) x22 + sin(2ϕ3 ) x32 ) A − (cos(2ϕ2 ) x22 + cos(2ϕ3 ) x32 ) B), (14.5) M16 = −2 (−i cos(ϕ1 − ϕ2 ) x1 x2 + sin(ϕ3 ) x0 x3 ) (cos(ϕ2 ) (cos(ϕ3 ) sin(ϕ1 ) x02 + cos(ϕ2 − ϕ3 ) sin(ϕ1 − ϕ2 ) x22 )

+sin(ϕ1 − ϕ3 ) (−sin(ϕ1 ) sin(ϕ1 − ϕ2 ) x12 + cos(ϕ2 − ϕ3 ) cos(ϕ3 ) x32 )) = −(−i cos(ϕ1 − ϕ2 ) x1 x2 + sin(ϕ3 ) x0 x3 ) (2sin(ϕ1 ) cos(ϕ2 ) cos(ϕ3 ) A − (cos(ϕ1 ) cos(ϕ2 − ϕ3 ) − sin(ϕ1 ) sin(ϕ2 + ϕ3 )) B), (14.6) M23 = 2 (i cos(ϕ1 ) x0 x1 + sin(ϕ2 − ϕ3 ) x2 x3 ) (−cos(ϕ1 − ϕ2 ) (cos(ϕ1 − ϕ3 ) sin(ϕ1 ) x12 + cos(ϕ2 − ϕ3 ) sin(ϕ2 ) x22 ) +sin(ϕ3 ) (sin(ϕ1 ) sin(ϕ2 ) x02 − cos(ϕ1 − ϕ3 ) cos(ϕ2 − ϕ3 ) x32 )) = (i cos(ϕ1 ) x0 x1 + sin(ϕ2 − ϕ3 ) x2 x3 ) (2sin(ϕ1 ) sin(ϕ2 ) sin(ϕ3 ) A − (cos(ϕ1 − ϕ2 ) cos(ϕ3 ) + sin(ϕ1 + ϕ2 ) sin(ϕ3 ))B), (14.7) M24 = 2 (sin(ϕ1 ) x0 x1 − i cos(ϕ2 − ϕ3 ) x2 x3 ) (cos(ϕ1 ) (cos(ϕ3 ) sin(ϕ2 ) x02 − cos(ϕ1 − ϕ3 ) sin(ϕ1 − ϕ2 ) x12 ) +sin(ϕ2 − ϕ3 ) (sin(ϕ1 − ϕ2 ) sin(ϕ2 ) x22 + cos(ϕ1 − ϕ3 ) cos(ϕ3 ) x32 )) = (sin(ϕ1 ) x0 x1 − i cos(ϕ2 − ϕ3 ) x2 x3 ) (2cos(ϕ3 ) cos(ϕ1 ) sin(ϕ2 ) A − (cos(ϕ3 − ϕ1 ) cos(ϕ2 ) − sin(ϕ3 + ϕ1 ) sin(ϕ2 )) B), (14.8) M25 = −2 i (cos(ϕ1 − ϕ2 ) x1 x2 − i sin(ϕ3 ) x0 x3 ) (cos(ϕ1 ) (−cos(ϕ3 ) sin(ϕ2 ) x02 + cos(ϕ1 − ϕ3 ) sin(ϕ1 − ϕ2 ) x12 )− sin(ϕ2 − ϕ3 ) (sin(ϕ1 − ϕ2 ) sin(ϕ2 ) x22 + cos(ϕ1 − ϕ3 ) cos(ϕ3 ) x32 )) = i (cos(ϕ1 − ϕ2 ) x1 x2 − i sin(ϕ3 ) x0 x3 ) (2cos(ϕ3 ) cos(ϕ1 ) sin(ϕ2 ) A − (cos(ϕ3 − ϕ1 ) cos(ϕ2 ) − sin(ϕ3 + ϕ1 ) sin(ϕ2 )) B), (14.9) M26 = i cos(ϕ1 − ϕ2 + ϕ3 ) (sin(2 ϕ1 ) x02 x12

+ sin(2 (ϕ1 − ϕ2 )) x12 x22 + (sin(2 ϕ3 ) x02 − sin(2 (ϕ2 − ϕ3 )) x22 ) x32 ) = i cos(ϕ1 − ϕ2 + ϕ3 ) ((sin(2ϕ1 ) x12 + sin(2ϕ3 ) x32 ) A − (cos(2ϕ1 ) x12 + cos(2ϕ3 ) x32 ) B), (14.10)

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M34 = −i cos(ϕ1 + ϕ2 − ϕ3 ) (sin(2 ϕ1 ) x02 x12 + sin(2 ϕ2 ) x02 x22 + (sin(2 (ϕ1 − ϕ3 )) x12 + sin(2 (ϕ2 − ϕ3 )) x22 ) x32 ) = −i cos(ϕ1 + ϕ2 − ϕ3 ) ((sin(2ϕ1 ) x12 + sin(2ϕ2 ) x22 ) A − (cos(2ϕ1 ) x12 + cos(2ϕ2 ) x22 ) B), (14.11) M35 = 2 i (i sin(ϕ2 ) x0 x2 + cos(ϕ1 − ϕ3 ) x1 x3 ) (cos(ϕ1 ) (−cos(ϕ2 ) sin(ϕ3 ) x02 + cos(ϕ1 − ϕ2 ) sin(ϕ1 − ϕ3 ) x12 ) +sin(ϕ2 − ϕ3 ) (cos(ϕ1 − ϕ2 ) cos(ϕ2 ) x22 + sin(ϕ1 − ϕ3 ) sin(ϕ3 ) x32 )) = −i (i sin(ϕ2 ) x0 x2 + cos(ϕ1 − ϕ3 ) x1 x3 ) (2cos(ϕ1 ) cos(ϕ2 ) sin(ϕ3 ) A − (cos(ϕ1 − ϕ2 ) cos(ϕ3 ) − sin(ϕ1 + ϕ2 ) sin(ϕ3 )) B), (14.12) M36 = −2 (sin(ϕ1 ) x0 x1 + i cos(ϕ2 − ϕ3 ) x2 x3 ) (cos(ϕ1 ) (−cos(ϕ2 ) sin(ϕ3 ) x02 + cos(ϕ1 − ϕ2 ) sin(ϕ1 − ϕ3 ) x12 ) +sin(ϕ2 − ϕ3 ) (cos(ϕ1 − ϕ2 ) cos(ϕ2 ) x22 + sin(ϕ1 − ϕ3 ) sin(ϕ3 ) x32 )) = (sin(ϕ1 ) x0 x1 + i cos(ϕ2 − ϕ3 ) x2 x3 ) (2cos(ϕ1 ) cos(ϕ2 ) sin(ϕ3 ) A − (cos(ϕ1 − ϕ2 ) cos(ϕ3 ) − sin(ϕ1 + ϕ2 ) sin(ϕ3 )) B), (14.13) M45 = −2 i (cos(ϕ2 ) x0 x2 − i sin(ϕ1 − ϕ3 ) x1 x3 ) (cos(ϕ1 ) (cos(ϕ3 ) sin(ϕ2 ) x02 − cos(ϕ1 − ϕ3 ) sin(ϕ1 − ϕ2 ) x12 ) +sin(ϕ2 − ϕ3 ) (sin(ϕ1 − ϕ2 ) sin(ϕ2 ) x22 + cos(ϕ1 − ϕ3 ) cos(ϕ3 ) x32 )) = −i (cos(ϕ2 ) x0 x2 − i sin(ϕ1 − ϕ3 ) x1 x3 ) (2cos(ϕ3 ) cos(ϕ1 ) sin(ϕ2 ) A − (cos(ϕ3 − ϕ1 ) cos(ϕ2 ) − sin(ϕ3 + ϕ1 ) sin(ϕ2 )) B), (14.14) M46 = 2 (−i cos(ϕ1 ) x0 x1 + sin(ϕ2 − ϕ3 ) x2 x3 ) (cos(ϕ2 ) (cos(ϕ3 ) sin(ϕ1 ) x02 + cos(ϕ2 − ϕ3 ) sin(ϕ1 − ϕ2 ) x22 ) +sin(ϕ1 − ϕ3 ) (−sin(ϕ1 ) sin(ϕ1 − ϕ2 ) x12 + cos(ϕ2 − ϕ3 ) cos(ϕ3 ) x32 )) = (−i cos(ϕ1 ) x0 x1 + sin(ϕ2 − ϕ3 ) x2 x3 ) (2sin(ϕ1 ) cos(ϕ2 ) cos(ϕ3 ) A − (cos(ϕ1 ) cos(ϕ2 − ϕ3 ) − sin(ϕ1 ) sin(ϕ2 + ϕ3 )) B), (14.15) M56 = 2 (sin(ϕ1 − ϕ2 ) x1 x2 − i cos(ϕ3 ) x0 x3 ) (cos(ϕ1 ) (cos(ϕ2 ) sin(ϕ3 ) x02 − cos(ϕ1 − ϕ2 ) sin(ϕ1 − ϕ3 ) x12 ) − sin(ϕ2 − ϕ3 ) (cos(ϕ1 − ϕ2 ) cos(ϕ2 ) x22 + sin(ϕ1 − ϕ3 ) sin(ϕ3 ) x32 )) = (sin(ϕ1 − ϕ2 ) x1 x2 − i cos(ϕ3 ) x0 x3 ) (2cos(ϕ1 ) cos(ϕ2 ) sin(ϕ3 ) A − (cos(ϕ1 − ϕ2 ) cos(ϕ3 ) − sin(ϕ1 + ϕ2 ) sin(ϕ3 )) B). (14.16) 15. Appendix 2: The Sixteen Theta Relations The sixteen theta relations are the following, with (w, x, y, z) = M (a, b, c, d),

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where ⎡

1 1 1 ⎢1 1 M := ⎣ 2 1 −1 1 −1

1 −1 1 −1

⎤ 1 −1 ⎥ . −1 ⎦ 1

ϑ2 (a) ϑ2 (b) ϑ2 (c) ϑ2 (d) + ϑ3 (a) ϑ3 (b) ϑ3 (c) ϑ3 (d) = ϑ2 (x) ϑ2 (y) ϑ2 (z) ϑ2 (w) + ϑ3 (x) ϑ3 (y) ϑ3 (z) ϑ3 (w) ϑ3 (a) ϑ3 (b) ϑ3 (c) ϑ3 (d) − ϑ2 (a) ϑ2 (b) ϑ2 (c) ϑ2 (d) = ϑ1 (x) ϑ1 (y) ϑ1 (z) ϑ1 (w) + ϑ4 (x) ϑ4 (y) ϑ4 (z) ϑ4 (w) ϑ1 (a) ϑ1 (b) ϑ1 (c) ϑ1 (d) + ϑ4 (a) ϑ4 (b) ϑ4 (c) ϑ4 (d) = ϑ3 (w) ϑ3 (x) ϑ3 (y) ϑ3 (z) − ϑ2 (w) ϑ2 (x) ϑ2 (y) ϑ2 (z) ϑ4 (a) ϑ4 (b) ϑ4 (c) ϑ4 (d) − ϑ1 (a) ϑ1 (b) ϑ1 (c) ϑ1 (d) = ϑ4 (w) ϑ4 (x) ϑ4 (y) ϑ4 (z) − ϑ1 (w) ϑ1 (x) ϑ1 (y) ϑ1 (z) ϑ1 (a) ϑ1 (b) ϑ2 (c) ϑ2 (d) + ϑ3 (c) ϑ3 (d) ϑ4 (a) ϑ4 (b) = ϑ1 (x) ϑ1 (w) ϑ2 (y) ϑ2 (z) + ϑ3 (y) ϑ3 (z) ϑ4 (x) ϑ4 (w) ϑ4 (a) ϑ4 (b) ϑ3 (c) ϑ3 (d) − ϑ1 (a) ϑ1 (b) ϑ2 (c) ϑ2 (d) = ϑ1 (y) ϑ1 (z) ϑ2 (x) ϑ2 (w) + ϑ3 (x) ϑ3 (w) ϑ4 (y) ϑ4 (z) ϑ1 (a) ϑ1 (b) ϑ3 (c) ϑ3 (d) + ϑ2 (c) ϑ2 (d) ϑ4 (a) ϑ4 (b) = ϑ1 (x) ϑ1 (w) ϑ3 (y) ϑ3 (z) + ϑ2 (y) ϑ2 (z) ϑ4 (x) ϑ4 (w) ϑ4 (a) ϑ4 (b) ϑ2 (c) ϑ2 (d) − ϑ1 (a) ϑ1 (b) ϑ3 (c) ϑ3 (d) = ϑ1 (y) ϑ1 (z) ϑ3 (x) ϑ3 (w) + ϑ2 (x) ϑ2 (w) ϑ4 (y) ϑ4 (z) ϑ2 (c) ϑ2 (d) ϑ3 (a) ϑ3 (b) + ϑ2 (a) ϑ2 (b) ϑ3 (c) ϑ3 (d) = ϑ2 (x) ϑ2 (w) ϑ3 (y) ϑ3 (z) + ϑ2 (y) ϑ2 (z) ϑ3 (x) ϑ3 (w) ϑ3 (a) ϑ3 (b) ϑ2 (c) ϑ2 (d) − ϑ2 (a) ϑ2 (b) ϑ3 (c) ϑ3 (d) = ϑ1 (x) ϑ1 (w) ϑ4 (y) ϑ4 (z) + ϑ1 (y) ϑ1 (z) ϑ4 (x) ϑ4 (w) ϑ1 (c) ϑ1 (d) ϑ4 (a) ϑ4 (b) + ϑ1 (a) ϑ1 (b) ϑ4 (c) ϑ4 (d) = ϑ3 (w) ϑ3 (x) ϑ2 (y) ϑ2 (z) − ϑ2 (w) ϑ2 (x) ϑ3 (y) ϑ3 (z)

(15.1)

(15.2)

(15.3)

(15.4)

(15.5)

(15.6)

(15.7)

(15.8)

(15.9)

(15.10)

(15.11)

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ϑ4 (a) ϑ4 (b) ϑ1 (c) ϑ1 (d) − ϑ1 (a) ϑ1 (b) ϑ4 (c) ϑ4 (d) = ϑ4 (w) ϑ4 (x) ϑ1 (y) ϑ1 (z) − ϑ1 (w) ϑ1 (x) ϑ4 (y) ϑ4 (z) ϑ2 (c) ϑ2 (d) ϑ3 (a) ϑ3 (b) + ϑ1 (c) ϑ1 (d) ϑ4 (a) ϑ4 (b) = ϑ2 (y) ϑ2 (z) ϑ3 (x) ϑ3 (w) + ϑ1 (y) ϑ1 (z) ϑ4 (x) ϑ4 (w) ϑ3 (a) ϑ3 (b) ϑ2 (c) ϑ2 (d) − ϑ4 (a) ϑ4 (b) ϑ1 (c) ϑ1 (d) = ϑ2 (x) ϑ2 (w) ϑ3 (y) ϑ3 (z) + ϑ1 (x) ϑ1 (w) ϑ4 (y) ϑ4 (z) ϑ1 (d) ϑ2 (b) ϑ3 (a) ϑ4 (c) + ϑ1 (c) ϑ2 (a) ϑ3 (b) ϑ4 (d) = ϑ1 (w) ϑ4 (x) ϑ2 (y) ϑ3 (z) − ϑ4 (w) ϑ1 (x) ϑ3 (y) ϑ2 (z) ϑ3 (a) ϑ2 (b) ϑ4 (c) ϑ1 (d) − ϑ2 (a) ϑ3 (b) ϑ1 (c) ϑ4 (d) = ϑ3 (w) ϑ2 (x) ϑ4 (y) ϑ1 (z) − ϑ2 (w) ϑ3 (x) ϑ1 (y) ϑ4 (z)

(15.12)

(15.13)

(15.14)

(15.15)

(15.16)

References 1. Abadie, B., Exel, R.: Hilbert C ∗ -bimodules over commutative C ∗ -algebras and an isomorphism condition for quantum Heisenberg manifolds. Rev. Math. Phys. 9(4), 411–423 (1997) 2. Abadie, B., Eilers, S., Exel, R.: Morita equivalence for crossed products by Hilbert C ∗ -bimodules. Trans. Amer. Math. Soc. 350(8), 3043–3054 (1998) 3. Artin, M., Schelter, W.F.: Graded algebras of global dimension 3. Adv. Math. 66, 171–216 (1987) 4. Artin, M., Tate, J., Van den Bergh, M.: Some algebras associated to automorphisms of elliptic curves. The Grothendieck Festschrift. Vol. I. Prog. Math., 86, 33–85 (1990) 5. Artin, M., Tate, J., Vanden Bergh, M.: Modules over regular algebras of dimension 3. Invent. Math. 106, 335–388 (1991) 6. Berger, R.: Koszulity for nonquadratic algebras. J. Alg. 239, 705–734 (2001) 7. Berger, R., Dubois-Violette, M., Wambst, M.: Homogeneous algebras. J. Alg. 261, 172–185 (2003) 8. Bourbaki, N.: Groupes et algèbres de Lie. Chapitres 4,5 et 6. Parts: Hermann, 1968 9. Bourbaki, N.: Groupes et algèbres de Lie. Chapitre 9. Parts: Masson, 1982 10. Connes, A.: C ∗ algèbres et géométrie différentielle. C.R. Acad. Sci. Paris, Série A 290, 599–604 (1980) 11. Connes, A.: Noncommutative differential geometry. Part I: The Chern character in K -homology; Part II: de Rham homology and noncommutative algebra. Preprints IHES M/82/53; M83/19, 1982/83 12. Connes, A.: Cyclic cohomology and the transverse fundamental class of a foliation. In: Geometric methods in operator algebras, Pitman Res. Notes in Math. 123 (1986), Harlow: Longman 1986, pp. 52–144 13. Connes, A.: Non-commutative differential geometry. Publ. IHES 62, 257–360 (1986) 14. Connes, A.: Non-commutative geometry. London New York: Academic Press, 1994 15. Connes, A., Dubois-Violette, M.: Noncommutative finite-dimensional manifolds. I.Spherical manifolds and related examples. Commun. Math. Phys. 230, 539–579 (2002) 16. Connes, A., Dubois-Violette, M.: Yang-Mills algebra. Lett. Math. Phys. 61, 149–158 (2002) 17. Connes, A., Dubois-Violette, M.: Moduli space and structure of noncommutative 3-spheres. Lett. Math. Phys. 66, 91–121 (2003) 18. Connes, A., Dubois-Violette, M.: Yang-Mills and some related algebras. In: Rigorous Quantum Field Theory. J. Alg. 261, 172–185 (2003) 19. Connes, A., Landi, G.: Noncommutative manifolds, the instanton algebra and isospectral deformations. Commun. Math. Phys. 221, 141–159 (2001) 20. Dubois-Violette, M.: Lectures on graded differential algebras and noncommutative geometry. In: Maeda Y., Moriyoshi H. editors, Noncommutative Differential Geometry and Its Applications to Physics, (Shonan, Japan, 1999), Dordrecht: Kluwer Academic Publishers, 2001, pp. 245–306 21. Dubois-Violette, M., Michor, P.W.: Dérivations et calcul différentiel non commutatif, II. C.R. Acad. Sci. Paris 319, 927–931 (1994) 22. Helgason, S.: Differential geometry, Lie groups and symmetric spaces. London: Academic Press, 1978 23. Majid, S.: Foundations of quantum group theory. Cambridge: Cambridge University Press, 1995

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24. Odesskii, A.V., Feigin, B.L.: Sklyanin elliptic algebras. Func. Anal. Appl. 23, 207–214 (1989) 25. Pimsner, M.V.: A class of C*-algebra generalizing both Cuntz-Krieger algebras and crossed products by Z . In: Free probablility Theory, Fields Inst. Comm., Providence, RI: Amer. Math. Soc. 12, 1997, pp. 189–212 26. Rieffel, M.: Deformation quantization of Heisenberg Manifolds. Commun. Math. Phys. 122, 531–562 (1989) 27. Sklyanin, E.K.: Some algebraic structures connected with the Yang-Baxter equation. Func. Anal. Appl. 16, 263–270 (1982) 28. Sklyanin, E.K.: Some algebraic structures connected with the Yang-Baxter equation. Representation of Quantum Algebras. Func. Anal. Appl. 17, 273–284 (1983) 29. Smith, S.P., Stafford, J.T.: Regularity of the four dimensional Sklyanin algebra. Compos. Math. 83, 259–289 (1992) 30. Tate, J., Vanden Bergh, M.: Homological properties of Sklyanin algebras. Invent. Math. 124, 619–647 (1996) Communicated by N.A. Nekrasov

Commun. Math. Phys. 281, 129–177 (2008) Digital Object Identifier (DOI) 10.1007/s00220-008-0481-x

Communications in

Mathematical Physics

A New Approach to the Modeling of Local Defects in Crystals: The Reduced Hartree-Fock Case Éric Cancès1 , Amélie Deleurence1 , Mathieu Lewin2 1 CERMICS, Ecole Nationale des Ponts et Chaussées (Paris Tech) & INRIA (Micmac Project),

6-8 Av. Pascal, 77455 Champs-sur-Marne, France. E-mail: [email protected]; [email protected] 2 CNRS & Laboratoire de Mathématiques UMR 8088, Université de Cergy-Pontoise, 2 Avenue Adolphe Chauvin, 95302 Cergy-Pontoise Cedex, France. E-mail: [email protected] Received: 20 February 2007 / Accepted: 19 December 2007 Published online: 26 April 2008 – © Springer-Verlag 2008

Abstract: This article is concerned with the derivation and the mathematical study of a new mean-field model for the description of interacting electrons in crystals with local defects. We work with a reduced Hartree-Fock model, obtained from the usual Hartree-Fock model by neglecting the exchange term. First, we recall the definition of the self-consistent Fermi sea of the perfect crystal, which is obtained as a minimizer of some periodic problem, as was shown by Catto, Le Bris and Lions. We also prove some of its properties which were not mentioned before. Then, we define and study in detail a nonlinear model for the electrons of the crystal in the presence of a defect. We use formal analogies between the Fermi sea of a perturbed crystal and the Dirac sea in Quantum Electrodynamics in the presence of an external electrostatic field. The latter was recently studied by Hainzl, Lewin, Séré and Solovej, based on ideas from Chaix and Iracane. This enables us to define the ground state of the self-consistent Fermi sea in the presence of a defect. We end the paper by proving that our model is in fact the thermodynamic limit of the so-called supercell model, widely used in numerical simulations. Describing the electronic state of crystals with local defects is a major issue in solid-state physics, materials science and nano-electronics [17,25,33]. In this article, we develop a theory based on formal analogies between the Fermi sea of a perturbed crystal and the polarized Dirac sea in Quantum Electrodynamics in the presence of an external electrostatic field. Recently, the latter model was extensively studied by Hainzl, Lewin, Séré and Solovej in the Hartree-Fock approximation [10– 13], based on ideas from Chaix and Iracane [6] (see also [1,7]). This was summarized in the review [14]. Using and adapting these methods, we are able to propose a new mathematical approach for the self-consistent description of a crystal in the presence of local defects. We focus in this article on the reduced Hartree-Fock (rHF) model in which the so-called exchange term is neglected. To further simplify the mathematical formulas, we do not explicitly take the spin variable into account and we assume that the host

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crystal is cubic with a single atom of charge Z per unit cell. The arguments below can be easily extended to the general case. In the whole paper, the main object of interest will be the so-called density matrix of the electrons. This is a self-adjoint operator 0 ≤ γ ≤ 1 acting on the one-body space L 2 (R3 ). When γ has a finite rank, it models a finite number of electrons. In the periodic 0 has an infinite rank (it describes infinitely many case, the ground state density matrix γper electrons) and commutes with the translations of the lattice. We will see in the sequel that the ground state density matrix of a crystal with a local defect can be written as 0 + Q, where Q is a compact perturbation of the periodic density matrix γ 0 of γ = γper per the reference perfect crystal. In each of the above three cases (finite number of electrons, perfect crystal, defective crystal), the ground state density matrix can be obtained by minimizing some nonlinear energy functional depending on a set of admissible density matrices. In the case of a crystal with a local defect, the perturbation Q is a minimizer of some nonlinear minimization problem set in the whole space R3 , with a possible lack of compactness at infinity. The main unusual feature compared to standard variational problems is that Q is a self-adjoint operator of infinite rank. This was already the case in [10–13]. The paper is organized as follows. In Sect. 1, we recall the definition of the reduced Hartree-Fock model for a finite number of electrons, which serves as a basis for the theories of infinitely many electrons in a (possibly perturbed) periodic nuclear distribution. Sect. 2 is devoted to the definition of the model for the infinite periodic crystal, following mainly [4,5] (but we provide some additional material compared to what was done in [4,5]). In Sect. 3, we define a model for the crystal with local defects which takes the perfect crystal as reference. In Sect. 4, we prove that this model is the thermodynamic limit of the supercell model. For the convenience of the reader, we have gathered all the proofs in Sect. 5. Often, the proofs follow the same lines as those in [10–13] and we shall not detail identical arguments. But there are many difficulties associated with the particular model under study which do not appear in previous works and which are addressed in detail here.

1. The Reduced Hartree-Fock Model for N Electrons We start by recalling the definition of the reduced Hartree-Fock model [31] for a finite number of electrons. Note that the reduced Hartree-Fock model should not be confused with the restricted Hartree-Fock model commonly used in numerical simulations (see e.g. [8]). We consider a system containing N nonrelativistic quantum electrons and a set of nuclei having a density of charge ρnuc . If for instance there are K nuclei of charges z 1 , ..., z K ∈ N \ {0} located at R1 , ..., R K ∈ R3 , then ρnuc (x) :=

K


z k m k (x − Rk ),

k=1

where m 1 , ..., m K are positive measures on R3 of total mass one. Point-like nuclei would correspond to m k = δ (the Dirac measure) but for convenience we shall deal with smeared nuclei in the sequel,  i.e. we assume that for all k = 1...K , m k is a smooth nonnegative function such that R3 m k = 1. The technical difficulties arising with point-like nuclei will be dealt with elsewhere.

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The energy of the whole system in the reduced Hartree-Fock model reads [5,31]    1  1 EρrHF γ + D ργ − ρnuc , ργ − ρnuc . (1.1) (γ ) = Tr − nuc 2 2 We have chosen a system of units such that
= m = e = 4π10 = 1, where m and e are respectively the mass and the charge of an electron,
is the reduced Planck constant and 0 is the dielectric permittivity of the vacuum. The first term in the right-hand side of (1.1) is the kinetic energy of the electrons and D(·, ·) is the classical Coulomb interaction, which reads for f and g in L 6/5 (R3 ) as     f (x) g(y) f (k) g (k) d x d y = 4π D( f, g) = dk, (1.2) 2 3 3 3 |x − y| |k| R R R where  f denotes the Fourier transform of f . In this mean-field model, the state of the N electrons is described by the one-body density matrix γ , which is an element of the following class:

√  √ P N = γ ∈ S(L 2 (R3 )) | 0 ≤ γ ≤ 1, Tr(γ ) = N , Tr −γ − < ∞ . Here and below, S(H) denotes the space of bounded √ self-adjoint √ operators acting on the Hilbert space H. Also we define Tr((−)γ ) := Tr( −γ −) which makes sense when γ ∈ P N . The set P N is the closed convex hull of the set of orthogonal projectors of rank N acting on L 2 (R3 ) and having a finite kinetic energy. Each such projector

N |ϕi ϕi | is the density matrix of a Hartree-Fock state γ = i=1  = ϕ1 ∧ · · · ∧ ϕ N

(1.3)

in the usual N -body space of fermionic wavefunctions with finite kinetic energy N 1 3 i=1 H (R ). The function ργ appearing in (1.1) is the density associated with the operator γ , defined by ργ (x) = γ (x, x), where γ (x, y) is the kernel of the trace class operator γ . √ Notice that for all γ ∈ P N , one has ργ ≥ 0 and ργ ∈ H 1 (R3 ), hence the last term of 1 3 3 (1.1) is well-defined, since ργ ∈ L (R ) ∩ L (R3 ) ⊂ L 6/5 (R3 ).

M It can be proved (see the appendix of [31]) that if N ≤ k=1 z k (neutral or positively charged systems), the variational problem   N IrHF (ρnuc , N ) = inf EρrHF (1.4) (γ ), γ ∈ P nuc has a minimizer γ and that the corresponding minimizing density ργ is unique. The Hartree-Fock model [21] is the variational approximation of the time-independent Schrödinger equation obtained by restricting the set of fermionic wavefunctions under consideration to the subset of functions of the form (1.3). The HF functional reads  |γ (x, y)|2 1 HF rHF Eρnuc (γ ) = Eρnuc (γ ) − d x d y, (1.5) 2 R6 |x − y| the last term being called the exchange energy. As the Hartree-Fock energy functional is nonconvex, there is little hope to obtain rigorous thermodynamic limits in this setting, at least with current state-of-the-art techniques. For this reason, the exchange term is often neglected in mathematical studies.

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2. The Reduced Hartree-Fock Model for a Perfect Crystal In this article, we clamp the nuclei on a periodic lattice, optimizing only over the state of the electrons. More precisely we are interested in the change of the electronic state of the crystal when a local defect is introduced. To this end, we shall rely heavily on the rHF model for the infinite perfect crystal (with no defect) which was studied by Catto, Le Bris and Lions in [4,5]. The latter can be obtained as the thermodynamical limit of the rHF model for finite systems which was introduced in the previous section. This will be explained in Sect. 4 below. Let = [−1/2, 1/2)3 be the unit cell. We denote by ∗ = [−π, π )3 the first Brillouin zone of the lattice, and by τk the translation operator on L 2loc (R3 ) defined by τk u(x) = u(x − k). We then introduce Pper = γ ∈ S(L 2 (R3 )) | 0 ≤ γ ≤ 1, ∀k ∈ Z3 , τk γ = γ τk ,  Tr L 2 ( ) ((1 − ξ )1/2 γξ (1 − ξ )1/2 ) dξ < ∞ , ∗

ξ

where (γξ )ξ ∈ ∗ is the Bloch waves decomposition of γ , see [5,27]:  1 γ = γξ dξ, γξ ∈ S(L 2ξ ( )), (2π )3 ∗   L 2ξ ( ) = u ∈ L 2loc (R3 ) | τk u = e−ik·ξ u, ∀k ∈ Z3 ⊕ which corresponds to the decomposition in fibers L 2 (R3 ) = ∗ L 2ξ ( )dξ . For any γ ∈ Pper , we denote by γξ (x, y) the integral kernel of γξ . The density of γ is then the nonnegative Z3 -periodic function of L 1loc (R3 ) ∩ L 3loc (R3 ) defined as  1 ργ (x) := γξ (x, x) dξ. (2π )3 ∗ Notice that for any γ ∈ Pper ,  ργ (x)d x =

1 (2π )3

 ∗

Tr L 2 ( ) (γξ ) dξ, ξ

i.e. this gives the number of electrons per unit cell. Later we shall add the constraint that the system is neutral and restrict to states γ ∈ Pper satisfying  ργ (x)d x = Z ,

where Z is the total charge of the nuclei in each unit cell. We also introduce the Z3 -periodic Green kernel of the Poisson interaction [22], denoted by G 1 and uniquely defined by ⎧ ⎛ ⎞ ⎪
⎪ ⎪ ⎨ −G 1 = 4π ⎝ δk − 1⎠ k∈Z3 ⎪ ⎪ ⎪ ⎩ G 1 Z3 -periodic, min G 1 = 0.

R3

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The Fourier expansion of G 1 is


G 1 (x) = c +

k∈2π Z3 \{0}

4π ik·x e |k|2

 with c = G 1 > 0. The electrostatic potential associated with a Z3 -periodic density ρ ∈ L 1loc (R3 ) ∩ L 3loc (R3 ) is the Z3 -periodic function defined as  G 1 (x − y) ρ(y) dy. (ρ G 1 )(x) :=

We also set for any Z3 -periodic functions f and g,   DG 1 ( f, g) := G 1 (x − y) f (x) g(y)d x d y.



Throughout this article, we will denote by χ I the characteristic function of the set I ⊂ R and by χ I (A) the spectral projector on I of the self-adjoint operator A. The periodic density of the nuclei is given by
µper (x) = Z m(x − R). (2.1) R∈Z3

We assumefor simplicity that m is a nonnegative function of Cc∞ (R3 ) with support in , and that R3 m(x)d x = 1. Hence µper (x)d x = Z , the total charge of the nuclei in each unit cell. The periodic rHF energy is then defined for γ ∈ Pper as 0 Eper (γ ) =

1 (2π )3



    1 1 Tr L 2 ( ) − γξ dξ + DG 1 ργ − µper , ργ − µper . ξ 2 2 ∗

(2.2)

Introducing Z Pper

 := γ ∈ Pper | ργ = Z ,

(2.3)

the periodic rHF ground state energy (per unit cell) is given by   0 0 Z . Iper = inf Eper (γ ), γ ∈ Pper

(2.4)

0 ∈ PZ It was proved by Catto, Le Bris and Lions in [5] that there exists a minimizer γper per to the minimization problem (2.4), and that all the minimizers of (2.4) share the same density ργper 0 . We give in Appendix 5.10 the proof of the following

Theorem 1. (Definition of the periodic rHF minimizer) Let Z ∈ N \ {0}. The minimiza0 . Denoting by tion problem (2.4) admits a unique minimizer γper 0 Hper := −

 + (ργper 0 − µper ) G 1 , 2

(2.5)

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0 is the solution to the the corresponding periodic mean-field Hamiltonian, γper self-consistent equation 0 0 = χ(−∞, F ] (Hper ), γper

(2.6)

where  F is a Lagrange multiplier called Fermi level, which can be interpreted as a chemical potential. 0 is the unique minimizer on Additionally, for any  F ∈ R such that (2.6) holds, γper Pper of the energy functional  0 ργ . γ → Eper (γ ) −  F

0 is unique, Theorem 1 contains three main results that were not present in [5]: first γper second it is a projector, and third it satisfies Eq. (2.6). These three properties are crucial for a proper construction of the model for the crystal with a defect. 2 3 It can easily be seen that (ργper 0 − µper ) G 1 belongs to L loc (R ). By a result of 0 is purely absolutely continuous. This Thomas [34] this implies that the spectrum of Hper 0 . Let (λ (ξ )) is an essential property for the proof of the uniqueness of γper n n≥1 denote 0 the nondecreasing sequence of the eigenvalues of (Hper )ξ . Then 0 σ (Hper )

=



∗

λn ( ),

0 Hper

n≥1

1 = (2π )3

 ∗

0 (Hper )ξ dξ.

0 represents the state of the Fermi sea, i.e. of the infinite system of The projector γper all the electrons in the periodic crystal. Of course, it is an infinite rank projector, meaning that
0 γper = |ϕk ϕk | k

should be interpreted as the one-body matrix of a formal infinite Slater determinant  = ϕ1 ∧ ϕ2 ∧ · · · ∧ ϕk ∧ · · · . 0 is additionally a spectral projector associated with the continuous The fact that γper spectrum of an operator leads to the obvious analogy with the Dirac sea which is the projector on the negative spectral subspace of the Dirac operator [10–13]. Most of our results will hold true for insulators (or semi-conductors) only. When necessary, we shall take Z ∈ N \ {0} and make the following assumption: + (A1) There is a gap between the Z th and the (Z + 1)st bands, i.e.  +Z <  − Z +1 , where  Z − th st and  Z +1 are respectively the maximum and the minimum of the Z and the (Z + 1) 0 . bands of Hper 0 of the nonWe emphasize that Assumption (A1) is a condition on the solution γper 0 = χ 0 linear problem (2.4). Note that under (A1), one has γper (−∞, F ] (Hper ) for any − +  F ∈ ( Z ,  Z +1 ).

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3. The Reduced Hartree-Fock Model for a Crystal with a Defect In this section, we define the reduced Hartree-Fock model describing the behavior of the Fermi sea and possibly of a finite number of bound electrons (or holes) close to a local defect. Our model is an obvious transposition of the Bogoliubov-Dirac-Fock model which was proposed by Chaix and Iracane [6] to describe the polarized Dirac sea (and a finite number of relativistic electrons) in the presence of an external potential. Our mathematical definition of the reduced energy functional follows mainly ideas from [10,11]. We shall prove in Sect. 4 that this model can be obtained as the thermodynamic limit of the so-called supercell model. An analogous result was proved in [13] for the Bogoliubov-Dirac-Fock (BDF) model. Assume that the periodic nuclear density µper defined in (2.1) is replaced by a locally perturbed nuclear density µper +ν. The defect ν can model a vacancy, an interstitial atom, or an impurity, with possible local rearrangement of the neighboring atoms. The main idea underlying the model is to define a finite energy by subtracting the infinite energy 0 defined in the previous section, from the infinite energy of of the periodic Fermi sea γper the perturbed system under consideration. For the BDF model, this was proposed first in [13]. Formally, one obtains for a test state γ ,

 rHF 0 0 0 EµrHF (γ ) − E (γ ) “ = ” Tr H (γ − γ ) µ per per per +ν +ν per per   ν(x)ρ   ρ 0 ] (y) 0 ] (x)ρ[γ −γ 0 ] (y) [γ −γper [γ −γper 1 per dx dy + d x d y. (3.1) − 3 3 3 3 |x − y| 2 |x − y| R R R R Of course the two terms in the left-hand side of (3.1) are not well-defined because µper 0 have infinite ranks, but we shall be able to give a is periodic and because γ and γper 0 mathematical meaning to the right-hand side, exploiting the fact that Q := γ − γper 0 . The formal computation (3.1) induces a small perturbation of the reference state γper will be justified by means of thermodynamic limit arguments in Sect. 4. 3.1. Definition of the reduced Hartree-Fock energy of a defect. We now define properly the reduced Hartree-Fock energy of the Fermi sea in the presence of the defect ν. We denote by S p the Schatten class of operators Q acting on L 2 (R3 ) having a finite p trace, i.e. such that Tr(|Q| p ) < ∞. Note that S1 is the space of trace-class operators, and that S2 is the space of Hilbert-Schmidt operators. Let  be an orthogonal projector on L 2 (R3 ) such that both  and 1− have infinite ranks. A self-adjoint compact operator Q is said to be -trace class (Q ∈ S 1 ) when Q ∈ S2 and Q, (1−)Q(1−) ∈ S1 . Its -trace is then defined as Tr  (Q) = Tr(Q + (1 − )Q(1 − )). Notice that if Q ∈ S1 , then Q ∈ S 1 for any  and Tr  (Q) = Tr(Q). See [10, Sect. 2.1] for general properties related to this definition. In the following, we use the shorthand notation 0 0 Q −− := γper Qγper ,

0 0 Q + + := (1 − γper )Q(1 − γper ),

   γ0 S01 := S1per = Q ∈ S2  Q + + ∈ S1 , Q −− ∈ S1 and Tr 0 (Q) := Tr γper 0 (Q). We also introduce the Banach space    Q = Q ∈ S01  Q ∗ = Q, |∇|Q ∈ S2 , |∇|Q + + |∇| ∈ S1 , |∇|Q −− |∇| ∈ S1 ,

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endowed with its natural norm

    ||Q||Q := ||Q||S2 +  Q + + S +  Q −− S 1   1   + |||∇|Q||S2 + |∇|Q + + |∇|S + |∇|Q −− |∇|S . 1

1

The convex set on which the energy will be defined is   0 0 . K := Q ∈ Q | − γper ≤ Q ≤ 1 − γper

(3.2)

(3.3)

Notice that K is the closed convex hull of states Q ∈ Q of the special form Q = 0 , γ being an orthogonal projector on L 2 (R3 ). Besides, the number Tr (Q) can γ − γper 0 be interpreted as the charge of the system measured with respect to that of the unperturbed Fermi sea. It can be proved [10, Lemma 2] that Tr 0 (Q) is always an integer if Q is a 0 , with γ an orthogonal Hilbert-Schmidt operator of the special form Q = γ − γper projector. Additionally, in this case, Tr 0 (Q) = 0 when Q < 1. 0 ≤ Q ≤ 1 − γ 0 in (3.3) is equivalent [1,10] to the Note that the constraint −γper per inequality Q 2 ≤ Q + + − Q −−

(3.4)

and implies in particular that Q + + ≥ 0 and Q −− ≤ 0 for any Q ∈ K. In order to define properly the energy of Q, we need to associate a density ρ Q with any state Q ∈ K. We shall see that ρ Q can in fact be defined for any Q ∈ Q. This is not obvious a priori since Q does not only contain trace-class operators. Additionally we need to check that the last two terms of (3.1) are well-defined. For this purpose, we introduce the so-called Coulomb space C := {ρ ∈ S  (R3 ) | D(ρ, ρ) < ∞},  where D( f, g) = 4π R3 |k|−2  f (k) g (k)dk was already defined before in (1.2). The dual   space of C is the Beppo-Levi space C  := V ∈ L 6 (R3 ) | ∇V ∈ L 2 (R3 ) . We now use a duality argument to define ρ Q : that Q ∈ Q. Then Proposition 1. (Definition of the density  ρ Q for Q ∈ Q) Assume  QV ∈ S01 for any V = V1 + V2 ∈ C  + L 2 (R3 ) ∩ L ∞ (R3 ) and moreover there exists a constant C (independent of Q and V ) such that |Tr 0 (QV )| ≤ C ||Q||Q (||V1 ||C  + ||V2 || L 2 (R3 ) ).   Thus the linear form V ∈ C  + L 2 (R3 ) ∩ L ∞ (R3 ) → Tr 0 (QV ) can be continuously extended to C  + L 2 (R3 ) and there exists a uniquely defined function ρ Q ∈ C ∩ L 2 (R3 ) such that 

   ∀V = V1 + V2 ∈ C  + L 2 (R3 ) ∩ L ∞ (R3 ) , ρ Q , V1 C ,C  + ρ Q V2 = Tr 0 (QV ). R3

The linear map Q ∈ Q → ρ Q ∈ C ∩ L 2 (R3 ) is continuous:     ρ Q  + ρ Q  2 3 ≤ C ||Q||Q . C L (R ) Eventually when Q ∈ S1 ⊂ S01 , then ρ Q (x) = Q(x, x), where Q(x, y) is the integral kernel of Q.

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The proof of Proposition 1 is given in Sect. 5.2. Assuming that (A1) holds true, we are now in a position to give a rigorous sense to 0 = Q ∈ K. In the sequel, we use the following the right-hand side of (3.1) for γ − γper notation for any Q ∈ Q: 0 0 0 Tr 0 (Hper Q) := Tr(|Hper − κ|1/2 (Q + + − Q −− )|Hper − κ|1/2 ) + κTr 0 (Q), (3.5)

where κ is an arbitrary real number in the gap ( +Z ,  − Z +1 ) (this expression will be proved to be independent of κ, see Corollary 1 below). Then we define the energy of any state Q ∈ K as 0 E ν (Q) := Tr 0 (Hper Q) − D(ρ Q , ν) +

1 D(ρ Q , ρ Q ). 2

(3.6)

The function ν is an external density of charge representing the nuclear charge of the defect. For the sake of simplicity, we shall assume that ν ∈ L 1 (R3 ) ∩ L 2 (R3 ) ⊂ C throughout the paper, although some of our results are true with a weaker assumption. We shall need the following Lemma 1. Assume that (A1) holds true. For any fixed κ in the gap ( +Z ,  − Z +1 ), there exist two constants c1 , c2 > 0 such that 0 − κ| ≤ c2 (1 − ) c1 (1 − ) ≤ |Hper

(3.7)

as operators on L 2 (R3 ). In particular  √    √    0  0 − κ|−1/2 (1 − )1/2  ≤ 1/ c1 . |Hper − κ|1/2 (1 − )−1/2  ≤ c2 , |Hper 0 − κ)(1 − )−1 and its inverse are bounded operators. Similarly, (Hper

The proof of the above lemma is elementary; it will be given in Sect. 5.1.1. By the definition of Q and Lemma 1, it is clear that the right-hand side of (3.5) is a well-defined quantity for any Q ∈ Q and any κ ∈ ( +Z ,  − Z +1 ). By Proposition 1 which states that ρ Q ∈ C for any Q ∈ Q, we deduce that (3.6) is a well-defined functional. We shall need the following space of more regular operators Qr := {Q ∈ Q | (−)Q 2 (−) ∈ S1 , (−)(Q + + − Q −− )(−) ∈ S1 }

(3.8)

and the associated convex set Kr := K ∩ Qr . The following result will be useful (its proof will be given below in Sect. 5.3): Lemma 2. The space Qr (resp. the convex set Kr ) is dense in Q (resp. in K) for the topology of Q.

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0 Q ∈ S0 . For any Corollary 1. Assume that (A1) holds true. When Q ∈ Qr , then Hper 1 0 Q) does not depend on κ ∈ ( + ,  − ). If Q ∈ Q, the expression (3.5) for Tr 0 (Hper Z +1 Z Q ∈ K, then

0 ≤ c1 Tr((1 − )1/2 Q 2 (1 − )1/2 ) ≤ c1 Tr((1 − )1/2 (Q + + − Q −− )(1 − )1/2 ) 0 ≤ Tr 0 (Hper Q) − κTr 0 (Q)

(3.9)

≤ c2 Tr((1 − )1/2 (Q + + − Q −− )(1 − )1/2 ), where c1 and c2 are given by Lemma 1. 0 + + = |H 0 − κ|Q + + = Proof. Let Q ∈ Qr and κ ∈ ( +Z ,  − per Z +1 ). Then ((Hper − κ)Q) 0 −1 + + |Hper − κ|(1 − ) (1 − )Q ∈ S1 by Lemma 1 and the definition of Qr . A similar 0 − κ)Q)−− proves that H 0 Q ∈ S0 . Then for any Q ∈ Q , (3.9) is argument for ((Hper r per 1 a straightforward consequence of (3.7) and (3.4). We conclude using the density of Qr in Q and the density of Kr in K.  

The following is an adaptation of [10, Thm 1]: Corollary 2. Let ν ∈ L 1 (R3 ) ∩ L 2 (R3 ), Z ∈ N \ {0} and assume that (A1) holds. For any κ ∈ ( +Z ,  − Z +1 ), one has for some d1 , d2 > 0, 

    ∀Q ∈ K, E (Q) − κTr 0 (Q) ≥ d1  Q + + S +  Q −− S 1 1 

 1     + |∇|Q + + |∇|S + |∇|Q −− |∇|S + d2 ||Q||2S2 + |||∇|Q||2S2 − D(ν, ν). 1 1 2 (3.10) ν

Hence E ν − κTr 0 is bounded from below and coercive on K. Additionally, when ν ≡ 0, Q → E 0 (Q) − κTr 0 (Q) is nonnegative, 0 being its unique minimizer. Proof. Inequality (3.10) is a straightforward consequence of (3.9) and the fact that D(·, ·) defines a scalar product on C. The rest of the proof is obvious.   0 + Q with respect Remark 1. The energy E ν (Q) measures the energy of a state γ = γper 0 . Thus the last statement of Corollary 2 is another way of expressing the to that of γper 0 fact that γper is the state of lowest energy of the periodic system when there is no defect.

3.2. Existence of minimizers with a chemical potential. In view of Corollary 2, it is natural to introduce the following minimization problem E νF := inf{E ν (Q) −  F Tr 0 (Q), Q ∈ K} > −∞

(3.11)

for any Fermi level  F ∈ ( +Z ,  − Z +1 ). The following result is proved in Sect. 5.5, following ideas from [11]:

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Fig. 1. Decomposition Q¯ = Q pol + γe− for not too strong a positively charged nuclear defect (ν ≥ 0)

Theorem 2 (Existence of minimizers with a chemical potential). Let ν ∈ L 1 (R3 ) ∩ L 2 (R3 ), Z ∈ N \ {0} and assume that (A1) holds. Then for any  F ∈ ( +Z ,  − Z +1 ), there exists a minimizer Q¯ ∈ K for (3.11). Problem (3.11) may have several minimizers, but they all share the same density ρ¯ = ρ Q¯ . Any minimizer Q¯ of (3.11) satisfies the self-consistent equation 

0 + δ, Q¯ = χ(−∞, F ) (H Q¯ ) − γper 0 + (ρ − ν) ∗ | · |−1 , H Q¯ = Hper Q¯

(3.12)

where δ is a finite rank self-adjoint operator satisfying 0 ≤ δ ≤ 1 and Ran (δ) ⊆ ker(H Q¯ −  F ). 0 , Remark 2. It is easily seen that (ρ Q¯ − ν) ∗ | · |−1 is a compact perturbation of Hper 0 ) = D(−) = H 2 (R3 ) and that σ (H ) = implying that H Q¯ is self-adjoint on D(Hper ess Q¯ 0 σ (Hper ). Thus the discrete spectrum of H Q¯ is composed of isolated eigenvalues of finite multiplicity, possibly accumulating at the ends of the bands.

Recall that the charge of the minimizing state Q¯ obtained in Theorem 2 is defined ¯ Similarly to [10,11], it can be proved by perturbation theory that for any as Tr 0 ( Q). fixed  F , there exists a constant C( F ) such that when D(ν, ν) ≤ C( F ), one has ¯ = 0, i.e. the minimizer of the energy with chemical ker(H Q¯ −  F ) = {0} and Tr 0 ( Q) potential  F is a neutral perturbation of the periodic Fermi sea. For a fixed external density ν and an adequately chosen chemical potential  F , one ¯ = 0 meaning either that electron-hole pairs have been created from can have Tr 0 ( Q) the Fermi sea, and/or that the system of lowest energy contains a finite number of bound electrons or holes close to the defect. In the applications, one will usually have for a positively charged nuclear defect (ν ≥ 0) that the spectrum of H Q¯ contains a sequence of eigenvalues converging to the bottom  − Z +1 of the lowest unfilled band (conduction band), and that  F is chosen such that exactly q eigenvalues are filled, corresponding to q bound electrons: 

 0 + χ[, F ) (H Q¯ ) + δ := Q pol + γe− , Q¯ = χ(−∞,) (H Q¯ ) − γper where we have chosen as a reference the center of the gap  :=

 +Z +  − Z +1 . 2

(3.13)

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For not too strong a defect density ν, one has ker(H Q¯ − ) = {0} and Tr 0 (Q pol ) = 0. Hence Tr(γe− ) = q. Let us assume for simplicity that δ = 0 and that q ∈ N \ {0}. Then γe− = χ[, F ) (H Q¯ ) =

q


|ϕn ϕn |,

(3.14)

n=1

where (ϕn ) are eigenfunctions of H Q¯ corresponding to its q first eigenvalues in [,  F ): H Q¯ ϕn = λn ϕn .

(3.15)

H Q¯ = −/2 + (ργe− − ν) ∗ | · |−1 + Vpol ,

(3.16)

Notice that

where −1 Vpol = (ργper 0 − µper ) G 1 + ρ Q pol ∗ | · |

is the polarization potential created by the self-consistent Fermi sea and seen by the q electrons. Thus the q electrons solve a usual reduced Hartree-Fock equation (3.15) in which the mean-field operator (3.16) additionally contains the self-consistent polarization of the medium. The interpretation given in the previous paragraph is different if the positive density of charge ν of the defect is strong enough to create an electron-hole pair from the Fermi sea. We end this section by specifying the regularity of solutions of (3.12). The proof is given in Sect. 5.4. Proposition 2. Let ν ∈ L 1 (R3 ) ∩ L 2 (R3 ), Z ∈ N \ {0} and assume that (A1) holds. Any Q ∈ K solution of the self-consistent equation (3.12) belongs to Kr . Remark 3. Notice that it is natural to wonder whether Q ∈ S1 , which would in particular imply that ρ Q ∈ L 1 (R3 ). This is known to be false for the Bogoliubov-Dirac-Fock model studied in [10–13]. We do not answer this question for our model in the present paper. 3.3. Existence of minimizers under a charge constraint. In the previous section, we stated the existence of minimizers for any chemical potential in the gap of the periodic 0 , but of course the total charge Tr ( Q) operator Hper 0 ¯ of the obtained solution was unknown a priori. Here we tackle the more subtle problem of minimizing the energy while imposing a charge constraint. Mathematically this is more difficult because although the energy E ν (Q) is convex on K and weakly lower semi-continuous (wlsc) for the weak-∗ 0 -trace functional topology of Q (as will be shown in the proof of Theorem 2), the γper Q ∈ K → Tr 0 (Q) is continuous but not wlsc for the weak-∗ topology of Q: in principle it is possible that a (positive or negative) part of the charge of a minimizing sequence for the charge-constrained minimization problem escapes to infinity, leaving at the limit a state of a different (lower or higher) charge. In fact, we can prove that a minimizer exists under a charge constraint, if and only if some binding conditions hold, the role of which being to prevent the lack of compactness. As explained above, imposing Tr 0 (Q) = q should intuitively lead (for a sufficiently weak defect density ν) to a system of q electrons coupled to a polarized Fermi sea.

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 Notice that we do not impose that q = R3 ν, i.e. our model allows a priori to treat defects with non-zero total charge. As usual in reduced Hartree-Fock theories, we consider the case of a real charge constraint q ∈ R: E ν (q) := inf{E ν (Q), Q ∈ K, Tr 0 (Q) = q}.

(3.17)

When no defect is present, E 0 (q) can be computed explicitly: Proposition 3. (Defect-free charge-constrained energy) Let Z ∈ N \ {0} and assume that (A1) holds. Then one has  −  Z +1 q when q ≥ 0 0 E (q) = when q ≤ 0.  +Z q The minimization problem (3.17) with ν ≡ 0 has no solution except when q = 0. The proof of Proposition 3 is given in Sect. 5.6. We now state the main result of this section, which is directly inspired from [12]: Theorem 3. (Existence of minimizers under a charge constraint) Let ν ∈ L 1 (R3 ) ∩ L 2 (R3 ), Z ∈ N\{0} and assume that (A1) holds. The following assertions are equivalent: ¯ (a) Problem (3.17) admits a minimizer Q; (b) Every minimizing sequence for (3.17) is precompact in Q and converges towards a minimizer Q¯ of (3.17); (c) ∀q  ∈ R \ {0}, E ν (q) < E ν (q − q  ) + E 0 (q  ). Assume that the equivalent conditions (a), (b) and (c) above are fulfilled. In this case, the minimizer Q¯ is not necessarily unique, but all the minimizers share the same density ¯ ρ¯ = ρ Q¯ . Besides, there exists  F ∈ [ +Z ,  − Z +1 ] such that the obtained minimizer Q is a global minimizer for E νF defined in (3.11). It solves Eq. (3.12) for some 0 ≤ δ ≤ 1 with Ran(δ) ⊆ ker(H Q¯ −  F ). The operator δ is finite rank if  F ∈ ( +Z ,  − Z +1 ) and trace-class if  F ∈ { +Z ,  − }. Z +1 Additionally the set of q’s in R satisfying the above equivalent conditions is a nonempty closed interval I ⊆ R. This is the largest interval on which q → E ν (q) is strictly convex.   ¯ Q¯ min. of E ν ,  F ∈ [ + ,  − ] . Hence I = ∅ Remark 4. One has I = Tr 0 ( Q), Z +1 Z F by Theorem 2. Theorem 3 is proved in Sect. 5.8. Many of the above statements are very common in reduced Hartree-Fock theories and not all the details will be given (see, e.g. [31]). The difficult part is the proof that (b) is equivalent to (c), for which we use ideas from [12]. Conditions like (c) appear classically when analyzing the compactness properties of minimizing sequences, for instance by using the concentration-compactness principle of P.-L. Lions [24]. They are also very classical for linear models in which the bottom of the essential spectrum has the form of the minimum with respect to q  of the right hand side of (c), as expressed by the HVZ Theorem [16,35,36]. Assume for simplicity that q > 0 and that Q¯ can be written as in (3.13) and (3.14). When 0 < q  ≤ q, (c) means that it is not favorable to let q  electrons escape to infinity, while keeping q −q  electrons near the defect. When q  < 0, it means that it is not favorable to let |q  | holes escape to infinity, while keeping q + |q  | electrons near the defect. When q  > q, it means that it

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is not favorable to let q  electrons escape to infinity, while keeping q  − q holes near the defect. In this article we do not show when (c) holds true. Proving (c) usually requires some decay property of the density of charge ρ Q¯ for a solution Q¯ of the nonlinear equation (3.12). In particular, knowing that ρ Q¯ ∈ L 1 (R3 ) would be very useful (see Remark 3). We plan to investigate more closely the decay properties of ρ Q¯ and the validity of (c) in the near future. At present, the validity of a condition similar to (c) for the BogoliubovDirac-Fock model was only proved in the nonrelativistic limit or in the weak coupling limit, see [12].

4. Thermodynamic Limit of the Supercell Model As mentioned before, we shall now justify the model of the previous section by proving that it is the thermodynamic limit of the supercell model. Let us emphasize that there are several ways of performing thermodynamic limits. In [5], the authors consider a box of size L,  L := [−L/2, L/2)3 , and assume that the nuclei are located on Z3 ∩  L . Then they consider the rHF model of Sect. 1 for N electrons living in the whole space, with N = Z L 3 chosen to impose neutrality. Denoting by ρ L the ground state electronic density of the latter problem, it is proved in 0 , and that the following holds: [5, Thm 2.2] that the energy per unit cell converges to Iper L −3




ρ L (x − k) →

!

ργper 0

(4.1)

k∈Z3 ∩ L p

1 (R3 ), strongly in L (R3 ) for all 2 ≤ p < 6 and almost everywhere on weakly in Hloc loc 3 0 and γ 0 are defined in Sect. 2. R when L → ∞. Let us recall that Iper per Another way for performing thermodynamic limits is to confine the nuclei and the electrons in a domain  L with | L | → ∞, by means of Dirichlet boundary conditions for the electrons. The latter approach was chosen for the Schrödinger model with quantum nuclei in the canonical and grand canonical ensembles [28] by Lieb and Lebowitz in the seminal paper [19] (see also [18]), where the existence of a limit for the energy per unit volume is proved. The crystal case in the Schrödinger model was tackled by Fefferman [9] in the same spirit. We do not know whether Fefferman’s proof can be adapted to treat the Hartree-Fock case. Another possibility, perhaps less satisfactory from a physical viewpoint but more directly related to practical calculations (see e.g. [8]), is to take  L =  L and to impose periodic boundary conditions on the box  L . Usually the Coulomb interaction is also replaced by a (LZ3 )-periodic Coulomb potential, leading to the so-called supercell model which will be described in detail below. This approach has the advantage of respecting the symmetry of the system in the crystal case. It was used by Hainzl, Lewin and Solovej in [13] to justify the Hartree-Fock approximation of no-photon Quantum Electrodynamics. The supercell limit of a linear model for photonic crystals is studied in [32]. Of course the conjecture is that the final results (the energy per unit cell and the ground state density of the crystal) should not depend on the chosen thermodynamic limit procedure. This is actually the case for the reduced Hartree-Fock model of the crystal. See [15] for a result in this direction for a model with quantum nuclei.

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Let us now describe the supercell model. For L ∈ N \ {0}, we introduce the supercell  L = [−L/2, L/2)3 and the Hilbert space   L 2per ( L ) = ϕ ∈ L 2loc (R3 ) | ϕ (LZ3 )-periodic . We also introduce the LZ3 -periodic Coulomb potential G L defined as the unique solution to ⎧ ⎛ ⎞ ⎪
⎪ 1 ⎨ −G L = 4π ⎝ δk − 3 ⎠ L ⎪ k∈L Z3 ⎪ ⎩ G L L Z3 -periodic, minR3 G L = 0. It is easy to check that G L (x) = L −1 G 1 (x/L) and that G L (x) =

c + L


3 k∈ 2π L Z \{0}

4π 1 ik·x e . |k|2 L 3

For any LZ3 -periodic function g, we define    g  L G L (x) := G L (x − y) g(y) dy, L

 DG L ( f, g) :=

L

 L

G L (x − y) f (x) g(y) d x d y.

An admissible electronic state is then described by a one-body density matrix γ in Psc,L = γ ∈ S1 (L 2per ( L )) | γ ∗ = γ , 0 ≤ γ ≤ 1, Tr L 2per ( L ) (−γ ) < +∞ . Any γ ∈ Psc,L has a well-defined density of charge ργ (x) = γ (x, x), where γ (x, y) is the kernel of the operator γ . Notice that γ (x + Lz, y + Lz  ) = γ (x, y) for any z, z  ∈ Z3 , which implies that ργ is LZ3 -periodic. Throughout this section, we use the subscript ‘sc’ to indicate that we consider the thermodynamic limit of the supercell model.

4.1. Thermodynamic limit without defect. Because our model with defect uses the defect-free density matrix of the Fermi sea as a reference, we need to start with the study of the thermodynamic limit without defect. We are going to prove for the supercell model a result analogous to [5, Thm 2.2]. The reduced Hartree-Fock energy functional of the supercell model is defined for γ ∈ Psc,L as     1 1 0 Esc,L (γ ) = Tr L 2per ( L ) − γ + DG L ργ − µper , ργ − µper 2 2

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3 3 where we recall that µper (x) = R∈Z3 Z m(x − R) is a Z - (thus LZ -) periodic function. The reduced Hartree-Fock ground state energy for a neutral system in the box of size L is then given by  0 0 Isc,L = inf Esc,L (γ ), γ ∈ Psc,L ,

 L

ργ =

L

µper = Z L 3 .

(4.2)

0 , γ 0 and H 0 are defined in Sect. 2. In Sect. 5.9, we prove the Let us recall that Iper per per following

Theorem 4 (Thermodynamic limit of the defect-free supercell model). Let Z > 0. 0 i) For all L ∈ N \ {0}, the minimizing problem Isc,L has at least one minimizer, and all the minimizers share the same density. This density is Z3 -periodic. Besides, there is one 0 minimizer γsc,L of (4.2) which commutes with the translations τk , k ∈ Z3 . ii) The following thermodynamic limit properties hold true: • (Convergence of the energy per unit cell). lim

L→∞

0 Isc,L

L3

0 = Iper ;

• (Convergence of the density). ! ! 1 ργ 0  ργper weakly in Hloc (R3 ), 0 sc,L

ργ 0

sc,L

(4.3)

p

→ ργper strongly in L loc (R3 ) for 1 ≤ p < 3 and a.e.; 0

• (Convergence of the mean-field Hamiltonian and its spectrum). Let 0 Hsc,L =−

 + (ργ 0 − µper ) G 1 , sc,L 2

0 0 is a seen as an operator acting on L 2 (R3 ). Then, for all L ∈ N \ {0}, Hsc,L − Hper bounded operator and    0 0  lim  Hsc,L − Hper  = 0. L→∞

0 ) for Denoting by (λnL (ξ ))n∈N\{0} the nondecreasing sequence of eigenvalues of (Hsc,L ξ ∗ ξ ∈ , one has     (4.4) lim sup sup λnL (ξ ) − λn (ξ ) = 0, L→∞ n∈N\{0} ξ ∈ ∗

0 ) introduced in Theorem 1. where (λn (ξ ))n≥1 are the eigenvalues of (Hper ξ iii) Assume in addition that (A1) holds. Fix some  F ∈ ( +Z ,  − Z +1 ). Then for L large 0 0 enough, the minimizer γsc,L of Isc,L is unique. It is also the unique minimizer of the following problem:   0 0 (4.5) Isc,L , F := inf Esc,L (γ ) −  F Tr L 2per ( L ) (γ ), γ ∈ Psc,L .

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Remark 5. Notice that some of the above assertions are more precise for the supercell model than for the thermodynamic limit procedure considered in [5, Thm 2.2] (compare for instance (4.3) with (4.1)). This is because the supercell model respects the symmetry 0 in the box of size L 3 which of the system, allowing in particular to have a minimizer γsc,L 0 3 is periodic for the lattice Z . For an insulator, the uniqueness of γsc,L for large L and the convergence properties of iii) are also very interesting for computational purposes.

4.2. Thermodynamic limit with defect. We end this section by considering the thermodynamic limit of the supercell model with a defect. Recall that ν ∈ L 1 (R3 )∩ L 2 (R3 ) ⊂ C is the density of charge of the defect. First we need to periodize this function with respect to the large box  L , for instance by defining ν L (x) :=




(1 L ν)(· − Lz).

z∈Z3

The reduced Hartree-Fock energy functional of the supercell model with defect is then defined for γ ∈ Psc,L as ν Esc,L (γ )

    1 1 = Tr L 2per ( L ) − γ + DG L ργ − µper − ν L , ργ − µper − ν L . 2 2

For  F ∈ ( +Z ,  − Z +1 ), we consider the following minimization problem   ν ν Isc,L , F = inf Esc,L (γ ) −  F Tr L 2per ( L ) (γ ), γ ∈ Psc,L .

(4.6)

0 is defined in Sect. 2, that E ν and Q ¯ are defined in Sect. 3.2, and that We recall that γper κ 0 Isc,L , F is defined in Sect. 4.1. In Sect. 5.10, we prove the

Theorem 5 (Thermodynamic limit of the supercell model with defect). Let Z ∈ N \ {0}. Assume that (A1) holds and fix some  F ∈ ( +Z ,  − Z +1 ). Then one has

lim

L→∞

ν Isc,L , F

−

0 Isc,L , F



=

E νF

 −

 1

D(ν, ν). (4.7) ν (ργper 0 − µper ) G 1 + 2 R3

ν Additionally, if γsc,L denotes a minimizer for (4.6), then one has, up to extraction of a subsequence, ν 0 ¯ (γsc,L − γsc,L )(x, y) → Q(x, y) 1 (R3 × R3 ) and strongly in L 2 (R3 × R3 ), where Q ¯ is a minimizer of weakly in Hloc loc (3.11), as obtained in Theorem 2. Besides, ν ργsc,L − ργ 0

sc,L

→ ρ¯

weakly in L 2loc (R3 ), where ρ¯ is the common density of all the minimizers of (3.11).

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Remark 6. In numerical simulations, the right-hand side of (4.7) is approximated by ν 0 Isc,L , F − Isc,L , F for a given value of L. This approach has several drawbacks. First, the values of L that lead to tractable numerical simulations are in many cases much too small to obtain a correct estimation of the limit L → ∞. Second, it is not easy to extend this method for computing E νF , to the direct evaluation of E ν (q) for a given q (i.e. the energy of a defect with a prescribed total charge). The formalism introduced in the present article (Problems (3.11) and (3.17)) suggests an alternative way for computing energies of defects in crystalline materials. A work in this direction was already started [3]. 5. Proof of the Main Results Unless otherwise stated, the operators used in the following proofs are considered as operators on L 2 (R3 ). 5.1. Useful estimates. We gather in this section some results which we shall need throughout the proofs. We start with the 0 = −/2 + V 5.1.1. Proof of Lemma 1 Recall that Hper 0 − µper ) per with Vper := (ργper

0 −κ| ≥ H 0 −κ ≥ −/2−C for some large enough constant G 1 ∈ L ∞ (R3 ). Thus |Hper per    0  C. On the other hand, as κ ∈ ( +Z ,  − Z +1 ), there exists α > 0 such that Hper − κ  ≥ α. This implies that 0 |Hper − κ| ≥ max(−/2 − C, α) ≥ c1 (1 − )

for some constant c1 > 0. The proof of the upper bound in (3.7) is straightforward. 0 − κ + c) = 1 + (−/2 + c)−1 (V Then (−/2 + c)−1 (Hper per − κ) is a bounded invertible operator for c large enough, since   V  + |κ| per L ∞   −1 . (−/2 + c) (Vper − κ) ≤ c 0 − κ + c)−1 (−/2 + c) is bounded for a well-chosen c  1, which clearly Thus (Hper implies that 0 (Hper − κ)−1 (− + 1) =

0 −κ +c Hper 0 −κ Hper

is also bounded, together with its inverse.

0 (Hper − κ + c)−1 (−/2 + c)

− + 1 −/2 + c

 

5.1.2. Some commutator estimates Throughout this paper, we shall use Cauchy’s 0 : formula to express the projector γper  1 0 =− (H 0 − z)−1 dz, (5.1) γper 2iπ C per where C is a fixed regular bounded closed contour enclosing the lowest Z bands of the 0 . spectrum of Hper

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147

0 − z)−1 with The following result will be a useful tool to replace the resolvent (Hper the operator (− + 1)−1 which will be easier to manipulate. Its proof is the same as the one of Lemma 1. 0 − z)−1 (− + 1) and its inverse are bounded Lemma 3. The operator B(z) := (Hper uniformly in z ∈ C .

The next result provides some useful properties of commutators: 0 , ] and (1 − ) [γ 0 , |∇|] (1 − ) are bounded. Lemma 4. The operators [γper per 0 , ] follows from (5.1) and from the fact that [(H 0 − Proof. The boundedness of [γper per z)−1 , ] is bounded uniformly in z ∈ C by Lemma 3. 0 − z)−1 , |∇|](1 − ) is Using again (5.1), it suffices to prove that (1 − )[(Hper 0 bounded uniformly in z ∈ C to infer that (1 − ) [γper , |∇|] (1 − ) is bounded. In 0 − z)−1 , |∇|](1 − ), we use order to prove the uniform boundedness of (1 − )[(Hper the formal equality

[(A − z)−1 , B] = −(A − z)−1 [A, B](A − z)−1 .

(5.2)

" # 0 (1 − )[(Hper − z)−1 , |∇|](1 − ) = B(z)∗ |∇|, Vper B(z).

(5.3)

We thus obtain

Using (5.1) and Lemma 3, we obtain  "  # 0 (1 − )[γper , |∇|](1 − ) ≤ C  |∇|, Vper  ≤ C ∇Vper  L ∞ (R3 ) .   0 , V ] ∈ S and Lemma 5. Let V = V1 + V2 with V1 ∈ C  and V2 ∈ L 2 (R3 ). Then [γper 2 there exists a positive real constant C such that 0 [γper , V ]S2 ≤ C(V1 C  + ||V2 || L 2 (R3 ) ).

Proof. Formulas (5.1) and (5.2) lead to  1 0 [γper , V2 ] = − B(z)(− + 1)−1 [, V2 ](− + 1)−1 B(z)∗ dz 4iπ C  1 =− B(z)((− + 1)−1 )V2 (− + 1)−1 B(z)∗ dz 4iπ C  1 + B(z)(− + 1)−1 V2 ((− + 1)−1 )B(z)∗ dz. 4iπ C As (− + 1)−1 and (− + 1)−1  are bounded operators, we obtain, using Lemma 3,         0 [γper , V2 ] ≤ C (− + 1)−1 V2  , S2

S2

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for some constant C independent of V2 . Likewise,  1 0 , V1 ] = − B(z)(− + 1)−1 [, V1 ](− + 1)−1 B(z)∗ dz [γper 4iπ C  3  ∂V


1 1 B(z) (− + 1)−1 ∂xi (− + 1)−1 B(z)∗ dz =− 4iπ C ∂ xi +

i=1 3
i=1

which implies

  1 ∂ V1

∂xi (− + 1)−1 B(z)∗ dz, B(z)(− + 1)−1 4iπ C ∂ xi

   0  [γper , V1 ]

S2

    ≤ C (− + 1)−1 ∇V1 

S2

for some constant C independent of V1 . We then use the Kato-Seiler-Simon inequality (see [29] and [30, Thm 4.1]) ∀ p ≥ 2, to infer

|| f (−i∇)g(x)||Sp ≤ (2π )−3/ p ||g|| L p (R3 ) || f || L p (R3 )    0  [γper , V2 ]     0 [γper , V1 ]

S2

(5.4)

≤ C  ||V2 || L 2 (R3 ) ,

(5.5)

≤ C  ||∇V1 || L 2 (R3 ) = C  ||V1 ||C  .

(5.6)

S2

  0 − κ|, |∇| ] is bounded for any κ in the gap ( + ,  − ). Lemma 6. The operator [ |Hper Z +1 Z 0 − κ| = −(H 0 − κ)γ 0 + (H 0 − κ)(1 − γ 0 ) and thus Proof. We have |Hper per per per per 0 0 0 0 [ |Hper − κ|, |∇| ] = −2(Hper − κ)[γper , |∇|] + [|∇|, Vper ](2γper − 1) 0 0 = −2(B(κ)∗ )−1 (1 − )[γper , |∇|] + [|∇|, Vper ](2γper − 1)     0 , |∇|] is which gives the result since [|∇|, Vper ] ≤ ∇Vper  L ∞ (R3 ) and (1 − )[γper bounded by Lemma 4.  

5.2. Proof of Proposition 1. Let V = V1 +V2 where V1 ∈ C  and V2 ∈ L 2 (R3 )∩L ∞ (R3 ), and Q ∈ Q. Notice that 0 ⊥ 0 0 ⊥ (QV )+ + = Q + + V (γper ) + Q +− [γper , V ](γper ) , 0 0 0 (QV )−− = Q −− V γper − Q −+ [γper , V ]γper ,

(5.7)

0 )⊥ = 1 − γ 0 . We only treat the (QV )−− term, the argument being the same where (γper per for (QV )+ + .

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149

0 0 and notice that First we write Q −− V γper = Q −− (1 + |∇|)(1 + |∇|)−1 V γper (1 + |∇|)−1 V is bounded since V2 ∈ L ∞ (R3 ) by assumption and     (1 + |∇|)−1 V1  ≤ C ||V1 || L 6 ≤ C  ||∇V1 || L 2 = C  ||V1 ||C 

S6

by the Kato-Seiler-Simon inequality (5.4) and the critical Sobolev embedding of H 1 (R3 ) 0 is a trace-class operator. Thus the following is in L 6 (R3 ). This proves that Q −− V γper true: 0 |Tr(Q −− V γper )| = |Tr(Q −− V )|

= |Tr((1 + |∇|)Q −− (1 + |∇|)(1 + |∇|)−1 V (1 + |∇|)−1 )|     ≤ ||Q||Q (1 + |∇|)−1 V (1 + |∇|)−1  . S∞

Then

    (1 + |∇|)−1 V1 (1 + |∇|)−1 

S∞

    ≤ (1 + |∇|)−1 V1  

S6

    (1 + |∇|)−1 

≤ C ||V1 || L 6 ≤ C ||V1 ||C  , and

    (1 + |∇|)−1 V2 (1 + |∇|)−1 

    ≤ (1 + |∇|)−1 |V2 |(1 + |∇|)−1  S∞ S∞  2    ≤ (1 + |∇|)−1 |V2 |1/2  ≤ C ||V2 || L 2 . S4

Hence, 0 |Tr(Q −− V γper )| ≤ C ||Q||Q (||V1 ||C  + ||V2 || L 2 ). 0 , V ]γ 0 ∈ For the second term of (5.7), we just use Lemma 5 which tells us that Q −+ [γper per −+ 0 S1 since Q ∈ S2 and [γper , V ] ∈ S2 . Additionally      0 0 0 0  |Tr(Q −+ [γper , V ]γper )| ≤ Q −+ [γper , V ]γper  ≤ C  Q −+ S2 (||V1 ||C  + ||V2 || L 2 ).

S1

The end of the proof of Proposition 1 is then obvious.

 

5.3. Proof of Lemma 2. Let Q ∈ Q. For  > 0, we introduce the following regularization operator 0 R := (1 + |Hper − |)−1

(5.8)

and set Q  := R Q R . Notice first that Q  ∈ Qr . Indeed, using the same notation as in Lemma 3, we obtain 0 (1 − )R = (1 − )(Hper − )−1

0 − Hper 0 − | 1 + |Hper

= B()∗

0 − Hper 0 − | 1 + |Hper

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which shows that (1−)R  ≤  −1 B()∗ . Similarly, R (1−) ≤  −1 B(). 0 , we infer As R commutes with γper −− R (1 − ) ∈ S1 . (1 − )Q −−  (1 − ) = (1 − )R Q

Likewise, (1−)Q + + (1−) ∈ S1 . Then we show that Q  ∈ Kr ⊂ K when Q ∈ K. To 0 ≤ Q ≤ 1 − γ 0 is equivalent to Q 2 ≤ Q + + − Q −− prove this, we use the fact that −γper per (see Sect. 3.1). As R  ≤ 1, we obtain (Q  )2 = R Q(R )2 Q R ≤ R Q 2 R ≤ R (Q + + − Q −− )R = (Q  )+ + − (Q  )−− ,

(5.9)

0 commutes with R . Hence, it only where we have used that (R )2 ≤ 1 and that γper  remains to prove that Q  → Q for the Q-topology as  → 0, for any fixed Q ∈ Q. We shall need the

Lemma 7. For any 1 ≤ p < ∞ and any fixed Q ∈ S p , one has lim ||R Q − Q||Sp = 0.

(5.10)

→0

Proof. Notice that R − 1 = −

0 − | |Hper 0 − | 1 + |Hper

satisfies R − 1 ≤ 1. Hence ||(R − 1)Q||Sp ≤ ||Q||Sp . By linearity and density of “smooth” finite rank operators in S p for any 1 ≤ p < ∞, it suffices to prove (5.10) for Q = | f  f | with f ∈ H 2 (R3 ). Then    0  ||(R − 1)| f  f |||S1 ≤ ||(R − 1) f || L 2 || f || L 2 ≤  |Hper − | f  2 || f || L 2 L

which converges to 0 as  → 0 and controls ||(R − 1)| f  f |||Sp for 1 ≤ p < ∞.

 

We are now able to complete the proof of Lemma 2. First, by (5.2) $ % 0 − |, |∇| R , [R , |∇|] = − R |Hper and therefore by Lemma 6 there exists a constant C > 0 such that  [R , |∇|]  ≤ C.

(5.11)

Hence, lim→0  [R , |∇|]  = 0. Compute now for instance |∇|(R Q R − Q)−− |∇| = |∇|((R − 1)Q −− R + Q −− (R − 1))|∇| = [|∇|, R ]Q −− [R , |∇|] + [|∇|, R ]Q −− |∇|R +(R − 1)|∇|Q −− [R , |∇|] + |∇|Q −− [R , |∇|] +(R − 1)|∇|Q −− |∇|R + |∇|Q −− |∇|(R − 1). Applying either (5.11) or Lemma 7 to each term of the previous expression allows to conclude that   −− lim |∇|(Q −− )|∇|S = 0.  −Q →0

1

The proof is the same for the other terms in the definition of ||·||Q .

 

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151

5.4. Proof of Proposition 2: Regularity of solutions. Let Q ∈ Q be of the form 0 0 + V ) − γper + δ, Q = χ(−∞, F ) (Hper 0 + V −  ) and where 0 ≤ δ ≤ 1 is a finite rank operator with Ran(δ) ⊆ ker(Hper F V = ρ ∗ | · |−1 for some ρ ∈ L 1 (R3 ) ∩ L 2 (R3 ) (in our case ρ = ρ Q − ν). Note that V ∈ C  ∩ L ∞ (R3 ) since   | ρ (k)| (k)|dk = C ||V || L ∞ ≤ (2π )−3/2 |V dk 2 3 3 R R |k| 1/2  1/2  | ρ (k)|2 (1 + |k|2 ) dk ≤C dk < ∞. 2 2 |k|2 R3 R3 |k| (1 + |k| ) 0 + V −  ) ⊆ D(H 0 + V ) = D(H 0 ) = H 2 (R3 ), it is clear that the Since ker(Hper F per per finite rank operator δ satisfies (1 − )δ(1 − ) ∈ S1 . Thus, up to a change of  F , we 0 + V −  ) = {0} and that δ = 0: can assume that ker(Hper F 0 0 + V ) − γper . Q = χ(−∞, F ) (Hper

Then we remark that Q 2 = Q + + − Q −− , hence we only have to prove that (1 − )Q ∈ S2 . 0 + V below  . Since Let C be a smooth curve enclosing the whole spectrum of Hper F ∞ 3 0 V ∈ L (R ) and |Hper + V − z| ≥ c > 0 uniformly in z ∈ C , we can mimic the proof of Lemma 3 and find that 0 + V − z)−1  < ∞. sup (1 − )(Hper

z∈C

(5.12)

We then use Cauchy’s formula (5.1) and iterate the resolvent formula 0 0 0 0 + V − z)−1 = (Hper − z)−1 − (Hper + V − z)−1 V (Hper − z)−1 (Hper

to obtain

   1 0 0 (Hper + V − z)−1 − (Hper − z)−1 dz 2iπ C = Q1 + Q2 + Q3

Q=−

with

 1 0 Q1 = (H 0 − z)−1 V (Hper − z)−1 dz, 2iπ C per Q2 = −

Q3 =

 1 0 0 (H 0 − z)−1 V (Hper − z)−1 V (Hper − z)−1 dz, 2iπ C per

 1 0 0 0 (H 0 + V − z)−1 V (Hper − z)−1 V (Hper − z)−1 V (Hper − z)−1 dz. 2iπ C per

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Notice that (1 − )Q 3 ∈ S2 by (5.12) and the estimate           0 − z)−1  ≤ V (1 − )−1   B(z)∗  ≤ C ||V || L 6 , V (Hper S6

S6

where we have used (5.4) and Lemma 3. Let us now prove that (1 − )Q 1 ∈ S2 . First we notice that  0 0 0 0 (Hper − z)−1 γper V γper (Hper − z)−1 dz C  0 0 ⊥ 0 ⊥ 0 = (Hper − z)−1 (γper ) V (γper ) (Hper − z)−1 dz = 0 C

by the residuum formula. Then we have  0 0 0 ⊥ 0 (1 − ) (Hper − z)−1 γper V (γper ) (Hper − z)−1 dz C  0 0 ⊥ 0 = B(z)∗ [γper , V ](γper ) (Hper − z)−1 dz C

which belongs to S2 by Lemmas 3 and 5. The proof is the same for Q 2 .

 

5.5. Proof of Theorem 2: Existence of a minimizer with chemical potential. Let (Q n )n∈N be a minimizing sequence for (3.11). It follows from (3.10) that (Q n )n∈N is bounded in Q. By Proposition 1, (ρ Q n )n∈N is bounded in C ∩ L 2 (R3 ). Up to extraction, we can assume that there exists Q¯ in the convex set K such that i) Q n  Q¯ and |∇|Q n  |∇| Q¯ weakly in S2 ; 0 −  |1/2 Q + + |H 0 −  |1/2  |H 0 −  |1/2 Q 0 −  |1/2 , ¯ + + |Hper ii) |Hper F F F F n per per 0 1/2 −− 0 1/2 0 1/2 −− 0 −  |1/2 |Hper −  F | Q n |Hper −  F |  |Hper −  F | Q¯ |Hper F for the weak-∗ topology of S1 ;

iii) ρ Q n  ρ Q¯ in C ∩ L 2 (R3 ). Recall that S1 is the dual of the space of compact operators [26, Thm VI.26]. Thus here An  A for the weak-∗ topology of S1 means Tr(An K ) → Tr(AK ) for any compact operator K . Then, as D(·, ·) defines a scalar product on C, D(ρ Q¯ − ν, ρ Q¯ − ν) ≤ lim inf D(ρ Q n − ν, ρ Q n − ν). n→+∞

0 −  |1/2 Q + + |H 0 −  |1/2 is also a nonnegative operator Now since Q +n + ≥ 0, |Hper F F n per for any n. Thus Fatou’s Lemma [30] yields 0 0 −  F |1/2 Q¯ + + |Hper −  F |1/2 ) Tr(|Hper 0 0 ≤ lim inf Tr(|Hper −  F |1/2 Q +n + |Hper −  F |1/2 ). n→∞

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The same argument for the term involving − Q¯ −− ≥ 0 yields ¯ −  F Tr 0 ( Q) ¯ ≤ lim inf (E ν (Q n ) −  F Tr 0 (Q n )) = E ν , E ν ( Q) F n→∞

i.e. Q¯ ∈ K is a minimizer. The proof that Q¯ satisfies the self-consistent equation (3.12) is classical: writing that ν ¯ for any Q ∈ Kr and t ∈ [0, 1], one deduces that Q¯ E ((1 − t) Q¯ + t Q) ≥ E ν ( Q) minimizes the following linear functional: 0 0 Q ∈ K → F(Q) := Tr(|Hper −  F |1/2 (Q + + − Q −− )|Hper

− F |1/2 ) + D(ρ Q¯ − ν, ρ Q ).

(5.13)

Notice that when Q ∈ Kr ⊆ K, one has F(Q) = Tr 0 ((H Q¯ −  F )Q), where we have used the definition of ρ Q in Proposition 1 to infer

  D(ρ Q¯ − ν, ρ Q ) = Tr 0 (ρ Q¯ − ν) | · |−1 Q , since (ρ Q¯ − ν) | · |−1 ∈ C  when ρ Q¯ − ν ∈ C. Minimizers of the functional (5.13) are easily proved to be of the form (3.12). They belong to Kr by Proposition 2.   5.6. Proof of Proposition 3: The value of E 0 (q). Clearly 0 Q), Q ∈ K, Tr 0 (Q) = q} := E˜ 0 (q) E 0 (q) ≥ inf{Tr 0 (Hper

(5.14)

since D(ρ Q , ρ Q ) ≥ 0. It can be easily proved that  −  Z +1 q when q ≥ 0 0 E˜ (q) =  +Z q when q ≤ 0, see, e.g., the proof of Lemma 13 in [12]. Thus it remains to prove that E 0 (q) ≤ E˜ 0 (q) which we do by a kind of scaling argument. Let us deal with the case q ≥ 0, the other case being similar. We can assume ∗ that  − Z +1 = minξ ∈ λ Z +1 (ξ ) = λ Z +1 (ξ0 ) since each λn (ξ ) is known to be continuous on ∗ . For simplicity, we also assume that ξ0 is in the interior of ∗ (the proof can be easily adapted if this is not the case). Let us denote by u Z +1 (ξ, ·) ∈ L 2ξ an eigenvector associated with the eigenvalue λ Z +1 (ξ ) for any ξ ∈ ∗ . It will be convenient to extend it on R3 ×R3 0 u by u Z +1 (ξ, x) = 0 when ξ ∈ R3 \ ∗ . Since Hper Z +1 (ξ, x) = λ Z +1 (ξ )u Z +1 (ξ, x) for ∗ any ξ ∈ , it is clear that sup ||u Z +1 (ξ, ·)|| L 2 ( ) < ∞,

ξ ∈R3

ξ

2 (R3 )) and u ∞ 3 3 3 i.e. u Z +1 ∈ L ∞ (R3 , Hloc Z +1 ∈ L (R × R ) by the Z -periodicity (resp. 3 −iξ ·x the 2π Z -periodicity) of e u Z +1 (ξ, x) with respect to x (resp. to ξ ).

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Consider now a fixed space V of dimension d = [q] + 1, consisting of C0∞ functions f : R3 → C with support in the unit ball B(0, 1) of R3 . For any λ ≥ 1, we introduce the following subspace of L 2 (R3 ):  Wλ := gλ := λ3/2 f (λ(ξ − ξ0 ))u Z +1 (ξ, ·)dξ, f ∈ V R3

which has the same dimension as V by the properties of the Bloch decomposition, when λ is large enough such that the ball B(ξ0 , λ−1 ) is contained in ∗ . Noting that for any gλ ∈ Wλ arising from some f ∈ V ,  −3/2 | f (ξ  )u Z +1 (ξ0 + λ−1 ξ  , x)|dξ  |gλ (x)| ≤ λ R3  ≤ λ−3/2 ||u Z +1 || L ∞ (R3 ×R3 ) | f (ξ )|dξ, B(0,1)

≤ Cλ

−3/2

||u Z +1 || L ∞ (R3 ×R3 ) || f || L 2 (R3 ) ,

we deduce by interpolation that ∀ 2 < p ≤ ∞,

lim

λ→∞

sup g∈Wλ

||g|| L p (R3 ) = 0.

||g|| L 2 (R3 ) =1

By construction one also has for any fixed f ∈ V with associated gλ ∈ Wλ ,  & ' − 0 3 gλ , (Hper −  Z +1 )gλ = λ | f (λ(ξ − ξ0 ))|2 (λ Z +1 (ξ ) −  − Z +1 )dξ →λ→∞ 0. R3

λ Take now an orthonormal basis (ϕ1λ , ..., ϕ[q]+1 ) of Wλ and introduce the operator

Q λ :=

[q]


|ϕnλ ϕnλ | + (q − [q])|ϕdλ ϕdλ |.

n=1 0 ϕ λ = 0 for any n = 1, ..., [q] + 1 and Tr (Q λ ) = Tr(Q λ ) = q, thus By construction γper 0 n λ Q ∈ K satisfies the charge constraint. Then 0 Tr 0 (Hper Qλ) =

[q] &


' & ' 0 λ 0 λ ϕnλ , Hper → − ϕnλ + (q − [q]) ϕ[q]+1 , Hper ϕ[q]+1 Z +1 q

n=1 1/2

as λ → ∞. Besides, (ρ Q λ )λ≥1 is a bounded family in H 1 (R3 ) which converges to 0 in L p (R3 ) for any p > 2. By the Hardy-Littlewood-Sobolev inequality [20], one has  2 D(ρ Q λ , ρ Q λ ) ≤ C ρ Q λ  L 6/5

(5.15)

which implies D(ρ Q λ , ρ Q λ ) → 0 as λ → ∞. Eventually E 0 (Q λ ) → E˜ 0 (q) and Proposition 3 is proved.  

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5.7. Density of finite rank operators in K. This section is devoted to the generalization of results in [12, Appendix B] concerning the density of finite rank operators, which will be useful for proving Theorem 3. 0 ∈K Lemma 8. For any Q ∈ K there exists an orthogonal projector P such that P −γper and a trace-class operator δ ∈ Q such that 0 ≤ δ ≤ 1, [P, δ] = 0 and 0 Q = P − γper + δ.

 

Proof. This is an easy adaptation of [12, Lemma 19]. We denote for simplicity H+ =

0 )L 2 (R3 ) (1 − γper

0 L 2 (R3 ). and H− = γper

0 ∈ K. Proposition 4. Let P be an orthogonal projector on L 2 (R3 ) such that Q = P−γper 1 3 N 0 − Denote by ( f 1 , ..., f N ) ∈ (H+ ∩ H (R )) an orthonormal basis of E 1 = ker(P −γper 0 ) ∩ ker(1 − P) and by (g , ..., g ) ∈ (H ∩ H 1 (R3 )) M an orthonormal 1) = ker(γper 1 M − 0 + 1) = ker(1 − γ 0 ) ∩ ker(P). Then there exist an basis of E −1 = ker(P − γper per orthonormal basis (vi )i≥1 ⊂ H+ ∩ H 1 (R3 ) of (E 1 )⊥ in H+ , an orthonormal basis (u i )i≥1 ⊂ H− ∩ H 1 (R3 ) of (E −1 )⊥ in H− , and a sequence (λi )i≥1 ∈ 2 (R+ ) such that

P=

N


| f n  f n | +

n=1

1− P = Additionally

M


|gm gm | +

m=1

∞
|u i + λi vi u i + λi vi | , 1 + λi2 i=1

2 2 i≥1 λi (∇u i  L 2

∞
|vi − λi u i vi − λi u i | . 1 + λi2 i=1

(5.16)

(5.17)

+ ∇vi 2L 2 ) < ∞.

Proof. This is an obvious corollary of [12, Theorem 7].

 

Corollary 3. Let Q ∈ K. Then there exists a sequence {Q k }k≥1 of finite rank operators belonging to Kr such that ||Q k − Q||Q → 0 as k → ∞ and for any k ≥ 1, Tr 0 (Q k ) = Tr 0 (Q). Proof. Taking λi = 0 for i > k in the decomposition of Proposition 4, one can approxi0 → P − γ 0 in Q as k → ∞ and mate P by another projector Pk such that Pk − γper per 0 is finite rank: Pk − γper 0 Pk − γper =

N


| f n  f n | −

n=1

+

k
i=1

M
m=1

|gm gm | +

k
i=1

 λi  |u i vi | + |vi u i | . 2 1 + λi

 λi2  |vi vi | − |u i u i | 2 1 + λi (5.18)

It then suffices to approximate each function in (5.18) by a smoother one, for instance R u and orthonormalizing these new functions, by defining for   1, u˜ i := ||R u i ||−1 L2  i where R was defined previously in Eq. (5.8). 0 + δ of the form given by Lemma 8, it remains to Then for any Q = P − γper approximate δ by a finite rank operator δk such that [Pk , δk ] = 0, which is done in the same way.  

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5.8. Proof of Theorem 3: Existence of minimizers under a charge constraint. The proof of Theorem 3 follows ideas of [12]. The proof that any minimizer solves (3.12) is the same as before and will be omitted. Step 1. Large HVZ-type inequalities. Let us start by the following result, which indeed shows that (b)⇒(c): Lemma 9. (Large HVZ-type inequalities) Let Z ∈ N \ {0}, ν ∈ C and assume that (A1) holds. Then, for every q, q  ∈ R, one has E ν (q) ≤ E ν (q − q  ) + E 0 (q  ). If moreover there is a q  = 0 such that E ν (q) = E ν (q − q  ) + E 0 (q  ), then there is a minimizing sequence of E ν (q) which is not precompact. Proof. Thanks to Corollary 3, the proof is exactly the same as [12, Prop. 6].

 

Step 2. A necessary and sufficient condition for compactness. The following Proposition is the analogue of [12, Lemma 8]: Proposition 5. (Conservation of charge implies compactness) Let Z ∈ N \ {0}, ν ∈ C, q ∈ R and assume that (A1) holds. Assume that (Q n )n≥1 is a minimizing sequence in Kr for (3.11) such that Q n  Q ∈ K for the weak-∗ topology of Q. Then Q n → Q for the strong topology of Q if and only if Tr 0 (Q) = q. Proof. Let (Q n )n≥1 ⊆ Kr be as stated and assume that Tr 0 (Q) = q. We know from the proof of Theorem 2 that E ν (Q) ≤ lim E ν (Q n ) = E ν (q), n→∞

(5.19)

hence Q ∈ K is a minimizer of E ν (q). Therefore Q satisfies the equation 0 +δ Q = χ(−∞, F ) (H Q ) − γper

for some  F ∈ ( +Z ,  − Z +1 ) and where δ is a finite rank operator satisfying 0 ≤ δ ≤ 1 and Ran(δ) ⊆ ker(H Q −  F ). In particular Q ∈ Kr by Proposition 2. We now introduce P := χ(−∞, F ) (H Q ),

P  := χ( F ,∞) (H Q ) and π := χ{ F } (H Q ).

Let us write E ν (Q n ) = E ν (Q) + Tr 0 (H Q (Q n − Q)) +

1 D(ρ Q n − ρ Q , ρ Q n − ρ Q ). 2

Now using [10, Lemma 1] and the hypothesis Tr 0 (Q n ) = Tr 0 (Q), we obtain Tr 0 (H Q (Q n − Q)) = Tr 0 ((H Q −  F )(Q n − Q)) = Tr P ((H Q −  F )(Q n − Q)) = Tr(|H Q −  F |(P  (Q n − Q)P  − P(Q n − Q)P)), where we recall that by definition Tr P (A) = Tr(P A P + (1 − P)A(1 − P)). We have −P − δ ≤ Q n − Q ≤ 1 − P − δ which yields P(Q n − Q)2 P + P  (Q n − Q)2 P  ≤ P  (Q n − Q)P  − P(Q n − Q)P, (5.20)

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157

and in particular P  (Q n − Q)P  ≥ 0 and P(Q n − Q)P ≤ 0, i.e. Tr 0 (H Q (Q n − Q)) ≥ 0. Since we know that limn→∞ E ν (Q n ) = E ν (Q), we infer lim Tr(|H Q −  F |P  (Q n − Q)P  ) = lim Tr(|H Q −  F |P(Q n − Q)P) = 0

n→∞

n→∞

(5.21) and from (5.20) lim Tr(|H Q −  F |P  (Q n − Q)2 P  ) = lim Tr(|H Q −  F |P(Q n − Q)2 P) = 0.

n→∞

n→∞

On the one hand, it is easy to see that P|H Q −  F |P ≥ c P|H Q − κ|P and P  |H Q −  F |P  ≥ c P  |H Q − κ|P  for some small enough constant c > 0 and some κ ∈ / σ (H Q ) close enough to  F . On the other hand, the weak convergence of (Q n ) and the fact that π is a “smooth” finite rank operator imply that lim Tr(|H Q − κ|π(Q n − Q)2 π ) = lim Tr(|H Q − κ|π(Q n − Q)π ) = 0.

n→∞

n→∞

(5.22)

It is then clear that this yields lim Tr(|H Q − κ|(Q n − Q)2 ) = 0.

n→∞

(5.23)

As we have chosen κ ∈ / σ (H Q ), we can mimic the proof of Lemma 3 and obtain that c1 (1 − ) ≤ |H Q − κ| ≤ c2 (1 − ).

(5.24)

Hence (5.23) shows that (1 − )1/2 (Q n − Q) → 0 in S2 . Writing now 0 0 (Q n − Q)−− = P(Q n − Q)P + (γper − P)(Q n − Q)γper 0 0 0 0 −(P − γper )(Q n − Q)(P − γper ) + γper (Q n − Q)(γper − P) (5.25)

and using (5.24), (5.21) and (5.22), we easily see that |H Q − κ|1/2 (Q n − Q)−− |H Q − κ|1/2 → 0 and (1 − )1/2 (Q n − Q)−− (1 − )1/2 → 0 in S1 . The proof is the same for (Q n − Q)+ + .   Step 3: Proof that (c)⇒(b). We argue by contradiction. Let (Q n )n≥1 ⊆ K be a minimizing sequence for E ν (q) which is not precompact for the topology of Q. By the density of Kr in K, we can further assume that each Q n ∈ Kr . The bound (3.9) on the energy tells us that (Q n )n≥1 is bounded in Q. Then, up to extraction and by Proposition 5, we can assume that Q n  Q ∈ K, where Tr 0 (Q) = q, and that ρ Q n  ρ Q weakly in C. We write Tr 0 (Q) = q − q  with q  = 0. We now prove that E ν (q) ≥ E ν (q − q  ) + E 0 (q  )

(5.26)

which will contradict (c). To this end, we argue like in the proof of [12, Thm. 3]: consider a smooth radial function χ with support in B(0, 1) such that 0 ≤ χ ≤ 1 and χ ≡ 1

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É. Cancès, A. Deleurence, M. Lewin

in B(0, 1/2); define χ R (x) := χ (x/R). Then let η R := following localization operators: 0 0 0 ⊥ 0 ⊥ Y R := γper η R γper + (γper ) η R (γper ) ,

Lemma 10. We have for all 3 < p ≤ ∞,    0  ] = O(R −1+3/ p ), [η R , γper

!

1 − χ R2 . Let us introduce the

XR =

!

1 − Y R2 .

||Y R − η R ||Sp = O(R −1+3/ p ).

Sp

(5.27)

    Moreover  X 2R − χ R2 S = O R −1+3/ p . p

Proof. By (5.1) and (5.2)  1 0 =− (H 0 − z)−1 [, η R ] (Hper − z)−1 dz 4iπ C per      0 ] which yields [η R , γper ≤ C (1 − )−1 (∇η R )S ≤ C  ||∇η R || L p by the Kato p 0 , ηR ] [γper

Sp

Seiler-Simon inequality and following the proof of Lemma 5. Eventually we notice 0 0 , η ] and thus (5.27) is proved since γ 0 is bounded. The [γper Y R − η R = 1 − 2γper R per  last inequality is a consequence of ||Y R || ≤ 1, ||η R || ≤ 1.  Lemma 11. One has $  %   0 0 − |1/2 |Hper − |−1/2  = O(R −1 ),  Y R , |Hper

(5.28)

where we recall that  = ( +Z +  − Z +1 )/2 is the middle of the gap. Proof. We use the well-known integral representation of the square root [2] 0 |Hper − |1/2 =

1 π



0 − | |Hper

∞

0

s

0 + |Hper

ds √ . − | s

(5.29)

0 η γ 0 + (γ 0 )⊥ η (γ 0 )⊥ . For shortness, we only detail the Recall that Y R = γper R per R per per 0 )⊥ η (γ 0 )⊥ . Using that |H 0 − | commutes estimation of the term involving (γper R per per 0 )⊥ and that (γ 0 )⊥ is bounded, we see that it suffices to estimate with (γper per 0 ⊥ ) (γper



1 =− 2

∞

(

0



∞

0

−|−1/2

ηR ,

0 − | |Hper

s

0 + |Hper

ds 0 ⊥ 0 − |−1/2 √ (γper ) |Hper − | s

0 )⊥ (γper 0 − | s + |Hper 0 )⊥ (γper 0 − | s + |Hper

)

0 [η R , −] |Hper

√ sds.

(5.30)

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159

   0 − |−1/2  = O(R −1 ), Then [η R , −] = (η R ) + 2(∇η R ) · ∇, hence [η R , −] |Hper  0 − |−1/2 is a bounded operator by Lemma 1. Then we where we have used that ∇|Hper 0 use that |Hper − | ≥  for some  > 0 to estimate the right hand side of (5.30) in the operator norm by    ∞ √sds  0 −1/2  , −] |H − | = O(R −1 ). [η R  × per 2 ( + s) 0

  Notice now that Y R Q n Y R ∈ K for all R ≥ 1 (the same is true for X R Q n X R but we shall actually not need it). To see this, notice for instance that $ % 0 −1/2 0 (1 + |∇|)(Y R Q n Y R ) = −(1 + |∇|)|Hper − | Y R , |Hper − |1/2 ×  0 −1/2 0 1/2 0 1/2 (5.31) ×|Hper − | |Hper − | Q n Y R − Y R |Hper − | Q n Y R % $ 0 − |1/2 Q ∈ S and Y , |H 0 − |1/2 |H 0 − which belongs to S2 since |Hper n 2 R per per |−1/2 is bounded by Lemma 11. The proof that (1 + |∇|)(Y R Q n Y R )+ + (1 + |∇|) and (1 + |∇|)(Y R Q n Y R )−− (1 + |∇|) are trace-class is similar. Eventually, the proof that 0 ≤ Y QY ≤ 1 − γ 0 is easy, using the equivalent condition (3.4) and the fact −γper R R per 0 commutes with Y . that γper R We are now able to prove (5.26) as announced. We write, following [11], 0 0 1/2 E ν (Q n ) = Tr(X R |Hper − |1/2 (Q +n + − Q −− X R) n )|Hper − | 0 0 1/2 +Tr(Y R |Hper − |1/2 (Q +n + − Q −− Y R ) + Tr 0 (X R Q n X R ) n )|Hper − | 1 1 (5.32) +Tr 0 (Y R Q n Y R ) + D(ρ Q n − ν, ρ Q n − ν) − D(ν, ν), 2 2 0 , X ] = [γ 0 , Y ] = 0 to infer Tr (Q ) = Tr (X Q X )+ where we have used that [γper R R 0 n 0 R n R per Tr 0 (Y R Q n Y R ). Then, by Lemma 11 and using the fact that (Q n )n≥1 is a bounded sequence in Q, we deduce that 0 0 1/2 Tr(Y R |Hper − |1/2 (Q +n + − Q −− YR ) n )|Hper − | 0 0 1/2 ≥ Tr(|Hper − |1/2 Y R (Q +n + − Q −− ) − C/R n )Y R |Hper − |

(5.33)

for some constant C > 0. Arguing similarly for the other terms, we obtain 0 0 1/2 E ν (Q n ) ≥ E˜ 0 (Y R Q n Y R ) + Tr(χ R |Hper − |1/2 (Q +n + − Q −− χR ) n )|Hper − | 1 +Tr(χ R (Q +n + + Q −− n )χ R ) + D(ρ Q n − ν, ρ Q n − ν) 2 1  − D(ν, ν) − C /R (5.34) 2

for some constant C  , where 0 0 E˜ 0 (Q) := Tr(|Hper − |1/2 (Q + + − Q −− )|Hper − |1/2 ) + Tr 0 (Q).

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Recall (Proposition 3) E 0 (q) = inf{E˜ 0 (Q),

Q ∈ K, Tr 0 (Q) = q}.

Thus, using q = Tr 0 (Q n ) = Tr 0 (Y R Q n Y R ) + Tr 0 (X R Q n X R ), and the fact that q → E 0 (q) is Lipschitz by Proposition 3, (5.34) yields 0 0 1/2 E ν (Q n ) ≥ Tr(χ R |Hper − |1/2 (Q +n + − Q −− χR ) n )|Hper − |   ++ −− 0 ++ −− +Tr(χ R (Q n + Q n )χ R ) + E q − Tr(χ R (Q n + Q n )χ R ) 1 1 + D(ρ Q n − ν, ρ Q n − ν) − D(ν, ν) − C  /R. 2 2

(5.35)

Let us now pass to the limit n → ∞. First we notice 0 0 1/2 lim inf Tr(χ R |Hper − |1/2 (Q +n + − Q −− χR ) n )|Hper − | n→∞

0 0 ≥ Tr(χ R |Hper − |1/2 (Q + + − Q −− )|Hper − |1/2 χ R ),

lim inf D(ρ Q n − ν, ρ Q n − ν) ≥ D(ρ Q − ν, ρ Q − ν) n→∞

(5.36)

(5.37)

by Fatou’s Lemma and the weak convergence ρ Q n  ρ Q in C. Then −− χR ) lim Tr(χ R Q +n + χ R ) = Tr(χ R Q + + χ R ), lim Tr(χ R Q −− n χ R ) = Tr(χ R Q

n→∞

n→∞

which is obtained by writing for instance Tr(χ R Q +n + χ R ) = Tr(χ R (1 + |∇|)−1 (1 + |∇|)Q +n + (1 + |∇|)(1 + |∇|)−1 χ R ) and using that χ R (1 + |∇|)−1 is compact (it belongs to S p for p > 3 by the Kato-SeilerSimon inequality) and that (1 + |∇|)Q +n + (1 + |∇|)  (1 + |∇|)Q + + (1 + |∇|) for the weak-∗ topology of S1 . Thus, 0 0 E ν (q) = lim inf E ν (Q n ) ≥ Tr(χ R |Hper − |1/2 (Q + + − Q −− )|Hper − |1/2 χ R ) n→∞   +Tr(χ R (Q + + + Q −− )χ R ) + E 0 q − Tr(χ R (Q + + + Q −− )χ R ) 1 1 (5.38) + D(ρ Q − ν, ρ Q − ν) − D(ν, ν) − C  /R. 2 2

Passing now to the limit as R → ∞, we obtain (5.26). This contradicts (3) and shows that (b)⇔(c) in Theorem 3.

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161

Step 4. Characterization of the q’s such that (c) holds. Because q → E ν (q) is a convex function, it is classical that the set I = {q ∈ R, (c) holds} is a closed interval of R, see ¯ for any minimizer Q¯ of E ν obtained e.g. [31]. It is non empty since it contains Tr 0 ( Q) F − + in Theorem 2, for any  F in the gap ( Z ,  Z +1 ). Additionally, q → E ν (q) is linear on the connected components of R \ I and I is the largest interval on which this function is strictly convex. Let us now state and prove Lemma 12. Let Z ∈ N\{0}, ν ∈ L 1 (R3 )∩ L 2 (R3 ), and assume that (A1) holds. Assume that Q 1 and Q 2 are respectively two minimizers of E ν (q1 ) and E ν (q2 ) with q1 = q2 . Then ρ Q 1 = ρ Q 2 and therefore E ν (tq1 + (1 − t)q2 ) ≤ E ν (t Q 1 + (1 − t)Q 2 ) < t E ν (q1 ) + (1 − t)E ν (q2 ). Proof. Assume by contradiction that ρ Q 1 = ρ Q 2 . It is classical that the operators Q 1 and Q 2 satisfy the self-consistent equations 0 Q 1 = χ(−∞,1 ) (H Q 1 ) − γper + δ1 ,

0 Q 2 = χ(−∞,2 ) (H Q 2 ) − γper + δ2 ,

where 0 ≤ δk ≤ 1 and Ran(δk ) ⊆ ker(H Q k − k ) for k = 1, 2. Necessarily 1 and 2 are in [ +Z ,  − Z +1 ] otherwise Q 1 and Q 2 would not be compact, which is not possible since every operator of K is compact. Since H Q 1 = H Q 2 has only a point spectrum in the gap, − + we deduce that if k ∈ ( +Z ,  − Z +1 ), then necessarily δk is finite rank. If k ∈ { Z ,  Z +1 }, then it can easily be proved that at least δk ∈ S1 . Hence Q 1 and Q 2 differ by a trace-class operator: Q 2 = Q 1 +δ, Tr|δ| < ∞. Now 0 = q2 −q1 = Tr(δ) = ρδ which contradicts our assumption that ρδ = ρ Q 1 − ρ Q 2 = 0. The rest follows from the strict convexity of ρ → D(ρ, ρ).   / I , the interval on which (c) holds. Corollary 4. There is no minimizer for E ν (q) if q ∈ Thus (a) implies (c). / I , for instance q1 > max I := Proof. Assume that there is a minimizer Q 1 for some q1 ∈ q2 . Applying Lemma 12 to q1 and q2 shows that E ν (·) cannot be linear on [q2 , q1 ] which contradicts the definition of I .   5.9. Proof of Theorem 4: Thermodynamic limit of the supercell model for a perfect crystal. Step 1. Let us first prove that lim sup L→+∞

1 0 0 I ≤ Iper . Starting from the Bloch decomL 3 sc,L

position 0 γper =

1 (2π )3

 ∗

0 (γper )ξ dξ

0 it is possible to construct a convenient test function * γsc,L for (4.2) as follows: of γper

γsc,L (x, y) = *

1 (2π )3  ×


ξ∈

2π L

eiξ x

Z3 ∩ ∗

ξ +[− 2πL η , 2π(1−η) )3 L

e

−iξ  x

0 (γper )ξ  (x,

y) e

iξ  y

dξ





e−iξ y ,

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with η = 0 if L is even and η = 1/2 if L is odd. It is indeed easy to check that * γsc,L is in Psc,L and satisfies ρ* 0 . In particular, γsc,L = ργper 

 L

ρ* γsc,L =

L

µper = Z L 3 ,

3 and, since both ργper 0 and µper are Z -periodic,

   3 DG L ρ* 0 − µper , ργ 0 − µper . γsc,L − µper , ρ* γsc,L − µper = L DG 1 ργper per Besides,      1 1 1 1 0  − * −  γper dξ γsc,L = Tr 2 Tr 2 ξ L 3 L per ( L ) 2 (2π )3 ∗ L ξ ( ) 2 
1 0 − |ξ − ξ  |2 Tr L 2 ( ) ((γper )ξ  ) dξ  3 2π η 2π(1−η) 3 ξ 2(2π ) ξ +[− L , L ) 2π ξ∈

−

i (2π )3

L

Z3 ∩ ∗






ξ∈

2π L

Z3 ∩ ∗

ξ +[− 2πL η , 2π(1−η) )3 L

0 (ξ − ξ  ) · Tr L 2 ( ) (−i∇(γper )ξ  ) dξ  . ξ

 0 ) ) dξ and from the inequaIt follows from the boundedness of ∗ Tr L 2 ( ) ((1 − )(γper ξ ξ lity | − 2i∇| ≤ (1 − ) that the last two terms of the above expression go to zero, hence that      1 1 1 1 0 − = − dξ. * γ (γ lim Tr Tr ) 2 2 sc,L ξ L per ( L ) per L→+∞ L 3 2 (2π )3 ∗ L ξ ( ) 2 0 (* 0 (γ 0 ) = I 0 and consequently Therefore lim L→+∞ L −3 Esc,L γsc,L ) = Eper per per

lim sup L→+∞

1 0 1 0 0 Isc,L ≤ lim Esc,L (* γsc,L ) = Iper . 3 L→+∞ L 3 L

1 0 0 Isc,L ≥ Iper . First, the existence of a miniL→+∞ L 3 0 mizer γ L for (4.2) and the uniqueness of the corresponding density ρsc,L follows from 0 the same arguments as in the proof of [5, Thm 2.1]. Note that, by symmetry, ρsc,L is 0 3 Z -periodic. We now define the operator γsc,L as

Step 2. Let us now establish that lim inf

0 γsc,L =

1 L3




τk∗ γ L τk .

k∈Z3 ∩ L

By simple periodicity arguments, it is clear that τk∗ γ L τk is also a minimizer for (4.2) for 0 . Besides, γ 0 all k ∈ Z3 . By convexity, so is γsc,L sc,L commutes with the translations τk 0 3 for all k ∈ Z so that its kernel γsc,L (x, y) satisfies 0 0 ∀(x, y, z) ∈ R3 × R3 × Z3 , γsc,L (x + z, y + z) = γsc,L (x, y).

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163

0 The optimality conditions imply that γsc,L can be expanded as follows:

1 L3

0 (x, y) = γsc,L







L iξ ·x L L (y)e−iξ ·y , n k,ξ e vk,ξ (x) vk,ξ

3 ∗ k≥1 ξ ∈ 2π L Z ∩

L 3 ∗ 2 where for any ξ ∈ 2π L Z ∩ , (vk,ξ )k≥1 is a Hilbert basis of L per ( ) consisting of eigenfunctions of the self-adjoint operator on L 2per ( ) defined by

1 1 0 − µper ) G 1 + |ξ |2 −  − iξ · ∇ + (ρsc,L 2 2 L are in associated with eigenvalues λ1L (ξ ) ≤ λ2L (ξ ) ≤ · · · . The occupation numbers n k,ξ the range [0, 1] and such that

1 L n k,ξ = Z. 3 L 2π ξ∈

L

Z3 ∩ ∗ k≥1

L = 1 whenever λ L (ξ ) <  L and Lastly, there exists a Fermi level  FL ∈ R such that n k,ξ k F L L L n k,ξ = 0 whenever λk (ξ ) >  F . One has

1 0 1 1 0 0 I = 3 Esc,L (γsc,L )= 3 L 3 sc,L L L


ξ ∈ 2π L

L
n k,ξ

Z3 ∩ ∗ k≥1

2

L 2 (−i∇ + ξ )vk,ξ L 2

per ( )

1 DG L (ργ 0 − µper , ργ 0 − µper ). (5.39) sc,L sc,L 2 L3 We now introduce 
1 0 L L L (x, y) = n k,β eiξ ·x vk,β (x) vk,β (y)e−iξ ·y dξ, γsc,L * L (ξ ) L (ξ ) L (ξ ) (2π )3 ∗ +

k≥1

2π η 2π(1−η) 3 3 ∗ ) . where β L (ξ ) is the unique element of 2π L Z ∩ such that ξ ∈ β L (ξ )+[− L , L 0 0 Z Z It is easy to check that γ˜sc,L ∈ Pper . Thus γ˜sc,L ∈ Pper can be used as a test function for (2.4). Therefore 0 0 0 Eper (γ˜sc,L ) ≥ Iper .

As ργ˜ 0

sc,L

= ργ 0

is Z3 -periodic, one has

sc,L

DG 1 (ργ˜ 0

− µper , ργ˜ 0

sc,L

(5.40)

sc,L

− µper ) =

1 DG L (ργ 0 − µper , ργ 0 − µper ). sc,L sc,L L3

(5.41)

Besides, 1 (2π )3



 ∗

Tr L 2 ( ) ξ

1 0  −  γsc,L ξ 2



1 dξ = (2π )3



L
n k,β ∗ k≥1

L (ξ )

2

L ×(−i∇ + ξ )vk,β 2 L (ξ ) L 2

per ( )

=

1 L3


ξ ∈ 2π L

Z3 ∩ ∗


k≥1

L n k,ξ

2

L 2 (−i∇ + ξ )vk,ξ L 2

per ( )

+ RL

(5.42)

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É. Cancès, A. Deleurence, M. Lewin

with 1 RL = (2π )3



L
n k,β

∗ k≥1

L (ξ )

2

L (−i∇ + ξ )vk,β 2 L (ξ ) L 2

per ( )



L −(−i∇ + β L (ξ ))vk,β 2 L (ξ ) L 2

per ( )

.

0 0 . As + R L ≥ Iper Putting (5.39)–(5.42) together, we end up with L13 Isc,L + ,1/2   0 6Z Isc,L 2π 3Z 2π 2 + |R L | ≤ , L3 L 2 L

we finally obtain lim inf

L→+∞

1 0 0 I ≥ Iper . L 3 sc,L

0 ) Step 3. Convergence of the density. A byproduct of Steps 1 and 2 is that (γ˜sc,L L∈N\{0} 0 . The convergence results on the density ρ is a minimizing sequence for Iper = ργ˜ 0 0 γsc,L sc,L immediately follow from the proof of [5, Thm 2.1]. 0 Step 4. Convergence of the mean-field Hamiltonian and its spectrum. One has Hsc,L − 0 Hper =  L , where  L solves the Poisson equation − L = 4π(ργ 0 − ργper 0 ) on sc,L with periodic boundary conditions. As it follows from Step 3 that (ργ 0 − ργper 0 ) sc,L

2 ( ), hence in converges to zero in L 2per ( ), we obtain that  L converges to zero in Hper L ∞ (R3 ). Consequently,    0 0  (5.43)  ≤ || L || L ∞ → 0  Hsc,L − Hper

as L → ∞. This clearly implies, via the min-max principle, that     sup sup λnL (ξ ) − λn (ξ ) ≤  L  L ∞ −→ 0, n≥1 ξ ∈ ∗

L→∞

0 (resp. where (λnL (ξ ))n≥1, ξ ∈ ∗ (resp. (λn (ξ ))n≥1, ξ ∈ ∗ ) are the Bloch eigenvalues of Hsc,L 0 Hper ). 0 for large values of L. In the remainder of the proof, we Step 5. Uniqueness of γsc,L 0 has a gap. assume that (A1) holds, i.e. that Hper 0 considered as an operator on L 2per ( L ) is given by The spectrum of Hsc,L   0 σ L 2per ( L ) (Hsc,L )= λnL (ξ ). n∈N\{0} ξ ∈ 2π Z3 ∩ ∗ L

It follows from Step 4 that there exists some L 0 ∈ N \ {0} such that for all L ≥ L 0 , the 0 lowest Z L 3 eigenvalues of Hsc,L (including multiplicities) are   λnL (ξ ). 1≤n≤Z ξ ∈ 2π Z3 ∩ ∗ L

Local Defects in Periodic Crystals

165

and there is a gap between the (Z L 3 )th and the (Z L 3 +1)st eigenvalues. As a consequence, 0 γsc,L is uniquely defined: it is the spectral projector associated with the lowest Z L 3 0 , considered as an operator on L 2 ( ). eigenvalues of Hsc,L L per 0 0 Step 6. Let  F ∈ ( +Z ,  − Z +1 ). For L large enough, γsc,L = χ(−∞, F ] (Hsc,L ) = 0 0 χ(−∞,0] (Hsc,L −  F ) as operators acting on L 2per ( L ). This means that γsc,L satis0 fies the Euler-Lagrange equation associated with Isc,L , F , see (4.5). As the functional 0 (γ ) −  Tr 0 γ → Esc,L (γ ) is convex on P , γ F L 2per ( L ) sc,L sc,L is a minimizer of this functional. Its uniqueness follows as usual from the uniqueness of the minimizing density 0 and from the fact that 0 is not in the spectrum of Hsc,L − F .

5.10. Proof of Theorem 5: Thermodynamic limit of the supercell model for a crystal with local defects. We follow the method of [13]. As in the previous section, we denote 0 by γsc,L the minimizer of (4.2), which is unique for L large enough and is also the unique minimizer of (4.5). Let K L be the set of operators Q L on L 2per ( L ) such that 0 + Q L ∈ Psc,L . In fact γsc,L  K L = Q L ∈ S1 (L 2per ( L )) | Q ∗L = Q L , |∇|Q L |∇| ∈ S1 (L 2per ( L )),  0 0 −γsc,L . ≤ Q L ≤ 1 − γsc,L ν ν We introduce Esc,L , F := Esc,L −  F Tr L 2per ( L ) . Let Q L ∈ K L , one has ν 0 0 0 0 Esc,L , F (γsc,L + Q L ) − Esc,L , F (γsc,L ) = Tr L 2per ( L ) (Hsc,L Q L ) − DG L (ρ Q L , ν L )

1 1 + DG L (ρ Q L , ρ Q L ) −  F Tr L 2per ( L ) (Q L ) − DG L (ν L , ργ 0 − µper ) + DG L (ν L , ν L ). sc,L 2 2 0 is considered as an operator on L 2 ( ). Using Note that in the above expression, Hsc,L L per Theorem 4, this equality can be rewritten, for L large enough, as ++,L ν 0 0 0 0 − Q −−,L )) Esc,L , F (γsc,L + Q L ) − Esc,L , F (γsc,L ) = Tr L 2per ( L ) (|Hsc,L −  F |(Q L L

1 + DG L (ρ Q L − ν L , ρ Q L − ν L ) − DG L (ν L , ργ 0 − µper ), sc,L 2 where we have set

(5.44)

0 0 0 0 Q ++,L = (1 − γsc,L )Q L (1 − γsc,L ) and Q −−,L = γsc,L Q L γsc,L . L L

It follows from (5.44) that

  ν 0 ν Isc,L , F − Isc,L , F = inf E sc,L (Q L ) −  F Tr L 2per ( L ) (Q L ), Q L ∈ K L −DG L (ν L , ργ 0

sc,L

− µper ) +

1 DG L (ν L , ν L ), 2

where 1 ν (Q L ) −  F Tr L 2per ( L ) (Q L ) := −DG L (ρ Q L , ν L ) + DG L (ρ Q L , ρ Q L ) E sc,L 2

 ++,L −−,L 0 1/2 0 +Tr L 2per ( L ) |Hsc,L −  F | (Q L − Q L )|Hsc,L −  F |1/2 .

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É. Cancès, A. Deleurence, M. Lewin

First, using ν being in L 1 (R3 ) ∩ L 2 (R3 ) and the convergence of  L = (ργ 0 − sc,L ∞ (see Step 4 of the proof of Theorem 4 in Sect. 5.9), we obtain ργper 0 ) G 1 to zero in L 1 −DG L (ν L , ργ 0 − µper ) + DG L (ν L , ν L ) −→ sc,L 2   1  D(ν, ν). − ν (ργper 0 − µper ) G 1 + 2 R3 Our goal is to prove that lim E νF ,L = E νF ,

(5.45)

  ν E νF ,L = inf E sc,L (Q L ) −  F Tr L 2per ( L ) (Q L ), Q L ∈ K L .

(5.46)

L→∞

where

Step 1. Preliminaries. In the proof of (5.45), we shall need several times to compare states living in L 2per ( L ) with states living in L 2 (R3 ). To this end, we introduce the map i L : L 2 (R3 ) → L 2per ( L )
ϕ → (1 L ϕ)(· − Lz). z∈Z3

Notice that (i L )∗ : L 2per ( L ) → L 2 (R3 ) is the operator which to any periodic function ϕ ∈ L 2per ( L ) associates the function 1 L ϕ ∈ L 2 (R3 ). Remark that i L (i L )∗ = I d L 2per ( L ) whereas (i L )∗ i L = 1 L . Hence i L defines an isometry from L 2 ( L ) to

L 2per ( L ). The equality (i L )∗ i L ϕ = ϕ is only true when ϕ ∈ L 2 (R3 ) has its support in  L . When ϕ ∈ H 1 (R3 ) satisfies Supp(ϕ) ⊂  L , then one also has ∂xi i L (ϕ) = i L (∂xi ϕ). Notice in addition that if A ∈ S1 (L 2per ( L )), then (i L )∗ Ai L ∈ S1 (L 2 ( L )) ⊆ S1 (L 2 (R3 )) and   Tr L 2per ( L ) (A) = Tr L 2 (R3 ) (i L )∗ Ai L .

Similarly if A ∈ S p (L 2per ( L )),

  ||A||Sp (L 2per ( L )) = (i L )∗ Ai L S

p (L

2 (R3 ))

.

(5.47)

Finally, we shall use that for any Z3 -periodic bounded function f , i L f = f i L , where we use the same notation f to denote the multiplication operator by the function f acting 0 either on L 2per ( L ) or on L 2 (R3 ). Similarly, the operator Hsc,L = −/2 + (ργ 0 − sc,L

µper ) G 1 can be seen as acting on L 2per ( L ) or on L 2 (R3 ) and we use the same notation in the two cases. Then we have for any ϕ ∈ C0∞ (R3 ) satisfying Supp(ϕ) ⊆  L , 0 0 Hsc,L i L ϕ = i L Hsc,L ϕ.

(5.48)

Notice that one can also define −i∇ on L 2 (R3 ) or on L 2per ( L ) and we shall adopt the same notation for these two operators. We gather some useful limits in the following

Local Defects in Periodic Crystals

167

Lemma 13. Let be ψ ∈ L 2 (R3 ) and ϕ ∈ C0∞ (R3 ). Then we have as L → ∞, (1) (2) (3) (4) (5)

0 i ψ → γ 0 ψ in L 2 (R3 ); (i L )∗ γsc,L L per 0 γ 0 i ϕ → H 0 γ 0 ϕ in L 2 (R3 ); (i L )∗ Hsc,L per per sc,L L 0 i ϕ → γ 0 ϕ in L 2 (R3 ); (i L )∗ γsc,L L per (i L )∗ (1 + |∇|)i L ϕ → (1 + |∇|)ϕ in L 2 (R3 ); 0 0 −  |1/2 ϕ in L 2 (R3 ) for any fixed  in the gap (i L )∗ |Hsc,L −  F |1/2 i L ϕ → |Hper F F ). ( +Z ,  − Z +1

0 i being uniformly bounded with respect to L, it suffices Proof. The operator (i L )∗ γsc,L L to prove the first assertion for a dense subset of L 2 (R3 ) like C0∞ (R3 ). Hence we may assume that ψ = ϕ ∈ C0∞ (R3 ). 0 . We are going to prove that Let K be a compact set in the resolvent set of Hper 0 0 lim (i L )∗ (Hsc,L − z)−1 i L ϕ → L→∞ (Hper − z)−1 ϕ

(5.49)

L→∞

in L 2 (R3 ), uniformly for z ∈ K . To this end, we first notice that by Theorem 4, K is 0 contained in the resolvent set of Hsc,L for L large enough and thus      0   0 0 0  − z)−1  2 ≤ C(K ) Hsc,L − Hper →0  2 (Hsc,L − z)−1 − (Hper B(L per ( L ))

B(L per ( L ))

0 0 (seen as an opeby (5.43). Hence it suffices to prove (5.49) with Hsc,L replaced by Hper 2 rator acting on L per ( L )). Then we use its Bloch decomposition (detailed in Appendix, see (A.2)) and compute, assuming L large enough for Supp(ϕ) ⊂  L , 
L −3 
∗ 0 −1 (i L ) (Hper − z) i L ϕ(x) = en (k, ·)ϕ 1 L (x)en (k, x), λn (k) − z R3 2π k∈

L

Z3 ∩ ∗ n≥1

 2   0 − z)−1 i L ϕ  2 (i L )∗ (Hper

L (R 3 )




=




3 ∗ n≥1 k∈ 2π L Z ∩

 2   L −3   . e (k, ·)ϕ n   2 |λn (k) − z| R3

0 − z)−1 i ϕ  (H 0 − z)−1 ϕ weakly in L 2 (R3 ) It is then easy to see that (i L )∗ (Hper L per (one cantake the scalar productagainst a function ψ ∈ C0∞ (R3 ) to identify the limit)   0  0 − z)−1 i ϕ  −1  and that (i L )∗ (Hper L  2 3 → (Hper − z) ϕ  2 3 , yielding the strong L (R )

L (R )

convergence in L 2 (R3 ). For the proof of (1), it then suffices to choose a curve C around the first Z bands of 0 and use the above convergence of the resolvent in the Cauchy formula. Assertion Hper (2) is an easy consequence of (1) and (5.48). Indeed, by Theorem 4, we know that   0 0 i is bounded, this implies that 0  lim L→∞  Hsc,L − Hper  2 3 = 0. Since (i L )∗ γsc,L L B(L (R ))

for L large enough such that Supp(ϕ) ⊆  L ,         0 0 0 0 0 0 (Hsc,L i L − i L Hper )ϕ  2 3 = (i L )∗ γsc,L i L (Hsc,L − Hper )ϕ  (i L )∗ γsc,L L (R )

L 2 (R 3 )

→ 0,

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É. Cancès, A. Deleurence, M. Lewin

0 ϕ ∈ L 2 (R3 ). Hence (1) implies that where we have used (5.48). Then we notice that Hper    0 i − γ 0 )H 0 ϕ  lim L→∞ ((i L )∗ γsc,L L per per  2 3 = 0. The argument is exactly the same for L (R )

the third assertion (3). Assertion (5) can be proved in the same way, using (5.48) and the integral representation of the square root (5.29). Finally, it remains to prove that (4) is true, which is done by computing explicitly, for L large enough such that Supp(ϕ) ⊆  L ,
(2π )3/2 (1 + |k|) ϕ (k)eik·x 1 L (x), L3 2π

(i L )∗ (1 + |∇|)i L ϕ =

k∈

  (i L )∗ (1 + |∇|)i L ϕ 2 2

L (R 3 )

=

L

Z3


(2π )3 |(1 + |k|) ϕ (k)|2 → L→∞ ||(1 + |∇|)ϕ||2L 2 (R3 ) . 3 L 2π

k∈

L

Z3

The strong convergence is obtained as above.

 

Lemma 14. Let V ∈ C0∞ (R3 ). We have as L → ∞, (i L )∗ (1 − )−1 i L (V )i L → (1 − )−1 V,

(5.50)

(i L )∗ (1 + |∇|)−1 i L (V )(1 + |∇|)−1 i L → (1 + |∇|)−1 V (1 + |∇|)−1

(5.51)

strongly in S2 (L 2 (R3 )). Proof. For L large enough, we have     (i L )∗ (1 − )−1 i L (V )i L  2

S2 (L (R3 ))

    = (1 − )−1 i L (V ) ⎛ ≤

S2 (L 2per ( L ))

⎞1/2

||V || L 2 (R3 ) ⎜
⎟ ⎝ 2 )2 ⎠ (2π )3/2 (1 + |k| 2π (2π/L)3

k∈

L

which shows that (i L )∗ (1 − )−1 i L (V )i L is bounded in S2 (L 2 (R3 )) since 
(2π/L)3 = |g( p)|2 dp, g( p) = (1 + | p|2 )−1 . lim 2 )2 3 L→∞ (1 + |k| R 2π k∈

L

,

Z3

(5.52)

Z3

Arguing as in the proof of the fourth assertion of Lemma 13, we can prove that (5.50) holds in the strong sense, hence the convergence holds weakly in S2 (L 2 (R3 )) towards (1 − )−1 V . Now   ||V || L 2 (R3 ) ||g|| L 2 (R3 )   = lim (i L )∗ (1 − )−1 i L (V )i L  L→∞ S2 (L 2 (R3 )) (2π )3/2      = (1 − )−1 V  2 3 S2 (L (R ))

and the limit holds strongly in S2 (L 2 (R3 )).

Local Defects in Periodic Crystals

169

The argument is the same for (5.51), noticing that 2    (i L )∗ (1 + |∇|)−1 i L (V )(1 + |∇|)−1 i L  S2 (L 2 (R3 ))

 = Tr L 2per ( L ) (1 + |∇|)−2 i L (V )(1 + |∇|)−2 i L (V )  −3/2 = (2π ) |h L (x − y)|2 V (x)V (y)d x d y ( L )2

→ L→∞

 2   (1 + |∇|)−1 V (1 + |∇|)−1 

S2 (L 2 (R3 ))

,

where we have used that


h L (x) :=

3 k∈ 2π L Z

(2π )3/2 eik·x L 3 (1 + |k|)2

converges to the Fourier inverse F −1 (h) of h( p) = (1 + | p|)−2 , strongly in L 2loc (R3 ).   Step 2. Upper bound. We prove here that lim sup L→∞ E νF ,L ≤ E νF . Let  > 0. Using Lemma 8, Proposition 4, Corollary 3, and the notation therein, one can find a finite rank operator Q ∈ Kr such that E νF ≤ E ν (Q) −  F Tr 0 (Q) ≤ E νF + ,

(5.53)

of the form Q=

−1


|vm vm | −

m=−M

+

k
i=0

−1


|u n u n | +

n=−N

k
i=0

 λi2  |vi vi | − |u i u i | 1 + λi2


  λi   |u v | + |v u | + δ with δ = n j |w j w j |. (5.54) i i i i 1 + λi2 j=1 J

η

η

η

Let 0 < η << 1. It is possible to choose a family of orthonormal functions u n , vm , w j in C0∞ (R3 ) such that η

η u ηn − u n  H 2 ≤ η, vm − vm  H 2 ≤ η, w j − w j  H 2 ≤ η

(5.55)

for all n = −N . . . k, m = −M . . . k and j = 1 . . . J . Let us define the Gram matrices ' ' & & η η η η 0 η η 0 (S+ )i, j := (1 − γper (S− )i, j := γper ui , u j , )vi , v j η

η

which, by (5.55) satisfy S+ = I d M+k+1 + o(1)η→0 and S− = I d N +k+1 + o(1)η→0 . We 0 u η , (1 − γ 0 )v η } and define also introduce the orthogonal projector η on Span{γper n per m ' & η η η η (Sw )i, j := (1 − η )wi , w j . Clearly Sw = I d J + o(1)η→0 . Now we introduce a new orthonormal system in L 2per ( L ), η

u i,L :=

k
n=−N

−1/2

η

0 (S−,L )i,n γsc,L i L u ηn , vi,L :=

k
m=−M

−1/2

0 η (S+,L )i,m (1 − γsc,L )i L vm ,

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' & η η 0 (S−,L )i, j = γsc,L i L ui , i L u j

L 2per ( L )

' & η η 0 , (S+,L )i, j = (1 − γsc,L )i L vi , i L v j

L 2per ( L )

.

η

Notice that by the first assertion of Lemma 13, lim L→∞ S±,L = S± . Finally, we introη η duce the projector  L on Span(u n,L , vm,L ) and define η

w j,L :=

J ' &
−1/2 η η η (Sw,L ) j, (1 −  L )i L w , (Sw,L )i, j = (1 −  L )i L wi , i L w j . =1

We now define a state in K L by η

QL =

−1


η

η

|vm,L vm,L | −

m=−M

+

k
i=0

−1


η

η

|u n,L u n,L | +

n=−N

k
i=0

λi2  η η η η  |vi,L vi,L | − |u i,L u i,L | 2 1 + λi


λi  η η η η  η η |u i,L vi,L | + |vi,L u i,L | + n j |w j,L w j,L |. 2 1 + λi j=1 J

By Lemma 13, we have η

η

η

η

η (i L )∗ u n,L → L→∞ u˜ ηn , (i L )∗ vm,L → L→∞ v˜m and (i L )∗ w j,L → L→∞ w˜ j

(5.56)

in L 2 (R3 ) ∩ L ∞ (R3 ), where the limits are defined by η

u˜ i :=

k


η −1/2

η

0 η (S− )i,n γper u n , v˜i :=

n=−N

k


η −1/2

0 η (S+ )i,m, (1 − γper )vm ,

m=−M

η w˜ j

:=

J
=1

−1/2

η

(Swη ) j, (1 − η )w .

By Lemma 13, we know that for any fixed ϕ, ψ ∈ C0∞ (R3 ), & ' & ' 0 0 0 0 0 0 lim i L∗ (Hsc,L −  F )γsc,L i L ϕ, (i L )∗ γsc,L i L ψ = (Hper −  F )γper ϕ, γper ψ .

L→∞

η

Hence, inserting the definition of Q L in the kinetic energy and using the convergence of the Gram matrices, we obtain



 η 0 0 lim Tr L 2per ( L ) (Hsc,L −  F )Q L = Tr L 2 (R3 ) (Hper −  F ) Q˜ η , L→∞

η η η η where Q˜ η is defined similarly as Q L but with the functions (u˜ n , v˜m , w˜ j ) instead of η η η (u n,L , vm,L , w j,L ). Let us now prove that

lim DG L (ρ Q η , ρ Q η ) = D(ρ Q˜ η , ρ Q˜ η ).

L→∞

L

L

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The convergence (5.56) implies that 1 L ρ Q η converges to ρ Q˜ η in particular in L 1 (R3 ) ∩ L

L 2 (R3 ). Notice the definition of DG L (·, ·) implies that

DG L (ρ, ρ) ≤ C ||ρ||2L 1

∀ρ ∈ L 1per ( L ) ∩ L 2per ( L ),

per ( L

+ ||ρ||2L 2 )



per ( L )

(5.57)

for a constant C independent of L. Let us now write ρ Q η = ρ1,L + ρ2,L , where ρ1,L is L the periodic function which equals 1 B(0,L/4) ρ Q η on  L . The convergence of 1 L ρ Q η L

L

towards ρ Q˜ η in L 1 (R3 ) ∩ L 2 (R3 ) and (5.57) give that   lim ρ2,L  L 1

per ( L )

L→∞

  = lim ρ2,L  L 2

per ( L )

L→∞

= lim DG L (ρ2,L , ρ2,L ) = 0. L→∞

Hence, it remains to show that lim DG L (ρ1,L , ρ1,L ) = D(ρ Q˜ η , ρ Q˜ η ).

L→∞

(5.58)

To this end we use the estimate [22]    1   = O(L −1 ), sup G L (x) − |x|  x∈ L to obtain  DG L (ρ1,L , ρ1,L ) =

( L )2

G L (x − y)ρ1,L (x)ρ1,L (y)d x d y

= D(1 L ρ1,L , 1 L ρ1,L ) + O(L −1 ),   where we have used that ρ1,L  L 1

per ( L )

is uniformly bounded and that x − y ∈  L

for any x, y ∈ B(0, L/4), the support of ρ1,L . The convergence of 1 L ρ1,L towards ρ Q˜ η in L 1 (R3 ) ∩ L 2 (R3 ) then proves (5.58). Using the same argument for the term DG L (ρ Q η , ν L ) we obtain L



  1 1

lim −DG L (ρ Q η , ν L ) + DG L ρ Q η , ρ Q η = −D(ρ Q˜ η , ν) + D ρ Q˜ η , ρ Q˜ η . L L L L→∞ 2 2 Finally η η ν lim E sc,L (Q L ) −  F Tr L 2per ( L ) (Q L ) = E ν ( Q˜ η ) −  F Tr( Q˜ η ).

L→∞

(5.59)

Passing to the limit as η → 0 using (5.55) and the convergence of the Gram matrices η η S± and Sw , we eventually obtain lim sup E νF ,L ≤ E ν (Q) −  F Tr(Q) ≤ E νF + . L→∞

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É. Cancès, A. Deleurence, M. Lewin

Step 3. Lower bound. We end the proof by showing that lim inf L→∞ E νF ,L ≥ E νF . As  L is bounded for any fixed L, the existence of a minimizer Q L of   ν (Q L ) −  F Tr L 2per ( L ) (Q L ), Q L ∈ K L inf E sc,L 0 , considered as an operator on is straightforward. In addition, the spectrum of Hsc,L L 2per ( L ), being purely discrete and bounded below, Q L is finite rank. Using (4.4) and reasoning as in the proof of Lemma 1 (see Sect. 5.1.1), we prove that 0 there exists a constant c > 0 (independent of L) such that |Hsc,L −  F | ≥ c(1 − ) on 2 L per ( L ), for L large enough. The following uniform bounds follow from Step 2: 0 0 −  F |1/2 (Q ++,L − Q −−,L )|Hsc,L −  F |1/2 ) ≤ C, Tr L 2per ( L ) (|Hsc,L L L

(5.60)

− Q −−,L )(1 + |∇|)) ≤ C, Tr L 2per ( L ) ((1 + |∇|)(Q ++,L L L

(5.61)

Tr L 2per ( L ) ((1 + |∇|)Q 2L (1 + |∇|))

≤ C,

(5.62)

DG L (ρ Q L − ν L , ρ Q L − ν L ) ≤ C,

(5.63)

with C independent of L. Consider now the sequence of operators Q˜ L := (i L )∗ Q L i L acting on L 2 (R3 ). It is bounded in S2 (L 2 (R3 )) by (5.62) and since Tr L 2 (R3 ) ( Q˜ 2L ) = Tr L 2per ( L ) (Q 2L ) by (5.47). Hence Q˜ L weakly converges, up to extraction, to some Q ∈ S2 (L 2 (R3 )). Similarly, the Hilbert-Schmidt operator R L := (i L )∗ Q L (1 + |∇|)i L weakly converges up to extraction to some R in S2 (L 2 (R3 )). Let ϕ and ψ be in C0∞ (R3 ) and assume that Supp(ϕ) ∪ Supp(ψ) ⊂  L . Then & '   (i L )∗ Q L (1 + |∇|)i L ϕ, ψ L 2 (R3 ) = Q˜ L (i L )∗ (1 + |∇|)i L ϕ, ψ 2 3 L (R )

→ L→∞ Q(1 + |∇|)ϕ, ψ L 2 (R3 ) , where we have used that Q˜ L  Q weakly in S2 and that (i L )∗ (1+|∇|)i L ϕ → (1+|∇|)ϕ strongly in L 2 (R3 ) by the third assertion of Lemma 13. Hence Q(1 + |∇|) = R ∈ S2 (L 2 (R3 )). Similarly, define the operator SL := (i L )∗ Q −−,L i L which is nonpositive and yields L a bounded sequence in S1 (L 2 (R3 )) by (5.61). Up to extraction, we may assume that (SL ) converges for the weak-∗ topology to some S ∈ S1 (L 2 (R3 )). To identify the limit S, we compute as above for ϕ, ψ ∈ L 2 (R3 ), & ' 0 0 SL ϕ, ψ L 2 (R3 ) = (i L )∗ γsc,L Q L γsc,L i L ϕ, ψ 2 3 L (R ) & ' ∗ 0 ∗ 0 ˜ = Q L (i L ) γsc,L i L ϕ, (i L ) γsc,L i L ψ 2 3 . L (R )

Using now the first assertion of Lemma 13 we obtain lim L→∞ SL ϕ, ψ L 2 (R3 ) =   −− Q ϕ, ψ L 2 (R3 ) . Hence Q −− = S ∈ S1 . The same arguments allow to conclude that in fact, Q ∈ K. 0 0 Now, let TL := (i L )∗ |Hsc,L −  F |1/2 Q −−,L |Hsc,L −  F |1/2 i L which also defines a L 2 3 bounded sequence in S1 (L (R )). Up to extraction, we may assume that TL  T for

Local Defects in Periodic Crystals

173

the weak-∗ topology of S1 . Arguing as above and using Lemma 13, we deduce that 0 −  |1/2 Q −− |H 0 −  |1/2 . Now, Fatou’s Lemma yields T = |Hper F F per

 0 0 lim inf Tr L 2per ( L ) |Hsc,L −  F |1/2 (−Q −−,L )|Hsc,L −  F |1/2 L L→∞

 0 0 = lim inf Tr L 2 (R3 ) (−TL ) ≥ Tr L 2 (R3 ) |Hper −  F |1/2 (−Q −− )|Hper −  F |1/2 . L→∞

This proves that

 0 0 lim inf Tr L 2per ( L ) (Hsc,L −  F )Q L ≥ Tr 0 (Hper Q) −  F Tr 0 (Q). L→∞

We now study the term involving the density ρ Q L . First, following the proof of Proposition 1 and using the bounds (5.60)–(5.62), we can prove that there exists a   constant C such that for all L large enough ρ Q L  L 2 ( ) ≤ C. Hence, up to extraction, per

L

we have 1 L ρ Q L  ρ weakly in L 2 (R3 ) for some function ρ ∈ L 2 (R3 ). We now introduce an auxiliary function ρ L ∈ L 2 (R3 ) defined in Fourier space as follows:


ρ / L :=

3 k∈ 2π L Z \{0}

ck,L (ρ Q L ) c0,L (ρ Q L ) 1 Bk + 1 B0 , |Bk |1/2 |B0 |1/2

where for any k ∈ (2π/L)Z3 \ {0}, Bk := B k + 1/|k  |

k

k , 1  10L|k|1 10L 0, 10L .

 which is chosen to ensure

that ≤ 1/|k| for any ∈ Bk , and B0 := B Notice that ρ L is bounded in L 2 (R3 ) as we have by definition   
2 ρ L2 = |/ ρ L |2 = |ck,L (ρ Q L )|2 = ρQ . L R3

R3

3 k∈ 2π L Z

L

On the other hand (up to extraction) ρ L  ρ weakly in L 2 (R3 ), the same weak limit as 1 L ρ Q L . This is easily seen by considering a scalar product against a fixed function ϕ ∈ C0∞ (R3 ). Now by the choice of the balls Bk , we also have for L  1,  |/ ρ L (k  )|2  dk ≤ DG L (ρ Q L , ρ Q L ) ≤ C. D(ρ L , ρ L ) = 4π |k  |2 Hence, up to extraction we may assume that ρ L  ρ weakly in C. Using the regularity of  ν, we also deduce that   1 1 lim inf −DG L (ρ Q L , ν L ) + DG L (ρ Q L , ρ Q L ) ≥ −D(ρ, ν) + D(ρ, ρ). L→∞ 2 2 What remains to be proved is that ρ = ρ Q , where Q is the weak limit of (i L )∗ Q L i L obtained above. This will clearly show lim inf E νF ,L ≥ E ν (Q) −  F Tr 0 (Q) ≥ E νF L→∞

and end the proof of Theorem 5. We identify the limit of 1 L ρ Q L using its weak convergence to ρ in L 2 (R3 ).

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We start with ρ Q ++,L and write, fixing some V ∈ C0∞ (R3 ) and assuming L large L enough for Supp(V ) ⊂  L ,  ρ Q ++,L V = Tr L 2per ( L ) (Q ++,L i L (V )) = Tr L 2 (R3 ) (A L B L ) with L L

L

(1 + |∇|)i L , A L := (i L )∗ (1 + |∇|)Q ++,L L

B L := (i L )∗ (1 + |∇|)−1 i L (V )(1 + |∇|)−1 i L .

The sequence (A L ) is bounded in S1 (L 2 (R3 )), hence in S2 (L 2 (R3 )), by (5.61) and converges (up to extraction) towards (1 + |∇|)Q + + (1 + |∇|) weakly in S2 (L 2 (R3 )) (we proceed as above to identify the weak limit using the fourth assertion of Lemma 13). By Lemma 14, B L converges towards (1 + |∇|)−1 V (1 + |∇|)−1 strongly in S2 (L 2 (R3 )). We thus obtain   lim ρ Q ++,L V = Tr L 2 (R3 ) (Q + + V ) = ρ Q + + V. L→∞  L

R3

L

Likewise, it can be proved that the weak limit of ρ Q −−,L is ρ Q −− . L Let us now treat ρ Q +−,L (the other case ρ Q −+,L being similar). Following the proof of L L Proposition 1, we write 

 0 ρ Q +−,L V = Tr L 2per ( L ) Q +−,L [γ , i (V )] L sc,L L L L 

 1 0 −1 0 −1 . =− dzTr L 2per( L ) Q +−,L (H − z) (i (V ) − i (V ))(H − z) L L sc,L sc,L L 4iπ C We only detail the argument to pass to the limit in

 0 −1 0 −1 Tr L 2per ( L ) Q +−,L (H − z) i (V )(H − z) L sc,L sc,L L

 0 0 = Tr L 2per ( L ) (Hsc,L − z)−1 Q +−,L (Hsc,L − z)−1 (1 − )(1 − )−1 i L (V ) L

 = Tr L 2 (R3 ) C L (i L )∗ (1 − )−1 i L (V )i L 0 − z)−1 Q +−,L (H 0 − z)−1 (1 − )i . One has, up to extracwith C L := (i L )∗ (Hsc,L L L sc,L 0 − z)−1 Q +− (H 0 − z)−1 (1−) weakly in S (L 2 (R3 )). To see this, tion, C L  (Hper 2 per 2 3 one first remarks that C L is bounded in S2 (L (R )) and then identifies the weak limit by passing to the limit in C L ϕ, ψ for some fixed ϕ, ψ ∈ C0∞ (R3 ), using the uniform convergence of the resolvent for z ∈ C , as shown in the proof of Lemma 13. Then by Lemma 14 we know that (i L )∗ (1 − )−1 i L (V )i L converges towards (1 − )−1 V strongly in S2 (L 2 ( L )), hence we can pass to the limit in the above expression, uniformly in z ∈ C . We conclude that   lim 1 L ρ Q L V = ρQ V L→∞ R3

for any V ∈ C0∞ (R3 ), thus ρ = ρ Q .

R3

 

Local Defects in Periodic Crystals

175

Appendix A. Proof of Theorem 1 Our proof uses classical ideas for Hartree-Fock theories. See [23, Sect. 4] for a very 0 of I 0 (it is known to exist by [5, similar setting. Let us consider a minimizer γper per Thm 2.1]). First we note that the periodic potential Vper := (ργper 0 − µper ) G 1 is in L 2loc (R3 ). Thus Vper defines a -bounded operator on L 2 (R3 ) with relative bound 0 = −/2 + V zero (see [27, Thm XIII.96]) and therefore Hper per is self-adjoint on 2 3 1 3 0 is purely D(−) = H (R ) with form domain H (R ). Besides, the spectrum of Hper absolutely continuous, composed of bands as stated in [34, Thm 1-2] and [27, Thm XIII.100]. The Bloch eigenvalues λk (ξ ), k ≥ 1, ξ ∈ ∗ are known to be real analytic in each fixed direction and cannot be constant with respect to the variable ξ . Hence the function
  {ξ ∈ ∗ | λk (ξ ) ≤ κ} C : κ → k≥1 0 being bounded from below, we is continuous and nondecreasing on R. The operator Hper ∗ have C ≡ 0 on (−∞, inf λk ( )) and it is known [34, Lemma A-2] that limκ→∞ C(κ) = ∞. We can thus choose a Fermi level  F such that
  {ξ ∈ ∗ | λk (ξ ) ≤  F } . (A.1) Z = C( F ) = k≥1 0 + tγ for any γ ∈ P Z and t ∈ [0, 1], we deduce Considering a variation (1 − t)γper per 0 that γper minimizes the following linear functional: 

 1 Z 0 (H → Tr ) γ γ ∈ Pper 2 per ξ ξ dξ, (2π )3 ∗ L ξ ( ) 0 is the mean-field operator defined in (2.5). We subtract the chemical potential where Hper  F defined above and introduce the functional 

 1 0 (H dξ. γ ∈ Pper → F(γ ) := Tr −  ) γ 2 F ξ ξ per (2π )3 ∗ L ξ ( )    Z , then γ 0 also miniNotice that since (2π1 )3 ∗ Tr L 2 ( ) γξ dξ = Z for any γ ∈ Pper per ξ

Z . mizes F on Pper For any ξ ∈ ∗ , we can find orthonormal functions ek (ξ ) ∈ L 2ξ ( ) such that 0 (Hper )ξ =




λk (ξ )|ek (ξ )ek (ξ )|,

(A.2)

k≥1

each function ξ → ek (ξ ) ∈ L 2ξ ( ) being measurable on ∗ . Let us now define γ 0 ∈ Pper by
1 if λk (ξ ) ≤  F (γ 0 )ξ = δk (ξ )|ek (ξ )ek (ξ )|, δk (ξ ) = 0 if λk (ξ ) >  F . k≥1

0 ). Notice  was chosen to ensure γ 0 ∈ P Z . Stating this differently γ 0 = χ(−∞, F ] (Hper F per

176

É. Cancès, A. Deleurence, M. Lewin

We now prove that γ 0 is the unique minimizer of the function F defined above, on Z , this will prove that γ 0 = γ 0 the set Pper without a charge constraint. Since γ 0 ∈ Pper per 0 and that γper is the unique minimizer of F on Pper . We write 

 0 Tr L 2 ( ) (Hper −  F )ξ (γ − γ 0 )ξ dξ F(γ ) − F(γ 0 ) = (2π )−3 ξ ∗ 
   = (2π )−3 (λk (ξ ) −  F ) γξ ek (ξ ), ek (ξ ) ξ − δk (ξ ) dξ, ∗

k≥1

where ·, ·ξ is the usual scalar product of L 2ξ ( ). Since 0 ≤ γ ≤ 1 in L 2 (R3 ), we have   that 0 ≤ γξ ≤ 1 on L 2ξ ( ) and thus γξ ek (ξ ), ek (ξ ) ∈ [0, 1], for almost every ξ ∈ ∗ . Hence, using the definition of δk (ξ ), 
   0 −3 (2π ) |λk (ξ ) −  F | ×  γξ ek (ξ ), ek (ξ ) − δk (ξ )dξ ≥ 0. F(γ ) − F(γ ) = k≥1

∗

Z . If now F(γ ) = F(γ 0 ), then necessarily This shows that γ 0 minimizes F on Pper   γξ ek (ξ ), ek (ξ ) = δk (ξ ) for almost every ξ ∈ ∗ and any k ≥ 1, the set {ξ ∈ ∗ | ∃k, λk (ξ ) =  F } having a Lebesgue measure equal to zero by [34, Lemma 2]. Using now that the operators γξ and (1 − γ )ξ are nonnegative, we infer that γξ ek (ξ ) = δk (ξ )ek (ξ ) for all k ≥ 1 and almost all ξ ∈ ∗ . Hence γ = γ 0 and γ 0 is the unique 0 = γ 0 , i.e. γ 0 solves the self-consistent equation minimizer of F. In particular γper per (2.6). 0 on P Z , we recall that ρ = Consider now another minimizer γ of the energy Eper γ per 0 0 ργper 0 as was shown in [5]. Hence the operators Hper and γ defined above do not depend 0 , on the chosen minimizer. The above argument applied to γ shows that γ = γ 0 = γper 0 i.e. γper is unique.  

Acknowledgement. We are thankful to Éric Séré for useful advice on the model, to Claude Le Bris and Isabelle Catto for helpful comments on the manuscript. We all acknowledge support from the INRIA project MICMAC and from the ACI program SIMUMOL of the French Ministry of Research. M.L. acknowledges support from the ANR project “ACCQUAREL”. A.D. has been supported by a grant from Région Ile-de-France.

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Commun. Math. Phys. 281, 179–202 (2008) Digital Object Identifier (DOI) 10.1007/s00220-008-0480-y

Communications in

Mathematical Physics

Approach to Equilibrium for the Phonon Boltzmann Equation Jean Bricmont1,
, Antti Kupiainen2,

1 UCL, FYMA, chemin du Cyclotron 2, B-1348 Louvain-la-Neuve, Belgium 2 Department of Mathematics, Helsinki University, P.O. Box 4, 00014 Helsinki, Finland.

E-mail: [email protected] Received: 21 February 2007 / Accepted: 3 December 2007 Published online: 26 April 2008 – © Springer-Verlag 2008

Abstract: We study the asymptotics of solutions of the Boltzmann equation describing the kinetic limit of a lattice of classical interacting anharmonic oscillators. We prove that, if the initial condition is a small perturbation of an equilibrium state, and vanishes at infinity, the dynamics tends diffusively to equilibrium. The solution is the sum of a local equilibrium state, associated to conserved quantities that diffuse to zero, and fast variables that are slaved to the slow ones. This slaving implies the Fourier law, which relates the induced currents to the gradients of the conserved quantities.

1. Introduction If a piece of solid is heated locally and left to cool, the initial temperature distribution will diffusively relax to a constant temperature. The process is accompanied by a heat flow that is proportional to the local temperature gradient, according to the Fourier law. A mathematical understanding of this phenomenon starting from a microscopic model of matter is a considerable challenge [2]. A simple classical mechanical system modeling heat transport in solids is given by a system of coupled oscillators organized on a d-dimensional cubic lattice Zd . The oscillators are indexed by lattice points x ∈ Zd , and carry momenta and coordinates ( px , qx ). In the simplest model px and qx take real values and their dynamics is generated by a Hamiltonian which is a perturbation of a harmonic system: H (q, p) =

1
2 1
λ
4 px + qx q y ω2 (x − y) + q , 2 x 2 xy 4 x x


Partially supported by the Belgian IAP program P6/02.

Partially supported by the Academy of Finland.

(1.1)

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J. Bricmont, A. Kupiainen

where ω2 denotes the Fourier transform of ω(k)2 , which will be given explicitly in (2.1, 2.2) below. In (1.1), the oscillators are coupled by harmonic forces generated by the coupling matrix ω2 which is taken short range. The parameter λ describing the strength of the anharmonicity is assumed to be small. The classical dynamics is given by Hamilton’s equations q˙ x = px , p˙ x = −

∂H . ∂qx

(1.2)

The dynamics (1.2) preserves the Gibbs measures in the phase space, formally given by Z −1 e−β H (q, p) dqdp

(1.3)

and describing an equilibrium state with constant inverse temperature β = 1/T . One expects initial states that agree with (1.3) “at infinity” to be attracted by (1.3) under the flow (1.2) (since one works here in infinite volume, one does not expect Poincaré’s recurrences to occur). While such a result is well beyond current techniques, one approach is to study this problem in an appropriate small coupling limit, the kinetic limit. In that limit, one takes λ2 = R, one rescales space and time by  and, in the limit as  → 0, one formally arrives at an evolution equation for the covariance of a Gaussian measure of the form [9]: 1   x W (x, k, t) = RC(W )(x, k, t), W˙ (x, k, t) + ∇ω(k) · ∇ 2π

(1.4)

where x ∈ Rd , k ∈ Td = Rd /2π Zd . The relation of W to the microscopic model is 
e−iky W (x, k, t)dk = lim ( ω(y − z)qx/+z qx/−z  + iqx/+y px/−y  ), (1.5) →0

z

where − is the state at time t/. W is real and positive. Since the function W (Rx, k, Rt) satisfies (1.4) with R = 1 we may, with no loss of generality, make that assumption. Equation (1.4) is a Boltzmann type equation where the collision term is given by  9π C(W )(x, k, t) = dk1 dk2 dk3 (ωω1 ω2 ω3 )−1 δ(ω + ω1 − ω2 − ω3 ) 4 T3d

δ(k + k1 − k2 − k3 )[W1 W2 W3 − W (W1 W2 + W1 W3 − W2 W3 )] (1.6) with ω = ω(k), W = W (x, k, t), ωi = ω(ki ), Wi = W (x, ki , t), i = 1, 2, 3. The sum k + k1 − k2 − k3 is mod (2π Z)d . We normalize Td dk = 1. Analogously to the collision term of the standard Boltzmann equation for gases, the collision term (1.6) describes a scattering process. The scattered “particles” are phonons, i.e. vibrational modes, and the scattering process in (1.6) involves four phonons. The phonons carry momenta, k, k1 , k2 , k3 and energies ω(k), ω(k1 ), ω(k2 ), ω(k3 ) and the scattering process conserves the total energy and momentum. We may pose the problem stated in the beginning, i.e. the question of relaxation to equilibrium, for Eq. (1.4). Thus, starting from some initial W that coincides “at infinity” with an equilibrium state we would like to prove that W (t) tends diffusively to equilibrium and the process is accompanied by heat fluxes governed by Fourier’s law. In this

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paper, we prove such a theorem (Theorem 1 in Sect. 3), for an initial condition that is close to equilibrium. Equation (1.4) becomes a coupled system for “fast” and “slow” variables (see (3.6), (3.7) below). The slow variables correspond to the temperature and the chemical potential of another conserved quantity, defined in (3.16). The slow variables flow under a nonlinear diffusion equation whereas the fast ones, that include the heat currents, are “slaved” to the slow ones by the Fourier law, which is discussed after stating Theorem 1 in Sect. 3. There is a rich literature on the derivation of hydrodynamics from the Boltzmann equation for gases. In particular in [1], it is proven that, in an appropriate limit, the Boltzmann equation gives rise to the Navier-Stokes equations (see also [6,5,7,11] for more results and background on the link between Boltzmann and Navier-Stokes). In Theorem 2 (Sect. 3), we consider the diffusive scaling limit, i.e. the limit where one scales the spatial variable by , and the time variable by  2 , and we show that a suitably rescaled solution of the Boltzmann equation (1.4) solves, in the limit  → 0, a nonlinear heat equation. Compared with the Boltzmann equation for gases, Eq. (1.4) has some nice features. In particular, its long time dynamics is described by hydrodynamic equations that are considerably simpler than the Navier-Stokes-Fourier system for the former (in the Euler scaling, where space and time are both scaled by , Eq. (1.4) reduces to the Fourier law). A mathematical theory of (1.4) however appears to be lacking. This paper should be considered as a step in that direction. For an attempt to prove the Fourier law for phonons, beyond the Boltzmann approximation, see [3,4]. 2. The Collision Term The collision term (1.6) involves integration over the subset determined by the constraints in the delta functions. Some assumptions on ω are needed for regularity of the resulting measure. The standard nearest neighbor coupling between the oscillators corresponds to ω = ω0 with ω0 (k)2 ≡ 2

d
(1 − cos k j ) + ν,

(2.1)

j=1

where ν ≥ 0 is the pinning parameter. The regularity properties stated in Proposition 2.1 should hold for this ω as long as ν > 0. The proof of this seems, however, quite tedious and, therefore, we take ω(k) = ω0 (k)2 ,

(2.2)

with ν > 0, which is simpler to analyze (see also the Remark after the proof of Proposition 2.1 below). Moreover, we need to take the spatial dimension d ≥ 2. From now on, we’ll make that assumption. For ν = 0 and ω(k) = ω0 (k) the regularity problems are more serious and we expect to need d ≥ 3. Moreover, the spectrum of the linearized collision operator has no gap and the nature of the flow of Eq. (1.4) is quite different since also the slaved modes are diffusive. Progress in that case is so far hampered by the complexity of the integrals appearing in (1.6). The basic properties of the collision term (1.6) are summarized in Proposition 2.1. Since the x variable plays no role in C(W ) we consider C as a map in B = C 0 (Td )

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equipped with the sup norm, denoted by · . We also need the Hilbert space H ≡ L 2 (Td , ω(k)2 dk),

(2.3)

whose norm will be denoted by · H . In this paper, we shall use C or c to denote constants that may vary from place to place. Proposition 2.1. For the map C defined in (1.6), with ω defined in (2.1, 2.2), we have: (a) C : B → B with C(W ) ≤ C W 3 , for some constant C < ∞. (b) The equation C(W ) = 0 has, for W > 0, exactly a two parameter family of solutions in B, WT,A =

T , ω(k) + A

where W > 0 for A > −ν. (c) For all W ∈ B,   dkC(W )(k) = dkω(k)C(W )(k) = 0.

(2.4)

(2.5)

(d) Let −L be the linearization of C(W ) around W0 = ω(k)−1 , i.e. L = −DC(W0 ).

(2.6)

Then, L is bounded in B and bounded and positive in H. It has two zero modes Lω−1 = Lω−2 = 0.

(2.7)

and the rest of its spectrum in B and H is contained in λ ≥ a > 0. Moreover L = M + K , where M is a multiplication operator by a strictly positive continuous function and K is an integral operator, compact in H and B, and satisfying  sup |K (k, k )|dk < ∞. (2.8) k

Proof. (b) We use an argument given in [9], for a similar kernel. Introduce the notation k0 = k, ω0 = ω(k0 ). Then, writing the [−] in (1.6) as 3 

  W (ki ) W (k0 )−1 − W (k3 )−1 − W (k2 )−1 + W (k1 )−1 ,

(2.9)

i=0

we have: 9π C(W )(k0 ) = 4

  3 T3d

i=1

dki

3 

(ωi−1 W (ki ))δ(ω0 + ω1 − ω2 − ω3 )

i=0

  ·δ(k0 + k1 − k2 − k3 ) W (k0 )−1 − W (k3 )−1 − W (k2 )−1 + W (k1 )−1 , (2.10)

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which vanishes, for W (ki ) > 0, if f (k) = W (k)−1 is a collisional invariant, i.e. a function satisfying f (k0 ) + f (k1 ) = f (k2 ) + f (k3 ),

(2.11)

on the set of vectors ki in the support of the delta functions in (2.10). In [10] it is proven that the only L 1 -solutions of (2.11) are of the form aω(k) + b. Hence, the functions in (2.4) satisfy C(W ) = 0. To show that these functions are the only solutions of C(W ) = 0, write, for g bounded,    3 9π (dki ωi−1 W (ki ))δ(ω0 + ω1 − ω2 − ω3 ) dkg(k)C(W )(k) = 4 T4d

i=0

·δ(k0 + k1 − k2 − k3 )g(k0 )   × W (k0 )−1 − W (k3 )−1 − W (k2 )−1 + W (k1 )−1 , (2.12) and use the symmetry 0 ↔ 1 of the rest of the integrand, to replace g(k0 ) in front of [−] in (2.12) by 21 (g(k0 ) + g(k1 )), and, using the antisymmetry of the rest of the integrand under the exchange 0 ↔ 3, 1 ↔ 2, we may replace 21 (g(k0 ) + g(k1 )) by − 12 (g(k3 ) + g(k2 )). This implies that (2.12) is proportional to   3 T4d

(dki ωi−1 W (ki ))δ(ω0 + ω1 − ω2 − ω3 )δ(k0 + k1 − k2 − k3 )

i=0

  · [g(k0 ) + g(k1 ) − g(k2 ) − g(k3 )] W (k0 )−1 − W (k3 )−1 − W (k2 )−1 + W (k1 )−1 . (2.13) W (k)−1 ,

Taking g(k) =   3 T4d

we obtain

(dki ωi−1 W (ki ))δ(ω0 + ω1 − ω2 − ω3 )δ(k0 + k1 − k2 − k3 )

i=0



· W (k0 )−1 − W (k3 )−1 − W (k2 )−1 + W (k1 )−1

2

,

(2.14)

which vanishes, for W (ki ) > 0, only if W (k)−1 is a collisional invariant, i.e., by [10], only if W is of the form (2.4). (c) Since dkg(k)C(W )(k) is proportional to (2.13), it vanishes, for any W , whenever g is a collisional invariant, in particular for g(k) = 1 and g(k) = ω(k). (d) Differentiating C(WT,A ) = 0 with respect to T and A at T = 1 and A = 0, we obtain Lω−1 = Lω−2 = 0.

(2.15)

Explicitly, we have (note the − sign in (2.6))   3 9π dki ωi−2 δ(ω0 + ω1 − ω2 − ω3 )δ(k0 + k1 − k2 − k3 ) L f (k0 ) = 4ω02 i=1 T3d   ω(k0 )2 f (k0 ) + ω(k1 )2 f (k1 ) − ω(k2 )2 f (k2 ) −ω(k3 )2 f (k3 ) . (2.16)

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J. Bricmont, A. Kupiainen

To obtain (2.16) from (1.6), write the [−] in (1.6) as in (2.9) and expand  −1 = ω(ki ) − ω(ki )2 f (ki ) + .... W (ki )−1 = ω(ki )−1 + f (ki )

(2.17)

If we take the first term in (2.17) in the [−], we get 0 because of the delta function in (1.6). So, the only linear terms in f correspond to taking the second term in (2.17) 3 3 W (ki ) by i=0 ωi−1 which implies (2.16). Using the inside the [−], and replacing i=0 symmetries that led to (2.13, 2.14), we see that the scalar product in H, ( f, L f ), defined in (2.3) (note the ω(k)2 there), is proportional to:   3 T4d

dki ωi−2 δ(ω0 + ω1 − ω2 − ω3 )δ(k0 + k1 − k2 − k3 )

i=0

 2 ω(k0 )2 f (k0 ) + ω(k1 )2 f (k1 ) − ω(k2 )2 f (k2 ) −ω(k3 )2 f (k3 ) ≥ 0.

(2.18)

Thus, the zero modes of L are of the form f = ω−2 g, where g are collisional invariants, i.e. necessarily of the form aω(k) + b (see part (b)) and therefore (2.7) gives the only zero modes in H and in B. We need to show that the zero modes are isolated from the rest of the spectrum of L. From (2.16), we get the decomposition L = M + K with 9π M(k) = 4ω02

  3 T3d

dki ωi−2 δ(ω0 + ω1 − ω2 − ω3 )δ(k0 + k1 − k2 − k3 )ω(k0 )2 (2.19)

i=1

(k = k0 ), and K is an integral operator  (K f )(k) = K (k, k ) f (k )ω(k )2 dk , Td

given by the last three terms in (2.16), i.e. K (k, k ) = −

9π (ω(k)ω(k ))−2 (I (1) (k, k ) + I (2) (k, k )), 4

(2.20)

where, with k = k0 , k = k2 or k = k3 , I (1) (k, k )  = 2 dk1 (ω(k1 )ω(k + k1 − k ))−2 δ(ω(k) + ω(k1 ) − ω(k ) − ω(k + k1 − k )), Td

(2.21) groups the last two terms in (2.16), and, with k = k0 ,

k

= k1 , k˜ = k2 ,

I (2) (k, k ) =  ˜ ˜ ˜ −2 δ(ω(k) + ω(k ) − ω(k) ˜ − ω(k + k − k)), ˜ − d k(ω( k)ω(k + k − k)) Td

is the remaining term.

(2.22)

Approach to Equilibrium for the Phonon Boltzmann Equation

185

We want to show that K defines a compact operator, both in B and in H, that the function M is strictly positive, and then use Weyl’s theorem on essential spectra to prove that the zero modes are isolated from the rest of the spectrum. To prove compactness, we need to see how singular the kernel of K is. The singularities of I (1) and I (2) are similar, and we will study I (1) first; let r = 2 (k − k ), 1

and change variables in (2.21) to k1 = q − r − ( π2 , . . . , π2 ), and write Recalling (2.1, 2.2), we get  (1)

I (k, k ) = dq δ( (q, r, k)) g(q, r )

π 2

for ( π2 , . . . , π2 ).

(2.23)

Td

with g = 2(ω(q − r − 21 π )ω(q + r − 21 π ))−2 , and

(q, r, k) = −ω(k − k + k1 ) + ω(k1 ) + ω(k) − ω(k ) =2

d
(sin(q j + r j ) − sin(q j − r j )) + ω(k) − ω(k − 2r ). (2.24) j=1

The summand equals 2 cos q j sin r j . We change variables to cos q j = s j (1 − y j ) with s j = 1 in the region q j ∈ [− π2 , π2 ], and s j = −1 in the region |q j | > π2 . In both cases, y j ∈ [0, 1]. Our integral becomes a sum of integrals of the form 
 −1 Is (k, k ) = δ(4 y j s j sin r j − m(k, r ))G(y, r ) (y j ) 2 h(y j )dy j , (2.25) j

[0,1]d

j

− 12

with h(y) = (2 − y)

, G smooth, bounded and strictly positive, and
s j sin r j . m(k, r ) = ω(k − 2r ) − ω(k) + 4

(2.26)

j

Let sin r j = 0 for all j, i.e. k j − k j = 0 modulo π and m(k, r ) = 0. Given k, the set of k such that these conditions hold is of full measure. Then, changing variables, y j → y j | sin r j |,  −1 Is (k, k ) = | sin r j | 2 Js (k, k ), (2.27) j

where Js (k, k ) =

j | sin r |

δ(4 0

j




y j s j − m(k, r ))G(yr , r )

j

with yr = y j /| sin r j | and s j = ±1.

 j

(y j )

− 21

j

h(yr )dy j ,

(2.28)

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Consider first d = 2. Note that G is bounded and strictly positive, and so is h(y), for y ∈ [0, 1]. So, integrating the delta function over the variable y 2 , we can bound from above and from below Js by the function 1 | sin r |

j (r, k, m) =

(sy + m ∈ [0, | sin r 2 |])dy , √ √ y sy + m

(2.29)

0

where (·) denotes the indicator function and where j is evaluated at m = s = −s 1 s 2 , s = s 2 , i.e. s, s = ±1. j is bounded by | j (r, k, m)| ≤ C(1 + | log |m||).

s m(k,r ) , 4

(2.30)

Since m(k, r ) is real analytic, it has at most polynomial singularities. Consider I (1) composed with itself:  (1) 2

(2.31) (I ) (k, k ) = dk I (1) (k, k )I (1) (k , k

). By Hölder’s inequality we can bound this by: ⎛ |(I (1) )2 (k, k

)| ≤ C ⎝



dk

 j

| sin r j |

− 2p



⎞1/ p | sin r j |

− 2p

⎠

,

(2.32)

j

with r as above and r = 21 (k − k

). This holds for any p > 1, because any power of p | log |m(k, r )| log |m(k , r )|| is integrable in k . Now j | sin r j |− 2 is in L q , as a function of r , for q < 2/ p (we choose p ∈ (1, 2)). The integral in (2.32) is a convolution of two functions in such an L q . Thus, as a function of k − k

, by Young’s inequality,

it is in L q , for q = q/(2 − q) which shows that (I (1) )2 (k, k

) is in L s (Td × Td ) for s = qp/(2 − q) < 2/(2 − 2/ p), for p arbitrarily close to 1, hence for s arbitrarily large. This means that the kernel (I (1) )2 defines a compact operator, and therefore so does I (1) , since I (1) (and I (2) ) are symmetric and thus define self-adjoint operators. We may repeat this analysis for I (2) , writing: I (2) (k, k + π )  ˜ ˜ − ω(k ) − ω(k˜ + k − k )), = − d k(ω( k˜ − π )ω(k˜ + k − k ))−2 δ(ω(k) + ω(k) Td

(2.33) where we changed variables k˜ → k˜ +(π, . . . , π ) in the integrand and used formula (2.1), (2.2), for ω(k) and cos(k j + π ) = − cos k j . Since the factor (ω(k˜ − π )ω(k˜ + k − k ))−2 is smooth, we can do exactly the same analysis as above and obtain the bound (2.30) for I (2) . So, K , given by (2.20), is compact in H. From (2.16), we see that L is self adjoint in H.

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To show compactness in B, we will use Hölder continuity properties of I (1) , I (2) . Consider first I (1) , i.e. Is (k, k ) given by (2.25). We want to bound  sup

dk

k,k˜

˜ k )| |Is (k, k ) − Is (k, , ˜α |k − k|

for some α > 0. We will analyze the dependence in k of each factor in (2.27), (2.28) separately. For u, v ∈ (0, 1] we have, for any n, |u

− 21

−v

− 12

1

1

| = |u 2n − v 2n ||

n


k

u − 2n v −

n+1−k 2n

1

n+1

n+1

| ≤ Cn |u − v| 2n max{u − 2n , v − 2n },

k=1

(2.34) 1

1

since u 2n ∈ C 2n ([0, 1]). We can use this, with n > 1, to show the Hölder continuity in −1 k of the factor j | sin r j | 2 in (2.27). For Js , let us, for simplicity, again study j. Let m˜ = m(k , r˜ ), with r˜ = 21 (k˜ − k ). 1 , Applying (2.34) to (2.29) we get, with α = 2n 1 | sin  r |

| 0

1 1 (sy + m ∈ [0, | sin r 2 |])( √ √ )dy −√  y sy + m y sy + m˜

˜ α (1 + max{|m|−α , |m| ˜ −α }) ≤ Cα |m − m| ˜ α (1 + max{|m|−α , |m| ≤ Cα |k − k| ˜ −α }),

(2.35)

where the last inequality holds because m is analytic in k.

−1 Since m is real analytic, its singularities are polynomial and j | sin r j | 2 |m(k, 21 (k− k ))|−α is integrable in k uniformly in k for α > 0 small enough (possibly smaller than 1 2n ). This fact and (2.35) prove the Hölder continuity of j, at least for its dependence on 1 k through √sy+m . Next, let | sin r˜ 1 | = | sin r 1 | + δ, assuming δ > 0. Then, using the fact that 1 | sin r |+δ

| sin r 1 |

δ (sy + m ∈ [0, | sin r 2 |])dy ≤ C min(1 + | log |m||, log(1 + )), √ √ y sy + m | sin r 1 |

we get, using | log(1 +

δ )| | sin r 1 |

1

1

≤ δ 2 if | sin r 1 | ≥ δ 2 , the following crude estimate: 1

1

| j (r 1 , r 2 , k, m) − j (˜r 1 , r 2 , k, m)| ≤ C(δ 2 + (| sin r 1 | ≤ δ 2 )(1 + | log |m||)). (2.36) A similar estimate holds for the dependence with respect to r 2 , and for the one with respect to m in (sy + m ∈ [0, | sin r 2 |]). 1 −1 Moreover, the integral of j | sin r j | 2 (| sin r 1 | ≤ δ 2 )(1 + | log |m(k, 21 (k − k )|) over k 1 is of order δ α , for some α small; this, combined with (2.36) proves the Hölder

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J. Bricmont, A. Kupiainen

continuity of j, for its dependence on k through | sin r 1 |, | sin r 2 |, and m in (sy + m ∈ [0, | sin r 2 |]). We get, by combining all these estimates, that for some α > 0 and all f ∈ B,  (2.37) Is (k, k ) f (k )dk α ≤ C f , where − α is the C α norm. We may repeat this analysis for I (2) , using (2.33), and obtain the bound (2.37) for I (2) . Then, the same bound holds for the kernel K , given by (2.20).This shows that K is a continuous map from B into the space C α of Hölder continuous functions. Since C α is compactly embedded in B (by Ascoli’s theorem), this proves that K is compact. Observe that, by symmetry, the integrals over k of the integrals in (2.21) and (2.22) are equal and, because of the factor 2 in (2.21) and the minus signs in (2.20) and (2.22), we get:  M(k) = − dk K (k, k )ω(k )2 , we also get M ∈ B (actually M is Hölder continuous) and that L is bounded (since K is compact). It remains to show that M > 0. Because of the symmetries mentioned above, it is enough to show that  dk I (1) (k, k )ω(k )2 > 0. Using the argument that led to (2.29), and integrating over r = 21 (k − k ) instead of k , we see that it is enough to show that  dr j (r, k, m) > 0, which in turn holds if we can satisfy the inequalities s m(k, r ) ≤ | sin r 2 |, 4 0 ≤ y ≤ | sin r1 |, 0 ≤ sy +

(2.38)

on a set of positive measure in the y, r variables, and for some choice of s, s = ±1. Consider first r1 = 0, y = 0. If we can show that the first inequalities in (2.38) hold in that situation with strict inequalities: 0<

s m(k, r ) < | sin r 2 |, 4

(2.39)

then, by continuity, (2.38) will hold on a set of positive measure in the y, r variables, with 0 ≤ y ≤ | sin r1 |. Since s = s 2 , we have, for r 1 = 0, using (2.26, 2.1, 2.2): s s m(k, r ) = (cos k 2 − cos(k 2 − 2r 2 )) + sin r 2 . 4 2

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If we take sin r 2 > 0 and s so that s (cos k 2 − cos(k 2 − 2r 2 )) < 0, the second inequality in (2.39) will be strict. To satisfy the first inequality, we must show that 1 | cos k 2 − cos(k 2 − 2r 2 )| < sin r 2 2 for some r 2 > 0. For k 2 = π/2 (mod π ), this holds for r 2 small enough, because then | sin k 2 | < 1. For k 2 = π/2, we need to show that | sin 2r 2 | < 2 sin r 2 , which again 3 holds for r 2 small enough and strictly positive, using sin x  x − x3! . So, the spectrum of the operator of multiplication by M is bounded away from zero, and, by Weyl’s theorem, adding a compact operator does not change the essential spectrum; hence the spectrum of L is bounded away from zero, apart from the two zero eigenvalues discussed above. Consider now d > 2. We start with the integration over the first two components of the vectors k, k etc., keeping the other components fixed; we proceed exactly as above and simply add the contributions of the other components to ω(k), ω(k ) . . . , to the function m in (2.26). This simply adds a constant to m and does not change the fact that m is real analytic, which is the only property of m that we used. Since we obtain the bounds (2.30, 2.37) after integrating over the first two components, for any fixed value of the other components, we obtain similar bounds after integrating over all the components. To prove strict positivity of M, it is enough to restrict the remaining components so that their contribution to m is small enough and that the arguments used for d = 2 still hold. (a) We proceed as in (d) with the delta functions and get  
 −1 3

|C(W )(k)| ≤ C W δ(4 y j s j sin r j−m(k, r )) (y j ) 2 h(y j )dy j dk j

[0,1]d

j

(2.40) which, using (2.27), (2.29), proves the claim.

 

Remark. If one takes ω(k) = ω0 (k), instead of ω0 (k)2 , see (2.1), (2.2), one expects to have similar singularities in d ≥ 2, but we cannot reduce the analysis to the one of the singularities of a simple integral, as we did here in the case of ω(k) = ω0 (k)2 . The identities (2.5) yield two conservation laws. To show this, define, for α = 1, 2, jα (x, t) = −

1 (ω−α , ∇ω W ), 2π

(2.41)

where the scalar product is in H, see (2.3); j1 is the thermal current and j2 can be called the phonon number current. Similarly, set Tα (x, t) = (ω−α , W ).

(2.42)

T1 is the temperature and T2 is related to the phonon chemical potential. Equations (1.4) and (2.5) give then the conservation laws T˙α = ∇ · jα .

(2.43)

Tα are the slow modes that will diffuse. The currents jα will be related to the gradients of Tα via the Fourier law, see (3.17) below.

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3. Results Let Wˆ ( p, k, t) denote the Fourier transform of W in the x variable. We shall look for solutions of (1.4) of the form: Wˆ ( p, k, t) ≡ W0 (k)δ( p) + w( p, k, t),

(3.1)

ω−1 .

where, we recall, W0 = In Theorems 1 and 2 below, we shall construct solutions with w( p, k, t) small, so that, due to the presence of the leading term W0 , W (x, k, t) will be everywhere strictly positive. Equation (1.4) becomes then, w˙ = −Dw + n(w),

(3.2)

where the linear operator is given by D=L+

i 2π

p · ∇ω(k),

(3.3)

and the nonlinear term is n(w) = C(W0 + w) + Lw = C(W0 + w) − DC(W0 )w. Written as an integral equation, (3.2) becomes w(t) = e

−t D

t w(0) +

dse−(t−s)D n(w(s)).

(3.4)

0

We need to decompose this in terms of the slow and the fast variables. Let P be the orthogonal projection in the Hilbert space H (defined by (2.3)) on E = span {ω−1 , ω−2 } and let Q be the one on the orthogonal complement of E. The identities (2.5) can then be written as: Pn = 0,

(3.5)

or n = Qn, since, by differentiation, (2.5) implies the same identities with C(W ) replaced by DC(W0 )w. Let Pw = T, Qw = v. Then, w(t) = T (t) + v(t), and Eq. (3.4) becomes T (t) = Pe

−t D

t w(0) +

ds Pe−(t−s)D Qn(w(s)),

(3.6)

ds Qe−(t−s)D Qn(w(s)).

(3.7)

0

v(t) = Qe−t D w(0) +

t 0

Define e( p, t) := (1 + (t + 1) p 2 )−n

(3.8)

with n > d/2 and let Et be the space of continuous functions f ( p, k) equipped with the norm f t ≡ sup e( p, t)−1 | f ( p, k)| = sup e( p, t)−1 f ( p, ·) , k, p

p

(3.9)

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191

and let E be the space of functions w(t) ∈ Et , with w E ≡ sup w(t) t .

(3.10)

t

Let κ : E → E be the linear operator κ = (2π )−2 Pω L −1 ω P,

(3.11)

∂ω . ∂k 1

κ is strictly positive by Proposition 2.1.d, since the set ω E forms a where ω = two dimensional subspace in E ⊥ (by symmetry: ω is odd in k, while the functions in E are even). Let T0 (t) = e−t p κ T (0) (| p| ≤ 1), 2

(3.12)

where T ( p, ·, 0) = Pw( p, ·, 0) ∈ E, and κ acts on E, and let v0 (t) =

−i −1 L p · ∇ωT0 (t), 2π

(3.13)

with v0 ∈ E ⊥ . Then, we have our main result: Theorem 1. There exists δ > 0 such that, for w(0) 0 ≤ δ, the Eq. (3.2) has a unique solution in E, w(t) = T (t) + v(t), satisfying, for t ≥ 1, −1

T (t) − T0 (t) t ≤ C(t)t 2 w(0) 0 , v(t) − v0 (t) t ≤ C(t)t −1 w(0) 0 ,

(3.14) (3.15)

where C(t) = C log(1 + t) for d = 2 and is constant for d > 2. Remark. The norm (3.9) and (3.14) imply that, in x space, T (x, t) is given by the d usual diffusive Gaussian term, which, for bounded x, decays like t − 2 , plus a correction d+1 bounded, uniformly in x, by O(t − 2 ). Equation (3.15) says that the fast mode v is slaved to the slow one T , i.e. it decays like an explicit term, the Fourier transform of d+2 v0 ( p, t), plus a correction uniformly bounded by O(C(t)t − 2 ). Equation (3.15) also implies the Fourier law for the leading terms of the solution. Write T in the basis
T ( p, k, t) = ω−β (k)T˜β ( p, t), (3.16) β=1,2

(since the basis is not orthogonal, T˜β does not coincide with the Fourier transform of Tβ in (2.42)). Then, in x-space, the currents (2.41) (where, by symmetry, only the v part of d+2 W contributes) become, up to terms O(C(t)t − 2 ), i.e. keeping only v0 ,
καβ ∇ T˜β , (3.17) jα = β

with the positive conductivity matrix καβ = (ω−α , κω−β ),

(3.18)

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with κ given by (3.11). To see this, observe that, by symmetry, (ω−α , (∂i ω)L −1 (∂ j ω)ω−β ) equals zero for i = j, and, for i = j, does not depend on i; so, one may choose i equal to 1, which shows that κ, given by (3.11), satisfies (3.18). Finally, we can also derive a nonlinear heat equation as the hydrodynamic scaling limit of the Boltzmann equation (1.4). We scale W˜ (x, k, t) = W (x, k,  2 t), where we define (with some abuse of notations) W˜ (x, k, t) as satisfying (1.4). Thus, we obtain, for W , the equation: W˙ (x, k, t) + (2π )−1 ∇ω(k) · ∇W (x, k, t) =  −2 C(W )(x, k, t).

(3.19)

We shall solve it with initial data W |t=0 = ω(k)−1 + w(x, k, 0) and w(x, k, 0) = T (x, k, 0) + v(x, k, 0),

(3.20)

with T (x, ·, 0) ∈ E, v0 (x, ·, 0) ∈ E ⊥ , ∀x. W˜ (x, k, 0) = W (x, k, 0) has spatial variations at scale  −1 . We have then Theorem 2. There exists δ > 0 such that, for all  ≤ 1 and T (0) 0 , v(0) 0 ≤ δ, Eq. (3.19) has a unique solution W = ω−1 + w  , with w  = T  + v  ∈ E. Moreover, T  (t) → T (t) and v  (t) → v(t) in Et , for all t > 0, as  → 0, where T and v are the unique solutions of DC(ω−1 + T )v = (2π )−1 ∇ω · ∇T, T˙ = −(2π )−1 P∇ω · ∇v.

(3.21) (3.22)

Remark. Since we have DC(ω−1 + T ) instead of DC(ω−1 ) = −L in (3.22), we get a nonlinear heat equation for T : T˙ = ∇ · (K(T )∇T )

(3.23)

K(T ) = −(2π )−2 Pω DC(ω−1 + T )−1 ω P,

(3.24)

with

(remembering that L is positive, we see that DC is negative and that K(T ) is a positive matrix). One can show that the long-time behavior of (3.23) is similar to the one in Theorem 1.

4. Proofs We start by proving some results on the spectrum of the operator D defined in (3.3). Recall that, by Proposition 2.1 d, (σ (L)\{0}) ≥ a.

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193

Proposition 4.1. There exist p0 > 0, and b > 0, such that, both in B and in H, the following holds: a) If | p| < p0 then σ (D) = {λ1 , λ2 } ∪ σ˜ with inf σ˜ > a/2 and λi = p 2 µi + O(| p|3 ).

(4.1)

where µi > 0 are the eigenvalues of the operator κ defined in (3.11). b) If | p| ≥ p0 then inf σ (D) ≥ b. Proof. a) Let L(β) = L + βρ, where ρ = (2π )−1 | p|−1 p · ∇ω(k) so that D = L(i| p|). ρ is a bounded operator in B and in H. Hence, for |β| < p0 small enough, the spectrum of L(β) consists of two eigenvalues close to the origin, with the rest of the spectrum having real part larger than a/2. Moreover, L(β) is self adjoint in H for β real and thus forms an analytic Kato family (see [8]). Therefore, there is a p0 such that, for |β| < p0 , the two eigenvalues of L(β), λ1 , λ2 , and the associated eigenfunctions ψi , are analytic in |β| < p0 . To compute their Taylor series, use the decomposition w = T + v and project the equation (L + βρ)w = λw onto QH and PH. Then, using PρT = 0 (since ρT is odd in k), we get a pair of equations: Lv + β Qρ(T + v) = λv, β Pρv = λT,

(4.2)

whereby, since (σ (L)\{0}) ≥ a, we get, for β small and λ = O(β), v = −β(L − λ + β Qρ)−1 QρT = −β L −1 QρT + O(β 2 )T,

(4.3)

where O(β 2 ) denotes a bound on the operator norm in B. Since QρT = ρT and T = P T , we get, inserting (4.3) in (4.2), −β 2 Pρ L −1 ρ P T + O(β 3 )T = λT. The 2×2 matrix Pρ L −1 ρ P is strictly positive, by Proposition 2.1 (d). The result follows by letting β = i| p|, and by observing that, since the functions in E are invariant under reflections and permutations of the coordinates of k, the matrix κ( p) = Pρ L −1 ρ P, where ρ depends on | p|−1 p, equals κ(e1 ), i.e. the one given in (3.11). b) Write D = M + i| p|ρ + K as a sum of a multiplication and a compact operator. Since σ (M + i| p|ρ) ≥ inf M > 0 we need only to show, by Weyl’s theorem on the essential spectrum, that all eigenvalues λ of D satisfy λ ≥ b, if | p| ≥ p0 . Let λ be an eigenvalue of D. Let us first show that, if λ < a2 , λ is real. We decompose H = He ⊕ Ho into even and odd subspaces. Then, since ρ is odd, and the operator L is parity-preserving, Lwe + i| p|ρwo = λwe , Lwo + i| p|ρwe = λwo .

(4.4)

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J. Bricmont, A. Kupiainen

Write λ = x + i y and suppose that x < a2 (where a = inf σ (L |Ho )). Then, since the zero eigenvectors are even, (L − λ) |Ho is invertible, and (4.4) becomes (L + p 2 ρ(L − λ)−1 ρ)we = λwe .

(4.5)

Let we H = 1. Then, taking the scalar product of (4.5) in H with we , and writing separately the real and imaginary parts, one gets: x = (we , (L + p 2 ρ(L − x)((L − x)2 + y 2 )−1 ρ)we ) a ≥ p 2 (we , ρ((L − x)2 + y 2 )−1 ρwe ), 2

(4.6)

since L ≥ 0 and, as operators in Ho (to which ρwe belongs, since ρ is odd), (L − x)((L − x)2 + y 2 )−1 ≥ a2 ((L − x)2 + y 2 )−1 . For the imaginary part, we get: y = yp 2 (we , ρ((L − x)2 + y 2 )−1 ρwe ).

(4.7)

If y = 0 (4.6) and (4.7) imply x ≥ a/2, against our assumption. Thus y = 0, i.e. all eigenvalues λ of D with λ < a2 are real, and, by (4.6), positive. Let us now define, motivated by (4.5), l(r, λ) = inf (we , (L + r 2 ρ(L − λ)−1 ρ)we ). we =1

Since, for λ ≤ a/2, (L − λ)−1 |Ho ≥ c > 0, and ρwe ∈ H0 , we get (we , (L + r 2 ρ(L − λ)−1 ρ)we ) ≥ (we , (L + cr 2 ρ 2 )we ). To bound this from below, observe that both terms in L + cr 2 ρ 2 are positive, that (we , Lwe ) ≥ c Qwe 2 , and that (Pwe , ρ 2 Pwe ) ≥ c1 Pwe 2 , for some c1 > 0, by inspection of the functions in PH and of ρ. So, if Qwe 2 ≥ c we 2 , we can use the first lower bound, while, if Qwe 2 ≤ c we 2 , i.e. Pwe 2 ≥ (1 − c ) we 2 , we can use the second one, if c is small enough (so that, if we expand we = Pwe + Qwe in (we , ρ 2 we ), the (Pwe , ρ 2 Pwe ) term dominates the sum), to get (L + cr 2 ρ 2 ) |He > c

r 2 for r < p0 , and p0 small enough. So, l( p, λ) ≥ c

p 2 , for | p| ≤ p0 , and λ ≤ a/2. Besides, l(r, λ) ≤ l(r , λ) if r < r . Since by (4.5) λ ≥ l(| p|, λ) we conclude λ ≥ c

p02 , for | p| ≥ p0 . Thus, we can take b = min( a2 , c

p02 ).   We are now ready to state the estimates for the heat kernels in Eqs. (3.6) and (3.7). Let R(z) = (z − D)−1 be the resolvent of the operator D and γ be a circle of radius a/4 around the origin. Then  dz ˜ P= R(z) 2πi , (4.8) γ

is, for | p| ≤ p0 , a projection onto the 2-dimensional eigenspace of D introduced in ˜ We have: Proposition 4.1 a. Let Q˜ = 1 − P.

Approach to Equilibrium for the Phonon Boltzmann Equation

195

Proposition 4.2. For all t ≥ 0 we have e−t D ≤ C(e−ct p + e−ct ), 2

(4.9)

where · is the operator norm in B. Moreover, there exists p0 > 0 so that for | p| ≤ p0 , Pe−t D Q + Qe−t D P ≤ C| p|(e−ct p + e−ct ), 2

Qe

−t D

e

−t D

2 −ct p 2

Q ≤ C( p e ˜ ≤ Ce−ct . Q

+e

−ct

),

(4.10) (4.11) (4.12)

Proof. Since D = L + 2πi p · ∇ω(k), we have, by the resolvent expansion:   i dz dz p · ∇ω(z − L)−1 2πi + O( p 2 ) P˜ = (z − L)−1 2πi + (z − L)−1 2π γ

γ

i i −1 L p · ∇ω P − P p · ∇ωL −1 Q + O( p 2 ) =P− 2π 2π = P + A P + P B Q + O( p 2 )

(4.13)

where, here and below, O( p 2 ) denotes a bound on the norm of operators in B, and we define A=−

i −1 L p · ∇ω, 2π

B=−

i p · ∇ωL −1 . 2π

(4.14)

(Note that the projectors in (4.13) and the oddness of ∇ω guarantee that L −1 acts on vectors on which it is well defined.) Hence, for | p| small, ˜ + Q˜ P + P˜ Q + Q P ˜ ≤ C| p|. P Q

(4.15)

˜ −t D P˜ + Qe ˜ −t D Q, ˜ e−t D = Pe

(4.16)

Now, write ˜ −t D P, ˜ ˜ and Q˜ project on invariant subspaces of D, we have e−t D P˜ = Pe (since P, −t D −t D ˜ ˜ and Q˜ = Qe Q), e  dz (4.17) e−t D Q˜ = e−t z R(z) 2πi , γ

where the curve γ goes around the part of the spectrum of D that lies in the complement of a ball of radius a/4, centered at the origin. Since L is bounded and | p| small, the length of γ is O(1), and we get (4.12). Then, the other claims follow, for | p| small, from (4.15), (4.16) and Proposition 4.1 a. To get (4.9) for | p| > p0 we note that by Proposition 4.1, e−t D is given by the right hand side of (4.17), where the curve γ can be chosen to be a rectangle in z > b, with vertical sides of length const | p|, and horizontal ones of length O(1) (since L is bounded). Hence, on the curve γ , we have |e−t z | ≤ e−bt , but the length of the contour of integration is not bounded as | p| → ∞. To control the integral, recall that D = M + i| p|ρ + K , and let R0 (z) = (z − M − i| p|ρ)−1 .

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J. Bricmont, A. Kupiainen

Then, by the resolvent formula, R = R0 + R0 K R0 + R0 K R0 K R.

(4.18)

The first term in (4.18), inserted in (4.17), gives the multiplication operator e−t (M+i| p|ρ) , which satisfies the bound (4.9) with c = inf M. For the second term, we use |e−t z | ≤ e−bt in the integral and write    1



, (4.19) | dk dz R0 (z, k)K (k, k )R0 (z, k )| ≤ dk |K (k, k )| dz |z − z 1 ||z − z 2 | γ

γ

where z 1 = (M + i| p|ρ)(k), z 2 = (M + i| p|ρ)(k ). Since |z − z i | ≥ c on γ , we have  1 dz ≤ C, (4.20) |z − z 1 ||z − z 2 | γ

 uniformly in k, k . Since, moreover, |K (k, k )|dk ≤ C, uniformly in k (which holds by (2.8)), we get the bound in (4.9) for the second term in (4.18). The third term in (4.18) is similar, since K R is bounded.   Proof of Theorem 1. Let us write Eq. (3.4) as w(t) = w (t) + N (w, t)

(4.21)

where w (t) is the solution of the linear problem: w ≡ e−t D w(0).

(4.22)

We will solve (4.21) in the space E. The linear term is bounded, using (4.9), and recalling the definition (3.8) of the weight function e, by w E ≤ C sup e( p, t)−1 e( p, 0)(e−ct p + e−ct ) w(0) 0 ≤ C w(0) 0 . 2

(4.23)

p,t

Consider then the nonlinear term in Eq. (3.4). By an easy extension of Proposition 2.1 a (see (2.40)), n(w(s)( p) ≤ C( w 2E e(s)∗2 ( p) + w 3E e(s)∗3 ( p)).

(4.24)

We need the simple estimate e(s)∗2 ( p) ≤ C(1 + s) which follows from ∗2

− d2

e( p, s),



e(s) ( p) ≤ 2

(4.25) 

dq e( p − q, s)e(q, s) ≤ Ce( p, s)

dq e(q, s),

(4.26)

| p−q|≥ 21 | p|

(the first inequality holds because either |q| ≥ 21 | p| or | p − q| ≥ 21 | p|). The last integral yields the power of 1 + s since n > 21 d in (3.8). The last term in (4.24) is smaller, for w E ≤ 1.

Approach to Equilibrium for the Phonon Boltzmann Equation

197

Thus, the second term in Eq. (3.6) and the second term in (3.7) are bounded by t R N (w, t, p) ≤ C w E 2

ds Re−(t−s)D Q (1 + s)

− d2

e( p, s),

(4.27)

0

where R equals P in (3.6) and Q in (3.7). Proposition 4.2 implies − 21 m − c (t−s) p 2 2

Re−(t−s)D Q ≤ C((1 + t − s)

e

c

with m = 1 if R = P and m = 2 if R = Q (using | p|m e− 2 p for t − s ≥ 1). Bounding c

e− 2 p

2 (t−s)

+ e−c(t−s) ), 2 (t−s)

(4.28)

≤ C(1 + t − s)

− 21 m

,

e( p, s) ≤ Ce( p, t)

and c

e− 2 (t−s) e( p, s) ≤ Ce( p, t), we may bound the remaining s-integral in (4.27), for m = 1, 2, i.e. for m ≤ d, by −1 m

C(t)(1 + t) 2 , where C(t) may be taken to be constant, except for d = 2, where C(t) = C log(1 + t). Hence we end up with R N (w, t) t ≤ C(t)(1 + t)

− 21 m

w 2E .

(4.29)

In particular, we have N (w) E ≤ C w 2E , so that N in (4.21) maps a ball of radius  in E into itself, for  small enough and is obviously a contraction there. The existence of a solution for Eq. (4.21) then follows from (4.23) and the Banach fixed point theorem and, since m = 1 if R = P, i.e. for (3.6), and m = 2 if R = Q, i.e. for (3.7), we obtain the bounds T (t) − T (t) t ≤ C(t)(1 + t)

− 21

v(t) − v (t) t ≤ C(t)(1 + t)

w(0) 20 ,

−1

w(0) 20 ,

(4.30) (4.31)

where we wrote w = Pw + Qw ≡ T + v .

(4.32)

To conclude the proof of Theorem 1 we need to relate T0 , v0 to T , v . For this we need to write the leading terms of w (t) more explicitly. Let us formulate this a little more generally, which will be useful also in the proof of Theorem 2. Let us denote K (t) = e−t p κ P, 2

(4.33)

where κ is given in Eq. (3.11). We have then the following lemma, proven at the end:

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Lemma 4.3. Let | p| ≤ p0 . Then the semigroup e−t D can be written with respect to the decomposition E ⊕ E ⊥ , as e−t D =



K (t) AK (t)

  2 K (t)B O(| p|e−ct p + p2 e−ct ) + 2 AK (t)B + R(t) O( p 2 e−ct p + | p|e−ct )

 2 O( p 2 e−ct p + | p|e−ct ) , 2 O(| p|3 e−ct p )

where O is with respect to the operator norm in B, ˜ −t D Q˜ Q, R(t) = Q Qe

(4.34)

and the operators A and B are defined in Eq. (4.14). Returning to the proof of Theorem 1 and recalling the definition (3.12) of T0 and (4.22), (4.32) of T , we get from Lemma 4.3, for | p| ≤ p0 , since B is of order | p|, see (4.14), T (t, p) − T0 (t, p) ≤ C| p|(e−ct p + e−ct )e( p, 0) w(0) 0 . 2

(4.35)

Since v0 , given in Eq. (3.13), equals v0 = AK (t)T (0) (see (4.14)), for | p| ≤ p0 ≤ 1, we have, from Lemma 4.3, v (t, p) − v0 (t, p) ≤ C( p 2 e−ct p + e−ct )e( p, 0) w(0) 0 , 2

(4.36)

where we used the bound, coming from (4.12), R(t) ≤ Ce−ct ,

(4.37)

and the fact that A and B are of order p. For | p| ≥ p0 , by (4.9), w0 (t, p) + w (t, p) ≤ Ce−ct e( p, 0) w(0) 0 . Using now the simple estimates e( p, t)−1 (e−ct p

2 /2

+ e−ct/2 ) ≤ Ce( p, 0)−1 ,

for t ≥ 1, e( p, t)−1 ≤ Ce( p, 0)−1 for t ≤ 1, and | p|m e−ct p

2 /2

+ e−ct/2 ≤ C(m)t

− 21 m

,

for t ≥ 1, m ≥ 0, we conclude that −1

T (t) − T0 (t) t ≤ C(1 + t) 2 w(0) 0 , v (t) − v0 (t) t ≤ C(1 + t)−1 w(0) 0 .

(4.38) (4.39)

The estimates (3.14) and (3.15) follow now from (4.30), (4.31), (4.38) and (4.39).

 

Approach to Equilibrium for the Phonon Boltzmann Equation

199

Proof of Theorem 2. The proof goes along the lines of the one of Theorem 1 and we will be brief. We expand 1 1 1 C(W ) = − 2 Lw + m(T, v) + n(T, v), 2    where m(T, v) is the term quadradic in T but linear in v, and n(T, v) collects the terms that are quadratic and cubic in v. Note that, since L T = 0, 12 Lw = 1 Lv and thus DC(ω−1 + T )v = −Lv + m(T, v).

(4.40)

We write (3.19) as w(t) = exp(−

t 1 D )w(0) + 2 

t ds exp(− 0

t −s D )(m(s) + n(s))ds, 2

where D = L + 

i p · ∇ω, 2π

(4.41)

i.e. it equals (3.3) evaluated at p. The decomposition w = T + v yields, as in (3.6), (3.7): T (t) = Pe−t D / P T (0) +  Pe−t D / Qv(0) t 1 2 + ds Pe−(t−s)D / Q(m(s) + n(s))  2

2

(4.42)

0

v(t) =

1 −t D / 2 2 Qe P T (0) + Qe−t D / Qv(0)  t 1 2 + 2 ds Qe−(t−s)D / Q(m(s) + n(s)). 

(4.43)

0

Consider first the case | p| ≤ p0 /. We use Lemma 4.3, with t replaced by t/ 2 and p by p. This leads to T = T ∗ + T1 , v = AT ∗ +

1 2

(4.44) t

˜ −(t−s)D / 2 Q˜ Qm(s) + v1 , ds Q Qe

(4.45)

0

where ∗

T =e

−t p 2 κ

t T (0) + 0

dse−(t−s) p κ P Bm(s), 2

(4.46)

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J. Bricmont, A. Kupiainen

and T1 =  Pe

−t D / 2

t Qv(0) +

ds Pe−(t−s)D / Qn(s) 2

0

t +

dsO(p 2 e−c(t−s) p + | p|e−c(t−s)/ )m(s), 2

2

0

v1 = Qe−t D / Qv(0) + 2

t +

1 

t

ds Qe−(t−s)D / Qn(s) 2

0

dsO(| p|3 e−c(t−s) p )m(s). 2

0

Then, following the proof of (4.38), (4.39), (4.30), (4.31), we get −1

T1 (t, p) ≤ C(1 + t) 2 e( p, t)( T (0) + v(0) ), v1 (t, p) ≤ C(t)(1 + t)−1 e( p, t)( T (0) + v(0) ),

(4.47) (4.48)

where C(t) = C max(t −1/2 , log(1 + t)) < ∞, for t > 0. We need to study the second term on the RHS of Eq. (4.45). Write m(s) = m(t) + (m(s) − m(t)) and consider the first term 1 2

t 0

˜ −(t−s)D / 2 Q˜ Qm(t) = 1 ds Q Qe 2

∞

˜ −s D / 2 Q˜ Qm(t) + O(e−ct/ 2 ) ds Q Qe

0

= Q Q˜ D−1 Q˜ Qm(t) + O(e−ct/ ) 2

= L −1 m(t) + O() + O(e−ct/ ), 2

(4.49)

where we used Q Q˜ D−1 Q˜ Q = Q L −1 Q + O() and where O(·) denotes a bound on the norm sup| p|≤ p0 / e( p, t)−1 · of the remainders, since sup| p|≤ p0 / e( p, t)−1 m(t) is bounded, uniformly in , because of the bounds on T , v, coming from (4.44)–(4.48). For the second term, we use Eqs. (4.42) and (4.43), to show that m(s) − m(t) t ≤ C|t − s|, for |t − s| small; whence 1 2

t

˜ −(t−s)D / 2 Q˜ Q(m(s) − m(t)) = O(). ds Q Qe

(4.50)

0

Altogether we obtain v(t) = AT ∗ (t) + L −1 m(t) + v2

(4.51)

sup e( p, t)−1 ( T1 + v2 ) → 0

(4.52)

and, for any t > 0, | p|≤ p0 /

Approach to Equilibrium for the Phonon Boltzmann Equation

201

as  → 0. For | p| > p0 / we have, from Proposition 4.2, e−t D / ≤ e−ct/ , 2

2

which implies that sup e( p, t)−1 ( T + v ) → 0,

(4.53)

| p|≥ p0 /

as  → 0. Hence, in the space Et , we have lim (T (t), v(t)) = (T ∗ (t), AT ∗ (t) + L −1 m(t)) ≡ (T ∗ (t), v ∗ (t)).

→0

(4.54)

Let us finally check that T ∗ and v ∗ satisfy Eqs. (3.21) and (3.22). From (4.40) and (4.54), we get AT ∗ = −L −1 DC(ω−1 + T ∗ )v ∗

(4.55)

which is (3.21) if we recall the definition (4.14) of A. From (4.46) and the definition (4.14) of B, we see that T ∗ satisfies i P p · ∇ωL −1 m. T˙ ∗ = − p 2 κ T ∗ − 2π

(4.56)

From (4.54) and the definition (4.14) of A, we have L −1 m = v ∗ − AT ∗ = v ∗ + i(2π )−1 L −1 p · ∇ωT ∗ . Substituting this into (4.56) and recalling the definition of κ in (3.11), which implies that − p 2 κ T ∗ + p 2 (2π )−2 P · ∇ωL −1 · ∇ωT ∗ = 0, we get Eq. (3.22).   Proof of Lemma 4.3. We need to calculate Re−t D R for R and R either P or Q. We use ˜ We need the following the representation (4.16) for e−t D and the formula (4.13) for P. expansions P P˜ = P + P B Q + O( p 2 ), P˜ Q = P B Q + O( p 2 ), ˜ = −P B Q + O( p 2 ), P Q˜ = P(1 − P) ˜ Q˜ Q = (1 − P)Q = Q − P B Q + O( p 2 ),

P˜ P = P + A P + O( p 2 ), Q P˜ = A P + O( p 2 ), Q˜ P = −Q A P + O( p 2 ), Q Q˜ = Q − A P + O( p 2 ),

where we used repeatedly P A P = 0 (A maps E into E ⊥ , because ∇ω is odd in k). Thus we get, using the identities on the left, ˜ −t D Q˜ Q + P Qe ˜ −t D Q˜ Q Pe−t D Q = P Pe = (P + P B Q)e−t D P B Q − P B Qe−t D (Q − P B Q) + O( p 2 e−ct p ) 2

= e−t p κ P B Q + O( p 2 e−ct p ) + O(| p|e−ct ), 2

2

(4.57)

where we use the bounds of Proposition 4.2, B = O(| p|), and λα = µα p 2 + O( p 3 ), with µα being the eigenvalues of κ, see (4.1). Similarly, we get ˜ −t D Q˜ Q + O(| p|3 e−ct p ) Qe−t D Q = Ae−t p κ P B Q + Q Qe 2

2

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and Pe−t D P = e−t p κ P + O(| p|e−ct p ) + O( p 2 e−ct ), 2

2

Qe−t D P = Ae−t p κ P + O( p 2 e−ct p ) + O(| p|e−ct ). 2

2

  Acknowledgement. We thank François Huveneers, Jani Lukkarinen and Herbert Spohn for useful discussions. J. B. thanks the Belgian Internuniversity Attraction Poles Program and A.K. thanks the Academy of Finland for financial support.

References 1. Bardos, C., Ukai, S.: The classical incompressible Navier-Stokes limit of the Boltzmann equation. Math. Models and Methods in Applied Sciences 1, 235–257 (1991) 2. Bonetto, F., Lebowitz, J.L., Rey-Bellet, L.: Fourier Law: A challenge to Theorists. In: Mathematical Physics 2000, London: Imp. Coll. Press, 2000, pp. 128–150 3. Bricmont, J., Kupiainen, A.: Towards a derivation of Fourier’s law for coupled anharmonic oscillators. Commun. Math. Phys. 274, 555–626 (2007) 4. Bricmont, J., Kupiainen, A.: Fourier’s law from closure equations. Phys. Rev. Lett. 98, 214301 (2007) 5. Esposito, R., Pulvirenti, M.: From particles to fluids. In: Handbook of Mathematical Fluid Dynamics, Vol. III, Friedlander, S., Serre, D. eds. Amsterdam: Elsevier Science, 2004 6. Golse, F., Saint-Raymond, L.: The Navier-Stokes limit of the Boltzmann equation for bounded collision kernels. Invent. Math. 155, 81–161 (2004) 7. Mouhot, C.: Rate of convergence to equilibrium for the spatially homogeneous Boltzmann equation with hard potentials. Commun. Math. Phys. 261, 629–672 (2006) 8. Reed, M., Simon, B.: Methods of Modern Mathematical Physics. Vol IV. Analysis of Operators, New York: Academic Press, 1978 9. Spohn, H.: The phonon Boltzmann equation, properties and link to weakly anharmonic lattice dynamics. J. Stat. Phys. 124, 1041–1104 (2006) 10. Spohn, H.: Collisional invariants for the phonon Boltzmann equation. J. Stat. Phys. 124, 1131–1135 (2006) 11. Villani, C.: A review of mathematical topics in collisional kinetic theory. In: Handbook of Mathematical Fluid Dynamics, Vol. I, Friedlander, S., Serre, D. eds. Amsterdam: Elsevier Science, 2002 Communicated by H. Spohn

Commun. Math. Phys. 281, 203–229 (2008) Digital Object Identifier (DOI) 10.1007/s00220-008-0484-7

Communications in

Mathematical Physics

Limit Correlation Functions for Fixed Trace Random Matrix Ensembles Friedrich Götze1,
, Mikhail Gordin2,

1 Fakultät für Mathematik, Universität Bielefeld, Postfach 100131, D-33501, Bielefeld, Germany.

E-mail: [email protected]

2 V.A. Steklov Mathematical Institute at St. Petersburg, 27 Fontanka emb., St. Petersburg,

191023, Russia. E-mail: [email protected] Received: 14 March 2007 / Accepted: 18 December 2007 Published online: 9 May 2008 – © Springer-Verlag 2008

Abstract: Universal limits for the eigenvalue correlation functions in the bulk of the spectrum are shown for a class of nondeterminantal random matrices known as the fixed trace or the Hilbert-Schmidt ensemble. These universal limits have been proved before for determinantal Hermitian matrix ensembles and for some special classes of the Wigner random matrices. 1. Introduction and the Statement of the Result
N Let H N be the set of all N × N (complex) Hermitian matrices, and let tr A = i=1 aii denote the trace of a square matrix A = (ai j )i,N j=1 . H N is a real Hilbert space of dimension N 2 with respect to the symmetric bilinear form (A, B)
→ tr AB. Let l N denote the unique Lebesgue measure on H N which satisfies the relation l N (Q) = 1 for every cube Q ⊂ H N with edges of length 1. A Gaussian probability measure on H N invariant with respect to all orthogonal linear transformations of H N is uniquely defined up to a scaling transformation. Such measures form a one-parameter family (µsN )s>0 , where the measure µsN is specified by its density   1 1 s 2 (1.1) tr A exp − g N (A) = √ 2 2s ( s2π ) N with respect to l N . Thus, for a random matrix X distributed according to µsN we have E µsN tr X 2 = s N 2 .

(1.2)


Research supported by Sonderforschungsbereich 701 “Spektrale Strukturen und Topologische Methoden in der Mathematik”.

Research supported by Sonderforschungsbereich 701 “Spektrale Strukturen und Topologische Methoden in der Mathematik,” and grants RFBR-05-01-00911, DFG-RFBR-04-01-04000, and NS-638.2008.1.

204

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The set H N endowed with the measure µsN is called the Gaussian Unitary Ensemble (GUE). Let X be a random N × N Hermitian matrix (that is a random variable taking values in H N ). We consider the eigenvalues λ1 , λ2 , . . . , λ N of the random matrix X as a finite sequence of exchangeable random variables. By definition, this means that their joint distribution PNX does not change under any permutation of these variables. Let for X denote the joint distribution of some n of these N variables. each n, 1 ≤ n ≤ N , Pn,N X Obviously, Pn,N is a permutation invariant probability measure in Rn . In particular, the X describes the distribution of a single eigenvalue. By definition, the n-point measure P1,N X of a random matrix X is a non-normalized measure defined correlation measure ρn,N by the relation X ρn,N =

N! PX . (N − n)! n,N

(1.3)

X (A) can be interpreted as the average For a measurable set A ⊂ Rn the quantity ρn,N X is absolutely number of n−tuples of eigenvalues in the set A. If the measure ρn,N n continuous with respect to the Lebesgue measure on R , its Radon-Nikodym derivative X is called the n-point correlation function of the random matrix X . In particular, Rn,N X has total mass N . For a measurable set E ⊂ R1 , the quantity ρ X (E) the measure ρ1,N 1,N expresses the expected number of the eigenvalues belonging to E. The corresponding density with respect to the Lebesgue measure in R1 , if it exists, is called the eigenvalue density or the density of states (caution: under the same names the normalized versions of the same measures are considered in the literature as well). Let X N be a random matrix XN XN GUE,s GUE,s with the distribution µsN . For n = 1, . . . , N we set Pn,N = Pn,N and ρn,N = ρn,N . A classical result for the GUE says that we have GUE,1/N

P1,N

→ W,

N →∞

(1.4)

where the measures converge in the weak sense, and W is the standard Wigner measure on [−2, 2] defined by the density  w(x) = (2π )−1 (4 − x 2 )+ , x ∈ R. (1.5) In terms of the correlation measures the same relation reads 1 GUE,1/N ρ → W. N →∞ N 1,N

(1.6)

For the n−point correlation measures we have a similar relation 1 GUE,1/N ρ → W × ··· × W, N →∞ N n n,N n times

(1.7)

which means that the eigenvalues become independent in the limit. However, for n ≥ 2, the study of a finer asymptotics near a point from the principal diagonal in the cube (−2, 2)n shows [10,11]: for every u ∈ (−2, 2) and t1 , . . . , tn ∈ R1 ,   1  GUE, N  sin π(ti − t j ) n 1 tn t1 u + = det R , . . . , u + . lim N w(u) N w(u) N →∞ (N w(u))n n,N π(ti − t j ) i. j=1

Limit Correlation Functions for Fixed Trace Random Matrix Ensembles

205

This limit relation presents a pattern for many other results, in particular, for that of the present paper. The right hand side of this relation represents an example of the correlation function of a so-called determinantal (or fermionic) random point process [15]. In general the n−point correlation function Rn of such a process is given by the formula  n R(u 1 , . . . , u n ) = det  K (u i , u j )i, j=1 , where K is the kernel of an integral operator on the line, which is trace-class having been restricted to any finite interval in R, and subject to some further conditions (see [15] for a detailed exposition). Moreover, in the asymptotic Hermitian random matrix theory (K N ) N ≥0 are the reproducing kernels of the subspaces of polynomials of degree ≤ N −1 with respect to some weight on the line. In this case we call the corresponding matrix ensemble determinantal. The GUE gives an example of such an ensemble. To a large extent the asymptotic study of determinantal ensembles reduces to that of the respective kernels [3,5]. In particular, the same local limit as in the case of GUE is established in [12,3] for two broad classes of determinantal matrix ensembles. It is expected that the sin-kernel limit (first discovered by F. Dyson) is rather common. This is known as the universality conjecture. Outside the class of determinantal Hermitian random matrices only very few results on the asymptotics of the correlation functions are known (see, for instance, [10], where a mixture of determinantal measures is considered). In the present paper we investigate the following non-determinantal ensemble of Hermitian random matrices. Let SrN = {A ∈ H N : tr A2 = r 2 }

√

(1.8)

be√the sphere in H N of the radius r > 0 centered at the origin. Set r = s N . The sphere sN S N carries a unique probability measure ν Ns invariant with respect to all orthogonal linear transformations in the space H N . We call this measure the fixed Hilbert-Schmidt norm ensemble (or just HSE) to reserve the term “the fixed trace ensemble” for more general ones (see [1]). Let Y N be a random matrix distributed according to ν Ns . We set YN YN HSE,s HSE,s for n = 1, . . . , N Pn,N = Pn,N and ρn,N = ρn,N . It is a well known result (see [11,14]) that HSE,1/N

P1,N

→ W,

(1.9)

N →∞

HSE,1/N

of like in the case of GUE. In this paper we prove that the correlation functions Rn,N arbitrary order n (1 ≤ n ≤ N ) near every point u ∈ (−2, 2), have the same determinantal limit with the kernel sin π(t1 −t2 )/π(t1 −t2 ) as the GUE correlation functions (for n = 1 the limit equals 1). More precisely, we establish in this paper the following result. ν,1/N

Theorem 1.1. Let Rn,N

be the n−point correlation function of the eigenvalues for a √

random matrix uniformly distributed on the sphere S N N . Then for every u ∈ (−2, 2) and t1 , t2 , . . . , tn ∈ R1 ,     sin π(ti − t j ) n 1 ν,1/N tn t1 u + N w(u) , . . . , u + N w(u) → det R (1.10) (N w(u))n n,N π(ti − t j ) i, j=1 as N → ∞. For every α ∈ (0, 1) and A > 0 the relation (3.1) holds uniformly in all u ∈ [−2 + α, 2 − α], and t1 ∈ [−A, A], . . . , tn ∈ [−A, A].

206

F. Götze, M. Gordin

The class of fixed trace matrix ensembles was studied in several recent publications. The authors are indebted to G. Akemann for drawing their attention, after the first version of the present work has appeared as a preprint, to the paper [2], he published jointly with G. Vernizzi. In this paper the local universality is studied for a class of ensembles containing the HSE. The authors provide heuristic arguments for the universality near zero based on the complex inversion of the Laplace transform. Our proof is based on the complex Laplace inversion as well (more precisely, we arrive at the Fourier transform after a sequence of translations of the origin and rescalings depending on N ). However, establishing bounds for orthogonal functions and related kernels in the complex domain allows us to rigorously conclude convergence near every point of the interval (−2, 2). In the following we will sketch the main steps of the proof. The guiding principle is that results for the Hilbert-Schmidt ensembles are deducible from the corresponding results for the GUE using the ’equivalence of ensembles’ or the ’concentration phenomenon’. In our setup a primitive form of the concentration is given by the law of large numbers for the squares of the Hilbert-Schmidt norms of the GUE random matrices. Supplemented by some estimates of the probabilities of large deviations, this is the main tool in [9], where the universality is shown near zero for the correlation measures (rather than the correlation functions) of the Hilbert-Schmidt ensemble; convergence was shown there in the weak topology determined by the continuous compactly supported functions. Such simple arguments seem to be insufficient for proving local results for correlation functions of eigenvalues (maybe, with exception for eigenvalues near zero), particularly, for stronger topologies. This agrees with M.L.Mehta’s doubts ([11], Sect. 27.1, p.490) concerning the deducibility of the local results for correlation functions of the Hilbert-Schmidt ensembles from the corresponding results for GUE by using the equivalence of ensembles. Solving this open problem in the present paper, we use a local form of concentration given by the local central limit theorem for the densities of the squared Hilbert-Schmidt norms. First we represent the Hilbert-Schmidt measure as a conditional measure of the GUE, given the Hilbert-Schmidt norm of the GUE random matrix. Starting with the disintegration of the GUE according to the level sets of the Hilbert-Schmidt norm, we arrive at formula (2.9) which is the crucial ingredient of the proof. Here we have to extend the scaling parameter to the complex domain. The formal Fourier inversion applied to this formula gives an ‘heuristic proof’ of the result. For an outline of it see Sect. 2. To make this sketch rigorous we need asymptotic estimates in the complex domain for the kernels related to the Hermite functions. This is done in Sect. 3, based on the results in [3–5]. Unfortunately, some of the results we need are contained in these papers not explicitly enough and have to be extracted from the proofs rather than from the statements (see the proof of Lemma 3.2). The main convergence result we need is Lemma 3.2. All necessary bounds are summarized in Proposition 3.1. Based on these two facts, we complete the proof. Note that the analytic part of the present paper may be viewed as a form of the Tauberian theorem. 2. Disintegration, a Fourier Transform Formula and the Sketch of the Proof In this section we discuss a disintegration representation of the GUE in terms of the HSE, and derive a Fourier transform formula involving these matrix ensembles. We suppress in this section the “spectral” arguments of the correlation functions and related quantities assuming that these arguments vary inside the domain described in Sect. 1. For every r > 0 denote by SrN the sphere of radius r in H N centered at the origin. . Set Let for s > 0, X N be a random matrix in H N distributed according to µGUE,s N

Limit Correlation Functions for Fixed Trace Random Matrix Ensembles

207

 TN = tr(X 2N /s) and Y N = N X N / tr X 2N . The random variable TN can be represented as a sum of N 2 squares of independent standard Gaussian random variables, hence it follows the familiar χ N2 2 distribution. Moreover, TN and Y N are independent, and Y N is uniformly distributed on the sphere S NN in H N , that is, Y N is distributed according to ν N1 in the notation of the previous section. Then X N can be represented as YN  XN = sTN (2.1) N with TN and Y N as above. Let γ N 2 denote the probability density of TN . Then it follows from (2.1) that ∞ u/N 2 µsN = ν N γ N 2 (s −1 u)s −1 du. (2.2) 0

As a consequence of (2.2), the correlation functions of GUE and HSE for 1 ≤ n ≤ N −1 satisfy the relation ∞ HSE,u/N 2 GUE,s Rn,N = Rn,N γ N 2 (s −1 u)s −1 du. (2.3) 0

Note that E TN = N 2 , DTN2 = E(TN − E TN )2 = 2N 2 , and, for every m > 0,



γm (u) =

(2m/2 Γ (m/2))−1 x (m/2)−1 e−x/2 , if u ≥ 0, 0, if u < 0.

(2.4)

Set for s > 0 γm,s (·) = s −1 γm (s −1 ·), so that γm,1 = γm (note that the same set of densities with a different parametrization appears in Lemma 3.7 as ( f a, p )a, p>0 ). Observe now that the probability density of sTN is given by γ N 2 ,s (·). Then (2.3) can be rewritten as ∞ HSE,u/N 2 GUE,s = Rn,N γ N 2 ,s (u)du. (2.5) Rn,N 0

In particular, we have GUE,1/N

Rn,N

= 0

∞

HSE,u/N 2

Rn,N

γ N 2 ,1/N (u)du. HSE,1/N

Our goal is to investigate the limiting behavior of Rn,N

(2.6)

when n is fixed, N → ∞,

HSE,1/N Rn,N

and the “spectral” arguments of vary within an /N −neighborhood of a point from the main diagonal in the cube (−2, 2)n . However, we prefer to consider the function HSE,u/N 2

u
→ Rn,N

HSE,1/N

rather than its value Rn,N

at N . Moreover, we will perform the study HSE,u/N 2

γ N 2 ,1/N (u). of the latter function indirectly, first dealing with the product u
→ Rn,N √ After the change of variable (u − N )/ 2 = v in (2.6) we obtain the relation ∞ GUE,1/N Rn,N = q N 2 (v)dv, (2.7) −∞

208

F. Götze, M. Gordin

where

√ √ HSE,1/N +v 2/N 2 √ q N 2 (v) = Rn,N 2γ N 2 ,1/N (N + v 2). (2.8) √ √ Notice that v
→ 2 γ N 2 ,1/N (N + v 2) is the probability density of the centered and √ normalized random variable (TN 2 − N 2 )/ 2N 2 , and it tends to the standard normal √ density ϕ : v
→ (1/ √2π ) exp (−v 2 /2) as N → ∞. Thus, the limit behavior of the density γ N 2 ,1/N (N + · 2) is well understood, and we have to study q N 2 (·). For every fixed u, due to the relation γ N 2 ,s (·) = s −1 γ N 2 (s −1 ·), we can analytically extend γ N 2 ,s (u) to the domain s > 0. Moreover, in the formula (2.5) the integral in the right-hand side can be analytically continued in s in accordance with the continuation of γ N 2 ,s (u) GUE,s so that (2.5) mentioned above. This leads to the corresponding continuation of Rn,N holds for s from the right half-plane. Now we will evaluate the Fourier transform of the (nonprobabilistic) density q N 2 , keeping in mind that by (2.7) and (2.8) it is a nonnegative integrable function.

Lemma 2.1.



∞ −∞

√ GUE,1/((1−i p 2/N )N )

exp (i pv)q N 2 (v)dv = φ N 2 ( p)Rn,N

,

(2.9)

where

√ √ 2 φ N 2 ( p) = exp (−i pN / 2)(1 − i p 2/N )−(N /2) √ is the characteristic function of the random variable (TN 2 − N 2 )/ 2N 2 .

(2.10)

Proof. Denoting by Cm the normalizing constant in formula (2.4), we have ∞ exp (i pv)q N 2 (v)dv −∞ ∞ √ √ HSE,1/N +v 2/N 2 √ = 2γ N 2 ,1/N (N + v 2)dv exp (i pv)Rn,N −∞   ∞ i p(u − N ) HSE,u/N 2 Rn,N = exp γ N 2 ,1/N (u)du √ 2 −∞  ∞  √ HSE,u/N 2 = exp −i√pN exp (i pu 2)Rn,N γ N 2 ,1/N (u)du 2

= exp



−i√pN 2





−∞

HSE,u/N 2

Rn,N

C N 2 N (N u)

 = exp −i√pN 1 −

√ − N 2 +1 2 ip 2 N

∞



2

−∞

HSE,u/N 2 Rn,N CN2 N

=



Nu 1 −



−i√pN 2

 1−

 N2 2 −1 exp − N u 2



1−

√  ip 2 du N

×

√  N 2 −1  2 ip 2 exp − N2u N

√ − N 2 ∞ 2 ip 2 N −∞ √ GUE,1/((1−i p 2/N )N ) . φ N 2 ( p)Rn,N

= exp



−∞ ∞



HSE,u/N 2

Rn,N

 1−

√  ip 2 du N

γ N 2 ,1/((1−i p√2/N )N ) (u)du (2.11)

Limit Correlation Functions for Fixed Trace Random Matrix Ensembles

209

In the following we will outline our approach. Write ∞ √ 1 1 1 GUE,1/((1−i p 2/N )N ) q (0) = φ N 2 ( p) Rn,N dp. N2 n n (N w(u)) 2π −∞ (N w(u)) Passing to the limit in the integral on the right-hand side (uniformly with respect to the spectral variables), the right-hand side has the same limit as ∞ 1 GUE,1/N 1 R φ 2 ( p)dp (N w(u))n n,N 2π −∞ N or, in view of the local Central Limit Theorem (CLT) , as 1 GUE,1/N Rn,N . √ 2π (N w(u))n On the other hand, it follows from (2.8) that 1 1 HSE,1/N √ q N 2 (0) = Rn,N 2γ N 2 ,1/N (N ). n n (N w(u)) (N w(u)) √ √ Again, the local CLT implies that 2γ N 2 ,1/N (N ) → 1/ 2π . Therefore,

(2.12)

1 HSE,1/N R (N w(u))n n,N tends to the same limit as 1 GUE,1/N R , (N w(u))n n,N and the conclusion follows. In the next section these heuristic arguments will be made rigorous. 3. Proofs For every α ∈ R, through the rest of the paper, we will denote by (·)α the function (·)α : C \ (−∞, 0] → C : z
→ exp α log z,

(3.1) √

where log denotes the principal branch of the logarithm. We use · as a notation for √ (·)1/2 extended to 0 by 0 = 0. The same symbol (for example, C, L , M, R, α, . . . , δ with or without indices, N0 , and so on) may denote different constants in the bounds obtained in the rest of the paper (even in the same proof). However, we will indicate explicitly the dependence of such constants on parameters. First we are going to state some definitions and basic formulas related to the Hermite polynomials and Hermite functions. Then we will formulate some results on the asymptotics of the Hermite polynomials and Hermite functions in the complex plane. The asymptotic behavior of the Hermite polynomials at the scale we are interested in was first established in 1922 by Plancherel and Rotach. For a convenient form of these (and much more general) results we refer to the monograph [5] and the papers [3],[4] (in particular, Appendix B), and [6]. These results will be employed later in this section.

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Let s > 0, and let for every N , N ≥ 0, p˜ N (·, s) be a polynomial of degree N with a positive leading coefficient such that for ( p˜ N (·, s)) N ≥0 the relations p˜ M (x, s) p˜ N (x, s) exp (−x 2 /(2s)) d x = δ M,N , M, N = 0, 1, . . . , R

hold. The (monic) Hermite polynomials ( H˜ N (·, s)) N =0,1,... are defined by the expansion exp (γ x − γ 2 s/2) =

∞  N =0

γ , H˜ N (x, s) N! N

(3.2)

and for M, N = 0, 1, . . . satisfy the relations √ H˜ M (x, s) H˜ N (x, s) exp (−x 2 /(2s)) d x = δ M,N 2πs (M+N +1)/2 N ! R

so that p˜ N (·, s) =

1 (2π )1/4 s (2N +1)/4 (N !)1/2

H˜ N (·, s), N = 0, 1, . . . .

The polynomials p˜ N (·, s) satisfy the difference equations √ √ x p˜ N (x, s) = s 1/2 N + 1 p˜ N +1 (x, s) + s 1/2 N p˜ N −1 (x, s), N = 1, 2, . . . . (3.3) The Hermite functions (ϕ˜ N (·, s)) N =0,1,... defined by ϕ˜ N (x, s) = p˜ N (x, s) exp (−x 2 /(4s)), N = 0, 1, . . . , form an orthonormal sequence of functions in L 2 (R, λ), where λ is the Lebesgue measure in R. Let us introduce the standardized Hermite polynomials and Hermite functions (corresponding to the weight exp (−x 2 /2)) by the relations H N (·) = H˜ N (·, 1), p N (·) = p˜ N (·, 1), ϕ N (·) = ϕ˜ N (·, 1) so that H˜ N (·, s) = s N /2 H N (· s −1/2 ), p˜ N (·, s) = s −1/4 p N (· s −1/2 ), ϕ˜ N (·, s) = s −1/4 ϕ N (· s −1/2 ).

(3.4)

The above relations for the Hermite polynomials imply that the Hermite functions satisfy, for every k ≥ 1, the following system of differential equations: √ x ϕk (x) = − ϕk (x) + kϕk−1 (x), 2 √ x (3.5) ϕk−1 (x) = − kϕk (x) + ϕk−1 (x). 2 The reproducing kernel K˜ N (·, ·, s) of the orthogonal projection in L 2 (R, λ) onto the linear span of ϕ˜0 (·, s), . . . , ϕ˜ N −1 (·, s) is given by K˜ N (x, y, s) =

N −1  k=0

ϕ˜k (x, s)ϕ˜k (y, s).

(3.6)

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Note that the correlation functions of the GUE can be expressed in terms of the latter kernel as  n GUE,s Rn,N (x1 , . . . , xn ) = det K˜ N (xi , x j , s) (3.7) i, j=1

(see [11], (6.2.7)). Further, setting K N (x, y) = K˜ N (x, y, 1), we obtain K˜ N (x, y, s) = s −1/2 K N (xs −1/2 , ys −1/2 ).

(3.8)

The Christoffel - Darboux formula for the kernels K˜ N and K N reads ϕ˜ N (x, s)ϕ˜ N −1 (y, s) − ϕ˜ N −1 (x, s)ϕ˜ N (y, s) K˜ N (x, y, s) = x−y

(3.9)

and K N (x, y) =

ϕ N (x)ϕ N −1 (y) − ϕ N −1 (x)ϕ N (y) . x−y

(3.10)

Note that the kernels K˜ N , K N enjoy the property K˜ N (−z 1 , −z 2 , s) = K˜ N (z 1 , z 2 , s), K N (−z 1 , −z 2 ) = K N (z 1 , z 2 ).

(3.11)

As to K N , this is a consequence of (3.10) and the fact that the functions (ϕ N (·)) N ≥0 are even or odd depending on N being even or odd; for K˜ N it follows from (3.8). The following integral representation of the reproducing kernel is a version of formula (4.56) in [8]: K N (x, y) = N ∞ (ϕ N (x + τ )ϕ N −1 (y + τ ) + ϕ N −1 (x + τ )ϕ N (y + τ ))dτ 2 0

(3.12)

so that K˜ N (x, y, s) =          (3.13) N ∞ ϕ N √xs + τ ϕ N −1 √ys + τ + ϕ N −1 √xs + τ ϕ N √ys + τ dτ. 2s 0 The relations (3.4), (3.8), and our extension of the function (·)α to C \ (−∞, 0) allow us to continue H˜ N (x, ·), p˜ N (x, ·), ϕ˜ N (x, ·) and K˜ N (x, y, ·) to this domain analytically in the parameter s. The relations (3.2) - (3.11) remain valid under these continuations. So does (3.13) whenever the integrals there are well-defined. Denote by w the analytic continuation of the standard Wigner density (1.5) to the domain C \ ((−∞, 2] ∪ [2, ∞)) . Let us define for every α ∈ (0, 2) and β > 0 the sets Sα,β and Sα,β by the relations Sα,β = {z ∈ C : | z| < 2 − α, |u| < β}

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and Sα,β = {z ∈ C : | z| ≤ 2 − α, |u| ≤ β}. √ Set for H ∈ R d(H ) = 1 + i H and observe that |d(H )| = (1 + H 2 )1/4 ,

√ 1 + H2 + 1 , d(H ) = 2

√ 1 + H2 − 1 , d(H ) = sgn H 2

(3.14)

( d(H ))2 − (d(H ))2 = 1.

(3.17)

| (ud(H ))| ≥ |(ud(H ))|

(3.18)

(3.15)

(3.16)

and

It is clear from (3.17) that

for every u ∈ R. Note that for every b ≥ 0 the equation |d(H )| = b has a unique nonnegative solution Hb =



(1 + 2b2 )2 − 1.

(3.19)

Lemma 3.1. Let α, β, and u be real numbers such that α ∈ (0, 1), 0 < β < α, and u ∈ (−2 + 2α, 2 − 2α). Then the following assertions hold true: 1. For every b ∈ [−∞, ∞] the relations |H | < H|b| and |(d(H ))| < |b| are equivalent; the relations |H | ≤ H|b| and |(d(H ))| ≤ |b| are also equivalent (we set here H∞ = ∞, (d(∞)) = ∞, and (d(−∞)) = −∞). 2. For every real H we have ud(H ) ∈ Sα,β whenever |(ud(H ))| ≤ β; 3. If for a certain real H the relation |H | ≤ |Hβ/2 | (or, equivalently, | (d(H )) | ≤ β/2) holds, it follows that ud(H ) ∈ Sα,β . Proof. It is clear from (3.19) and the definition of Hb that for b ∈ [0, ∞] and H ∈ [0, ∞] the functions b
→ Hb and H
→ |d(H )| are strictly increasing mutually inverse mappings. This shows equivalence of the conditions in point 1 of the lemma for the functions in question restricted to [0, ∞]. The same is true for b, H ∈ [−∞, ∞] since both functions are even. This proves assertion 1. For every H such that | (ud(H ))| ≤ β

(3.20)

| (ud(H ))| < 2 − α

(3.21)

we have

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√ since it follows from the relation (3.17) and the assumptions β ≤ α, u ∈ (−2+2α, 2 − 2α) that    2 2 | (ud(H ))| = u + (u (d(H ))) ≤ (2 − 2α)2 + β 2 < (2 − 2α)2 + α 2 < 2 − α. In view of the definition of Sα,β , the second assertion is proved. The third assertion is an immediate consequence of the second one since |u| ≤ 2.

 Corollary 3.1. Let α, β, and u satisfy the assumptions of Lemma 3.1, and assume that the relation ud(H ) ∈ / Sα,β holds for some real H. Then we have |(ud(H ))| > β. Moreover, the relation |(ud(H ))| > β holds for every real H such that |H | ≥ |H |. Proof. The corollary follows immediately from the second and the first assertions of Lemma 3.1 combined with the fact that, by (3.16), |d(H )| is a strictly increasing function of |H |.

 In the next lemma the main result on convergence is formulated. Lemma 3.2. For every number α ∈ (0, 1) there exists a positive number H = H (α) such that for every A > 0 the relation      t1 t2 1 −1 ˜ u+ d(H ), u + d(H ), N KN N w(u) N w(u) N w(u) (3.22) sin (π(t1 − t2 )d(H )w(ud(H ))/w(u)) −→ N →∞ π(t1 − t2 )d(H ) holds uniformly for all real numbers u, t1 , t2 , and H provided that u ∈ [−2+2α, 2−2α], |t1 | ≤ A, |t2 | ≤ A, and |H | ≤ H . Proof. Note that for every α ∈ (0, 1) there exists β = β (α) > 0 such that for every A > 0 the relation   z1 z2 1 1 sin π(z 1 − z 2 ) ,v + , → (3.23) K˜ N v + N w(v) N w(v) N w(v) N N →∞ π(z 1 − z 2 ) holds uniformly for all complex numbers v ∈ Sα,β (α) , z 1 , and z 2 such that |z 1 | ≤ A, |z 2 | ≤ A. This assertion (and more general ones involving some class of weights) is implicitly contained in the papers [3] and [4] (see also the monograph [5]). Actually, Lemma 6.1 in [3] establishes the desired result for real u, z 1 and z 2 . However, the same reasoning applies to the complex u, z 1 , and z 2 satisfying the assumptions just made provided that β is chosen to be sufficiently small in accordance with α. The key ingredient of the proof in these references, the boundedness property of a certain derivative, is established in [3] and [4] for some complex neighborhood of an arbitrary real point u ∈ [−2 + 2α, 2 − 2α] (see relation (4.122) in [3]) which allows to bound this derivative in a rectangular strip Sα,β (α) . Lowering β if necessary, we can substitute β (α) by some β(α) ≤ α/4. Note that the function w(·) has no zeroes in C \ ((−∞, 2] ∪ [2, ∞)) ⊃ Sα,β(α) . Picking a point v ∈ Sα,β(α) and setting in (3.23) z i = z i w(v)/w(v ) (i = 1, 2), we obtain     sin π(z 1 − z 2 )w(v)/w(v ) z 1 z 2 1 1 ˜ ,v + , → . KN v + N w(v ) N w(v ) N w(v ) N N →∞ π(z 1 − z 2 ) (3.24)

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Note that for v ∈ S α,β(α) we have C −1 ≤ |w(v)| ≤ C with some C > 1. This fact and (3.23) imply that for every A > 0 (3.24) holds uniformly in v, v ∈ Sα,β(α) and |z 1 | ≤ A, |z 2 | ≤ A. According to assertion 3 of Lemma 3.1, for every u ∈ [−2 + 2α, 2 − 2α] and every H ∈ [−Hβ(α)/2 , Hβ(α)/2 ] we have ud(H ) ∈ Sα,β(α) . Moreover, it is clear from −1

< |d(H )| < C with some (3.14) that for H ∈ [−Hβ(α)/2 , Hβ(α)/2 ] we have C positive constant C > 1 depending on α. Then, setting v = u ∈ [−2 + 2α, 2 − 2α], v = ud(H ) ∈ Sα,β(α) , z i = ti d(H ) (i = 1, 2), we obtain (3.22).

 Remark 3.1. It is sometimes convenient to use the following reformulation of (3.23) and the related assumptions: For every number α ∈ (0, 1) there exists a real number β = β(α) ≤ α/4 such that for every A > 0 the relation     1 sin (N π(v1 − v2 )w(v))   K˜ N v1 , v2 , N −1 − 0 N  N−→ →∞ N π(v1 − v2 ) holds uniformly for all complex numbers v, v1 , v2 satisfying the constraints v ∈ Sα,β(α) , |v1 − v| ≤ A/N , |v2 − v| ≤ A/N .

 We will now derive upper bounds for the modulus of the kernel  1 ˜  K N ·, ·, N −1 N valid for various values of its arguments. We will use asymptotics of two types for the Hermite functions: one of them is valid in a narrow strip around the interval [−2 + α, 2 − α]; another one applies in the complement to some neighborhood of [−2, +2]. With this asymptotics, after obtaining certain intermediate bounds for the quantity  −1 −1 ˜ |N K N u + t1 /N )d(H ), u + t2 /N )d(H ), N | , we will finally determine for this quantity an upper estimate which does not depend on H ∈ R. This bound and analogous bounds for the correlation functions will be derived in Proposition 3.1. In the next two lemmas we present upper estimates for the Hermite functions. These estimates are immediate consequences of the Plancherel-Rotakh-type asymptotics. First we consider a bound valid in the strip S α,β(α) , where β(α) was defined in the proof of Lemma 3.2. Lemma 3.3. For any α ∈ (0, 1) there exist a constant C(α) > 0 such that we have   √    (3.25) ϕ N z N  ≤ (C(α)) N and

  √    ϕ N −1 z N  ≤ (C(α)) N −1

(3.26)

for every N ∈ N and z ∈ S α,β(α) . Proof. First we recall a known result about the Plancherel-Rotach asymptotics for the Hermite functions in a rectangular strip (Theorem 2.2, part (ii), in [4]; the particular case of the Hermite functions for the measure exp (−x 2 )d x is considered there in Appendix

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215

B). This strip in the complex plane contains, up to some neighborhoods of the ends, the interval where the zeroes of the Hermite functions asymptotically concentrate. Set ψ : C \ ((−∞, −1] ∪ [1, ∞)) → C : z
→

1 (1 − z)1/2 (1 + z)1/2 . 2π

The function arcsin is defined as the inverse function of  π → C \ ((−∞, −1] ∪ [1, ∞)). sin : z ∈ C : | (z)| < 2 In our notation, the statement in [4] reads: for every α ∈ (0, 1) uniformly in z ∈ {z ∈ C : | (z )| ≤ 1 − α, |(z )| ≤ α} we have √ ϕ˜ N ( 2N z, 1/2)

2 = √ (1 − z)−1/4 (1 + z)−1/4 π 2N      z 1 1 1+O × cos N π ψ(y)dy + arcsin z 2 N 1     z 1 1 . (3.27) + sin N π ψ(y)dy − arcsin z O 2 N 1 z The analytic functions z
→ 1 ψ(y)dy and arcsin are bounded on the set {z ∈ C : | (z )| ≤ 1 − α, |(z )| ≤ α}. Representing cos and sin as the sum and the difference of the exponentials, it follows from (3.27) (omitting the multiplier N −1/4 ) that the modulus of the left-hand side of (3.27) is bounded above by the N th power of a positive number depending on α. For every α ∈ (0, 2), due to the relations   z√  √ z √ −1/4 ϕ N (z N ) = 2 ϕ˜ N √ N , 1/2 = 2−1/4 ϕ˜ N 2N , 1/2 , 2 2 √ a similar bound by the N th power of some number depending on α holds for ϕ N (z N ) uniformly for z ∈ S α,α . For     √ ϕ N −1 (z N ) = ϕ N −1 z (N − 1) N /(N − 1) , √ we also have a bound of such type in S α,α/4 ⊂ S α/2,α/2 . Indeed, set ρ N = N /(N − 1), and let N0 = N0 (α) be defined by the equation ρ N0 = (2 − α/2)/(2 − α). Then we have ρ N ≤ 2 and ρ N S α,α/4 ⊂ S α/2,α/2 whenever N ∈ N satisfies N ≥ N0 . This means that √ for z ∈ S α,α/4 both z ∈ S α/2,α/2 and z N /(N − 1) ∈ S α/2,α/2 hold for every N ≥ N0 . Therefore, the conclusion of the lemma is valid for such values of N . Choosing a larger constant C(α) if necessary and recalling that β(α) ≤ α/4, the proof is completed.

 We will obtain now some estimates valid for the Hermite functions in the domain |z| ≥ δ > 0. Lemma 3.4. For every δ > 0 there exist constants C(δ) and M(δ) such that the inequalities  √   (3.28) ϕ N (z N ) ≤ C(δ)N −1/4 M N (δ) |z| N

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and

  √    ϕ N −1 z N  ≤ C(δ)(N )−1/4 M N −1 (δ) |z| N −1

(3.29)

hold for every N ∈ N and every z satisfying | z| ≥ |z| ≥ δ. Proof. Again we start with stating some known result about the Plancherel-Rotach asymptotics for the Hermite polynomials in the complex plane (Theorem 7.185 in [5], see also [18] and the references therein).√In our √ notation, uniformly in z from every compact set contained in (C ∪ {∞}) \ [− 2, 2], we have √    1 U (z) exp (N (z − z 2 − 2)2 /4) −1 ˜ , (3.30) 1+O H N (z, (2N ) ) = √ 2 N 2 N (z − z − 2) where

 U (z) :=

z − 2 1/4 z+2 ) +( z+2 z−2

1/4 .

From this relation we obtain for the monic orthogonal Hermite polynomials (H N ) N ≥0 with respect to the weight exp (−x 2 /2) that √ √ H N (z N ) = (2N ) N /2 H˜ N (z/ 2, (2N )−1 ) √    1 exp (N (z − z 2 − 4)2 /8) 1+O (3.31) = 2 N −1 N N /2 U (z) √ 2 N N (z − z − 4) uniformly for z from every compact set contained in (C ∪ {∞}) \ [−2, 2]. Passing to the normalized polynomials we see that √ p N (z N ) =

1 (2π )1/4 (N !)1/2

√ H N (z N )

√    1 2 N N N /2 U (z) exp (N (z − z 2 − 4)2 /8) 1 + O , (3.32) = √ 1/4 1/2 2(2π ) (N !) N (z − z 2 − 4) N √ 1 and, in view of the Stirling formula N ! = 2π N N + 2 e−N (1 + O(1/N )), we obtain √    √ U (z) exp (N ( 21 + log 2 + (z − z 2 − 4)2 /8)) 1 1+O p N (z N ) = √ 1/2 1/4 2 N N 2(2π ) (N ) (z − z − 4) (3.33) so that we have

√    U (z) exp ( N2 (1 + ((z − z 2 − 4)/2)2 )) 1 p N (z N ) =  N 1 + O √ N 2(2π )1/2 (N )1/4 (z − z 2 − 4)/2 √

(3.34)

uniformly in z from any compact subset of (C ∪ {∞}) \ [−2, 2]. For every complex number z ∈ C \ [−2, 2], let x = x(z) be the root of the equation x + x −1 = z satisfying |x| < 1. Then we have    x = x(z) = z − z 2 − 4 /2,

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√ where z
→ κ(z) = z 2 − 4 is, by definition, a univalent analytic function in C\[−2, 2] satisfying the relations κ 2 (z) = z 2 − 4 and κ(t) > 0 for t > 2. For every r ∈ (0, 1) the equation |x(z)| = r and the inequality |x(z)| ≤ r define the ellipse ( z)2 /(r −1 + r )2 + (z)2 /(r −1 − r )2 = 1

(3.35)

with focal points −2, 2 and the closed exterior domain Er of this ellipse, respectively. Thus for z ∈ Er we have       |x(z)| =  z − z 2 − 4 /2 ≤ r < 1, and  2  (z − z 2 − 4)/2 ≤ r 2 < 1.

(3.36)

Note that for every r ∈ (0, 1), min |z| = r −1 − r,

|x(z)|=r

which for z ∈ Er implies 2 1   = = |z − x(z)| ≤ |z| + 1 ≤ (1 + |z|−1 )|z| √   |x(z)| 2 − z − 4 z    1 |z|. ≤ 1 + −1 r −r

(3.37)

Further, for z ∈ Er (0 < r < 1) we can write      |z − 2| = (x(z) − 1) + x −1 (z) − 1  = 2|x(z) − 1||x −1 (z) − 1| = 2|x(z)|−1 |1 − x(z)|2 ≥ 2r −1 |1 − x(z)|2 ≥ 2r −1 |1 − r |2 = 2(r −1/2 − r 1/2 )2 , and analogously |z + 2| ≥ 2(r −1/2 − r 1/2 )2 . This implies that the function    z − 2 1/4  z + 2 1/4    |U (z)| =  +    z+2 z−2 is bounded above on Er by a constant depending on r ∈ (0, 1). Applying this estimate along with (3.36) and (3.37) to the relation (3.34) we arrive, for a fixed r ∈ (0, 1) and every z ∈ Er , at  √   (3.38)  p N (z N ) ≤ C1 (r )N −1/4 L N (r )|z| N .

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√ Now we set z n = z N /(N − 1) and note that, by convexity of C \ Er , z N ∈ Er if z ∈ Er . Hence, applying (3.38) to p N −1 , we see that for every z ∈ Er ,   √    √       p N −1 z N  =  p N −1 z N N − 1  ≤ C1 (r )(N − 1)−1/4 L N −1 (r )|z N | N −1 = C1 (r ) (1 + 1/(N − 1))(2N −3)/4 N −1/4 L N −1 (r )|z| N −1 . Since (1 + 1/(N − 1))(2N −3)/4 → e1/2 as N → ∞, we conclude that   √     p N −1 z N  ≤ C2 (r )(N )−1/4 L N −1 (r )|z| N −1 .

(3.39)

Observe that for every r ∈ (0, 1) the inequality z ≥ r −1 − r implies, by Eq. (3.35), that z ∈ Er . Finally set

δ δ2 r =− + +1 2 4 so that r is the solution of the equation r −1 − r = δ satisfying 0 < r < 1. We can now conclude from (3.38) and (3.39) that for every δ > 0 with some constants C(δ) and M(δ) the inequalities  √   (3.40)  p N (z N ) ≤ C(δ)N −1/4 M N (δ)|z| N and

  √     p N −1 z N  ≤ C(δ)(N )−1/4 M N −1 (δ)|z| N −1

(3.41)

hold for every natural N and every z with |z| ≥ δ. We pass now from Hermite polynomials to Hermite functions. Note that for every z satisfying | z| ≥ |z| we have   | exp (−z 2 /4)| ≤ exp (− 2 z + 2 z)/4 ≤ 1 (3.42) and, therefore, |ϕk (z)| ≤ | pk (z)| for every k ∈ N. Applying (3.40) and (3.41), the lemma follows.

(3.43) 

Corollary 3.2. For every real number δ > 0 and A > 0 there exist such constants C(δ, A) and M(δ) that the inequalities   √   (3.44) ϕ N (u + t/N ) d(H ) N  ≤ C(δ, A)M N (δ)|H | N and

  √   ϕ N −1 (u + t/N ) d(H ) N  ≤ C(δ, A)M N −1 (δ)|H | N −1

(3.45)

hold for every number u ∈ [−2, 2], t ∈ [−A, A], N ∈ N and H ∈ R such that | ((u + t/N )d(H ))| ≥ δ.

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Proof. Set v = (u + t/N ) d(H ). Since   √ |(d(H ))| = 1 + H 2 − 1 /2 ≤ |H |/2, we have that |(v)| ≤ (2 + (A/N ))|(d(H ))| ≤ (1 + (A/2N )) |H | and |(v)| N ≤ (1 + (A/2N )) N |H | N ≤ C(A)|H | N , since (1 + (A/2N )) N → exp (A/2). N →∞

Thus, we proved that |(v)| N ≤ C(A)|H | N .

(3.46)

In view of (3.18), | v| ≥ |v|, and we can apply Lemma 3.4. Omitting N −1/4 in the bound, the proof is completed.

 Now we will derive upper bounds for |N −1 K˜ N (·, ·, N −1 )| valid for various combinations of the values of its arguments. Lemma 3.5. For every real α ∈ (0, 1) and A > 0 there exists a constant R1 (α, A) > 0 such that for every real u ∈ [−2 + α, 2 − α], t1 , t2 ∈ [−A, A], H, and every N ∈ N we have   |N −1 K˜ N (u + t1 /N )d(H ), (u + t2 /N )d(H ), N −1 | ≤ (R1 (α, A))2N (3.47) whenever |(u + ti /N )d(H )| ≤ β(α/4)

(3.48)

holds for i = 1, 2. Proof. Set u i = u + ti /N , vi = (u + ti /N )d(H ) for i = 1, 2 and u = (u 1 + u 2 )/2, v = (v1 + v2 )/2. Let the number N0 = N0 (α, A) be large enough so that the inequality A/N ≤ α/2

(3.49)

holds for every N ≥ N0 . Observe that by (3.49) u i ∈ [−2 + α/2, 2 − α/2], i = 1, 2, for N ≥ N0 . Then it follows from (3.48) and Corollary 3.1 that v1 , v2 ∈ Sα/4,β(α/4) . Therefore, v ∈ Sα/4,β(α/4) , too.

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First assume now that we have |v1 − v2 | ≤ 1/N .

(3.50)

Then Remark 3.1 (with A = 1) applies, and it is clear that, for v1 , v2 , v = (v1 + v2 )/2 ∈ Sα/4,β(α/4) with v1 , v2 satisfying (3.50), the function | sin (N π(v1 − v2 )w(v)) / (N π(v1 − v2 )) | is bounded above by a constant depending on α and A. Therefore, by Remark 3.1, this applies to the left hand side of (3.47) as well. Now, instead of (3.50), we assume that the inequality |v1 − v2 | ≥ 1/N holds. Then, by the Christoffel-Darboux formula (3.9), we have   √  √ |N −1 K˜ N v1 , v2 , N −1 | = N 1/2 |K N N v1 , N v2 |  √  √ √ √ √ ≤ N |ϕ N ( N v1 )||ϕ N −1 ( N v2 )| + |ϕ N ( N v2 )||ϕ N −1 ( N v1 )| . Applying Lemma 3.3 and enlarging, if necessary, the constant in the bound, the lemma is proved.

 Lemma 3.6. For every α ∈ (0, 1) and A > 0 there exists a constant R2 (α, A) such that for every u ∈ [−2 + α, 2 − α], t1 , t2 ∈ [−A, A], H ∈ R and N ∈ N we have the relation   |N −1 K˜ N (u + t1 /N )d(H ), (u + t2 /N )d(H ), N −1 | ≤ Γ (N ) (R2 (α, A)) N H N (3.51) whenever none of the following pairs of inequalities | ((u + ti /N )d(H ))| ≤ β(α/4), i = 1, 2, | ((u + ti /N )d(H ))| ≥ β(α/4)/2, i = 1, 2,

(3.52)

holds. Proof. Set u i = (u + ti /N ), vi = u i d(H ), i = 1, 2. Since (3.52) does not hold we have |(vi )| < β(α/4)/2 and |(v j )| > β(α/4) for some choice of i, j ∈ {1, 2}, i = j. This implies |v2 − v1 | ≥ β(α/4)/2, and by the Christoffel-Darboux formula (3.9) we have 1 N

   √  √  √ ˜    N v1 , N v2   K N v1 , v2 , N −1  = N K N √  √ √ √ √ 2 N  ≤ |ϕ N ( N v1 )||ϕ N −1 ( N v2 )| + |ϕ N ( N v2 )||ϕ N −1 ( N v1 )| . β(α/4)

Assume now that, for example, |(v1 )| < β(α/4)/2 and |(v2 )| > β(α/4). In addition we have that u 1 ∈ [−2 + α/2, 2 − α/2] for N ≥ N (α, A). Then, by assertion 2 of Lemma 3.1, v1 ∈ S α/4,β(α/4)/2 ⊆ S α/4,β(α/4) , and we may apply Lemma 3.3 (with the √ √ parameter value α/4 instead of α) to bound ϕ N ( N v1 ) and ϕ N −1 ( N v1 ). Furthermore,

Limit Correlation Functions for Fixed Trace Random Matrix Ensembles

221

√ √ we apply Corollary 3.2 with δ = β(α/4) to bound ϕ N ( N v2 ) and ϕ N −1 ( N v2 ). Therefore, we arrive at the inequality 1 ˜ | K N ((u + t1 /N )d(H ), (u + t2 /N )d(H ), 1/N ) | N √ ≤ C(α, A) N Γ (N )M N (β(α/4))H N . Replacing the quantity M(β(α/4)) by an appropriate larger constant, we reduce this inequality to the form (3.51). Moreover, this way the bound can be made valid for all N ∈ N, which proves the lemma.

 Lemma 3.7. For every δ > 0 and certain constants C(δ), M(δ) the inequality 1 ˜  | K N z1, z2 , N

1 N



| ≤ C(δ)Γ (N )R 2N (δ) ((z 1 )) N ((z 2 )) N

(3.53)

holds for every complex number z 1 , z 2 such that |z i | ≥ δ and | z i | ≥ |z i | (i = 1, 2), and every natural number N ≥ δ −2 . Proof. First we consider the case when z 1 z 2 > 0. Let f a, p (x) =

1 a p x p−1 e−ax , p > 0, x > 0, Γ ( p)

be the gamma density [7] with the parameters a > 0, p > 0, and let P ≥ 0 and N ≥ 1 be some integers. Then for every complex number z satisfying z ≥ |z| > 0 we have ∞ ∞ N 2 2 |z + θ |2P e−N (z+θ) /2 dθ = |ξ + iz|2P e− 2 (ξ +iz) dξ 0 z ∞ ∞ N N 2 2 2 2P − 2 (ξ +iz) ≤ |ξ + iz| e dξ = (ξ 2 + (z)2 ) P e− 2 (ξ −(z) ) dξ |z|

=e

N (z)2



∞

|z|

2 P e N (z) N P+1 |z|

2

≤

|z|

(ξ 2 + (z)2 ) P e



∞ |z|



− N2

(ξ 2 +(z)2 )

dξ

  P N 2     2 N ξ 2 + (z)2 /2 e− 2 (ξ +(z) ) d N ξ 2 + (z)2 /2

∞ 2 2 P e N (z) Γ (P + 1) = P+1 η e dη = P+1 f 1,P (η)dη N |z| N (z)2 N |z| N (z)2   2 N (z)2 2 P e N (z) [N (z)2 ] P −N (z)2 1+ = P+1 Γ (P + 1)e + ··· + N |z| 1! P!   P 2 2 P N (z) [N (z) ] 2 Γ (P + 1) 1+ + ··· + , = P+1 N |z| 1! P! 2P

2 e N (z)



∞

P −η

(3.54) where we used a well known formula ([7], p. 11) while integrating f 1,P . If |z| ≥ δ, then N (z)2 ≥ 1 for N ≥ δ −2 , and the bound (3.54) yields ∞ 2 |z + θ |2P e−N (z+θ) /2 dθ ≤ e2 P N −1 Γ (P + 1)|(z)|2P−1 (3.55) 0

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whenever z ≥ |z| > 0. In particular, for P = N and P = N − 1 we have ∞ 2 |z + θ |2N e−N (z+θ) /2 dθ ≤ e2 N N −1 Γ (N + 1)|(z)|2N −1

(3.56)

0

and

∞

|z + θ |2(N −1) e−N (z+θ)

2 /2

dθ ≤ e2 N −1 N −1 Γ (N )|(z)|2N −3 .

(3.57)

0

Thus, in view of representation (3.12) and the bounds in Lemma 3.4, for z 1 , z 2 such that |z i | ≥ δ and z i ≥ |z i | (i = 1, 2) we have   √  √ N z1, N z2 | |N −1 K˜ N z 1 , z 2 , N −1 | = |N −1/2 K N  ∞  √ √  −1/2  = (2N ) ϕ N ( N z 1 + τ )ϕ N −1 ( N z 2 + τ )  0   √ √ + ϕ N −1 ( N z 1 + τ )ϕ N ( N z 2 + τ ) dτ   ∞  √ √  −1/2  =2 ϕ ( N (z + θ ))ϕ ( N (z 2 + θ )) N 1 N −1  0   √ √ + ϕ N −1 ( N (z 1 + θ ))ϕ N ( N (z 2 + θ )) dθ  ∞ 2 2 2 −1/2 2N −1 ≤ C (δ)N |z 1 + θ | N e−N (z 1 +θ) /4 |z 2 + θ | N −1 e−N (z 2 +θ) /4 M (δ) 0  2 N −1 −N (z 1 +θ)2 /4 + |z 1 + θ | e |z 2 + θ | N e−N (z 2 +θ) /4 dθ ≤ C 2 (δ)N −1/2 M 2N −1 (δ)  ∞ 2 |z 1 + θ |2N e−N (z 1 +θ) /2 dθ × 0



∞

+ 0

∞

|z 2 + θ |2(N −1) e−N (z 2 +θ)

2 /2

1/2 dθ

0

|z 1 + θ |

2(N −1) −N (z 1 +θ)2 /2

e

dθ

∞

(|z 2 + θ |

2N −N (z 2 +θ)2 /2

e

1/2  dθ

.

0

(3.58) Combining this bound with relations (3.56) and (3.57) we find that   | K˜ N z 1 , z 2 , N −1 | e21/2 C 2 (δ)M 2N −1 (δ)2 N Γ (N )|(z 1 )| N |(z 2 )| N ≤ . (3.59) N δ 2 N 3/2 The bound (3.59) applies as well when z 1 , z 2 satisfy |z i | ≥ δ and z i ≤ −|z| (i = 1, 2). Indeed, this follows from (3.11). Finally, we consider z 1 , z 2 such that | z i | ≥ |z| ≥ δ (i = 1, 2) and z 1 z 2 < 0. In this case we have 1 1 1 ≤ ≤ ≤ (2δ)−1 |z 2 − z 1 | | (z 1 )| + | (z 2 )| |(z 1 )| + |(z 2 )|

Limit Correlation Functions for Fixed Trace Random Matrix Ensembles

223

and, by (3.9),  √  √ 1 1 ˜  | K N z 1 , z 2 , N −1 | = √ |K N N z1, N z2 | N N  √ √ √ √ 2  ≤ √ |ϕ N ( N z 1 )||ϕ N −1 ( N z 2 )| + |ϕ N ( N z 2 )||ϕ N −1 ( N z 1 )| . δ N Applying Lemma 3.4 we obtain   |N −1 K˜ N z 1 , z 2 , N −1 | ≤ C(δ)N −1 M 2N −1 (δ)2 N Γ (N ) ((z 1 )) N ((z 2 )) N .

(3.60)

√ Setting R(δ) = M(δ) 2 and comparing 3.59 and 3.60 with the factor N −1 omitted, the proof is completed.

 Corollary 3.3. Let α ∈ (0, 1) and A be positive numbers. Then there exist positive constants C(α, A) and R2 (α) such that for every N ∈ N and real numbers u ∈ [−2, 2], t1 , t2 ∈ [−A, A], and H we have   1       K˜ N u + t1 d(H ), u + t2 d(H ), 1  ≤ Γ (N ) (R3 (α, A))2N H 2N (3.61) N N N N whenever the inequalities | ((u + ti /N )d(H ))| ≥ β(α/4)/2, i = 1, 2, hold. Proof. Set vi = (u + ti /N ) d(H ) for i = 1, 2. From (3.18) we conclude that | vi | ≥ |vi |, i = 1, 2. It follows from (3.46) that |(vi )| N ≤ C(A)|H | N , i = 1, 2. Then by Lemma 3.7 with δ = β(α/4)/2 we have   1       K˜ N u + t1 d(H ), u + t2 d(H ), 1  ≤ C(β(α/4)/2, A)Γ (N )R 2N (β(α/4)/2)H 2N N N N N for every N ≥ (β(α/4)/2)−2 . Substituting R in the latter inequality by a proper larger constant depending also on A, we obtain (3.61) which is valid for all N .

 Proposition 3.1. For every α ∈ (0, 1) and A > 0 there exists a constant R(α, A) such that for every u ∈ [−2 + α, 2 − α], t1 , . . . , tn ∈ [−A, A] and H ∈ R we have the relations    N 1       K˜ N u + t1 d(H ), u + t2 d(H ), 1  ≤ Γ (N ) R(α, A)(1 + H 2 ) (3.62) N N N N and 1 GUE, (1+i H )N

|Rn,N

(u + Nn

t1 N,...,u

+

tn N )|

≤ n! (Γ (N ))n (R(α, A))n N (1 + H 2 )

  1 n N+ 4 .

(3.63)

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F. Götze, M. Gordin

Proof. For every α ∈ (0, 1) and A > 0, every combination of t1 ∈ [−A, A], t2 ∈ [−A, A], H ∈ R, and N ∈ N satisfies the assumptions of at least one of the assertions in Lemma 3.5, Lemma 3.6 and Corollary 3.3. Hence, at least one of these   upper bounds is valid. Writing R(α, A) = max (R1 (α, A))2 , R2 (α, A), (R3 (α, A))2 , we obtain   1       K˜ N u + t1 d(H ), u + t2 d(H ), 1  N N N N    ≤ max (R1 (α, A))2N , Γ (N ) (R2 (α, A)) N H N , Γ (N ) (R3 (α, A))2N H 2N   ≤ (R(α, A)) N max 1, Γ (N )H N , Γ (N )H 2N   < (R(α, A)) N max 1, Γ (N )(1 + H 2 ) N /2 , Γ (N )(1 + H 2 ) N = Γ (N ) (R(α, A)) N (1 + H 2 ) N , and the bound (3.62) is proved. GUE,1/((1+i H )N ) Now we will derive a bound for the correlation function Rn,N . In view of formulas (3.7) and (3.8) we have GUE,σ s GUE,s −1/2 Rn,N (x1 , . . . , xn ) = σ −n/2 Rn,N (σ x1 , . . . , σ −1/2 xn ).

It follows from (2.5) that



GUE,σ s (x1 , . . . , xn ) = Rn,N

where

γm,s (u) =

∞ 0

HSE,u/N 2

Rn,N

γ N 2 ,σ s (u)du,

(2m/2 s −m/2 Γ (m/2))−1 u (m/2)−1 exp (−u/(2s)), if u ≥ 0, 0, if u < 0.

(3.64)

(3.65)

(3.66)

GUE,s (x1 , . . . , xn ) is an entire function, formula (3.64) can be Since (x1 , . . . , xn )
→ Rn,N GUE,σ s (x1 , . . . , xn ) in the parameter σ to used to analytically continue the function Rn,N the domain σ > 0. On the other hand, an analytic continuation can be obtained from formulas (3.65) and (3.66) (we omit the check of this fact requiring routine bounds for γm,s in the parameter s ∈ C). Since these continuations obviously agree, both (3.64) and (3.65) are valid for complex s with s > 0. In particular, we have the identity GUE,1/((1+i H )N )

Rn,N

GUE,1/N

(x1 , . . . , xn ) = d n (H )Rn,N

(x1 d(H ), . . . , xn d(H )). (3.67)

Along with the determinantal representation (3.7) (which also admits an analytic continuation) and inequality (3.62), this leads to GUE,1/((1+i H )N )

N −n |Rn,N = ≤

(u + t1 /N , . . . , u + tn /N )|

GUE,1/N ) N |d(H )| Rn,N ((u + t1 /N )d(H ), . . . , (u + t1 /N )d(H ))| n!|d(H )|n max0≤i< j≤n |N −1 K˜ ((u + ti /N )d(H ), (u + t j /N )d(H ))|n  n n N 2 N −n

≤ n!|d(H )|

n

Γ (N ) (R(α, A)) (1 + H )

= n! (Γ (N ))n (R(α, A))n N (1 + H 2 )n(N +1/4) , since |d(H )| = (1 + H 2 )1/4 by (3.14).



(3.68)

Limit Correlation Functions for Fixed Trace Random Matrix Ensembles

225

Proof of Theorem. As explained in the end of Sect. 2 we need to show that   ∞ √ 1 t1 tn GUE,1/((1−i h 2/N )N ) u+ ,...,u + dh φ 2 (h)Rn,N (N w(u))n −∞ N N w(u) N w(u) n  sin π(ti − t j ) 1 −→ √ det (3.69) N →∞ π(ti − t j ) i, j=1 2π uniformly in u, t1 , . . . , tn subject to the conditions formulated in the the statement of the theorem. Throughout the proof we will assume that these conditions are satisfied and GUE,· omit the arguments of Rn,N whenever that is possible. The existence of the integral in the left hand side of relation (3.69) will be a consequence of our estimates. GUE,s GUE,s (−u 1 , . . . , −u n ) = Rn,N (u 1 , . . . , u n ) since K N (−u 1 , −u 2 ) = Note that Rn,N K N (u 1 , u 2 ) by (3.11). In view of this property it suffices to prove (3.69) for u ≥ 0 only. Observe that for real h, φ N 2 (−h) = φ N 2 (h) and √ GUE,1/((1+i h 2/N )N )

Rn,N

√ GUE,1/((1−i h 2/N )N )

(u 1 , . . . , u n ) = Rn,N

(u 1 , . . . , u n )

for u 1 , . . . , u n ∈ R which shows that (3.69) is established if we prove ∞  √  1 GUE,1/((1+i h 2/N )N ) dh φ 2 (h)Rn,N N (N w(u))n 0 n  sin π(ti − t j ) 1 −→ √ det . N →∞ 2 2π π(ti − t j ) i, j=1

(3.70)

It follows from the central limit theorem for densities that ∞   1 φ N 2 (h) dh −→ √ . (3.71) N →∞ 2 2π 0 √ Here φ N 2 (·) is a prelimiting characteristic function and 1/2 2π is half the value of the limiting standard Gaussian density at 0. Thus, to prove (3.70) it suffices to establish the following relation: ∞  √  1 GUE,1/((1+i h 2/N )N ) dh φ (h)R 2 N n,N (N w(u))n 0 n  ∞   sin π(ti − t j ) (3.72) φ N 2 (h) dh − det π(ti − t j ) i, j=1 0 =: I (u, N ) − J (N ) −→ 0 N →∞

under the same uniformity constraints as above. Omitting the arguments t1 , . . . , tn , we set    n sin π(ti − t j )d(H )w(ud(H ))/w(u) Sn (u, H ) = det π(ti − t j )d(H ) i, j=1

226

F. Götze, M. Gordin

and

 Sn = Sn (u, 0) = det

sin π(ti − t j ) π(ti − t j )

n . i, j=1

For a given α ∈ (0, 1), let H (α) > 0 be a number whose existence is guaranteed by Lemma 3.2. Let > 0 be an arbitrary positive number, and H0 = H0 (α, A, ) ∈ (0, H (α)) be small enough to ensure that √ |Sn (u, H ) − Sn | ≤ 2 2/π (3.73) for all u ∈ [−2 + α, 2 − α], |H | ≤ H0 and t1 , . . . , tn satisfying |t1 | ≤ A, . . . , |tn | ≤ A. Such a number exists because of continuity of the sin-kernel √ and continuity near 0 of the function d(·). Performing the change of variable H = h 2/N , we may write ∞  √  1 GUE,1/((1+i h 2/N )N ) dh φ (h)R I (u, N ) = 2 N n,N n (N w(u)) 0 ∞   √ N GUE,1/((1+i H )N ) dH = √ φ N 2 (N H/ 2)Rn,N 2(N w(u))n 0 H0 ∞ N N = √ ... dH + √ ... dH 2(N w(u))n 0 2(N w(u))n H0 (3.74) =: I1 ( , u, N ) + I2 ( , u, N ). Quite analogously, we have ∞ √    N Sn ∞  φ N 2 (h) dh = √ φ N 2 (N H/ 2) d H J (N ) = Sn 2 0 0 √  √  N Sn ∞  N Sn H0  φ N 2 (N H/ 2) d H + √ φ N 2 (N H/ 2) d H = √ 2 0 2 H0 =: J1 ( , N ) + J2 ( , N ). (3.75) With this notation we need to show that I1 ( , u, N ) − J1 ( , N ) −→ 0,

(3.76)

I2 ( , u, N ) −→ 0,

(3.77)

N →∞

N →∞

and J2 ( , N ) −→ 0 N →∞

(3.78)

uniformly in u ∈ [−2 + α, 2 − α] and ti ∈ [−A, A], i = 1, . . . , n. By Proposition 3.1, for every , we obtain |I2 ( , u, N )| ∞   N   n! (Γ (N ))n (R(α, A))n N (1 + H 2 )n(N +1/4) φ N 2 ( N√H ) d H. ≤√ 2 2(w(u))n H0 (3.79)

Limit Correlation Functions for Fixed Trace Random Matrix Ensembles

227

Since √ 2 |φ N 2 (N H/ 2)| = (1 + H 2 )−(N /4) and

∞

(1 + H 2 )−K d H <

L

1 (1 + L 2 ) K −1



∞

(1 + H 2 )d H =

0

π (3.80) 2(1 + L 2 ) K −1

for every K > 1, we have, in particular, that

∞ L

√ |φ N 2 (N H/ 2)|d H <

π 2(1 + L 2 ) N

2 /4−1

.

(3.81)

 Note that N 2 /4 − n N − 1/4 > 1 whenever N > N0 = 2n + 4n 2 + 5/4. For every N > N0 we obtain from (3.80) with K = N 2 /4 − n N − 1/4 and L = H0 that 1 πN I2 ( , u, N ) ≤ √ n! (Γ (N ))n (R(α, A))n N  .  2 /4−n N −5/4 N n 2 2(w(u)) 1 + H02 (3.82) Since inf

u∈[−2+ ,2− ]

|w(u)| = w(2 − )

(3.83)

and, by the Stirling formula, (1 + H02 ) N /4 tends to infinity more rapidly than any fixed power of Γ (N ) (or, moreover, the N th power of any positive number), we conclude from (3.82) that 2

I2 ( , u, N ) −→ 0 N →∞

(3.84)

uniformly in u ∈ [−2 + α, 2 − α] and ti ∈ [−A, A], i = 1, . . . , n. The check of (3.78) is even more straightforward. Indeed, using (3.80) with K = N 2 /4 and L = H0 , we have N Sn |J2 ( , N )| = √ 2 N Sn = √ 2 and (3.78) follows.

 ∞   √  √    N Sn ∞    φ φ 2 (N H/ 2) d H ≤ √ 2 (N H/ 2) d H  N N   2 H0 H ∞0 π N Sn 2 (1 + H 2 )−(N /4) d H = √ , 2 2 2(1 + L 2 ) N /4−1 H0

228

F. Götze, M. Gordin

Now we complete the proof by establishing (3.76). Recall that we omit the arguments GUE,1/N of Rn,N . All bounds and limit transitions are uniform with respect to u ∈ [−2 + α, 2 − α] and t1 , t2 , . . . , tn satisfying |t1 | ≤ A, . . . , |tn | ≤ A. We have |I1 ( , u, N ) − J1 ( , N )|   H0  GUE,1/((1+i H )N ) √  N  Rn,N − Sn φ N 2 (N H/ 2) d H ≤√   (N w(u))n 2 0    H0  GUE,1/((1+i H )N ) √  N  Rn,N ≤√ − Sn (u, H ) φ N 2 (N H/ 2) d H   (N w(u))n 2 0  H0  √  N  +√ (Sn (u, H ) − Sn ) φ N 2 (N H/ 2) d H 2 0 (1)

(2)

=: D1 ( , u, N ) + D2 ( , u, N ).

(3.85)

It follows from the determinantal formula (3.7) and Lemma 3.2 that  GUE,1/((1+i H )N )  R   n,N  sup  − S (u, H )  −→ 0 n n   N →∞ (N w(u)) 0≤H ≤H0

(3.86)

uniformly in u ∈ [−2 + α, 2 − α], and |ti | ≤ A (i = 1, . . . , n). Therefore, using (3.81), we obtain (1)

D1 ( , u, N )  GUE,1/((1+i H )N )  R  ∞   n,N  φ N 2 (h) dh −→ 0 ≤ sup  − S (u, H )  n n  0 N →∞ (N w(u)) 0≤H ≤H0  since

 φ N 2 (h) dh −→

∞ 0

N →∞

1 √ . 2 2π

(3.87)

Finally, we see from (3.73) and (3.87) that D1(2) ( , u, N ) ≤ . Since is arbitrary small, this completes the proof.



Acknowledgements. The authors are indebted to the anonymous referees for careful reading of the manuscript. Moreover, the main theorem in the first version of the paper required excluding of a neighborhood of zero (this case has been considered in the note [9]). Stimulating questions of one of the referees and the editor led us to a revision of the proof which removes this restriction. Finally, we would like to thank Justine Swierkot for her assistance in the preparation of the text.

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References 1. Akemann, G., Cicuta, G.M., Molinari, L., Vernizzi, G.: Compact support probability distributions in random matrix theory. Phys. Rev. E (3) 59(2, part A), 1489–1497 (1999) 2. Akemann, G., Vernizzi, G.: Macroscopic and microscopic (non-)universality of compact support random matrix theory. Nucl. Phys. B 583(3), 739–757 (2000) 3. Deift, P., Kriecherbauer, T., McLaughlin, K.T.-R., Venakides, S., Zhou, X.: Uniform asymptotics for polynomials orthogonal with respect to varying exponential weights and applications to universality questions in random matrix theory. J. Comm. Pure Appl. Math. 52(11), 1335–1425 (1999) 4. Deift, P., Kriecherbauer, T., McLaughlin, K.T.-R., Venakides, S., Zhou, X.: Strong asymptotics for polynomials orthogonal with respect to exponential weights. J. Comm. Pure Appl. Math. 52(12), 1491– 1552 (1999) 5. Deift, P. A.: Orthogonal polynomials and random matrices: a Riemann-Hilbert approach. Courant Lecture Notes in Mathematics, Vol. 3. N. Y.: New York University/Courant Institute of Mathematical Sciences, 1999 6. Deift, P., Kriecherbauer, T., McLaughlin, K.T.-R., Venakides, S., Zhou, X.: A Riemann-Hilbert approach to asymptotic questions for orthogonal polynomials. J. Comput. Appl. Math. 133(1–2), 47–63 (2001) 7. Feller, W.: An introduction to probability theory and its applications. Vol. II. Second edition. New York: John Wiley & Sons Inc., 1971 8. Forrester, P.J.: Log-gases and random matrices. Book in progress. http://www.ms.unimelb.edu.au/ ~matpjf/matpjf.html, 2005 9. Götze, F., Gordin, M., Levina, A.: Limit correlation function at zero for fixed trace random matrix ensembles.(Russian) Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 341, 68–80 (2007); translation in J. Math. Sci. (N.Y.) 147(4), 6884–6890 (2007) 10. Johansson, K.: Universality of the local spacing distribution in certain ensembles of Hermitian Wigner matrices. Commun. Math. Phys. 215(3), 683–705 (2001) 11. Mehta, M. L.: Random matrices. Third edition. Pure and Applied Mathematics (Amsterdam), 142. Amsterdam: Elsevier/Academic Press, 2004 12. Pastur, L., Shcherbina, M.: Universality of the local eigenvalue statistics for a class of unitary invariant random matrix ensembles. J. Stati. Phys. 86(1–2), 109–147 (1997) 13. Petrov, V.V.: Sums of independent random variables. N.Y.: Springer, 1975 14. Rosenzweig, N.: Statistical mechanics of equally likely quantum systems. In: Statistical physics (Brandeis Summer Institute, 1962, Vol. 3), N. Y.: W. A. Benjamin, (1963), pp. 91–158 15. Soshnikov, A.: Determinantal point random fields. Russ. Math. Surv. 55(5), 923–975 (2000) 16. Szego, G.: Orthogonal polynomials. Fourth edition. Providence, R.I.: Amer. Math. Soc., 1975 17. Tracy, C.A., Widom, H.: Correlation functions, cluster functions, and spacing distributions for random matrices. J. Stat. Phys. 92(5–6), 809–835 (1998) 18. Van Assche, W., Geronimo, J.S.: Asymptotics for orthogonal polynomials with regularly varying recurrence coefficients. Rocky Mountain J. Math. 19(1), 39–49 (1989) Communicated by H. Spohn

Commun. Math. Phys. 281, 231–249 (2008) Digital Object Identifier (DOI) 10.1007/s00220-008-0487-4

Communications in

Mathematical Physics

Stability of Viscous Shocks in Isentropic Gas Dynamics
Blake Barker1 , Jeffrey Humpherys1 , Keith Rudd2 , Kevin Zumbrun3 1 Department of Mathematics, Brigham Young University, Provo, UT 84602, USA.

E-mail: [email protected]; [email protected]

2 Department of Eng. Sci. and App. Math., Northwestern University, Evanston,

IL 60208-3125, USA. E-mail: [email protected]

3 Department of Mathematics, Indiana University, Bloomington, IN 47402, USA.

E-mail: [email protected] Received: 25 March 2007 / Accepted: 21 December 2007 Published online: 6 May 2008 – © Springer-Verlag 2008

Abstract: In this paper, we examine the stability problem for viscous shock solutions of the isentropic compressible Navier–Stokes equations, or p-system with real viscosity. We first revisit the work of Matsumura and Nishihara, extending the known parameter regime for which small-amplitude viscous shocks are provably spectrally stable by an optimized version of their original argument. Next, using a novel spectral energy estimate, we show that there are no purely real unstable eigenvalues for any shock strength. By related estimates, we show that unstable eigenvalues are confined to a bounded region independent of shock strength. Then through an extensive numerical Evans function study, we show that there are no unstable spectra in the entire right-half plane, thus demonstrating numerically that large-amplitude shocks are spectrally stable up to Mach number M ≈ 3000 for 1 ≤ γ ≤ 3. This strongly suggests that shocks are stable independent of amplitude and the adiabatic constant γ . We complete our study by showing that finite-difference simulations of perturbed large-amplitude shocks converge to a translate of the original shock wave, as expected. 1. Introduction Consider the isentropic compressible Navier-Stokes equations in one spatial dimension expressed in Lagrangian coordinates, also known as the p-system with real viscosity: vt − u x = 0,
u  x , u t + p(v)x = v x

(1)

where, physically, v is the specific volume, u is the velocity, and p(v) is the pressure law. We assume that p(v) is adiabatic, that is, a γ -law gas satisfying p(v) = a0 v −γ for  This work was supported in part by the National Science Foundation award numbers DMS-0607721 and DMS-0300487.

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some constants a0 > 0 and γ ≥ 1. In the thermodynamical rarified gas approximation, γ > 1 is the average over constituent particles of γ = (n + 2)/n, where n is the number of internal degrees of freedom of an individual particle [3]: n = 3 (γ = 5/3) for monatomic, n = 5 (γ = 1.4) for diatomic gas. More generally, γ is usually taken within 1 ≤ γ ≤ 3 in physical approximations of gas-dynamical flow [27]. This system is an important and widely studied gas-dynamical model (see for example [27] and references within), and yet little is presently known about the stability of its large-amplitude viscous shock wave solutions. Over two decades ago, Matsumura and Nishihara [24] used a clever energy estimate to show that small-amplitude shocks are stable under zero-mass perturbations. The linear portion of their analysis is equivalent to proving spectral stability, which through the more recent work of Zumbrun and collaborators [32,21,23,22,31,17], implies asymptotic orbital stability, hereafter called nonlinear stability. We remark that the results of [23,22,31] hold for shocks of arbitrary amplitude, and thus nonlinear stability is implied by spectral stability. Hence, for largeamplitude shocks, spectral stability is the missing piece of the stability puzzle and the subject of our present focus. In this paper, we first improve upon the work in [24] slightly by extending the known parameter regime for which small-amplitude viscous shocks are provably spectrally stable, using an optimized version of the same method. We also show that this method cannot be extended any further to larger amplitudes. Using a novel spectral energy estimate, however, we are able to show that there are no purely real unstable eigenvalues for any shock strength. We note that this result is stronger than that which could be given by the Evans function stability index (sometimes called the orientation index), which only measures the parity of unstable real eigenvalues, see [4,14,15,31]. A consequence of this result (which follows also by the Evans function computations used to determine the stability index [31]) is that if an instability were to occur for large-amplitude viscous shocks, its onset, or indeed any change in the number of unstable eigenvalues, would be associated with a Hopf-like bifurcation in which one or more conjugate pairs of eigenvalues cross through the imaginary axis; see [28–30] for further discussion of this scenario. Continuing our investigations, we appeal to numerical Evans function computation to explore the large-amplitude regime through the use of winding number calculations via the argument principle. Before doing so, however, we rule out high-frequency instability through the combination of two spectral energy estimates, showing that unstable eigenvalues are confined to a bounded region independent of shock amplitude. This reduces the problem to investigation, feasible by numerics, of a compact set. Then by checking the low-frequency regime by repeating several Evans function computations, we determine whether or not a particular viscous shock is spectrally stable. As a final verification, we use a finite-difference method to simulate perturbed large-amplitude viscous shocks, and check whether they converge, as expected, to a translate of the original profile. Conclusions and results of numerical investigations. We carry out our numerical experiments far into the hypersonic shock regime, exploring up through Mach number M ≈ 3000 for 1 ≤ γ ≤ 3. Particular attention is given to the monatomic and diatomic cases, γ = 5/3 and γ = 7/5, respectively. In all cases, our results are consistent with spectral stability (hence also linear and nonlinear stability [22,23,31]). This strongly suggests that viscous shock profiles in an isentropic are spectrally stable independent of both amplitude and γ . Our bounds on the size of unstable eigenvalues, which are independent of shock strength, may be viewed as a first step in establishing such a result analytically.

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Extensions and open questions. The present study is not a numerical proof. However, as discussed in [6], it could be converted to numerical proof by the implementation of interval arithmetic and a posteriori error estimates for numerical solution of ODE. This would be an interesting direction for future investigation. A crucial step in carrying out numerical proof by interval arithmetic is by analytical estimates special to the problem at hand to sufficiently reduce the computational domain to make the computations feasible in realistic time. This we have carried out by the surprisingly strong estimates of Sect. 5 and Appendices B–C. A second very interesting direction would be to establish stability in the largeamplitude limit by a singular-perturbation analysis, an avenue we intend to follow in future work. Together, these two projects would give a complete, rigorous proof of stability for arbitrary-amplitude viscous shock solutions of (1) on the physical range γ ∈ [1, 3]. 2. Background By a viscous shock profile of (1), we mean a traveling wave solution v(x, t) = v(x ˆ − st), u(x, t) = u(x ˆ − st), moving with speed s and having asymptotically constant end-states (v± , u ± ). As an alternative, we can translate x → x − st, and consider instead stationary solutions of vt − svx − u x = 0,
u  x . u t − su x + (a0 v −γ )x = v x

(2)

Under the rescaling (x, t, v, u) → (−εsx, εs 2 t, v/ε, −u/(εs)), where ε is chosen so that 0 < v+ < v− = 1, our system takes the form vt + vx − u x = 0,
u  x , u t + u x + (av −γ )x = v x

(3)

where a = a0 ε−γ −1 s −2 . Thus, the shock profiles of (3) satisfy the ordinary differential equation v  − u  = 0,    u , u  + (av −γ ) = v subject to the boundary conditions (v(±∞), u(±∞)) = (v± , u ± ). This simplifies to    v  −γ  . v + (av ) = v By integrating from −∞ to x, we get the profile equation v  = v(v − 1 + a(v −γ − 1)),

(4)

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where a is found by setting x = +∞, thus yielding the Rankine-Hugoniot condition a=−

v+ − 1

−γ v+

−1

γ

= v+

1 − v+ γ. 1 − v+

(5) γ

Evidently, a → γ −1 in the weak shock limit v+ → 1, while a ∼ v+ in the strong shock limit v+ → 0. Remark 2.1. Since the profile equation (4) is first order scalar, it has a monotone solution. Since v+ < v− , we have that vˆ x < 0 for all x ∈ R. By linearizing (3) about the profile (v, ˆ u), ˆ we get the eigenvalue problem λv + v  − u  = 0,      h(v) ˆ u v = , λu + u  − vˆ γ +1 vˆ

(6)

h(v) ˆ = −vˆ γ +1 + a(γ − 1) + (a + 1)vˆ γ .

(7)

where

We say that a shock profile of (1) is spectrally stable if the linearized system (6) has no spectrum in the closed deleted right half-plane P = { e(λ) ≥ 0} \ {0}, that is, there are no growth or oscillatory modes for (6). We remark that a traveling wave profile always has a zero-eigenvalue associated with its translational invariance. This generally negates the possibility of good uniform bounds in energy estimates, and so we employ the standard technique (see [16,32]) of transforming into integrated coordinates. This goes as follows: Suppose that (v, u) is an eigenfunction of (6) with a non-zero eigenvalue λ. Then  u(x) ˜ =

x −∞

 u(z)dz, v(x) ˜ =

x

−∞

v(z)dz,

and their derivatives decay exponentially as x → ∞. Thus, by substituting and then integrating, (u, ˜ v) ˜ satisfies (suppressing the tilde) λv + v  − u  = 0, λu + u  −

(8a) u 

h(v) ˆ  . v = γ +1 vˆ vˆ

(8b)

This new eigenvalue problem differs spectrally from (6) only at λ = 0, hence spectral stability of (6) is implied by spectral stability of (8). Moreover, since (8) has no eigenvalue at λ = 0, one can expect to have a better chance of developing a successful spectral energy method to prove stability. We demonstrate this in the following section.

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3. Stability of Small-Amplitude Shocks In this section we further the work in [24] by extending slightly the known parameter regime for which small-amplitude viscous shocks are provably spectrally stable. We also show that this method cannot be extended any further for larger amplitudes. Theorem 3.1. ([24]) Viscous shocks of (1) are spectrally stable whenever 

γ +1

v+ aγ

2



γ +1

v + 2(γ − 1) + aγ

 − (γ − 1) ≥ 0.

(9)

In particular, as v+ → 1 (hence aγ → 1), the left-hand side of (9) approaches γ and so the inequality is satisfied. Therefore, small-amplitude viscous shocks of (1) are spectrally stable. Proof. We note that h(v) ˆ > 0. By multiplying (8b) by both the conjugate u¯ and vˆ γ +1 / h(v) ˆ and integrating along x from ∞ to −∞, we have   γ +1    γ λu u¯ vˆ γ +1 u u¯ vˆ u u¯ vˆ + − . v  u¯ = ˆ ˆ ˆ R h(v) R h(v) R R h(v)



Integrating the last three terms by parts and appropriately using (8a) to substitute for u  in the third term gives us   γ +1    γ   γ 2 vˆ λ|u|2 vˆ γ +1 u u¯ vˆ vˆ |u | + + =− v(λv + v  ) + u  u. ¯ h(v) ˆ ˆ ˆ ˆ R R h(v) R R h(v) R h(v)



We take the real part and appropriately integrate by parts to get   e(λ)

  vˆ γ +1 2 vˆ γ 2 |u| + |v|2 + |u  |2 = 0, g(v)|u| ˆ + h( v) ˆ h( v) ˆ R R R

where 1 g(v) ˆ =− 2



vˆ γ +1 h(v) ˆ



 +

vˆ γ h(v) ˆ

 

.

Thus, to prove stability it suffices to show that g(v) ˆ ≥ 0 on [v+ , 1]. By straightforward computation, we obtain identities: γ h(v) ˆ − vh ˆ  (v) ˆ = aγ (γ − 1) + vˆ γ +1 and γ −1 vˆ vˆ x = aγ − h(v). ˆ

(10) (11)

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(a)

(b)

Fig. 1. In (a), we have a graph of g(v) ˆ against vˆ for v+ = 1 × 10−4 and γ = 2.0. Note that g(v) ˆ dips down below zero on the left-hand size. Hence, the energy estimate will not generalize beyond the small-amplitude regime. In (b) we graph the stability boundaries (dark lines) given by (9) and (15), where γ and v+ are the horizontal and vertical axes, respectively. We see that in our scaling (15) is only a modest improvement over (9). The dotted lines correspond from top to bottom as the parameter regimes for Mach numbers 2, 5, and 10 (see the Appendix to see how to determine the Mach number). Hence, this energy estimate does not even hold for shocks at Mach 2 and γ > 1.084

Using (10) and (11), we abbreviate a few intermediate steps below:  (γ + 1)vˆ γ h(v) ˆ − vˆ γ +1 h  (v) ˆ ˆ − vˆ γ h  (v) ˆ d γ vˆ γ −1 h(v) + vˆ x h(v) ˆ 2 d vˆ h(v) ˆ 2



  ˆ − vh ˆ  (v) ˆ ˆ − vh ˆ  (v) ˆ d γ h(v) vˆ x vˆ γ (γ + 1)h(v) + (aγ − h(v)) ˆ =− 2 h(v) ˆ 2 d vˆ h(v) ˆ 2

vˆ x g(v) ˆ =− 2

=−



a vˆ x vˆ γ −1 × 2h(v) ˆ 3  γ 2 (γ + 1)vˆ γ +2 − 2(a + 1)γ (γ 2 − 1)vˆ γ +1 + (a + 1)2 γ 2 (γ − 1)vˆ γ  + aγ (γ + 2)(γ 2 − 1)vˆ − a(a + 1)γ 2 (γ 2 − 1)

=−



2 a vˆ x vˆ γ −1 [(γ + 1)vˆ γ +2 + vˆ γ (γ − 1) (γ + 1)vˆ − (a + 1)γ 3 2h(v) ˆ

(12)

+aγ (γ 2 − 1)(γ + 2)vˆ − a(a + 1)γ 2 (γ 2 − 1)] a vˆ x vˆ γ −1 [(γ + 1)vˆ γ +2 + aγ (γ 2 − 1)(γ + 2)vˆ − a(a + 1)γ 2 (γ 2 − 1)] 2h(v) ˆ 3 ⎡ ⎤  γ +1   γ +1 2 v+ γ 2 a 3 vˆ x (γ + 1) ⎣ v+ + 2(γ − 1) ≥− − (γ − 1)⎦ . (13) 2h(v) ˆ 3 v+ aγ aγ ≥−

Thus from (9), we have spectral stability.



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We note that the hypothesis in (9) is not sharp. Indeed, one can show from (12), that a stronger condition could be given as

2 (γ + 1)vˆ γ +2 + vˆ γ (γ − 1) (γ + 1)vˆ − (a + 1)γ (14) +aγ (γ 2 − 1)(γ + 2)vˆ − a(a + 1)γ 2 (γ 2 − 1) ≥ 0, which is sharp in the following sense: When this condition fails to be true, then g(v) ˆ is no longer nonnegative, and thus the energy method fails. In Fig. 1(a), we see that g(v) ˆ dips on the left-hand side when this inequality is compromised. We remark further that near v+ , the left-hand side of (14) is monotone increasing in v. ˆ Thus, (14) holds if and only if γ +2

γ

(γ + 1)v+ + v+ (γ − 1) ((γ + 1)v+ − (a + 1)γ )2 +aγ (γ 2 − 1)(γ + 2)v+ − a(a + 1)γ 2 (γ 2 − 1) ≥ 0.

(15)

We see from Fig. 1(b) that (15) is only a marginal improvement over (9). However, since (15) is sharp, we cannot hope to prove large-amplitude spectral stability using this approach. Instead, we proceed by a combined analytical and numerical approach as in [6–8], first showing that unstable eigenvalues can occur only in a bounded set, then searching for eigenvalues in this set by computing the Evans function numerically. Before doing so, however, we show in the following section that real unstable eigenvalues do not exist, even for large-amplitude viscous shocks. 4. No Real Unstable Eigenvalues In this section we use a novel spectral energy estimate to show that there are no purely real unstable eigenvalues for any shock strength. We note that this result is stronger than that which could be given by the Evans function stability index (sometimes called the orientation index), which only measures the parity of unstable real eigenvalues, see [4,15, 31]. The fact that this holds for all shock strengths is interesting because it is among the strongest statements about large-amplitude spectral stability that has been proven to date. Theorem 4.1. Viscous shocks of (1) have no unstable real spectra. Proof. We multiply (8b) by the conjugate v¯ and integrate along x from ∞ to −∞. This gives      h(v)v ˆ  v¯ u v¯  . λu v¯ + u v¯ − = γ +1 R R R vˆ R vˆ Notice that on the real line, λ¯ = λ. Using (8a) to substitute for λv in the first term and for u  in the last term, we get     h(v)v ˆ  v¯ (λv  + v  )v¯ . u(u¯  − v¯  ) + u  v¯ − = γ +1 vˆ R R R vˆ R Separating terms and simplifying gives        h(v)v ˆ  v¯ v v¯ v v¯ + . u u¯  + 2 u  v¯ − = λ γ +1 R R R vˆ R vˆ R vˆ

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We further simplify by substituting for u  in the second term and integrating the last terms by parts to give        h(v)v ˆ  v¯ v v¯ vˆ x  |v  |2   + . u u¯ + 2 (λv + v )v¯ − = λ v v ¯ − γ +1 2 R R R vˆ R vˆ R vˆ R vˆ Combining the third and fifth terms using (11) and simplifying, we have        v  v¯ |v  |2 v v¯  2  =λ . u u¯ + 2λ |v| + 2 v v¯ − aγ + γ +1 v ˆ v ˆ R R R R R R vˆ By taking the real part, we arrive at      λ vˆ x vˆ x |v  |2 aγ (γ + 1) 2 2 4 − 2 |v| − = 0. |v| + γ +2 2 R vˆ 2 R vˆ R vˆ This is a contradiction when λ ≥ 0.



We remark that the absence of positive real eigenvalues limits the admissible onset of instability to Hopf-like bifurcations where a pair of conjugate eigenvalues crosses the imaginary axis. In the following section, we give an upper bound on the spectral frequencies that are admissible. In other words, we show that if a pair of conjugate eigenvalues cross the imaginary axis, they must do so within these bounds. 5. High-Frequency Bounds In this section, we prove high-frequency spectral bounds. This is an important step because it gives a ceiling as to how far along both the imaginary and real axes that one must explore for spectrum when doing Evans function computations. Indeed if we want to check for roots of the Evans function in the unstable half-plane, say using the argument principle, then we need only compute within these bounds. If no roots are found therein, then we have strong numerical evidence that the given shock is spectrally stable. (Indeed, at the expense of further effort, such a calculation may be used as the basis of numerical proof, as described in [6,7].) In this section we show that the high-frequency bounds are quite strong, only allowing unstable eigenvalues to persist in a relatively small triangle adjoining the origin. Moreover, these bounds are independent of the shock amplitude. Lemma 5.1. The following holds for eλ ≥ 0:    1 2 2 ( e(λ) + | m(λ)|) v|u| ˆ − vˆ x |u| + |u  |2 2 R R R  √  h(v) ˆ  ≤ 2 |v ||u| + v|u ˆ  ||u|. γ v ˆ R R

(16)

Proof. We multiply (8b) by vˆ u¯ and integrate along x. This yields     h(v) ˆ  λ v|u| ˆ 2+ vu ˆ  u¯ + |u  |2 = v u. ¯ γ v ˆ R R R R We get (16) by taking the real √ and imaginary parts and adding them together, and noting  that | e(z)| + | m(z)| ≤ 2|z|.

Stability of Viscous Shocks in Isentropic Gas Dynamics

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Lemma 5.2. The following identity holds for eλ ≥ 0:      h(v) ˆ |v  |2 1 aγ  2 2 2 + |u | = 2 e(λ) |v| + e(λ) + |v  |2 , (17) 2 R vˆ γ +1 vˆ γ +1 R R R vˆ Proof. We multiply (8b) by v¯  and integrate along x. This yields      h(v) ˆ 2 1   1 u (λv  + v  )v¯  . u  v¯  − |v | = v ¯ = λ u v¯  + γ +1 R R R vˆ R vˆ R vˆ Using (8a) on the right-hand side, integrating by parts, and taking the real part gives       h(v) ˆ |v  |2 vˆ x     2 |v . e λ u v¯ + u v¯ = + | + e(λ) γ +1 2vˆ 2 R R R vˆ R vˆ The right hand side can be rewritten as       1 h(v) ˆ |v  |2 aγ     2 e λ u v¯ + |v . u v¯ = + | + e(λ) 2 R vˆ γ +1 vˆ γ +1 R R R vˆ

(18)

Now we manipulate the left-hand side. Note that         ¯ λ u v¯ + u v¯ = (λ + λ) u v¯ − u(λ¯ v¯  + v¯  ) R R R R   = −2 e(λ) u  v¯ − u u¯  R R    = −2 e(λ) (λv + v )v¯ + |u  |2 . R

R

Hence, by taking the real part we get      u  v¯  = |u  |2 − 2 e(λ)2 |v|2 . e λ u v¯  + R

R

R

R

This combines with (18) to give (17).

 Remark 5.3. Lemma 5.2 is a special case of the high-frequency bounds given in [22,31]. We also note that (17) follows from a “Kawashima-type” estimate as described in [22,31]. Lemma 5.4. For h(v) ˆ as in (7), we have    h(v) ˆ  1 − v+  sup  γ  = γ γ ≤ γ. vˆ 1 − v+ v∈[v ˆ + ,1]

(19)

Proof. Defining H (v) ˆ := h(v) ˆ vˆ −γ = −vˆ + a(γ − 1)vˆ −γ + (a + 1),

(20)

we have H  (v) ˆ = −1 − aγ (γ − 1)vˆ −γ −1 < 0 for 0 < v+ ≤ vˆ ≤ v− = 1, hence the maximum of H on vˆ ∈ [v+ , v− ] is achieved at vˆ = v+ . Substituting (5) into (20) and simplifying yields (19).

 We complete this section by proving our high-frequency bounds.

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B. Barker, J. Humpherys, K. Rudd, K. Zumbrun

Theorem 5.5. Any eigenvalue λ of (8) with nonnegative real part satisfies 1 √ e(λ) + | m(λ)| ≤ ( γ + )2 . 2

(21)

Proof. Using Young’s inequality twice on right-hand side of (16) together with (19), we get    1 ( e(λ) + | m(λ)|) v|u| ˆ 2− vˆ x |u|2 + |u  |2 2 R R R  √  h(v) ˆ  ≤ 2 |v ||u| + v|u ˆ  ||u| γ R vˆ R √     h(v) ˆ  2 ( 2)2 h(v) ˆ 1 2  2 ≤θ |v | + v|u| ˆ +  v|u ˆ | + v|u| ˆ 2 γ +1 γ 4θ 4 R R vˆ R vˆ R     1 h(v) ˆ 2 γ  2 + <θ |v | +  |u | + v|u| ˆ 2. γ +1 2θ 4 R R vˆ R Assuming that 0 <  < 1 and θ = (1 − )/2, this simplifies to   ( e(λ) + | m(λ)|) v|u| ˆ 2 + (1 − ) |u  |2 R R    1 h(v) ˆ 2 1− γ + < |v | + v|u| ˆ 2. γ +1 2 2θ 4 R R vˆ Applying (17) yields 



1 γ + ( e(λ) + | m(λ)|) v|u| ˆ < 1 −  4 R



2

R

v|u| ˆ 2,

or equivalently, ( e(λ) + | m(λ)|) < √ Setting  = 1/(2 γ + 1) gives (21).

(4γ − 1) − 1 . 4(1 − )



In the following section, we do an extensive Evans function study to explore numerically the rest of the right-half plane in an effort to locate the presence of unstable complex eigenvalues. 6. Evans Function Computation In this section, we numerically compute the Evans function to locate any unstable eigenvalues, if they exist, in our system. The Evans function D(λ) is analytic to the right of the essential spectrum and is defined as the Wronskian of decaying solutions of the eigenvalue equation for the linearized operator (8) (see [1,9–13]). In a spirit similar to the characteristic polynomial, we have that D(λ) = 0 if and only if λ is in the point spectrum of the linearized operator (8). While the Evans function is generally too complex to compute explicitly, it can readily be computed numerically, even for large systems [19].

Stability of Viscous Shocks in Isentropic Gas Dynamics

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Since the Evans function is analytic in the region of interest, we can numerically compute the winding number in the right-half plane. This allows us to systematically locate roots (and hence eigenvalues) within. As a result, spectral stability can be determined, and in the case of instability, one can produce bifurcation diagrams to illustrate and observe its onset. This approach was first used by Evans and Feroe [13] and has been advanced further in various directions (see for example [2,5–7,19,25,26]). We begin by writing (8) as a first-order system W  = A(x, λ)W , where ⎛ ⎞ ⎛ ⎞ 0 λ 1 u ⎠ , W = ⎝v ⎠ ,  = d , 1 A(x, λ) = ⎝ 0 0 (22) dx λvˆ λvˆ f (v) ˆ −λ v and f (v) ˆ = vˆ − vˆ −γ h(v), ˆ with h as in (7). Note that eigenvalues of (8) correspond to nontrivial solutions of W (x) for which the boundary conditions W (±∞) = 0 are satisfied. We remark that since vˆ is asymptotically constant in x, then so is A(x, λ). Thus at each end-state, we have the constant-coefficient system W  = A± (λ)W.

(23)

Hence solutions that satisfy the needed boundary condition must emerge from the unstable manifold W1− (x) at x = −∞ and the stable manifold W2+ (x) ∧ W3+ (x) at x = ∞. In other words, eigenvalues of (8) correspond to the values of λ for which these two manifolds intersect, or more precisely, when D(λ) := det(W1− W2+ W3+ )|x=0 is zero. As an alternative, we consider the adjoint formulation of the Evans function [4,26], where instead of computing the 2-dimensional stable manifold, we find the  + (x) coming from the unstable manifold of single trajectory W 1   = −A(x, λ)∗ W  W

(24)

 + (x) is orthogonal to both W + (x) and W + (x) since at x = ∞. Note that W 1 2 3 ∗    + is orthogonal to that of W + (W (x) W (x)) = 0 for all x and the initial data of W 1 2  + and W − are orthogonal, and and W3+ . Hence, the original manifolds intersect when W 1 1  + · W − )|x=0 . therefore, the adjoint Evans function takes the form D+ (λ) := (W 1 1 To further improve the numerical efficiency and accuracy of the shooting scheme,  to remove exponential growth/decay at infinity, and thus eliminate we rescale W and W − potential problems with stiffness. Specifically, we let W (x) = eµ x V (x), where µ− is the growth rate of the unstable manifold at x = −∞, and we solve instead V  (x) = (A(x, λ) − µ− I )V (x). We initialize V (x) at x = −∞ to be the eigenvector of A− (λ)  (x) at corresponding to µ− . Similarly, it is straightforward to rescale and initialize W x = ∞. Numerically, we use a finite domain for x, replacing the end states x = ±∞ with x = ±L, for sufficiently large L. Some care needs to be taken, however, to make sure that we go out far enough to produce good results. In Appendices B and C, we explore the decay rates of the profile vˆ and A(x, λ) and combine our analysis with numerical convergence experiments to conclude that our domain [−L , L] is sufficiently large. Our experiments, described below, were primarily conducted using L = 12, with relative error bounds ranging mostly between 10−3 and 10−4 . Larger choices of L were needed on the high end of the Mach scale, going up to L = 18 in some cases, to get the relative errors down to 10−4 . In Table 1, we provide a sample of relative errors in D(λ) for large-amplitude shocks.

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Table 1. Relative errors in D(λ) are computed by taking the maximum relative error for 60 contour points evaluated along the semicircle in Fig. 2(a). Samples were taken for varying L and γ , leaving v+ fixed at v+ = 10−4 (Mach M ≈ 1669). We used L = 8, 10, 12, 14, 16 and γ = 1.2, 1.4, 1.666, 1.8. Relative errors were computed using the next value of L as the baseline L

γ = 1.2

γ = 1.4

γ = 1.666

γ = 1.8

8 10 12 14

1.23(−1) 2.07(−2) 2.00(−3) 6.90(−4)

1.16(−1) 1.46(−2) 1.40(−3) 5.31(−4)

1.08(−1) 1.75(−2) 9.85(−4) 4.73(−4)

1.04(−1) 1.78(−2) 7.20(−4) 4.71(−4)

We remark also that in order to produce analytically varying Evans function output, (L), must be chosen analytically. The method of Kato [20, the initial data V (−L) and V p. 99], also described in [8], does this well by replacing the eigenvectors of (23) with analytically defined spectral projectors (see also [5,18]). Before we can compute the Evans function, we first need to compute the traveling wave profile. We use both Matlab’s ode45 routine, which is the adaptive fourthorder Runge-Kutta-Fehlberg method (RKF45), and Matlab’s bvp4c routine, which is an adaptive Lobatto quadrature scheme. Both methods work well and produce essentially equivalent profiles. Our experiments were carried out on the range (γ , M) ∈ [1, 3] × [1.6, 3000]. Recall, for γ ∈ [1, 3], that shocks are known to be stable for M ∈ [1, 1.6], by (15), Sect. 3, hence this completes the study of the range (γ , M) ∈ [1, 3] × [1, 3000] from minimum Mach number M = 1 far into the hypersonic shock regime, and encompassing the entire physically relevant range of γ . For each value of γ , the Mach number M was varied on logarithmic scale with regular mesh from M = 1.6 up to M = 3000. We used 50 mesh points for γ = 1.0 + 0.1k, where k = 1, 2, . . . , 20. For the special values γ = 1.4 and 1.666 (monatomic and diatomic cases), we did a refined study with 400 mesh points. For each value of (γ , M), we computed the Evans function along semi-circular contours that contain the triangular region found in the previous section via our high-frequency bounds; see Fig. 2(a). The ODE calculations for individual λ were carried out using Matlab’s ode45 routine, which is the adaptive 4th -order RungeKutta-Fehlberg method (RKF45), and after scaling out the exponential decay rate of the constant-coefficient solution at spatial infinity, as described in [5–8,19]. This method is known to have excellent accuracy [5,19]; in addition, the adaptive refinement gives automatic error control. Typical runs involved roughly 300 mesh points, with error tolerance set to AbsTol = 1e-6 and RelTol = 1e-8. Values of λ were varied on a contour with 60 points. As a check on winding number accuracy, it was tested a posteriori that the change in argument of D for each step was less than π/25. Recall, by Rouché’s Theorem, that accuracy is preserved so long as the argument varies by less than π along each mesh interval. In all the cases that we examined, the winding number was zero. This indicates that the shocks we considered are spectrally stable, and in view of [21–23], nonlinear stability follows. Moreover, in light of the large Mach numbers considered, this is highly suggestive of stability for all shock strengths.

Stability of Viscous Shocks in Isentropic Gas Dynamics

(a)

243

(b)

Fig. 2. The graph of the contour and its mapping via the Evans function. In (a), we have a contour, which is a large semi-circle aligned on the imaginary axis on the right-half plane (horizontal axis real, vertical axis imaginary). In (b) we have the image of the contour mapped by the Evans function. Note that the winding number of this graph is zero. Hence, there are no unstable eigenvalues in the semi-circle. Together with the high-frequency bounds, this implies spectral stability. Our computation was carried out for γ = 5/3 and v+ = 1 × 10−4 . This corresponds to a monatomic gas with Mach number M ≈ 1669 (see Appendix A)

7. Numerical Evolution of Strong Shocks In this section, we use a standard finite-difference method to simulate perturbed large-amplitude viscous shocks, and show that they converge, as expected, to a translate of the original profile. We do this with a nonlinear Crank-Nicholson scheme with a Newton solver to deal with the nonlinearity. This method provides second-order accuracy and will allow for larger time steps than a naive explicit scheme. By differentiating the viscosity term in (3), we have vt + vx − u x = 0, u x vx uxx − 2 . u t + u x − aγ v −γ −1 vx = v v By implementing the Crank-Nicholson averaging, we obtain − v nj v n+1 j

1 n n + (v n+1 − v n+1 j−1 + v j+1 − v j−1 ) t 4x j+1 1 n n (u n+1 − u n+1 − j−1 + u j+1 − u j−1 ) = 0 4x j+1

and u n+1 − u nj j

1 n n + (u n+1 − u n+1 j−1 + u j+1 − u j−1 ) t 4x j+1 a n+1 n n − γ (v nj )−γ −1 (v n+1 j+1 − v j−1 + v j+1 − v j−1 ) 4x 1 n n n − (u n+1 − 2u n+1 + u n+1 j j−1 + u j+1 − 2u j + u j−1 ) 2(x)2 v nj j+1

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Fig. 3. Snapshots of the evolution of a perturbed viscous shock wave solution generated by our extended Crank-Nicholson scheme (top curve: v against x; bottom curve: u against x, with time t fixed and increasing from figure to figure). The parameters used were γ = 1.4 and v+ = 9 × 10−6 . This corresponds to a diatomic gas with Mach number M ≈ 2877 (see Appendix A). As expected, the wave converges to a translate of the original shock

+

1 n n (u n+1 − u n+1 j−1 + u j+1 − u j−1 ) 16(x)2 (v nj )2 j+1

n+1 n n ×(v n+1 j+1 − v j−1 + v j+1 − v j−1 ) = 0,

where n and j are, respectively, the discretized temporal and spacial indices. To cope with the nonlinearities, we use the Newton solver −1   n+1   n+1    U U Du F Dv F F(U n+1 , V n+1 ) = − , , Du G Dv G k G(U n+1 , V n+1 ) k V n+1 k+1 V n+1 k where F(U n+1 , V n+1 ) and G(U n+1 , V n+1 ) are the above finite difference schemes, and Du F, Dv F, Du G, and Dv G are their corresponding partial derivatives. Hence, we use the previous time step as our initial guess in Newton’s method and then iterate until convergence. In Fig. 3, we see the evolution of a perturbed viscous shock of extremely high Mach number. We used a perturbation with a positive mass so that we could observe the convergence to a translate of the original profile, thus numerically demonstrating nonlinear stability. We remark that perturbed profiles as predicted by the nonlinear theory [22,23] exhibit also a “diffusion wave”, or approximate Gaussian signal, of constant

Stability of Viscous Shocks in Isentropic Gas Dynamics

245

mass, leaving the shock wave on the right and convecting with outgoing characteristic speed, which absorbs the portion of the mass not accounted for by shock shift. This is clearly visible for small- or intermediate-amplitude shocks, but for large-amplitude −1/2 shocks travels with characteristic speed ∼ v+ → +∞, leaving the viewing frame essentially instantaneously.

Appendix A. Mach Number for the p-System The Mach number is defined as M=

u+ − σ , c+

where u + is the downwind velocity, σ is the shock speed, and c+ is the downwind sound speed, all in Eulerian coordinates. By considering the conservation of mass equation, we have ρt + (ρu)x = 0. Hence, the jump condition is given by σ [ρ] = [ρu], which implies, in the original scaling for (1), that σ =

u + v− − u − v+ . v− − v+

Hence, M=

v+ [u] v+ v+ (u − − u + ) u+ − σ = = −s = c+ c+ (v− − v+ ) c+ [v] c+

or  M = 2

u+ − σ c+

2

γ +3

=

s 2 v+2 s 2 v+2 v = = + −γ −1  − p (v+ ) γ γ a0 v+

s2 . a0

To express this in our scaling, which is given in (3), we need to swap the pluses and minuses. Noting that 0 < v+ < v− = 1, we simplify to get γ

M2 =

1 1 − v+ 1 . = γ γa γ v+ 1 − v+

γ

Recalling that a ∼ v+ as v+ → 0, we find that v+ ∼ (γ M 2 )

− γ1

(25)

as M → ∞. In particular, | log v+ | ∼ γ −1 (log γ + 2 log M).

(26)

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Appendix B. Profile Bounds Denote profile equation (4) as v  = H (v, v+ ) := v(v − 1 + a(v −γ − 1)). Lemma B.1. For γ ≥ 1, 0 ≤ x < 1, 1≤

1 − xγ ≤ γ. 1−x

(27)

Proof. By convexity of f (x) = x γ , the difference quotient (27) is increasing in x, bounded above by f  (1) = γ , and below by its value at x = 0.

 Proposition B.2. For γ ≥ 1 and v+ ≤ v ≤ 16 , v+ ≤

1 12 ,

3 − γ (v − v+ ) ≤ H (v, v+ ) ≤ − (v − v+ ). 4 For γ ≥ 1, v+ ≤

1 4γ

, and

3 4

(28)

≤ v ≤ v− = 1,

1 (v − v− ) ≤ H (v, v+ ) ≤ (v − v− ). 2

(29)

Proof. By (5),
(v+ − 1)(v −γ − 1)  H (v, v+ ) = v (v − 1) − −γ v+ − 1
1 − v 
v 
+ + γ −1 = v (v − v+ ) + γ v 1 − v+

1 − v 
1 − v+ γ  + v . = (v − v+ ) v − γ 1 − vv+ 1 − v+ Applying (27) with x =

v+ v ,

we obtain (28) from

v−γ ≤

H (v, v+ ) ≤ v − (1 − v+ ). v − v+

Similarly, we may obtain (29) by the calculation

v γ
1 − v 
1 − v γ  H (v, v+ ) 3 1 + + =v 1− ≥ v − γ v+ ≥ − . γ v−1 v 1−v 4 4 1 − v+ 

Corollary B.3. For γ ≥ 1, 0 < v+ ≤ satisfies

1 12 ,

and v(0) ˆ := v+ +


1  3x e− 4 x ≥ 0, 12
1  x+12 |v(x) ˆ − v− | ≤ e 2 x ≤ 0. 4 |v(x) ˆ − v+ | ≤

1 12 ,

the solution vˆ of (4)

(30a) (30b)

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1 Proof. Observing that (vˆ − v+ )(0) = 12 , we obtain (30a) by Proposition B.2 and the comparison principle for first-order scalar ODE. Likewise, (30b) follows by (29) together with the observation that, by convexity of H , |H | is bounded below by estimates obtained 1 1 1 3 at vˆ = v+ + 12 and vˆ = 43 of 16 and 18 , respectively, so that vˆ traverses [v+ + 12 , 4 ] over 3/4  an x-interval of length ≤ 1/16 = 12.

Remark B.4. From Proposition B.2 and Corollary B.3, we obtain the remarkable fact that, in the scaling we have chosen, vˆ decays up to first derivative to endstates v± at a uniform exponential rate independent of shock strength, despite the apparent singularity at v = v+ . Appendix C. Initialization Error Lemma C.1. For A as in (22), | · | the Euclidean (2 ) matrix operator norm,
2|λ| + 1 + γ 2 (γ − 1)v −1  3x + e− 4 , x ≥ 0, 12
2|λ| + 1 + 2γ 3 (γ − 1)  x+12 e 2 , x ≤ 0. |A(x, λ) − A− (λ)| ≤ 4 |A(x, λ) − A+ (λ)| ≤

(31a) (31b)

ˆ − f (v± )|. As computed in the Proof. By (22), |A(x, λ) − A± (λ)| ≤ 2λ|vˆ − v± | + | f (v) proof of Lemma 5.4, f  (v) ˆ = −1 − aγ (γ − 1)vˆ −γ −1 . Applying (27) to the expression γ γ for a in (5), we obtain v+ ≤ a ≤ γ v+ , so that | f  (v)| ˆ ≤ 1 + γ 2 (γ − 1)vˆ −1 ≤ 1 + γ 2 (γ − 1)v+−1 , yielding (31a) by the Mean Value Theorem and (30a). Bound (31b) follows similarly.

 + and  µ+ be a left eigenvector Theorem C.2. (Simplified Gap Lemma [6,7,15]) Let V + and associated eigenvalue of −A (λ) and suppose that |e(−A

+ − µ+ )x

ˆ | ≤ C1 e−ηx x ≤ 0,

|(A − A+ )(x)| ≤ C2 e−ηx x ≥ 0,

(32)

µ+ x V (x, λ) of (24) with with 0 ≤ ηˆ < η. Then, there exists a solution W = e

(x, λ) − V + (λ)| C1 C2 e−ηx |V x≥L ≤ + (λ)| (η − η)(1 ˆ − ) |V

(33)

provided (η− η) ˆ −1 C1 C2 e−ηL ≤ . Similar estimates hold for solutions of (22) on x ≤ 0. (−A+ −  (−A + A+ ) and imposing the limiting behavior  = V µ+ ) + V Proof. Writing V +   V (+∞, λ) = V , we obtain by Duhamel’s Principle,  +∞ + µ+ )(x−y)   (y)e(−A+ − V (x) = V − (A − A+ )dy, V x

from which the result follows by a straightforward Contraction Mapping argument using (32). See [6,7,15] for details.



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Lemma C.3. For λ satisfying (21), γ ∈ [1, 3], and v+ ≤ η = 1/2γ , ηˆ = 1/4γ , and C1 = 104 , and similarly for x ≤ 0.

1 12 ,

(32)(a) holds with

Proof. Using the inverse Laplace transform representation  1 (−A+ − µ+ )x e = e zt (z + A+ +  µ+ )−1 dz, 2πi where is a contour enclosing the eigenvalues of (−A+ −  µ+ ) and distance ηˆ away, and + + −1 estimating the resolvent norm |(z + A +  µ ) | by Kramer’s rule, we obtain the stated crude bound, and similarly for x ≤ 0.

 For error tolerance θ := 10−k (| log θ | ∼ 2k), define
 L − (θ ) := 2 | log 10−4 | + | log(2γ + 7 + 2γ 3 (γ − 1))| + | log θ | + 12,  4
| log 10−4 | + | log(2γ + 7 + γ 2 (γ − 1)v+−1 )| + | log θ | . L + (θ, v+ ) := 3 Corollary C.4. For λ satisfying (21), we have relative error bounds −

|W1− (−L − , λ) − V − eµ x | , − |V − eµ x |

µ x| + e  + (L + , λ) − V |W 1 ≤ θ. µ+ x | + e |V

Proof. This follows by (33) with (31), Lemma C.3, and (21).

+

(34)



Remark C.5. Combining (34) with (26), we find that L+ ∼

4 (2 log M + 4 + | log 10−4 | + | log θ |) 3

suffices to obtain relative initialization error less than θ for γ ∈ [1, 3] and λ in the computational region (21). For θ = 10−3 , we thus obtain θ tolerance up to M = 3, 000 ∼ 103 for L + ∼ 40. This is conservative, as we have made little effort to optimize bounds, but still within the realm of our experiments. Our numerical convergence studies indicate that L ± = 18 in fact suffices for 10−3 accuracy.

References 1. Alexander, J., Gardner, R., Jonesm, C.: A topological invariant arising in the stability analysis of travelling waves. J. Reine Angew. Math. 410, 167–212 (1990) 2. Alexander, JC, Sachs, R.: Linear instability of solitary waves of a Boussinesq-type equation: a computer assisted computation. Nonlinear World 2(4), 471–507 (1995) 3. Batchelor, G.K.: An introduction to fluid dynamics. Cambridge Mathematical Library. Cambridge: Cambridge University Press, paperback edition, 1999 4. Benzoni-Gavage, S., Serre, D., Zumbrun, K.: Alternate Evans functions and viscous shock waves. SIAM J. Math. Anal. 32(5), 929–962 (electronic) (2001) 5. Bridges, T.J., Derks, G., Gottwald, G.: Stability and instability of solitary waves of the fifth-order KdV equation: a numerical framework. Phys. D 172(1–4), 190–216 (2002) 6. Brin, L.Q.: Numerical testing of the stability of viscous shock waves. PhD thesis, Indiana University, Bloomington, 1998 7. Brin, L.Q.: Numerical testing of the stability of viscous shock waves. Math. Comp. 70(235), 1071–1088 (2001)

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8. Brin, L.Q., Zumbrun, K.: Analytically varying eigenvectors and the stability of viscous shock waves. In: Seventh Workshop on Partial Differential Equations, Part I (Rio de Janeiro, 2001), P.L. Dias, D. Marchesin, A. Nelson, C. Tomei, eds. Mat. Contemp. 22, 19–32, (2002) 9. Evans, J.W.: Nerve axon equations. I. Linear approximations. Indiana Univ. Math. J. 21, 877–885 (1971) 10. Evans, J.W.: Nerve axon equations. II. Stability at rest. Indiana Univ. Math. J. 22, 75–90 (1972) 11. Evans, J.W.: Nerve axon equations. III. Stability of the Nerve Impulse. Indiana Univ. Math. J. 22, 577–593 (1972) 12. Evans, J.W.: Nerve axon equations. IV. The stable and the unstable impulse. Indiana Univ. Math. J. 24(12), 1169–1190 (1974) 13. Evans, J.W., Feroe, J.A.: Traveling waves of infinitely many pulses in nerve equations. Math. Biosci. 37, 23–50 (1977) 14. Gardner, R., Jones, C.K.R.T.: A stability index for steady state solutions of boundary value problems for parabolic systems. J. Differ. Eqs. 91(2), 181–203 (1991) 15. Gardner, R.A., Zumbrun, K.: The gap lemma and geometric criteria for instability of viscous shock profiles. Comm. Pure Appl. Math. 51(7), 797–855 (1998) 16. Goodman, J.: Nonlinear asymptotic stability of viscous shock profiles for conservation laws. Arch. Rat. Mech. Anal. 95(4), 325–344 (1986) 17. Howard, P., Zumbrun, K.: Pointwise estimates and stability for dispersive-diffusive shock waves. Arch. Rat. Mech. Anal. 155(2), 85–169 (2000) 18. Humpherys, J., Sandstede, B., Zumbrun, K.: Efficient computation of analytic bases in Evans function analysis of large systems. Numer. Math. 103(4), 631–642 (2006) 19. Humpherys, J., Zumbrun, K.: An efficient shooting algorithm for evans function calculations in large systems. Physica D 220(2), 116–126 (2006) 20. Kato, T.: Perturbation theory for linear operators. Classics in Mathematics. Berlin: Springer-Verlag, 1995 (Reprint of the 1980 edition) 21. Mascia, C., Zumbrun, K.: Pointwise Green function bounds for shock profiles of systems with real viscosity. Arch. Rat. Mech. Anal. 169(3), 177–263 (2003) 22. Mascia, C., Zumbrun, K.: Stability of large-amplitude viscous shock profiles of hyperbolic-parabolic systems. Arch. Rat. Mech. Anal. 172(1), 93–131 (2004) 23. Mascia, C., Zumbrun, K.: Stability of small-amplitude shock profiles of symmetric hyperbolic-parabolic systems. Commun. Pure Appl. Math. 57(7), 841–876 (2004) 24. Matsumura, A., Nishihara, K.: On the stability of travelling wave solutions of a one-dimensional model system for compressible viscous gas. Japan J. Appl. Math. 2(1), 17–25 (1985) 25. Pego, R.L., Smereka, P., Weinstein, M.I.: Oscillatory instability of traveling waves for a KdV-Burgers equation. Phys. D 67(1–3), 45–65 (1993) 26. Pego, R.L., Weinstein, M.I.: Eigenvalues, and instabilities of solitary waves. Philos. Trans. Roy. Soc. London Ser. A 340(1656), 47–94 (1992) 27. Smoller, J.: Shock waves and reaction-diffusion equations. New York: Springer-Verlag, Second edition, 1994 28. Texier, B., Zumbrun, K.: Relative Poincaré-Hopf bifurcation and galloping instability of traveling waves. Methods Appl. Anal. 12(4), 349–380 (2005) 29. Texier, B., Zumbrun, K.: Galloping instability of viscous shock waves. http://arxiv.org/list/math.Ap/ 0609331, 2006 30. Texier, B., Zumbrun, K.: Hopf bifurcation of viscous shock waves in compressible gas- and magnetohydrodynamics. Toappear Arch. Rat. Mech. Anal., doi:10.1007/s00205-008-0012-x 31. Zumbrun, K.: Multidimensional stability of planar viscous shock waves. Lecture notes, TMR summer school, 1999 32. Zumbrun, K., Howard, P.: Pointwise semigroup methods and stability of viscous shock waves. Indiana Univ. Math. J. 47(3), 741–871 (1998) Communicated by P. Constantin

Commun. Math. Phys. 281, 251–261 (2008) Digital Object Identifier (DOI) 10.1007/s00220-008-0445-1

Communications in

Mathematical Physics

Nonlinear Dirac Operator and Quaternionic Analysis Andriy Haydys
Fakultät für mathematik, Universität Bielefeld, Postfach 100131, 33501 Bielefeld, Germany. E-mail: [email protected] Received: 28 March 2007 / Accepted: 11 September 2007 Published online: 28 March 2008 – © Springer-Verlag 2008

Abstract: Properties of the Cauchy–Riemann–Fueter equation for maps between quaternionic manifolds are studied. Spaces of solutions in case of maps from a K3–surface to the cotangent bundle of a complex projective space are computed. A relationship between harmonic spinors of a generalized nonlinear Dirac operator and solutions of the Cauchy–Riemann–Fueter equation are established. 1. Introduction Nonlinear generalizations of the Dirac operator were known to physicists long ago [3] and appeared in the framework of the σ –model. Much later they were considered by mathematicians [19,22] in the realm of the Seiberg–Witten theory. The basic idea of generalization is to replace the fibre of the spinor bundle C2 ∼ = H, i.e. the simplest hyperKähler manifold, by an arbitrary hyperKähler manifold with suitable symmetries. On the other hand, Anselmi and Fre [1] generalized quaternionic analysis in the form of Fueter [13,20] for maps between arbitrary hyperKähler manifolds. It is natural to expect that there is a connection between these two approaches, since in case of the flat source manifold harmonic spinors are exactly solutions to the Cauchy–Riemann–Fueter equation ∂u ∂u ∂u ∂u −i −j −k = 0, x ∈ H, u : H → H. ∂ x0 ∂ x1 ∂ x2 ∂ x3 One of the purposes of this paper is to establish a link between the generalized Dirac operator and quaternionic analysis. The paper is organized as follows. In Sect. 2 we study properties of the Cauchy– Riemann–Fueter equation for maps between quaternionic manifolds. In particular we show that its solutions are exactly those maps, whose differential has vanishing
This paper is based upon part of the author’s thesis;he was partially supported by the grant “Gauge theory and exceptional geometry” (Universität Bielefeld) while preparing the final form.

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quaternion–linear component. Therefore we call such maps aholomorphic. Usually a hyperKähler manifold comes equipped with some symmetries. It turns out that certain symmetries of the target manifold force aholomorphic maps to be (anti)holomorphic (in the usual complex sense). This in turn allows to apply a well–developed technique of algebraic geometry to compute certain spaces of aholomorphic maps. In the last section we show that the nonlinear Dirac operator can be regarded as an analogue of the ∂–operator in complex geometry. Harmonic spinors are shown to be twisted (in an appropriate sense) aholomorphic maps. 2. Aholomorphic Maps Algebraic preliminaries. Denote by H the R–algebra of quaternions and by Sp(1) the group of quaternions of unitary length. Let (U, J1 , J2 , J3 ) and (V, I1 , I2 , I3 ) be quaternionic vector spaces. One can regard U and V as natural real Sp(1)–representations. Further, denote by W the standard Sp(1)–representation given by the left multiplication on the space of quaternions H. As usual, sp(1) denotes the adjoint representation of Sp(1). Proposition 1. Let dimH U = m, dimH V = n. We have the following decomposition into irreducible components: H om R (U, V ) ∼ = 4mn sp(1) ⊕ R4mn ,

(1)

where R4mn denotes the trivial 4mn–dimensional representation. Proof. First observe, that a choice of quaternionic basis gives an isomorphism U∼ = W ⊗ Rn . Since W ∗ ∼ = W we get = W ⊗ Rm and similarly V ∼ H om R (U, V ) ∼ = W ⊗R W ⊗ Rmn . = U∗ ⊗ V ∼ ∼ WC ⊗C WC = ∼ W ⊗ W ⊗ C2 ∼ Further, since W¯ ∼ = (S 2 W ⊕ = W we have (W ⊗R W )C = 2 2 ∼ C) ⊗ C = (sp(1)C ⊕ C) ⊗ C and the statement follows. Our next aim is to find subspaces B± ⊂ H om R (U, V ) that give the decomposition (1). Consider a linear map C : H om R (U, V ) → H om R (U, V ), C(A) = I1 A J1 + I2 A J2 + I3 A J3 .

(2)

A direct computation shows that C satisfies the equation C 2 +2C −3 = 0. Consequently, C has two eigenvalues 1 and −3 and we have the decomposition H om R (U, V ) = B+ ⊕ B− , where B+ (resp. B− ) denotes the eigenspace corresponding to the eigenvalue 1 (resp. −3). Observe that since C is Sp(1)–invariant, the subspaces B± are also Sp(1)–invariant. The trivial Sp(1)–subrepresentation of H om R (U, V ) is by definition the space H om H (U, V ) of quaternion–linear maps. It is straightforward to check the inclusions H om H (U, V ) ⊂ B− , H om H (U, V ) ⊗ R3 ⊂ B+ , where the second one is given by A1 ⊗ e1 + A2 ⊗ e2 + A3 ⊗ e3 → I1 A1 + I2 A2 + I3 A3 .

Nonlinear Dirac Operator and Quaternionic Analysis

253

By dimension counting we conclude that these inclusions are in fact isomorphisms. Since B+ is Sp(1)–invariant and complementary to the trivial 4mn–dimensional representation, it must be isomorphic to 4mn sp(1). We summarize the above considerations in the following proposition. Proposition 2. The eigenspace B− of the linear map C (2) corresponding to the eigenvalue −3 consists of quaternion–linear maps and we have the following decomposition:

H om R (U, V ) ∼ = H om H (U, V ) ⊕ B+ . Definition 1. We say that a linear map A ∈ H om R (U, V ) between two quaternionic vector spaces (U, J1 , J2 , J3 ) and (V, I1 , I2 , I3 ) is aquaternionic if I1 A J1 + I2 A J2 + I3 A J3 = A. Corollary 1. An R–linear map is aquaternionic if and only if its quaternion–linear component vanishes.

Aholomorphic maps. Let (X, J1 , J2 , J3 ) and (M, I1 , I2 , I3 ) be almost hypercomplex manifolds and u : X → M be a smooth map. Then the differential u ∗ is pointwise an R–linear map between quaternionic vector spaces T.X and T.M. Definition 2. We say that a map u : X → M is aholomorphic, if it satisfies the Cauchy– Riemann–Fueter equation I1 u ∗ J1 + I2 u ∗ J2 + I3 u ∗ J3 = u ∗ .

(3)

Theorem 1. A map u : X → M is aholomorphic if and only if the quaternion–linear component of its differential vanishes at each point.

Aholomorphic maps were studied under a variety of different names. They naturally arise in the supersymmetric σ –model and appeared in physical literature [1,11] for the first time as “triholomorphic maps” or “hyperinstantons”. Such maps naturally appear in higher–dimensional gauge theory [9] and were also studied by Chen [7], Chen and Li [8] (“quaternionic maps”), Wang [23] (“triholomorphic maps”). Joyce [16] (“q–holomorphic functions”) considered the case of the flat target manifold H. Equation (3) was known long ago and was introduced in 1934 by Fueter [13](“regular functions”) for the simplest case of maps u : H → H in his attempts to construct a quaternionic analogue of the theory of complex holomorphic maps. An extensive exposition of this theory can be found in [20]. In the author’s opinion the proposed term aholomorphic map better reflects the properties of maps satisfying Eq. (3), namely the fact that the differential of solutions to (3) has vanishing quaternion–linear component. A reader can find examples of aholomorphic maps in the above mentioned sources. Other examples will appear below. Observe that the Cauchy–Riemann–Fuether equation (3) is elliptic only in the case when a source manifold X is four–dimensional. We will concentrate on this case below. We now consider aholomorphic maps between hyperKähler manifolds. A Riemannian 4n–dimensional manifold M is called hyperKähler if the holonomy group is a subgroup of Sp(n). In other words, a Riemannian manifold (M, g) is hyperKähler if it admits three covariantly constant complex structures I1 , I2 , I3 with quaternionic relations I1 I2 = −I2 I1 = I3 , I12 = I22 = I32 = −id, compatible with the Riemannian structure: g(Il ·, Il ·) = g(·, ·), l = 1, 2, 3. Let ωl denote the Kähler 2–form corresponding to Il .

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Proposition 3. Suppose X and M are both hyperKähler manifolds. If X is also compact and 4–dimensional (i.e. X is a torus or a K3 surface), then for any smooth map u : X → M the following identity holds:
1 1 u ∗ 2L 2 = u ∗ − C(u ∗ )2L 2 − 2 4 3



ωlX ∧ u ∗ ωlM .

(4)

l=1 X

This proposition was essentially proven by Chen and Li [8, Propo. 2.2]. Formula (4) immediately follows from the result of Chen and Li, once you observe that each Kähler form on X is self–dual and that the induced scalar product on Λ2 R4 is given by the sequence ∗ ⊗id · ∧· Λ2 R4 ⊗ Λ2 R4 −−−−→ Λ2 R4 ⊗ Λ2 R4 −−→ Λ4 R4 ∼ = R.

Corollary 2 (Vanishing theorem). Let X and M be as in Proposition 3. If the cohomology class of each Kähler form ωlM on M vanishes, then any aholomorphic map u : X → M is constant.

Let I = I M denote the trivial 3–dimensional subspace of Γ (End(T M)) spanned by I1 , I2 and I3 naturally identified with Im H = sp(1). Definition 3. An isometric action of the group Sp(1) (or S O(3)) on a hyperKähler manifold M is called permuting if the subspace I is preserved and the induced action on I is the adjoint one. The hypothesis of Corollary 2 is automatically satisfied for hyperKähler manifolds that admit a permuting action of Sp(1) or S O(3) [6]. Such actions will play a crucial role in Sect. 3. The class of hyperKähler manifolds that admit a permuting action is quite wide and includes a lot of interesting examples: Hn with the flat metric and its hyperKähler reductions with respect to the zero value of momentum map; different moduli spaces, obtained as infinite–dimensional hyperKähler reductions and in particular moduli spaces of framed instantons [18] over R4 and monopoles [2]. For any quaternionic Kähler manifold with positive scalar curvature Swann [21] constructed a hyperKähler manifold with permuting action of H∗ ⊃ Sp(1). Let us consider a lager class of target hyperKähler manifolds. Namely suppose that (M, I1 , I2 , I3 ) admits only an action of S 1 that fixes one complex structure, say I1 , and rotates the other two, i.e. (L z )∗ I1 = I1 (L z )∗ ,

(L z )∗ Iw (L z¯ )∗ = Izw ,

(5)

where L z : M → M denotes the left shift by z ∈ S 1, w = a + bi is a complex number of unitary length, Iw = a I2 + bI3 . Proposition 4. Let X be a compact hyperKähler 4–dimensional manifold. Assume that the target hyperKähler manifold M admits an isometric action of S 1 that fixes I1 while rotating I2 and I3 . Then a map u : X → M is aholomorphic if and only if it is (J1 , I1 )– antiholomorphic.

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Proof. First notice that existence of an isometric S 1 –action that fixes I1 and rotates the other two complex structures implies that ω2M and ω3M are exact [15]. Indeed, denote by K the Killing vector field of the S 1 –action and by L K the Lie derivative. It follows from (5) that L K ω2M = ω3M . Applying the Cartan formula, one gets ω3M = L K ω2M = d(ı K ω2M ). Therefore equality (4) takes the following form:  1 1 2 2 u ∗  L 2 = u ∗ − C(u ∗ ) L 2 − ω1X ∧ u ∗ ω1M . (6) 2 4 X

Further, since the action of E(u) =

S1

1 u ∗ 2L 2 2

is isometric, the energy functional  1 = u ∗ 2 dvol X , u:X→M 2

(7)

X

is S 1 –invariant. Suppose now that u is aholomorphic. In particular, u is an absolute minimum of the energy functional (7) within its homotopy class α = [u]. Then for each z ∈ S 1 the map u z = L z ◦ u lies in the same homotopy class α and is also an absolute minimum of the energy functional within α. From Eq. (6) we conclude that u z must be aholomorphic: I1 u ∗z J1 + I2 u ∗z J2 + I3 u ∗z J3 = u ∗z . The above equation can be rewritten as I1 u ∗ J1 + (L z¯ )∗ I2 (L z )∗ u ∗ J2 + (L z¯ )∗ I3 (L z )∗ u ∗ J3 = u ∗ . In particular, for z = −1 we get I1 u ∗ J1 − I2 u ∗ J2 − I3 u ∗ J3 = u ∗ . Since u satisfies the Cauchy–Riemann–Fueter equation (3), we obtain I1 u ∗ J1 = u ∗ , i.e. u is (J1 , I1 )– antiholomorphic. On the other hand, it easily follows from the definition that any antiholomorphic map is also aholomorphic. An example of the circle action preserving one complex structure and rotating the other two is the standard fiberwise action on the cotangent bundle T∗ Pn of the complex projective space Pn equipped with the Calabi metric. More generally, Kaledin [17] and independently Feix [10] constructed such metrics on a neighborhood of the zero section in T ∗ Z for real–analytic Kähler manifolds Z . Example 1 (Aholomorphic maps from a K3–surface into T∗ Pn ). It follows from Proposition 4 that any aholomorphic map u : X → T∗ Pn must be (J1 , I1 )–antiholomorphic. The standard antiholomorphic automorphism of Pn induces an I1 –antiholomorphic automorphism on T∗ Pn and therefore we have a natural bijection between the spaces of holomorphic and antiholomorphic maps into T∗ Pn . Further, each holomorphic map u : X → T∗ Pn naturally decomposes into the projection ϕ : X → Pn and a holomorphic section s ∈ Γ (ϕ ∗ T∗ Pn ). Each nonconstant holomorphic map ϕ into the projective space can be obtained as a morphism associated to a linear system of a positive base point free divisor D (see [14] for example). Then ϕ ∗ OPn (1) = O X (D) and the pull–back to X of the short exact sequence dual to the Euler one has the following form: 0 → ϕ ∗ T∗ Pn → (n + 1)O X (−D) → O X → 0. It follows that the lift s must be the zero section. Therefore our problem reduces to description of the space of holomorphic maps X → Pn .

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From now on we assume that X is a K3–surface. Let a map ϕ = ϕ D : X → Pn be given by the complete linear system |D| = P(H 0 (O X (D))) with empty base locus of a positive divisor D. Since D 2 is the degree of ϕ we may assume D 2 ≥ 0. It turns out that base point free divisors D on a K3–surface admit a purely numeric characterization at least if D is big (D 2 > 0). Indeed, for a big divisor D the associated morphism ϕ : X → Pn is generically finite. It follows that dim ϕ(X ) = 2 and by the Bertini’s theorem [4] the divisor D is linearly equivalent to a smooth irreducible curve C with pa (C) = 1 + C 2 /2 = 1 + D 2 /2 > 1. It follows that D is nef and by assumption D is also big (D 2 > 0). On the other hand, the complete linear system |D| of a nef and big divisor D has a non–empty base locus if and only if D = k E + R, where R is a smooth rational curve (R 2 = −2), E is a smooth elliptic curve (E 2 = 0), E · R = 1 and k ≥ 2 [12]. In case when D is big, nef and base point free the associated morphism is a map into the complex projective space of dimension N = 1 + D 2 /2. Now fix a positive integer n. Let ϕ : X → Pn be a holomorphic map and k be the dimension of the projective span of ϕ(X ), that is ϕ decomposes as i ◦ ψ, where i : Pk → Pn is a standard embedding. Then ψ is a morphism given by a projective basis of a linear k–dimensional system of D with k ≤ N = dim |D| = 1 + D 2 /2. One can describe ψ equivalently as the composition of a morphism ϕ D : X → P N , given by a projective basis of the complete linear system |D|, and a projection π : P N \ P N −k−1 → Pk with respect to a subspace P N −k−1 → P N that does not intersect the image of ϕ D . Denote pr by PV N −k (C N +1 ) the projective Stiefel manifold, PV N −k (C N +1 ) −−→ Gr N −k (C N +1 ). Then the space (D, k) of all holomorphic maps ψ : X → Pk that can be obtained by a choice of k + 1 linearly independent sections of O X (D) without base locus is the Zariski–open set pr −1 { V ∈ Gr N −k (C N +1 ) | [V ] ∩ ϕ D (X ) = ∅ } ⊂ PV N −k (C N +1 ).

(8)

In particular, (D, k) is connected. Observe also that P N \ P N −k−1 is the total space of a vector bundle over Pk → P N . Therefore π ◦ ϕ D is homotopic to ϕ D . It follows that the spaces (D, k) and (D  , k  ) lie in the same component of Map(X, Pn ) if and only if D = D  . Summing up the above considerations we get the following result. Proposition 5. Let X be a K3–surface and n be a positive integer. Then the space of all nonconstant aholomorphic maps u : X → T∗ Pn (equivalently, the space of all holomorphic maps ϕ : X → Pn ) is the stratified space  (D, k) × Grk+1 (Cn+1 ), D, k

where D denotes a divisor class on X such that the base locus of the complete linear system |D| is empty; k is an integer, 1 ≤ k ≤ n; (D, k) is a Zariski–open subset (8) of the projective Stiefel manifold PV N −k (C N +1 ), N = 1 + D 2 /2. The subsets (D, k) × Grk+1 (Cn+1 ) and (D  , k  ) × Grk+1 (Cn+1 ) lie in the same component of Map(X, T∗ Pn ) if and only if D = D  .

Observe that any holomorphic map X → Pn can be obtained from a morphism ϕ D , given by a primitive divisor D, composing it with holomorphic maps between complex projective spaces. In the examples below we will indicate only the primitive divisor classes.

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For a K3–surface X the Euclidean lattice (H 2 (X ; Z), ∪) is isomorphic to −2E 8 ⊕3H and is of rank 22. Notice that by the Torelli theorem a K3–surface can be specified by its period point (see [4] for details). The natural map c1 : Pic(X ) → H 2 (X, Z) is injective and the rank ρ of its image, the Neron–Severi group N S(X ), can take any integer value between 0 and 20. We shall consider some examples for small values of ρ (the examples are taken from [5]). 1) ρ = 1, N S(X ) ∼ = 2k 2 . We have just one primitive divisor class D with D 2 = 2k 2 . The complete linear system is base point free and we get a regular holomorphic map 2 ϕ D : X → P1+k . 2) ρ = 2, N S(X ) ∼ = H . In this case the two generators E 1 and E 2 are elliptic curves. The complete linear system |E i | is base point free [12] and defines a regular map ϕ Ei : X → P1 . Any other positive primitive divisor class has the form D = pE 1 + q E 2 , where p and q are positive coprime integers. The base locus of D is empty and therefore we get a map ϕ D : X → P2 pq .   2 1 3) ρ = 2, N S(X ) ∼ = 1 −2 . Let C and R be generators, C 2 = 2, R 2 = −2, C · R = 2. It is easy to check that there are no divisors with vanishing self–intersection number. A divisor D = pC + q R is nef iff p ≥ q. In this case D is also big. Consequently, for coprime p and q, p ≥ q the divisor D is primitive with empty 2 2 base locus and we get a holomorphic map ϕ D : X → P1+ p +2 pq−q . 3. Harmonic Spinors as Twisted Aholomorphic Maps In this section we establish connection between aholomorphic maps and (generalized) harmonic spinors. Before proceeding let us briefly outline the definition of the (generalized) Dirac operator in a form suitable for our purposes. Details can be found in the paper of Pidstrygach [19], where the Dirac operator is defined in a slightly more general context. Algebraic preliminaries. Recall that the group Spin(4) is isomorphic to the product of two copies of Sp(1): Spin(4) = Sp+ (1) × Sp− (1), where we use subscripts “±” to distinguish between different copies. The isomorphism Spin(4)/ ± 1 ∼ = S O(4) is given by [q+ , q− ] → Bq+ ,q− : H → H,

Bq+ ,q− h = q− h q¯+ .

(9)

Recall that W denotes the Sp(1)–representation on H by multiplication on the left. Further, R4 denotes the standard S O(4)–representation. The following homomorphism of Spin(4) = Sp+ (1) × Sp− (1)–representations R4 ⊗ W + → W − ,

h 1 ⊗ h 2 → h 1 h 2 ,

is called the Clifford multiplication, where W ± denote the representations induced by Sp± (1). Let (V, I1 , I2 , I3 ) be a quaternionic vector space. We identify the subspace span (I1 , I2 , I3 ) ⊂ EndR (V ) with Im H = sp(1).

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Lemma 1. Let V be a real representation of the group Sp(1) such that the induced action on EndR (V ) preserves span(I1 , I2 , I3 ). Assume that the induced representation on span(I1 , I2 , I3 ) coincides with the adjoint one. Then VC ∼ = W ⊗ E,

(10)

where E is an Sp(1)–representation. The proof is a straightforward calculation of the actions of i, j, k ∈ Sp(1) on C⊗R V ∼ = C2 ⊗C VI1 , where VI1 denotes the complex vector space (V, I1 ). From (10) one obtains a variant of the Clifford multiplication: R4C ⊗ VC ∼ = R4C ⊗ W + ⊗ E → W − ⊗ E.

(11)

Observe that the homomorphism (11) induces an homomorphism between real parts: R4 ⊗ V → [W − ⊗ E]r . The nonlinear Dirac operator. From now on X 4 is a smooth closed oriented Riemannian manifold. We also assume that X is a spin manifold with a Spin(4)–principal bundle π : P → X . The assumption that X is spin is not essential, since the bundle P can be replaced by a Spin c (4)–bundle, however the exposition becomes a bit clearer in the case of spin manifolds. Let (M, I1 , I2 , I3 ) be a hyperKähler manifold with permuting action of Sp(1) (see Definition 3). Then the group Spin(4) acts on M via the homomorphism Spin(4) → Sp+ (1) and we get the associated fibre bundle: M = P × Spin(4) M. Observe that in case M = H with the standard action of Sp(1) one gets the usual positive spinor bundle W + . Therefore sections of M are called (generalized) spinors. Notice that the space of spinors can be naturally identified with the space of equivariant maps: Γ (M) ∼ = Map Spin(4) (P, M). The Levi–Civita connection on P determines the covariant derivative: ∇v u = u ∗ (ˆv),

u ∈ Map Spin(4) (P, M),

where vˆ denotes the horizontal lift of v ∈ T X . In other words we get a map ∇ : Map Spin(4) (P, M) → Γ (T ∗ X ⊗ π! (u ∗ T M)), where π! (u ∗ T M) → X denotes the factor of u ∗ T M → P by the group action. Remark 1. Strictly speaking, the operator ∇ is not well–defined, since its range depends on the element of the domain. However one can define ∇ as a section of a certain vector bundle as follows. Consider the evaluation map ev : Map Spin(4) (P, M) × P → M, (u, p) → u( p). Then one gets the following diagram: ev ∗ T M ⏐ ⏐ 

−−−−→ T M ⏐ ⏐  ev

Map Spin(4) (P, M) × P −−−−→ M.

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Dividing the first column by the group action one gets a vector bundle E over the infinite dimensional space Map Spin(4) (P, M) × X ; the restriction of E to {u} × X coincides with π! (u ∗ T M). Then ∇ is well–defined as a section of E → Map Spin(4) (P, M) × X . In order to keep the exposition clear, we will not keep to the above formalism of vector bundles over infinite–dimensional spaces. The tangent space of M has a natural structure of quaternionic vector space at each point. Since we have a permuting action of Sp(1) on M, we get from Proposition 1 that ˜ where E˜ is a complex vector bundle over M with an action of Sp(1) TC M ∼ = W + ⊗ E, ( E˜ coicides with (T M, I1 ) as a complex vector bundle, however the action of Sp(1) is different from the induced one). Consequently, we get the Clifford multiplication Cl : T ∗ X ⊗ π! (u ∗ T M) → [W − ⊗ E]r ,

(12)

˜ and W − is the negative spinor bundle over X . where E = π! (u ∗ E) Definition 4 ([19]). The first order differential operator D defined by the sequence  Cl  ∇ D : Γ (M) −−→ Γ T ∗ X ⊗ π! (u ∗ T M) −−→ Γ [W − ⊗ E]r is called a (generalized) Dirac operator. Definition 5. A spinor u such that D u = 0 is called harmonic. Remark 2. The Dirac operator is well–defined as a section of a vector bundle over an infinite–dimensional space similarly to the covariant derivative (see Remark 1) and it is a Fredholm section [19]. In the case when the fibre M of the spinor bundle is a copy of quaternions with the standard action of Sp(1) one recovers the usual linear Dirac operator. Remark 3. Notice that if the target hyperKähler manifold admits a permuting action of S O(3) rather than Sp(1) one needs just the principal S O(4)–bundle of orthonormal frames rather than its Spin(4)–lifting to define the Dirac operator. A well–known example in classical theory is d + + d ∗ : 1 (X ) → 2+ (X ) ⊕ 0 (X ). One can also define a Dirac operator with the help of a Spin c (4)–structure. In this case the target manifold M is required to carry a triholomorphic action of S 1 commuting with the permuting action of Sp(1). The Levi–Civita connection splits T P into horizontal and vertical bundles: T P ∼ = H ⊕ V. Since we have a natural projection pr : P → PS O onto the principal bundle of orthonormal frames of X , the horizontal bundle H has a natural quaternionic structure (J1 , J2 , J3 ), which is defined as follows: a point p ∈ P determines an orthonormal basis v = pr ( p) = (v0 , v1 , v2 , v3 ) of Tπ( p) X and consequently a basis vˆ of H p ; then (J1 , J2 , J3 ) is defined as the unique quaternionic structure1 such that vˆ l = −Jl vˆ 0 for l = 1, 2, 3. Theorem 2. For a map u ∈ Map Spin(4) (P, M) denote by u ∗h the restriction of the differential u ∗ to the horizontal subbundle H ⊂ T P. Then the spinor u is harmonic if and only if the Cauchy–Riemann–Fueter–type equation holds: I1 u ∗h J1 + I2 u ∗h J2 + I3 u ∗h J3 = u ∗h . 1 The basis v provides an isomorphism T X ∼ H; then the quaternionic structure on H, compatible with x = the isomorphism (9), is the right one.

260

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Proof. Pick a point p ∈ P and denote by v = pr ( p) the basis of T X at the point x = π( p) as before. Let u ∈ Map Spin(4) (P, M) be a harmonic spinor, m = u( p) ∈ M. The pull–back of the Clifford multiplication (12) to P can be described by the following sequence(see (10), (11)):

 ∼ R4 ⊗ u ∗ T M → R4 ⊗ u ∗ T M = R4C ⊗ u ∗ TC M C (13) ∼ =(W − ⊗ W + ) ⊗ (W + ⊗ T M) → W − ⊗ T M. Notice that the image of the above map lies automatically in the real part of W − ⊗ T M. Further, R4 and W ± are isomorphic to H as vector spaces. Then the homomorphism (13) is the following map: h ⊗ w → h · 1 ⊗ w − h · j ⊗ I2 w,

h ∈ H, w ∈ T M.

Observe that h = 1 (resp. i, j, k) corresponds to the horizontal lift of v0 (resp. v1 , v2 , v3 ). According to the definition of the Dirac operator take h = 1, w = ∇vˆ 0 u = u ∗ (ˆv0 ) = u ∗h (ˆv0 ); h = i, w = u ∗h (ˆv1 ) . . . and sum up the result. It follows that u is harmonic at the point x if and only if the following equation holds:



 1 ⊗ u ∗h (ˆv0 ) − j ⊗ I2 u ∗h (ˆv0 ) + i ⊗ u ∗h (ˆv1 ) − k ⊗ I2 u ∗h (ˆv1 ) +



 + j ⊗ u ∗h (ˆv2 ) + 1 ⊗ I2 u ∗h (ˆv2 ) + k ⊗ u ∗h (ˆv3 ) + i ⊗ I2 u ∗h (ˆv3 ) = 0. After a simplification one gets

 1 ⊗ u ∗h (ˆv0 ) + I1 u ∗h (ˆv1 ) + I2 u ∗h (ˆv2 ) + I3 u ∗h (ˆv3 )

 + j ⊗ −I2 u ∗h (ˆv0 ) + I3 u ∗h (ˆv1 ) + u ∗h (ˆv2 ) − I1 u ∗h (ˆv3 ) = 0.

(14)

It is easy to see that (14) is equivalent to the single equation u ∗h (ˆv0 )+ I1 u ∗h (ˆv1 )+ I2 u ∗h (ˆv2 )+ I3 u ∗h (ˆv3 ) = 0. Recalling that vˆ l = −Jl vˆ 0 , l = 1, 2, 3 we get ˜ ∗h ) vˆ 0 = u ∗h vˆ 0 − I1 u ∗h J1 vˆ 0 − I2 u ∗h J2 vˆ 0 − I3 u ∗h J3 vˆ 0 = 0. C(u ˜ ∗h ) vˆ 0 = 0 and C(u ˜ ∗h ) Jl vˆ 0 = 0 are Observe that for any fixed l = 1, 2, 3 the equations C(u equivalent. It remains to note that vˆ 0 , J1 vˆ 0 , J2 vˆ 0 and J3 vˆ 0 span the horizontal subspace ˜ ∗h ) = 0. at the point p ∈ P and therefore C(u Corollary 3. A spinor u is harmonic if and only if the horizontal part u ∗h of its differential has no quaternion–linear component.

The above corollary reveals a deep analogy between the Dirac operator and the ∂–operator of complex geometry. Indeed, let (Y, IY ) and (Z , I Z ) be complex manifolds. Assume that a Lie group G acts holomorphically on Z and pick a G–principal bundle π G : PG → Y with a connection. Then the operator ∂ can be defined in the usual way, namely   ∇ ∂ : Γ (PG ×G Z ) −−→ 1 (Y ) ⊗ Γ π!G (u ∗ T Z ) → 1,0 (Y ) ⊗ Γ π!G (u ∗ T Z ) . Further, we also have a splitting of the tangent bundle of PG into vertical and horizontal parts: T PG = V ⊕ H. Observe that the bundle H inherits a complex structure from T Y . Then an equivariant map u satisfies the equation ∂u = 0 if and only if I Z u ∗h IY = u ∗h or, in other words, if and only if the horizontal component has no complex–linear component.

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Acknowledgement. I would like to thank my supervisor Prof. V. Pidstrygach for his constant encouragement and helpful discussions.

References 1. Anselmi, D., Fré, P.: Topological σ -models in four dimensions and triholomorphic maps. Nucl. Phys. B 416(1), 255–300 (1994) 2. Atiyah, M., Hitchin, N.: The geometry and dynamics of magnetic monopoles. M. B. Porter Lectures. Princeton, NJ: Princeton University Press, 1988 3. Bagger, J., Witten, E.: Matter couplings in N = 2 supergravity. Nucl. Phys. B 222(1), 1–10 (1983) 4. Barth, W.P., Hulek, K., Peters, C.A.M., Van de Ven, A.: Compact complex surfaces, Volume 4 of Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics]. Berlin: Springer-Verlag, second edition, 2004 5. Belcastro, S.-M.: Picard lattices of families of K 3 surfaces. Comm. Alg. 30(1), 61–82 (2002) 6. Boyer, C.P., Galicki, K., Mann, B.M.: Quaternionic reduction and Einstein manifolds. Comm. Anal. Geom. 1(2), 229–279 (1993) 7. Chen, J.: Complex anti-self-dual connections on a product of Calabi-Yau surfaces and triholomorphic curves. Commun. Math. Phys. 201(1), 217–247 (1999) 8. Chen, J., Li, J.: Quaternionic maps between hyperkähler manifolds. J. Differ. Geom. 55(2), 355–384 (2000) 9. Donaldson, S.K., Thomas, R.P.: Gauge theory in higher dimensions. In: The geometric universe (Oxford, 1996), Oxford: Oxford Univ. Press 1998, pp. 31–47 10. Feix, B.: Hyperkahler metrics on cotangent bundles. J. Reine Angew. Math. 532, 33–46 (2001) 11. Figueroa-O’Farrill, J.M., Köhl, C., Spence, B.: Supersymmetric Yang-Mills, octonionic instantons and triholomorphic curves. Nucl. Phys. B 521(3), 419–443 (1998) 12. Friedman, R.: Algebraic surfaces and holomorphic vector bundles. Universitext. New York: SpringerVerlag, 1998 13. Fueter, R.: Die Funktionentheorie der Differentialgleichungen ∆u = 0 und ∆∆u = 0 mit vier reellen Variablen. Comment. Math. Helv. 7(1), 307–330 (1934) 14. Griffiths, P., Harris, J.: Principles of algebraic geometry. New York: Wiley-Interscience [John Wiley & Sons], 1978 15. Hitchin, N.J.: The self-duality equations on a Riemann surface. Proc. Lond. Math. Soc., III. Ser. 55, 59–126 (1987) 16. Joyce, D.: Hypercomplex algebraic geometry. Quart. J. Math. Oxford Ser. (2) 49(194), 129–162 (1998) 17. Kaledin, D.: Hyperkaehler structures on total spaces of holomorphic cotangent bundles. http://arxiv.org/ list/alg-geom/9710026, 1997 18. Maciocia, A.: Metrics on the moduli spaces of instantons over Euclidean 4-space. Commun. Math. Phys. 135(3), 467–482 (1991) 19. Pidstrygach, V.Ya.: Hyper-Kähler manifolds and the Seiberg-Witten equations. Tr. Mat. Inst. Steklova, 246(Algebr. Geom. Metody, Svyazi i Prilozh.), 263–276, (2004) 20. Sudbery, A.: Quaternionic analysis. Math. Proc. Camb. Philos. Soc. 85, 199–225 (1979) 21. Swann, A.: HyperKahler and quaternionic Kahler geometry. Math. Ann. 289(3), 421–450 (1991) 22. Taubes, C.H.: Nonlinear generalizations of a 3-manifold’s Dirac operator. In: Trends in mathematical physics (Knoxville, TN, 1998), Volume 13 of AMS/IP Stud. Adv. Math., Providence, RI: Amer. Math. Soc. pp. 475–486 1999 23. Wang, C.: Energy quantization for triholomorphic maps. Calc. Var. Partial Differ. Eqs. 18(2), 145–158 (2003) Communicated by G.W. Gibbons

Commun. Math. Phys. 281, 263–286 (2008) Digital Object Identifier (DOI) 10.1007/s00220-008-0491-8

Communications in

Mathematical Physics

Mott Law as Upper Bound for a Random Walk in a Random Environment A. Faggionato1 , P. Mathieu2 1 Dipartimento di Matematica “G. Castelnuovo”, Università “La Sapienza”,

P.le Aldo Moro 2, 00185 Roma, Italy. E-mail: [email protected]

2 Centre de Mathématiques et d’Informatique, Université de Provence,

39 rue Joliot Curie, 13453 Marseille cedex 13, France. E-mail: [email protected] Received: 2 April 2007 / Accepted: 8 January 2008 Published online: 6 May 2008 – © Springer-Verlag 2008

Abstract: We consider a random walk on the support of an ergodic simple point process on Rd , d ≥ 2, furnished with independent energy marks. The jump rates of the random walk decay exponentially in the jump length and depend on the energy marks via a Boltzmann–type factor. This is an effective model for the phonon–induced hopping of electrons in disordered solids in the regime of strong Anderson localization. Under some technical assumption on the point process we prove an upper bound for the diffusion matrix of the random walk in agreement with Mott law. A lower bound for d ≥ 2 in agreement with Mott law was proved in [8]. 1. Introduction 1.1. Physical motivations. Phonon-assisted electron transport in disordered solids in which the Fermi level (set equal to 0 below) lies in a region of strong Anderson localization can be modeled by Mott variable–range hopping of the following interacting particle system in a random environment [17]. The environment is given by ξ := ({xi }, {E i }), where {xi } is an infinite and locally finite set of points in Rd such that each point xi is labeled by an energy mark E i belonging to some finite interval. Given ξ , particles can lie only at points of {xi } and perform random walks on {xi } with hard–core interaction. The probability rate for a jump of a particle at x to the vacant site y, with x = y in {xi }, is given by 
(1.1) r x,y (ξ ) = r0 exp −2|x − y|/∗ − β{E y − E x }+ , where E z is defined by E i for z = xi and {E y − E x }+ = max{E y − E x , 0}. Above | · | denotes the Euclidean norm in Rd , β = 1/(kT ), ∗ denotes the localization radius of wavefunctions and r0 is a constant depending on the material but which depends only weakly on β, |x − y| and the energies E x , E y . Without loss of generality, in what follows we set ∗ = 2, r0 = 1.

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The disorder of the solid is modeled by the randomness of the environment. The points {xi } correspond to the impurities of the disordered solid and the electron Hamiltonian has exponentially localized quantum eigenstates with localization centers xi if the corresponding energies E i are close to the Fermi level. The random set {xi } is a stationary simple point process with finite intensity (i.e. with finite mean number of points in finite boxes) and its stationarity reflects the homogeneity of the medium. Conditioned to {xi }, the energy marks {E i } are supposed to be i.i.d. random variables taking value in a finite interval (set in what follows equal to [−1, 1]) with common distribution ν, such that ν(d E) ∼ c |E|α d E, |E|  1, for a suitable nonnegative exponent α. The independence of the energy marks is compatible with Poisson level statistics, which is a general rough indicator for the localization regime and has been proven to hold for an Anderson model [13]. The exponent α allows to model a possible Coulomb pseudogap in the density of states [17]. In particular, the physically relevant possible values of α are 0 and d − 1, the latter corresponding to the Coulomb pseudogap. The DC conductivity matrix σ (β), measuring the linear response of the solid to a uniform external electric field, would vanish if it were not for the lattice vibrations (phonons) at nonzero temperature. For an isotropic solid at low temperature and with dimension d ≥ 2, Mott law [14,15] predicts that the conductivity decays exponentially as   α+1 σ (β) ∼ exp −c β α+1+d I, β  1, (1.2) where c is a β–independent positive constant and I denotes the identity matrix. The original derivation of (1.2) is based on an heuristic optimization argument, alternative and more robust derivations have been proposed after that (see [1,17] and references therein). Finally, we point out that due to the Einstein relation between σ (β) and the bulk diffusion matrix Dbulk (β) [18], (1.2) is equivalent to the asymptotic behavior   α+1 Dbulk (β) ∼ exp −c β α+1+d I, β  1. (1.3) A mean field version of Mott variable–range hopping at small temperature is given ξ by the continuous–time random walk X t on {xi } such that the jump from x to y, x = y in {xi }, has probability rate given by
 (1.4) cx,y (ξ ) = exp −|x − y| − β(|E x − E y | + |E x | + |E y |)/2 . ξ

We will call X t Mott variable–range random walk. The choice of the transition rates cx,y (ξ ) comes from the fact that, at small temperature, cx,y (ξ ) = r x,y (ξ ) µ(ηx = 1) µ(η y = 0) (1 + o(1)),

β  1,

(1.5)

where ηx is the particle number at site x in the above particle system,  while µ is the Gibbs measure of the particle system w.r.t. the Hamiltonian H = i E i ηxi with zero Fermi level, i.e. µ is the product measure on {0, 1}{xi } such that µ(ηxi = 1) = e−β Ei /(1 + e−β Ei ). In (1.5), the error term is negligible as β → ∞ uniformly in x, y belonging to a finite volume (as in realistic solids), for a typical environment ξ . Another derivation of the

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mean field version (1.4) from the original Mott variable range hopping (1.1) has been obtained in [12] (see also [1, Sect. IV]) by reduction to a random resistor network. The analogue of Mott law (1.3) for Mott variable–range random walk is given by   α+1 β  1, (1.6) D(β) ∼ exp −c β α+1+d I, where D(β) denotes the diffusion matrix of the random walk. For dimension d ≥ 2, a lower bound of D(β) in agreement with (1.6) has been recently proven in [8]. The present work addresses the problem of rigorously deriving an upper bound of D(β) in agreement with (1.6). The one dimensional case presents special features, and rigorous results on the corresponding Mott law have been obtained in [5].

1.2. Model and results. Let us give a precise definition of Mott variable–range random ξ walk X t , generalizing the choice of jump rates (1.4). The environment ξ = ({xi }, {E i }) is defined as follows. Let {xi } be a simple point process, i.e. a random locally finite subset of Rd , whose law is the Palm distribution Pˆ 0 associated to the law Pˆ of a stationary simple point process on Rd with finite intensity. Given {xi }, the energy marks {E i } are i.i.d. random variables having value in [−1, 1] and common law ν (restrictions on ν will be specified later). The law P0 of the environment is the so called ν–randomization of ˆ the Palm distribution Pˆ 0 associated to P. The reason for considering Palm distributions is the following: in order to make the random walk start at a fixed point, taken equal to the origin below, we condition to contain the origin the stationary point process of impurities introduced in the previous subsection. As discussed in Sect. 2, if the stationary process has finite intensity and law ˆ the law of the resulting process is given by the Palm distribution Pˆ 0 associated to P. ˆ P, d In what follows, we write ξ for a generic locally finite subset of R × [−1, 1] such that ξ has at most one point in each fiber {x} × [−1, 1], and we write ξˆ for a generic d ˆ locally finite subset of   R . Both ξ and ξ can be identified with the counting measures δ and δ respectively. Moreover, when there is no ambiguity, given x (x,E) ˆ (x,E)∈ξ x∈ξ d ξ its spatial projection on R will be denoted by ξˆ . We finally recall that the κ–moment ˆ κ > 0, is defined as ρκ of P,   ρκ := EPˆ |ξˆ ∩ [0, 1]d |κ . ˆ Then ρ := ρ1 is called the intensity of the process P. ξ Given a realization of the environment ξ , Mott variable–range random walk X t is defined as the continuous–time random walk on ξˆ = {xi } starting at the origin and jumping from x to y, x = y in {xi }, with probability rate
 cx,y (ξ ) = exp −|x − y| − βu(E x , E y ) , (1.7) where the function u satisfies κ1 (|E x | + |E y |) ≤ u(E x , E y ) ≤ κ2 (|E x | + |E y |)

(1.8)

for some positive constants κ1 ≤ κ2 . To simplify the notation, it is convenient to set cx,x (ξ ) ≡ 0 for all x ∈ {xi }.

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The law Pξ of X t is characterized by the following identities: ξ

Pξ (X 0 = 0) = 1, ξ

ξ

t ≥ 0, x = y, Pξ (X t+dt = y | X t = x) = cx,y (ξ )dt + o(dt),  ξ ξ cx,z (ξ )dt + o(dt), t ≥ 0. Pξ (X t+dt = x | X t = x) = 1 − z ξ

Equivalently, the dynamics of X t can be described as follows: after arriving at site x the particle waits an exponential time with parameter  λx (ξ ) = cx,z (ξ ), (1.9) z

and then jumps to site y, y = x, with probability cx,y (ξ ) . λx (ξ )

(1.10)

By standard methods (see e.g. [3,8, Appendix A]), one can check that the random walk ξ X t is well–defined for P0 –a.a. ξ as soon as Pˆ is ergodic w.r.t. spatial translations and EP0 (λ0 (ξ )) < ∞. As proven in [8] (see also Lemma 1 below) this last condition is equivalent to require that ρ2 < ∞. Assuming Pˆ to be ergodic, ρ12 to be finite and under some additional technical assumption on the law of the environment P0 , in [8] the authors prove that the diffusively rescaled random walk X ξ converges in P0 –probability to a Brownian motion whose covariance matrix coincides with D(β), where D(β) is the diffusion matrix of X ξ defined as the unique symmetric matrix such that    1 ξ EP0 EPξ (X t · a)2 , a ∈ Rd . (a · D(β)a) = lim (1.11) t→∞ t 0 Moreover, they prove the following variational characterization of D(β): P0 (ξ ) ξˆ (d x) c0,x (ξ ) (a · x − ∇x f (ξ ))2 , inf (a, D(β) a) = f ∈L ∞ (P0 )

a ∈ Rd , (1.12)

ˆ where the set ξˆ is defined as the spatial projection of ξ , i.e. ξ = {xi }, and is identified with the counting measure i δxi . Moreover, the gradient ∇x f (ξ ) is defined as ∇x f (ξ ) = f (Sx ξ ) − f (ξ ),

Sx ξ := {(xi − x, E i )} .

In addition, for d ≥ 2 and assuming that ν([−E, E]) ≥ c0 |E|α+1 ,

∀E ∈ [−1, 1],

(1.13)

for some positive constant c0 and some exponent α ≥ 0, the authors prove a lower bound on D(β): for β large enough it holds   α+1 (1.14) D(β) ≥ c1 β −c2 exp −C β α+1+d I,

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where c1 , c2 , C are positive constants independent of β. The above bound is in agreement with Mott law (1.6). We note that the requirement α > 0 in [8, Theorem 1] is due to a typing error. Moreover it is simple to check that, although in [8] the transition rates cx,y (ξ ) are defined as in (1.4), all the results in [8] remain true for transition rates cx,y (ξ ) defined as in (1.7) and assuming only that α > −1 in (1.13). Our main result consists in an upper bound for D(β) in agreement with Mott law. Roughly, we claim that if (1.13) holds with inverted sign, then also (1.14) remains valid with inverted sign. In order to precisely state our technical assumptions we fix some notation. ˆ its p–thinning is the simple Given p ∈ [0, 1] and a simple point process with law P, ˆ point process obtained as follows: for each realization ξ of the process with law Pˆ erase each point independently with probability 1 − p. We will write Pˆ ( p) for the law of the ˆ p–thinning of P. Finally, we recall that the Poisson point process on Rd with intensity ρ > 0 is a random locally finite subset ξˆ ⊂ Rd such that (i) for any A ⊂ Rd Borel and bounded, the cardinality ξˆ (A) = |ξˆ ∩ A| is a Poisson random variable with expectation ρ (A), where (A) is the Lebesgue measure of A; (ii) for any disjoint Borel subsets A1 , . . . , An ⊂ Rd , ξˆ (A1 ), . . . , ξˆ (An ) are independent random variables. We denote by Pˆ ρ the law of the Poisson point process with intensity ρ. The process Pˆ ρ is stationary. We can finally state our main result: Theorem 1. Let P0 be the ν–randomization of the Palm distribution Pˆ 0 associated to a stationary simple point process on Rd , d ≥ 2, with law Pˆ and finite intensity ρ, and let the following conditions be satisfied: • (i) For some constants c0 > 0 and α > −1, 0 < ν([−E, E]) ≤ c0 |E|α+1 ,

∀E ∈ (0, 1] ;

(1.15)

• (ii) There exist positive constants ρ  , K and there exists p ∈ (0, 1] such that, setting K (x) = x + [−K /2, K /2)d and defining the random field Y as Y = {Y (x) : x ∈ K Zd },

Y (x) := ξˆ ( K (x)) ,

(1.16)

ˆ is stochasthen the law of Y when ξˆ is chosen with law Pˆ ( p) (the p–thinning of P) ˆ ˆ tically dominated by the law of Y when ξ is chosen with law Pρ  (the Poisson point process with density ρ  ). Then the d × d symmetric matrix D(β) solving the variational problem (1.12) admits the following upper bound for β large enough:   α+1 (1.17) D(β) ≤ c1 β c2 exp −C β α+1+d I, for suitable β–independent positive constants c1 , c2 , C.

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We recall that due to the Strassen theorem the stochastic domination assumption in condition (ii) above is equivalent to the fact that one can construct on the same probability space processes Y1 = {Y1 (x) : x ∈ K Zd } and Y2 = {Y2 (x) : x ∈ K Zd } in such a way that Y1 (x) ≤ Y2 (x),

∀x ∈ K Zd ,

(1.18)

the law of Y1 equals the law of Y when ξˆ is chosen with law Pˆ ( p) and the law of Y2 equals the law of Y when ξˆ is chosen with law Pˆ ρ  . We have that D(β) ≥ e−Cβ I since |E| is assumed to be bounded from above. In case |E| is bounded from below by a positive constant (which violates the left condition in  (1.15)), then the reverted inequality holds: D(β) ≤ e−C β I. Indeed, due to (1.8) we can −|x−y|−2κ 2 β and, if ν([−E, E]) = 0 bound the transition rates cx,y (ξ ) from below by e for some E > 0, then we can bound cx,y (ξ ) from above by e−|x−y|−2κ1 Eβ . Using the variational characterization (1.12) and a similar formula for the diffusion matrix D˜ of the random walk on ξˆ with probability rate for a jump from x to y = x given by e−|x−y| , it is easy to derive from the above estimate that D(β) ≥ e−2κ2 β D˜ and, if ν([−E, E]) = 0 ˜ for some E > 0, D(β) ≤ e−2κ1 Eβ D. Theorem 1 applies to the case that Pˆ is or is dominated by a stationary Poisson point process Pˆ ρ  , i.e. when one can define random sets (ξ, ξ  ) such that ξ ⊂ ξ  almost surely and ξ, ξ  have marginal distributions Pˆ and Pˆ ρ  respectively. An example is given by Gibbsian random point fields with repulsive interactions (cf. [9] and [4, Sect. 5]). Moreover, the above theorem covers also the case of thinnings of point processes with uniform bounds on the local density, as for example diluted crystals. Note that in this case the point process Pˆ is not stochastically dominated by any stationary Poisson point process. We refer to Sect. 6 for a more detailed discussion. Finally we note that (1.14) and (1.17) may both hold. Assume for instance that points are distributed according to a Poisson point process and suppose that both conditions (1.13) and (1.15) are satisfied for the energy marks (the constants c0 may be different in (1.13) and (1.15)). Then the two bounds (1.14) and (1.17) both hold. In such a situation, one might wonder whether α+1

lim β − α+1+d log D(β)

β↑∞

exists. This question seems to be out of reach of the techniques used here and in [8], and seems to require more sophisticated arguments. 1.3. Overview. In Sect. 2 we recall some definitions and results about point processes (see [6,8] for more details) and state some technical results needed later on. The proof of Theorem 1 is based on the variational formula (1.12) since for each fixed function f ∈ L ∞ (P0 ) the r.h.s. in (1.12) gives an upper bound on (a, D(β)a). In Sect. 3 we derive an upper bound on D(β) by taking special test functions f in the r.h.s. of (1.12). The choice of such test functions is inspired by [16] and is related to the percolation approach of [1,8]. In Sect. 4 we first show that the above upper bound together with some scaling arguments leads to (1.17) if P0 is the ν-randomization of the Palm distribution associated to a stationary Poisson point process. In Sect. 5 we extend the proof to general point processes satisfying assumption (ii).

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As conjectured in the introduction of [8], the leading contribution to the conductivity of the medium as β ↑ ∞ comes from a subset of impurities that converges to a Poisson point process under suitable space rescaling. Due to the relevance of Poisson point processes in Mott’s law and since in the Poissonian case the proof of Theorem 1 is more transparent and simple, we have preferred to treat the Poissonian case and the general case separately. Finally, in Sect. 6 we show that Theorem 1 can be applied to thinnings of point processes with uniform bounds on the local density as diluted crystals. For a more detailed discussion about mathematical aspects and physical motivations of the Mott random walk we refer to [8,17] and references therein. 2. Simple Point Processes In this section we recall some basic definitions and results about simple point processes, referring to [6,8] for more details. In what follows, given a topological set Y we write B(Y ) for the σ –algebra of its Borel subsets. We denote Nˆ the space of simple counting measures ξˆ on Rd , i.e. integer– valued measures such that ξˆ (B) < ∞ for all bounded B ∈ B(Rd ), and ξˆ (x) ∈ {0, 1} for  all x ∈ Rd . One can show that ξˆ ∈ Nˆ if and only if ξˆ = j δx j , where {x j } ⊂ Rd is a locally finite set. Trivially, a simple counting measure ξˆ can be identified with its sup port. Given x ∈ Rd the translated counting measure Sx ξˆ is defined as Sx ξˆ = j δx j −x  if ξˆ = j δx j . The space Nˆ is endowed with the σ –algebra of measurable subsets generated by the maps Nˆ  ξˆ → ξˆ (B) ∈ N,

B ∈ B(Rd ) bounded.

A simple point process (on Rd ) is a measurable map from a probability space into Nˆ . With abuse of notation, we identify a simple point process with its distribution Pˆ on ˆ ˆ x A), for all x ∈ Rd and A ⊂ Nˆ Nˆ . Moreover, one calls it stationary if P(A) = P(S measurable. In this case, we define the κ–moment ρκ as   κ  , κ > 0. (2.1) ρκ := EPˆ ξˆ [0, 1]d Then ρ = ρ1 is the so–called intensity of the process. The Palm distribution Pˆ 0 associated to a stationary simple point process Pˆ on Rd with finite intensity ρ is the probability measure on the measurable subset Nˆ 0 ⊂ Nˆ ,   Nˆ 0 := ξˆ ∈ Nˆ : ξˆ (0) = 1 , characterized by the Campbell identity: 1 ˆ ˆ ˆ ξˆ (d x) χ A (Sx ξˆ ), P0 (A) = P(d ξ ) ρ K d Nˆ QK

∀A ⊂ Nˆ 0 measurable, (2.2)

where Q K = [−K /2, K /2]d , K > 0 (since Pˆ is stationary, the r.h.s. in Campbell identity does not depend on K ). As discussed in [6], the point process Pˆ 0 can be thought of

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as obtained from the point process Pˆ by conditioning the latter to give positive mass at the origin. ˆ Pˆ  one says that Pˆ is stochastically dominated Given two simple point processes P,   ˆ Pˆ  such that almost surely ξˆ ⊂ ξˆ  , by Pˆ , shortly Pˆ  Pˆ , if there exists a coupling of P, with (ξˆ , ξˆ  ) denoting the coupled random sets with marginal distributions given by Pˆ and Pˆ  respectively. We refer to [9] for more details on stochastic domination between point processes. We denote N the space of (marked) simple counting measures ξ on Rd ×[−1, 1] such that ξ(B × [−1, 1]) < ∞ for all B ∈ B(Rd ) bounded, and ξ({x} × [−1, 1]) ∈ {0, 1} for all x ∈ R. One can show that ξ ∈ N if and only if ξ = j δ(x j ,E j ) , where d {(x j , E j )} ⊂ R × [−1, 1] is a locally finite set such that for each x ∈ R there is at most one couple (x j , E j ) with x j = x. Trivially, a simple counting measure ξ can be identified with its support. The value E j is called the mark at x j . For physical reasons, we call it the energy mark. Given ξ ∈ N we write ξˆ for the simple counting measure on Rd defined as ξˆ (B) = ξ(B × [−1, 1]), for all B ∈ B(Rd ) bounded. Given x ∈ Rd the translated  simple counting measure Sx ξ is defined as Sx ξ = j δ(x j −x,E j ) if ξ = j δ(x j ,E j ) . The space N is endowed with the σ –algebra of measurable subsets generated by the maps N  ξ → ξ(B) ∈ N,

B ∈ B(Rd × [−1, 1]) bounded.

A marked simple point process on Rd is a measurable map from a probability space into N . Again, with abuse of notation, we will identify it with its distribution P on N . Moreover, it is called stationary if P(A) = P(Sx A), for all x ∈ Rd and A ⊂ N measurable. In this case, we define the κ–moment ρκ as   κ  ρκ := EP ξˆ [0, 1]d , κ > 0, and call ρ := ρ1 the intensity of the process. The Palm distribution P0 associated to a stationary marked simple point process P with finite intensity ρ is the probability measure on the measurable subset N0 ⊂ N ,   N0 := ξ ∈ N : ξˆ (0) = 1 , characterized by the Campbell identity 1 ξˆ (d x) χ A (Sx ξ ), P(dξ ) P0 (A) = ρKd N QK

∀A ⊂ N0 measurable, K > 0. (2.3)

A standard procedure for obtaining a marked simple point process from a given simple point process on Rd is the ν–randomization, where ν is a probability measure on [−1, 1]: given a realization of the simple point process on Rd , its points are marked by i.i.d. random variables with common law ν. It is simple to check that the ν-randomization of the Palm distribution associated to a given stationary simple point process coincides with the Palm distribution associated to the ν–randomization of the stationary simple point process. We conclude this section recalling some technical results derived in [8]. In particular, point (i) of the following lemma follows from [8, Lemma 1, (i)] and the Monotone Convergence Theorem, while the proof of point (ii) is similar to the proof of [8, Lemma 2]:

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Lemma 1. [8]. Let P0 be the Palm distribution associated to a stationary marked simple point process. function which is non negative or such (i) Let f : N0 × N0 → R be a measurable

ˆ ˆ that ξ (d x) | f (ξ, Sx ξ )| and ξ (d x) | f (Sx ξ, ξ )| are in L 1 (P0 ). Then P0 (dξ ) ξˆ (d x) f (ξ, Sx ξ ) = P0 (dξ ) ξˆ (d x) f (Sx ξ, ξ ) . (ii) Let n be a nonnegative integer such that ρn+1 < ∞. Then n  ξˆ (d x)e−γ |x| < ∞ P0 (dξ ) for any γ > 0. 3. Upper Bounds via Special Test Functions In this section we let P0 be the Palm distribution associated to the ν–randomization of a stationary simple point process with finite intensity and obtain upper bounds on D(β) by choosing special test functions f ∈ L ∞ (P0 ) in the r.h.s. of (1.12). We suppose that ν is not concentrated in a unique value, i.e. ν is not of the form ν = δ E . This implies that for P0 –a.a. ξ , Sx ξ = S y ξ if x, y ∈ ξˆ and x = y. Note that the above assumption is satisfied whenever (1.15) is fulfilled, moreover in the case ν = δ E the β–dependence of the diffusion matrix is trivial. Given ξ ∈ N0 , let E β (ξ ) be a family of non-oriented links in ξˆ , i.e.   (3.1) E β (ξ ) ⊂ {x, y} : x, y ∈ ξˆ and x = y , covariant w.r.t. space translations, i.e. E β (Sx ξ ) = E β (ξ ) − x,

∀ξ ∈ N0 , x ∈ ξˆ .

Consider the graph G β (ξ ) with vertices set V β (ξ ) and edges set E β (ξ ), where   V β (ξ ) = x ∈ ξˆ : ∃y ∈ ξˆ with {x, y} ∈ E β (ξ ) .

(3.2)

(3.3)

Given x, y in V β (ξ ) we say that they are connected if there exists a path in G β (ξ ) going β from x to y. Moreover, given x ∈ ξˆ , we denote by C x (ξ ) the connected component in β β β G (ξ ) containing x if x ∈ V (ξ ), while we set C x (ξ ) = ∅ if x ∈ ξˆ \V β (ξ ). It is clear β that C x (ξ ) depends on the edge set E β (ξ ). Proposition 1. Let P0 be the Palm distribution associated to the ν–randomization of a stationary simple point process with ρ2 < ∞ and ν = δ E for any  E ∈ [−1, 1]. Suppose that for each β > 0 a random graph G β (ξ ) = V β (ξ ), E β (ξ ) , satisfying (3.1), (3.2) β and (3.3), is assigned such that the function N0  ξ → C0 (ξ ) ∈ N is measurable. In addition, suppose that the following assumption (A1) is fulfilled: • (A1) There exists a positive function (β) such that |x − y| > (β) ⇒ {x, y} ∈ E β (ξ ),

∀x, y ∈ ξˆ , ∀ξ ∈ N0 .

(3.4)

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Then, for all i = 1, . . . , d and for all β,   2 β Di,i (β) ≤ 6 P0 (ξ ) ξˆ (d x)c0,x (ξ ) x (i) + (β)2 |C0 (ξ )|2 Ix∈C β (ξ ) , (3.5) 0

where x (i) denotes the i th coordinate of x and Ix∈C β (ξ ) is the characteristic function of 0

β

the event {x ∈ C0 (ξ )}. Moreover, suppose that the following additional assumptions (A2) and (A3) are fulfilled: • (A2) For some ε > 0,

  β lim sup EP0 |C0 (ξ )|2+ε < ∞,

(3.6)

ρ2(1+ε)/ε+1 < ∞

(3.7)

β↑∞

holds where a denotes the smallest integer larger than a. • (A3) There exists a positive function C(β) such that {x, y} ∈ E β (ξ ) ⇒ cx,y (ξ ) ≤ C(β),

∀x, y ∈ ξˆ , ∀ξ ∈ N0 .

(3.8)

Then, for all i = 1, . . . , d, κ ∈ (0, 1) and β large enough, Di,i (β) ≤ c(κ) C(β)κ (1 + (β)2 ),

(3.9)

for a suitable positive constant c(κ) depending on κ, but not on β. In particular, D(β) ≤ d c(κ) C(β)κ (1 + (β)2 ) I.

(3.10)

The proof of the above proposition is obtained by plugging suitable test functions f in the variational formula (1.12). Our test functions are similar to the ones used in [16, Proof of Theorem 3.12]. Remark 1. As one can easily deduce from the proof of the above proposition, the results (3.9) and (3.10) hold for all β > 0 if condition (3.6) is satisfied with lim supβ↑∞ replaced by supβ>0 . Proof. Assume (A1) to be satisfied and, given a positive integer N , consider the test β function f N : N0 → R≥0 defined as follows:    β β − min z (i) : z ∈ C0 (ξ ) if 1 ≤ |C0 (ξ )| ≤ N , β f N (ξ ) = (3.11) 0 otherwise, β

where z (i) denotes the i th coordinate of z. Due to (A2), f N is measurable, while due to (A1), β

β

0 ≤ f N (ξ ) ≤ |C0 (ξ )|(β),

∀ξ ∈ N0 .

(3.12)

In particular,    2 2 β β β x (i) − ∇x f N (ξ ) ≤ 3 x (i) + (β)2 |C0 (ξ )|2 + (β)2 |C0 (Sx ξ )|2 , ∀ξ ∈ N0 . (3.13)

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Due to (1.12),  2 β (1) (2) Di,i (β) ≤ P0 (ξ ) ξˆ (d x)c0,x (ξ ) x (i) − ∇x f N (ξ ) = I N (β) + I N (β), (3.14) where



(1)

I N (β) = (2) I N (β)

=



P0 (ξ ) P0 (ξ )



 2 β ξˆ (d x)c0,x (ξ ) x (i) − ∇x f N (ξ ) Ix∈C β (ξ ) , ξˆ (d x)c0,x (ξ )



0

2 β x (i) − ∇x f N (ξ ) Ix∈C β (ξ ) , 0

and I A denotes that characteristic function of the event A. (1)

β

β

β

• Estimate of I N (β). If x ∈ C0 (ξ ), then C0 (Sx ξ ) = C0 (ξ ) − x and in particular β β β β β |C0 (Sx ξ )| = |C0 (ξ )|. Hence, f N (Sx ξ ) = x (i) + f N (ξ ), i.e. x (i) − ∇x f N (ξ ) = 0, thus implying that (1)

I N (β) = 0.

(3.15)

• Estimate of I N(2) (β). Due to (3.13) we can bound   2 β β (2) I N (β) ≤ 3 P0 (ξ ) ξˆ (d x)c0,x (ξ ) x (i) + (β)2 |C0 (ξ )|2 + (β)2 |C0 (Sx ξ )|2 Ix∈C β (ξ ) .

(3.16)

0

We claim that, due to Lemma 1 (i), β P0 (ξ ) ξˆ (d x)c0,x (ξ )|C0 (ξ )|2 Ix∈C β (ξ ) 0 β = P0 (ξ ) ξˆ (d x)c0,x (ξ )|C0 (Sx ξ )|2 Ix∈C β (ξ ) holds.

(3.17)

0

In order to prove the above claim we define the function f on N0 × N0 as  β c0,x (ξ )|C0 (ξ )|2 Ix∈C β (ξ ) if ζ = Sx ξ, x ∈ ξˆ , 0 f (ξ, ζ ) = 0 otherwise. Note that the above definition is well posed P0 –a.s. since due to the choice ν = δ E we have that Sx ξ = S y ξ if x, y ∈ ξˆ , x = y, for P0 –a.a. ξ . Moreover, given x ∈ ξˆ , we can write β

f (ξ, Sx ξ ) = c0,x (ξ )|C0 (ξ )|2 Ix∈C β (ξ ) , 0

while, using the translation invariance, see (3.2), we get that β

f (Sx ξ, ξ ) = f (Sx ξ, S−x (Sx ξ )) = c0,−x (Sx ξ )|C0 (Sx ξ )|2 I−x∈C β (S 0

=

xξ)

β c0,x (ξ )|C0 (Sx ξ )|2 Ix∈C β (ξ ) . 0

At this point, (3.17) follows by applying Lemma 1 (i) to the above function f . Due to (3.17) we can conclude that   2 (2) (i) 2 β 2 ˆ I N (β) ≤ 6 P0 (ξ ) ξ (d x)c0,x (ξ ) x + (β) |C0 (ξ )| Ix∈C β (ξ ) . (3.18) 0

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• Conclusions. The bound (3.5) follows from (3.14), (3.15) and (3.18). Suppose now β that also assumptions (A2) and (A3) are valid. In particular, if x ∈ C0 (ξ ) then {0, x} ∈ E β (ξ ) and therefore c0,x (ξ ) ≤ C(β). In particular c0,x (ξ ) ≤ C(β)κ e−(1−κ)|x| , for all κ ∈ (0, 1). Due to (3.5) we get   β Di,i (β) ≤ 6 C(β)κ P0 (ξ ) ξˆ (d x)e−(1−κ)|x| |x|2 + (β)2 |C0 (ξ )|2 . (3.19)



Due to Lemma 1 (ii), the integral P0 (ξ ) ξˆ (d x)e−(1−κ)|x| |x|2 is finite. In order to bound the remaining contribution to the integral in the r.h.s. of (3.19), we apply the Hölder inequality: given p, q > 1 with 1/ p + 1/q = 1, β P0 (ξ ) ξˆ (d x)e−(1−κ)|x| (β)2 |C0 (ξ )|2   p 1/ p  1/q β ≤ (β)2 EP0 |C0 (ξ )|2q EP0 holds. ξˆ (d x)e−(1−κ)|x| Choosing 2q = 2+ε (hence p = (2+ε)/ε) we have that, due to (3.6), (3.7) and Lemma 1 (ii), both the expectations in the r.h.s. are uniformly bounded in β, for large enough β. The above observations, together with (3.19), imply (3.9). We claim that this last bound implies (3.10). Indeed, since D(β) is symmetric, there exists an orthonormal basis v1 , v2 , . . . , vd of Rd such that vi is an eigenvector of D(β), i.e. D(β)vi = λi vi . Since D(β) is a nonnegative matrix due to (1.12), all the eigenvalues λi must be nonnegative. Hence, given a ∈ Rd , we can estimate (a, D(β)a) =

d  i=1

 λi (a, vi ) ≤ 2

max λi

1≤i≤d

 d

(a, vi ) = (a, a) max λi ≤ (a, a) 2

1≤i≤d

i=1

d 

λi .

i=1

(3.20) The last term is the trace of the matrix U −1 D(β)U , where U is the d × d orthogonal matrix whose i th column coincides with vi . Note that the matrix U −1 D(β)U is a diagonal matrix with diagonal entries λ1 , λ2 , . . . , λd . Since T r (A B) = T r (B A) for generic matrixes A, B, d 

λi = T r (U −1 D(β)U ) = T r (D(β)) =

i=1

Finally, (3.10) follows from (3.9), (3.20) and (3.21).

d 

D(β)i,i holds.

(3.21)

i=1

 

4. Proof of Theorem 1 in the Poissonian Case In this section we prove Theorem 1 in the special case that P0 is the Palm distribution associated to the ν–randomization of the Poisson point process with intensity ρ > 0. In the next section, we extend the proof to the general case. We write Pˆ ρ for the Poisson point process with density ρ, Pρ for its ν–randomization, Pˆ 0,ρ for the Palm distribution associated to Pˆ ρ , and P0,ρ for the Palm distribution associated to Pρ . Equivalently, P0,ρ is the ν–randomization of Pˆ 0,ρ . In what follows, we write Q L for the cube [−L/2, L/2]d .

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The proof is based on Proposition 1, scaling arguments and continuum percolation. We recall some results of continuum percolation referring to [11] for a more detailed discussion. Given r > 0 we write Br (x) for the closed ball centered at x ∈ Rd with radius r . If x = 0 we simply write Br . We define the occupied region of the Boolean model with radius r driven by ξˆ ∈ Nˆ as Xr (ξˆ ) := ∪x∈ξˆ Br (x) = {y ∈ Rd : d(y, ξˆ ) ≤ r }, where d(·, ·) denotes the Euclidean distance. The connected components in the occupied region will be called occupied components. For A ⊂ Rd , we denote by Wr (A) = Wr (A)[ξˆ ] the union of all occupied components having non–empty intersection with r A. Given A ⊂ Rd and B ⊂ Rd we write A ←→ B if there exists a path inside Xr (ξˆ ) connecting A and B. The following results hold for stationary Poisson point processes [11]: there exists a positive density ρc such that for all bounded subsets A ⊂ Rd , A = ∅,   ρc = inf ρ > 0 : Pˆ ρ [ diam W1 (A) = ∞] > 0 ,   1 Pˆ ρ A ←→ ∂ Q L ≤ e−c(ρ,A) L , ∀L > 0, ∀ρ < ρc

(4.1) (4.2)

for a suitable positive constant c(ρ, A) depending on ρ, A. Above ∂ Q L denotes the border of the cube Q L . Moreover, one can prove that all occupied components in X1 are bounded Pˆ ρ –a.s. for ρ < ρc , while there exists a unique unbounded occupied component in X1 Pˆ ρ –a.s. for ρ > ρc . Note that the function Rd  x → x/r ∈ Rd maps Pˆ ρ in Pˆ ρ r d , namely if ξˆ has law Pˆ ρ then {x/r : x ∈ ξˆ } has law Pˆ ρ r d . This scaling property allows to restate the above results for a fixed density ρ and varying radius r : the positive constant rc (ρ) := (ρc /ρ)1/d = ρ −1/d rc (1)

(4.3)

satisfies   rc (ρ) = inf r > 0 : Pˆ ρ [ diam Wr (A) = ∞] > 0 ,   r Pˆ ρ A ←→ ∂ Q L ≤ e−c(r,ρ,A) L , ∀L > 0, ∀r < rc (ρ),

(4.4) (4.5)

for all bounded subsets A ⊂ Rd , A = ∅, and for a suitable positive constant c(r, ρ, A) depending on r, ρ, A. Moreover, all occupied components in Xr are bounded Pˆ ρ –a.s. for r < rc (ρ), while there exists a unique unbounded occupied component in Xr Pˆ ρ –a.s. for r > rc (ρ). We point out a simple consequence of (4.5): Lemma 2. If r < rc (ρ) then for all bounded sets A ⊂ Rd with A = ∅ and for all s > 0 EPˆ

ρ

  s  < ∞. ξˆ Wr (A)[ξˆ ]

(4.6)

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Proof. Due to the stationarity of Pˆ ρ , we can assume that 0 ∈ A without loss of generality. Fixed p, q > 1 with 1/ p + 1/q = 1, by the Hölder inequality and (4.5) we get ∞     s   ˆ ˆ r EPˆ ξ Wr (A)[ξ ] ≤ EPˆ ξˆ (Q L )s I{A←→∂ Q L−1 } ρ ρ L=1

≤

∞  L=1

 EPˆ

ρ

ξˆ (Q L )s p

1/ p

∞  1/q   1/ p r Pˆ ρ A ←→ ∂ Q L−1 ≤ EPˆ ξˆ (Q L )s p e−cL . L=1

ρ

(4.7) The thesis then follows by observing that, since ξˆ (Q L ) is a Poisson random variable with expectation ρ L d ,   EPˆ ξˆ (Q L )n ≤ c(ρ)L d n , ∀L > 0, n ∈ N, ρ

thus implying that the last member in (4.7) is summable.

 

We can now give the proof of Theorem 1 for Poisson point processes: Proof of Theorem 1. for P0 = P0,ρ . Given ξ ∈ N0 we set d

E(β) := β − α+1+d , ρ(β) := ρ ν ([−E(β), E(β)]) , α+1

(β) := λ β α+1+d ,   E β (ξ ) := {x, y} : x, y ∈ ξˆ , x = y, |E x | ≤ E(β), |E y | ≤ E(β) and |x − y| ≤ (β) , where λ is a β–independent positive constant that will be chosen at the end. Note that, due to assumption (1.15), d(α+1)

0 < ρ(β) ≤ c0 ρ β − α+1+d holds,

(4.8)

while, due to (1.8), {x, y} ∈ E β (ξ )

  α+1 ⇒ cx,y (ξ ) ≤ exp {−(β) ∧ [κ1 β E(β)]} = exp −(λ ∧ κ1 )β α+1+d =: C(β) holds. (4.9)

Then, due to (3.10), in order to conclude the proof of Theorem 1 it is enough

to check that the assumptions of Proposition 1 are fulfilled when the graph G β (ξ ) = V β (ξ ), E β (ξ ) is defined via (3.3). Conditions (3.1) and (3.2) are trivially satisfied. Assumption (A1) is obvious, (A3) has already been checked: it therefore only remains to consider (A2). Since Poisson point processes have finite moments, the only non-trivial condition to be checked is given by (3.6), which can be justified by means of scaling arguments and Lemma 2 as follows.

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As discussed in [6], the process ξˆ with law Pˆ 0,ρ can be constructed by setting ˆξ := ωˆ ∪ {0}, where ωˆ is a Poisson point process with law Pˆ ρ . Let ω be the ν–randomization of the process ωˆ and let E 0 be a random variable with law ν, independent from ω. Then ξ := ω ∪ {(0, E 0 )} has law P0,ρ . Setting
 ωˆ β = x ∈ ωˆ : |E x | ≤ E(β) , r (β) = (β)/2, we get

  β |C0 (ξ )| ≤ 1 + ωˆ β Wr (β) Br (β) [ωˆ β ] .

(4.10)

Above we have used the standard the set ωˆ β with the counting

convention

 to identify   measure x∈ωˆ β δx . Hence, ωˆ β Wr (β) Br (β) [ωˆ β ] is the cardinality of the set obtained

 as intersection of Wr (β) Br (β) [ωˆ β ] with ωˆ β . Moreover, we note that the process ωˆ β , obtained by thinning the Poisson process ωˆ with density ρ, has law Pˆ ρ(β) . We now consider the space rescaling Rd  x → x ρ(β)1/d ∈ Rd . Since the above function maps Pˆ ρ(β) onto Pˆ 1 and points at distance r (β) into points at distance r (β)ρ(β)1/d , the random variable

  (4.11) ωˆ β Wr (β) Br (β) [ωˆ β ] has the same law as

  ωˆ ∗ Wr∗ (β) Br∗ (β) [ωˆ ∗ ] ,

(4.12)

where ωˆ ∗ is a Poisson point process on Rd with intensity 1, i.e. ωˆ ∗ has law Pˆ 1 , and r∗ (β) := r (β)ρ(β)1/d . Since due to (4.8), r∗ (β) ≤ λ(c0 ρ)1/d /2, taking λ such that λ(c0 ρ)1/d /2 ≤ rc (1)/2, the random variable in (4.12) is bounded from above by

  ωˆ ∗ Wrc (1)/2 Brc (1)/2 [ωˆ ∗ ] . (4.13) This last random variable is β–independent and has finite moments due to Lemma 2. Hence all moments of (4.11) are β–independent and finite. Due to (4.10), (3.6) is satisfied and we can apply Proposition 1.   5. Proof of Theorem 1 in the General Case We now explain how one can derive Theorem 1 in the general (non Poissonian) case, under the domination assumption (ii). First we observe that due to assumption (ii) Pˆ has finite moments ρκ for all κ ≥ 0. In fact, assumption (ii) trivially implies that Pˆ ( p) has finite moments. Hence, denoting by X n a generic binomial variable with parameters n, p, we have  ∞ n      n j p (1 − p)n− j Pˆ ξˆ ([0, 1]d ) = n ∞ > EPˆ ( p) ξˆ ([0, 1]d )κ = jκ j n=1

=

∞  n=1

  Pˆ ξˆ ([0, 1]d ) = n E(X nκ ).

j=0

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Since for a suitable positive constant C it holds that C −1 n κ ≤ E(X nκ ) ≤ Cn κ for any n, κ ∈ N+ , this implies that ρκ =

∞ 

  Pˆ ξˆ ([0, 1]d ) = n n κ < ∞,

∀κ ∈ N.

n=1

Define d

E(β) := β − α+1+d .

(5.1)

Since E(β) ↓ 0 as β ↑ ∞, due to (1.15) we can find β∗ such that γ := ν ([−E(β∗ ), E(β∗ )]) ≤ p. Since the γ –thinning Pˆ (γ ) is stochastically dominated by the p–thinning Pˆ ( p) , assumption (ii) in Theorem 1 remains valid with p replaced by γ . Hence, without loss of generality, we can assume that γ = p in assumption (ii), i.e. p = ν ([−E(β∗ ), E(β∗ )])

(5.2)

for some β∗ . In what follows we take β ≥ β∗ and define α+1

(β) := λβ α+1+d , (5.3)   E β (ξ ) := {x, y} : x, y ∈ ξˆ , x = y, |E x | ≤ E(β), |E y | ≤ E(β) and |x − y| ≤ (β) , (5.4) where the positive β–independent constant λ will be fixed at the end.  We want to apply Proposition 1 where, given E β (ξ ), the graph G β (ξ ) = V β (ξ ), E β (ξ ) is defined by (3.3). Trivially, the set E β (ξ ) satisfies (3.1) and (3.2), and condition (A1) is fulfilled. Moreover, due to (1.8),   α+1 {x, y} ∈ E β (ξ ) ⇒ cx,y (ξ ) ≤ exp {−(β) ∧ [κ1 β E(β)]} ≤ exp −c β α+1+d =: C(β), for some β–independent positive constant c. Therefore, condition (A3) is satisfied. Moreover, as already observed, ρκ < ∞ for all κ > 0. Hence, in order to obtain the bound (3.10), which corresponds to (1.17), we only need to verify (3.6). We will prove that   β lim sup EP0 |C0 (ξ )|3 < ∞. (5.5) β↑∞

Due to the Campbell identity (2.3), which holds also with Q K replaced by K (0) [6], we can write    1 β β 3 ˆ EP0 |C0 (ξ )|3 = ξ (d x)|C , E (S ξ )| x P 0 ρKd K (0) ˆ where P denotes the ν–randomization of P. Let us define the conditioned measure ν∗ := ν (· | |E 0 | ≤ E(β∗ )) .

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Then, due to (5.2), we have the identity ν ([−E(β), E(β)]) = p ν∗ ([−E(β), E(β)]) .   Hence, for β ≥ β∗ , the random set x ∈ ξˆ : |E x | ≤ E(β) with ξ chosen with law P   and the random set x ∈ ξˆ : |E x | ≤ E(β) with ξ chosen with law P∗ , defined as the

β (ξ ) = V β (ξ ), ν∗ –randomization of P ( p) , have the same distribution. Since the graph G    E β (ξ ) is univocally determined by the set x ∈ ξˆ : |E x | ≤ E(β) , we conclude that   β EP0 |C0 (ξ )|3 =

1 EP ρKd ∗

 K (0)

β ξˆ (d x)|C0 (Sx ξ )|3 .

(5.6)

In order to bound the r.h.s. of (5.6) using the domination assumption (ii), we consider the partition of Rd in the cubes K (x), x ∈ K Zd , and to each A ⊂ Rd we associate the sets   VK (A) := x ∈ K Zd : K (x) ∩ A = ∅ ,  √  E K (A) := {x, y} : x, y ∈ VK (A), x = y, |x − y| ≤ (β) + d K . We define S K (A) as the connected cluster in the graph G K (A) = (VK (A), E K (A)) containing the origin if 0 ∈ VK (A), and as the empty set if 0 ∈ VK (A). Finally, we set C K (A) := ∪x∈SK (A) ( K (x) ∩ A) .   β If A = x ∈ ξˆ : |E x | ≤ E(β) we will simply write C K (ξ ) for C K (A).

(5.7)

Lemma 3. For all x ∈ K (0), β

β

C0 (Sx ξ ) + x ⊂ (ξ ∩ K (0)) ∪ C K (ξ ) holds. β

(5.8)

β

Proof. Due to the covariant property (3.2), C0 (Sx ξ ) + x = C x (ξ ). If this set is empty β the thesis is trivially true. Otherwise suppose that z ∈ C x (ξ ). If z ∈ K (0) then z belongs to the r.h.s. of (5.8). If z ∈ K (u) for some u ∈ K Zd \ {0}, by following the path connecting x to z in G β (ξ ) we can define a sequence of distinct points u 0 = 0, u 1 , u 2 , . . . , u n−1 , u n = u in K Zd such that for each i, 0 ≤ i ≤ n −1, there exist points ai ∈ K (u i ), bi ∈ K (u i+1 ) with {ai , bi } ∈ E β (ξ ). Hence {u i , u i+1 } ∈ E K (A) for all 0 ≤ i ≤ n − 1, where A = {x ∈ ξˆ : |E x | ≤ E(β)}. This proves that 0 and u are connected in the graph G K (A). In particular, u ∈ S K (A) and therefore β  z ∈ K (u) ∩ A ⊂ C K (ξ ).  Due to the above lemma we get the bound    c β 3 3 ˆ ˆ ξ (d x) ξ ( r.h.s. of (5.6) ≤ E (0)) + |C (ξ )| K P K ρ K d ∗ K (0)     β = c p EP∗ ξˆ ( K (0))3 + c p EP∗ |C K (ξ )|3

(5.9)

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(recall that P∗ is the ν∗ –randomization of P ( p) ). Since     EP∗ ξˆ ( K (0))3 = EP ( p) ξˆ ( K (0))3 < ∞, in order to conclude the proof we only need to prove that   β lim sup EP∗ |C K (ξ )|3 < ∞. β↑∞

Let us derive from the domination assumption (ii) that     β β EP∗ |C K (ξ )|3 ≤ EP∗,ρ  |C K (ξ )|3 ,

(5.10)

(5.11)

where P∗,ρ  is the ν∗ –randomization of the Poisson point process Pˆ ρ  . To this aim, we define   ξˆ ∈ N .

K (ξˆ ) = ξˆ ( K (x)) : x ∈ K Zd , We claim that, given a marked point process Q obtained as ν∗ –randomization of a stationary simple point process, the conditional expectation    β (5.12) EQ |C K (ξ )|3  K is an increasing function in K that does not depend on Q. In order to prove this statement, we write Bin(N , p) for a generic binomial variable with parameters N , p and recall that Bin(N , p) is stochastically dominated by Bin(N  , p) if N ≤ N  . Given 

d

K (ξˆ ), the random variables ax (ξ ), x ∈ K Z defined as   ax (ξ ) = {z ∈ ξˆ ∩ K (x) : |E z | ≤ E(β)} are independent binomial r.v.’s with parameters ξˆ( K (x)), ν∗ [−E(β), E(β)]. In par ticular, the conditional law of ax (ξ ), x ∈ K Zd given K does not depend on Q  β  and it increases with K . Besides, the cardinality C K (ξ ) is an increasing function

 of ax (ξ ), x ∈ K Zd . Since the composition of increasing functions is increasing, we conclude that the conditional expectation (5.12) is an increasing function of K , independent of Q. This proves our claim. Since by assumption (ii) the random field K (ξˆ ) with ξˆ chosen with law Pˆ ( p) is stochastically dominated by the random field K (ξˆ ) with ξˆ chosen with law Pˆ ρ  , we obtain that      β β EP∗ |C K (ξ )|3 = EP∗ EP∗ |C K (ξ )|3 | K (ξˆ )       β β = EPˆ ( p) EP∗ |C K (ξ )|3 | K (ξˆ ) ≤ EPˆ  EP∗ |C K (ξ )|3 | K (ξˆ ) ρ      β β 3 ˆ = EPˆ  EP∗,ρ  |C K (ξ )| | K (ξ ) = EP∗,ρ  |C K (ξ )|3 , ρ

thus concluding the proof of (5.11).

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Due to (5.11), in order to derive (5.10) and complete the proof of Theorem 1 we only need to show that   β lim sup EP∗,ρ  |C K (ξ )|3 < ∞. (5.13) β↑∞

To this aim we will use scaling and percolation arguments as in the previous section.   ˆ The random set A = x ∈ ξ : |E x | ≤ E(β) , where ξ is chosen with law P∗,ρ  , is a Poisson point process with intensity µ(β) = ρ  ν∗ ([−E(β), E(β)]) . β

Hence, by definition of C K (ξ ) (see (5.7)), we have     β EP∗,ρ  |C K (ξ )|3 = EPˆ |C K (ξˆ )|3 . µ(β)

Let

(5.14)

√ r (β) := 2 d(β)/3.

Recall that Bs denotes the closed ball of radius s centered at the origin. Using the same notation as in the previous section, for β large enough we can bound     3  3 ˆ ˆ ˆ . (5.15) |C K (ξ )| ≤ EPˆ EPˆ ξ Wr (β) (Br (β) )[ξ ] µ(β)

µ(β)

By the scaling invariance of the Poisson point process, for each γ > 0,     3  3  ˆ ˆ ˆ ˆ EPˆ = EPˆ γ d ξ Wr (β)/γ (Br (β)/γ )[ξ ] . ξ Wr (β) (Br (β) )[ξ ] µ(β)

µ(β)

Taking

−1/d

γ := µ(β)−1/d = ρ  ν∗ [−E(β), E(β)] , we get EPˆ

µ(β)

    3  3  ξˆ Wr (β) (Br (β) )[ξˆ ] = EPˆ ξˆ Wr (β)/γ (Br (β)/γ )[ξˆ ] . 1

(5.16)

Due to the definition of ν∗ , E(β) and assumption (1.15) we get that α+1

γ ≥ c β α+1+d for some positive constant c depending only on p, ρ  , d and on the constant c0 appearing α+1 in (1.15). Since (β) = λβ α+1+d , then √ √ √ α+1 r (β)/γ = 2 d(β)/(3γ ) ≤ 2 d(β)β − α+1+d /(3c) = 2 dλ/(3c). (5.17) It is enough to choose λ such that √ 2 dλ/(3c) ≤ rc (1)/2.

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With this choice (5.16) and (5.17) imply that     3  3  ξˆ Wr (β) (Br (β) )[ξˆ ] ≤ EPˆ ξˆ Wrc (1)/2 (Brc (1)/2 )[ξˆ ] . (5.18) sup EPˆ β>0

µ(β)

1

The r.h.s. in the above inequality is finite due to Lemma 2. This observation together with (5.14), (5.15) and (5.18) implies (5.13). This concludes the proof of Theorem 1. Remark 2. In the case that Pˆ is stochastically dominated by the Poisson point process Pˆ ρ  one can give a much simpler proof of Theorem 1, that we describe in what follows. Denote by P and Pρ  the ν–randomization of Pˆ and Pˆ ρ  , respectively. Due to the definition of stochastic domination given in Sect. 2, one can exhibit a coupling between P and Pρ  such that ξ ⊂ ξ  almost surely, with (ξ, ξ  ) denoting the random sets with marginal distributions given by Pˆ and Pˆ ρ  , respectively. Consider the graph G β (ξ ) introduced in Sect. 4. Due to the Campbell identity (2.3), we can write    

 1 β β ξˆ (d x) |C0 (Sx ξ )|3 . Fβ (ξ ) := EP0 |C0 (ξ )|3 = EP Fβ (ξ ) , ρ Q1 Since Fβ (ξ ) ≤ Fβ (ξ  ) if ξ ⊂ ξ  and due to the above coupling between P and Pρ  , we can conclude that

  EP Fβ (ξ ) ≤ EPρ  Fβ (ξ ) . Due to the Campbell identity (2.3), the r.h.s. in the above expression equals ρ  EP0,ρ    β |C0 (ξ )|3 which, as proven in Sect. 4, is bounded from above uniformly in β. Hence we have that   ρ   β β sup EP0 |C0 (ξ )|3 ≤ sup EP0,ρ  |C0 (ξ )|3 < ∞. (5.19) ρ β>0 β>0 At this point it is enough to apply Proposition 1: condition (3.6) is fulfilled due to (5.19), while all other conditions can be easily checked. 6. Point Processes with Uniform Bounds on the Local Density Let us prove that the conditions of Theorem 1 are fulfilled by stationary point processes with uniform bounds: Proposition 2. Let Pˆ be a stationary simple point process such that, for suitable positive constants K and N , ξˆ ( K (x)) ≤ N ,

∀x ∈ K Zd ,

Pˆ a.s.,

(6.1)

where K (x) = x + [−K /2, K /2)d . Then, for each q ∈ (0, 1], the q–thinning Pˆ (q) of Pˆ satisfies condition (ii) in Theorem 1. In order to prove the above statement, we use a standard result of stochastic domination (cf. Lemma 1.1 in [10]):

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Lemma 4. Let p ∈ [0, 1] and let σ = σx : x ∈ Zd be a random field s.t. σx ∈ {0, 1} for all x ∈ Zd . Suppose that   P σx = 1 | σ = ζ on Zd \{x} ≤ p, (6.2) for almost all ζ ∈ {0, 1}Z and for all x ∈ Zd . Then σ is stochastically dominated by the Bernoulli site percolation with parameter p, i.e. one can define random fields (σ  , ω) with d

σx ≤ ωx

∀x ∈ Zd ,

a.s., 

σ  having the same law of σ and ω being given by ω = ωx : x ∈ Zd , where ωx are i.i.d. random variables taking value 1 with probability p and value 0 with probability 1 − p. Proof of Proposition 2.

random fields σ = σx as  1 σx (ξˆ ) = 0

If q < 1 set p = q, otherwise fix some p ∈ (0, 1). Consider the 

 d : x ∈ Zd , τ = τx : x ∈ Zd having value in {0, 1}Z defined  if ξˆ ( K (x)) ≥ 1, 1 if ξˆ ( K (x)) ≥ N , τx (ξˆ ) = otherwise ; 0 otherwise.

Then, given ζ ∈ {0, 1}Z , we have   Pˆ ( p) σx = 1 | σ = ζ on Zd \{x} ≤ 1 − (1 − p) N =: p  . d

On the other hand, the random field τ with ξˆ chosen with Poisson law Pˆ ρ  is a Bernoulli site percolation with parameter p˜ := P(Z ≥ N ), where Z is a Poisson variable with mean ρ  K d . Since p  < 1 we can choose ρ  large enough so that p  ≤ p. ˜ Due to the ˆ previous lemma, we can conclude that the random field σ with ξ chosen with law Pˆ ( p) is stochastically dominated by the random field τ with ξˆ chosen with Poisson law Pˆ ρ  . Recall the definition of the random field Y from Theorem 1. Due to the transitivity of stochastic domination and since Y (x)[ξˆ ] ≤ N σx (ξˆ ), N τx (ξˆ ) ≤ Y (x)[ξˆ ],

∀x ∈ Zd , for Pˆ ( p) –a.a. ξˆ , ∀x ∈ Zd , ∀ξˆ ,

(the first inequality follows from (6.1)), we can conclude that the random field Y with ξˆ chosen with law Pˆ ( p) is stochastically dominated by the random field Y with ξˆ chosen with law Pˆ ρ  .   As a corollary of Proposition 2, we obtain that Theorem 1 can be applied to crystals or diluted crystals. In order to be more precise, let us start with a crystal, i.e. (cf. [2]) a locally finite set  ⊂ Rd such that for a suitable basis v1 , v2 , . . . , vd of Rd , −x =

∀x ∈ G := {z 1 v1 + z 2 v2 + · · · + z d vd : z i ∈ Z ∀i} holds.

Let  be the elementary cell  := {t1 v1 + t2 v2 + · · · + td vd : 0 ≤ ti < 1 ∀i} . Note that both the group G and the cell  depend on the basis v1 , v2 , . . . , vd .

(6.3)

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Let ω = (ωx : x ∈ ) be a site Bernoulli percolation on  with parameter p ∈ (0, 1] and let V be a random vector independent of ω, chosen in the elementary cell  with uniform distribution. Then consider the simple point process  ζˆ := ωx δV +x , x∈

obtained from the set  by a spatial randomization and a p–thinning, and call Pˆ its law. The following holds: Proposition 3. The simple point process ζˆ with law Pˆ is stationary and does not depend on the specific basis v1 , v2 , . . . , vd satisfying (6.3). Moreover, its Palm distribution Pˆ 0 is given by Pˆ 0 =

 1 Pu , | ∩ |

(6.4)

u∈∩

where Pu is the law of the simple point process  ζˆu := δ0 + ωx δx−u . x∈\{u}

ˆ the above result implies that Pˆ 0 Since the Palm distribution Pˆ 0 depends only on P, does not depend on the specific basis v1 , v2 , . . . , vd . If p = 1, the Palm distribution corresponds to choosing a point u of the crystal inside the elementary cell  with uniform probability and translating the crystal of −u. If p < 1, in addition to the previous step one erases points different from the origin independently with probability 1 − p. The resulting simple point process is what we called p–diluted crystal obtained from . Trivially, it has uniform bounds in the local density. Hence, due to Proposition 2, it fulfills the assumptions of Theorem 1. Proof. Due to the translation invariance (6.3) one easily proves that Pˆ is stationary. Let us prove that it does not depend on the specific basis v1 , v2 , . . . , vd . Given a Borel subset A ⊂ Rd with  finite positive Lebesgue measure, we define Pˆ A as the law of the simple point process x∈ ωx δW +x , where ω is a site Bernoulli percolation on  with parameter p and W is a vector independent from ω and chosen in A with uniform probability. Note that Pˆ = Pˆ  . By means of (6.3) one can easily check that Pˆ  = Pˆ A if A is a union of sets of the form  + x, x ∈ G. Now, consider the elementary cell  associated to another basis v1 , v2 , . . . , vd satisfying (6.3). By the previous observation, Pˆ  = Pˆ N  for each positive integer N . Define A N as the union of all the sets of the form  + x, x ∈ G, included in N  . Then Pˆ  = Pˆ A N . Finally observe that, given a bounded measurable function f : Nˆ → R,     EPˆ  ( f ) − EPˆ ( f ) ≤ C(,  , f )/N holds. N

AN

Hence the same estimate holds with Pˆ N  and Pˆ A N replaced by Pˆ  and Pˆ  , respectively. Taking N ↑ ∞, we conclude that Pˆ  = Pˆ  . Hence the law Pˆ does not depend on the specific basis v1 , v2 , . . . , vd satisfying (6.3).

Mott Law as Upper Bound for a Random Walk in a Random Environment

285

We prove the characterization (6.4) of the Palm distribution Pˆ 0 by means of the Campbell identity (2.2). Take a bounded measurable function f : Nˆ 0 → R and denote ˆ It is simple to check that by ρ the intensity of P. ρ = p| ∩ |/||,

(6.5)

where | ∩ | denotes the cardinality of  ∩  and || denotes the Lebesgue volume of . Since (2.2) holds also with Q K replaced by , we get that EPˆ ( f ) = 0

1 Eˆ ρ|| P

 

ξˆ (x) f (Sx ξˆ ) .

(6.6)

 Let us define ω as the support of the measure x∈ ωx δx . Denoting by Eω the expectation w.r.t. the site Bernoulli percolation ω we can write ⎛ ⎞⎤  1 ξˆ (x) f (Sx ξˆ ) = Eω ⎣ dy ⎝ f (ω + y − z)⎠⎦ ||   z∈∩(ω +y) ⎛ ⎞⎤ ⎡  1 = Eω ⎣ dy ⎝ f (ω − u)⎠⎦ ||  u∈(−y)∩ω ⎛ ⎞⎤ ⎡  1 = Eω ⎣ dy ⎝ ωu f ((ω − u) ∪ {0})⎠⎦ ||  u∈(−y)∩ ⎡ ⎛ ⎞⎤  1 = pEω ⎣ dy ⎝ f ((ω − u) ∪ {0})⎠⎦ ||  u∈(−y)∩  p = dy Eω [ f ((ω − u) ∪ {0})] . || 

 EPˆ



⎡

u∈(−y)∩

(6.7) Due to (6.3), it is simple to check that the last summation does not depend on y. Hence from the above identities (6.5), (6.6) and (6.7) we get that   p p Eω [ f ((ω − u) ∪ {0})] = E Pu ( f ) ρ|| ρ|| u∈∩ u∈∩  1 E Pu ( f ), = | ∩ |

EPˆ ( f ) = 0

(6.8)

u∈∩

thus concluding the proof of (6.4).

 

Acknowledgement. One of the authors, A.F., thanks the Centre de Mathématiques et d’Informatique, Université de Provence, for the kind hospitality and acknowledges the financial support of GREFI–MEFI.

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References 1. Ambegoakar, V., Halperin, B.I., Langer, J.S.: Hopping conductivity in disordered systems. Phys. Rev. B 4(8), 2612–2620 (1971) 2. Ashcroft, N.W., Mermin, N.D.: Solid state physics. Philadelphia PA: Saunders College, Publishing, 1976 3. Breiman, L.: Probability, Reading, MA: Addison–Wesley, 1953 4. Caputo, P., Faggionato, A.: Isoperimetric inequalities and mixing time for a random walk on a random point process. Ann. Appl. Probab. nos. 5/6, 1707–1744 (2007) 5. Caputo, P., Faggionato, A.: Diffusivity in one-dimensional generalized Mott variable-range hopping models. http://arxiv.org/list/math.PR/07101253, 2007 6. Daley, D.J., Vere–Jones, D.: An Introduction to the theory of point processes. New York: Springer, 1988 7. De Masi, A., Ferrari, P.A., Goldstein, S., Wick, W.D.: An invariance principle for reversible Markov processes. Applications to Random Motions in Random Environments. J. Stat. Phys. 55, 787–855 (1989) 8. Faggionato, A., Schulz–Baldes, H., Spehner, D.: Mott law as lower bound for a random walk in a random environment. Commun. Math. Phys. 263, 21–64 (2006) 9. Georgii, H.-O., Küneth, T.: Stochastic comparison of point random fields. J. Appl. Probab. 34, 868–881 (1997) 10. Liggett, T.M., Schonmann, R.H., Stacey, A.M.: Domination by product measures. The Annals of Probability 25, 71–95 (1997) 11. Meester, R., Roy, R.: Continuum Percolation. Cambridge: Cambridge University Press, 1996 12. Miller, A., Abrahams, E.: Impurity Conduction at Low Concentrations. Phys. Rev. 120, 745–755 (1960) 13. Minami, N.: Local fluctuation of the spectrum of a multidimensional Anderson tight binding model. Commun. Math. Phys. 177, 709–725 (1996) 14. Mott, N.F.: Conduction in non-crystalline materials. III. Localized states in a pseudogap and near extremities of conduction and valence bands. Phil. Mag. 19, 835–852 (1969) 15. Mott, N.F.: Charge transport in non-crystalline semiconductors. Festkörperprobleme 9, 22–45 (1969) 16. Piatnitski, A., Remy, E.: Homogenization of elliptic difference operators. SIAM J. Math. Anal. 33, 53–83 (2001) 17. Shklovskii, B.I., Efros, A.L.: Electronic Properties of Doped Semiconductors. Berlin: Springer, 1984 18. Spohn, H.: Large Scale Dynamics of Interacting Particles. Berlin: Springer, 1991 Communicated by M. Aizenman

Commun. Math. Phys. 281, 287–347 (2008) Digital Object Identifier (DOI) 10.1007/s00220-008-0496-3

Communications in

Mathematical Physics

Pointwise Convergence to Knudsen Layers of the Boltzmann Equation Shijin Deng1 , Weike Wang1 , Shih-Hsien Yu2 1 Department of Mathematics, Shanghai Jiaotong University, Shanghai, PRC.

E-mail: [email protected]; [email protected]

2 Department of Mathematics, National University of Singapore, 2 Science Drive 2, Singapore,

Singapore. E-mail: [email protected] Received: 4 April 2007 / Accepted: 16 December 2007 Published online: 6 May 2008 – © Springer-Verlag 2008

Abstract: In this paper Green’s functions for the Boltzmann equation around a global Maxwellian are used to construct the non-characteristic nonlinear Knudsen layers as well as their time-asymptotic stability. Furthermore, the detailed pointwise structures, nonlinear wave couplings, and wave interactions with boundary are studied.

Contents 1. 2.

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . Tools: Green’s Functions for Initial Value Problem and Initial-Boundary Value Problem . . . . . . . . . . . . . . 2.1 Basic notions and the macro-micro decomposition . . . . 2.2 Upwind-Downwind decomposition and energy estimates 2.3 Grad’s lemma on collision operator . . . . . . . . . . . . 2.4 Green’s functions . . . . . . . . . . . . . . . . . . . . . 3. Existence of Knudsen’s Layers (Time-asymptotic Approach) 3.1 Green’s functions for the damped equation . . . . . . . . 3.2 Energy estimates on the boundary data . . . . . . . . . . 3.3 Nonlinear term . . . . . . . . . . . . . . . . . . . . . . 4. Coupling Waves . . . . . . . . . . . . . . . . . . . . . . . . 5. Stability of Knudsen’s Layers for M > 1 . . . . . . . . . . . 5.1 Energy estimates on the boundary data . . . . . . . . . . 5.2 Nonlinear term . . . . . . . . . . . . . . . . . . . . . . 6. Stability of Knudsen’s Layers for −1 < M < 1(M
= 0) . . . Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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297 297 299 300 302 306 308 310 315 318 323 324 325 335 345 345

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1. Introduction There are two purposes of the present paper. The first one is to conclude the nonlinear time-asymptotic stability of the Knudsen layers constructed in [33] after the serial preparation works, the Green’s functions for the Boltzmann equation in [24–26]. The second one is to initiate the methodology: to use the Green’s function of a linearized equation around a constant state to solve the nonlinear problem around other non-uniform nonlinear wave patterns such as boundary layers, shock layers, etc. The Knudsen layers are in an important class of wave pattens for the Boltzmann equation:
∂t F(x, t, ξ ) + ξ 1 ∂x F(x, t, ξ ) = K1n Q(F), (x, t, ξ ) ∈ R+ × R+ × R3 , F(0, t, ξ )|ξ 1 >0 , F(x, 0, ξ ) : imposed. Here for simplicity, the collision operator is assumed to be the model for hard sphere molecules for which the collision operator is ⎧ ⎪ Q(F) ≡ B(F, F), ⎪  ⎪ ⎪ ⎪ ⎨B(g, h)(ξ ) ≡ 1 2 R3 × S2 (ξ −ξ∗ )·≥0 ⎪ ⎪ ⎪ (−g(ξ )h(ξ∗ ) − h(ξ )g(ξ∗ ) + g(ξ  )h(ξ∗ ) + h(ξ  )g(ξ∗ )) ⎪ ⎪ ⎩ · |(ξ − ξ∗ ) · | dξ∗ d,  ξ  = ξ − [(ξ − ξ∗ ) · ] , ξ∗ = ξ∗ + [(ξ − ξ∗ ) · ] , and the parameter K n > 0 is the Knudsen number which is the ratio of the mean free path to the physical dimension. The precise definition of a Knudsen layer  is a boundary layer solution of the Boltzmann equation for a half-space problem with an incoming data b+ (ξ )|ξ 1 >0 at boundary x = 0 and with a boundary value as a global Maxwellian state M at x = ∞, i.e. ⎧ 1 3 ⎪ ⎨ξ ∂x  = Q() for x > 0, ξ ∈ R , (0, ξ )|ξ 1 >0 = Fb (ξ )|ξ 1 >0 , (1.1) ⎪ ⎩ lim (x, ξ ) = M(ξ ), x→∞

where the global Maxwellian states M = M[ρ,u,θ] , (ρ, u, θ ) constant, ⎧ −u|2 ρ − |ξ2Rθ ⎪ ⎪ M = e , [ρ,u,θ] ⎪ ⎪ (2π Rθ )3/2 ⎨ ρ>0: bulk density, ⎪ 1 ⎪ u = (u , 0, 0) : bulk fluid velocity, ⎪ ⎪ ⎩ θ >0: bulk temperature, satisfy Q(M) = 0 and are constant solutions of the Boltzmann equation. The Knudsen layer is also an important subject in the study of condensation-vaporization problem for rarefied gas. See Sone [29] for a complete reference on this issue. Knudsen layers are the central components in the generalized Hilbert-Grad expansion. A Knudsen-layer correction to the Grad-Hilbert expansion was introduced by Y. Sone to fit the kinetic boundary data after the results on the linear Knudsen layers by [1,7,5,10].

Pointwise Convergence to Knudsen Layers of the Boltzmann Equation

289

This Knudsen layer correction opened an new chapter on the asymptotic expansion theories on the field of rarefied gas. Since the generalized Grad-Hilbert expansion, theories on thermal transpiration flow, condensation-vaporization, ghost-effect, etc. had been developed by Y. Sone, K. Aoki, and their coworkers on the base of the generalized Grad-Hilbert expansion. See [29] for comprehensive and detailed description of the asymptotic theories. The works of Y. Sone initiated the mathematical interest on the Knudsen layers. Here, the Grad-Hilbert expansion was proposed by H. Grad to develop a general asymptotic theory for the Boltzmann equation with small Knudsen number K n in [12], but the Grad-Hilbert expansion did not have sufficient freedom to fit kinetic boundary conditions imposed. The existence of nonlinear Knudsen layers with small magnitudes for the hard sphere collision model was obtained in [33] under the condition that the boundary is noncharacteristic , i.e. the speed of the boundary and the Maxwellian state M = M[ρ,u,θ] at x = ∞ satisfy that ⎧ 0
∈ {u 1 − c, u 1 , u 1 + c}, (speed of the boundary=0), ⎪ ⎪ ⎨ u = (u 1 , 0, 0) : bulk fluid velocity, ⎪ ⎪ ⎩c = 5θ : sound speed at rest. 3 This condition can be rephrased as |M |
= 0 or 1, where M ≡

u1 , (Mach Number). c

The nonlinear time-asymptotic stability of the Knudsen layers was obtained under the assumption that the boundary is negative supersonic, u 1 < −c, (M < −1) in [34]. However, since the boundary layer is a physical relevant issue, the condition M < −1 for the time-asymptotic stability is only a technical assumption. The methodologies in [33,34] are almost identical, though they solved different types of problems. (One is for the existence of the nonlinear Knudsen layers, and the other is for the time-asymptotic stability.) They are weighted energy estimates motivated by the Macro-Micro decomposition developed in [23]. Due to the constraint from the energy estimates one will need high order energy estimates and the Sobolev inequality together to close the a priori assumption on the smallness on the super norms of the lower terms which were used in the nonlinear terms for obtaining the high order energy estimates. However, it is not necessary for the solution of the Boltzmann equation with a kinetic boundary condition to be continuous at the boundary. This gives rise to a conflict with the possibility to perform the high order energy estimates. The condition M < −1 yields that an exponentially decaying energy estimate in time and this exponential decaying property compensates the discontinuity presence at the boundary. One still can obtain the nonlinear stability of the Knudsen layer. When M > −1, the exponential decaying property no longer exists. One needs to introduce a new methodology which does not require any regularity property in the solution to overcome the difficulty due to discontinuities at the boundary. Denote F and  to be the solution of the Boltzmann equation and the Knudsen layer respectively. Follow the custom way in [6] to rewrite the Boltzmann equation for the planar wave solution and the equation for the Knudsen layer as follows: 1

1

F = M 2 f + M,  = M 2 ψ + M, so that the Boltzmann equation becomes

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1

∂t f + ξ 1 ∂x f = Lf + (f), Lf ≡ 2M− 2 B(M, M 2 f), (f) ≡ M−1/2 Q(M1/2 f), while the equation for the Knudsen layer turns to be 1

1

ξ 1 ∂x ψ = Lψ + (ψ), Lψ ≡ 2M− 2 B(M, M 2 ψ), (ψ) ≡ M−1/2 Q(M1/2 ψ), where the definition the linearized collision operator L = −ν + K will be given later in Sect. 2. Set w = f − ψ. We consider the time stability problem of the Knudsen layer  in the sense of the following initial boundary value problem: ⎧ 1 ⎪ ⎨∂t w + ξ ∂x w − Lw = L δ w + (w), (1.2) w(0, t, ξ )|ξ 1 >0 = 0, ⎪ ⎩w(x, 0, ξ ) = w (x, ξ ), 0 where

⎧ −1/2 B(M, M1/2 w), ⎪ ⎨ Lw = 2M L δ w = 2M−1/2 B( − M, M1/2 w), ⎪ ⎩ (w) = M−1/2 Q(M1/2 w),

and w0 (x, ξ ) satisfies sup |w0 (x, ξ )|(1 + |ξ |)3 ≤ e−x

ξ ∈R3

with 1.

The stability of  is equivalent to the problem when in the initial data is sufficiently small, and the question whether the solution w(x, t, ξ ) tends to zero as t tends to infinity. Consider the backward Green’s function Gi (x, y, t, σ, ξ, ξ∗ ) for the initial value problem, ⎧ 1 3 ⎪ ⎨−∂s Gi − ξ∗ ∂ y Gi − LGi = 0 for 0 < s < t, x, y ∈ R ξ, ξ∗ ∈ R , 3 Gi (x, y, s, s, ξ, ξ∗ ) = δ(x − y)δ (ξ − ξ∗ ), ⎪ ⎩ lim Gi (x, y, t, s, ξ, ξ∗ ) = 0, y→∞

and the backward Green’s function Gb (x, y, t, σ, ξ, ξ∗ ) for the initial boundary value problem, ⎧ ⎪ −∂s Gb − ξ∗1 ∂ y Gb − LGb = 0 for 0 < s < t, x, y > 0, ξ, ξ∗ ∈ R3 , ⎪ ⎪ ⎪ ⎨Gb (x, y, s, s, ξ, ξ∗ ) = δ(x − y)δ 3 (ξ − ξ∗ ), Gb (x, 0, t, s, ξ, ξ∗ )|ξ∗1 <0 = 0, ⎪ ⎪ ⎪ ⎪ ⎩ lim Gb (x, y, t, s, ξ, ξ∗ ) = 0. y→∞

Then the solution w(x, t, ξ ) of (1.2) can be represented as follows:  ∞ Gi (x, y, t, 0, ξ, ξ∗ )w(y, 0, ξ∗ )dξ∗ dy w(x, t, ξ ) = R3 0  t + Gi (x, 0, t, s, ξ, ξ∗ )ξ∗1 f(0, s, ξ∗ d)dξ∗ ds 0

R3

Pointwise Convergence to Knudsen Layers of the Boltzmann Equation

 t + 

∞

0 0 ∞

w(x, t, ξ ) =

R3

0

R3

Gi (x, 0, t, s, ξ, ξ∗ ) (w)dξ∗ dyds,

291

(1.3)

Gb (x, y, t, 0, ξ, ξ∗ )w(y, 0, ξ∗ )dξ∗ dy

 t

+  + 

0

Gb (x, 0, t, s, ξ, ξ∗ )ξ∗1 f(0, s, ξ∗ d)dξ∗ ds R3 t  ∞

0 0 ∞

=

R3

Gb (x, 0, t, s, ξ, ξ∗ ) (w)dξ∗ dyds

Gb (x, y, t, 0, ξ, ξ∗ )w(y, 0, ξ∗ )dξ∗ dy R3 ∞

0

 t

+ 0

0

R3

Gb (x, 0, t, s, ξ, ξ∗ ) (w)dξ∗ dyds.

(1.4)

These representations do not require any regularity property to obtain the solution. It avoids the difficulty in obtaining the high order energy estimates, namely the discontinuities at the boundary. The main result of the paper is: Theorem 1.1 (Main Theorem for stability when M > −1 ∩ {|M |
= 1, 0}). Suppose that M > −1 and |M |
= 0, 1. Then, the Knudsen layer  is nonlinear stable, i.e. when is sufficiently small, the solution of (1.2), w(x, t, ξ ) satisfies that (x−λ j t)2

3 e− C(1+t) (1 + |ξ |)3 |w(x, t, ξ )| ≤ C + C e−(|x|+t)/C √ t + 1 j=1 ⎧ 2 ⎪ ⎪ 1 ⎪ ⎪ for x ∈ (λ1 t, λ3 t), √ ⎨ (|x − λl t| + 1)(|x − λl+1 t| + 1) +C l=1 (1.5) ⎪ ⎪ ⎪ ⎪ ⎩0 for x ∈ (0, λ1 t) ∪ (λ3 t, ∞),

where (λ1 , λ2 , λ3 ) = (u 1 − c, u 1 , u 1 + c), and C > 0 is an universal constant. The linearized equation around the Knudsen layer can be rewritten as a perturbation of the linearized problem around a global Maxwellian state. The perturbation term can be regarded as a small linear coupling term, but it takes some effort to justify that the effect from the small linear coupling remains uniformly small in all time. Then, we use the Green’s function of the linearized equation around a global Maxwellian state to rewrite it as an integral equation to generate the ansatz. As a result of the length calculation, the ansatz created is still bounded by the ansatz created by the nonlinear problem around a global Maxwellian state. This leads to an example to reduce a problem on non-uniform background to a problem around an uniform background. The proof of Theorem 1.1 is considered for the cases M > 1 and 0 < |M | < 1 respectively. The proof for the case M > 1 is also an interesting new methodology. In this case one can make a priori assumption on the smallness of the sup norm to obtain a priori estimate on the full boundary data. Then, the estimates on the boundary data combining with the Green’s function Gi and the representation (1.3) yields the ansatz. To combine the boundary a priori estimate with the Green’s function Gi is subtle. One

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needs to use the particlelike-wavelike decomposition to decompose the Green’s function Gi in order to resolve some singularities due to slow particles. One also needs to have a priori estimates on the small linear coupling terms. Finally, the ansatz will justify the smallness a priori assumption. Thus, the nonlinear problem is closed without a high order energy estimate. In contrast to this, for the case 0 < |M | < 1, one does not need any a priori estimates on the boundary data, since we will use the representation (1.4) and the Green’s function Gb for the initial-boundary value problem already takes care of the singularities which arise from the boundary. The second purpose of this paper is to reconsider some large-time stability problems by using the Green’s function. In fact, for the case M < −1, by using the Green’s function it’s more convenient to get the results stated in [33] and [34]. Consider the time-asymptotic stability property of an upwind-damped Boltzmann equation, ⎧ 1 + 1 ⎪ ⎨∂t f + ξ ∂x f − Lf = −γ P0 ξ f + (f), γ > 0, f(0, t, ξ )|ξ 1 >0 = b+ (ξ )|ξ 1 >0 , ⎪ ⎩f(x, 0, ξ ) = f (x, ξ ). 0 The stability of the solution of the damped equation together with the fact that limt→∞ f(x, t, ξ ) is the solution of  ξ 1 ∂x ψ − Lψ = −γ P+0 ξ 1 ψ + (ψ), γ > 0, ψ(0, t, ξ )|ξ 1 >0 = b+ (ξ )|ξ 1 >0 , yields the existence of the boundary layer: 1 Theorem 1.2 (Existence).

For1 given √ M = M[1,u,θ] , u = (u , 0, 0), with the condition on the Mach number M ≡ u / 5θ/3
∈ {−1, 0, 1}, there exist a map, 

3 2 3  : h(ξ )|ξ 1 >0 sup(1 + |ξ |) |h(ξ )| < ∞, h(ξ ) : Even function in ξ and ξ −→ Rd , ξ

  δ, γ , and C > 0 such that for any h ∈  −1 (0) ∩ supξ (1 + |ξ |)3 |h(ξ )| ≤ δ there is a Knudsen layer  connecting the kinetic boundary data Fb |ξ 1 >0 = (M + M1/2 h)|ξ 1 >0 to the Maxwellian state M, and the Knudsen layer  also satisfies that sup(1 + |ξ |)3 M−1/2 | − M| ≤ δe−γ x/C for x > 0, ξ

where     d = the number of elements in u 1 − 5θ/3, u 1 , u 1 + 5θ/3 ∩ R+ . When M < −1, the operator P+0 is a null operator. Thus stability for the case M < −1 follows: Theorem 1.3 (Stability for M < −1). When M < −1, the Knudsen layer  given in Theorem 1.2 is exponentially stable in time, i.e. the solution of (1.2), w(x, t, ξ ) converges to zero exponentially fast.

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293

The Green’s function G Di for the damped Boltzmann equation is used for the existence of the Knudsen layer and the stability for M < −1; the Green’s function Gi for the Cauchy problem is used for the stability for M > 1; and the Green’s function Gb for the initial boundary value problem is used for the stability when |M | < 1(M
= 0). The technology used depends on the Mach number M . However, it’s clear that we can deal with the existence and stability of the boundary layer by the Green’s function. In the above statement, we find that one needs more detailed information on the Green’s functions in order to obtain the pointwise estimate of the nonlinear problem. This seems a down side of this approach. Indeed, a full detailed understanding of the Green’s function would satisfy H. Grad’s requirement as a qualitative tool, “The main qualitative feature of all problem in kinetic theory is the transition from free flow to continuum flow“, Grad [15]. For the purpose, Grad’s project, to understand the detailed structure of the Green’s functions, it leads to the studies, [24–26], on the Boltzmann equation linearized around a global Maxwellian. The Green’s functions for an initial value problem for planar wave solutions are, [24]:  ∂t Gi (x, t, ξ, ξ0 )+ξ 1 ∂x Gi (x, t, ξ, ξ0 )−LGi (x, t, ξ, ξ0 ) = 0 for x ∈ R, t > 0, ξ, ξ0 ∈ R3, Gi (x, 0, ξ, ξ0 ) = δ(x)δ 3 (ξ − ξ0 ), for 3-D solutions are, [25]:  ∂t Gi (x, t, ξ, ξ0 ) + ξ · ∇x Gi (x, t, ξ, ξ0 )− LGi (x, t, ξ, ξ0 ) = 0 for t > 0, x, ξ, ξ0 ∈ R3, Gi (x, 0, ξ, ξ0 ) = δ 3 (x)δ 3 (ξ − ξ0 ), and for an initial-boundary value problem for 1-D case is, ⎧ 1 3 ⎪ ⎨∂t Gb + ξ ∂x Gb − LGb = 0 for 0 < s < t, x, y > 0, ξ, ξ∗ ∈ R , Gb (x, y, s, s, ξ, ξ∗ ) = δ(x − y)δ 3 (ξ − ξ∗ ), ⎪ ⎩G (x, 0, t, s, ξ, ξ )| 1 = 0. b ∗ ξ >0 In [9,20,28,31] the Green’s functions for an initial value problem had been studied in terms of the Fourier variable η:    1 Gi (x, t, ξ, ξ∗ )e−iη·x d x . e(−iη·ξ +L)t ≡ 3 π R3 This representation is also regarded as a semi-group. By using this semi-group representation in the Fourier variable η S. Ukai obtained the first global existence theory around a global Maxwellian state in [31]. The free flow feature is rather unclear in the semi-group approach due to the lack of a precise spectrum property of the operator iηξ 1 + L in the high wave number regime, |η|  1. Thus, the semi-group approach alone is insufficient to observe the transition from the free flow to the continuum flow, and it alone can not be a good qualitative tool for the Boltzmann equation. T.-P. Liu and S.-H. Yu started to analyze to the Green’s function instead of the semigroup and tried to understand the transition from the free flow to the continuum flow in [24]. They began from the following reduced problem:  ∂t f + ξ 1 ∂x f = Lf for (x, t, ξ ) ∈ R × R+ × R3 , f(x, 0, ξ ) = gin (x, ξ ), gin (x, ξ ) = 0 for |x| ≥ 1 and sup |gin (x, ξ )|(1 + |ξ |)3 ≤ 1. x,ξ

(1.6)

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In order to resolve the problem due to insufficient information on the high wave number spectrum, two complementary decompositions, namely the long wave-short wave decomposition and particlelike-wavelike decomposition, were introduced. Long wave-short wave decomposition. The long wave-short wave decomposition for the solution of (1.6) is: ⎧ ⎪ f = fL + fS ,  ⎪ ⎪ ⎪ 1 ⎪ ⎪ ⎪ ei xη+(−iηξ +L)t gˆ in (η, ξ )dη, f L (x, t, ξ ) ≡ ⎪ ⎨  − 1 (x, t, ξ ) ≡ ei xη+(−iηξ +L)t gˆ in (η, ξ )dη, f ⎪ S ⎪ ⎪ |η|> ⎪  ⎪ ⎪ 1 ⎪ ⎪ e−i xη gin (x, ξ )d x, ⎩gˆ in (η, ξ ) ≡ π R where the parameter > 0 is an universal constant related to the spectrum gap of the operator −iξ η + L around the origin η = 0. The long wave component f L catches the continuum flow as the semi-group does. Furthermore, by inverse Fourier transformation and complex analysis one can obtain exponentially sharp pointwise estimates of the long wave component f L (x, t, ξ ) within a finite Mach number region |x| < αc(t +1) for some α > 2, 

R

(x−λ j t) 3 − C(1+t) e 2 f L (x, t, ξ ) dξ ≤ C √ 3 1+t

2

for |x| ≤ αc(1 + t),

j=1

where λ1 = u 1 − c, λ2 = u 1 , λ3 = u 1 + c, and the regularity in the x variable is assured for this long wave component in the following sense:   |∂xk f L (x, t, ξ )|2 dξ d x ≤ Ok , R R3

where Ok is independent of t. In the same time, the short wave component f S dissipates to zero exponentially in the L 2x (L 2ξ )-sense, i.e.   |f S (x, t, ξ )|2 dξ d x ≤ Ce− 0 t , R R3

where 0 > 0 is also an universal constant related to the spectrum gap of −iηξ 1 + L near η = 0. This energy estimate on the short-wave component is still not qualitative enough to reveal pointwise structure due to the integration over x. Here, to obtain the estimates on the long wave component f L in a finite Mach number region one needs to show the spectrum is an analytic function around η = 0. Such an analytic property on the spectrum is not necessarily true for a hard potential model, see [21]. Particlelike-wavelike decomposition (Wave Remainder decomposition in [24]). The particlelike-wavelike decomposition for the solution of (1.6) is based on the kinetic nature of (1.6). Rewrite (1.6) into the form to emphasize the feature of kinetic equation:  ∂t f + ξ 1 ∂x f + νf = Kf, f(x, 0, ξ ) = gin (x, ξ ).

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295

For any given i > 0 by the mixture lemma in [24] and by the solution operator e(−ξ ∂x +ν)t one can have two functions Wi (x, t, ξ ) (particlelike component) and Ri (x, t, ξ ) (wavelike component) such that ⎧ f(x, t, ξ ) = Wi (x, t, ξ ) + Ri (x, t, ξ ), ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ Wi (x, 0, ξ ) = gin (x, ξ ), ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨Ri (x, 0, ξ ) ≡ 0, (1.7) |Wi (x, t, ξ )| ≤ Ce−(|x|+t)/C , ⎪ ⎪ ⎪    ⎪  2 ⎪ ⎪ i ⎪ 1 ⎪ ∂ ∂ + ξ ∂ + ν −K Wi dξ d x ≤ Ce−t/C (due to the mixture lemma), t x ⎪ x ⎪ 3 ⎪ ⎪ ⎪ R R 2 ⎩ 2 i R3 ∂x Ri + |Ri | dξ d x ≤ C, 1

where the constants C and C > 0 depend on i but not on t. Here, the mixture lemma is about the conversion of the microscopic regularity in ξ variable to the regularity property to the macroscopic variable x variable. It will be stated in Sect. 2 later. Eventually, by combining these two decompositions one has that f L − Ri = Wi − f S . This gives that ⎧  ⎪ |f S − Wi |2 dξ d x = Ce−t/C1 , ⎪ ⎪ ⎨ R R3   ⎪ ⎪ ⎪ ⎩

 

R R3

|∂x (f S − Wi )|2 dξ d x =

R R3

|∂x (f L − Ri )|2 dξ d x = C.

From the above and Sobolev inequality one can see that  sup |f S − Wi |2 dξ ≤ O(1)e−t/C2 x∈R R3

for some C2 > 0. Thus, from this and the information on Wi together one can obtain the structure of f S (x, t) without acquiring the whole spectrum of the operator −iηξ 1 + L . The most important fact is about the wavelike component Ri :  sup |f L − Ri |2 dξ ≤ Ce−t/C2 . x∈R R3

From this estimate one can conclude that the wavelike component Ri approaches the continuum structure exponentially fast within the region |x| ≤ αc(1 + t) : 

 R3

|Ri (x, t, ξ )|2 dξ ≤ C 0

t

e−τ/C dτ

 3 j=1

(x−λ j t)2

e− C(1+t) . √ 1+t

(1.8)

Thus, this and (1.7) give a clear picture about the transition from the free flow to the continuum flow in a finite Mach number region |x| ≤ αc(1 + t).

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Waves in the high Mach number region, (|x| > αc(1 + t)), are not necessarily related to the continuum flow. It is sufficient to use the particlelike-wavelike decomposition and energy estimates together to show all wave structures decay exponentially fast in x and t in the region |x| ≥ αc(1 + t) :  |Ri (x, t, ξ )|2 dξ ≤ Ce−(|x|+t)/C . (1.9) R3

The same framework for the planar wave solution can be applied to a 3-D wave, but one needs to apply a complicated S O(3)-symmetry to yield exponentially sharp estimates on the inverse Fourier transformation for the long-wave component, see [25]. When the Mach number is subsonic, (|M | < 1), one has to develop the Green’s function Gb (x, y, t, s, ξ, ξ∗ ) for the initial-boundary value problem, ⎧ 1 + 3 ⎪ ⎨∂t f + ξ ∂x f = Lf, (x, t, ξ ) ∈ R × R × R , f(0, t, ξ )|ξ 1 >0 = b+ (t, ξ ), (imposed boundary data,), ⎪ ⎩f(x, 0, ξ ) = f (x, ξ ), (imposed initial data), 0 since one can not have such a priori exponentially decaying estimate on the boundary data  ∞   ∞  εt 1 2 e |ξ |f(0, t, ξ ) dξ dt ≤ C eεt |ξ 1 |b+ (t, ξ )2 dξ dt for some ε > 0, 0

R3 ξ 1 <0

0

R3

ξ 1 >0

because u 1 − c, u 1 , and u 1 + c are not in the same signs in contrast to the case |M | > 1. To construct Gb the Fourier variable is no longer valid to approximate the continuum flow in a half space problem, but the Green’s function of the initial value problem remains valid in terms of the physical parameter (x, t, ξ ). The Green’s function for the initial value problem Gi (x, t, ξ, ξ∗ ) is used to approximate the continuum flow structure in Gb (x, y, t, s, ξ, ξ∗ ) in [26]. The basic tools in the construction of Gb are Gi and a upwind-damped Boltzmann equation, ⎧ 1 + 1 ⎪ ⎨∂t ˜f + ξ ∂x ˜f − L ˜f = −γ P0 ξ ˜f, γ > 0, ˜f(0, t, ξ )| 1 = b+ (t, ξ )| 1 , ξ >0 ξ >0 ⎪ ⎩˜f(x, 0, ξ ) = f (x, ξ ), 0 to approximate the boundary data f(0, t, ξ )|ξ 1 <0 by assuming γ 1. On the other hand with the extra upwind-damping −γ P+0 ξ 1˜f, one can have such an exponentially decaying a priori estimate on the boundary data:  ∞   ∞  eεt |ξ 1 |˜f(0, t, ξ )2 dξ dt ≤ C eεt |ξ 1 |b+ (t, ξ )2 dξ dt for some ε > 0. 0

R3 ξ 1 <0

0

R3

ξ 1 >0

(1.10) Using the boundary data ˜f(0, t, ξ ) and Gi one can obtain a solution of the Boltzmann equation, ˜f:

Pointwise Convergence to Knudsen Layers of the Boltzmann Equation

˜f(x, t, ξ ) =

297

 t

Gi (x, t − s, ξ, ξ∗ )ξ∗1˜f(0, s, ξ∗ )dξ∗ ds 0  ∞ Gi (x − y, t, ξ, ξ∗ )f0 (y, 0, ξ∗ )dξ∗ dy. + R3

0

R3

Then, one uses (1.7), (1.8), (1.9), and (1.10) to conclude the difference f(0, t, ξ )ξ 1 >0 − ˜f(0, t, ξ ) 1 is of the order γ 1/4 e−Cγ t , where C is a positive constant. By repeating the ξ >0 approximation procedure one can obtain the exponential shape estimates on the Green’s function Gb in [26]. With this Green’s function Gb the nonlinear stability of a global Maxwellian state M for a half space problem is proved. The estimates on the Green’s function Gb will be stated in Sect. 2. After the completion of the analysis on the Green’s functions Gi and Gb , the study of the Knudsen layer can be outlined as follows. In Sect. 2 mainly we will review the results on the Green’s functions for both the initial-value problems and the boundary-value problems as well as the necessary preliminaries for this paper. In Sect. 3 we obtain the existence of the Knudsen layer under the time-asymptotic approach. This approach also yields the stability of the boundary layer for the case M < −1 In Sect. 4 we prepare the technical integrals for estimating the nonlinear coupling wave in studying the case M > −1. Finally, we complete the nonlinear stability problems in Sects. 5 and 6 for the cases M > 1 and 0 < |M | < 1 respectively. 2. Tools: Green’s Functions for Initial Value Problem and Initial-Boundary Value Problem In this section we will summarize the Green’s functions developed for the initial value problem and the initial-boundary value problem developed in [24,26] for linearized Boltzmann equations around a global Maxwellian M = M[1,u,1] , u = (u 1 , 0, 0):  ∂t h + ξ 1 ∂x h = Lh for x ∈ R, t > 0, (2.1) h(x, 0) = h0 (x), ⎧ 1 ⎪ ⎨∂t j + ξ ∂x j = Lj for x, t > 0, (2.2) j(x, 0) = j0 (x), ⎪ ⎩j(0, t)| 1 = 0. ξ >0 2.1. Basic notions and the macro-micro decomposition. We denote L 2ξ the restricted Hilbert spaces with given inner products: L 2ξ ≡ {h ∈ L 2 (R3)|h(ξ ) : an even function in ξ 2 and ξ 3 .} ⎧ ⎨(h, g) ≡ h(ξ )g(ξ )dξ for h, g ∈ L 2ξ , R3 ⎩h 2 = (h, h). L ξ

Two auxiliary inner products (·, ·)− and (·, ·)+ on L 2ξ are introduced due to the presence of a physical boundary:

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(g, h)− ≡

g(ξ )h(ξ )dξ, (g, h)+ ≡ R3 ∩{ξ 1 ≤0}

g(ξ )h(ξ )dξ, R3 ∩{ξ 1 ≥0}

with the corresponding space L 2ξ,+ ≡ L 2 (R3+ ), R3+ ≡ {(ξ 1 , ξ 2 , ξ 3 ) : ξ 1 ≥ 0, ξ 2 , ξ 3 ∈ R}. We still use the same sets of notions in [26]: ⎧  1/2 ⎪ 2 ⎪ ⎪ g L 2 ≡ g(ξ ) dξ , ⎪ ⎪ ξ ⎪ R3 ⎪ ⎪ 2 ⎪ (L ξ ,  ·  L 2 ) : a Hilbert space with inner ⎪ ⎪ ξ ⎪ ⎪ ⎨ product  Measures locally in (x, t) : ⎪ h1 (ξ )h2 (ξ ) dξ, (h1 , h2 ) ≡ ⎪ ⎪ ⎪ R3 ⎪ ⎪ ⎪ ⎪ g L ∞ ≡ sup |g(ξ )|(1 + |ξ |)β , ⎪ ξ,β ⎪ ⎪ ξ ∈R3 ⎪ ⎪ ⎩ g ∞ ≡ sup 3 (1 + |ξ |β )|g(ξ )|, L ξ,β+ ξ ∈R+ ⎧ h L ∞ (L 2 ) ≡ supx∈R+ h(x, ·) L 2 , ⎪ ⎪ x ξ ξ ⎪   1/2 ⎪ ⎪ ⎪ ⎪ 2 ⎪ h ≡ h(x, ξ ) dξ d x , 2 ⎪ L 2x (L ξ ) ⎪ 3 ⎪ R R + ⎪ ⎛ ⎞1/2 ⎨   i Measures locally in t: j ⎪ h H i (L 2 ) ≡ ⎝ |∂x h(x, ξ )|2 dξ d x ⎠ , ⎪ ⎪ x ξ ⎪ R+ R3 j=0 ⎪ ⎪ ⎪ ⎪ h ∞ ∞ ≡ sup ⎪ , ⎪ L x (L ξ,β ) x∈R+ h(x, ·) L ∞ ⎪ ξ,β ⎪ ⎩ ∞ ∞ |||h||| ≡ h L x (L ξ,3 ) . The null space of L in the restricted Hilbert space L 2ξ is a three-dimensional vector space with orthogonal basis χi , i = 0, 1, 4: ker(L) ≡ span{χ0 , χ1 , χ4 }, ⎧ ⎪ ≡ M1/2 , χ 0 ⎨ χ1 ≡ (ξ 1 − u 1 )M1/2 , ⎪ ⎩χ4 ≡ √1 (|ξ − u|2 − 3)M1/2 . 6

The macro-micro decomposition (P0 , P1 ) on L 2ξ is given as follows: for any g ∈ L 2ξ , ⎧ ⎪ ⎨g ≡ P0 g + P1 g(≡ g0 + g1 ), P0 g ≡ (χ0 , g)χ0 + (χ1 , g)χ1 + (χ4 , g)χ4 , ⎪ ⎩P g ≡ g − P g. 1 0 The characteristic information of the Euler equations ∂t P0 f + ∂x P0 (ξ 1 P0 f) = 0

(2.3)

Pointwise Convergence to Knudsen Layers of the Boltzmann Equation

299

is computed from the flux operator P0 ξ 1 on P0 L 2ξ , [23]: dim(P0 L 2ξ ) = 3, P0 ξ 1 Ei = λi Ei for i = 1, 2, 3,   λ1 = −c + u 1 , λ2 = u 1 , λ3 = c + u 1 ,   ⎧  ⎪ 3 5 ⎪ E1 ≡ ⎪ 2 χ0 − 2 χ1 + χ 4 , ⎪ ⎪    ⎪ ⎪ ⎪ ⎨E ≡ − 2 χ + χ , 2 4 3 0    ⎪ ⎪ ⎪ 3 5 ⎪ E3 ≡ ⎪ 2 χ0 + 2 χ1 + χ 4 , ⎪ ⎪ ⎪ ⎩ i (Ei , E j ) = δ j (Kronecker’s delta function), (2.4) where c=



5/3

is the speed of sound. Lemma 2.1. There exist positive constants ν0 such that, for any h ∈ L 2ξ , ⎧ 3 ⎪ ⎪ ⎪ (P h, P h) = (E j , h)2 , ⎪ 0 0 ⎪ ⎪ ⎪ j=1 ⎨ 3 ⎪ (P0 h, ξ 1 P0 h) = λ j (E j , h)2 , ⎪ ⎪ ⎪ ⎪ ⎪ j=1 ⎪ ⎩ (P1 h, LP1 h) ≤ −ν0 (P1 h, (1 + |ξ |)P1 h).

2.2. Upwind-Downwind decomposition and energy estimates. When the fluid velocity u 1 > −c as well as not characteristic, i.e. u 1
∈ {−c, 0, c}, one has the eigendecomposition of the operator P0 ξ 1 | Range(P0 ) : ⎧ P0 | Range(P0 ) = P+0 | Range(P0 ) + P− ⎪ 0 | Range(P0 ) , ⎪ ⎪ − 1 ⎨P ξ 1 | + 1 0 Range(P0 ) = P0 ξ | Range(P0 ) ⊕ P0 ξ | Range(P0 ) , P+0 ξ 1 | Range(P+0 ) : positive-definite, ⎪ ⎪ ⎪ ⎩P− ξ 1 | − Range(P− ) : negative-definite or P0 ≡ 0. 0 0

The explicit expression of

and P− 0 is ⎧ +  P0 = Ej ⊗ Ej , ⎪ ⎪ ⎨ λ j >0  − ⎪ P = Ej ⊗ Ej . ⎪ 0 ⎩

P+0

λ j <0

Here, the operator g ⊗ h| is given by

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g ⊗ h| f ≡ (h, f)g for f ∈ L 2ξ . Lemma 2.2. If u 1 ∈ (−c, ∞) and it is not characteristic, then there exists δ0 > 0 such that whenever λ ∈ (0, δ0 ), 1 − γ (f, ξ 1 f) + γ (f, P+0 ξ 1 f) − (f, Lf) ≥ γ (P0 f, P0 f) + γ (P1 f, (1 + |ξ |)P1 f) holds, 2 (2.5) where ≡

1 min |λ j |. 8

2.3. Grad’s lemma on collision operator. For the hard sphere model we consider, the linearized collision operator L is of the following form, [16]: ⎧ Lg(ξ ) = −ν(ξ − u)g(ξ ) + Kg(ξ ), ⎪ ⎪  ⎪ ⎪ ⎪ Kg(ξ ) ≡ R3 K (ξ − u, ξ∗ − u)g(ξ∗ ) dξ∗ , ⎪ ⎪   ⎪ ⎪ 2 |ξ − ξ∗ |2 (|ξ |2 − |ξ∗ |2 )2 ⎪ ⎪ ) ≡ − exp − K (ξ, ξ √ ⎪ ∗ ⎪ 2 ⎨ 8|ξ − ξ 8  2π|ξ − ξ∗ | ∗| (2.6) (|ξ |2 + |ξ∗ |2 ) |ξ − ξ∗ | ⎪ exp − , − ⎪ ⎪ 4  ⎪ ⎪    |ξ |  2 ⎪ 2 2 ⎪ 1 1 ⎪ − |ξ2| − u2 ⎪ ν(ξ ) ≡ √ 2e + 2 |ξ | + e du , ⎪ ⎪ ⎪ |ξ | 0 2π ⎪ ⎩ ν(ξ ) ∼ 1 + |ξ |. This lemma follows from direct computations and Carleman’s theory, [4], on the negative definiteness of the operator L on Range(P1 ). Lemma 2.3 (Grad, [13]). For any given β ≥ 0 there exist positive constants C(β) and C1 such that  ≤ C(β)j L ∞ , K j L ∞ ξ,β ξ,β+1 ≤ C1 j L 2 , Kj L ∞ ξ,0 ξ

where

 K j (ξ ) ≡

R3

K (ξ, ξ∗ )j(ξ∗ )dξ∗ .

The proof of this lemma in [6] for β = 0 can easily be generalized for any β ∈ R. Lemma 2.4. For the hard sphere model, there exists a constant C > 0 such that for any α ≥ −1,        (1 + |ξ |)1+α f   (1 + |ξ |)1+α g   (1 + |ξ |)α B(f, g)        1 1 1   ∞ ≤C   ∞  ∞ , (2.7) M2 M2 M2 Lξ Lξ Lξ       1+α   1+α    (1 + |ξ |)α B(f, g)   ≤ C  (1 + |ξ |) f   (1 + |ξ |) g   (2.8) 1 1 1   2  2  2 .  M2 M2 M2 Lξ Lξ Lξ

Pointwise Convergence to Knudsen Layers of the Boltzmann Equation

301

The proof of (2.7) for α = −1 in [11] and that of (2.8) for α = − 21 quoted from [23] can be generalized for α ≥ −1. Proof. Since (ξ  , ξ∗ ) which satisfies 

ξ  = ξ − [(ξ − ξ∗ ) · ], ξ∗ = ξ∗ + [(ξ − ξ∗ ) · ],

(2.9)

conserve the kinetic energy and momentum after collision, |ξ |2 + |ξ∗ |2 = |ξ  |2 + |ξ∗ |2 . As a result, M(ξ )M(ξ∗ ) = M(ξ  )M(ξ∗ ).

(2.10)

Consider B(f, g) =

1 (B1 (f, g) + B1 (g, f) − B2 (f, g) − B2 (g, f)) , 2

(2.11)

where  B1 (f, g) =

R3 × S 2

 B2 (f, g) =

R3 × S 2

f(ξ  )g(ξ∗ ) |(ξ − ξ∗ ) · | dξ∗ d, f(ξ )g(ξ∗ ) |(ξ − ξ∗ ) · | dξ∗ d. 1

1

We consider B1 first. Without losing generality, we set f = M 2 u, g = M 2 v, (1+|ξ |)α B (f, g) (1+|ξ |)α  1 1 1 M(ξ ) 2 M(ξ∗ ) 2 |u(ξ  )||v(ξ∗ )| |(ξ −ξ∗ ) · | dξ∗ d ≤ 1 1 2 R3 × S M(ξ ) 2 M2 (1 + |ξ |)α+1 v L ∞ ≤ (1 + |ξ |)α (1 + |ξ |)α+1 u L ∞ ξ ξ 1  M(ξ∗ ) 2 |(ξ − ξ∗ ) · | × dξ∗ d.  α+1 (1 + |ξ  |)α+1 R3 × S2 (1 + |ξ |) ∗

(2.12)

From (2.9), we can get 1 + |ξ  | = 1 + |ξ − [(ξ − ξ∗ ) · ]| ≤ C(2 + |ξ | + |ξ∗ |), 1 + |ξ∗ | = 1 + |ξ∗ + [(ξ − ξ∗ ) · ]| ≤ C(2 + |ξ | + |ξ∗ |).

(2.13) (2.14)

Similarly, 1 + |ξ | ≤ C(2 + |ξ  | + |ξ∗ |), 1 + |ξ∗ | ≤ C(2 + |ξ  | + |ξ∗ |). This suggests 1 + |ξ | + |ξ∗ | ∼ 1 + |ξ  | + |ξ∗ |. As a result,

α+1 (1 + |ξ  |)α+1 (1 + |ξ∗ |)α+1 = (1 + |ξ  |)(1 + |ξ∗ |)

≥ (1 + |ξ  | + |ξ∗ |)α+1 ≥ C(1 + |ξ | + |ξ∗ |)α+1 ≥ C(1 + |ξ |)α+1 .

(2.15)

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 1 This together with the fact that R3 × S2 M(ξ∗ ) 2 |(ξ − ξ∗ ) · | dξ∗ d ≤ C(1 + |ξ |) yields that 

1

M(ξ∗ ) 2 |(ξ − ξ∗ ) · | 1 dξ∗ d ≤ C 2 (1 + |ξ  |)α+1 (1 + |ξ  |)α+1 3 (1 + |ξ |)α+1 R ×S ∗  1 1 × M(ξ∗ ) 2 |(ξ − ξ∗ ) · | dξ∗ d ≤ C . (1 + |ξ |)α R3 × S 2

Thus (2.16) and (2.12) result in (1 + |ξ |)α B (f, g) 1 (1 + |ξ |)α+1 v L ∞ . ≤ C(1 + |ξ |)α+1 u L ∞ 1 ξ ξ M(ξ ) 2

(2.16)

(2.17)

The proof for B2 is similar but much simpler. Now, we complete the proof of (2.7). By using the Hölder inequality and the analysis applied for (2.7), we can also get (2.8).   2.4. Green’s functions. Denote Gi (x, t, ξ, ξ∗ ) and Gb (x, y, t, s, ξ, ξ∗ ) is the Green’s function for the initial value problem (2.1) and for the initial-boundary value problem (2.2) respectively. The equations for Gi and Gb are
∂t Gi + ξ 1 ∂x Gi = LGi f or x ∈ R, t > 0ξ, ξ∗ ∈ R3 , (2.18) Gi (x, 0, ξ, ξ∗ ) = δ(x)δ 3 (ξ − ξ∗ ), ⎧ ⎨ ∂t Gb + ξ 1 ∂x Gb = LGb f or x, y > 0, t > s, ξ, ξ∗ ∈ R3 , (2.19) G (x, y, s, s, ξ, ξ∗ ) = δ(x − y)δ 3 (ξ − ξ∗ ), ⎩ b Gb (0, y, t, s, ξ, ξ∗ )|ξ 1 >0 = 0. The Green’s functions in the above also satisfy the backward equations:  (−∂s − ξ∗ ∂ y − L)Gi (x − y, t − s, ξ, ξ∗ ) = 0, Gi (x − y, 0, ξ, ξ∗ ) = δ 1 (y − x)δ 3 (ξ∗ − ξ ), x, y ∈ R, ξ, ξ∗ ∈ R3 , ⎧ ⎪ ⎨(−∂s − ξ∗ ∂ y − L)Gb (x, y, t, s, ξ, ξ∗ ) = 0, Gb (x, y, 0, ξ, ξ∗ ) = δ 1 (y − x)δ 3 (ξ∗ − ξ ), x, y > 0, ξ, ξ∗ ∈ R3 , ⎪ ⎩G (x, 0, t, s, ξ, ξ )| 1 = 0. b ∗ ξ∗ <0

(2.20)

(2.21)

We still keep the following abbreviations: For given (x, t) ∈ R × R+ , Gi(b) (x, t) : h ∈ L 2ξ −→ Gi(b) (x, t)h ∈ L 2ξ , Gi(b) (x, t)h(ξ ) ≡ Gi(b) (x, t, ξ, ξ∗ )h(ξ∗ )dξ∗ . R3

Let gin be a function satisfying  gin (x, ξ ) ≡ 0 for |x| ≥ 1, ∞ < ∞. |||gin ||| ≡ gin  L ∞ x (L ξ,3 )

(2.22)

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2.4.1. Green’s function for an initial value problem We recall the following theorems in Liu-Yu [24] on the Green’s function Gi for the initial value problem. Theorem 2.1. ([24], Theorem 1.1). Green’s function Gi (x, t, ξ ; ξ0 ) contains mainly the particlelike waves e−ν(ξ0 )t δ(x − ξ 1 t)δ 3 (ξ − ξ0 ) + j1 (x, t, ξ ; ξ0 ) + j2 (x, t, ξ ; ξ0 ) propagating with microscopic velocity, and fluidlike waves 3 k=1

−

(x−λ¯ k t)2

e 4 Ak (t+1) Ek (ξ )Ek (ξ0 ) √ 4 Ak π(t + 1)

propagating along Euler characteristics. Precisely, for some positive constant C1 , Gi (x, t, ξ ; ξ0 ) = G0i (x − u 1 t, t, ξ − u; ξ0 − u) satisfies   G0i (x, t, ξ ; ξ0 ) − e−ν(ξ0 )t δ(x − ξ 1 t)δ 3 (ξ − ξ0 ) − j1 (x, t, ξ ; ξ0 ) − j2 (x, t, ξ ; ξ0 )  ⎤ ⎡  (x−λ¯ k t)2 (x−λ¯ i t)2 − − 3 3  e 4 Ak (t+1)  ⎥ ⎢ e C0 (t+1) + e−(|x|+t)/C1 ⎦ , Ek (ξ )Ek (ξ0 ) = C ⎣ − √  (t + 1) 4 A π(t + 1) k k=1 k=1  2 Lξ

where {λ¯ 1 , λ¯ 2 , λ¯ 3 } = {−c, ⎧ 0, c}, 1 1 ⎪ ⎨0 if (x − ξ0 t)(ξ t − x) < 0, −ν(ξ )(t−t )−ν(ξ )t 1 1 0 1 K (ξ, ξ ) j (x, t, ξ ; ξ0 ) = e 0 ⎪ else if x − ξ0 t
= 0, ⎩ 1 (ξ 1 − ξ0 ) where t1 = t − j2 (x, t, ξ ; ξ

0)

(x−ξ01 t) , ξ 1 −ξ01

= O(1)e

−

ν0 (t+|x|) 3



|ξ − ξ0 |−1 (1 + |ξ |)−1 for |ξ − ξ0 | ≤ 1, (1 + |ξ |)−1 for |ξ − ξ0 | ≥ 1.

Here the parameters A j , j = 1, 2, 3, are given below, and the constant C0 satisfies 

A j ≡ −(P1 ξ 1 E j , L −1 P1 ξ 1 E j ) > 0, j = 1, 2, 3, C0 > 4 max(A1 , A2 , A3 ).

(2.23)

The dissipation parameters A j , j = 1, 2, 3, are the same as the viscosity and the heat conductivity coefficients for the Navier-Stokes equations obtained through the Chapman-Enskog expansion. Theorem 2.2. ([24], Theorem 5.6). For bounded initial data gin with compact support in x, (2.22), and for the constant C0 satisfying (2.23), there exists a positive constant C1 independent of gin such that

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for |x − u 1 t| ≤ 2ct, ⎞ ⎛ ⎧ |x−λi t|2 ⎪ − C (t+1) 3 ⎪ e 0 ⎪ ⎟ ⎪ Gt g (x) 2 = O(1) |||g ||| ⎜ ⎪ + e−(|x|+t)/C1 ⎠ , √ in ⎝ ⎪ i in Lξ ⎪ (t + 1) ⎪ ⎪ i=1 ⎪ ⎪ ⎪ ⎪ ⎪ Git P1 gin (x) L 2 , P1 Git gin (x) L 2 ⎪ ⎪ ξ ⎪ ⎞ ⎛ ξ ⎪ ⎪ |x−λi t|2 ⎪ − 3 ⎪ ⎪ ⎟ ⎜ e C0 (t+1) ⎪ ⎪ = O(1) |||gin ||| ⎝ + e−(|x|+t)/C1 ⎠ , ⎪ ⎪ ⎪ (t + 1) ⎪ i=1 ⎨ ⎞ ⎛ |x−λi t|2 − C (t+1) 3 ⎪ ⎪ (2.24) e 0 ⎟ ⎜ ⎪ t ⎪ + e−(|x|+t)/C1 ⎠ , ⎪ ⎪ P1 Gi P1 gin (x) L 2ξ = O(1) |||gin ||| ⎝ 3/2 ⎪ (t + 1) ⎪ i=1 ⎪ ⎪ ⎪ ⎪ t ⎪ ⎪ | E Gi Ek ⊗ k gin (x) L 2 ⎪ ξ ⎪ ⎞ ⎛ ⎪ ⎪ ⎪ |x−λk t|2 |x−λi t|2 ⎪ ⎪ e− C0 (t+1) ⎪ ⎟ ⎜e− C0 (t+1) ⎪ ⎪ ⎪ + e−(|x|+t)/C1⎟ + = O(1) |||gin ||| ⎜ √ ⎪ ⎠, ⎝ ⎪ (t + 1) 1≤i≤3 (t + 1) ⎪ ⎩ i
=k

for |x − u t| ≥ 2ct 1

Git gin (x) L 2 ≤ C |||gin ||| e−(|x|+t)/C1 , ξ

where

 Git gin (x) ≡ R Gi (x − y, t)gin (y)dy. Theorem 2.3 ([24], Theorem 7.2). For bounded, compact-supported initial data gin , (2.22), there exist positive constants C1 and C0 such that for all x ∈ R the main fluid part satisfies  [E j ⊗ E j Git Ek ⊗ Ek | gin ](x) L ∞ ξ,3 ⎞ ⎛ (x−λk t)2 (x−λi t)2 − C (t+1) − C (t+1) 3 0 0 e ⎟ ⎜ e + + e−(|x|+t)/C1 ⎠ , (2.25) = O(1)|||gin ||| ⎝δ kj √ (1 + t) (1 + t) i=1

and the non-fluid parts have higher rate of decay in time: ⎞ ⎛ (x−λi t)2 − C (t+1) 3 e 0 ⎟ ⎜ + e−(|x|+t)/C1 ⎠ , [Git P1 gin (x) L ∞ = O(1)|||gin ||| ⎝ ξ,3 (1 + t) i=1

P1 Git gin (x) L ∞ ξ,3

⎞ ⎛ (x−λi t)2 − 3 ⎟ ⎜ e C0 (t+1) + e−(|x|+t)/C1 ⎠ , = O(1)|||gin ||| ⎝ (1 + t) i=1

P1 Git P1 gin (x) L ∞ ξ,3

⎞ ⎛ (x−λi t)2 − C (t+1) 3 e 0 ⎟ ⎜ = O(1)|||gin ||| ⎝ + e−(|x|+t)/C1 ⎠ . (1 + t)3/2 i=1

Pointwise Convergence to Knudsen Layers of the Boltzmann Equation

305

2.4.2. Green’s function for the initial boundary value problem for the case −c < u 1 < c and u 1
= 0 Suppose −1 < M ≡ u 1 /c < 1(M
= 0). The following theorem which gives the properties of Green’s function Gb of the initial boundary value problem can be found in [26]: Theorem 2.4 ([26], Theorem 1.1). Suppose that the Euler characteristics may leave or enter the boundary x = 0. For definiteness, assume that λ1 < λ2 < 0 < λ3 . Then p Green’s function Gb (x, y, t, s, ξ, ξ∗ ), (2.19), consists of particlelike waves Gb , fluidlike f waves Gb and the remaining term Grb : p

f

Gb (x, y, t, s, ξ, ξ∗ ) = Gb + Gb + Grb

f or x, y, t − s > 0.

(2.26)

The particle like-waves consist of a δ-function propagating from the initial data and damped under the operator ν and non-smooth functions jkb , k = 1, 2 resulting from the action of the operator K in linear collision operator L, on the δ function: Gb = e−ν(ξ −u)(t−s) δ(x − y − ξ 1 (t − s))δ 3 (ξ − ξ∗ ) + j1b (x, y, t − s, ξ, ξ∗ ) p

+ j2b (x, y, t − s, ξ, ξ∗ ),

(2.27)

j1b (x, ⎧

y, t, ξ, ξ∗ ) 0 for (x − y − ξ∗1 t)(ξ 1 t − x + y) < 0, ⎪ ⎪ ⎪ ⎨0 for (x − y − ξ 1 t)(ξ 1 t − x + y) > 0, − y + x < t, ξ 1 > 0, −ξ 1 > 0, ∗ ∗ ξ∗1 ξ1 = (2.28) −ν(ξ −u)(t−t1 )−ν(ξ∗ −u)t1 K (ξ − u, ξ − u) ⎪ e ⎪ ∗ ⎪ otherwise, ⎩ (ξ 1 − ξ∗1 ) (x−y−ξ∗1 t) where t1 = t − ξ 1 −ξ 1 , ∗  ν (t+|x−y|) |ξ − ξ∗ |−1 (1 + |ξ |)−1 for |ξ − ξ∗ | ≤ 1, 2 − 0 3 |jb (x, y, t, ξ, ξ∗ ) = O(1)e (1 + |ξ |)−1 for |ξ − ξ∗ | ≥ 1. (2.29) The fluidlike waves propagate with Euler characteristic speeds and diffuses like the heat kernel, that is, for some positive constant C, ⎛

 (x−y−λ j t)2 − Ct

3 ⎜ e ⎜ f Gb (x, y, t, ξ, ξ∗ ) L 2 = O(1) ⎜ √ ξ ⎝ t +1 j=1

+

 λ y 2 x−λ3 t− λ3 j Ct

⎞

2 ⎟ e− ⎟ √ ⎟. ⎠ (t + 1)(y + 1) j=1

(2.30)

The remaining term is a continuous function and decays exponentially: Grb (x, y, t, ξ, ξ∗ ) L 2 = e−O(1)(|x|+|y|+t)/C . ξ

(2.31)

Furthermore, except for the explicitly given particlelike waves, the decay rates increase when the Green’s function acts on the non-fluid functions. Precisely, for any h in the

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S. Deng, W. Wang, S.-H. Yu

range of the non-fluid projection P1 = I − P0 and in L 2ξ ,      Gb (x, y, t, ξ, ξ∗ )h(ξ∗ )dξ∗    R3

L 2ξ

⎛



 λ y 2 x−λ3 t− λ3 j Ct

⎞

( ) 3 2 ⎜ ⎟ e− e− ⎜ ⎟ + = O(1)h L 2 ⎜ + e−O(1)(|x|+|y|+t)/C ⎟ . √ ξ ⎝ ⎠ t +1 (t + 1)(y + 1) j=1 j=1 x−y−λ j t Ct

2

(2.32) Remark 2.1. The construction of the Green’s function for the initial-boundary value is based on the Green’s function for the initial value problem through a complicated procedure to obtain. Thus, when the analysis can avoid to use the Green’s function for the initial-boundary value problem, one could consider it as a simple proof. 3. Existence of Knudsen’s Layers (Time-asymptotic Approach) The Knudsen’s layer  is a stationary solution of the Boltzmann equation with an imposed incoming boundary data: ⎧ 1 + 3 ⎪ ⎨ξ ∂x  = Q(), (x, ξ ) ∈ R × R , (0, ξ )|ξ 1 >0 = Fb (ξ ), (3.1) ⎪ ⎩ lim (x, ξ ) = M(ξ ). x→∞

We denote ψ(x, ξ ) as the quotient of  − M to ψ(x, ξ ) ≡

√

M:

(x, ξ ) − M . √ M

The equation for ψ(x, ξ ) is ⎧ 1 + 3 ⎪ ⎨ξ ψx − Lψ = (ψ), (x, ξ ) ∈ R × R , ψ(0, ξ )|ξ 1 >0 = b+ (ξ ), ⎪ ⎩ lim ψ(x, ξ ) = 0,

(3.2)

x→∞

where

√ √ Mψ) + B( Mψ, M) Q( Mψ) Lψ ≡ , (ψ) ≡ , √ √ M M (Fb − M) b+ = √ 1 . M ξ >0 B(M,

√

The existence of a solution of the boundary value problem (3.2) is subject to the solvability of the incoming boundary data b+ , [33]. This problem was solved by considering the following system which is (3.2) with an extra upwind damping −γ P+0 ξ 1 ψ, γ > 0: ⎧ 1 + 1 + 3 ⎪ ⎨ξ ψx − Lψ = (ψ) − γ P0 ξ ψ, (x, ξ ) ∈ R × R , ψ(0, ξ )|ξ 1 >0 = b+ (ξ ), (3.3) ⎪ ⎩ lim ψ(x, ξ ) = 0. x→∞

Pointwise Convergence to Knudsen Layers of the Boltzmann Equation

307

The methodology in solving (3.3) requires regularities on ψ(x, ξ ) for non-small |ξ 1 | due to energy estimates. Here, we implement the time-asymptotic approach to obtain the stationary solution, since this methodology will imply the time-asymptotic stability of Knudsen’s layer when M < −1. This is because P+0 ξ 1 = 0 when M < −1. We consider the time evolution equation: ⎧ ft + ξ 1 fx − Lf = (f) − γ P+0 ξ 1 f, (x, t, ξ ) ∈ R+ × R+ × R3 , ⎪ ⎪ ⎪ ⎨f(x, 0, ξ ) = f0 (x, ξ ), (3.4) f(0, t, ξ )|ξ 1 >0 = b+ (ξ ), ⎪ ⎪ ⎪ ⎩ lim f(x, 0, ξ ) = 0, x→∞

where the initial data f0 (x, ξ ) satisfies sup(1 + |ξ |)4 |f0 (x, ξ )| ≤ Cδe−|x|/C ,

(3.5)

sup(1 + |ξ |)4 |(f0 )x (x, ξ )| ≤ Cδe−|x|/C ,

(3.6)

ξ

and ξ

for some constant C > 0. Then sup(1 + |ξ |)3 |ft (x, t, ξ )|t=0 | ξ

= sup(1 + |ξ |)3 −ξ 1 fx (x, 0, ξ ) + Lf(x, 0, ξ ) + (f)(x, 0, ξ ) − γ P+0 ξ 1 f(x, 0, ξ ) ξ

≤ C sup(1 + |ξ |)4 (f0 (x, ξ ) + f0x (x, ξ )) ≤ Cδe−|x|/C . ξ

(3.7)

Lemma 3.1. Suppose that M
∈ {−1, 0, 1}. Then, there exist γ > 0 and δ > 0 such that whenever sup (1 + |ξ |)3 |b+ (ξ )| ≤ δ,

(3.8)

ξ 1 >0

the limit limt→∞ f(x, t, ξ ) exists and satisfies     e−γ x/C  lim f(x, t) ∞ ≤ Cb+  L ∞ ξ,3+ t→∞

(3.9)

L ξ,3

for some C > 0. Here, the constants C depend on M . Furthermore, ψ(x, ξ ) ≡ limt→∞ f(x, t, ξ ) solves (3.3). The proof of this lemma will be completed in the rest of this section. When P+0 ξ 1 ψ(x, ξ ) ≡ 0, the solution of (3.3) is also the solution of the original system (3.2). In [33], it has been proved that this condition can be reduced to P+0 ξ 1 ψ(0, ξ ) ≡ 0, where ψ(0, ξ ) is determined by the incoming boundary data b+ (ξ ) uniquely. That is, there exists a map,

3 −1/2 2 3  : b+ (ξ )|ξ 1 >0 sup(1+|ξ |) M |b+ (ξ )| < ∞, b+ (ξ ) : Even function in ξ and ξ  −→Rd , ξ 

such that when b+ (ξ ) ∈  −1 (0), the damped system (3.3) is just the original system (3.2). Thus, Theorem 1.2 is a direct consequence of Lemma 3.1.

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3.1. Green’s functions for the damped equation. We denote G Di (x, t, ξ, ξ∗ ) to be the Green’s function of  ft + ξ 1 fx − Lf + γ P+0 ξ 1 f = 0, (x, t, ξ ) ∈ R+ × R+ × R3 , (3.10) f(x, 0, ξ ) = f0 (x, ξ ). Then, the solution of (3.4) can be represented through the Green’s function G Di as follows:  ∞ f(x, t, ξ ) = G Di (x − y, t, ξ, ξ∗ )f0 (y, ξ∗ )dξ∗ dy R3 0  t + G Di (x, t − s, ξ, ξ∗ )ξ∗1 f(0, s, ξ∗ )dξ∗ ds 0 R3  t  ∞ + G Di (x − y, t − s, ξ, ξ∗ ) (f)(y, s, ξ∗ )dξ∗ dyds. (3.11) 0

R3

0

For convenience in writing, (3.11) can be written as follows without specifying the dependence on ξ :  ∞  t f(x, t) = G Di (x − y, t)f0 (y)dy + G Di (x, t − s)[ξ 1 f](0, s)ds 0 0  t ∞ + G Di (x − y, t − s) (f)(y, s)dyds. (3.12) 0

0

∞ In the representation (3.11), the only clear term is 0 R3 G Di (x − y, t, ξ, ξ∗ ) f0 (y, ξ∗ )dξ∗ dy which comes from the initial data. To get the estimate of this term, the properties of the Green’s function G Di are needed. In order to apply the framework in [24] to obtain the pointwise structure of the Green’s function G Di , we need the long wave information of the operator −iηξ 1 + L − γ P+0 ξ 1 . The following lemma can be proved by a straightforward calculation. Lemma 3.2. There exist κ0 > 0 and κ1 > 0 such that the discrete spectrum σ (η) of the operator −iηξ 1 + L − γ P+0 ξ 1 satisfies the following properties: for |η| > κ0 , σ (η) ⊂ {z ∈ C : Re(z) ≤ −κ1 };

(3.13)

for |η| ≤ κ0 , σ (η) = {σi (η), i = 1, 2, 3}, where the functions σi (η) are analytic functions satisfying ⎧ σ j (η) = −χ ( j)γ λ j − i(λ j + χ ( j)A j )η − B j |η|2 + O(1)|η|3 , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪  ⎪ ⎪ ⎪ ⎪ 1, for λ j > 0 ⎪ ⎪ ⎨χ ( j) = 0, for λ < 0 , j ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ A j = (E j , γ P+0 ξ 1 L−1 P1 ξ 1 E j ), ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ B j = −(P1 ξ 1 E j , L−1 P1 ξ 1 E j ) > 0, and λ j are the eigenvalues of the operator P0 ξ 1 P0 | Range(P0 ) .

(3.14)

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309

Following the method in [24] to get the properties of G Di , we only have to check ˜ = Kf − γ P+ ξ 1 f. Since the operator whether the mixture lemma still holds. Denote Kf 0 ˜ K˜ ξ and K˜ ξ are P+0 ξ 1 is an integration operator, it is straightforward to prove that K, ∗ 2 bounded L operators. Thus the mixture lemma still works. From the spectrum of the operator −iηξ 1 + L − γ P+0 ξ 1 stated in (3.14), we find that the damping term P+0 ξ 1 forces some branches to detach from the origin and lie in the interior of the left complex plane Re(z) ≤ −α0 γ for some positive constant α0 which is of the order O(1) minλ j >0 λ j . This fact results in the extra decaying factor e−α0 γ t before

)

−

(x−λ j t)2

e √C(t+1) λ j >0 t+1

.

Theorem 3.1. For the function gin specified in (2.22) and for the constant C0 satisfying (2.23), there exists a positive constant C1 independent of gin such that for |x − u 1 t| ≤ 2ct, ⎧ t ⎪ G Di gin (x) L 2 ⎪ ξ ⎛ ⎪ ⎞ ⎪ ⎪ |x−λi t|2 |x−λi t|2 ⎪ − − ⎪ C (t+1) C (t+1) ⎪ e 0 e 0 ⎪ ⎟ ⎜ ⎪ ⎪ = O(1) |||gin ||| ⎝ + e−α0 γ t + e−(|x|+t)/C1 ⎠ , √ √ ⎪ ⎪ (t + 1) (t + 1) ⎪ ⎪ λi <0 λi >0 ⎪ ⎪ ⎪ ⎪ ⎪ GtDi P1 gin (x) L 2 , P1 GtDi gin (x) L 2 ⎪ ⎪ ξ ξ ⎞ ⎛ ⎪ ⎨ |x−λi t|2 |x−λi t|2 − − C (t+1) C (t+1) 0 0 e e ⎟ ⎜ ⎪ + e−α0 γ t + e−(|x|+t)/C1 ⎠ , = O(1) |||gin ||| ⎝ ⎪ ⎪ (t + 1) (t + 1) ⎪ ⎪ λi <0 λi >0 ⎪ ⎪ ⎪ ⎪ t ⎪ P1 G Di P1 gin (x) L 2 ⎪ ⎪ ⎪ ⎞ ⎛ ξ ⎪ ⎪ |x−λi t|2 |x−λi t|2 ⎪ − − ⎪ C (t+1) C (t+1) ⎪ e 0 e 0 ⎪ ⎟ ⎜ ⎪ = O(1) |||gin ||| ⎝ + e−α0 γ t + e−(|x|+t)/C1 ⎠ , ⎪ ⎪ 3/2 3/2 ⎪ (t + 1) (t + 1) ⎩ λi <0

λi >0

for |x − u 1 t| ≥ 2ct GtDi gin (x) L 2 ≤ C |||gin ||| e−(|x|+t)/C1 , ξ

(3.15)

where  GtDi gin (x) ≡ R G Di (x − y, t)gin (y)dy. From Theorem 3.16 and (3.5),  ∞      G (x − y, t)f (y)dy Di 0   0

 ≤ 0

⎛ ∞

L 2ξ

⎞ |x−y−λi t|2 |x−y−λi t|2 − C (t+1) − C (t+1) 0 0 e e ⎜ ⎟ + e−α0 γ t + e−(|x−y|+t)/C1 ⎠ √ √ ⎝ (t + 1) (t + 1) λ <0 λ >0 i

i

×f0 (y) L 2 dy ≤ Ce−γ (|x|+t)/C . ξ

(3.16)

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S. Deng, W. Wang, S.-H. Yu

3.2. Energy estimates on the boundary data. In the representation (3.11), the full boundary data are needed while only f(0, t, ξ )|ξ 1 >0 are given. We shall get the information about f(0, t, ξ )|ξ 1 ≤0 by the energy method. The basic energy estimates for the solution f(x, t, ξ ) of (3.4) and its boundary data f(0, t, ξ ) follow from the standard method, cf. [1,33]. To control the nonlinear term (f), we make a priori smallness assumption: f L 2 1. ξ

Lemma 3.3. Suppose f L 2 < γ 2 .

(3.17)

ξ

Then there exists δ1 > 0 such that for any γ ∈ (0, δ1 ),  t  t e−γ (t−s) (f, |ξ 1 |f)− ds ≤ e−γ (t−s) (b+ , |ξ 1 |b+ )+ ds + e−γ t x=0 0 0  ∞ × eγ x (f0 , f0 )d x. (3.18) 0

Proof. From (3.4), we get  ∞   eγ x f · ∂t f + ξ 1 ∂x f − Lf − (f) + γ P+0 ξ 1 f dξ d x 0= R3 0  ∞  ∞  γ  1 γx d e (f, f)d x + eγ x − (f, ξ 1 f) + (f, γ P+0 ξ 1 f) − (f, Lf)−(f, (f)) d x = dt 0 2 2 0 1 . (3.19) − (f, ξ f) x=0

Lemma 2.4 and macro-micro decomposition yield the estimate of the nonlinear term 1

1

− (f, (f)) ≥ −(1 + |ξ |) 2 f L 2 (1 + |ξ |)− 2 (f) L 2 ξ

≥

ξ

−Cf L 2 (P1 f, (1 + |ξ |)P1 f) − Cf3L 2 . ξ ξ

(3.20)

A priori smallness assumption and (2.5) yield that γ − (f, ξ 1 f) + (f, γ P+0 ξ 1 f) − (f, Lf) − (f, (f)) 2 γ γ γ (P0 f, P0 f) + (P1 f, (1 + |ξ |)P1 f) > (f, f), ≥ 2 2 2 where  is the constant in Lemma 2.2. Equations (3.21) and (3.19) give that for any t > 0,  ∞  ∞ d eγ x (f, f)d x + γ eγ x (f, f)d x − (f, ξ 1 f) ≤ 0. x=0 dt 0 0 This inequality gives the estimate for any t > 0,  ∞  t eγ t eγ x (f, f)d x + eγ s (f, −ξ 1 f) 0

0

 x=0

∞

ds ≤

(3.22)

eγ x (f0 , f0 )d x. (3.23)

0

Therefore we get the estimate (3.18) for the full boundary data.

(3.21)

 

Pointwise Convergence to Knudsen Layers of the Boltzmann Equation

311

With the full boundary data f(0, t, ξ ), we can study the propagation of it into the interior through the integration 

t

I D (x, t) ≡

G Di (x, t − s)[ξ 1 f](0, s)ds.

0

Similar to the decomposition of Gi in Theorem 2.1,   G Di (x, t, ξ, ξ∗ ) ≡ δ 1 (x − ξ∗1 t)δ 3 (ξ − ξ∗ )e−ν(ξ∗ )t + j1D (x, t, ξ, ξ∗ ) + j2D (x, t, ξ, ξ∗ ) +r D (x, t, ξ, ξ∗ ),

(3.24)

where ⎧ 1 1 ⎪ ⎨0 if (x − ξ∗ t)(ξ t − x) < 0, ˜ j1D (x, t, ξ ; ξ∗ ) = e−ν(ξ )(t−t1 )−ν(ξ∗ )t1 K(ξ, ξ∗ ) ⎪ else if x − ξ∗ t
= 0, ⎩ 1 1 (ξ − ξ0 ) (x−ξ∗1 t) ˜ + 1 where t1 = t − 1 , K(ξ, ξ∗ ) = K(ξ, ξ∗ ) − γ P ξ , 0 ∗

ξ 1 −ξ0

j2D (x, t, ξ ; ξ∗ )

r D (x, t) L 2 ξ

= O(1)e

−

ν0 (t+|x|) 3

⎛ ) ≤ C ⎝ λ <0 i

−

 |ξ − ξ∗ |−1 (1 + |ξ |)−1 for |ξ − ξ∗ | ≤ 1, (1 + |ξ |)−1 for |ξ − ξ∗ | ≥ 1,

|x−λi t|2 C0 (t+1)

e√

(t+1)

+ e−α0

) γt

−

λi >0

|x−λi t|2 C0 (t+1)

e√

(t+1)

(3.25)

⎞ + e−(|x|+t)/C1 ⎠ .

This decomposition is obtained by Picard’s iteration:  (∂t + ξ 1 ∂x + ν(ξ ))h0D = 0, h0D (x, 0, ξ ) = δ 1 (x)δ 3 (ξ − ξ∗ ), 





˜ 0, (∂t + ξ 1 ∂x + ν(ξ ))j1D = Kh D j1D (x, 0, ξ ) = 0, ˜ 1, (∂t + ξ 1 ∂x + ν(ξ ))j2D = Kj D 2 j D (x, 0, ξ ) = 0, ˜ 2, (∂t + ξ 1 ∂x − L)r D = Kj D r D (x, 0, ξ ) = 0.

˜ we can get By solving the equations directly and from the properties of the operator K, (3.25). We omit the details. Thus the function I D (x, t) can be decomposed as follows: I D (x, t) = I D0 (x, t) + I D1 (x, t) + I D2 (x, t) + R D (x, t).

(3.26)

312

S. Deng, W. Wang, S.-H. Yu

For x, t > 0, I D0 (x, t, ξ ) ≡

I D1 (x, t, ξ ) ≡ I D2 (x, t, ξ ) ≡ R D (x, t, s, ξ ) ≡

⎧ 1 ⎪ ⎨0 if ξ x < 0, −ν(ξ ) 1 ξ b (ξ ) if {0 < x < ξ 1 t} ∩ {ξ 1 > 0}, e + ⎪ ⎩ 0 if {x > ξ 1 t} ∩ {ξ 1 > 0 }.  t j1D (x, t − s, ξ, ξ∗ )ξ∗1 f(0, s, ξ∗ )dξ∗ ds, 0 R3  t j2D (x, t − s, ξ, ξ∗ )ξ∗1 f(0, s, ξ∗ )dξ∗ ds, 0 R3  t r D (x, t − s, ξ, ξ∗ )ξ∗1 f(0, s, ξ∗ )dξ∗ ds. 0

R3

(3.27)

From (3.27), we have

 −ν(ξ ) x    ξ1 b  I D0 (x, t) L 2 ≤ e + ξ

L 2ξ

≤ Ce−ν0 |x| b+  L 2 ≤ Cδe−|x|/C . ξ

(3.28)

Also from (3.27), I D1 (x, t, ξ )  t = 0

(x−ξ 1 (t−s))(x−ξ∗1 (t−s))<0

˜ e−ν(ξ )(t−s−τ1 )−ν(ξ∗ )τ1 K(ξ, ξ∗ ) 1 ξ∗ f(0, s, ξ∗ )dξ∗ ds, 1 1 ξ − ξ∗ (3.29)

where x − ξ∗1 (t − s) . ξ 1 − ξ∗1 In the special domain for integration, ξ 1 − ξ∗1 has a non-zero lower bound: x − ξ 1 (t − s) x 1 1 1 = − ξ . ξ − ξ∗ ≥ t −s t −s τ1 = t − s −

(3.30)

(3.31)

Note that from (3.30), x = ξ 1 (t − s − τ1 ) + ξ∗1 τ1 . Thus from ν(ξ ) ∼ 1 + |ξ |, we have ν(ξ )(t − s − τ1 ) + ν(ξ∗ )τ1 ≥ ν0 ((1 + |ξ |)(t − s − τ1 ) + (1 + |ξ∗ |)τ1 ) ≥ ν0 (t − s + |x|), (3.32) which implies e−ν(ξ )(t−s−τ1 )−ν(ξ∗ )τ1 ≤ e−ν0 (t−s+|x|) .

(3.33)

We divide the domain {(ξ∗ , s) : s ∈ (0, t), (x − s))(x − s)) < 0} into two regions:
* 1 H1 = (ξ∗ , s) : s ∈ (0, t), (x − ξ 1 (t − s))(x − ξ∗1 (t − s)) < 0, |ξ 1 − ξ∗1 | ≥ |ξ∗1 | , 2 − ξ 1 (t

− ξ∗1 (t

Pointwise Convergence to Knudsen Layers of the Boltzmann Equation

313


* 1 H2 = (ξ∗ , s) : s ∈ (0, t), (x − ξ 1 (t − s))(x − ξ∗1 (t − s)) < 0, |ξ 1 − ξ∗1 | < |ξ∗1 | , 2 and denote   I1D1 =

˜ e−ν(ξ )(t−s−τ1 )−ν(ξ∗ )τ1 K(ξ, ξ∗ ) 1 ξ∗ f(0, s, ξ∗ )dξ∗ ds, 1 1 ξ − ξ∗ ˜ e−ν(ξ )(t−s−τ1 )−ν(ξ∗ )τ1 K(ξ, ξ∗ ) 1 ξ∗ f(0, s, ξ∗ )dξ∗ ds. 1 1 ξ − ξ∗

H1

  I2D1 =

H2

Then

In H1 ,

|I D1 (x, t, ξ )| ≤ I1D1 (x, t, ξ ) + I2D1 (x, t, ξ ) . |ξ∗1 | |ξ 1 −ξ∗1 |

(3.34)

≤ 2,  

I1D1 (x, t, ξ )2

˜ e−ν(ξ )(t−s−τ1 )−ν(ξ∗ )τ1 K(ξ, ξ∗ ) 1 ξ∗ f(0, s, ξ∗ )dξ∗ ds 1 1 ξ − ξ∗

≤ H1

  ≤

2

e−ν(ξ )(t−s−τ1 )−ν(ξ∗ )τ1 ˜ |K(ξ, ξ∗ )ξ∗1 |dξ∗ ds 3 1 1 H1 |ξ − ξ∗ | 2   −ν(ξ )(t−s−τ1 )−ν(ξ∗ )τ1 |K(ξ, ˜ e ξ∗ )| 1 2 · |ξ∗ f (0, s, ξ∗ )|dξ∗ ds . 1 H1 |ξ 1 − ξ∗1 | 2 (3.35) ˜

∗) After a straightforward calculation, we can find that √K(ξ,ξ ∈ L 1ξ∗ . Thus 1 1

|ξ −ξ∗ |

 

e−ν(ξ )(t−s−τ1 )−ν(ξ∗ )τ1 ˜ |K(ξ, ξ∗ )ξ∗1 |dξ∗ ds 3 1 1 2 H1 |ξ − ξ∗ |   ˜ |K(ξ, ξ∗ )| ≤ e−ν0 (t−s) dξ∗ ds ≤ C. 1 1 H1 |ξ − ξ∗1 | 2

(3.36)

˜

∗) The L 1 property of √K(ξ,ξ together with (3.35), (3.36) and (3.18) yields that 1 1

|ξ −ξ∗ |

 R3

I1D1 (x, t, ξ )2 dξ

≤ Ce

−ν0 |x|



t

e



t

≤C 0

2    e−ν0 (t−s+|x|)  |ξ 1 |f(0, s) 2 ds

−γ (t−s)

(b+ , |ξ |b+ )+ ds + e 1

−γ t

0 2 −ν0 |x|

≤ Cδ e and so



Lξ

∞

 e (f0 , f0 )d x γx

0

,

(3.37)    1 I D1 (x, t)

L 2ξ

≤ Cδe−|x|/C .

(3.38)

314

S. Deng, W. Wang, S.-H. Yu

In the domain H2 , |ξ∗1 | − |ξ 1 | ≤ |ξ 1 − ξ∗1 | ≤ 21 |ξ∗1 | ⇒ |ξ∗1 | ≤ 2|ξ 1 |. Since K˜ gains decaying rate in ξ variable,   e−ν(ξ )(t−s−τ1 )−ν(ξ∗ )τ1 ˜ |K(ξ, ξ∗ )ξ∗1 |dξ∗ ds 3 H2 |ξ 1 − ξ∗1 | 2  2|ξ 1 | ≤ e−ν0 (t−s+|x|) ds 1 {s |s∈(0,t)∩|x−ξ 1 (t−s)|≤1 } |x − ξ 1 (t − s)| 2  1 + e−ν0 (t−s+|x|) ds 1 1 1 {s |s∈(0,t)∩|x−ξ (t−s)|≥1 } |x − ξ (t − s)| 2  2|ξ 1 | ≤ e−ν0 |x| ds 1 {s |s∈(0,t)∩|x−ξ 1 (t−s)|≤1 } |x − ξ 1 (t − s)| 2  + e−ν0 (t−s+|x|) ds ≤ Ce−ν0 |x| . (3.39) {s |s∈(0,t)∩|x−ξ 1 (t−s)|≤1 } ˜ K(ξ,ξ ∗)

The L 1 property of   2  I D1 (x, t)

L 2ξ

1

⎛  =⎝  ≤

|ξ 1 −ξ∗1 | 2

together with (3.39) yields that

 

R3

H2

 

R3

H2

 

˜ ξ∗ ) 1 e−ν(ξ )(t−s−τ1 )−ν(ξ∗ )τ1 K(ξ, ξ∗ f(0, s, ξ∗ )dξ∗ ds ξ 1 − ξ∗1

1

H2

≤ Ce

−ν0 |x|

2

dξ ⎠

e−ν(ξ )(t−s−τ1 )−ν(ξ∗ )τ1 ˜ |K(ξ, ξ∗ )ξ∗1 |dξ∗ ds 3 |ξ 1 − ξ∗1 | 2

˜ e−ν(ξ )(t−s−τ1 )−ν(ξ∗ )τ1 |K(ξ, ξ∗ )|

·

⎞1

2

   R3

|ξ 1 − ξ∗1 | 2

1 2

|ξ∗1 f2 (0, s, ξ∗ )|dξ∗ ds

˜ e−ν(ξ )(t−s−τ1 )−ν(ξ∗ )τ1 |K(ξ, ξ∗ )| 1

|ξ 1 − ξ∗1 | 2

H2

≤ Cδe−|x|/C .

dξ 1 2

|ξ∗1 f2 (0, s, ξ∗ )|dξ∗ dsdξ (3.40)

From (3.38) and (3.40),

    I D1 (x, t) L 2 ≤ I1D1 (x, t) ξ

L 2ξ

    + I2D1 (x, t)

L 2ξ

≤ Cδe−|x|/C .

(3.41)

1 Since j2D doesn’t contain the strong singularity |ξ 1 −ξ 1 | and only contains an integrable

singularity

1 |ξ −ξ∗ | ,

∗

it also gains decay in ξ variable. By direct computations, 

I D2 (x, t) L 2 ≤ C ξ

t

e

 21  2  1 ≤ Cδe−|x|/C .  |ξ |f(0, s) 2 ds

−ν0 (t−s+|x|) 

Lξ

0

(3.42)

r D doesn’t contain any singularity while contains an extra decaying rate in ξ . Thus from (3.18) and the Schwartz inequality, we get  t  (x−λ j (t−s))2  1   R D (x, t) L 2 ≤ C e− C(t−s+1)  |ξ 1 |f(0, s) 2 ds e−α0 γ (t−s) √ ξ Lξ t − s + 1 0 λ >0 j

≤ Cδe

−γ |x|/C

.

(3.43)

Pointwise Convergence to Knudsen Layers of the Boltzmann Equation

315

Equations (3.28), (3.41), (3.42) and (3.43) result in the estimate of the function I D (x, t): I D (x, t) L 2 ξ

 t    1  =  G Di (x, t − s)[ξ f](0, s)ds   0

L 2ξ

≤ Cδe−γ |x|/C .

(3.44)

3.3. Nonlinear term. From (3.16) and (3.44), we already have the estimates for the initial and boundary terms. There exists some constant A0 > 0 such that  ∞    A0 −γ |x|/C  G Di (x − y, t)f0 (y)dy  ,   2 ≤ 3 δe 0 Lξ  t     G Di (x, t − s)[ξ 1 f](0, s)ds  ≤ A0 δe−γ |x|/C .   2 3 0

(3.45) (3.46)

Lξ

Thus we make ansatz assumptions: f(x, t) L 2 ≤ C0 δe−γ |x|/C ,

(3.47)

f(x, t) L ∞ ≤ C1 δe−γ |x|/C , ξ,3

(3.48)

ξ

where C0 > A0 and C1 > 0 depends on C0 . Under such assumptions, the nonlinear term (f) can be controlled by  (f) L 2 ≤ Cδ 2 e−γ |x|/C ,

(3.49)

ξ

where C is independent of δ. The representation (3.12) of f(x, t, ξ ) and (3.45), (3.46) yield that   f(x, t) L 2 =   ξ

∞ 0

  G Di (x − y, t)f0 (y)dy  

 t   + 

L 2ξ

 t    1  + G (x, t − s)[ξ f](0, s)ds Di   0

 ∞  G Di (x − y, t − s) (f)(y, s)dyds   2 0 0 Lξ  t  ∞    2 A0 −γ |x|/C  δe ≤ + G Di (x − y, t − s) (f)(y, s)dyds   2. 3 0 0 L

L 2ξ

ξ

(3.50) To verify (3.47), we just have to proof  t     0

0

∞

  G Di (x − y, t − s) (f)(y, s)dyds  

L 2ξ



A0 −γ |x|/C . δe 3

(3.51)

316

S. Deng, W. Wang, S.-H. Yu

t ∞ Substitute (3.49) into 0 0 Gi (x − y, t − s) (f)(y, s)dyds,   t  ∞    G Di (x − y, t − s) (f)(y, s)dyds    0

≤C

L 2ξ

0

 t 0

∞

e−α0 γ (t−s)

0



t

≤ Cδ 2

e−α0 γ (t−s)

0

e

(x−y−λ j (t−s))2 − C(t−s+1)

√

λ j >0



t −s+1

δ 2 e−γ |y|/C dyds

e−γ |x−λ j (t−s)|/C ds ≤

λ j >0

C 2 −γ |x|/C A0 −γ |x|/C

, δ e δe γ 3 (3.52)

since δ is sufficiently small and γ is independent of δ. Thus we’ve proved (3.51) and justified the ansatz (3.47). To verify (3.48), we need another representation of f(x, t, ξ ),  t    ˜ + (f) x − ξ 1 (t − s), s, ξ ds, f(x, t, ξ ) = e−ν(ξ )t f0 (x − ξ 1 t, ξ )+ e−ν(ξ )(t−s) Kf 0

(3.53) ˜ = Kf − γ P+ ξ 1 f. Since K˜ gains decay in the ξ variable, where Kf 0  t  t ˜ 1 ˜ e−ν(ξ )(t−s) Kf(x ≤C (t − s), s)ds e−ν(ξ )(t−s) Kf(x − ξ 1 (t − s), s) ds − ξ 0 0  t −ν(ξ )(t−s) 1 ≤C e f(x − ξ (t − s), x) L 2 ds ξ 0  t δ −γ |x|/C 1 e ≤ CC0 δ e−ν(ξ )(t−s) e−γ |x−ξ (t−s)|/C ds ≤ C . (3.54) ν(ξ ) 0 Thus, consider sup eγ |x|/C f(x, t, ξ ) x,t

 t γ |x|/C−ν(ξ )t 1 γ |x|/C−ν(ξ )(t−s) ˜ 1 ≤ sup e f0 (x − ξ t, ξ ) + sup e Kf(x −ξ (t −s), s)ds x,t x,t 0  t + sup eγ |x|/C−ν(ξ )(t−s) (f)(x − ξ 1 (t − s), s)ds x,t 0  t Cδ Cδ −ν(ξ )(t−s)+γ |ξ 1 |(t−s) γ |x−ξ 1 (t−s)| 1 ≤ +sup + e δ|ξ |e f(x −ξ (t −s), s)ds (1+|ξ |)3 (1+|ξ |) x,t 0  t Cδ Cδ 1 γ |z|/C ≤ + δ sup + sup f(z, s, ξ ) e−ν(ξ )(t−s)+γ |ξ |(t−s) |ξ |ds. e 3 (1 + |ξ |) (1 + |ξ |) x,t z,s 0 (3.55) Since ν(ξ ) ∼ 1 + |ξ | and γ is a small parameter, we can get Cδ Cδ γ |x|/C e + f(x, t, ξ ) sup eγ |x|/C f(x, t, ξ ) ≤ + δ sup , (3.56) (1 + |ξ |)3 (1 + |ξ |) x,t x,t

Pointwise Convergence to Knudsen Layers of the Boltzmann Equation

317

which results in sup eγ |x|/C f(x, t, ξ ) ≤ x,t

2Cδ 2Cδ . + (1 + |ξ |)3 (1 + |ξ |)

(3.57)

Repeat the same argument for another two times. One will get sup eγ |x|/C f(x, t, ξ ) ≤ x,t

Cδ . (1 + |ξ |)3

(3.58)

Thus the ansatz assumption (3.48) holds. Until now, we’ve proved the a priori smallness assumption (3.17) and the fact that f(x, t, ξ ) is bounded under the norms || · || L 2 and || · || L ∞ . In the rest of this section, ξ,3 ξ we will check whether ft has a decaying rate fast enough to ensure the existence of limt→∞ f (x, t, ξ ). From (3.4),   0 = ft + ξ 1 fx − Lf − (f) + γ P+0 ξ 1 f = ftt + ξ 1 fxt − Lft − ( (f))t + γ P+0 ξ 1 ft . t

Set g = ft , then we get the equation about g, ⎧ gt + ξ 1 gx − Lg = ( (f))t − γ P+0 ξ 1 g, (x, t, ξ ) ∈ R+ × R+ × R3 , ⎪ ⎪ ⎪ ⎨g(x, 0, ξ ) = g0 (x, ξ ), g(0, t, ξ )|ξ 1 >0 = 0, ⎪ ⎪ ⎪ ⎩ lim g(x, t, ξ ) = 0,

(3.59)

x→∞

where  ( (f))t ≡

√ Q( Mf) √ M

= t

2B

√ √  Mf, Mg . √ M

From (3.7), the initial data g0 (x, ξ ) = ft (x, t, ξ )|t=0 satisfies sup(1 + |ξ |)3 |g0 | ≤ Cδe−|x|/C . ξ

Lemma 2.4 yields that the nonlinear term ( (f))t can be controlled by     ( (f))t  2 ≤ C f 2 (1 + |ξ |)g 2 + (1 + |ξ |)f 2 g 2 ≤ Cδe−γ |x|/C g L ∞ . L L L L L ξ,3 ξ

ξ

ξ

ξ

ξ

(3.60) Similar to what we have done with (3.4), then one can get that there exists some constant C which is independent of δ such that ||g(x, t)|| L 2 ≤ Cδe−γ (|x|+t)/C ,

(3.61)

||g(x, t)|| L ∞ ≤ Cδe−γ (|x|+t)/C . ξ,3

(3.62)

ξ

318

S. Deng, W. Wang, S.-H. Yu

t Therefore f(x, t, ξ ) = f0 (x, ξ ) + 0 g(x, s, ξ )ds gains exponential decaying rate in t. As a result, f(x, t, ξ ) does converge to ψ(x, ξ ) as t tends to infinity. In other words, the boundary layer solution exists and satisfies ||ψ(x)|| L 2 ≤ Cδe−γ |x|/C ,

(3.63)

≤ Cδe−γ |x|/C . ||ψ(x)|| L ∞ ξ,3

(3.64)

ξ

Therefore we have proved Lemma 3.1. When M < −1, the projection operator P+0 ξ 1 ≡ 0. Therefore, the damped system is just the original system. Then the fact that the solution f(x, t, ξ ) of (3.4) converges to the boundary layer solution ψ(x, ξ ) shows the stability of the boundary layer. 4. Coupling Waves We shall introduce some integrals which will be used in the following two chapters. Denote  A1 (x, t, ξ ; λ j ) ≡

t

√

0

1 s+1

e−

(x−ξ 1 (t−s)−λ j s)2 C(s+1)

e−ν(ξ )(t−s) ds,

A2 (x, t, ξ ; λk , λk+1 )  ≡ {s |s∈(0,t),x−ξ 1 (t−s)∈(λk s,λk+1 s) } e−ν(ξ )(t−s) ds, × (x − ξ 1 (t − s) − λk s + 1)(λk+1 s − x + ξ 1 (t − s) + 1)

(4.1)

(4.2)

for j = 1, 2, 3 and k = 1, 2. Lemma 4.1. For x, t > 0, 2

(x−λ j t) 1 1 e− C(t+1) , √ ν(ξ ) t + 1

A1 (x, t; λ j ) ≤ C

(4.3)

for j = 1, 2, 3. Proof. Consider A1 (x, t, ξ ; λ j ):  A1 (x, t, ξ ; λ j ) ≤

{s ||x−λ j s |>2|

ξ1



+

|(t−s) }

{s ||x−λ j s |<2|ξ 1 |(t−s) }

1

e−

(x−λ j s)2 4C(t+1)

e−ν(ξ )(t−s) ds s+1 |x−λ j s| ν(ξ )(t−s) 1 e− 4 e− 2 ds √ s+1

√

2

(x−λ j t) 1 1 ≤ e− C(t+1) . √ ν(ξ ) t + 1

Thus we get the estimate (4.3).

 

(4.4)

Pointwise Convergence to Knudsen Layers of the Boltzmann Equation

319

Lemma 4.2. For x, t > 0, A2 (x, t; λk , λk+1 ) ⎛ ⎞ (x−λ j t)2 3 − C(1+t) 1 ⎜ e ⎟ ≤C + e−(|x|+t)/C ⎠ √ ⎝ ν(ξ ) t +1 j=1 ⎧ 2 ⎪ 1 1 ⎨ for x ∈ (λ1 t, λ3 t), √ +C (|x − λl t| + 1)(|x − λl+1 t| + 1) l=1 ν(ξ ) ⎪ ⎩ 0 for x ∈ (0, λ1 t) ∪ (λ3 t, ∞).

(4.5)

for k = 1, 2. Proof. The estimate of A2 (x, t, ξ ; λk , λk+1 ) should be considered in different regions of x. When λk t < x < λk+1 t,  t−5logt e−ν(ξ )(t−s) ds A2 (x, t, ξ ; λk , λk+1 ) ≤ 0  t 1 −ν(ξ )(t−s) + ds e √ (x − λ t + 1)(λ t−5logt k k+1 t − x + 1) 1 1 ≤ . √ ν(ξ ) (x − λk t + 1)(λk+1 t − x + 1)

(4.6)

When x > λk+1 t, in the special integral domain of A2 (x, t, ξ ; λk , λk+1 ), 0 > x − ξ 1 (t − s) − λk+1 s > λk+1 (t − s) − ξ 1 (t − s). Thus |x − ξ 1 (t − s) − λk+1 s| < C|ξ |(t − s) which implies  t (x−λk+1 t)2 ν(ξ )(t−s) 1 1 1 1 A2 (x, t, ξ ; λk , λk+1 ) ≤ √ e− 2 −|x−ξ (t−s)−λk+1 s|/C ds ≤ e− C(t+1) . √ ν(ξ ) t +1 0 t +1 When x < λk t, in the special integral domain of A2 (x, t, ξ ; λk , λk+1 ), 0 < x − ξ 1 (t − s) − λk s < λk (t − s) − ξ 1 (t − s). Thus |x − ξ 1 (t − s) − λk s| < C|ξ |(t − s) which implies  t (x−λk t)2 ν(ξ )(t−s) 1 1 1 1 A2 (x, t, ξ ; λk , λk+1 ) ≤ √ e− 2 −|x−ξ (t−s)−λk s|/C ds ≤ e− C(t+1) . √ ν(ξ ) t +1 t +1 0 Then we complete the proof.

 

320

S. Deng, W. Wang, S.-H. Yu

Denote J1 (x, t; λi , λ j ) ≡

 t 0

∞

0

 t

2

(x−y−λi (t−s))2 j s) 1 1 − (y−λ e− C(t−s+1) e C(s+1) dyds, (4.7) t −s+1 s+1

λk+1 s

2

(x−y−λ j (t−s)) 1 e− C(t−s+1) (t − s + 1) 0 λk s 1 × dyds, (4.8) (|y − λk s| + 1)(|y − λk+1 s| + 1)  t ∞ (y−λ j s)2 (x−y−λi (t−s))2 1 1 e−γ y/C e− C(s+1) dyds, e− C(t−s+1) √ J1 (x, t; λi , λ j ) ≡ s+1 0 0 t −s +1 (4.9)  t  λ3 s (x−y−λ j (t−s))2 1 J2 (x, t; λ j , λk , λk+1 ) ≡ e− C(t−s+1) (t − s + 1) 0 0 e−Cγ y ×√ dyds, (4.10) (|y − λk s| + 1)(|y − λk+1 s| + 1)

J2 (x, t; λ j , λk , λk+1 ) ≡

for i, j = 1, 2, 3 and k = 1, 2. The estimates of J1 (x, t; λi , λ j ) and J2 (x, t; λ j , λk , λk+1 ) can be gotten from [26]. Lemma 4.3. J1 (x, t; λi , λ j ), J2 (x, t; λ j , λk , λk+1 ) ⎛ ⎞ (x−λ j t)2 3 − C(1+t) e ⎜ ⎟ ≤ C ⎝ + e−(|x|+t)/C ⎠ √ t +1 j=1 ⎧ 2 ⎪ 1 ⎨ √ for x ∈ (λ1 t, λ3 t), +C (|x − λl t| + 1)(|x − λl+1 t| + 1) l=1 ⎪ ⎩ 0 for x ∈ (0, λ1 t) ∪ (λ3 t, ∞).

(4.11)

for i, j = 1, 2, 3 and k = 1, 2. The following lemmas give explicit estimates for J1 (x, t; λi , λ j ) and J2 (x, t; λ j , λk , λk+1 ): Lemma 4.4. There exists some constant C > 0 which depends on γ such that ⎞ ⎛ (x−λ j t)2 3 − C(1+t) e ⎟ ⎜ J1 (x, t; λi , λ j ) ≤ C ⎝ + e−(|x|+t)/C ⎠ , √ t +1 j=1 for i, j = 1, 2, 3.

(4.12)

Pointwise Convergence to Knudsen Layers of the Boltzmann Equation

321

Proof. J1 (x, t; λi , λ j ) can be rewritten to be  t  λ j s/2 (x−y−λi (t−s))2 1 J1 (x, t; λi , λ j ) ≤ C e−Cγ y e− C(t−s+1) e−Cs dyds √ (t − s + 1) s + 1 0 0  t ∞ (x−y−λi (t−s))2 1 +C e−Cγ y/2+Cγ λ j s/4 e− C(t−s+1) dyds √ 0 λ j s/2 (t −s +1) s + 1  t ∞ (x−y−λi (t−s))2 1 ≤C e−Cγ |y|−Cγ s− C(t−s+1) dyds √ (t − s + 1) s + 1 0 0  t 1 e−Cγ |x−λi t| e−Cγ s−Cγ |x−λi (t−s)| ds ≤ C √ ≤C √ (t − s + 1)(s + 1) t +1 0 (x−λi t)2

e− C(t+1) . ≤C √ t +1

(4.13)

  Consider K α,β,σ (x, t; 0, t; λ, D) ≡

 t 0

∞ −∞

×e− Set α (t) ≡

 0

t

(t − s)(−β+σ )/2 (t − s + 1)−σ/2

(x−y−λ(t−s))2 D(t−s)

(s + 1)−α/2 e−

4γ |y| D

dyds.

(4.14)

⎧ ⎪ ⎨1 for α > 2, −α/2 (s + 1) ds = O(1) log(t + 1) for α = 2, ⎪ ⎩(t + 1)(2−α)/2 for α < 2. (4.15)

The following lemma is quoted from [22] and will be used to get the estimate of J2 (x, t; λ j , λk , λk+1 ). Lemma 4.5. Suppose that α ≥ 0, 3 > β −σ ≥ 0, σ ≥ 0 and λ a positive constant. Then there exists a positive constant C such that for any fixed positive constant E > D+O(1)γ and a bound O(1) independent of γ , K α,β,σ (x, t; 0, t; λ, D) = O(1) √ ⎧ (x + 1)(−β+2)/2 (λt − x)−α/2 (1 + γ x + 1)−1 + β−1 (γ −1 )(t + 1)−α/2 e−Cγ x ⎪ ⎪ √ ⎪ ⎪ ⎪ + α (γ −1 )(t + 1)(−β+1)/2 e−Cγ |λt−x| for 0 < x ≤ λ(t + 1) − t + 1, ⎪ √ √ ⎪ ⎪ ⎪ (t + 1)(−β+1)/2 [ α (t + 1)e−Cγ t + α ( t + 1)(1 + γ t)−1 ] ⎪ ⎪ √ ⎪ ⎪ ⎨+(t + 1)−α/2 β−1 (t + 1)e−Cγ t for |x − λ(t + 1)| ≤ t + 1, (x−λt)2 √ √ · ⎪ (t + 1)(−β+1)/2 [ α ( t + 1)(1 + γ t)−1 e− D(t+1) +(t +1)1/2 (x −λt)−α/2 e−Cγ (x−λt) ⎪ ⎪ ⎪ 4γ (x−λt) 2 ⎪ ⎪ + α (γ −1 )e−Cγ (x−λt) e−Cγ t + γ (α−2)/2 e− D ] ⎪ ⎪ ⎪ ⎪ ⎪ +(x − λt)(−α−2β+3)/2 e−Cγ (x−λt) + (t + 1)−α/2 ( β−1 (γ −1 ) + β−1 (t + 1)) ⎪ ⎪ √ 4γ (x−λt) ⎩ for x > λ(t + 1) + t + 1. e−Cγ t e− D (4.16)

322

S. Deng, W. Wang, S.-H. Yu

Lemma 4.6. There exists some constant C > 0 which depends on γ such that J2 (x, t; λ j , λk , λk+1 ) ⎛ ⎞ (x−λ j t)2 3 − ⎜ e C(1+t) ⎟ ≤C⎝ + e−(|x|+t)/C ⎠ √ t + 1 j=1 ⎧ 2 ⎪ 1 ⎨ √ for x ∈ (0, λ3 t), +C (|x − λ t| + 1)(|x − λl+1 t| + 1) l l=1 ⎪ ⎩ 0 for x ∈ (λ3 t, ∞).

(4.17)

for j = 1, 2, 3 and k = 1, 2. Proof. Similar to J1 (x, t; λi , λ j ), we’ll find J2 (x, t; λ j , λk , λk+1 )  t  λ3 s/2 (x−y−λ j (t−s))2 1 1 −Cγ y e− C(t−s+1) e ≤C dyds t − s + 1 s + 1 0 0  t  λ3 s (x−y−λ j (t−s))2 1 e− C(t−s+1) e−Cγ y/2−Cγ λ3 s/4 dyds +C 0 λ3 s/2 t − s + 1  t ∞ (x−y−λ j (t−s))2 1 e− C(t−s+1) e−Cγ y dyds. ≤C (t − s + 1)(s + 1) 0 0

(4.18)

When λ j > 0, J2 (x, t; λ j , λk , λk+1 ) ≤ O(1)K 2,2,2 (x, t; 0, t; λ j , C) ⎛ ⎞ (x−λ j t)2 3 − C(1+t) e ⎜ ⎟ ≤C⎝ + e−(|x|+t)/C ⎠ √ t +1 j=1 ⎧ 2 ⎪ 1 ⎨ √ for x ∈ (0, λ3 t), +C (|x − λl t| + 1)(|x − λl+1 t| + 1) l=1 ⎪ ⎩ 0 for x ∈ (λ3 t, ∞). When λ j < 0,

(4.19)



t e−Cγ |x−λ j (t−s)| e−Cγ |x| J2 (x, t; λ j , λk , λk+1 ) ≤ C ds ≤ C √ t +1 t − s + 1(s + 1) 0 ⎞ ⎛ (x−λ j t)2 3 − ⎟ ⎜ e C(1+t) ≤C⎝ + e−(|x|+t)/C ⎠ √ t + 1 j=1

+C  

⎧ 2 ⎪ ⎨ √

1 for x ∈ (0, λ3 t), (|x − λl t| + 1)(|x − λl+1 t| + 1)

⎪ ⎩ l=1 0 for x ∈ (λ3 t, ∞).

(4.20)

Pointwise Convergence to Knudsen Layers of the Boltzmann Equation

323

5. Stability of Knudsen’s Layers for M > 1 When M > −1, the damped system is no longer the original system. To study the stability problem, we treat the problem in the perturbation form. Set w(x, t, ξ ) = f(x, t, ξ ) − ψ(x, ξ ). Then w(x, t, ξ ) satisfies ⎧ wt + ξ 1 wx − Lw = L δ w + (w), (x, t, ξ ) ∈ R+ × R+ × R3 , ⎪ ⎪ ⎪ ⎨w(x, ξ ) = w0 (x, ξ ) = f0 (x, ξ ) − ψ(x, ξ ), w(0, t, ξ )|ξ 1 >0 = 0, ⎪ ⎪ ⎪ ⎩ lim w(x, t, ξ ) = 0,

(5.1)

x→∞

where

Lδw ≡

B

√

Mw,

√

 √  √ Mψ + B Mψ, Mw , √ M

and the initial data w0 (x, ξ ) satisfies sup(1 + |ξ |)3 |w0 (x, ξ )| ≤ C e−|x|/C , ξ

(5.2)

for some positive constant C independent of . As stated in Sect. 1, we’ll prove Theorem 1.1 under the assumption M > 1 in this section. Suppose M > 1. Then w(x, t, ξ ) can be represented directly by the Green’s function Gi (x, t) of the initial value problem:  w(x, t) =

 t Gi (x − y, t)w0 (y)dy + Gi (x, t − s)[ξ 1 w](0, s)ds 0 0  t ∞ + Gi (x − y, t − s)(L δ w + (w))(y, s)dyds. ∞

0

(5.3)

0

From Theorem 2.2,    

  Gi (x − y, t)w0 (y)dy   2 0 Lξ ⎛ ⎞ 2 (x−y−λ t)  ∞ 3 − C(t+1)j e ⎜ ⎟ ≤ C + e−(|x−y|+t)/C ⎠ e−|y|/C dy √ ⎝ t + 1 0 j=1 ⎞ ⎛ (x−λ j t)2 3 − C(t+1) e ⎟ ⎜ ≤ C ⎝ + e−(|x|+t)/C ⎠ . √ t +1 j=1 ∞

(5.4)

324

S. Deng, W. Wang, S.-H. Yu

5.1. Energy estimates on the boundary data. Similar to that in Sect. 3.2, we apply the weighted energy method to get the full boundary data. The weighted function is chosen to be e−βx , where β is a positive and sufficiently small constant. Lemma 5.1. Suppose the solution w(x, t, ξ ) of (5.1) satisfies w L 2 < β 2 ,

(5.5)

ξ

then there exists δ2 > 0 such that for any β ∈ (0, δ2 ),  t  ∞ eβt e−βx (w, w)d x + eβs (w, |ξ 1 |w)− ds x=0 0 0  ∞ e−βx (w0 , w0 )d x. ≤ 0

Proof. Following the process in Sect. 3.2, we only have to check whether L δ w)d x is small enough. Since     1 1     − (w, L δ w) ≥ −C (1 + |ξ ) 2 w 2 (1 + |ξ |)− 2 L δ w 2 Lξ

∞ 0

(5.6) e−βx (w,

Lξ

≥ −Cδ ((P1 w, (1 + |ξ |)P1 w) + (P0 w, P0 w)) and δ is independent of β, one can choose δ small enough and complete the process. Thus from the energy estimate, we have the full boundary data w(0, t, ξ ) and also the estimate of w(x, t, ξ ).   With the full boundary data, we now study how it propagates into the interior through the boundary integration  t I(x, t) ≡ Gi (x, t − s)[ξ 1 w](0, s)ds. 0

From Theorem 2.1,   Gi (x, t, ξ, ξ∗ ) ≡ δ 1 (x − ξ∗1 t)δ 3 (ξ − ξ∗ )e−ν(ξ∗ )t + j1 (x, t, ξ, ξ∗ ) + j2 (x, t, ξ, ξ∗ ) +r(x, t, ξ, ξ∗ ),

(5.7)

and the function I(x, t) can be decomposed as follows: I(x, t) = I0 (x, t) + I1 (x, t) + I2 (x, t) + R(x, t).

(5.8)

I0 (x, t, ξ ) ≡ 0,  t j1 (x, t − s, ξ, ξ∗ )ξ∗1 w(0, s, ξ∗ )dξ∗ ds, I1 (x, t, ξ ) ≡ 3 0 R  t I2 (x, t, ξ ) ≡ j2 (x, t − s, ξ, ξ∗ )ξ∗1 w(0, s, ξ∗ )dξ∗ ds, 3 0 R  t R(x, t, ξ ) ≡ r(x, t − s, ξ, ξ∗ )ξ∗1 w(0, s, ξ∗ )dξ∗ ds.

(5.9)

For x, t > 0,

0

R3

Pointwise Convergence to Knudsen Layers of the Boltzmann Equation

325

Similar to the analysis applied to I D1 (x, t, ξ ), I D2 (x, t, ξ ) and R D (x, t, ξ ) in (3.37), (3.40), (3.42) and (3.43), we get  Ii (x, t) L 2 ≤ C

t

e

ξ

 2  21  1 , for i = 1, 2, (5.10)  |ξ |w(0, s) 2

−ν0 (|x|+t−s) 

Lξ

0

R(x, t) L 2 ≤ C ξ



t

+C

e

3 

t

(x−λ j (t−s))2 C(t−s+1)

e− √

j=1 0

t −s+1

     |ξ 1 |w(0, s)

L 2ξ

 21  2  1  |ξ |w(0, s) 2 ds .

−ν0 (|x|+t−s) 

(5.11)

Lξ

0

ds

Equations (5.6), (5.10) and (5.11) together with the Schiwartz inequality yield that Ii (x, t) L 2 ≤ Ce

−ν0 |x|

ξ

⎜ ≤C⎝ 3

ξ

t

e

0 −βt−ν0 |x|

≤ C e ⎛ R(x, t) L 2





t

j=1 0

 ·

t

eβs

0

−β(t−s)

(w, |ξ |w)−

, for i = 1, 2, (x−λ j (t−s))2

 21 x=0

x=0

ds (5.12)

⎞1

e− C(t−s+1) −βs ⎟ e ds ⎠ t −s+1 (w, |ξ 1 |w)−

 21

1

ds

2

≤ C

3 

t

0

(x−λ j t)2 C(t+1)

t +1

j=1 0

where C depends on β but is independent of . Therefore we get the estimate of I(x, t, ξ ),  t    1  I(x, t) L 2 =  G (x, t − s)[ξ w](0, s)ds i   ξ

e

−

,

(5.13)

L 2ξ

⎞ ⎛ (x−λ j t)2 3 − C(t+1) e ⎟ ⎜ ≤ C ⎝ + e−(|x|+t)/C ⎠ , √ t +1 j=1

(5.14)

where the constant C is independent of . 5.2. Nonlinear term. From (5.4) and (5.14), one has that for some constant A0 > 0, ⎛ ⎞ (x−λ j t)2  ∞  3 − C(t+1)   e A0 ⎜ ⎟  (5.15) Gi (x − y, t)w0 (y)dy  + e−(|x|+t)/C ⎠ , √   2 ≤ 4 ⎝ t + 1 0 Lξ j=1 ⎞ ⎛ (x−λ j t)2  t  3 −   e C(t+1) ⎟  Gi (x, t − s)[ξ 1 w](0, s)ds  ≤ A0 ⎜ + e−(|x|+t)/C ⎠ . (5.16) √ ⎝   2 4 t +1 0

Lξ

j=1

326

S. Deng, W. Wang, S.-H. Yu

Take a Picard iteration several times: ⎧ ∞ t 1 ⎪ ⎨w1 (x, t) = 0 Gi (x − y, t)w0 (y)dy + 0 Gi (x, t − s)[ξ w](0, s)ds, ⎪ ⎩w (x, t) = w (x, t) +  t  ∞ G (x − y, t − s) (w )(y, s)dyds for k ≥ 2. k 1 i k−1 0 0 (5.17) Then we’ll get wk (x, t) L 2 ξ ⎞ ⎛ (x−λ j t)2 3 − C(1+t) e ⎟ ⎜ ≤ C ⎝ + e−(|x|+t)/C ⎠ √ t +1 j=1

+C

⎧ 2 ⎪ ⎨ √

1 for x ∈ (λ1 t, λ3 t), (|x − λl t| + 1)(|x − λl+1 t| + 1)

⎪ ⎩ l=1 0 for x ∈ (0, λ1 t) ∪ (λ3 t, ∞),

(5.18)

for k ≥ 2. Thus we make ansatz assumptions: w(x, t) L 2 ξ ⎞ ⎛ (x−λ j t)2 3 − ⎟ ⎜ e C(1+t) ≤ C0 ⎝ + e−(|x|+t)/C ⎠ √ t + 1 j=1 ⎧ 2 ⎪ 1 ⎨ √ for x ∈ (λ1 t, λ3 t), +C0 (|x − λ t| + 1)(|x − λl+1 t| + 1) l l=1 ⎪ ⎩ 0 for x ∈ (0, λ1 t) ∪ (λ3 t, ∞). w(x, t) L ∞ ξ,3 ⎞ ⎛ (x−λ j t)2 3 − ⎟ ⎜ e C(1+t) ≤ C1 ⎝ + e−(|x|+t)/C ⎠ √ t + 1 j=1 +C1

⎧ 2 ⎪ ⎨ √

1 for x ∈ (λ1 t, λ3 t), (|x − λl t| + 1)(|x −λl+1 t| + 1)

⎪ ⎩ l=1 0 for x ∈ (0, λ1 t) ∪ (λ3 t, ∞).

(5.19)

(5.20)

for some C > 0, C0 > A0 and C1 > 0. Then Lemma 2.4 yields that (w) and L δ w can be controlled by

Pointwise Convergence to Knudsen Layers of the Boltzmann Equation

327

⎛

 (w) L 2 ≤ C(1 + |ξ |)w2L 2 ξ

ξ

+C 2

L δ w L 2 ξ

⎧ 2 ⎪ ⎨

⎞ (x−λ j t)2 3 − C(1+t) e ⎜ ⎟ + e−(|x|+t)/C ⎠ ≤ C 2 ⎝ t +1 j=1

1 for x ∈ (λ1 t, λ3 t), (|x − λl t| + 1)(|x − λl+1 t| + 1)

⎪ ⎩ l=1 0 for x ∈ (0, λ1 t) ∪ (λ3 t, ∞),   ≤ C (1 + |ξ |)w L 2 (1 + |ξ |)ψ L 2 ξ ξ ⎛ ⎞ (x−λ j t)2 3 − ⎜ e C(1+t) ⎟ + e−(|x|+t)/C ⎠ ≤ C δe−γ |x|/C ⎝ √ t + 1 j=1

+C δe

⎧ 2 ⎪ ⎨ √

−γ |x|/C

(5.21)

1 for x ∈ (λ1 t, λ3 t), (|x −λl t|+1)(|x − λl+1 t|+1)

⎪ ⎩ l=1 0 for x ∈ (0, λ1 t) ∪ (λ3 t, ∞),

(5.22) for some C > 0 independent of and δ. From the representation (5.3) of w(x, t) and (5.15), (5.16), we just have to prove the following fact to verify (5.19):   Gi (x − y, t − s)L δ w(y, s)dyds   2 0 0 Lξ ⎛ ⎞ 2 (x−λ j t) 3 − A0 −γ |x|/C ⎜ e C(1+t) ⎟ e

+ e−(|x|+t)/C ⎠ √ ⎝ 4 t + 1 j=1

 t    

∞

⎧ 2 ⎪ 1 A0 −γ |x|/C ⎨ for x ∈ (λ1 t, λ3 t), √ (5.23) + e (|x − λ t| + 1)(|x − λl+1 t| + 1) l l=1 ⎪ 4 ⎩ 0 for x ∈ (0, λ1 t) ∪ (λ3 t, ∞),   t  ∞    Gi (x − y, t − s) (w)(y, s)dyds   2  0 0 Lξ ⎛ ⎞ 2 (x−λ j t) 3 − A0 −γ |x|/C ⎜ e C(1+t) ⎟ e

+ e−(|x|+t)/C ⎠ √ ⎝ 4 t + 1 j=1 ⎧ 2 ⎪ 1 A0 −γ |x|/C ⎨ for x ∈ (λ1 t, λ3 t), √ (5.24) + e (|x − λ t| + 1)(|x − λl+1 t| + 1) l l=1 ⎪ 4 ⎩ 0 for x ∈ (0, λ1 t) ∪ (λ3 t, ∞).

328

S. Deng, W. Wang, S.-H. Yu

Linear coupling wave t ∞ Consider ˜I(x, t, ξ ) = 0 0 Gi (x − y, t − s)L δ w(y, s)dyds. To deal with the singularities, we decompose it into four parts like that in Sect. 5.1: for x, t > 0, ˜ ˜I(x, t) = ˜I0 (x, t) + ˜I1 (x, t) + ˜I2 (x, t) + R(x, t), (5.25)  t  ∞ ˜I0 (x, t, ξ ) ≡ δ 1 (x − y − ξ∗1 (t − s))δ 3 (ξ − ξ∗ )L δ w(y, s, ξ∗ )dξ∗ dyds, 0

˜I1 (x, t, ξ ) ≡

 t 0

˜I2 (x, t, ξ ) ≡

∞

(5.26)

∞

j (x − y, t − s, ξ, ξ∗ )L δ w(y, s, ξ∗ )dξ∗ dyds, 2

R3

0

 t 0

j1 (x − y, t − s, ξ, ξ∗ )L δ w(y, s, ξ∗ )dξ∗ dyds,

R3

0

 t 0

˜ R(x, t, ξ ) ≡

R3

0

∞

r(x − y, t − s, ξ, ξ∗ )L δ w(y, s, ξ∗ )dξ∗ dyds.

R3

0

To avoid complex computations, we treat the different parts of ˜I(x, t, ξ ) differently. Since r(x, t, ξ, ξ∗ ) doesn’t contain any singularity and gains are extra decaying rate in ˜ ξ , we substitute the result (5.6) of the energy estimate into R(x, t, ξ ),    ˜ R(x, t)

L 2ξ

≤C

 t 0

3 ∞ e−

√

0

j=1

⎛



t

≤ Cδ 0



⎜ ⎝



∞

×

(x−y−λ j (t−s))2 C(1+t−s)

e

0

t −s+1

3 ∞ e− j=1

−Cγ |y|

    (1 + |ξ |)−1 L δ w(y, s)

(x−y−λ j (t−s))2 C(1+t−s)

t −s+1

L 2ξ

dyds

⎞1 2

⎟ e−Cγ |y| dy ⎠

1 (w, w)dy

2

ds

0 2

2

(x−λ j (t−s)) (x−λ j t)  t 3 3 e− C(1+t−s) −Cγ s e− C(1+t) ≤ C δ e ds ≤ C δ , √ √ t −s+1 t +1 0 j=1 j=1

(5.27)

where C depends on γ but is independent of , δ. To deal with the singular parts ˜I0 (x, t, ξ ), ˜I1 (x, t, ξ ), ˜I2 (x, t, ξ ), we apply the ansatz assumptions directly. From (5.26),    ˜ I0 (x, t)

L 2ξ

 t    −ν(ξ )(t−s) 1  = e L δ w(x − ξ (t − s), s)ds   0

L 2ξ

.

(5.28)

Pointwise Convergence to Knudsen Layers of the Boltzmann Equation

329

For the convenience of representation, denote (x−λ j t)2

3 e− C(1+t) ϒ(x, t) = + e−(|x|+t)/C √ t + 1 j=1 ⎧ 2 ⎪ 1 ⎨ √ for x ∈ (λ1 t, λ3 t), + (|x − λ t| + 1)(|x − λl+1 t| + 1) l l=1 ⎪ ⎩ 0 for x ∈ (0, λ1 t) ∪ (λ3 t, ∞).

Consider  t      1 1 ˜  −ν(ξ )(t−s) 1  L e w(x − ξ (t − s), s)ds I0 (x, t) 2 ≤ sup  δ   2 Lξ ϒ(x, t) x,t ϒ(x, t) x,t 0 Lξ   t 1   1  e−ν(ξ )(t−s) 1 ϒ(x − ξ (t − s), s) ds  . L δ w(x, t) L ∞ ≤ sup  2 ξ,1  ϒ(x, t) 1 + |ξ | ϒ(x, t) x,t 0 L sup

ξ

(5.29) From Lemma 4.1 and Lemma 4.2, 

t

e−ν(ξ )(t−s) ϒ(x − ξ 1 (t − s), s)ds =

0

+

3 

t

√

j=1 0

1 s+1

e−

(x−ξ 1 (t−s)−λ j s)2 C(s+1)

e−ν(ξ )(t−s) ds

2  k=1

{s |s∈(0,t),x−ξ 1 (t−s)∈(λk s,λk+1 s) }

e−ν(ξ )(t−s) ds, × (x − ξ 1 (t − s) − λk s + 1)(λk+1 s − x + ξ 1 (t − s) + 1) =

3 j=1

A1 (x, t; λ j ) +

2 k=1

A2 (x, t; λk , λk+1 ) ≤

C ϒ(x, t). ν(ξ )

(5.30)

Then    t  1     1  < C, (5.31)  e−ν(ξ )(t−s) 1 ϒ(x − ξ (t − s), s) ds  ≤     1 + |ξ | ϒ(x, t) ν(ξ )(1 + |ξ |)  L 2 0 L2 ξ

ξ

and so sup x,t

  1 1 ˜  L δ w(x, t) L ∞ I0 (x, t) 2 ≤ sup ξ,1 Lξ ϒ(x, t) x,t ϒ(x, t) 1 ψ(x, t) L ∞ w(x, t) L ∞ ≤ sup ≤ C δ. ξ,2 ξ,2 x,t ϒ(x, t) (5.32)

330

S. Deng, W. Wang, S.-H. Yu

Thus    ˜ I0 (x, t)

L 2ξ

⎞ ⎛ (x−λ j t)2 3 − C(1+t) e ⎟ ⎜ ≤ C δ ⎝ + e−(|x|+t)/C ⎠ √ t +1 j=1

⎧ 2 ⎪ 1 ⎨ √ for x ∈ (λ1 t, λ3 t), +C δ (|x − λl t| + 1)(|x − λl+1 t| + 1) l=1 ⎪ ⎩ 0 for x ∈ (0, λ1 t) ∪ (λ3 t, ∞).

(5.33)

˜I1 (x, t, ξ ) is defined by ˜I1 (x, t, ξ ) =

 t 0

×

∞ (x−y−ξ 1 (t−s))(x−y−ξ∗1 (t−s))<0

0

K(ξ, ξ∗ ) −ν(ξ )(t−s−τ1 )−ν(ξ∗ )τ1 ˜ e Lw(y, s, ξ∗ )dξ∗ dyds, |ξ 1 − ξ∗1 |

(5.34)

x−y−ξ 1 (t−s)

∗ where τ1 = t − s − . In the domain for integration, |ξ 1 − ξ∗1 | has a non-zero ξ 1 −ξ∗1 1 lower bound x−y t−s − ξ . From the formula of τ1 , we also have

ν(ξ )(t −s −τ1 )+ν(ξ∗ )τ1 ≥ ν0 ((1+|ξ |)(t − s − s1 )+(1+|ξ∗ |)s1 ) ≥ ν0 (t − s + |x − y|). Similar to the analysis in dealing with I D1 (x, t, ξ ), we divide the integration domain into two parts, ˜ 1 = {(ξ∗ , y, s) : s ∈ (0, t), y ∈ (0, ∞), (x − y − ξ 1 (t − s))(x − y − ξ 1 (t − s)) H ∗ < 0, |ξ 1 − ξ∗1 | ≥ 1},

˜ 2 = {(ξ∗ , y, s) : s ∈ (0, t), y ∈ (0, ∞), (x − y − ξ 1 (t − s))(x − y − ξ 1 (t − s)) H ∗ < 0, |ξ 1 − ξ∗1 | < 1}, and denote ˜I1 (x, t, ξ ) = 1 ˜I2 (x, t, ξ ) = 1

     

˜1 H ˜1 H

K(ξ, ξ∗ ) −ν(ξ )(t−s−τ1 )−ν(ξ∗ )τ1 ˜ Lw(y, s, ξ∗ )dξ∗ dyds, e |ξ 1 − ξ∗1 | K(ξ, ξ∗ ) −ν(ξ )(t−s−τ1 )−ν(ξ∗ )τ1 ˜ e Lw(y, s, ξ∗ )dξ∗ dyds. |ξ 1 − ξ∗1 |

Pointwise Convergence to Knudsen Layers of the Boltzmann Equation

331

By a straightforward calculation,  t ∞    ˜1 e−ν0 (t−s+|x−y|) Lδ w(y, s) L 2 dyds I1 (x, t) 2 ≤ C ξ Lξ 0 0 ⎛ (y−λ j s)2  t ∞ 3 − C(s+1) e ⎜ dyds e−ν0 (t−s+|x−y|) e−Cγ y ≤ C δ ⎝ √ s+1 0 0 j=1  t +

λ3 s

e−ν0 (t−s+|x−y|) e−Cγ y

λ1 s

0

2 k=1

⎛ ⎞ (x−λ j t)2 3 − C(1+t) e ⎜ ⎟ ≤ C δ ⎝ + e−(|x|+t)/C ⎠ √ t + 1 j=1

√

1 dyds (|y −λk s|+1)(|y −λk+1 s|+1)

⎧ 2 ⎪ ⎨ +C δ

1 for x ∈ (λ1 t, λ3 t), √ (|x − λ t| + 1)(|x − λl+1 t| + 1) l ⎪ l=1 ⎩ 0 for x ∈ (0, λ1 t) ∪ (λ3 t, ∞).

(5.35)

˜ 2 , an analysis like (3.39) and (3.40) yields that In the domain H       K(ξ, ξ∗ ) −ν0 (t−s−|x−y|) ˜2  e dξ∗ dyds I1 (x, t) 2 ≤ C 3 3 Lξ ˜ H2 |ξ 1 − ξ∗1 | 2 R 1    2 K(ξ, ξ∗ ) −ν0 (t−s−|x−y|) 2 · e L δ w(y, s) dξ∗ dyds dξ . 1 ˜ 2 |ξ 1 − ξ 1 | 2 H ∗ (5.36) From

     ≤

K(ξ, ξ∗ )

˜2 H

|ξ 1

3

− ξ∗1 | 2

e−ν0 (t−s−|x−y|) dξ∗ dyds e−ν0 (t−s−|x−y|) 1

 ≤

{(y,s)|s∈(0,t),y∈(0,∞),|x−y−ξ 1 (t−s)|≤1 } |x − y − ξ 1 (t − s)| 2 t

e−ν0 (t−s) ds ≤ C

(5.37)

0 ∗) and the L 1 property of √K(ξ,ξ , we get 1 1

   ˜2 I1 (x, t)

L 2ξ

≤C

|ξ −ξ∗ |

 t 0

∞

0

e−ν0 (t−s+|x−y|) Lδ w(y, s) L 2 dyds

⎛  t ⎜ ≤ C δ ⎝ 0

 t + 0

λ3 s

λ1 s

dyds

ξ

(y−λ j s)2

∞

e

−ν0 (t−s+|x−y|) −Cγ y

e

0

e−ν0 (t−s+|x−y|) e−Cγ y

3 e− C(s+1) dyds √ s + 1 j=1

332

S. Deng, W. Wang, S.-H. Yu

×

2 k=1

1 dyds (|y − λk s| + 1)(|y − λk+1 s| + 1) ⎞ 2

⎛ (x−λ j t) 3 − ⎜ e C(1+t) ⎟ ≤ C δ ⎝ + e−(|x|+t)/C ⎠ √ t +1 j=1

⎧ 2 ⎪ 1 ⎨ √ for x ∈ (λ1 t, λ3 t), +C δ (|x − λl t| + 1)(|x − λl+1 t| + 1) l=1 ⎪ ⎩ 0 for x ∈ (0, λ1 t) ∪ (λ3 t, ∞). (5.38) Thus    ˜ I1 (x, t)

L 2ξ

    ≤ ˜I11 (x, t)

L 2ξ

    + ˜I21 (x, t)

⎞ (x−λ j t)2 3 − C(1+t) e ⎟ ⎜ ≤ C δ ⎝ + e−(|x|+t)/C ⎠ √ t + 1 j=1 ⎛

L 2ξ

⎧ 2 ⎪ 1 ⎨ √ for x ∈ (λ1 t, λ3 t), +C δ (|x − λ t| + 1)(|x − λl+1 t| + 1) l l=1 ⎪ ⎩ 0 for x ∈ (0, λ1 t) ∪ (λ3 t, ∞). ˜I2 (x, t, ξ ) only contains an integrable singularity in ξ . Therefore by a straight calculation, we get   ˜  I2 (x, t)

L 2ξ

1 |ξ −ξ∗ |

(5.39)

and gains extra decaying rate

⎞ ⎛ (x−λ j t)2 3 − C(1+t) e ⎟ ⎜ ≤ C δ ⎝ + e−(|x|+t)/C ⎠ √ t +1 j=1 ⎧ 2 ⎪ 1 ⎨ √ for x ∈ (λ1 t, λ3 t), +C δ (|x − λl t| + 1)(|x − λl+1 t| + 1) l=1 ⎪ ⎩ 0 for x ∈ (0, λ1 t) ∪ (λ3 t, ∞). (5.40)

Since   ˜  I(x, t)

L 2ξ

    ≤ ˜I0 (x, t)

L 2ξ

we get the following conclusion.

    + ˜I1 (x, t)

L 2ξ

    + ˜I2 (x, t)

L 2ξ

  ˜  + R(x, t)

L 2ξ

,

Pointwise Convergence to Knudsen Layers of the Boltzmann Equation

333

Lemma 5.2.

 t  ∞     ˜I(x, t) L 2 =  G (x − y, t − s)L w(y, s)dyds i δ   2 ξ 0 0 Lξ ⎞ ⎛ (x−λ j t)2 3 − ⎟ ⎜ e C(1+t) ≤ C δ ⎝ + e−(|x|+t)/C ⎠ √ t +1 j=1

+C δ

⎧ 2 ⎪ ⎨ √

1 for x ∈ (λ1 t, λ3 t), (|x − λl t| + 1)(|x − λl+1 t| + 1)

⎪ ⎩ l=1 0 for x ∈ (0, λ1 t) ∪ (λ3 t, ∞).

(5.41) where C is independent of δ and . Thus when δ is sufficiently small, (5.23) holds. t ∞ Nonlinear term Substitute (5.21) into 0 0 Gi (x − y, t − s) (w(y, s)dyds.  t  ∞      G (x − y, t − s) (w)(y, s)dyds i   2 0 0 Lξ ⎛   3 2 t ∞ (x−y−λi (t−s))2 j s) 1 1 − (y−λ e− C(t−s+1) e C(s+1) dyds ≤ C 2 ⎝ t −s+1 s+1 0 0 i, j=1

⎞ (x−y−λ j (t−s))2 1 1 − C(t−s+1) + e dyds⎠ (t −s + 1) (|y −λk s|+1)(|y −λk+1 s|+1) s 0 λ k j=1 k=1 ⎞ ⎛ 3 3 3 2 ≤ C 2 ⎝ J1 (x, t; λi , λ j ) + J2 (x, t; λ j , λk , λk+1 )⎠ . (5.42) 3 2  t

i=1 j=1

λk+1 s

j=1 k=1

Estimates

⎞ ⎛ (x−λ j t)2 3 − C(1+t) e ⎟ ⎜ |J1 |≤ C ⎝ + e−(|x|+t)/C ⎠ √ t + 1 j=1 ⎧ 2 ⎪ ⎪ ⎪ ⎪ ⎨

+C

√

⎪ l=1 ⎪ ⎪ ⎪ ⎩0 for

+C

√

⎪ l=1 ⎪ ⎪ ⎪ ⎩0 for

Lemma

4.3

1 for x ∈ (λ1 t, λ3 t), (|x − λl t| + 1)(|x − λl+1 t| + 1)

⎛ x ∈ (0, λ1 t)2 ∪ (λ3 t, ∞). ⎞ (x−λ j t) 3 − ⎟ ⎜ e C(1+t) |J2 |≤ C ⎝ + e−(|x|+t)/C ⎠ √ t + 1 j=1 ⎧ 2 ⎪ ⎪ ⎪ ⎪ ⎨

Reference

1 for x ∈ (λ1 t, λ3 t), (|x − λl t| + 1)(|x − λl+1 t| + 1) x ∈ (0, λ1 t) ∪ (λ3 t, ∞).

(5.43) Lemma

4.3

334

S. Deng, W. Wang, S.-H. Yu

Table (5.43) and (5.42) yield that  t  ∞      G (x − y, t − s) (w)(y, s)dyds i   0

0

⎛

⎞

(x−λ j t)2

L 2ξ

3 e− C(1+t) ⎟ ≤ C ⎝ + e−(|x|+t)/C ⎠ √ t + 1 j=1 2⎜

⎧ 2 ⎪ 1 ⎨ √ for x ∈ (λ1 t, λ3 t), 2 +C (|x − λl t| + 1)(|x − λl+1 t| + 1) l=1 ⎪ ⎩ 0 for x ∈ (0, λ1 t) ∪ (λ3 t, ∞).

(5.44)

Thus we justify (5.24) and so the ansatz assumption (5.19) holds. To verify (5.20), we only have to consider w(x, t, ξ ) = e−ν(ξ )t w0 (x − ξ 1 t, ξ )  t e−ν(ξ )(t−s) (Kw + L δ w + (w)) (x − ξ 1 (t − s), s, ξ )ds. +

(5.45)

0

From Lemma 2.3 and the ansatz (5.19),  t  t    −ν(ξ )(t−s)  1 e−ν(ξ )(t−s) Kw(x −ξ 1 (t − s), s)ds ≤ C − ξ e (t − s), s) w(x  0

≤C

3  j=1 0

+C

0

t

(x−ξ 1 e− √ s+1

2 

1 (t−s)−λ s)2 j C(s+1)

L 2ξ

ds

e−ν(ξ )(t−s) ds



1 k=1 {s∈(0,t)}∩{s x−ξ (t−s)∈(λk s,λk+1 s)} e−ν(ξ )(t−s)

ds, × (x − ξ 1 (t − s) − λk s + 1)(λk+1 s − x + ξ 1 (t − s) + 1)

=

3

A1 (x, t; λ j ) +

j=1

2 k=1

A2 (x, t; λk , λk+1 ) ≤

C ϒ(x, t). ν(ξ )

(5.46)

Consider sup x,t

|w(x, t)| ϒ(x, t)

 t e−ν(ξ )t w0 (x −ξ 1 t, ξ ) 1 −ν(ξ )(t−s) 1 e Kw(x − ξ (t − s), s)ds ≤ sup +sup ϒ(x, t) x,t x,t ϒ(x, t) 0  t 1 −ν(ξ )(t−s) 1 + sup e (L δ w + (w)) (x − ξ (t − s), s)ds ϒ(x, t) x,t 0

Pointwise Convergence to Knudsen Layers of the Boltzmann Equation

≤

|(δ + )w(x, t)| C C + sup + (1 + |ξ |)3 1 + |ξ | x,t ϒ(x, t)



t

335

e−ν(ξ )(t−s) (1 + |ξ |)

0

ϒ(x − ξ 1 (t − s), s) × ds. ϒ(x, t)

(5.47)

From 

t

e−ν(ξ )(t−s) ϒ(x − ξ 1 (t − s), s)ds =

0

3 

t

j=1 0

+

√

1 s+1

e−

(x−ξ 1 (t−s)−λ j s)2 C(s+1)

e−ν(ξ )(t−s) ds

2  1 k=1 {s |s∈(0,t),x−ξ (t−s)∈(λk s,λk+1 s) } e−ν(ξ )(t−s)

ds, × (x − ξ 1 (t − s) − λk s + 1)(λk+1 s − x + ξ 1 (t − s) + 1) =

3

A1 (x, t; λ j ) +

j=1

2

A2 (x, t; λk , λk+1 ) ≤

k=1

C ϒ(x, t), ν(ξ )

(5.48)

one has 

ϒ(x − ξ 1 (t − s), s) ds e−ν(ξ )(t−s) (1 + |ξ |) ϒ(x, t) 0  t (1 + |ξ |) = e−ν(ξ )(t−s) ϒ(x − ξ 1 (t − s), s)ds ≤ C. ϒ(x, t) 0 t

(5.49)

Thus (5.47) and (5.49) yield that sup x,t

2C |w(x, t)| 2C ≤ , + ϒ(x, t) (1 + |ξ |)3 1 + |ξ |

(5.50)

when , δ 1. Repeat the process for another two times. Since the operator K gains the decaying rate in the ξ variable, we have sup x,t

C |w(x, t)| ≤ . ϒ(x, t) (1 + |ξ |)3

(5.51)

Thus the ansatz assumption (5.20) is justified. 6. Stability of Knudsen’s Layers for −1 < M < 1(M
= 0) When −1 < M < 1(M
= 0), we can no longer get full boundary data by a priori energy estimate. If we use the Green’s function Gi for the initial value problem, the incomplete boundary data can’t be complemented. We have to use the Green’s function Gb for the initial-boundary value problem to get the representation of w(x, t, ξ ):

336

S. Deng, W. Wang, S.-H. Yu



∞

w(x, t) =

Gb (x, y, t)w0 (y)dy +

0

 t 0

0

∞

Gb (x, y, t −s)(L δ w+ (w))(y, s)dyds. (6.1)

In the following part of this paper, we will assume −1 < M < 0 for definiteness. That is λ1 < λ2 < 0 < λ3 . The case of 0 < M < 1 is almost the same. From Theorem 2.4, one has ⎞ ⎛ (x−λ j t)2  ∞  3 − C(t+1)   e ⎟ ⎜  (6.2) Gb (x, y, t)w0 (y)dy  + e−(|x|+t)/C ⎠ , √   2 ≤ A0 ⎝ t + 1 0 Lξ j=1 for some constant A0 > 0. Make ansatz assumptions from the anastz of the perturbation around a global Maxwellian in the half-space problem in [26]: w(x, t) L 2 ξ ⎞ ⎛ (x−λ j t)2 3 − ⎟ ⎜ e C(1+t) ≤ C0 ⎝ + e−(|x|+t)/C ⎠ √ t + 1 j=1 +C0

⎧ 2 ⎪ ⎨ √

1 for x ∈ (0, λ3 t), (|x − λl t| + 1)(|x − λl+1 t| + 1)

(6.3)

1 for x ∈ (0, λ3 t), (|x − λl t| + 1)(|x − λl+1 t| + 1)

(6.4)

⎪ ⎩ l=1 0 for x ∈ (λ3 t, ∞), w(x, t) L ∞ ξ,3 ⎞ ⎛ (x−λ j t)2 3 − ⎟ ⎜ e C(1+t) ≤ C1 ⎝ + e−(|x|+t)/C ⎠ √ t + 1 j=1 +C1

⎧ 2 ⎪ ⎨ √

⎪ ⎩ l=1 0 for x ∈ (λ3 t, ∞),

for some positive constant C, C0 > 2 A0 and C1 > 0. Then Lemma 2.4 yields that (w) and L δ w can be controlled by  (w) L 2 ξ

+C 2

⎛ ⎞ (x−λ j t)2 3 − C(1+t) e ⎜ ⎟ + e−(|x|+t)/C ⎠ ≤ C 2 ⎝ t +1

⎧ 2 ⎪ ⎨

j=1

1 for x ∈ (0, λ3 t), (|x − λl t| + 1)(|x − λl+1 t| + 1)

⎪ ⎩ l=1 0 for x ∈ (λ3 t, ∞),

(6.5)

Pointwise Convergence to Knudsen Layers of the Boltzmann Equation

337

⎛

L δ w L 2 ξ

⎞ (x−λ j t)2 3 − C(1+t) e ⎜ ⎟ ≤ C δe−γ |x|/C ⎝ + e−(|x|+t)/C ⎠ √ t +1 j=1

⎧ 2 ⎪ 1 ⎨ √ for x ∈ (0, λ3 t), −γ |x|/C (6.6) +C δe (|x − λl t| + 1)(|x − λl+1 t| + 1) l=1 ⎪ ⎩ 0 for x ∈ (λ3 t, ∞).

1 of Gi P1 plays an important role in closing the nonlinThe algebraic decaying rate t+1 earity in the initial value problem. However, for Gb P1 one can not get such a global decaying rate. As a result, one needs rather detailed  t  ∞ spatial information on Gb P1 in order to close the nonlinearity. We shall consider 0 0 Gb (x, y, t−s) (L δ w + (w)) (y, s)dyds in the following different regions. Case 1. When 0 < x < λ3 t/2, we use the forward Green’s function. From (2.32) 1 1 in Theorem 2.4, the algebraic decaying rate of Gb P1 is t+1 + √(t+1)(y+1) . This suggests

that when y ≤ Ct, the dominant decaying rate is 1 t+1

get:

√

1 (t+1)(y+1)

while

1 t+1

+

√

1 (t+1)(y+1)

∼

when y ≥ Ct. Thus substitute (2.32) and (6.6) into the linear coupling wave to

 t     0

  Gb (x, y, t − s)L δ w(y, s)dyds  

∞ 0

⎛

 t

3

≤ C δ ⎝

i, j=1 0

+

0

3 2  t j=1 k=1 0

⎛

λ3 s 0

2

k=1 0

⎛ = C δ ⎝

0

0

3

⎜ +C δ ⎝ 2

J1 (x, t; λi , λ j ) +

k=1 0

3 2

λ1 (s−t) 0

L 2ξ

dyds

2 λ x− λ3 y−λ3 (t−s) 2 C(t−s+1)

e−Cγ y dyds √ (|y − λk s| + 1)(|y − λk+1 s| + 1) ⎞

e−Cγ y ⎟ dyds ⎠ √ (|y − λk s| + 1)(|y − λk+1 s| + 1) ⎞

J2 (x, t; λ j , λk , λk+1 )⎠

1 e− √ (t − s + 1)(y + 1) 

λ2 (s−t) 0

2 λ x− λ3 y−λ3 (t−s) 1 C(t−s+1)



 t

2  t

(y−λ j s)2 − C(s+1)

j=1 k=1

k=1 0

+

1 e− √ (t −s +1)(y +1)

1 e− √ (t − s + 1)(y + 1)

i, j=1

⎛

0

  Gb (x, y, t − s)P1 L δ w(y, s)dyds  

⎞ (x−y−λ j (t−s))2 1 e−Cγ y − C(t−s+1) e dyds ⎠ √ (t − s + 1) (|y − λk s| + 1)(|y − λk+1 s| + 1)

λ1 (s−t)

λ2 (s−t)

∞

(x−y−λi (t−s))2 1 e−Cγ y e− C(t−s+1) e √ (t − s + 1) s + 1



2  t k=1 0

0



 t

⎜ + C δ ⎝

+

∞

L 2ξ

 t   = 

√

1 e− (t −s +1)(y +1)

2 λ x− λ3 y−λ3 (t−s) 1 C(t−s+1)

2 λ x− λ3 y−λ3 (t−s) 2 C(t−s+1)

√

e−Cγ y dyds (|y −λk s|+1)(|y −λk+1 s|+1) ⎞ e−Cγ y

⎟ dyds ⎠ . √ (|y −λk s|+1)(|y −λk+1 s|+1)

(6.7)

338

S. Deng, W. Wang, S.-H. Yu Estimates |J1 |≤

⎛

⎜ C⎝

e

(x−λ j t)2 − C(1+t)

j=1

⎧ 2 ⎪ ⎪ ⎪ ⎪ ⎨ +C

3

√

⎪ l=1 ⎪ ⎪ ⎪ ⎩0 for

√ t +1

Reference

⎞ ⎟ + e−(|x|+t)/C ⎠

Lemma

4.4

1 for x ∈ (0, λ3 t), (|x − λl t| + 1)(|x − λl+1 t| + 1)

x ∈ (λ3 t, ∞). ⎞ (x−λ j t)2 3 e− C(1+t) ⎜ ⎟ + e−(|x|+t)/C ⎠ |J2 |≤ C ⎝ √ t + 1 j=1

(6.8)

⎛

⎧ 2 ⎪ ⎪ ⎪ ⎪ ⎨

+C

√

⎪ l=1 ⎪ ⎪ ⎪ ⎩0 for

Lemma

4.6

1 for x ∈ (0, λ3 t), (|x − λl t| + 1)(|x − λl+1 t| + 1) x ∈ (λ3 t, ∞).

From Table (6.8), we can conclude that ⎛ C δ ⎝

3

J1 (x, t; λi , λ j ) +

i, j=1

3 2

⎞ J2 (x, t; λ j , λk , λk+1 )⎠

j=1 k=1

⎛ ⎞ (x−λ j t)2 3 − C(1+t) e ⎜ ⎟ ≤ C δ ⎝ + e−(|x|+t)/C ⎠ √ t + 1 j=1 ⎧ 2 ⎪ 1 ⎨ √ for x ∈ (0, λ3 t). + C δ (|x − λ t| + 1)(|x − λl+1 t| + 1) l l=1 ⎪ ⎩ 0 for x ∈ (λ3 t, ∞).

(6.9)

Consider 2  t  λ1 (s−t)

1 − e √ (t − s + 1)(y + 1)

λ (x− λ3 y−λ3 (t−s))2 1 C(t−s+1)



e−Cγ y

(|y −λk s|+1)(|y −λk+1 s|+1)  t−x/λ3 1 1 −Cγ y e ds = O(1) √ √ λ1 y + 1 (|y − λ s| + 1)(|y − λ s| + 1) 0 1 2 y= λ (x−λ3 (t−s)) 3  t−x/λ3 1 1 −Cγ y e ds +O(1) √ √ λ1 y + 1 (|y − λ s| + 1)(|y − λ s| + 1) 0 2 3 y= λ (x−λ3 (t−s)) 3 ⎛ λ3  2 −λ1 1 1 ⎜ λ3 −λ1 (t−x/λ3 ) e−Cγ |x−λ3 t|  ds ≤C⎝ λ1 λ3 t − x 0 | λ (x − λ3 (t − s)) − λ3 s| 3 ⎞  t−λ3 x λ 1 1 −Cγ λ1 (x−λ3 (t−s)) ⎠ 3  + λ3 −λ e ds 1 2 λ t − x λ1 (x − λ (t − s)) + 1 λ −λ (t−x/λ3 ) 3 k=1 0

0

3

1

λ3

dyds

3

1 1 1 1 ≤C√ ≤C√ √ . γ λ3 t − x γ (|x − λ2 t| + 1)(|x − λ3 t| + 1)

(6.10)

Pointwise Convergence to Knudsen Layers of the Boltzmann Equation

339

Similarly, 2  t k=1 0

λ2 (s−t)

0

λ3

2

(x− λ y−λ3 (t−s)) 2 1 e−Cγ y C(t−s+1) e− dyds √ √ (t −s +1)(y +1) (|y −λk s|+1)(|y −λk+1 s|+1)

1 1 ≤ C√ √ . γ (|x − λ2 t| + 1)(|x − λ3 t| + 1)

(6.11)

Therefore (6.7), (6.9), (6.10) and (6.11) yield that the linear coupling wave satisfies  t     0

∞ 0

  Gb (x, y, t − s)L δ w(y, s)dyds  

L 2ξ

1 ≤ C δ √ (|x − λ2 t| + 1)(|x − λ3 t| + 1) (6.12)

when 0 < x < λ3 t/2. Then consider the nonlinear term:  t  ∞     Gb (x, y, t − s) (w)(y, s)dyds    2 0 0 Lξ  t  ∞     = Gb (x, y, t − s)P1 (w)(y, s)dyds   0

0

⎛

3  t

≤ C 2 ⎝

+

i, j=1 0

0

2  t 3

λ3 s

j=1 k=1 0

⎛

0

2  t ⎜ + C 2 ⎝ k=1 0

×

+

∞

2

(y−λ j s) (x−y−λi (t−s))2 1 e− C(t−s+1) e− C(s+1) dyds (t − s + 1)(s + 1)

1 e (t − s + 1)

(x−y−λ j (t−s))2 − C(t−s+1)



λ1 (s−t) 0

1 e− √ (t − s + 1)(y + 1)



λ2 (s−t) 0

1 e− √ (t − s + 1)(y + 1)

λ x− λ3 y−λ3 (t−s) 1 C(t−s+1)

λ x− λ3 y−λ3 (t−s) 2 C(t−s+1)

2

 1 dyds × (|y − λk s| + 1)(|y − λk+1 s| + 1) ⎞ ⎛ 3 3 2 = C 2 ⎝ J1 (x, t; λi , λ j ) + J2 (x, t; λ j , λk , λk+1 )⎠ i, j=1

j=1 k=1

⎞

1 dyds⎠ (|y −λk s| + 1)(|y −λk+1 s|+1)

1 dyds (|y − λk s| + 1)(|y − λk+1 s| + 1)

2  t k=1 0

L 2ξ

2

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S. Deng, W. Wang, S.-H. Yu

⎛ 2  t ⎜ + C 2 ⎝ k=1 0

0

1 e− √ (t − s + 1)(y + 1)

2 λ x− λ3 y−λ3 (t−s) 1 C(t−s+1)

1 dyds (|y − λk s| + 1)(|y − λk+1 s| + 1)

×

+



λ1 (s−t)



2  t k=1 0

λ2 (s−t)

0

1 e− √ (t − s + 1)(y + 1)

2 λ x− λ3 y−λ3 (t−s) 2 C(t−s+1)

 1 dyds . × (|y − λk s| + 1)(|y − λk+1 s| + 1)

Estimates

⎛

⎧ 2 ⎪ ⎪ ⎪ ⎪ ⎨ +C

Reference

⎞

(x−λ j t)2 − C(1+t)

⎟ ⎜ e C⎝ + e−(|x|+t)/C ⎠ √ t +1 j=1 3

|J1 |≤

Lemma

4.3

1 for x ∈ (0, λ3 t), √ (|x − λl t| + 1)(|x − λl+1 t| + 1)

⎪ l=1 ⎪ ⎪ ⎪ ⎩0 for

⎞ ⎛ x ∈ (λ3 t, ∞). (x−λ j t)2 3 − C(1+t) e ⎟ ⎜ |J2 |≤ C ⎝ + e−(|x|+t)/C ⎠ √ t +1 j=1 ⎧ 2 ⎪ ⎪ ⎪ ⎪ ⎨

+C

(6.13)

(6.14) Lemma

4.3

1 for x ∈ (0, λ3 t), √ (|x − λl t| + 1)(|x − λl+1 t| + 1)

⎪ l=1 ⎪ ⎪ ⎪ ⎩0 for

x ∈ (λ3 t, ∞).

From Table (6.14), we can conclude that ⎞ ⎛ 3 3 2 C 2 ⎝ J1 (x, t; λi , λ j ) + J2 (x, t; λ j , λk , λk+1 )⎠ i, j=1

j=1 k=1

⎛ ⎞ (x−λ j t)2 3 − C(1+t) e ⎜ ⎟ ≤ C 2 ⎝ + e−(|x|+t)/C ⎠ √ t +1 j=1 ⎧ 2 ⎪ 1 ⎨ √ for x ∈ (0, λ3 t), 2 + C (|x − λl t| + 1)(|x − λl+1 t| + 1) l=1 ⎪ ⎩ 0 for x ∈ (λ3 t, ∞).

(6.15)

Consider 2  t k=1 0

0

λ1 (s−t)

λ3

2

(x− λ y−λ3 (t−s)) 1 1 1 C(t−s+1) dyds e− √ (|y −λk s|+1)(|y −λk+1 s|+1) (t −s +1)(y +1)

Pointwise Convergence to Knudsen Layers of the Boltzmann Equation

341

1 1 = O(1) ds √ y + 1 (|y − λ1 s| + 1)(|y − λ2 s| + 1) y= λ1 (x−λ3 (t−s)) 0 λ3  t−x/λ3 1 1 +O(1) ds √ y + 1 (|y − λ2 s| + 1)(|y − λ3 s| + 1) y= λ1 (x−λ3 (t−s)) 0 λ3 ⎛ λ  23 −λ1 (t−x/λ3 ) 1 1 ⎜ λ3 −λ1 ds ≤C⎝ 3 λ1 0 (λ3 t − x) 2 | λ3 (x − λ3 (t − s)) − λ3 s| ⎞  t−λ3 x 1 1  + λ3 −λ ds ⎠ 1 2 (λ3 t − x)2 λ1 (x − λ (t − s)) + 1 λ −λ (t−x/λ3 ) 

3

≤C

t−x/λ3

λ3

1

log(λ3 t − x) (λ3 t − x)

3 2

≤ C√

3

1 , (|x − λ2 t| + 1)(|x − λ3 t| + 1)

(6.16)

and 2  t k=1 0

0

λ2 (s−t)

λ3

(x− λ y−λ3 (t−s)) 2 1 C(t−s+1) e− √ (t − s + 1)(y + 1)

2

1 dyds (|y − λk s| + 1)(|y − λk+1 s| + 1) 1 ≤ C√ . (|x − λ2 t| + 1)(|x − λ3 t| + 1) ×

(6.17)

Equations (6.13), (6.15), (6.16) and (6.17) result in  t     0

0

∞

  Gb (x, y, t − s) (w)(y, s)dyds  

L 2ξ

≤ C 2 √

1 (|x − λ2 t| + 1)(|x − λ3 t| + 1) (6.18)

when 0 < x < λ3 t/2. Thus, (6.12) and (6.18) together with (6.2) verify the ansatz assumption (6.3) for x ∈ (0, λ3 t/2). Case 2. When λ3 t/2 < x < λ3 t, we will fix (x, t) and use the backward Green’s function. By a symmetry consideration as stated in [26], one has ⎧ (x−y−λ j (t−s+1))2 ⎪ 3 − ⎪ C(t−s+1) ⎪ e ⎪ ⎪ + e−(|x−y|+t−s)/C ⎪ ⎪ ⎪ (t − s + 1) ⎪ ⎪ j=1 ⎪ ⎪ ⎨ for s ∈ (t − x/λ3 , t), λ x Gb (x, y, t, s)P1  L 2 = O(1) (6.19) (y−λ j (s−t)+ λi ))2 ξ ⎪ 3 ⎪ 2 − ⎪ C(t−s+1) ⎪ e ⎪ ⎪ + e−(|x−y|+t−s)/C √ ⎪ ⎪ (t − s + 1)x ⎪ ⎪ j=1 ⎪ ⎪ ⎩ for s ∈ (0, t − x/λ ). 3

342

S. Deng, W. Wang, S.-H. Yu

t ∞ We substitute (6.5), (6.6) and (6.19) into the representation 0 0 Gb (x, y, t − s) t ∞ L δ w(y, s)dyds and 0 0 Gb (x, y, t − s) (w)(y, s)dyds respectively:  t   

  t ∞       Gb (x, y, t − s)L δ w(y, s) =  Gb (x, y, t −s)P1 L δ w(y, s)dyds   2 0 0 0 0 L 2ξ Lξ ⎛ 3  t ∞ (y−λ j s)2 (x−y−λi (t−s))2 1 ≤ C δ ⎝ e−Cγ y e− C(t−s+1) e− C(s+1) dyds √ (t − s + 1) s + 1 i, j=1 0 0 +

∞

3 2  t j=1 k=1 0

λ3 s 0

2

(x−y−λ j (t−s)) 1 e− C(t−s+1) (t − s + 1) ⎞

e−Cγ y dyds ⎠ (|y − λk s| + 1)(|y − λk+1 s| + 1) ⎛ λ 2  t  λ1 (s−t) (x− λ3 y−λ3 (t−s))2 1 1 − C(t−s+1) ⎝ + C δ e √ (t − s + 1)(x + 1) k=1 0 0

× √

e−Cγ y dyds (|y − λk s| + 1)(|y − λk+1 s| + 1) λ 2  t  λ2 (s−t) (x− λ3 y−λ3 (t−s))2 2 1 − C(t−s+1) e + √ (t − s + 1)(x + 1) k=1 0 0  −Cγ y e × √ dyds (|y − λk s| + 1)(|y − λk+1 s| + 1) ⎞ ⎛ 3 3 2 J1 (x, t; λi , λ j ) + J2 (x, t; λ j , λk , λk+1 )⎠ = C δ ⎝ × √

i, j=1

⎛ + C δ ⎝

j=1 k=1

2  t k=1 0



λ1 (s−t)

0

λ3

(x− λ y−λ3 (t−s)) 1 1 C(t−s+1) e− √ (t − s + 1)(x + 1)

e−Cγ y dyds (|y − λk s| + 1)(|y − λk+1 s| + 1) λ 2  t  λ2 (s−t) (x− λ3 y−λ3 (t−s))2 2 1 − C(t−s+1) e + √ (t − s + 1)(x + 1) k=1 0 0  −Cγ y e × √ dyds . (|y − λk s| + 1)(|y − λk+1 s| + 1)

2

× √

The following estimates: 2  t k=1 0

0



λ1 (s−t)

1 e− √ (t − s + 1)(x + 1)

λ x− λ3 y−λ3 (t−s) 1 C(t−s+1)

2

(6.20)

Pointwise Convergence to Knudsen Layers of the Boltzmann Equation

343

e−Cγ y dyds (|y − λk s| + 1)(|y − λk+1 s| + 1)  t−x/λ3 1 1 −Cγ y = O(1) √ e ds √ (|y − λ1 s| + 1)(|y − λ2 s| + 1) y= λ1 (x−λ3 (t−s)) x +1 0 λ3  t−x/λ3 1 1 +O(1) √ e−Cγ y √ ds (|y − λ2 s| + 1)(|y − λ3 s| + 1) y= λ1 (x−λ3 (t−s)) x +1 0 λ3 ⎛ λ  23 −λ1 1 ⎜ λ3 −λ1 (t−x/λ3 ) −Cγ |x−λ3 t| 1  ≤ C√ e ds ⎝ √ x +1 0 λ3 t − x | λλ13 (x − λ3 (t − s)) − λ3 s| ⎞  t−λ3 x λ 1 1 −Cγ λ (x−λ3 (t−s)) ⎟ 3 e + λ3 −λ ds ⎠ 1 2 λ t − x 3 λ −λ (t−x/λ3 ) ×√

3

1

1 1 1 1 ≤C √ √ √ √ γ λ3 t − x x + 1 γ λ3 t − x t + 1 1 1 ≤C √ , γ (|x − λ2 t| + 1)(|x − λ3 t| + 1) λ 2  t  λ2 (s−t) (x− λ3 y−λ3 (t−s))2 2 1 − C(t−s+1) e √ (t − s + 1)(y + 1) k=1 0 0 ≤C

(6.21)

e−Cγ y ×√ dyds (|y − λk s| + 1)(|y − λk+1 s| + 1) 1 1 , ≤C √ γ (|x − λ2 t| + 1)(|x − λ3 t| + 1) together with Table (6.8) result in   t  ∞    Gb (x, y, t − s)L δ w(y, s)dyds    0

0

(6.22)

L 2ξ

⎛ ⎞ (x−λ j t)2 3 − C(1+t) e ⎜ ⎟ ≤ C δe−γ |x|/C ⎝ + e−(|x|+t)/C ⎠ √ t +1 j=1 ⎧ 2 ⎪ 1 ⎨ √ for x ∈ (0, λ3 t), −γ |x|/C (6.23) +C δe (|x − λl t| + 1)(|x − λl+1 t| + 1) l=1 ⎪ ⎩ 0 for x ∈ (λ3 t, ∞). The estimate for the nonlinear term is as follows:  t  ∞     Gb (x, y, t − s) (w)(y, s)dyds    2 0 0 Lξ  t  ∞     = Gb (x, y, t − s)P1 (w)(y, s)dyds   0

0

L 2ξ

344

S. Deng, W. Wang, S.-H. Yu

⎛ ≤ C

2⎝

3  t

i, j=1 0 3 2  t

∞

2

(y−λ j s) (x−y−λi (t−s))2 1 e− C(t−s+1) e− C(s+1) dyds (t − s + 1)(s + 1)

0 λ3 s

2

(x−y−λ j (t−s)) 1 e− C(t−s+1) (t − s + 1) j=1 k=1 0 0 ⎞ 1 dyds ⎠ × (|y − λk s| + 1)(|y − λk + 1s| + 1) ⎛ λ 2  t  λ1 (s−t) (x− λ3 y−λ3 (t−s))2 1 1 − 2⎝ C(t−s+1) + C e √ (t − s + 1)(x + 1) k=1 0 0

+

1 dyds (|y − λk s| + 1)(|y − λk+1 s| + 1) λ 2  t  λ2 (s−t) (x− λ3 y−λ3 (t−s))2 2 1 C(t−s+1) + e− √ (t − s + 1)(x + 1) k=1 0 0 ×

1 dyds (|y − λk s| + 1)(|y − λk+1 s| + 1) ⎛ ⎞ 3 3 2 = C 2 ⎝ J1 (x, t; λi , λ j ) + J2 (x, t; λ j , λk , λk+1 )⎠ ×

i, j=1

⎛

+ C

2⎝

j=1 k=1

2  t k=1 0

λ1 (s−t) 0

λ3

(x− λ y−λ3 (t−s)) 1 1 C(t−s+1) e− √ (t − s + 1)(x + 1)

1 dyds (|y − λk s| + 1)(|y − λk+1 s| + 1) λ 2  t  λ2 (s−t) (x− λ3 y−λ3 (t−s))2 2 1 − C(t−s+1) + e √ (t − s + 1)(x + 1) k=1 0 0 ⎞ 1 dyds ⎠ . × (|y − λk s| + 1)(|y − λk+1 s| + 1)

2

×

(6.24)

From Table (6.14) and 2  t



λ1 (s−t)

1 e− √ (t −s +1)(x +1)  t−x/λ3

λ x− λ3 y−λ3 (t−s) 1 C(t−s+1)

2

1 dyds (|y −λk s|+1)(|y −λk+1 s|+1) k=1 0 0 1 1 = O(1) √ ds (|y − λ1 s| + 1)(|y − λ2 s| + 1) y= λ1 (x−λ3 (t−s)) x +1 0 λ3  t−x/λ3 1 1 + O(1) √ ds (|y − λ2 s| + 1)(|y − λ3 s| + 1) y= λ1 (x−λ3 (t−s)) x +1 0 λ3

Pointwise Convergence to Knudsen Layers of the Boltzmann Equation

⎛ ≤ C√  +

1 x +1

⎜ ⎝



λ3 2 −λ1 λ3 −λ1 (t−x/λ3 )

0

t−λ3 x λ3 2 −λ1 λ3 −λ1 (t−x/λ3 )

≤C

345

1 1 ds (λ3 t − x) | λλ1 (x − λ3 (t − s)) − λ3 s| 3

1 ds (λ3 t − x)2

log(λ3 t − x) log(λ3 t − x) 1 ≤C ≤ C√ , √ √ (|x − λ2 t| + 1)(|x − λ3 t| + 1) (λ3 t − x) x + 1 (λ3 t − x) t + 1 (6.25) 

2  t k=1 0

λ2 (s−t) 0

√

1 e− (t − s + 1)(y + 1)

λ x− λ3 y−λ3 (t−s) 2 C(t−s+1)

1 dyds (|y − λk s| + 1)(|y − λk+1 s| + 1) 1 ≤ C√ , (|x − λ2 t| + 1)(|x − λ3 t| + 1)

2

×

(6.26)

we find that   t  ∞     G (x, y, t − s) (w)(y, s)dyds b  2  0 0 Lξ ⎛ ⎞ (x−λ j t)2 3 − C(1+t) e ⎜ ⎟ ≤ C 2 e−γ |x|/C ⎝ + e−(|x|+t)/C ⎠ √ t + 1 j=1 ⎧ 2 ⎪ 1 ⎨ √ for x ∈ (0, λ3 t), 2 −γ |x|/C + C e (|x − λl t| + 1)(|x − λl+1 t| + 1) l=1 ⎪ ⎩ 0 for x ∈ (λ3 t, ∞). (6.27) Thus (6.2), (6.23) and (6.27) verify the ansatz assumption (6.3) in the region x ∈ (λ3 t/2, λ3 t). The justification of the ansatz (6.4) is same to that in Sect. 5. Thus we complete the proof of Theorem 1.1. Acknowledgements. The research of the first and the second authors are supported by National Science Foundation of China 10531020. The research of the third author is supported by the National University of Singapore start-up grant R-146-000-108-133.

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3. Carleman, T.: Sur La Théorie de l’Équation Intégrodifférentielle de Boltzmann. Acta Mathematica 60, 91–142 (1933) 4. Carleman, T.: Problémes mathématiques dans la théorie cinétique des gaz. (French) Publ. Sci. Inst. Mittag-Leffler. 2, Uppsala, Almqvist & Wiksells Boktryckeri Ab, 1957 5. Cercignani, C.: Half-space problem in the kinetic theory of gases. In: Kroner, E., Kirchgassner, K. eds., Trends in Applications of Pure Mathematics to Mechanics, Berlin: Springer-Verlag, 1986, p. 3550 6. Cercignani, C., Illner, R., Pulvirenti, M.: The mathematical theory of dilute gases. Applied Mathematical Sciences. 106. New York: Springer-Verlag, 1994 7. Coron, F., Golse, F., Sulem, C.: A classification of well-posed kinetic layer problems. Commun. Pure Appl. Math. 41, 409–435 (1988) 8. DiPerna, R.J., Lions, P.-L.: On the Cauchy problem for Boltzmann equations: global existence and weak stability. Ann. of Math. 130(2), 321–366 (1989) 9. Ellis, R., Pinsky, M.: The first and second fluid approximations to the linearized Boltzmann equation. J. Math. Pures Appl. 54(9), 125–156 (1975) 10. Golse, F., Poupaud, F.: Stationary solutions of the linearized Boltzmann equation in a half-space. Math. Meth. Appl. Sci. 11, 483–502 (1989) 11. Grad, H.: Asymptotic equivalence of the Navier-Stokes and nonlinear Boltzmann equations. Proc. Symp. Appl. Math. 17(ed. R.Finn), Providence, RI: Amer. Math. Soc., 1965, pp. 154–183 12. Grad, H.: Asymptotic theory of the Boltzmann equation. Phys. Fluids 6, 147–181 (1963a) 13. Grad, H.: Asymptotic theory of the Boltzmann equation. In: Rarefied Gas Dynamics, Laurmann, J. A. ed. 1, New York: Academic Press, 1963, pp. 26–59 14. Grad, H.: Singular and nonuniform limits of solutions of the Boltzmann equation. Transport Theory (Proc. Sympos. Appl. Math., New York, 1967), SIAM-AMS Proc., Vol. I Providence, RI: Amer. Math. Soc. 1969, pp. 269–308 15. Grad, H.: High Frequency sound according to the Boltzmann equation. J. SIAM Appl. Math. 14(4), 935–955 (1966) 16. Hilbert, D.: Grundzüge einer Allgemeinen Theorie der Linearen Integralgleichungen. Leipzig, Teubner Chap. 22, 1912 17. Kawashima, S., Matsumura, A., Nishida, T.: On the fluid-dynamical approximation to the Boltzmann equation at the level of the Navier-Stokes equation. Commun. Math. Phys. 70(2), 97–124 (1979) 18. Kaniel, S., Shinbrot, M.: The Boltzmann equation. I. Uniqueness and local existence. Commun. Math. Phys. 58(1), 65–84 (1978) 19. Kato, T.: Perturbation theory for linear operators. Die Grundlehren der mathematischen Wissenschaften, Band 132, New York: Springer-Verlag, Inc., 1966 20. Kawashima, S., Matsumura, A., Nishida, T.: On the fluid-dynamical approximation to the Boltzmann equation at the level of the Navier-Stokes equation. Commun. Math. Phys. 70(2), 97–124 (1979) 21. Lee, M.-Y., Liu, T.-P., Yu, S.-H.: Large-Time Behavior of Solutions for the Boltzmann Equation with Hard potentials. Commun. Math. Phys. 246(1), 133–179 (2006) 22. Liu, T.-P.: Pointwise Convergence to Shock Waves for Viscous Conservation Laws. Commun. Pure and Appl. Math. 50(11), 1113–1182 (1997) 23. Liu, T.-P., Yu, S.-H.: Boltzmann Equation: Micro-Macro Decompositions and Positivity of Shock Profiles. Commun. Math. Phys. 246(1), 133–179 (2004) 24. Liu, T.-P., Yu, S.-H.: Green’s function and large-Time Behavior of Solutions for One-Dimensional Boltzmann Equation. Commun. Pure Appl. Math. 57, 1543–1608 (2004) 25. Liu, T.-P., Yu, S.-H.: Green’s Function of Boltzmann Equation, 3-D waves. Bulletin, Inst. Math. Academia Sinica (N.S.) 1(1), 1–78 (2006) 26. Liu, T.-P., Yu, S.-H.: Initial-Boundary Value Problem for One-Dimensional Wave Solutions of the Boltzmann Equation. Comm. Pure Appl. Math. 60(3), 295–356 (2007) 27. Liu, T.-P., Zeng, Y.: Large time behavior of solutions for general quasilinear hyperbolic-parabolic systems of conservation laws. Mem. Amer. Math. Soc. 125, 599 (1997) 28. Nishida, T.: Fluid dynamical limit of the nonlinear Boltzmann equation to the level of the compressible Euler equation. Commun. Math. Phys. 61(2), 119–148 (1978) 29. Sone, Y.: Kinetic Theory and Fluid Dynamics. Basel-Boston: Birkhauser, 2002 30. Sone, Y., Ohwada, T., Aoki, K.: Evaporation and condensation on the plane condense phase: Numerical analysis of the linearized Boltzmann equation for hard-sphere molecules. Phys. Fluids A 1(8), 1398–1405 (1989) 31. Ukai, S.: On the existence of global solutions of mixed problem for non-linear Boltzmann equation. Proc. Japan Acad. 50, 179–184 (1974) 32. Ukai, S.: Les solutions globales de l’équation de Boltzmann dans l’espace tout entier et dans le demiespace. C.R.Acad. Sci. Paris Ser. A-B 282(6), A317–A320 (1976) 33. Ukai, S., Yang, T., Yu, S.-H.: Nonlinear Boundary Layers of the Boltzmann Equation: I. Existence. Commun. Math. Phys. 236, 373–393 (2003)

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Commun. Math. Phys. 281, 349–367 (2008) Digital Object Identifier (DOI) 10.1007/s00220-008-0489-2

Communications in

Mathematical Physics

The BCS Functional for General Pair Interactions Christian Hainzl1 , Eman Hamza2 , Robert Seiringer3 , Jan Philip Solovej4 1 Departments of Mathematics and Physics, UAB, 1300 University Blvd,

Birmingham, AL 35294, USA. E-mail: [email protected]

2 Department of Mathematics, UAB, 1300 University Blvd, Birmingham, AL 35294, USA.

E-mail: [email protected]

3 Department of Physics, Princeton University, Princeton, NJ 08542-0708, USA.

E-mail: [email protected]

4 Department of Mathematics, University of Copenhagen, Universitetsparken 5,

2100 Copenhagen, Denmark. E-mail: [email protected] Received: 5 April 2007 / Accepted: 21 November 2007 Published online: 1 May 2008 – © The Authors 2008

Abstract: The Bardeen-Cooper-Schrieffer (BCS) functional has recently received renewed attention as a description of fermionic gases interacting with local pairwise interactions. We present here a rigorous analysis of the BCS functional for general pair interaction potentials. For both zero and positive temperature, we show that the existence of a non-trivial solution of the nonlinear BCS gap equation is equivalent to the existence of a negative eigenvalue of a certain linear operator. From this we conclude the existence of a critical temperature below which the BCS pairing wave function does not vanish identically. For attractive potentials, we prove that the critical temperature is non-zero and exponentially small in the strength of the potential.

1. Introduction Over the last few years, significant attention has been devoted to ultra-cold gases of fermionic atoms, both from the experimental and the theoretical point of view. Because of the ability to magnetically tune Feshbach resonances it has become possible to vary interatomic interactions via external magnetic fields, and hence to control the strength of the interactions among the atoms. This method was used in remarkable experiments to study the crossover from the regime of Bose-Einstein condensation (BEC) of tightly bound diatomic molecules to the Bardeen-Cooper-Schrieffer (BCS) regime of weakly bound Cooper pairs in ultra-cold gases of fermionic atoms. For a recent review on this topic we refer to [5,7]. The main purpose of this paper is to give a mathematically precise study of the BCS regime. This regime is usually described by the BCS functional, derived by Leggett [10] based on the original work of Bardeen-Cooper-Schrieffer [3]. We note that while in the original BCS model simple phonon-induced non-local interaction potentials were considered, the interaction potentials are local for atomic Fermi gases. In order to allow © 2008 by the authors. This paper may be reproduced, in its entirety, for non-commercial purposes.

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for a wide range of applications, we will consider here the BCS functional for general pair interaction potentials V . One purpose of our paper is to obtain necessary and sufficient conditions on V for the system to display superfluid behavior. Recall that in the BCS model superfluidity (or, rather, superconductivity in the original BCS case, where electrons in a crystal structure have been considered) is related to the non-vanishing of the pairing wave function, describing particle pairs with opposite spin and zero total momentum. That is, superfluidity occurs if the paired state is energetically favored over the normal state described by the Fermi-Dirac distribution. More precisely, for arbitrary temperature T = 1/β ≥ 0 a system is in a superfluid state if there exists a non-trivial solution  to the BCS gap equation
E(q) 1 (q) tanh dq, (1.1) ( p) = − Vˆ ( p − q) 3/2 (2π ) E(q) 2T R3  with E( p) = ( p 2 − µ)2 + |( p)|2 . Here, µ denotes the chemical potential. Notice that the gap equation is highly nonlinear. In Theorem 1 below we will show that the existence of a non-trivial solution to the gap equation is equivalent to the existence of a negative eigenvalue of the corresponding linear operator K β,µ + V,

K β,µ = ( p 2 − µ)

eβ( p

2 −µ)

2 eβ( p −µ)

+1 −1

.

(1.2)

The operator K β,µ is understood as a multiplication operator in momentum space, i.e., p 2 equals − in configuration space. Observe that in the limit T → 0 this operator reduces to the Schrödinger type operator | p 2 − µ| + V.

(1.3)

This linear characterization of the existence of solutions to the nonlinear BCS equation represents a considerable simplification of the analysis. In particular, it proves the existence of solutions to (1.1) for a wide range of interaction potentials V , and hence generalizes previous results [4,13,18,19] valid only for non-local V ’s under suitable assumptions. Moreover, the monotonicity of K β,µ in β guarantees the existence of a critical temperature Tc , with 0 ≤ Tc < ∞, such that superfluidity occurs whenever T < Tc and normal behavior for T ≥ Tc . In particular, there is a phase transition at Tc . For positive values of µ, we shall show that Tc is strictly positive whenever the negative part of V , denoted by V− , is non-vanishing and the positive part V+ is small enough in a suitable sense. Moreover, we prove that Tc is exponentially small in the strength of the interaction potential. More precisely, if the potential is given by λV for some fixed V ∈ L 1 ∩ L 3/2 , then Tc
e−1/λ as λ → 0. This generalizes the corresponding result in [3, Eq. (3.29)] to a very large class of interaction potentials. For negative potentials, also a corresponding lower bound will be shown to hold. In a forthcoming work [9], the precise asymptotics of Tc for λ → 0 will be investigated. At zero temperature, the existence of a non-trivial solution of the BCS gap equation is also related to the non-vanishing of the energy gap between the ground state and the first excited state in the effective quadratic BCS Hamiltonian on Fock space. For very specific attractive interaction potentials, it has been argued that this gap is non-vanishing, see e.g. [10,14,15]. We note, however, that the existence of such an energy gap is not

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necessarily implied by the non-vanishing of the solution to the BCS equation. In fact, the presence of a gap is not necessary for our analysis. We consider general potentials where such an energy gap need not be present, a priori. We do prove that the gap is non-vanishing for a certain class of interaction potentials, however. We note that our results are relevant for actual experiments on cold atomic gases as long as the description via the BCS functional is applicable. There is an extensive literature on this subject, see e.g. [1,5–7,14–17]. According to Leggett [10] this is the case for weak interactions as long as the range of the potential V is much smaller than µ−1/2 (in appropriate units), i.e., for weakly coupled dilute gases. In particular, “weakly coupled” means that the corresponding Schrödinger operator p 2 + V typically does not allow for a negative energy bound state since otherwise the system is in the BEC regime of tightly bound bosonic molecules. We refer to [10] for a detailed discussion on this question. 2. Model and Main Results The precise definition of the BCS functional considered in this paper is given as follows. For the convenience of the reader, we describe in the Appendix the motivation and physical background; the discussion there follows Leggett’s derivation in [10]. See also [2] for a detailed study of the BCS approximation to the Hubbard model. We use the  standard convention α( ˆ p) = (2π )−3/2 R3 α(x)e−i px d x for the Fourier transform. Definition 1. Let D denote the set of pairs of functions (γ , α), with γ ∈ L 1 (R3 , (1 + p 2 )dp), 0 ≤ γ ( p) ≤ 1, and α ∈ H 1 (R3 , d x), satisfying |α( ˆ p)|2 ≤ γ ( p)(1 − γ ( p)). 3/2 3 Let V ∈ L (R , d x) be real-valued and µ ∈ R. For T = 1/β ≥ 0 and (γ , α) ∈ D, the energy functional Fβ is defined as

1 (2.1) Fβ (γ , α) = ( p 2 − µ)γ ( p)dp + |α(x)|2 V (x)d x − S(γ , α), β where


S(γ , α) = −

     s( p) ln s( p) + 1 − s( p) ln 1 − s( p) dp,

with s( p) determined by s(1 − s) = γ (1 − γ ) − |α| ˆ 2. As explained in the Appendix, (2.1) has the physical interpretation of − 21 (2π )3/2 times the pressure of a system of spin 1/2 fermions at temperature T interacting via a pair potential given by 2V . It is natural to require V ∈ L 3/2 (R3 ), as we do here, since this guarantees that V is relatively form-bounded with respect to p 2 (see, e.g., Sect. 11 in [11]). In fact, it would be enough to require the negative part of V to be in L 3/2 + L ∞  , and allow for more general positive parts of V . For simplicity of the presentation, we will not pursue this generalization here. In the absence of a potential V , it is easy to see that Fβ is minimized on D by the 2 choice α ≡ 0 and γ ( p) = γ0 ( p) = [eβ( p −µ) + 1]−1 . We refer to this state as the normal state of the system. We will be concerned with the question of whether a non-vanishing α can lower the energy for given potential V .

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Note that at T = 0 it is natural to formulate the model in terms of a functional that depends only on α and not on γ . Namely, for fixed |α( ˆ p)|2 , the optimal choice of γ ( p) is   1 (1 + 1 − 4|α( ˆ p)|2 ) for p 2 < µ γ ( p) = 21 (2.2) ˆ p)|2 ) for p 2 > µ 2 (1 − 1 − 4|α( in this case. Similarly, also at T > 0 it is enough to minimize Fβ only over pairs (γ , α) satisfying a certain relation independent of V , which is displayed in Eq. (3.4) below. According to the usual interpretation of the BCS theory, the normal state is unstable if the energy can be lowered by the formation of Cooper pairs, i.e., if Fβ can be lowered by choosing α = 0. The following theorem, which is the main result of this paper, shows that this property is equivalent to the existence of a negative eigenvalue of a certain linear operator. Theorem 1. Let V ∈ L 3/2 (R3 ), µ ∈ R, and ∞ > T = statements are equivalent:

1 β

≥ 0. Then the following

(i) The normal state (γ0 , 0) is unstable under pair formation, i.e., inf

(γ ,α)∈D

Fβ (γ , α) < Fβ (γ0 , 0).

(ii) There exists a pair (γ , α) ∈ D, with α = 0, such that ( p) = −

p2 − µ γ ( p) −

1 2

α( ˆ p)

(2.3)

satisfies the BCS gap equation

  βE  = −Vˆ ∗ tanh , with E( p) = ( p 2 − µ)2 + |( p)|2 . (2.4) E 2 (iii) The linear operator K β,µ + V, K β,µ = ( p − µ) 2

eβ( p

2 −µ)

2 eβ( p −µ)

+1 −1

,

(2.5)

has at least one negative eigenvalue. Here, ∗ denotes convolution in the BCS gap equation (2.4). In order to avoid having to write factors of 2π , we find it convenient to define convolution with a factor (2π )−3/2  in front, i.e., ( f ∗ g)( p) := (2π )−3/2 R3 f ( p − q)g(q)dq. Remark 1. The operator (2.5) is obtained by taking the second derivative of the functional Fβ with respect to α at the normal state (γ0 , 0) (see Eq. (3.10) below). Since the normal state is a critical point of Fβ , it is then easy to prove that (iii) implies (i). The opposite direction is not immediate, however. It says that if (γ0 , 0) is not a minimizer, it can also not be locally stable. Remark 2. As mentioned above, a minimizer of Fβ satisfies the V -independent relation (2.2) at T = 0, and (3.4) at T > 0. It is not necessary to assume that these equations hold in part (ii) of Theorem 1, however.

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Note that in the zero temperature case T = 0 the gap equation (2.4) takes the form   = −Vˆ ∗ , E (compare with [3,8,10,12]) and the operator in (2.5) is given by | p 2 − µ| + V . For any potential V we shall prove the existence of a critical temperature 0 ≤ Tc = 1 β c < ∞ at which a phase transition from a normal to superfluid state takes place. More precisely, we shall show that the gap equation (2.4) has a non-trivial solution for all T < Tc , whereas it has no non-trivial solution if T ≥ Tc . This follows from the next theorem, in combination with Theorem 1. Theorem 2. For any V ∈ L 3/2 (R3 ) there exists a critical temperature 0 ≤ Tc = β1 < ∞ c such that inf spec (K β,µ + V ) < 0 K β,µ + V ≥ 0

if T < Tc if T ≥ Tc .

With the aid of the Birman-Schwinger principle, it is possible to characterize the value of the critical temperature Tc , at least if Tc > 0. In order to do this, it is convenient to decompose V into its positive and negative parts, i.e., V = V+ − V− , with V+ ≥ 0, V− ≥ 0 and V− V+ = 0. We shall show in Proposition 4 that Tc > 0 satisfies the equation 1/2 1 1/2 (2.6) V V 1= − . − K βc ,µ + V+ Note that since K β,µ ≥ 2/β, this leads immediately to the general rough upper bound Tc ≤ 21 V− ∞ .

(2.7)

For negative potentials V and a positive chemical potential µ, we shall show that the critical temperature is always positive, i.e., that | p 2 − µ| + V has a negative energy bound state. We can even say more, namely we can guarantee superfluidity for small temperature whenever the positive part V+ is not too large in a suitable sense. Theorem 3. Let V ∈ L 3/2 (R3 ) be not identically zero, and let µ > 0. (i) If V ≤ 0, then the corresponding critical temperature is positive, Tc = β1c > 0. (ii) Let V = V+ − V− , and let β > βc (−V− ), where βc (−V− ) is the critical inverse temperature for the potential −V− . Then the system at temperature 1/β is still in 1/2 −1 1/2 a superfluid state if V+ K β,µ V+ is small enough. In particular, this theorem states that for any negative potential one can add a sufficiently small positive part such that the critical temperature Tc is still strictly positive. The next question concerns the magnitude of Tc . The rough upper bound given in (2.7) turns out to be much too rough, in general. In fact, the following theorem shows that Tc is always exponentially small in the strength of V . To be able to state the bound precisely, let a := inf t>0 t (et + 1)/((t + 2)(et − 1)). Numerically, a ≈ 0.654. Let further


∞ 1 1 1 2 . (2.8) f (t) := dp p − 2π 2 0 | p 2 − 1| + t p2 + 1 + t Note that f is a positive and monotone decreasing function, with f (t) ∼ ln(1/t) as t → 0. We obtain the following upper bound on the critical temperature Tc .

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Theorem 4. Let f −1 be the inverse of the function f in (2.8). Let µ > 0, and let  4/3 V = V+ −V− , with V ∈ L 3/2 , V− ∈ L 1 ∩L 3/2 , and with 13 π2

V− 3/2 < a ≈ 0.654. Then the critical temperature satisfies the upper bound

  4/3

V− 3/2 µ −1 a − 13 π2 Tc ≤ f . (2.9) 2 µ1/2 V− 1 Note that, in particular, f −1 (t) ∼ e−t for large t. Hence the critical temperature is exponentially small in the potential for V− ∈ L 1 ∩ L 3/2 . For negative potentials, one can show that the bound given in Theorem 4 is indeed optimal. More precisely, if the potential is given by λV for some fixed negative V ∈ L 3/2 , we shall show in Proposition 5 the lower bound Tc ≥ gµ,V (λ), where gµ,V is some positive function depending only on µ and V satisfying gµ,V (t) ∼ e−1/t for small t. Combining both bounds then implies that for negative potentials V the critical temperature is exponentially small in the coupling parameter. This generalizes well-known calculations in the physics literature, see e.g. [3, Eq. (3.29)], to a very large class of interaction potentials. Note that, in particular, this result implies that it is not possible to calculate the critical temperature via a perturbative expansion in V . Finally, we comment on the continuity properties of minimizers of Fβ . In fact, for positive temperature T > 0 the minimizing γ and αˆ can be shown to be continuous functions. (We show this in Prop. 3 below.) This may not be surprising, since already 2 the minimizer without potential, γ0 ( p) = 1/(eβ( p −µ) + 1), is continuous. But in the case of T = 0 the situation is not so clear, since the normal state is a step function with jump at p 2 = µ for µ > 0. The continuity of a minimizer γ at T = 0 is closely related to the existence of an energy gap, given by  := inf E( p), p

(2.10)

with E defined in (2.4). As explained in the Appendix, E( p) has the interpretation of the dispersion relation for quasi-particle excitations. The following shows that in case V has a strictly negative Fourier transform, γ is continuous even at T = 0 and the gap  is non-vanishing. Proposition 1. Assume that inf spec (| p 2 − µ| + V ) < 0, and that Vˆ is strictly negative. Let (γ , α) ∈ D be a minimizer of F∞ . Then  > 0 and γ is a continuous function. This result implies, in particular, that  > 0 for negative V with strictly negative Fourier transform. We note that this property need not necessarily be true for all potentials V , however. We conclude this section with two remarks. Remark 3. Our results can be generalized to non-local potentials given by an integral kernel V (x, y). Exactly such a non-local potential was used in the original paper of ˆ p) = 1 if BCS [3] who considered V of the form V (x, y) = −V0 φ(x)φ(y), with φ( 2 | p − µ| ≤ h and 0 otherwise. This models an effective potential arising from electronphonon interactions in crystals. In this case, the minimizer of Fβ can be evaluated explicitly (see, e.g., [12]).

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In particular, our analysis can be used to generalize previous results [4,13,18,19] on the existence of solutions of the gap equation, as already mentioned in the Introduction. None of this previous work allows for an interaction kernel Vˆ (k, k ) of the form Vˆ (k−k ), however, since such a kernel does not satisfy appropriate L p conditions. Such conditions were necessary for utilization of fixed point arguments in the previous investigations of this problem. Remark 4. In [6,15] the BCS-BEC crossover is studied numerically for certain potentials V that are negative and have negative Fourier transform. These potentials satisfy all the assumptions of our results stated above. In the following Sects. 3 and 4, the proofs of the claims of this section will be given. 3. Properties of the Minimizer of Fβ In this section we shall give the proof of Theorem 1 and Proposition 1 stated above. We start by showing the existence of a minimizer of Fβ . Proposition 2. There exists a minimizer of Fβ in D. Proof. We first show that Fβ dominates both the L 1 (R3 , (1 + p 2 )dp) norm of γ and the H 1 (R3 , d x) norm of α. Hence any minimizing sequence will be bounded in these norms. We have

3 2 Fβ (γ , α) ≥ C1 + ( p − µ)γ ( p)dp + |α(x)|2 V (x)d x, 4 where C1 =





β 1 1 1 2 ( p 2 − µ)γ ( p)dp − S(γ , α) = − ln(1 + e− 4 ( p −µ) )dp. β β (γ ,α)∈D 4 inf

Since V ∈ L 3/2 by assumption, it is relatively bounded with respect to − (in the sense of quadratic forms), and hence C2 = inf spec ( p 2 /4+V ) is finite. Using |α( ˆ p)|2 ≤ γ ( p), we thus have that


1 2 2 p γ ( p)dp + V (x)|α(x)| d x ≥ C2 γ ( p)dp. 4 Using again that |α( ˆ p)|2 ≤ γ ( p) ≤ 1, it follows that 1 1 Fβ (γ , α) ≥ −A + α 2H 1 (R3 ,d x) + γ L 1 (R3 ,(1+ p2 )d p) , 8 8 where A = −C1 −




p 2 /4 − 3µ/4 − 1/4 + C2

(3.1)

 −

dp,

with [ · ]− = min{ ·, 0} denoting the negative part. To show that a minimizer of Fβ exists in D, we pick a minimizing sequence (γn , αn ) ∈ D, with Fβ (γn , αn ) ≤ 0. From (3.1) we conclude that αn 2H 1 ≤ 8A,

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and hence we can find a subsequence that converges weakly to some α˜ ∈ H 1 . Since V ∈ L 3/2 (R3 ), this implies that

2 lim V (x)|αn (x)|2 d x = V (x)|α(x)| ˜ dx n→∞ R3

R3

[11, Thm. 11.4]. It remains to show that the remaining part of the functional, 
    1 1 0 2 Fβ (γ , α) = ( p − µ)γ ( p)dp+ s( p) ln s( p)+ 1 − s( p) ln 1 − s( p) dp, β β (3.2) is weakly lower semicontinuous. Note that Fβ0 is jointly convex in (γ , α), and that its domain D is a convex set. We already know that αn α˜ weakly in H 1 (R3 ). Moreover, since γn is uniformly bounded in L 1 (R3 ) ∩ L ∞ (R3 ), we can find a subsequence such that γn γ˜ weakly in L p (R3 ) for some 1 < p < ∞. We can then apply Mazur’s Theorem [11, Theorem 2.13] to construct a new sequence as convex combinations of the old one, which now converges strongly to (γ˜ , α) ˜ in L p (R3 ) × L 2 (R3 ). By going to a subsequence, we can also assume that γn → γ and αˆ n → αˆ˜ pointwise [11, Theorem 2.7]. Because of convexity of Fβ0 , this new sequence is again a minimizing sequence. Note that the integrand in (3.2) is bounded from below independently of γ and α 2 by −β −1 ln(1 + e−β( p −µ) ). Since this function is integrable, we can apply Fatou’s Lemma [11, Lemma 1.7], together with the pointwise convergence, to conclude that lim inf Fβ0 (γn , αn ) ≥ Fβ0 (γ˜ , α). ˜ We have thus shown that ˜ ≤ lim inf Fβ (γn , αn ). Fβ (γ˜ , α) n→∞

It is easy to see that (γ˜ , α) ˜ ∈ D, hence it is a minimizer. This proves the claim.

 

Next, we establish a few auxiliary results that will be needed below in the proof of Theorem 1. Lemma 1. Let (γ , α) ∈ D be a minimizer of Fβ for 0 < β < ∞. Then, for a.e. p ∈ R3 , α( ˆ p) , 2γ ( p) − 1 

2γ ( p) − 1 ( p 2 − µ) = − f 2 (γ ( p) − 21 )2 + |α( ˆ p)|2 , β

(Vˆ ∗ α)( ˆ p) = ( p 2 − µ)

where f (a) :=

1 a

ln

1+a 1−a

(3.3) (3.4)

for 0 ≤ a < 1.

ˆ p)|2 + (γ ( p) − 1/2)2 ≤ 1/4 − }. Let Proof. For  > 0, let A := { p ∈ R3 : |α( 1 3 1 3 g ∈ H (R ) ∩ L (R ), with Fourier transform supported in A . Then (γ , α + tg) ∈ D for small enough |t|. By the Lebesgue dominated convergence theorem, it follows that 
 d Fβ (γ , α + tg) = 2 Re V (x)g(x)α(x) d x dt t=0


2 ˆ p)α( + Re (3.5) g( ˆ p) f 2 (γ ( p) − 21 )2 + |α( ˆ p)|2 dp. β

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ˆ α) ∈ D for small |t|, and we get Similarly, for gˆ real, (γ + t g, 
 d ˆ α) ˆ p)dp Fβ (γ + t g, = ( p 2 − µ)g( (3.6) dt t=0 


1 ˆ p)(2γ ( p) − 1) f 2 (γ ( p) − 21 )2 + |α( ˆ p)|2 dp. + g( β Since (γ , α) minimize Fβ by assumption, the expressions in (3.5) and (3.6) vanish. It follows that for a.e. p ∈ A we have    1 ˆ p) f 2 (γ ( p) − 21 )2 + |α( ˆ p)|2 , (Vˆ ∗ α)( ˆ p) = − α( β    1 (3.7) ( p 2 − µ) = − (2γ ( p) − 1) f 2 (γ ( p) − 21 )2 + |α( ˆ p)|2 . β To conclude that (3.7) holds a.e. in R3 , it remains to show that B := R3 \ ∪>0 A has zero By definition, if p ∈ B we have γ ( p)(1 − γ ( p)) = |α( ˆ p)|2 , i.e.  measure.  s( p) s( p) − 1 = 0. It is then easy to see that if B has non-zero measure, Fβ (γ , α) can be lowered by modifying γ and αˆ on B (since t → t ln t has a diverging derivative at zero). This contradicts the fact that (γ , α) is a minimizer, and hence B has zero measure.   Remark 5. A similar strategy as in the proof of Lemma 1 leads to the conclusion that Eq. (3.3) is also valid at T = 0. The relation between γ and α, given in Eq. (3.4) for T > 0, has to be replaced by (2.2) at T = 0, however. We omit the details. Note that Eq. (3.4) and the positivity of f immediately imply that γ ( p) ≥ 1/2 for p 2 < µ, and γ ( p) ≤ 1/2 for p 2 > µ. Of greater importance below will be the following monotonicity property, which bounds γ in the opposite direction. Lemma 2. Assume that (γ , α) ∈ D satisfies Eq. (3.4), and let γ0 ( p) = [eβ( p −µ) + 1]−1 . For a.e. p ∈ R3 , we have γ ( p) ≤ γ0 ( p) if p 2 < µ, while for p 2 > µ we have γ ( p) ≥ γ0 ( p). Moreover, these inequalities are strict if α( ˆ p) = 0. In particular, p2 − µ p2 − µ < on the support of α. ˆ 1 − 2γ0 ( p) 1 − 2γ ( p) 2

Proof. Let p 2 > µ such that α( ˆ p) = 0 and assume by way of contradiction that γ ( p) ≤ γ0 ( p). We then have  1 − γ0 ( p) < |α( ˆ p)|2 + ( 21 − γ ( p))2 . 2 Since f is a strictly monotone increasing function, we obtain

   1 2 2 f 1 − 2γ0 ( p) < f 2 |α( ˆ p)| + ( 2 − γ ( p)) and hence   (1 − 2γ0 ( p)) f 1 − 2γ0 ( p) < (1 − 2γ ( p)) f

 1 2 2 2 |α( ˆ p)| + ( 2 − γ ( p)) .

(3.8)

This, however, contradicts the fact that both (γ0 , 0) and (γ , α) are solutions of (3.4), and hence both sides of (3.8) are equal to β( p 2 − µ). The case p 2 < µ is treated similarly.  

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We now have all the necessary prerequisites at our disposal to give the proof of Theorem 1. Proof of Theorem 1. [i] ⇒ [ii]: Consider first the case T > 0. According to Lemma 1 a minimizing pair (γ , α) ∈ D with α = 0 satisfies Eqs.  (3.3) and (3.4). Using the definitions of  and E in (2.3) and (2.4) we see that (γ ( p) − 1/2)2 + |α( ˆ p)|2 = 1/2−γ ( p) E( p). Plugging this into Eq. (3.4) leads to p 2 −µ 1 p2 − µ =− f 2γ ( p) − 1 β



1 − 2γ ( p) E( p) p2 − µ





2 1 − 2γ ( p) p2 − µ tanh−1 E( p) , β E( p) 2γ ( p) − 1 p2 − µ

= or

tanh

β E( p) 1 − 2γ ( p) = E( p). 2 p2 − µ

(3.9)

( p) β E( p) Therefore, we can rewrite the relation between αˆ and  as α( ˆ p) = 2E( p) tanh 2 . Together with Eq. (3.3) this gives the required form of the gap equation. The case T = 0 is treated similarly (cf. Remark 5). [ii] ⇒ [iii]: Given T ≥ 0 and a  satisfying (2.3) for some (γ , α) ∈ D, we first show that there exists a (γ˜ , α) ˜ ∈ D yielding the same  but satisfying in addition Eq. (2.2) at ˆ˜ p) = m( p)α( T = 0 or Eq. (3.4) at T > 0, respectively. Namely, we can let α( ˆ p) and γ˜ ( p) = 1/2 + m( p)(γ ( p) − 1/2) without changing . If we choose

m( p) = sgn

p2 − µ 1 − 2γ ( p)



tanh(β E( p)/2)  , 2 (γ ( p) − 21 )2 + |α( ˆ p)|2

with E( p) as in (2.4), it is easy to see that (3.9) is satisfied. Using in addition the gap equation, the same reasoning as in the previous part of the proof leads to the conclusion that Eqs. (3.3) and (2.2) (respectively (3.4)) are satisfied for this (γ˜ , α). ˜ Hence there exists a (γ , α) ∈ D satisfying (3.3) as well as (2.2) (at T = 0) or (3.4) (at T > 0). Using (3.3) and the definition of K β,µ , we have that  α, (K β,µ + V )α = α,

 1 1 − ( p 2 − µ)α . 1 − 2γ0 ( p) 1 − 2γ ( p)

It follows from Lemma 2 that this expression is negative whenever α is not identically zero. [iii] ⇒ [i]: Consider first the case T > 0. Showing that the existence of a negative eigenvalue of the operator (2.5) implies that the minimizing α is not identically zero is equivalent to proving that α ≡ 0 implies that inf spec (K β,µ + V ) ≥ 0. First, we note that, for any δ ∈ C0∞ , (tδ)2 + (γ0 − 21 )2 ∞ < 41 for all small |t|. With the aid of the 2 ˆ exists for Lebesgue dominated convergence theorem it is easy to see that d 2 Fβ (γ0 , t δ) dt

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small t, and is given by

(γ0 ( p) − 21 )2 d2 2 2 2 ˆ ˆ F (γ , t δ) = 2 V (x)| δ(x)| d x + |δ( p)| β 0 dt 2 β |tδ( p)|2 + (γ0 ( p) − 21 )2

 × f 2 |tδ( p)|2 + (γ0 ( p) − 21 )2 dp +

1 β




t 2 |δ( p)|4 [|tδ( p)|2 + (γ0 ( p) −

1 2 2 2 ) ][γ0 ( p)(1 − γ0 ( p)) − |tδ( p)| ]

dp.

Assuming that the minimizing α is identically zero, it follows that the corresponding d2 ˆ γ = γ0 and hence dt 2 Fβ (γ0 , t δ)|t=0 ≥ 0. But 

 d2 2  ˆ ˆ F (γ , t δ) = 2 V (x)| δ(x)| d x + 2 |δ( p)|2 K β,µ ( p) dp, β 0  dt 2 t=0

(3.10)

and hence δ, (K β,µ + V )δ ≥ 0 for all δ ∈ C0∞ . This proves the statement. In the case T = 0, it is sufficient to minimize Fβ over (γ , α) ∈ D with γ of the form (2.2), as remarked earlier. For simplicity, we abuse the notation slightly and denote the resulting functional by F∞ (α). Subtracting its value for α = 0, it is given by

 1 F∞ (α) − F∞ (0) = ˆ p)|2 )dp + V (x)|α(x)|2 d x. | p 2 − µ|(1 − 1 − 4|α( 2 (3.11) Assume that | p 2 − µ| + V has a negative eigenvalue. We can then find an α ∈ C0∞ (R3 ) such that α, (| p 2 − µ| + V )α < 0. For 0 <   1 small enough, we then have F∞ (α) − F∞ (0) =  2 α, (| p 2 − µ| + V )α + O( 4 ) < 0. This implies that a minimizer α does not vanish identically.

(3.12)

 

In the following, we investigate the continuity properties of a minimizing pair (γ , α) ∈ D. Proposition 3. If (γ , α) ∈ D is a minimizer of Fβ for 0 < β < ∞, then both γ and αˆ are continuous functions. Proof. Since V ∈ L 3/2 and α ∈ H 1 , we know that Vˆ ∈ L 3 and αˆ ∈ L 6/5+ for all  > 0. Hence Vˆ ∗ αˆ is continuous [11, Thm. 2.20]. From Eq. (3.3) and the definition of  and E given in (2.3) and (2.4), we observe that Vˆ ∗ αˆ = − 21  and hence both  and E are continuous functions. As in Eq. (3.9) in the proof of Thm. 1, Eq. (3.4) can be rewritten in the form p2 − µ eβ E( p) + 1 = − β E( p) E( p). 2γ ( p) − 1 e −1

(3.13)

The right side of this equation is continuous and does not vanish, not even when E( p) = 0 (which could happen, in principle, for p 2 = µ). Hence γ is continuous. Using again Eq. (3.3), this also implies continuity of α. ˆ  

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The situation is different at zero temperature, however. In the limit β → ∞, the right side of Eq. (3.13) vanishes in case E( p) = 0. Hence γ could possibly be discontinuous if p 2 = µ. In fact, this is what happens in the case of vanishing potential. On the other hand, a non-vanishing gap  = inf p E( p) > 0 implies continuity of γ . At T = 0, the gap equation (2.4) takes the form E( p)α( ˆ p) = −(Vˆ ∗ α)( ˆ p), with E( p) = 

(3.14)

| p 2 − µ|

. Hence the gap  is guaranteed to be non-zero if the right 1 − 4|α( ˆ p)|2 side of (3.14) vanishes nowhere on the sphere p 2 = µ. This can be easily shown in case Vˆ is strictly negative, which is the content of our Proposition 1. Proof of Proposition 1. Since Vˆ ≤ 0,

α( ˆ Vˆ ∗ α) ˆ ≥ |α|( ˆ Vˆ ∗ |α|). ˆ

(3.15)

Moreover, since Vˆ is assumed to be strictly negative, there can be equality in (3.15) only if α( ˆ p) = eiκ |α( ˆ p)| for some constant κ ∈ R. This implies that this is the case for a minimizer of F∞ , and hence Vˆ ∗ αˆ vanishes nowhere. The statement then follows from Eq. (3.14).   4. The Critical Temperature In this section we investigate the relation between properties of the interaction potential V and the critical temperature Tc at which the phase transition takes place. In particular, we shall prove Theorems 2–4. Proof of Theorem 2. Let βc = inf β {β > 0 : inf spec(K β,µ + V ) < 0}. It is easy to see that βc > 0, since V is relatively form-bounded with respect to | p 2 − µ| and K β,µ ≥ c(| p 2 − µ| + 1/β) for some c > 0 (cf. Lemma 3 below). From the definition of K β,µ in (2.5) one easily sees that K β1 ,µ ≤ K β2 ,µ

if β1 ≥ β2 .

(4.1)

It follows that inf spec(K β,µ + V ) < 0 for all β > βc . Moreover, since K β,µ depends continuously on β (in a norm resolvent sense), we have that K βc ,µ + V ≥ 0. Setting βc = ∞ in the case inf spec (K β,µ + V ) ≥ 0 for all β and letting Tc = 1/βc finishes the proof.   Next, we prove the first assertion of Theorem 3, namely that for any negative potential the critical temperature is strictly positive. Proof of Theorem 3(i). In light of Theorem 2 it suffices to prove that inf spec(| p 2 − µ| + V ) < 0. According to the Birman-Schwinger principle, this is equivalent to the fact that for some e > 0, the Birman-Schwinger operator Be := |V |1/2 (| p 2 − µ| + e)−1 |V |1/2

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has an eigenvalue 1. Notice that lime→∞ Be = 0 since V ∈ L 3/2 is relatively bounded with respect to p 2 (in the sense of quadratic forms). Because of continuity in e, it thus suffices to find an e > 0 such that Be > 1. Hence we have to show that there exists a φ ∈ L 2 (R3 ) with φ 2 = 1 such that φ, Be φ ≥ 1 for an appropriate e. For this purpose, we choose a normalized function
φ in such a way that φ|V |1/2 does not vanish in a neighborhood of some point p with
p 2 = µ. We can, for instance, choose φ to be rapidly decaying, in which case |φ|V |1/2 | is continuous and hence strictly positive on some small ball. This ball can then be made to overlap with the sphere p 2 = µ by simply multiplying φ by a factor eiq x for appropriate
q, which translates the function φ|V |1/2 by q. In this way,
1
φ, Be φ = |φ|V dp → ∞ as e → 0. |1/2 ( p)|2 2 | p − µ| + e Thus there exists e > 0 such that φ, Be φ > 1.

 

Before proceeding with the proof of the second part of Theorem 3, we introduce an alternative characterization for the critical temperature Tc in the case Tc > 0, which we already referred to in Eq. (2.6). Proposition 4. For all V ∈ L 3/2 (R3 ), such that V = V+ − V− , with V+ ≥ 0, V− ≥ 0 and V− V+ = 0, the critical temperature Tc = 1/βc is characterized by 1/2 1 1/2 V V − = 1, − K βc ,µ + V+ whenever Tc = 0. Recall that K β,µ ≥ 2/β, hence K β,µ + V+ has a bounded inverse. β

Proof. Let the operator Be be defined as Beβ = V−

1/2

1 1/2 V . K β,µ + V+ + e −

This operator is compact (in fact, Hilbert-Schmidt) and depends continuously on β and e. β Moreover, Be is strictly decreasing in e and 1/β. Since Tc > 0 by assumption, we know from Theorem 2 that K β,µ + V has a negative β eigenvalue for T < Tc . Hence the Birman-Schwinger principle implies that Be > 1 β for 1/β and e small enough. On the other hand, B0 → 0 as β → 0. Thus there exists β˜ a unique β˜ such that B0 = 1. ˜ Beβ > 1 for e small enough, and hence K β,µ + V has a negative energy For β > β, ˜ however, Beβ < 1 for all e ≥ 0, and hence K β,µ + V ≥ 0. bound state. For β < β, This shows that β˜ = βc .   From Prop. 4 it is then easy to conclude that the critical temperature stays strictly positive under small positive perturbations to negative potentials.

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Proof of Theorem 3(ii). According to Prop. 4, it suffices to prove that for β > βc (−V− ) 1/2 1/2 and for λ > 0 small enough, V− (K β,µ + λV+ )−1 V− > 1. In order to simplify the 1/2 1/2 notation let Bλ = V− (K β,µ + λV+ )−1 V− . It follows that Bλ = ≥

1/2 V−

1

1



−1/2 −1/2 1 + λK β,µ V+ K β,µ   −1/2 −1/2 −1 B0 1 + λ K β,µ V+ K β,µ

. 1/2 K β,µ

1 1/2 K β,µ

1/2

V−

Since V+ ∈ L 3/2 by assumption, it is relatively form-bounded with respect to K β,µ , and −1/2 −1/2 1/2 −1 1/2 hence K β,µ V+ K β,µ = V+ K β,µ V+ is finite. Therefore Bλ > 1 if B0 > 1 and λ is small enough.   Next, we shall derive upper and lower bounds on the critical temperature. These bounds are similar to the ones found in the literature, but generalize these results from simple rank 1 potentials as in [3] to very general pair interactions. We start with the upper bound given in Theorem 4. To this aim we first prove two auxiliary lemmas. Lemma 3. Let K β,µ ( p) = ( p 2 − µ)(eβ( p −µ) + 1)(eβ( p −µ) − 1)−1 . Then there is a constant a > 0 such that



2 2 ≤ K β,µ ( p) ≤ | p 2 − µ| + . (4.2) a | p 2 − µ| + β β 2

2

Proof. Let g(t) = t (et + 1)/((t + 2)(et − 1)) for t ≥ 0. It is easy to see that g(t) ≤ 1. Moreover, g(0) = 1 and limt→∞ g(t) = 1, hence a := inf t>0 g(t) > 0. In fact, a ≈ 0.654 numerically. Since K β,µ ( p) = (| p 2 − µ| + 2/β)g(β| p 2 − µ|), this implies the statement.   Lemma 4. Assume that V ∈ L 1 ∩ L 3/2 and that µ > 0. Then 1/2 |V |



1 1 2 4/3 1/2 1/2 ≤ µ |V |

V

f (e/µ) +

V 3/2 , 1 | p 2 − µ| + e 3 π

(4.3)

where f is defined in (2.8). Proof. Define ke,µ ( p) ≥ 0 by 1 1 = 2 + ke,µ ( p). | p 2 − µ| + e p +µ+e Then 1/2 |V |

(4.4)

1/2 1 1 1/2 1/2 1/2 1/2 |V | | ≤ + |V | |V | k ( p)|V | |V . e,µ | p 2 − µ| + e p2 + µ + e (4.5)

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363

The last term is bounded by 1/2 |V | ke,µ ( p)|V |1/2 = sup ψ|V |1/2 , kˆe,µ ∗ (ψ|V |1/2 )

ψ 2 =1

≤ (2π )−3/2 kˆe,µ ∞ V 1 ≤ (2π )−3 ke,µ 1 V 1 . Moreover, ke,µ 1 = (2π )3 µ1/2 f (e/µ), with f given in (2.8). For the first term on the right side of (4.5), the use of the Hardy-Littlewood-Sobolev inequality and Hölder’s inequality yields

  1/2 1 2 4/3 1 1/2 1/2 1 1/2 |V | |V | ≤ sup ψ|V | , 2 ψ|V | ≤

V 3/2 . p2 + µ + e p 3 π

ψ 2 =1   Equipped with these two lemmas we are now able to prove the upper bound on the critical temperature stated in Theorem 4. Proof of Theorem 4. Assume that Tc > 0, for otherwise there is nothing to prove. According to Proposition 4, the critical temperature Tc = 1/βc is determined by the equation 1/2 1 1/2 V V− − K = 1. βc ,µ + V+ Using Lemmas 3 and 4 as well as V+ ≥ 0, we thus have 1 1 1/2 1/2 V 1 ≤ V− a | p 2 − µ| + 2/βc −

 1 2 4/3 µ1/2 V− 1  ≤ f 2/(βc µ) +

V− 3/2 . a 3a π Since f is monotone decreasing, this implies the statement.

 

Finally, we turn to a lower bound for the critical temperature. We assume that the interaction potential is given by λV , with V ≤ 0 not identically zero, and λ > 0 a coupling parameter. Proposition 5. Let V ∈ L 3/2 , with V ≤ 0 not identically zero, and let µ > 0. Then there exists a monotone increasing strictly positive function gµ,V , with gµ,V (t) ∼ e−1/t as t → 0, such that the critical temperature for the potential λV satisfies the lower bound Tc ≥ gµ,V (λ)

(4.6)

for λ > 0. Proof. We have already shown in Thm. 3 that Tc > 0 for positive λ. By Prop. 4, the fact that V is non-positive and (4.2) we have that 1/2 1/2 1 1 1/2 1/2 |V | ≥ |V | |V | . 1/λ = |V | 2 K βc ,µ | p − µ| + 2/βc

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For any function ψ ∈ L 2 with ψ 2 = 1 we therefore obtain
 2 1  
|1/2 ( p) dp. 1/λ ≥ ψ|V | p 2 − µ| + 2/βc The latter integral is monotone increasing in βc and diverges logarithmically as βc → ∞
for appropriate ψ such that ψ|V |1/2 ( p) does not vanish identically if p 2 = µ. (See the proof of Thm. 3(i) for an argument concerning the existence of such a ψ.) This implies the statement.   Acknowledgements. The authors would like to thank George Bruun, Thomas Hoffmann-Ostenhof and Michael Loss for valuable suggestions and remarks. C.H., E.H., and R.S. thank the Erwin Schrödinger Institute in Vienna for the kind hospitality during part of this work. R.S. acknowledges partial support by U.S. National Science Foundation grant PHY-0353181 and by an Alfred P. Sloan Fellowship.

Appendix A. Motivation and Physical Background In this Appendix, we shall briefly explain the derivation of the BCS functional defined in Definition 1. Our presentation follows the one of Leggett in [10]. Similar derivations can be found in [8,12]. The BCS functional is obtained from several approximations and assumptions on the underlying many-body problem. We assume the reader to be familiar with the usual Fock space formalism in quantum statistical mechanics. Consider a system of spin 21 fermions confined to a cubic box  ⊂ R3 , with periodic boundary conditions, interacting via a two-body potential V . Assume, for simplicity, that V ∈ L 1 (R3 ) so that it has a bounded Fourier transform, denoted by Vˆ ( p) = (2π )−3/2 V (x)e−i px d x. The many-body Hamiltonian on Fock space is then given as H=

 σ,k

† k 2 ak,σ ak,σ +

(2π )3/2 2||



† Vˆ (k)a †p−k,σ aq+k,ν aq,ν a p,σ .

(A.1)

k, p,q,σ,ν

† and ak,σ denote the usual creation and annihilation operators for a particle Here, ak,σ of momentum k and spin σ , satisfying the canonical anticommutation relations. The momentum sums run over k ∈ (2π/L)Z3 , where L is the side length of the cube , whereas the spin sums are over σ ∈ {↑, ↓}. Units are chosen such that  = 2m = 1, m denoting the particle mass. Following [3,10] we restrict H to BCS type states. These are quasi-free states that do not have a fixed number of particles. For details on the formalism, we refer the reader to [2]. Quasi-free states are linear maps ρ from bounded operators on the Fock space F to the complex numbers, satisfying ρ(A† A) ≥ 0, ρ(I ) = 1, and

ρ(e1 e2 e3 e4 ) = ρ(e1 e2 )ρ(e3 e4 ) − ρ(e1 e3 )ρ(e2 e4 ) + ρ(e1 e4 )ρ(e2 e3 ), † . whenever ei is either ak,σ or ak,σ Assuming both translation invariance and SU (2) invariance for rotations of the spins, the state ρ is completely determined by the quantities † † ak,↑ ) = ρ(ak,↓ ak,↓ ), ϕ(k) = ρ(a−k,↑ ak,↓ ) = −ρ(a−k,↓ ak,↑ ). γ (k) = ρ(ak,↑

All other expectation values of quadratic expressions are zero under these symmetry assumptions. Note that 0 ≤ γ ( p) ≤ 1 and |ϕ( p)|2 ≤ γ ( p)(1 − γ (− p)) by Schwarz’s

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365

inequality and the canonical anticommutation relations. Since also ϕ( p) = ϕ(− p), this implies that |ϕ( p)|2 ≤ γ ( p)(1 − γ ( p)). The energy expectation value can be conveniently expressed in terms of these quantities: 

(2π )3/2  ˆ V (k)ϕ(k − p)ϕ( p) || k k, p

2  2(2π )3/2 ˆ (2π )3/2  ˆ V (0) + V (k)γ ( p − k)γ ( p). (A.2) γ ( p) − || || p

ρ(H) = 2

k 2 γ (k) +

k, p

The first term in the second line is just a constant times the square of the number of particles. The last term is the exchange terms. Both these terms will be dropped in the sequel. This truncations can be justified on physical grounds for interaction potentials V that are of short range such that s-wave scattering is dominant [10]. The von-Neumann entropy of the quasi-free state ρ under consideration here, S(ρ) = −Tr F ρ ln ρ, can be conveniently expressed in terms of the 2 × 2 matrix

γ ( p) ϕ( p) ( p) = . (A.3) ϕ( p) 1 − γ ( p) Note that 0 ≤ ( p) ≤ 1 as an operator on C2 . The entropy is then given by (compare with [2, Eq. (2.c.10)])  S(ρ) = −2 Tr C2 [( p) ln ( p)] . p

The factor 2 results from the 2 spin components. The thermodynamic pressure, P, of a state at a fixed temperature T = chemical potential µ is defined as P(ρ) =

1 β

≥ 0 and

1 [T S(ρ) − ρ(H − µN)] . ||

 † Here, N = σ,k ak,σ ak,σ denotes the number operator. Dropping the last two terms in (A.2) and taking a formal infinite volume limit, one thus obtains
  2 2 T Tr P(γ , α) := lim P(ρ) = − 2 ( p) ln ( p) + ( p − µ)γ ( p) dp C (2π )3 R3 →R3
1 |α(x)|2 V (x) d x. − (2π )3 R3  Here, α denotes the inverse Fourier transform of ϕ, α(x) = (2π )−3/2 ϕ( p)ei px dp. If we set Fβ = − 21 (2π )3 P and replace V by 2V to slightly simplify the notation, we are thus led to the definition of the BCS model given in Definition 1. Finally, we will briefly explain in which sense E( p) is an effective dispersion relation and how it is related to an energy gap between the ground state and the first excited state of the system.

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With ϕ = αˆ the Fourier transform of a maximizer of the pressure functional at T = 0, we introduce the following quadratic Hamiltonian as an approximation to the full Hamiltonian H: Hqu =

 σ,k



† k 2 ak,σ ak,σ +

(2π )3/2  ˆ V (k) || k, p

 † × ϕ(k − p)a− p,↑ a p,↓ + ϕ( p)a †p−k,↑ ak− p,↓ .

(A.4)

The Hamiltonian Hqu − µN can be diagonalized in a standard way (see, e.g., [12]) via a Bogoliubov transformation. Using the fact that ϕ  satisfies the gap equation E(k)ϕ(k) =  − 21 (2π )3/2 k Vˆ (k − k )ϕ(k ), where E(k) = (k 2 − µ)2 + |(k)|2 and  is determined from ϕ via (2.2)–(2.4), this leads to      (A.5) Hqu − µN − inf spec Hqu − µN = E(k) ck† ck + dk† dk . k

The operators ck , ck† , dk and dk† satisfy the canonical anticommutation relations, and both ck and dk annihilate the BCS ground state. Equation (A.5) thus explains why E(k) is interpreted as the quasi-particle dispersion relation and  = inf k E(k) as the energy gap of the system. References 1. Andrenacci, N., Perali, A., Pieri, P., Strinati, G.C.: Density-induced BCS to Bose-Einstein crossover. Phys. Rev. B 60, 12410 (1999) 2. Bach, V., Lieb, E., Solovej, J.: Generalized Hartree-Fock theory and the Hubbard model. J. Stat. Phys. 76, 3–89 (1994) 3. Bardeen, J., Cooper, L., Schrieffer, J.: Theory of Superconductivity. Phys. Rev. 108, 1175–1204 (1957) 4. Billard, P., Fano, G.: An existence proof for the gap equation in the superconductivity theory. Commun. Math. Phys. 10, 274–279 (1968) 5. Bloch, I., Dalibard, J., Zwerger, W.: Many-Body Physics with Ultracold Gases. http://arxiv.org/abs/:0704. 3011, 2007, to appear in Rev. Mod. Phys. 6. Carlson, J., Chang, S.-Y., Pandharipande, V.R., Schmidt, K.E.: Superfluid Fermi Gases with Large Scattering Length. Phys. Rev. Lett. 91, 0504011 (2003) 7. Chen, Q., Stajic, J., Tan, S., Levin, K.: BCS–BEC crossover: From high temperature superconductors to ultracold superfluids. Phys. Rep. 412, 1–88 (2005) 8. Fetter, A., Walecka, J.D.: Quantum theory of many-particle systems. New-York: McGraw-Hill, 1971 9. Frank, R.L., Hainzl, C., Naboko, S., Seiringer, R.: The critical temperature for the BCS equation at weak coupling. J. Geom. Anal. 17, 559–568 (2007) 10. Leggett, A.J.: Diatomic Molecules and Cooper Pairs. Modern trends in the theory of condensed matter, J. Phys. (Paris) Colloq, C7–19 Bertin-Heidelberg-New York: Springer, 1980 11. Lieb, E., Loss, M.: Analysis. Providence RI: Amer. Math. Soc., 2001 12. Martin, P.A., Rothen, F.: Many-body problems and Quantum Field Theory. Berlin-Heidelberg-New York: Springer, 2004 13. McLeod, J.B., Yang, Y.: The uniqueness and approximation of a positive solution of the Bardeen-CooperSchrieffer gap equation. J. Math. Phys. 41, 6007–6025 (2000) 14. Nozières, P., Schmitt-Rink, S.: Bose Condensation in an Attractive Fermion Gas: From Weak to Strong Coupling Superconductivity. J. Low Temp. Phys. 59, 195–211 (1985) 15. Parish, M., Mihaila, B., Timmermans, E., Blagoev, K., Littlewood, P.: BCS-BEC crossover with a finiterange interaction. Phys. Rev. B 71, 0645131–0645136 (2005) 16. Randeria, M.: In: Bose-Einstein Condensation, Griffin, A., Snoke, D.W., Stringari, S. eds., Cambridge: Cambridge University Press, 1995 17. Tiesinga, E., Verhaar, B.J., Stoof, H.T.C.: Threshold and resonance phenomena in ultracold ground-state collisions. Phys. Rev. A 47, 4114 (1993)

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18. Vansevenant, A.: The gap equation in superconductivity theory. Physica 17D, 339–344 (1985) 19. Yang, Y.: On the Bardeen-Cooper-Schrieffer integral equation in the theory of superconductivity. Lett. Math. Phys. 22, 27–37 (1991) Communicated by H. Spohn

Commun. Math. Phys. 281, 369–385 (2008) Digital Object Identifier (DOI) 10.1007/s00220-008-0492-7

Communications in

Mathematical Physics

Electromagnetic Wormholes via Handlebody Constructions Allan Greenleaf1 , Yaroslav Kurylev2 , Matti Lassas3 , Gunther Uhlmann4 1 2 3 4

Department of Mathematics, University of Rochester, Rochester, NY 14627, USA Department of Mathematical Sciences, Loughborough Univ., Loughborough, LE11 3TU, UK Helsinki University of Technology, Institute of Mathematics, P.O.Box 1100, 02015 Helsinki, Finland Department of Mathematics, University of Washington, Seattle, WA 98195, USA. E-mail: [email protected]

Received: 10 April 2007 / Accepted: 24 December 2007 Published online: 16 May 2008 – © Springer-Verlag 2008

Abstract: Cloaking devices are prescriptions of electrostatic, optical or electromagnetic parameter fields (conductivity σ (x), index of refraction n(x), or electric permittivity (x) and magnetic permeability µ(x)) which are piecewise smooth on R3 and singular on a hypersurface , and such that objects in the region enclosed by  are not detectable to external observation by waves. Here, we give related constructions of invisible tunnels, which allow electromagnetic waves to pass between possibly distant points, but with only the ends of the tunnels visible to electromagnetic imaging. Effectively, these change the topology of space with respect to solutions of Maxwell’s equations, corresponding to attaching a handlebody to R3 . The resulting devices thus function as electromagnetic wormholes.

1. Introduction There has recently been considerable interest, both theoretical [3,13,16,18,19] and experimental [20], in invisibility (or “cloaking”) from observation by electromagnetic (EM) waves. (See also [17] for a treatment of cloaking in the context of elasticity.) Theoretically, cloaking devices are given by specifying the conductivity σ (x) (in the case of electrostatics), the index of refraction n(x) (for optics in the presence of polarization, where one uses the Helmholtz equation), or the electric permittivity (x) and magnetic permeability µ(x) (for the full system of Maxwell’s equations). In the constructions to date, the EM parameter fields ( σ ; n;  and µ ) have been piecewise smooth and anisotropic. (See, however, [5, Sect.4] for an example that can be interpreted as shielding with respect to Helmholtz by an isotropic negative index of refraction material). Furthermore, the EM parameters have singularities, with one or more eigenvalues of the tensors going to zero or infinity as one approaches, from one or both sides, the cloaking surface , which encloses the region within which objects may be hidden from external observation. Such constructions might have remained theoretical curiosities, but the advent of

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metamaterials [1] allows one, within the constraints of current technology, to construct media with fairly arbitrary (x) and µ(x). It thus becomes an interesting mathematical problem with practical significance to understand what other new phenomena of wave propagation can be produced by prescribing other arrangements of  and µ. Geometrically, cloaking can be viewed as arising from a singular transformation of R3 . Intuitively, for a spherical cloak [6,7,18], it is as if an infinitesimally small hole in space has been stretched to a ball D; an object can be inserted inside the hole so created and is then invisible to external observations. On the level of the EM parameters, homogeneous, isotropic parameters , µ are pushed forward to become inhomogeneous, anisotropic and singular as one approaches  = ∂ D from the exterior. There are then two ways, referred to as the single and double coating in [3], of continuing , µ to within D so as to rigorously obtain invisibility with respect to locally finite energy waves. We refer to either process as blowing up a point. As observed in [3], one can use the double coating to produce a manifold with a different topology, but with the change in topology invisible to external measurements. To define the solutions of Maxwell’s equations rigorously in the single coating case, one has to add boundary conditions on . Physically, this corresponds to the lining of the interior of the single coating material, e.g., in the case of blowing up a point, with a perfectly conducting layer, see [3]. We point out here that in the very interesting preprint [21], the single coating construction is supplemented with selfadjoint extensions of Maxwell operators in the interior of the cloaked regions; these implicitly impose interior boundary conditions on the boundary of the cloaked region, similar to the PEC boundary condition suggested in [3]. For the case of an infinite cylinder, the Soft-and-Hard surface (SHS) interior boundary condition is used in [3] to guarantee cloaking of active objects, and is needed even for passive ones. In this paper, we show how more elaborate geometric constructions, corresponding to blowing up a curve, enable the description of tunnels which allow the passage of waves between distant points, while only the ends of the tunnels are visible to external observation. These devices function as electromagnetic wormholes, essentially changing the topology of space with respect to solutions of Maxwell’s equations. Our proofs are valid of all positive frequencies; however current metamaterial designs which might be used to implement electromagnetic wormholes are fairly narrowband. Potential applications and further discussion can be found in [4]. We form the wormhole device around an obstacle K , a tubular neighborhood of a portion of a cylindrical surface in R3 as follows (see Fig. 1). First, one surrounds K with metamaterials, corresponding to a specification of EM parameters
ε and
µ. Secondly, one lines the surface of K with material implementing the SHS boundary condition from antenna theory [8,10,11]; this condition arose previously [3] in the context of cloaking an infinite cylinder. The EM parameters, which become singular as one approaches K , are given as the pushforwards of nonsingular parameters ε and µ on an abstract threemanifold M, described in Sect. 2. For a curve γ ⊂ M, we construct the diffeomorphism F from M\γ to the wormhole device in Sect. 3. For the resulting EM parameters
ε and
µ, we have singular coefficients of Maxwell’s equations at K , and so it is necessary to formulate an appropriate notion of locally finite energy solutions (see Def. 4.1). In Theorem 4.2, we then show that there is a perfect correspondence between the external measurements of EM waves propagating through the wormhole device and those propagating on the wormhole manifold, with the same boundary measurements. It was shown in [3] that the cloaking constructions are mathematically valid at all frequencies k. However, both cloaking and the wormhole effect studied here should be

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371

Fig. 1. The obstacle K without surrounding metamaterial

considered as essentially monochromatic, or at least very narrow-band, using current technology, since, from a practical point of view the metamaterials needed to implement the constructions have to be fabricated and assembled with a particular wavelength in mind, and theoretically are subject to significant dispersion [18]. Thus, as for cloaking in [3,16,18], here we describe the wormhole construction relative to electromagnetic waves at a fixed positive frequency k. We point out that the metamaterials used in the experimental verification of cloaking [20] should be readily adaptable to yield a physical implementation, at microwave frequencies, of the wormhole device described here. See Remark 1 in Sect. 4.2 for further discussion. The results proved here were announced in [4]. 2. The Wormhole Manifold M First we explain, somewhat informally, what we mean by a wormhole. The concept of a wormhole is familiar from general relativity [9,22], but here we define a wormhole as an object obtained by gluing together pieces of Euclidean space equipped with certain anisotropic EM parameter fields. We start by describing this process heuristically; later, we explain more precisely how this can be effectively realized vis-a-vis EM wave propagation using metamaterials. We first describe the wormhole as an abstract manifold M, see Fig. 2; in the next section we will show how to realize this concretely in R3 , as a wormhole device N . Start by making two holes in the Euclidean space R3 = {(x, y, z)|x, y, z ∈ R}, say by removing the open ball B1 = B(O, 1) with center at the origin O and of radius 1, and also the open ball B2 = B(P, 1), where P = (0, 0, L) is a point on the z-axis having the distance L > 3 to the origin. We denote by M1 the region so obtained, M1 = R3 \(B1 ∪ B2 ), which is the first component we need to construct a wormhole. Note that M1 is a 3dimensional manifold with boundary, the boundary of M1 being ∂ M1 = ∂ B1 ∪ ∂ B2 , the union of two 2-spheres. Thus, ∂ M1 can be considered as a disjoint union S2 ∪ S2 , where we will use S2 to denote various copies of the two-dimensional unit sphere. The second component needed is a 3−dimensional cylinder, M2 = S2 × [0, 1]. This cylinder can be constructed by taking the closed unit cube [0, 1]3 in R3 and, for each value of 0 < s < 1, gluing together, i.e., identifying, all of the points on the boundary of the cube with z = s. Note that we do not identify points at the top of the boundary,

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Fig. 2. Schematic figure: a wormhole manifold is glued from two components, the “handle” and space with two holes. Note that in the actual construction, the components are three dimensional

at z = 1, or at the bottom, at z = 0. We then glue together the boundary ∂ B(O, 1) ∼ S2 of the ball B(O, 1) with the lower end (boundary component) S2 × {0} of M2 , and the boundary ∂ B(P, 1) with the upper end, S2 × {1}. In doing so we identify the point (0, 0, 1) ∈ ∂ B(O, 1) with the point N P × {0} and the point (0, 0, L − 1) ∈ ∂ B(P, 1) with the point N P × {1}, where N P is the north pole on S2 . The resulting manifold M no longer lies in R3 , but rather is the connected sum of the components M1 and M2 and has the topology of R3 with a 3−dimensional handle attached. Note that adding this handle makes it possible to travel from one point in M1 to another point in M1 , not only along curves lying in M1 but also those in M2 . To consider Maxwell’s equations on M, let us start with Maxwell’s equations on R3 at frequency k ∈ R, given by ∇ × E = ik B, ∇ × H = −ik D,

D(x) = ε(x)E(x),

B(x) = µ(x)H (x).

Here E and H are the electric and magnetic fields, D and B are the electric displacement field and the magnetic flux density, ε and µ are matrices corresponding to permittivity and permeability. As the wormhole is topologically different from the Euclidean space R3 , we use a formulation of Maxwell’s equations on a manifold, and as in [3], do this in the setting of a general Riemannian manifold, (M, g). For our purposes, as in [3,14] it suffices to use ε, µ which are conformal, i.e., proportional by scalar fields, to the metric g. In this case, Maxwell’s equations can be written, in the coordinate invariant form, as d E = ik B, d H = −ik D,

D =  E, B = µH in M,

where E, H are 1-forms, D, B are 2-forms, d is the exterior derivative, and  and µ are scalar functions times the Hodge operator of (M, g), which maps 1-forms to the corresponding 2-forms [2]. In local coordinates these equations are written in the same form as Maxwell’s equations in Euclidean space with matrix valued ε and µ. Although not necessary, for simplicity one can choose a metric on the wormhole manifold M which is Euclidean on M1 , and on M2 is the product of a given metric g0 on S2 and the standard metric of [0, 1]. More generally, can also choose the metric on M2 to be a warped product. Even the simple choice of the product of the standard metric of S2 and the metric δ 2 ds 2 , where δ is the “length” of the wormhole, gives rise to interesting ray-tracing effects for rays passing through the wormhole tunnel. For δ << 1, the image through one end of the wormhole (of the region beyond the other end) would resemble the image in a a fisheye lens; for δ
1, multiple images and greater distortion occur. (See [4, Fig. 2]).

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The proof of the wormhole effect that we actually give is for yet another variation, where the balls that form the ends have their boundary spheres flattened; this may be useful for applications, since it allows for there to be a vacuum (or air) in a neighborhood of the axis of the wormhole, so that, e.g., instruments may be passed through the wormhole. We next show how to construct, using metamaterials, a device N in R3 that effectively realizes the geometry and topology of M, relative to solutions of Maxwell’s equations at frequency k, and hence functions as an electromagnetic wormhole. 3. The Wormhole Device N in R3 We now explain how to construct a “device” N in R3 , i.e., a specification of permittivity ε and permeability µ, which affects the propagation of electromagnetic waves in the same way as the presence of the handle M2 in the wormhole manifold M. What this means is that we prescribe a configuration of metamaterials which make the waves behave as if there were an invisible tube attached to R3 , analogous to the handle M2 in the wormhole manifold M. In other words, as far as external EM observations of the wormhole device are concerned, it appears as if the topology of space has been changed. We use cylindrical coordinates (θ, r, z) corresponding to a point (r cos θ, r sin θ, z) in R3 . The wormhole device is built around an obstacle K ⊂ R3 . To define K , let S be the two-dimensional finite cylinder {θ ∈ [0, 2π ], r = 2, 0 ≤ z ≤ L} ⊂ R3 . The open region K consists of all points in R3 that have distance less than one to S and has the shape of a long, thick-walled tube with smoothed corners. Let us first introduce a deformation map F from M to N = R3 \K or, more precisely, from M\γ to N \, where γ is a closed curve in M to be described shortly and  = ∂ K . We will define F separately on M1 and M2 denoting the corresponding parts by F1 and F2 . To describe F1 , let γ1 be the line segment on the z−axis connecting ∂ B(O, 1) and ∂ B(P, 1) in M1 , namely, γ1 = {r = 0, z ∈ [1, L −1]}. Let F1 (r, z) = (θ, R(r, z), Z (r, z)) be such that (R(r, z), Z (r, z)), shown in Fig. 3, transforms in the (r, z) coordinates the semicircles AB and C D in the left picture to the vertical line segments A B  = {r ∈ [0, 1], z = 0} and C  D  = {r ∈ [0, 1], z = L} in the right picture and the cut γ1 on the left picture to the curve B  C  on the right picture. This gives us a map F1 : M1 \γ1 → N1 \, where the closed region N1 in R3 is obtained by rotation of the region exterior to the curve A B  C  D  around the z−axis. We can choose F1 so that it is the identity map in the domain U = R3 \{−2 ≤ z ≤ L + 2, 0 ≤ r ≤ 4}. To describe F2 , consider the line segment, γ2 = {N P} × [0, 1] on M2 . The sphere without the north pole can be “flattened” and stretched to an open disc with radius one which, together with stretching [0, 1] to [0, L], gives us a map F2 from M2 \γ2 to N2 \. The region N2 is the 3−dimensional cylinder, N2 = {θ ∈ [0, 2π ], r ∈ [0, 1],

Fig. 3. The map (R(r, z), Z (r, z)) in cylindrical coordinates (z, r )

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z ∈ [0, L]}. When flattening S2 \N P, we do it in such a way that F1 on ∂ B(O, 1) and ∂ B(P, 1) coincides with F2 on (S2 \N P) × {0} and (S2 \N P) × {1}, respectively. Thus, F maps M\γ , where γ = γ1 ∪ γ2 is a closed curve in M, onto N \; in addition, F is the identity on the region U . Now we are ready to define the electromagnetic material parameter tensors on N . We define the permittivity to be  (D F)(x)· ε(x)· (D F(x))t 
ε = F∗ ε(y) = ,  det (D F) x=F −1 (y) where D F is the derivative matrix of F, and similarly the permeability to be
µ = F∗ µ. These deformation rules are based on the fact that permittivity and permeability are conductivity type tensors, see [14]. Maxwell’s equations are invariant under smooth changes of coordinates. This means that, by the chain rule, any solution to Maxwell’s equations in M\γ , endowed with material parameters ε, µ becomes, after transformation by F, a solution to Maxwell’s equations in N \ with material parameters
ε and
µ, and vice versa. However, when considering the fields on the entire spaces M and N , these observations are not enough, due to the singularities of
ε and
µ near ; the significance of this for cloaking was observed and analyzed in [3]. In the following, we will show that the physically relevant class of solutions to Maxwell’s equations, namely the (locally) finite energy solutions, remains the same, with respect to the transformation F, in (M; ε, µ) and (N ;
ε,
µ). One can analyze the rays in M and N endowed with the electromagnetic wave propagation √ √ metrics g = εµ and
g=
ε
µ, respectively. Then the rays on M are transformed by F into the rays in N . As almost all the rays on M do not intersect with γ , therefore, almost all the rays on N do not approach . This was the basis for [16,18] and was analyzed further in [19]; see also [17] for a similar analysis in the context of elasticity. Thus, heuristically one is led to conclude that the electromagnetic waves on (M; ε, µ) do not feel the presence of γ , while those on (N ;
ε,
µ) do not feel the presence of K , and these waves can be transformed into each other by the map F. Although the above considerations are mathematically rigorous, on the level both of the chain rule and of high frequency limits, i.e., ray tracing, in the exteriors M\γ and N \, they do not suffice to fully describe the behavior of physically meaningful solution fields on M and N . However, by carefully examining the class of the finite-energy waves in M and N and analyzing their behavior near γ and , respectively, we can give a complete analysis, justifying the conclusions above. Let us briefly explain the main steps of the analysis using methods developed for theory of invisibility (or cloaking) at frequency k > 0 [3] and at frequency k = 0 in [6,7]. The details will follow. First, to guarantee that the fields in N are finite energy solutions and do not blow up near , we have to impose at  the appropriate boundary condition, namely, the Soft-and-Hard surface condition, see [8,11], eθ · E| = 0, eθ · H | = 0, where eθ is the angular direction. Secondly, the map F can be considered as a smooth coordinate transformation on M\γ ; thus, the finite energy solutions on M\γ transform under F into the finite energy solutions on N \, and vice versa. Thirdly, the curve γ in M has Hausdorff dimension equal to one. This implies that the possible singularities of the finite energy electromagnetic fields near γ are removable [12], that is, the finite energy fields in M\γ are exactly the restriction to M\γ of the fields defined on all of M.

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Combining these steps we can see that measurements of the electromagnetic fields on (M; ε, µ) and on (R3 \K ;
ε,
µ) coincide in U . In other words, if we apply any current on U and measure the radiating electromagnetic fields it generates, then the fields on U in the wormhole manifold (M; ε, µ) coincide with the fields on U in (R3 \K ;
ε,
µ), 3-dimensional space equipped with the wormhole device construction. Summarizing our construction, the wormhole device consists of the metamaterial coating of the obstacle K . This coating should have the permittivity
ε and permeability
µ. In addition, we need to impose the SHS boundary condition on , which may be realized by fabricating the obstacle K from a perfectly conducting material with parallel corrugations on its surface [8,11]. In the next section, the permittivity
ε and and permeability
µ are described in a rather simple form. (As mentioned earlier, in order to allow for a tube around the axis of the wormhole to be a vacuum or air, we deal with a slightly different construction than was described above, starting with flattened spheres). It should be possible to physically implement an approximation to this mathematical idealization of the material parameters needed for the wormhole device, using concentric rings of split ring resonators as in the experimental verification of cloaking obtained in [20]. 4. Rigorous Construction of the Wormhole Here we present a rigorous model of a typical wormhole device and justify the claims above concerning the behavior of the electromagnetic fields in the wormhole device in R3 in terms of the fields on the wormhole manifold (M, g). 4.1. The wormhole manifold (M, g) and the wormhole device N . Here we prove the wormhole effect for a variant of the wormhole device described in the previous sections. Instead of using a round sphere S2 as before, we present a construction that uses a deformed sphere S2flat that is flat near the south and north poles, S P and N P. This makes it possible to have constant isotropic material parameters near the z-axis located inside the wormhole. For possible applications, see [4]. We use following notations. Let (θ, r, z) ∈ [0, 2π ] × R+ × R be the cylindrical coordinates of R3 , that is the map X : (θ, r, z) → (r cos θ, r sin θ, z) that maps X : [0, 2π ] × R+ × R → R3 . In the following, we identify [0, 2π ] and the unit circle S 1 . Let us start by removing from R3 two “deformed” balls which have flat portions near the south and north poles. More precisely, let M1 = R3 \(P1 ∪ P2 ), where in the cylindrical coordinates P1 = {X (θ, r, z) : −1 ≤ z ≤ 1, 0 ≤ r ≤ 1} ∪{X (θ, r, z) : (r − 1)2 + z 2 ≤ 1}, P2 = {X (θ, r, z) : 1 ≤ z − L ≤ 1, 0 ≤ r ≤ 1} ∪{X (θ, r, z) : (r − 1)2 + (z − L)2 ≤ 1}. We say that the boundary ∂ P1 of P1 is a deformed sphere with flat portions, and denote it by S2flat . We say that the intersection points of S2flat with the z-axis are the north pole, N P, and the south pole, S P.

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Fig. 4. A schematic figure on the map G 0 , considered in the (r, z) coordinates. Later, we impose the SHS boundary condition on the portion of the boundary, colored red in [4, Fig. 5]

Let g1 be the metric on M1 inherited from R3 , and let γ1 be the path γ1 = {X (0, 0, z) : 1 < z < L − 1} ⊂ M1 . Set A1 = M1 \V1/4 , Vt = {X (θ, r, z) : 0 ≤ r ≤ t, 1 < z < L − 1}, 0 < t < 1, and consider a map G 0 : M1 \γ1 → A1 ; see Fig. 4. G 0 defined as the identity map on M1 \V1/2 and, in cylindrical coordinates, as   1 r G 0 (X (θ, r, z)) = X θ, + , z , (θ, r, z) ∈ V1/2 . 4 2 Clearly, G 0 is C 0,1 −smooth. Let U (x) ∈ R3×3 , x = X (θ, r, z), be the orthogonal matrix that maps the standard unit vectors e1 , e2 , e3 of R3 to the Euclidean unit vectors corresponding to the θ , r , and z directions, that is, U (x)e1 = (− sin θ, cos θ, 0), U (x)e2 = (cos θ, sin θ, 0), U (x)e3 = (0, 0, 1). Then the differential of G 0 in the Euclidean coordinates at the point x ∈ V1/2 is the matrix ⎛1 1 r ⎞ r (4 + 2) 0 0 1 ⎠ U (x)−1 , x = X (θ, r, z), y = G (x). (1) DG 0 (x) = U (y) ⎝ 0 0 2 0 0 01 Later we impose on part of the boundary, 0 = ∂ A1 ∩ {1 < z < L − 1}, the SHS boundary condition. Next, let (θ, z, τ ) = (θ (x), z(x), τ (x)) be the Euclidean boundary normal coordinates associated to 0 , that is, τ (x) = distR3 (x, 0 ) and (θ (x), z(x)) are the θ and z-coordinates of the closest point of 0 to x. Denote by (G 0 )∗ g1 the push forward of the metric g1 in G 0 , that is, the metric obtained from g1 using the change of coordinates G 0 , see [2]. The metric (G 0 )∗ g1 coincides with g1 in A1 \V1/2 , and in the Euclidean boundary normal coordinates of 0 , on A1 ∩ V1/2 , the metric (G 0 )∗ g1 , has the length element ds 2 = 4τ 2 dθ 2 + dz 2 + 4dτ 2 .

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Fig. 5. Map G 1 in (r, z)-coordinates

Next, let

 q3 = conv {(r, z) : (r − 2)2 + (z − (−2))2 ≤ 1}  ∪{(r, z) : (r − 2) + (z − (L + 2)) ≤ 1} , 2

2

q4 = {(r, z) : 0 ≤ r ≤ 1, −1 ≤ z ≤ L + 1}, where conv(q) denotes the convex hull of the set q. Let N1 P3 P4 1

= = = =

R3 \(P3 ∪ P4 ), {X (θ, r, z) : (r, z) ∈ q3 }, {X (θ, r, z) : (r, z) ∈ q4 }, ∂ N1 \∂ P4 .

We can find a Lipschitz smooth map G 1 : A1 → N1 , see Fig. 5, of the form G 1 (X (θ, r, z)) = X (θ, R(r, z), Z (r, z)) such that it maps 0 to 1 , and in A1 near 0 it is given by G 1 (x + tν0 ) = G 1 (x) + tν1 .

(2)

Here, x ∈ 0 , ν0 is the Euclidean unit normal vector of 0 , ν1 is the Euclidean unit normal vector of 1 , and 0 < t < 41 . Moreover, we can find a G 1 so that it is the identity map near the z-axis, that is, G 1 (x) = x, x ∈ A1 ∩ {0 ≤ r <

1 } 4

(3)

and such that G 1 is also the identity map in the set of points with the Euclidean distance 4 or more from P1 ∪ P2 . Note that we can find such a G 1 such that both G 1 and its inverse G −1 1 are Lipschitz smooth up to the boundary. Thus the differential DG 1 of G 1 at x ∈ A1 in Euclidean coordinates is   a (r, z) 0 U (x)−1 , x = X (θ, r, z), y = G 1 (x), DG 1 (x) = U (y) 11 0 A(r, z) where c0 ≤ a11 (r, z) ≤ c1 and A(r, z) is a symmetric (2 × 2)-matrix satisfying c0 I ≤ A(r, z) ≤ c1 I with some c0 , c1 > 0.

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The map F1 (x) = G 1 (G 0 (x)) then maps F1 : M1 \γ1 → N1 . Let
g1 = (F1 )∗ g1 be metric on N1 . From the above considerations, we see that the differential D F1 of F1 at x ∈ M1 \γ1 near 0 , in Euclidean coordinates, is given by   b (θ, r, z) 0 U (x)−1 , D F1 (x) = U (y) 11 0 B(r, z) (4) c11 (r, z) x = X (θ, r, z), y = F1 (x), b11 (θ, r, z) = distR3 (X (θ, r, z), 0 ) where c0 ≤ c11 (r, z) ≤ c1 , and B(r, z) is a symmetric (2 × 2)-matrix satisfying c0 I ≤ B(r, z) ≤ c1 I, for some c0 , c1 > 0. Note that ∂ P4 ∩ {r < 1} consists of two two-dimensional discs, B2 (0, 1) × {−1} and B2 (0, 1) × {L + 1}. Below, we will use the map f 2 = F1 |∂ P1 \N P : ∂ P1 \N P → B2 (0, 1) × {−1} ⊂ ∂ N1 . The map f 2 can be considered as the deformation that “flattens” S2flat \N P to a two dimensional unit disc. To describe f 2 , consider S2flat as a surface in Euclidean space and define on it the θ coordinate corresponding to the θ coordinate of R3 \{z = 0}. Let then s(y) be the intrinsic distance of y ∈ S2flat to the south pole S P. Then (θ, s) define coordinates in S2flat \{S P, N P}. We denote by y(θ, s) ∈ S2flat \{S P, N P} the point corresponding to the coordinates (θ, s). By the above construction, the map f 2 has the form, with respect to the coordinates used above, f 2 (y(θ, s)) = X (θ, R(s), −1) ∈ B2 (0, 1) × {−1}, where 1 R(s) = s, for 0 < s < , 4 1 1 R(s) = 1 − [(π + 4) − s], for (π + 4) − < s < (π + 4), 2 4

(5)

cf. formulae (2) and (3). In the following we identify B2 (0, 1) × {−1} with the disc B2 (0, 1). Let h 1 be the metric on ∂ P1 \N P inherited from (M1 , g1 ). Let h 2 = ( f 2 )∗ h 1 be the metric on B2 (0, 1). We observe that the metric h 2 makes the disc B2 (0, 1) isometric to S2flat \N P, endowed with the metric inherited from R3 . Thus, let M2 = S2flat × [−1, L + 1]. On M2 , let the metric g2 be the product of the metric of S2flat inherited from R3 and the metric α2 (z)dz 2 , α2 > 0 on [−1, L + 1]. Let γ2 = {N P} × [−1, L + 1] be a path on M2 . Define N2 = P4 = {X (θ, r, z) : 0 ≤ r < 1, −1 ≤ z ≤ L + 1} ⊂ R3 , 2 = ∂ N2 ∩ {r = 1}, and let F2 : M2 \γ2 → N2 be the map of the form F2 (y, z) = ( f 2 (y), z) ∈ R3 , (y, z) ∈ (S2flat \N P) × [−1, L + 1]. Let
g2 = (F2 )∗ g2 be the resulting metric on N2 .

(6)

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Fig. 6. Left: The set N2 in the (r, z) coordinates. Right: The set N = N1 ∪ N2 ⊂ R3 having the complement K , presented in the (r, z) coordinates. Later, the SHS boundary condition is imposed on all of ∂ K

Denote by M 1 = M1 ∪ ∂ M1 the closure of M1 and let (M, g) = (M 1 , g1 )#(M2 , g2 ) be the connected sum of M 1 and M2 , that is, we glue the boundaries ∂ M1 and ∂ M2 . The set N = N1 ∪ N2 ⊂ R3 is open, and its boundary ∂ N is  = 1 ∪ 2 . Let F be the map F : M\γ → N defined by the maps F1 : M1 \γ1 → N1 and F2 : M2 \γ2 → N2 , and finally, let γ = γ1 ∪ γ2 and
g = F∗ g. t,
θ ), where Let K = R3 \N . On the surface  = ∂ K we can use local coordinates (

θ is the θ -coordinate of the ambient space R3 and
t is either the r or z -coordinate of the ambient space R3 restricted to . Denote also
τ =
τ (x) = distR3 (x, ∂ K ). Then by formula (2) we see that in N1 , in the Euclidean boundary normal coordinates (
θ ,
t,
τ ) associated to the surface 1 , the metric
g has the length element τ 2 + α1 (
t) d
t 2 + 4
τ 2 d
θ 2, 0 <
τ< ds 2 = 4d


1 , c0−1 ≤ α1 (
t) ≤ c0 , c0 ≥ 1. 4

The construction of F2 yields that in N2 , in the Euclidean boundary normal coordinates (
θ ,
t,
τ ) with
t = z, associated to the surface 2 = ∂ K ∩ ∂ N2 , the metric
g has the length element, near 2 , τ 2 + α2 (
t)d
t 2 + 4
τ 2 d
θ 2, 0 <
τ< ds 2 = 4d


1 . 4

Here, near ∂ N1 ∩ ∂ N2 , we use
t = z on 1 . Choosing the map G 1 in the construction of the map F1 appropriately, we have α2 (−1) = α1 (−1), α2 (L + 1) = α1 (L + 1), and the resulting map is Lipschitz. On M1 , N1 , and N2 that are subsets of R3 we have the well defined cylindrical coordinates (θ, r, z). Similarly, M2 = S2flat × [−1, L + 1] and we define the coordinates (θ, s, z), where (θ, s) are the above defined coordinates on S2flat \{S P, N P}. We can also consider on N ⊂ R3 also the Euclidean metric, denoted by g e . In Euclidean coordinates, (g e )i j = δ jk . Consider next the above defined Euclidean boundary normal coordinates (
θ ,
t,
τ ) associated to ∂ K . They are well defined in a neighborhood of ∂ K . We define the vector fields

ξ = ∂
η = ∂
τ,
t θ , ζ = ∂
on N near ∂ K . These vector fields are orthogonal with respect to the metric
g and to the metric g e . On M near γ , we use coordinates (θ, t, τ ). On M1 , near γ1 in the terms of the cylindrical coordinates they are (θ, t, τ ) = (θ, z, r ). On M2 , they are the coordinates

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(θ, t, τ ) = (θ, z, s), where s is the intrinsic distance to the north pole N P. We define also the vector fields ξ = ∂τ , η = ∂θ , ζ = ∂t on M\γ near γ . These vector fields are orthogonal with respect to the metric g. In the sequel, we consider the differential of F as the linear map D F : (Tx M, g) → (Ty N , g e ), y = F(x), x ∈ M\γ . Using formula (4) in M1 and formulas (5), (6) in M2 , we see that D F −1 (x) at x ∈ N near ∂ N is a bounded linear map that satisfies |(η, D F −1 (x)
η)g | ≤ C
τ (x), −1
(η, D F (x)ζ )g = 0,

(ζ, D F −1 (x)
η)g = 0, −1
|(ζ, D F (x)ζ )g | ≤ C,

(η, D F −1 (x)
ξ )g = 0,

|(ζ, D F −1 (x)
ξ )g | ≤ C,

(ξ, D F −1 (x)
η)g = 0, −1 |(ξ, D F (x)
ζ )g | ≤ C, (7) −1
|(ξ, D F (x)ξ )g | ≤ C,

where C > 0 and (· , · )g is the inner product defined by the metric g. Moreover, we obtain similar estimates for D F in terms of the Euclidean metric g e , (
η, D F(y)ζ )ge = 0,

(
ζ , D F(y)η)ge = 0,
|(ζ , D F(y)ζ )ge | ≤ C,

(
ξ , D F(y)η)ge = 0, |(
ξ , D F(y)ζ )ge | ≤ C,

(
η, D F(y)ξ )ge = 0,

|(
ζ , D F(y)ξ )ge | ≤ C,

|(
ξ , D F(y)ξ )ge | ≤ C

|(
η, D F(y)η)ge | ≤ C τ (y)−1 ,

(8)

for y ∈ M\γ near γ with C > 0. Next, consider D F(y) at y ∈ M\γ . Recall that the singular values s j (y), j = 1, 2, 3 of D F(y) are the square roots of the eigenvalues of (D F(y))t D F(y), where (D F)t is the transpose of D F. By (7), the singular values s j = s j (y), j = 1, 2, 3, of D F(y), numbered in increasing order, satisfy c1 ≤ s1 (y) ≤ c2 , c1 ≤ s2 (y) ≤ c2 , c1 c2 ≤ s3 (y) ≤ , τ (y) τ (y) where c1 , c2 > 0. The determinant of the matrix D F(y) can be computed in terms of its singular values by det (D F) = s1 s2 s3 . Later, we need the norm of the matrix det (D F(y))−1 D F(y). It satisfies by formula (8) 3 



−1 −1

det (D F(y)) D F(y) = sk sk−1 ≤ c1−2 . diag (s1 , s2 , s3 ) = max k=1

1≤ j≤3

k= j

(9) 4.2. Maxwell’s equations on the wormhole with the SHS lining. Let d V0 (x) denote the Euclidean volume element on N ⊂ R3 . Recall that N ⊂ R3 is an open set with boundary ∂ N = . Let d Vg be the Riemannian volume on (M, g). We consider below the map F : M\γ → N as a coordinate deformation. The map F induces for any differential
on N a form E = F ∗ E
in M\γ called the pull back of E
in F, see [2]. form E Next, we consider Maxwell equations with degenerate material parameters
ε and
µ on N with SH boundary conditions on . On M and N we define the permittivity and permeability by setting

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381

ε jk = µ jk = det (g)1/2 g jk , on M,

(10)

µ jk = det (
g )1/2
g jk , on N .
ε jk =


Here, and below, the matrix [g jk (x)] is the representation of the metric g in local coordinates, [g jk (x)] is the inverse of the matrix [g jk (x)], and det (g) is the determinant of [g jk (x)]. We note that the metric
g is degenerate near , and thus
ε and
µ, represented as matrices in the Euclidean coordinates, have elements that tend to infinity at , that is, the matrices
ε and
µ have a singularity near . Remark 1. Modifying the above construction by replacing M2 with M2 = S2flat × [l1 , l2 ] for appropriate l1 , l2 ∈ R and choosing F1 in an appropriate way, we can use local coordinates (
θ ,
t) on  such that the Euclidean distance along  of points (
θ ,
t1 ) and

(θ , t2 ) is proportional to |
t1 −
t2 |, and the metric
g in the Euclidean boundary normal coordinates (
θ ,
t,
τ ) associated to ∂ K has the form τ 2 + d
t 2 + 4
τ 2 d
θ 2, 0 <
τ< ds 2 = 4d


1 . 4

4.3. Finite energy solutions of Maxwell’s and the equivalence theorem. In the equations
=
=
x j and H
x j in the Euclidfollowing, we consider 1-forms E j E j d
j H j d
x 2,
x 3 ) of N ⊂ R3 . In the sequel, we use Einstein’s summation ean coordinates (
x 1,
convention and omit the sum signs. We use the Euclidean coordinates as we want to consider N with the differential structure inherited from the Euclidean space. We say
j and H
j are the (Euclidean) coefficients of the forms E
and H
, correspondingly. that E p We say that these coefficients are in L loc (N , d V0 ), 1 ≤ p < ∞, if

|E j (x)| p d V0 (x) < ∞, for all bounded measurable sets W ⊂ N . W


and H
are finite energy solutions of Maxwell’s Definition 1. We say that the 1-forms E equations in N with the soft-and-hard (SH) boundary conditions on  and the frequency k = 0,
= ik

, ∇ × H
= −ik

+ J
on N , ∇×E µ(x) H ε(x) E


η · E| = 0,
η · H | = 0,
and H
and 2-forms D
=

and

in N have coefficients in if 1-forms E εE B =
µH L 1loc (N , d V0 ) and satisfy

2

j E
k d V0 (x) < ∞,

E L 2 (W,|
=
ε jk E g |1/2 d V0 ) W


2 2
j H
k d V0 (x) < ∞ =
µ jk H

H 1/2 L (W,|
g|

d V0 ))

W

for all bounded measurable sets W ⊂ N , and finally,


− ik

) d V0 (x) = 0, ((∇ ×
h) · E h ·
µ(x) H N


+

− J
)) d V0 (x) = 0, ((∇ ×
e) · H e · (ik
ε(x) E N

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for all 1-forms
e and
h with coefficients in C0∞ (N ) that satisfy
η ·
e| = 0,
η ·
h| = 0,

(11)

where
η = ∂θ is the angular vector field that is tangential to . Below, we use for 1-forms E = E j d x j and H = H j d x j , given in local coordinates on M, the notations

(x 1 , x 2 , x 3 )

∇ × E = d H, ∇· (εE) = d ∗ E, ∇· (µH ) = d ∗ H, where d is the exterior derivative and ∗ is the Hodge operator on (M, g), cf. formula (10). We have the following “equivalent behavior of electromagnetic fields on N and M” result, analogous to the results of [3] for cloaking.
and H
be 1-forms with coefficients Theorem 1. Let E and H be 1-forms on M\γ and E 1 ∗ ∗


in L loc (N , d V0 ) such that E = F E, H = F H . Let J and J = F ∗ J
be 2-forms with smooth coefficients in N and M\γ that are supported away from  and γ . Then the following are equivalent:
and H
satisfy Maxwell’s equations with SHS boundary condi1. On N , the 1-forms E tions in the sense of Definition 1. 2. On M, the forms E and H can be extended on M so that they are classical solutions E and H of Maxwell’s equations, ∇ × E = ikµH, in M, ∇ × H = −ikεE + J, in M. Proof. Assume first that E and H satisfy Maxwell’s equations on M with source J supported away from γ . Then E and H are C ∞ smooth near γ .
H
and 2-form J
on N by Using F −1 : N → M\γ we define the 1-forms E,
= (F −1 )∗ E, H
= (F −1 )∗ H , and J
= (F −1 )∗ J. These fields satisfy Maxwell’s E equations in N ,
= ik

, ∇ × H
= −ik

+ J
in N . ∇×E µ(x) H ε(x) E

(12)

Now, writing E = E j (x)d x j on M near γ , we see using the transformation rule for
= (F −1 )∗ E is in local coordinates differential 1-forms that the form E
= E
j (
x )d
x j, E


j (
E x ) = (D F −1 )kj (
x ) E k (F −1 (
x )),
x ∈ N.

(13)


H
are Using the smoothness of E and H near γ on M and formulae (7), we see that E, forms on N with L 1loc (N , d V0 ) coefficients. Moreover,

ε(x) E(x) = det (D F(y))−1 D F(y)ε(y)D F(y)t (D F(y)t )−1 E(y) = det (D F(y))−1 D F(y)ε(y)E(y),
=

and

are ε E, B=
µH where x ∈ N , y = F −1 (x) ∈ M\γ . Formula (9) shows that D 2-forms on N with L 1loc (N , d V0 ) coefficients. Let (t) ⊂ N be the t-neighbourhood of  in the
g -metric. Note that for small t > 0 the set (t) is the Euclidean (t/2)-neighborhood of ∂ K . Denote by ν be the unit exterior
ge =

Euclidean normal vector of ∂(t) and the Euclidean inner product by (
η, E) η · E.

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Formulas (7) and (13) imply that the angular components satisfy
≤ Ct, x ∈ ∂(t), |
η · E| and
≤ C, x ∈ ∂(t) |
ζ · E| with some C > 0. Thus denoting by d S the Euclidean surface area on ∂(t), Stokes’ formula, formula (12), and the identity ν ×
ξ = ±
η yield


− ik

) d V0 (x) ((∇ ×
h) · E h ·
µH N


− ik

) d V0 (x) = lim ((∇ ×
h) · E h ·
µH t→0 N \(t)


·
= − lim (ν × E) h d S(x) t→0 ∂(t)


η + (


= − lim ν × ((
η · E)
ζ · E) ζ) ·
h d S(x) =0

t→0 ∂(t)

for a test function
h satisfying formula (11).
shows that 1-forms E
and H
satisfy Maxwell’s equations with Similar analysis for H SH boundary conditions in the sense of Definition 1.
and H
form a finite energy solution of Maxwell’s equations on Next, assume that E (N , g) with a source J
supported away from , implying in particular that
j E
k ∈ L 1 (W, d V0 ),

j H
k ∈ L 1 (W, d V0 )
ε jk E µ jk H where W = F(U \γ ) ⊂ N and U ⊂ M is a relatively compact open neighbourhood of
H = F∗ H
, and J = F ∗ J
on M\γ . Therefore γ , supp ( J
) ∩ W = ∅. Define E = F ∗ E, we conclude that ∇ × E = ikµ(x)H, ∇ × H = −ikε(x)E + J, in M\γ and ε jk E j E k ∈ L 1 (U \γ , d Vg ), µ jk H j Hk ∈ L 1 (U \γ , d Vg ). As representations of ε and µ, in local coordinates of M, are matrices that are bounded from above and below, these imply that ∇ × E ∈ L 2 (U \γ , d Vg ), ∇ × H ∈ L 2 (U \γ , d Vg ), ∇· (εE) = 0, ∇· (µH ) = 0, in U \γ . Let E e , H e ∈ L 2 (U, d Vg ) be measurable extensions of E and H to γ . Then ∇ × E e − ikµ(x)H e = 0, in U \γ , ∇ × E e − ikµ(x)H e ∈ H −1 (U, d Vg ), ∇ × H e + ikε(x)E e = 0, in U \γ , ∇ × H e + ikε(x)E e ∈ H −1 (U, d Vg ),

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where H −1 (U, d Vg ) is the Sobolev space with smoothness (−1) on (U, g). Since γ is a subset with (Hausdorff) dimension 1 in the 3-dimensional domain U , it has zero capacitance. Thus, the Lipschitz functions on U that vanish on γ are dense in H 1 (U ), see [12]. Therefore, there are no non-zero distributions in H −1 (U ) supported on γ . Hence we see that ∇ × E e − ikµ(x)H e = 0, ∇ × H e + ikε(x)E e = 0 in U. This also implies that ∇· (εE e ) = 0, ∇· (µH e ) = 0 in U, which, by elliptic regularity, imply that E e and H e are C ∞ smooth in U . In summary, E and H have unique continuous extensions to γ , and the extensions are classical solutions to Maxwell’s equations.   Acknowledgements. A.G. was supported by NSF, M.L. by CoE-program 213476 of Academy of Finland, G.U. was partially supported by the NSF and a Walker Family Endowed Professorship.

References 1. Eleftheriades, G., Balmain, K.: Negative-Refraction Metamaterials. New York IEEE Press (Wiley-Interscience), 2005 2. Frankel, T.: The geometry of physics. Cambridge University Press, Cambridge, 1997 3. Greenleaf, A., Kurylev, Y., Lassas, M., Uhlmann, G.: Full-wave invisibility of active devices at all frequencies. Commun. Math. Phys. 275(3), 749–789 (2007) 4. Greenleaf, A., Kurylev, Y., Lassas, M., Uhlmann, G.: Electromagnetic wormholes and virtual magnetic monopoles from metamaterials. Phys. Rev. Lett. 99, 183901 (2007) 5. Greenleaf, A., Lassas, M., Uhlmann, G.: The Calderón problem for conormal potentials, I: Global uniqueness and reconstruction. Comm. Pure Appl. Math 56(3), 328–352 (2003) 6. Greenleaf, A., Lassas, M., Uhlmann, G.: Anisotropic conductivities that cannot detected in EIT. Physiological Measurement (special issue on Impedance Tomography), 24, 413–420 (2003) 7. Greenleaf, A., Lassas, M., Uhlmann, G.: On nonuniqueness for Calderón’s inverse problem. Math. Res. Let. 10(5–6), 685–693 (2003) 8. Hänninen, I., Lindell, I., Sihvola, A.: Realization of generalized Soft-and-Hard Boundary. ProgrIn Electromag. Res. PIER 64, 317–333 (2006) 9. Hawking, S., Ellis, G.: The Large Scale Structure of Space-Time. Cambridge: Cambridge Univ. Press, 1973 10. Kildal, P.-S.: Definition of artificially soft and hard surfaces for electromagnetic waves. Electron. Lett. 24, 168–170 (1988) 11. Kildal, P.-S.: Artificially soft-and-hard surfaces in electromagnetics. IEEE Trans. Ant. and Propag. 10, 1537–1544 (1990) 12. Kilpeläinen, T., Kinnunen, J., Martio, O.: Sobolev spaces with zero boundary values on metric spaces. Potential Anal. 12(3), 233–247 (2000) 13. Kohn, R., Shen, H., Vogelius, M., Weinstein, M.: Cloaking via change of variables in electrical impedance tomography. Inverse Problems 24, 015016 (2008) 14. Kurylev, Y., Lassas, M., Somersalo, E.: Maxwell’s equations with a polarization independent wave velocity: Direct and inverse problems. J. Math. Pures Appl. 86, 237–270 (2006) 15. Lassas, M., Taylor, M., Uhlmann, G.: On determining a non-compact Riemannian manifold from the boundary values of harmonic functions. Comm. Geom. Anal. 11, 207–222 (2003) 16. Leonhardt, U.: Optical Conformal Mapping. Science 312, 1777–1780 (2006) 17. Milton, G., Briane, M., Willis, J.: On cloaking for elasticity and physical equations with a transformation invariant form. New J. Phys. 8, 248 (2006) 18. Pendry, J.B., Schurig, D., Smith, D.R.: Controlling Electromagnetic Fields. Science 312, 1780–1782 (2006) 19. Pendry, J.B., Schurig, D., Smith, D.R.: Calculation of material properties and ray tracing in transformation media. Optics Express 14, 9794 (2006)

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20. Schurig, D., Mock, J., Justice, B., Cummer, S., Pendry, J., Starr, A., Smith, D.: Metamaterial electromagnetic cloak at microwave frequencies. Science 314, 977–980 (2006) 21. Weder, R.: A rigorous time-domain analysis of full–wave electromagnetic cloaking (Invisibility). http://arxiv.org.abs/0704.0248, 2007 22. Visser, M.: Lorentzian Wormholes. New York: AIP Press, 1997 Communicated by P. Constantin

Commun. Math. Phys. 281, 387–400 (2008) Digital Object Identifier (DOI) 10.1007/s00220-008-0475-8

Communications in

Mathematical Physics

Tangent Euler Top in General Relativity
José Natário Centro de Análise Matemática, Geometria e Sistemas Dinâmicos, Departamento de Matemática, Instituto Superior Técnico, 1049-001 Lisboa, Portugal. E-mail: [email protected] Received: 11 April 2007 / Accepted: 9 October 2007 Published online: 25 April 2008 – © Springer-Verlag 2008

Abstract: We define the tangent Euler top in General Relativity through a constrained Lagrangian on the orthonormal frame bundle. The corresponding motions are studied to various degrees of approximation, the lowest of which is shown to yield the Mathisson-Papapetrou equations. 0. Introduction As is well known [Born09,Her10,Noe10,Ros47,ST54,NJ59,PW62,Boy65,WE66, WE67] there are two separate sets of problems preventing the consideration of generic rigid bodies in General Relativity. The first is the lack of a global notion of simultaneity, which renders the notion of distance between two particles undefined. Associated to this is the inexistence of arbitrarily fast signals, which would be necessary for a rigid body to keep its exact shape. The second problem is that even if a family of spacelike slices is somehow singled out to define a notion of simultaneity, it is in general impossible to keep the mutual distances of more than four particles constant, due to the varying curvature of the slices. Nevertheless, it is many times useful to consider approximate notions of rigidity in General Relativity [Dix70,ER77,TG83,Voi88,XTW04]. These are usually derived from multipole or post-Newtonian expansions, which can be quite formidable. A simpler method for obtaining the (approximate) motion of a small (approximately) rigid body from a finite-dimensional Lagrangian, mirroring what is done in Newtonian mechanics, would therefore be desirable. A possibility (suggested by Turakulov [Tur] in the context of Newtonian mechanics on a curved space) is to approximate the motion of a rigid body by the motion of
This work was partially supported by the Fundação para a Ciência e a Tecnologia through the Program POCI 2010/FEDER and by the grant POCI/MAT/58549/2004.

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an orthonormal frame (determined by a naturally defined Lagrangian). This avoids the curvature problem. The simultaneity problem can be overcome by adding the natural constraint that the frame’s timelike unit vector should be tangent to the motion of the base point (related ideas in flat spacetime can be found in [HR74,Ell82,Tak82]). The consequences of this model are explored in the present paper. Section 1 begins by briefly reviewing the Euler top, in order to set up the notation. The Newtonian tangent Euler top on a curved space, which is a toy model for its relativistic counterpart, is investigated in Sect. 2. This generalizes [Tur], where the discussion was restricted to spherical tops. The relativistic tangent Euler top is defined in Sect. 3, as well as a simpler Lagrangian determining the spinning motion only. An approximation scheme for these is devised in Sect. 4, where the quadratic and cubic approximations are examined. The first (but not the second) is seen to lead to the usual Euler top motion with respect to a Fermi-Walker transported frame. Section 5 studies the exact spinning motion when the base point is moving along a geodesic. The corresponding Lagrangian is shown to yield a (new) completely integrable system on S O(3). Finally, Sect. 6 analyzes the full Lagrangian in the quadratic approximation. It is found that the motion of the frame’s base point is determined by the Mathisson-Papapetrou equation of motion subject to the Mathisson-Pirani spin supplementary condition [Mat37,Pap51,Pir56,MTW73,Sem99,Ste04]. The present paper thus succeeds in deriving the Mathisson-Papapetrou equation from a Lagrangian, defined on the (finite-dimensional) orthonormal frame bundle, which is the natural generalization of the Lagrangian for a free rigid body in Newtonian mechanics. This is quite different from other approaches inspired on variational principles [K72,YB93,Por06], where the authors start with the Mathisson-Papapetrou equation and proceed to construct an action (generally with a much less clear physical meaning). Moreover, this Lagrangian formulation allows for the study of the rotational motion (which can be analyzed for arbitrary motions of the center of mass), instead of considering only the evolution of the angular momentum tensor. This leads to interesting new mechanical systems on S O(3). We use the conventions of [MTW73] (including the Einstein summation convention), except for the meaning of the indices, which is the following: latin indices will always be associated to an orthonormal frame; they will be spacetime indices (ranging from 0 to 3) if they are from the beginning of the alphabet (a, b, . . .), and space indices (ranging from 1 to 3) if the are from the middle of the alphabet (i, j, . . .). Similarly, greek indices will always be associated to a coordinate system; they will be spacetime indices if the are from the beginning of the alphabet (α, β, . . .), and space indices if the are from the middle of the alphabet (µ, ν, . . .)1 . As is usual, we will not worry about the vertical position of space indices on orthonormal frames. 1. Euler top We shall use this brief review of the Euler top to set up our notation. For simplicity we will consider only the case where the Euler top spins about its center of mass. Definition 1.1. The reference configuration of an Euler top is a positive finite Borel measure m on R3 , not supported on any 1-dimensional subspace, and such that

ξ dm = 0,
ξ
2 dm < +∞. R3

R3

1 This complication is unfortunate but inevitable, as we work in the orthonormal frame bundle and often distinguish between space and time.

Tangent Euler Top in General Relativity

389

The mass of the Euler top is M = m(R3 ), and its Euler tensor is the symmetric matrix
ξ i ξ j dm. Ii j = R3

A motion of the Euler top is an Euler-Lagrange curve for the Lagrangian L : T S O(3) → R given by2
1 ˙ ˙ , Sξ ˙  dm = 1 S˙i j I jk S˙ik = 1 tr( S˙ I S˙ t ). L(S, S) =  Sξ 2 R3 2 2 As is well known [Ar97,MR99], the Euler top is completely integrable: in addition to the total energy H = L, one can obtain commuting first integrals from the conservation of angular momentum. In fact, the Euler-Lagrange equations themselves are equivalent to the conservation of angular momentum. Theorem 1.2. If S : R → S O(3) is a motion of the Euler top, then the angular momentum matrix σ = S(I A + AI )S t ∈ so(3) is constant, where A : R → so(3) is such that S˙ = S A. Proof. S O(3) acts on itself by left multiplication, and L = − 21 tr(AI A) is clearly S O(3)-invariant. By Noether’s theorem, the function FL(X B ) is conserved along the motion for all B ∈ so(3), where FL is the fiber derivative of the Lagrangian and X B ∈ X(S O(3)) is the infinitesimal action of B ∈ so(3). Now  d  (X B ) S = exp(t B)S = B S = SS t B S, dt t=0 and hence 1 1 1 FL(X B ) = − tr(S t B S I A) − tr(AI S t B S) = − tr(S(I A + AI )S t B). 2 2 2 The formula C, D = − 21 tr(C D) defines an inner product on so(3). Since σ, B is conserved for all B ∈ so(3), so is σ .   Remark 1.3. (1) The matrix σ is related to the angular momentum vector σ through σξ = σ × ξ for all ξ ∈ R3 , i.e. σik = εi jk σ j . (2) We have σ = S S t with  = I A + AI ∈ so(3). Conservation of σ is equivalent to the Euler equation [Ar97,MR99,Oli02] ˙ =  A − A.  2 Here we use the usual convention that S˙ can either mean the time derivative of a curve S : R → S O(3) or a tangent vector on the tangent bundle T S O(3).

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2. Tangent Euler top on a curved space Let O M be the bundle of positive orthonormal frames on an oriented3 3-dimensional Riemannian manifold (M, ·, ·). A curve E : R → O M is given by E(t) = (c(t), E 1 (t), E 2 (t), E 3 (t)), where c : R → M is a curve on M and E 1 , E 2 , E 3 : R → T M are vector fields along c which form a positive orthonormal frame at c(t) for each t ∈ R. Definition 2.1. A motion of a tangent Euler top on (M, ·, ·) is an Euler-Lagrange curve for the Lagrangian L : T O M → R given by
  ˙ =1 L(E, E) c˙ + ξ i ∇c˙ E i , c˙ + ξ j ∇c˙ E j dm, 2 R3 where ∇ is the Levi-Civita connection and m is the reference configuration of an Euler top. Remark 2.2. (1) Notice that if (M, ·, ·) is R3 with the Euclidean metric then this Lagrangian is just the Lagrangian for a free rigid body. (2) Physically, one can expect a tangent Euler top to represent the motion of an approximately rigid extended body in the limit in which its size is much smaller than the local radius of curvature. Theorem 2.3. The motions of a tangent Euler top on (M, ·, ·) are as follows: the frame {E 1 , E 2 , E 3 } rotates exactly as an Euler top with respect to any frame { Eˆ 1 , Eˆ 2 , Eˆ 3 } which is parallel-transported along the motion of the base point; the base point moves according to the equation4 M

 D c˙ 1  ˆ i j σi j = 0, + ι(c) ˙

dt 2

(1)

ˆ i j are the curvature forms for the frame { Eˆ 1 , Eˆ 2 , Eˆ 3 },  : T ∗ M → T M is the where

isomorphism induced by the metric and σ ∈ so(3) is the angular momentum matrix. ∼ U × S O(3) by choosing a local orthoProof. We define a local trivialization O M|U = normal frame { Eˆ 1 , Eˆ 2 , Eˆ 3 } ⊂ X(U ) on a sufficiently small open set U ⊂ M. For this trivialization we have E i (t) = S ji (t)( Eˆ j )c(t) ⇒ ∇c˙ E i = S˙ ji Eˆ j + S ji ωˆ k j (c) ˙ Eˆ k , j

where ωˆ i are the connection forms associated to our local frame. The Lagrangian can therefore be written as L = T + K + C + F, 3 The hypothesis of orientability is not crucial, as one can always pass to the orientable double cover. 4 As is usual, D = ∇ is the covariant derivative along the curve c : R → M. c˙ dt

Tangent Euler Top in General Relativity

391

where


1 1 ˙ c; ˙ c, ˙ c ˙ dm = M c, 2 R3 2
1 1 K = ξ i S˙ki ξ j S˙l j  Eˆ k , Eˆ l  dm = tr( S˙ I S˙ t ); 2 R3 2
C= ξ i S˙ki ξ j Sl j ωˆ ml (c) ˙ Eˆ k , Eˆ m  dm = tr( S˙ I S t ωˆ t (c)); ˙ R3
1 1 F= ˆ c)S ˙ I S t ωˆ t (c)). ξ i Ski ωˆ l k (c)ξ ˙ j Sm j ωˆ nm (c) ˙ Eˆ l , Eˆ n  dm = tr(ω( ˙ 2 R3 2 T =

Our task is now to write the Euler-Lagrange equations. For the S O(3) part, we notice ˆ c) ˙ = 0, that if the frame { Eˆ 1 , Eˆ 2 , Eˆ 3 } happens to be parallel transported along c then ω( and hence only terms coming from the rotational kinetic energy K survive in the EulerLagrange equations. Therefore {E 1 , E 2 , E 3 } rotates with respect to { Eˆ 1 , Eˆ 2 , Eˆ 3 } exactly as an Euler top with reference configuration m. For the motion of the base point, there will be terms coming from both the translational kinetic energy T and the Coriolis term C (but not from the frame term F, which is quadratic in ω( ˆ c)). ˙ As is well known,   ∂T d ∂T D x˙ ν − µ = M gµν , µ dt ∂ x˙ ∂x dt where gµν is the matrix of the metric. Since the trace of the product of symmetric and anti-symmetric matrices is zero, we can write ˙ = C = tr(S AI S t ωˆ t (c)) Therefore d dt



∂C ∂ x˙ µ

 −

1 1 1 tr(S(AI − I At )S t ωˆ t (c)) ˙ = tr(σ ωˆ t (c)) ˙ = σi j ωˆ i jµ x˙ µ . 2 2 2

∂C 1 1 ˆ i jνµ x˙ ν , = σi j (∂ν ωˆ i jµ − ∂µ ωˆ i jν )x˙ ν = σi j

µ ∂x 2 2

ˆ i = d ωˆ i + ωˆ i ∧ ωˆ k are the curvature forms associated to { Eˆ 1 , Eˆ 2 , Eˆ 3 }. Conwhere

j j k j sequently, the equations of motion for the base point are M

D x˙ µ 1 ˆ µ x˙ ν = 0. + σi j

i jν dt 2

  Remark 2.4. (1) If (M, ·, ·) is R3 with the Euclidean metric then the curvature forms vanish and we obtain the familiar result that the center of mass of a free rigid body moves with constant velocity. (2) We can write Eq. (1) as M

D x˙ µ 1 µ = R νκλ x˙ ν s κλ , dt 2

where s = σi j Eˆ i ⊗ Eˆ j

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is the angular momentum tensor. As noticed by Turakulov [Tur], this equation is similar to the Mathisson-Papapetrou equation for a spinning particle in General Relativity [Mat37,Pap51]. The fact that the angular momentum matrix is constant on the parallel transported frame { Eˆ 1 , Eˆ 2 , Eˆ 3 } is translated into the fact that the angular momentum tensor satisfies the parallel transport equation Ds µν = 0. dt 3. Tangent Euler top in General Relativity Let O M be the bundle of positive future-pointing orthonormal frames on an oriented, time-oriented5 4-dimensional Lorentzian manifold (M, ·, ·). An admissible curve E : R → O M is given by E(t) = (c(t), E 0 (t), E 1 (t), E 2 (t), E 3 (t)), where c : R → M is a future-directed timelike curve on M parameterized by its proper time and E 0 , E 1 , E 2 , E 3 : R → T M are vector fields along c which form a positive orthonormal frame at c(t) satisfying E 0 (t) = c(t) ˙ for each t ∈ R. Definition 3.1. A motion of a tangent Euler top on (M, ·, ·) is a (constrained) EulerLagrange curve for the Lagrangian L : T O M → R given by
  1 2  ˙ = L(E, E)  c˙ + ξ i ∇c˙ E i , c˙ + ξ j ∇c˙ E j  dm, R3

where ∇ is the Levi-Civita connection and m is the reference configuration of an Euler top, chosen among all admissible curves, with unspecified final time. Remark 3.2. (1) This is the natural generalization of the Lagrangian for a (Newtonian) tangent Euler top on a curved space: one simply replaces the squared velocity by its relativistic length. (2) Restricting the action to admissible curves means considering bodies which are rigid as seen by an observer comoving with the base point. (3) We must leave the final time unspecified because we insisted that admissible curves should be parameterized by the proper time of the base point, which needs not be the same for all admissible curves with given endpoints. (4) A tangent Euler top in General Relativity can therefore be physically interpreted as the motion of an extended body which is approximately rigid as seen by an observer placed at its center of mass, in the limit in which the body’s size is much smaller than the local radius of curvature. A simpler problem is obtained by focussing on the spinning motion. To do this, we regard the motion c : R → M of the base point as given (which may or may not be the one arising from Definition 3.1) and look for the positive orthonormal frame {c(t), ˙ E 1 (t), E 2 (t), E 3 (t)} along c which maximizes the action. Choosing a positive orthonormal Fermi-Walker transported frame {c(t), ˙ Eˆ 1 (t), Eˆ 2 (t), Eˆ 3 (t)} along c, so that ∇c˙ c(t) ˙ = aˆ i (t) Eˆ i (t),

∇c˙ Eˆ i (t) = aˆ i (t)c(t), ˙

5 Again this is not a crucial hypothesis, as one can always pass to the time-orientable double cover.

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we have E i (t) = S ji (t)( Eˆ j )c(t) ⇒ ∇c˙ E i = S˙ ji Eˆ j + S ji a j c˙ for some curve S : R → S O(3). Therefore   c˙ + ξ i ∇c˙ E i , c˙ + ξ j ∇c˙ E j = −1 + S˙ki ξ i S˙k j ξ j − aˆ k Ski ξ i aˆ l Sl j ξ j − 2aˆ j S ji ξ i . Definition 3.3. A spinning motion of a tangent Euler top on (M, ·, ·) is an (unconstrained) Euler-Lagrange curve for the Lagrangian L : T S O(3) × R → R given by
 1  ˙ , Sξ ˙  + Sξ , aˆ (t)2 + 2Sξ , aˆ (t) 2 dm, ˙ t) = L(S, S, 1 −  Sξ R3

where m is the reference configuration of an Euler top and aˆ : R → R3 represents the proper acceleration on a Fermi-Walker transported frame. 4. Approximate Spinning Motions Expanding the integrand in the Lagrangian for a spinning motion in power series of ξ and integrating we can write L = M + L (2) + L (3) + . . . , where L (n) is the integral of the term of order n in ξ . Definition 4.1. An approximate spinning motion of order n of a tangent Euler top is an Euler-Lagrange curve of the truncated Lagrangian L = M + L (2) + . . . + L (n) . Theorem 4.2. The approximate spinning motions of order 2 of a tangent Euler top are exactly the motions of an Euler top with the same reference configuration. Proof. One just needs to check that L (2) = −  


1 ˙ , Sξ ˙  dm.  Sξ 2 R3

Remark 4.3. Notice that the proper acceleration aˆ does not appear in the expression of L (2) : in this approximation, the spinning motion completely decouples from the translational motion. This is similar to what happens in the Newtonian case (and is, of course, the principle underlying the operation of a gyroscope). Proposition 4.4. Generically, the approximate spinning motions of order 3 of a tangent Euler top are not the motions of an Euler top with the same reference configuration. Proof. One just needs to check that
1 ˙ , Sξ ˙ Sξ , aˆ  dm, L (3) =  Sξ 2 R3 which generically is not a total time derivative. L (3)

 

Remark 4.5. can be thought of as (minus) the potential energy corresponding to the weight of the rotational kinetic energy in the constant “gravitational field” −ˆa induced by the acceleration of the center of mass.

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5. Spinning Motions Along Geodesics If the spinning motion is along a timelike geodesic, we have aˆ = 0, and the Lagrangian becomes S O(3)-invariant. In this case, we can apply Noether’s Theorem. Theorem 5.1. If S : R → S O(3) is a spinning motion of a tangent Euler top along a timelike geodesic of (M, ·, ·), then σ = S S t ∈ so(3) is constant, where
=

(Aξ )ξ t − ξ (Aξ )t

R3

1

|1 − Aξ , Aξ | 2

dm ∈ so(3)

and A : R → so(3) is such that S˙ = S A. Proof. We can write the Lagrangian as



 1 t 2  ˙ ˙ ˙ 1 − tr ( Sξ )( Sξ ) dm = L(S, S) = R3

R3



 1 1 − tr Aξ ξ t At  2 dm.

By Noether’s theorem, FL(X B ) is conserved along the motion for all B ∈ so(3), where X B ∈ X(S O(3)) is the infinitesimal action of B. Since


tr S t B Sξ ξ t At + Aξ ξ t S t B t S 1 B dm FL(X ) = − 

 1 2 R3 1 − tr Aξ ξ t At  2  
1 Aξ ξ t − ξ ξ t At = tr B S dm S t , 1 3 2 R |1 − Aξ , Aξ | 2 we see that − 21 tr(Bσ ) = B, σ  is conserved for all B ∈ so(3), and so is σ .

 

Remark 5.2. (1) The angular momentum vector σ , related to the matrix σ through σξ = σ × ξ for all ξ ∈ R3 , can be seen to have the familiar expression
σ =

˙ ) (Sξ ) × ( Sξ dm. ˙ , Sξ ˙ | 21 R3 |1 −  Sξ

(2) This system is completely integrable: in addition to the total energy
1 dm, K = −H = −FL(A) + L = 1 3 R |1 − Aξ , Aξ | 2 one can obtain commuting first integrals from the conservation of angular momentum. In fact, the Euler-Lagrange equations themselves are equivalent to the conservation of angular momentum. The images of the motions A : R → so(3) can be obtained from the intersections of the level surfaces of the functions K (A) and (A), (A) = − 21 tr((A)2 ).

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(3) We have L(A) = M −

1 1 Ii j Aki Ak j − Ii jkl Ami Am j Ank Anl − . . . , 2 8

where


Ii jkl =

R3

ξ i ξ j ξ k ξ l dm, . . . .

Therefore the Euler tensor is not enough to characterize a relativistic tangent Euler top: the higher multipoles are also required (and become more important as the spinning motion becomes more relativistic). 6. Quadratically Approximate Motions We can extend Definition 4.1 to the general motion of a tangent Euler top. Definition 6.1. A quadratically approximated motion of a tangent Euler top on (M, ·, ·) is an Euler-Lagrange curve for the Lagrangian L : T O M → R given by ˙ =M− L(E, E)

1 Ii j ∇c˙ E i , E k ∇c˙ E j , E k , 2

chosen among all admissible curves, with unspecified final time. Theorem 6.2. Quadratically approximated motions of a tangent Euler top on (M, ·, ·) are as follows: {E 1 , E 2 , E 3 } rotates exactly as an Euler top with respect to { Eˆ 1 , Eˆ 2 , Eˆ 3 }, where { Eˆ 0 , Eˆ 1 , Eˆ 2 , Eˆ 3 } : R → O M is Fermi-Walker transported along the motion of the base point and satisfies Eˆ 0 = c; ˙ the base point moves according to the equation 

 d D c˙ ˆ D c˙ 1  ˆ i j σi j + Eˆ i σi j (2) + ι(c) ˙

, E j = 0, (M + K ) dt 2 dt dt ˆ ab are the curvature forms for the frame where K is the rotational kinetic energy,

 ∗ ˆ ˆ ˆ ˆ { E 0 , E 1 , E 2 , E 3 }, : T M → T M is the isomorphism induced by the metric and σ ∈ so(3) is the angular momentum matrix. Proof. The spinning motion with respect to a Fermi-Walker transported frame was already shown to be that of an Euler top in Sect. 4. To compute the motion of the base point we define a local trivialization O M|U ∼ = U × S O ↑ (3, 1) by choosing a local orthonormal frame { Eˆ 0 , Eˆ 1 , Eˆ 2 , Eˆ 3 } ⊂ X(U ) on a sufficiently small open set U ⊂ M. For this trivialization we have E i (t) = S ai (t)( Eˆ a )c(t) ⇒ ∇c˙ E i = S˙ ai Eˆ a + S ai ωˆ ba (c) ˙ Eˆ b , where ωˆ ba are the connection forms associated to our local frame, and hence the Lagrangian can be written as   1  L = M − Ii j ωˆ ab (c)S ˙ ak S bi + Aki ωˆ cd (c)S ˙ ck S d j + Ak j , 2

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where A = (Aab ) ∈ so(3, 1) is defined by S˙ = S A ⇔ S˙ ab = S ac Ac b . Notice that the conditions S ∈ O(3, 1) and A ∈ so(3, 1) are conveniently written by raising and lowering indices as (S −1 )ab = Sb a and Aab = −Aba . The best way to deal with the constraint while avoiding introducing coordinates on S O ↑ (3, 1) is to to set up our problem as an optimal control problem (see for instance [Law63,Pin93]). As state variables we choose the local coordinates x α of the base point, the components Sa b of the frame vectors and the action θ . As control variables we simply choose 6 independent components of the anti-symmetric matrix (Aab ). The state variables depend on the control variables through the solution of the Cauchy problem ⎧ α ˆα a ⎪ ⎨x˙ = E a S 0 a S˙ b = S ac Ac b    ⎪ ⎩θ˙ = M − 1 I ωˆ α S f Sd Se + A α Sc Sa Sb + A ˆ ˆ ω ˆ E E abα c ik deα f 0 j k jk . 0 i k 2 ij Our objective is to maximize θ1 = θ (t1 ), i.e. minimize  = −θ1 , where the final time t1 is not fixed. To do so, we introduce the multipliers λα , ab , λ (one for each differential equation) and set up the Hamiltonian function H = λα Eˆ aα S a0 + ab S ac Ac b    1  β f +λ M − Ii j ωˆ abα Eˆ cα S c0 S ai S bk + Aik ωˆ deβ Eˆ f S 0 S d j S ek + A jk . 2 According to the Pontryagin Maximum Principle, we must select the values of the control variables which maximize H . The minimizing curve will then be obtained by solving the Hamilton equations ⎧ ∂H ˙ ⎪ ⎨λ α = − ∂ x α ˙ ab = − ∂ Ha  ∂S b ⎪ ⎩˙ λ = −∂H ∂θ

together with the Cauchy problem. The multiplier λ is clearly constant; since θ1 is not specified, we must select λ(t1 ) = −

∂ =1 ∂θ1

(hence λ ≡ 1). Moreover, H is constant along the solution. Since t1 is not fixed, we must have H ≡ 0. Because we already know what the spinning motion will be, we do not have to write out all the minimum equations. Also, we shall assume that the trivializing frame { Eˆ 0 , Eˆ 1 , Eˆ 2 , Eˆ 3 } happens to be Fermi-Walker transported along the minimizing curve, ˙ As a consequence, we will have with Eˆ 0 = c. ωˆ i jα Eˆ 0α = 0,

ωˆ i0α Eˆ 0α = aˆ i ,

S a0 = δ a0

along this curve. In particular, S˙ a0 = 0 ⇔ S ac Ac 0 = 0 ⇔ Aa0 = 0 ⇔ Aa0 = −A0a = 0 along the minimizing curve.

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We now write the relevant minimum conditions. Maximizing H with respect to the control variables A0i yields ∂H = 0 ⇔ ai S a0 − a0 S ai = 0 ⇔ 0i − a0 S ai = 0. ∂ A0i

(3)

We have the Hamilton equations ˙l0 = − 

∂H ∂ Sl 0

= −λα Eˆ lα + l b A0b    β f −Ii j ωˆ abα Eˆ lα S ai S bk ωˆ deβ Eˆ f S 0 S d j S ek + A jk = −λα Eˆ lα

(4)

and ˙ 0l = − 

   ∂H β f d e b l α c b ˆ ˆ ω ˆ ω ˆ = − A − I S S S S S + A E E l j 0bα deβ jk c 0 0 k b f 0 j k ∂ S 0l    β f −Ii j ωˆ a0α Eˆ cα S c0 S ai ωˆ deβ Eˆ f S 0 S d j S el + A jl = 0i Ali + Il j aˆ m S mk A jk − Ii j aˆ m S mi A jl .

(5)

Differentiating (3) with respect to t and using (4), (5), and again (3) yields λα Eˆ αj S ji = (Ii j A jk + Ai j I jk )Smk aˆ m , from which one readily obtains λα Eˆ lα = σlm aˆ m .

(6)

We must also consider the Hamilton equation λ˙ µ = −

  ∂H ˆ 0α + Ii j ∂µ ωˆ abα Eˆ 0α S a S b A jk = −λ ∂ E α µ i k ∂xµ   +Ii j ωˆ abα ∂µ Eˆ 0α S a S b A jk . i

k

(7)

We now use our freedom in the choice of local coordinates to select Fermi normal coordinates (t, x 1 , x 2 , x 3 ), defined by the parameterization ϕ(t, x 1 , x 2 , x 3 ) = expc(t) (x 1 Eˆ 1 + x 2 Eˆ 2 + x 3 Eˆ 3 ), where exp p : T p M → M is the geodesic exponential map. To first order in x we can then choose6 ∂ Eˆ 0 = (1 − aˆ i x i ) , ∂t

∂ Eˆ i = i ∂x

6 Here we use latin indices i, j, . . . for the coordinates, as they are associated to the Fermi-Walker transported frame. By the same token, the coordinate t coincides with the proper time t along the minimizing curve.

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(see [Poi04]). In particular, ∇ Eˆ i Eˆ j = 0 for x = 0, from where we can deduce that ωˆ i j = 0 and ωˆ 0i ( Eˆ j ) = 0 along the minimizing curve. Equation (7) then becomes   λ˙ n = λt aˆ n + Ii j ∂n ωˆ lmα Eˆ 0α Sli Smk A jk . (8) The quantity λt in this equation can be obtained from H = 0 ⇔ λt + k j Ski Ai j + M −

1 Ii j Aik A jk = 0. 2

(9)

Maximizing H with respect to Ai j (with i < j, say) yields further relations: ∂H j = 0 ⇔ a S ai − ai S a j − Iik Ak j + I jk Aki = 0 ∂ Ai j ⇔ k j Ski − ki Sk j = Iik Ak j − I jk Aki . These can be used to rewrite the second term in (9), which is tr(S At ) =

1 1 1 tr(S At + At S t ) = tr(t S A − S t A) = tr((t S − S t )A). 2 2 2

After substitution we obtain λt = −M −

1 Ii j Aik A jk = −(M + K ), 2

where K is the rotational kinetic energy of the Euler top. Because ∂n ωˆ lmα Eˆ 0α is anti-symmetric in l, m, the quantity multiplying it in (8) can be replaced by 1 1 1 (Ii j Sli Smk A jk − Ii j Smi Slk A jk ) = Sli (Ii j A jk − Ik j A ji )Smk = σlm . 2 2 2 Consequently, Eq. (8) can be written as 1 λ˙ n = −(M + K )aˆ n + ∂n ωˆ lmα Eˆ 0α σlm . 2

(10)

The curvature forms are given by ˆ ba = d ωˆ ba + ωˆ bc ∧ ωˆ ca ⇒

ˆ i j = d ωˆ i j − ωˆ i0 ∧ ωˆ 0 j + ωˆ ik ∧ ωˆ k j .

In components, ˆ i jnt = ∂n ωˆ i jt − ∂t ωˆ i jn − ωˆ i0n ωˆ 0 jt + ωˆ i0t ωˆ 0 jn + ωˆ ikn ωˆ k jt − ωˆ ikt ωˆ k jn .

Recalling that with our choices ωˆ i j = 0 and ωˆ 0i ( Eˆ j ) = 0 along the minimizing curve, we have ˆ i jnt = ∂n ωˆ i jt

on this curve. Therefore (10) is equivalent to 1 ˆ lmnt σlm . λ˙ n = −(M + K )aˆ n +

2

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399

On the other hand, (6) can be written as λl = σlm aˆ m . From these two equations we finally deduce d aˆ 1 ˆ jkit σ jk + σi j j = 0, (M + K )aˆ i −

2 dt which are the nontrivial components of Equation (2).

 

Remark 6.3. (1) As one would expect, the remaining minimum conditions lead to the Euler equation. (2) We can write Eq. (2) as (M + K )

D x˙ α 1 D 2 x˙ β = R αβγ δ x˙ β s γ δ − s αβ , dt 2 dt 2

where s = σi j Eˆ i ⊗ Eˆ j is the angular momentum tensor7 . This equation is exactly the Mathisson-Papapetrou equation subject to the Mathisson-Pirani spin supplementary condition [Mat37, Pap51,Pir56,MTW73,Sem99,Ste04]. Notice that the total mass appearing in this equation is the sum of the Euler top’s rest mass M with its rotational kinetic energy K (which is constant along the motion). The fact that the angular momentum matrix is constant on the Fermi-Walker transported frame { Eˆ 1 , Eˆ 2 , Eˆ 3 } is translated into the fact that the angular momentum tensor satisfies the Fermi-Walker transport equation Ds αβ D 2 x˙ γ β D 2 x˙ γ α = x˙ α sγ + x˙ β s . 2 dt dt dt 2 γ References [Ar97] [Born09] [Boy65] [Dix70] [ER77] [Ell82] [K72] [HR74] [Her10]

Arnold, V.I.: Mathematical methods of classical mechanics. Berlin-Heidelberg-New York: Springer, 1997 Born, M.: Die theorie des starren elektrons in der kinematik des relativitätsprinzips. Ann. der Physik 30, 1–56 (1909) Boyer, R.: Rigid frames in general relativity. Proc. R. Soc. Lond. A 283, 343–355 (1965) Dixon, W.: Dynamics of extended bodies in general relativity i. Momentum and Angular Momentum. Proc. R. Soc. London A 314, 499 (1970) Ehlers, J., Rudolph, E.: Dynamics of extended bodies in general relativity – center-of-mass description and quasi-rigidity. Gen. Rel. Grav. 8, 197–217 (1977) Ellis, J.: Motion of a classical particle with spin. i. the canonical theory of multipliers. J. Phys. A: Math. Gen. 15, 409–427 (1982) Künzle, H.: Canonical dynamics of spinning particles in gravitational and electromagnetic fields. J. Math. Phys. 13, 739–744 (1972) Hanson, A., Regge, T.: The relativistic spherical top. Ann. Phys. 87, 498–566 (1974) Herglotz, G.: Über den vom standpunkt des relativitätsprinzips als “starr” zu bezeichnenden körper. Ann. der Physik 31, 393–415 (1910)

7 Notice however that many authors define the angular momentum tensor to be  s = −s, corresponding to the alternative relation  σi j = εi jk σ k between the angular momentum matrix and the angular momentum vector.

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[Law63] [MR99]

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Lawden, D.: Optimal trajectories for space navigation. Oxford: Butterworths, 1963 Marsden, J., Ratiu, T.: Introduction to mechanics and symmetry. Berlin-Heidelberg-New York: Springer, 1999 [Mat37] Mathisson, M.: Neue mechanik materieller systeme. Acta Phys. Polon. 6, 163–200 (1937) [MTW73] Misner, C., Thorne, K., Wheeler, J.A.: Gravitation. Newyark: Freeman, 1973 [NJ59] Newman, E., Janis, A.: Rigid frames in relativity. Phys. Rev. 116, 1610–1614 (1959) [Noe10] Noether, F.: Zur kinematik des starren körpers in der relativtheorie. Ann. der Physik 31, 919–944 (1910) [Oli02] Oliva, W.: Geometric mechanics. Berlin-Heidelberg-New York: Springer, 2002 [Pap51] Papapetrou, A.: Spinning test particles in general relativity i. Proc. R. Soc. London A 209, 248 (1951) [Pin93] Pinch, E.: Optimal control and the calculus of variations. Oxford, Oxford University Press, 1993 [Pir56] Pirani, F.: On the physical significance of the riemann tensor. Acta Phys. Polon. 15, 389– 405 (1956) [PW62] Pirani, F., Williams, G.: Rigid motion in a gravitational field. Semin. Mec. analytique Mec. celeste M. Janet 5(8-9), 1–16 (1961/62) [Poi04] Poisson, E.: The motion of point particles in curved spacetime. Living Rev. Relativity 7 (2004), available at http://www.livingreviews.org/lrr-2004-6 [Por06] Porto, R.: Post-newtonian corrections to the motion of spinning bodies in nonrelativistic general relativity. Phys. Rev. D 73, 104031 (2006) [Ros47] Rosen, N.: Notes on rotation and rigid bodies in relativity theory. Phys. Rev. 71, 54–58 (1947) [ST54] Salzman, G., Taub, A.: Born-type rigid motion in relativity. Phys. Rev. 95, 1659–1669 (1954) [Sem99] Semerák, O.: Spinning test particles in a kerr field - i. Mont. Not. Astron. Soc. 308, 863–875 (1999) [Ste04] Stephani, H.: Relativity. Cambridge: Cambridge University Press, 2004 [Tak82] Takabayasi, T.: Theory of relativistic top and electromagnetic interaction. Progress of Theoretical Physics 67, 357–374 (1982) [TG83] Thorne, K., Gürsel, Y.: The free precession of slowly rotating neutron stars: rigid-body motion in general relativity. Mon. Not. R. Astr. Soc. 205, 809–817 (1983) [Tur] Turakulov, Z.: Classical mechanics of spinning particle in a curved space, http://arxiv.org/list/ dg-ga/9703008, 1997 [Voi88] Voinov, A.: Motion and rotation of celestial bodies in the post-newtonian approximation. Celest. Mech. 42, 293–307 (1988) [WE66] Wahlquist, H., Estabrook, F.: Rigid motions in einstein spaces. J. Math. Phys. 7, 894–905 (1966) [WE67] Wahlquist, H., Estabrook, F.: Herglotz-noether theorem in conformal space-time. J. Math. Phys. 8, 919 (1967) [XTW04] Xu, C., Tao, J., Wu, X.: Post newtonian rigid body. Phys. Rev. D 69, 024003 (2004) [YB93] Yee, K., Bander, M.: Equations of motion for spinning particles in external electromagnetic and gravitational fields. Phys. Rev. D 48, 2797–2799 (1993)

Communicated by G.W. Gibbons

Commun. Math. Phys. 281, 401–444 (2008) Digital Object Identifier (DOI) 10.1007/s00220-008-0495-4

Communications in

Mathematical Physics

Vanishing of Vacuum States and Blow-up Phenomena of the Compressible Navier-Stokes Equations Hai-Liang Li1,3 , Jing Li2,3 , Zhouping Xin1,3 1 Department of Mathematics, Capital Normal University, Beijing, P. R. China.

E-mail: [email protected]

2 Institute of Applied Mathematics, AMSS, Academia Sinica, Beijing 100080, P. R. China.

E-mail: [email protected]

3 Institute of Mathematical Science, The Chinese University of Hong Kong, Shatin, Hong Kong.

E-mail: [email protected] Received: 12 April 2007 / Accepted: 3 December 2007 Published online: 22 May 2008 – © Springer-Verlag 2008

Abstract: The Navier-Stokes systems for compressible fluids with density-dependent viscosities are considered in the present paper. These equations, in particular, include the ones which are rigorously derived recently as the Saint-Venant system for the motion of shallow water, from the Navier-Stokes system for incompressible flows with a moving free surface [14]. These compressible systems are degenerate when vacuum state appears. We study initial-boundary-value problems for such systems for both bounded spatial domains or periodic domains. The dynamics of weak solutions and vacuum states are investigated rigorously. First, it is proved that the entropy weak solutions for general large initial data satisfying finite initial entropy exist globally in time. Next, for more regular initial data, there is a global entropy weak solution which is unique and regular with well-defined velocity field for short time, and the interface of initial vacuum propagates along the particle path during this time period. Then, it is shown that for any global entropy weak solution, any (possibly existing) vacuum state must vanish within finite time. The velocity (even if regular enough and well-defined) blows up in finite time as the vacuum states vanish. Furthermore, after the vanishing of vacuum states, the global entropy weak solution becomes a strong solution and tends to the non-vacuum equilibrium state exponentially in time.

1. Introduction The compressible isentropic Navier-Stokes equations, which are the basic models describing the evolution of a viscous compressible fluid, read as follows:
ρt + div(ρu) = 0, (1.1) (ρu)t + div(ρu ⊗ u) − 2div(µD(u)) − ∇(ξ divu) + ∇ p(ρ) = 0,

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H.-L. Li, J. Li, Z. Xin

where x ∈  ⊂ R N , t ∈ (0, T ), D(u) = (∇u + (∇u)T )/2, and p(ρ) = aρ γ , a > 0, γ ≥ 1, the viscosity coefficients µ, ξ are assumed to satisfy µ ≥ 0 and ξ + 2µ/N ≥ 0. If µ and ξ are both constants, there is huge literature on the studies of the global existence and behavior of solutions to (1.1). For instance, the one-dimensional (1D) problems were addressed by Kazhikhov et al [24] for sufficiently smooth data, and by Serre [46,47] and Hoff [18] for discontinuous initial data, where the data are uniformly away from the vacuum; the multidimensional problems (1.1) were investigated by Matsumura et al [35–37], who proved global existence of smooth solutions for data close to a non-vacuum equilibrium, and later by Hoff for discontinuous initial data [19], and more recently, by Danchin [11], who obtained existence and uniqueness of global solutions in a functional space invariant by the natural scaling of the associated equations; and for the existence of solutions for arbitrary data (which may include vacuum states), Lions [29–31] (see also Feireisl et al [13]) obtained global existence of weak solutions - defined as solutions with finite energy - when the exponent γ is suitably large, where the only restriction on initial data is that the initial energy is finite, so that the density is allowed to vanish. Despite the important progress, the regularity, uniqueness and behavior of these weak solutions remain largely open. As emphasized in many papers related to compressible fluid dynamics [9,10,18,20,21,24,27,45,46,48,50,52], the possible appearance of vacuum is one of the major difficulties when trying to prove global existence and strong regularity results. Hoff and Smoller [21] proved that weak solutions of the compressible Navier-Stokes equations (1.1) in one space dimension do not exhibit vacuum states in a finite time provided that no vacuum is present initially under fairly general conditions on the data. Such a result was extended to the spherically symmetric case in [54] recently. On the other hand, the results of Xin [52] showed that there is no global smooth solution to Cauchy problem for (1.1) with a nontrivial compactly supported initial density, which gives results for finite time blow-up in the presence of vacuum. It is also proved in [27] that for bounded domain even one point initial vacuum shall cause the global strong solutions to 1D (1.1) to blow up as time goes to infinity provided the initial data satisfies some compatibility conditions. The independence of viscosities on density makes it possible to trace either the particle path or the trajectory of vacuum state. This, however, leads to the failure of continuous dependence of weak solutions containing vacuum state on initial data [20]. For the case that the density changes continuously across the interfaces separating the gas and vacuum, the global existence and uniqueness of weak solutions was obtained in [33], where the authors obtained that velocity is smooth enough up to the interfaces which are particle paths separating the gas from the vacuum and that the support of gas density expands outside as well as interface connecting gas and vacuum moves at an algebraic rate. Thus, viscous compressible fluids near vacuum should be better modeled by the compressible Navier-Stokes equations with density-dependent viscosities, as was derived in the fluid-dynamical approximation of Boltzmann equation for dilute gases. Further, as was first pointed out and investigated by Liu-Xin-Yang in [32], in the derivation of the compressible Navier-Stokes equations from the Boltzmann equation by the ChapmanEnskog expansions, the viscosity depends on the temperature,which is translated into the dependence of the viscosity on the density for isentropic flows. Moreover, it should be emphasized that a one-dimensional compressible flow model, called the viscous Saint-Venant system for laminar shallow water, derived rigorously from incompressible Navier-Stokes system with a moving free surface by Gerbeau-Perthame recently in [14],

Vanishing of Vacuum States and Blow-up Phenomena

has the form:




ρt + (ρu)x = 0, (ρu)t + (ρu 2 )x − a(ρu x )x + (ρ 2 )x = 0,

403

(1.2)

with the viscosity coefficients given by µ(ρ) = ξ(ρ) = aρ/3 for a given positive constant a. Indeed, such models appear naturally and often in geophysical flows [2,6,7]. In the case of the one-dimensional problem with µ = ξ = aρ α for some positive constants a and α, the well-posedness of the Cauchy problem has been studied by many authors for not only initially compact-supported density but initially non-vacuum density. Indeed, for the initially compact-supported density case, the local (in time) well-posedness of weak solutions to this problem was first established by Liu-XinYang in [32], where the initial density was assumed to be connected to vacuum with discontinuities. This property, as shown in [32], can be maintained for some finite time. And the global existence of weak solutions, together with α ∈ (0, 1) and the density function connecting to vacuum with discontinuities, was considered by many authors, see [23,43,55] and the references therein. It is noticed that the above analysis is based on the uniform positive lower bound of the density with respect to the construction of the approximate solutions. On the other hand, if the density function connects to vacuum continuously, there is no positive lower bound for the density and the viscosity coefficient vanishes at vacuum. This degeneracy in the viscosity coefficient gives rise to new difficulties in analysis because of the less regularizing effect on the solutions. Yang et al [56] first obtained a local existence result for this case under the free boundary condition with α > 1/2. The authors in [12,51,57] obtained the global existence of weak solutions with α ∈ (0, 1/2). However, almost all the above results concern mainly free boundary problems, and for the global existence results, the choices of viscosity do not fit the important physical model, the shallow water equation (1.2) with µ(ρ) = ρ (namely α = 1). For the constant viscosity case, one has known not only that the vacuum state will not develop later on time if there is no vacuum state initially [21], but also that the separate two initial vacuum states shall not meet together in a finite time [54], and that one point initial vacuum causes strong solutions to blow up [27] at infinity as well. However, little is known on the dynamics of the vacuum states of weak solutions to the compressible Navier-Stokes equations (1.1) with density-dependent viscosity on bounded domain. And in particular, it is not clear yet how the vacuum states evolve with respect to time and whether the initial vacuum states shall exist all the time or not for weak solutions. The study of these important dynamical problems about vacuum states is rather difficult because the nonlinear diffusion is degenerate as the vacuum appears, which is quite different from the case of constant viscosity. This causes the loss of information about the velocity and makes it difficult to trace the evolution of vacuum states in general. Across the interface (or vacuum boundary), it is usually difficult to obtain enough information about velocity even if considering specific cases such as point vacuum or continuous vacuum of one piece. It is important to get enough information about the velocity since the flow particles transport usually along particle path, and all interfaces of vacuum, such as free boundaries [23,33] which can be observed and dealt with, also move along the trajectories determined by velocity field. For the multidimensional case, Vaigant et al [49] first proved that for the 2D case and for the case µ is a constant and ξ(ρ) = aρ β , with a > 0, β > 3, (1.1) with periodic boundary condition has a unique strong and classical solution with density away from vacuum. More recently, Bresch and Desjardins [2,3,6,7] (see also [38]) have made

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important progress. Under the condition that ξ(ρ) = 2(µ (ρ)ρ − µ(ρ)), they establish a new Bresch-Desjardins (BD) entropy inequality which can not only be applied to the vacuum case but also used to get the compactness results for (1.1) which extended the compactness results due to Lions [29–31] to the case γ ≥ 1. On the other hand, the constructions of the approximation solutions do not seem routine in the general case of the appearance of vacuum. However, it should be noted that recently Bresch et al [4–6] have made significant progress on the construction of approximate solutions and existence of global weak solutions to the multi-dimensional compressible Navier-Stokes equations and the 2D shallow water model in the case that there is either a drag friction (for barotropic compressible flows) or a cold pressure (for viscous and heat conducting flows). In the case that there is neither drag friction nor cold pressure included, GuoJiu-Xin [16] recently have shown how to construct approximate smooth solutions and obtain the global existence of weak solutions via the BD entropy to the (2D and 3D) barotropic compressible Navier-Stokes for the spherical symmetric initial data. There are also other recent interesting applications of the BD entropy to one-dimensional compressible Navier-Stokes equations with degenerate viscosities, for instance, on the global existence and long time behavior of weak solutions for the free boundary problem [15,17], or the existence and uniqueness of the global strong solution away from vacuum in real line [39]. We study mainly the initial-boundary-value problem (IBVP) for (1.1), where µ = ρ α with α > 1/2, on spatial one-dimensional bounded spatial domains or periodic domains. This contains the physical important model for shallow water equations (1.2). The choice of α > 1/2 is necessary in order to consider the dynamics of vacuum states since it allows the existence of the initial vacuum in Eulerian coordinates as one can see later. We first establish the global existence of entropy weak solutions for the compressible Navier-Stokes equations (1.1), with pressure p = ρ γ and γ ≥ α/2, for general initial data with finite entropy and vacuum. The key in our analysis is the construction nonvacuum approximate solutions so that we can make use of the stability analysis in [38], where the Bresch-Desjardins (BD) entropy inequality (see [2,3,6,7]) was used to obtain the compactness results. Our construction of the approximate solutions is strongly motivated by the previous work of Jiang-Xin-Zhang [23] about the existence of global weak solutions to one-dimensional compressible Navier-Stokes with free boundary as vacuum interface. In general, it seems rather difficult to investigate the dynamics of vacuum states due to the degeneracy of nonlinear diffusion and the density function connecting to vacuum continuously. Therefore, we further consider the cases of more regular initial data containing point the vacuum or continuous vacuum of one piece, and we show that there is a global entropy weak solution which is unique and regular with well-defined velocity field at least for a short time, and the vacuum states remain for the short time. Then, we use some ideas due to [22,27,28] to prove that any possible vacuum state in such global weak solutions which satisfy the BD entropy must vanish within finite time. This shows that such short time structure and vacuum states of weak solutions can not be maintained all the time. And as the vacuum states vanish, the spatial derivative of velocity (if it exists) has to blow up even if the velocity is regular enough and welldefined before. After the vanishing of vacuum states, we can redefine the velocity field and recover the nonlinear diffusion term in terms of density and velocity. In addition, the global entropy weak solution is shown to become a strong solution and tends to the non-vacuum equilibrium state exponentially in time. This phenomena, applied to the compressible shallow water equations (1.2), seems to have never been observed for the compressible Navier-Stokes equations before.

Vanishing of Vacuum States and Blow-up Phenomena

405

The rest of the paper is as follows. In Sect. 2, the main results about the vanishing of vacuum states and blow-up phenomena of global entropy weak solutions for the compressible Navier-Stokes equations are stated. The global existence of entropy weak solutions for general large initial data with vacuum states allowed is proven in Sect. 3. The short time structure of global entropy weak solution with initial one point vacuum state or initial continuous vacuum states of one piece are investigated in Sect. 4. In Sect. 5, we show the vanishing of vacuum states and blow-up phenomena of any global entropy weak solution within finite time and analyze the regularity and large time asymptotic behavior of global entropy weak solutions after the vanishing of vacuum states. 2. Main Results We consider the initial-boundary-value-problem (IBVP) for the 1D compressible NavierStokes equations with density-dependent viscosity ρt + (ρu)x = 0,

(2.1)

(ρu)t + (ρu + p(ρ))x − (µ(ρ)u x )x = 0,

(2.2)

2

with ρ ≥ 0 the density, ρu the momentum. The pressure and viscosity are assumed to have the form: p(ρ) = a1 ρ γ , µ(ρ) = a2 ρ α , where γ ≥ 1, a1 > 0, a2 > 0, and α > 1/2 are constants, and for simplicity we set a1 = a2 = 1. The initial data is given for the density ρ and the momentum ρu, ρ(x, 0) = ρ0 (x) ≥ 0, ρu(x, 0) = m 0 (x), x ∈ ,

(2.3)

where the domain  is chosen as the unit interval denoting the spatial domain (0, 1) or periodic domain with period length one, and throughout the present paper the initial data is assumed to satisfy
α−1/2 ρ0 ≥ 0 a.e. in , ρ0 ∈ L 1 (), (ρ0 )x ∈ L 2 (), (2.4) |m 0 |2 m 0 = 0, a.e. on {x ∈  | ρ0 (x) = 0}, ρ0 ∈ L 1 (). Remark 2.1. Note here that the condition (2.4) implies ρ0 ∈ L ∞ (), ρ0 log+ ρ0 ∈ L 1 ().

(2.5)

It should be clear that a large class of initial data satisfy the conditions in (2.4). In particular, the assumptions (2.4) are satisfied for the following initial data: ρ0 (x) = (|x − x0 |2 )1/(2α−1) , m 0 (x) = 0, x ∈ . Without the loss of generality, the total initial mass is renormalized to be one throughout the present paper, i.e.,  ρ0 (x)d x = 1. 

The boundary conditions are one of the boundary conditions of Dirichlet type and periodic type for Eqs. (2.1)-(2.2) imposed as

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H.-L. Li, J. Li, Z. Xin

(1) Dirichlet case: ρu(0, t) = ρu(1, t) = 0, t ≥ 0,

(2.6)

ρ, u are periodic in x of period 1,

(2.7)

(2) Periodic case: where we consider Eqs. (2.1)–(2.2) on R × (0, ∞). Remark 2.2. For the case of the Dirichlet boundary (2.6), the boundary is given by the physical observable momentum instead of velocity. It is natural to employ such boundary condition as considering the dynamics of (global in time) weak solutions to the IBVP for the compressible Navier-Stokes equations with possible vacuum states included, since it is usually the case that, at vacuum states, the momentum is zero and is observed and controllable, but almost nothing is known yet for the velocity for weak solutions. Note here that for any (weak) solution away from the vacuum at the boundary, the boundary condition (2.6) reduces to u(0, t) = u(1, t) = 0, t ≥ 0. In order to define the weak solutions to the IBVP for the compressible Navier-Stokes equations (2.1)–(2.2) with initial data (2.3) and boundary condition (2.6) or (2.7), we define the set of test functions as follows:
C0∞ ( × [0, T )) for the Dirichlet case (2.6), 
∞ (R × [0, T )) for the periodic case (2.7), Cper and


C0∞ ( × [0, T )) 
∞ (R × [0, T )) Cper

for the Dirichlet case (2.6), for the periodic case (2.7),

∞ (R × [0, T )) defined by with Cper   ∞ Cper (R × [0, T )) = ϕ ∈ C ∞ (R × [0, T ))| ϕ is periodic in x of period 1 .

We define the weak solutions to the IBVP for the compressible Navier-Stokes Equations (2.1)–(2.2) as follows. Definition 2.3 (Global Weak Solutions). For any T > 0, (ρ, u) is said to be a weak solution to Eqs. (2.1)–(2.2) with initial data (2.3) and boundary value (2.6) or (2.7) in  × (0, T ), if ⎧ ∞ 1 γ α−1/2 ) ∈ L ∞ (0, T ; L 2 ()), ⎪ x ⎨0 ≤ ρ ∈ L (0, T ; L () ∩ L ()), (ρ (2.8) ⎪ ⎩√ρu ∈ L ∞ (0, T ; L 2 ()), ρ α u ∈ L 2 (0, T ; W −1,1 ()), x loc and (ρ, u) satisfies   ρ0 ψ(x, 0)d x + 

T 0



 

ρψt d xdt + 0

for any ψ ∈ , and   m 0 ϕ(x, 0)d x + 

0



T

+ 0

T

 

T

 

√ √ ρ ρuψx d xdt = 0

√ √ ρ( ρu)ϕt d xdt



√ ( ρu)2 + ρ γ ϕx d xdt − ρ α u x , ϕx = 0 

(2.9)

(2.10)

Vanishing of Vacuum States and Blow-up Phenomena

407

for all ϕ ∈ . The nonlinear diffusion term ρ α u x is defined as 

α

T

ρ u x , ϕ = −



√ ρ α−1/2 ρuϕx d xdt

0

2α − 2α − 1



T



√ (ρ α−1/2 )x ρuϕd xdt

(2.11)

0

for any ϕ ∈ , where √ ρ ∈ L ∞ ( × (0, T )) due to (2.8). Moreover, for the spatial periodic case (2.7), (ρ, ρu) is also periodic. Remark 2.4. For the Dirichlet case, (2.9), together with the fact ρ ∈ L ∞ ( × (0, T )) due to (2.8), implies that (ρ, u) satisfies Eq. (2.1) in the sense of distribution and justifies the boundary condition (2.6) in the sense that, for any time interval I ⊂ [0, T ], ε−1

  I

ε

√ √ ρ ρu(x, s)d xds → 0, ε−1

  I

0

ε

√ √ ρ ρu(x, s)d xds → 0

1−ε

√ √ as ε → 0+ . If further ρ ρu ∈ L p (0, T ; W 1,q ()) for some p ≥ 1, q ≥ 1, then (2.9) √ √ 1,q yields that ρ ρu ∈ L p (0, T ; W0 ()), that is, the Dirichlet boundary condition (2.6) is satisfied in the sense of trace. Definition 2.5 (Global Entropy Weak Solutions). Let (ρ, u) be a global weak solution (in the sense of Definition 2.3) to (2.1)–(2.2) with initial data (2.3) and boundary value (2.6) or (2.7) in  × (0, T ). Then, (ρ, u) is said to be a global entropy weak solution if there exists some function ∈ L 2 ( × (0, T )) satisfying (2.11), i.e., 

T





T

ϕd xdt = −

0



√ ρ α−1/2 ρuϕx d xdt

0

−

2α 2α − 1



T



√ (ρ α−1/2 )x ρuϕd xdt

(2.12)

0

for any ϕ ∈ , and the following uniform entropy inequality holds: 

√ | ρu|2 + |(ρ α−1/2 )x |2 + π(ρ) (x, t) d x sup 0≤t≤T  T

+ 

0

≤C0







(|(ρ (γ +α−1)/2 )x |2 + 2 )(x, t) d xdt 2



α−1/2

( |mρ00| + |(ρ0

)x |2 + π+ (ρ0 ))(x) d x

(2.13)

with C0 > 0 independent of T, and





ρ log ρ, if γ = 1, π(ρ)
1 γ γ −1 ρ , if γ > 1,

π+ (ρ)


ρ log+ ρ, if γ = 1, 1 γ if γ > 1. γ −1 ρ ,

(2.14)

We have the following result on the existence of global entropy weak solutions.

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H.-L. Li, J. Li, Z. Xin

Theorem 2.1. (Global Existence). Assume that α>

α 1 , γ > , 2 2

and that the initial data (ρ0 , m 0 ) satisfies (2.4) and

(2.15) |m 0 |2+ν ρ01+ν

∈ L 1 () for some positive

constant ν. Then for any T > 0, there exists a global entropy weak solution (ρ, u) to the IBVP for the compressible Navier-Stokes equations (2.1)–(2.3) with boundary condition (2.6) or (2.7) in  × (0, T ) in the sense of Definition 2.5. Moreover, for the case of Dirichlet boundary condition (2.6), if α ∈ (1/2, 3/2) √ and ν satisfies (2.17) (see Remark 2.6 below), then in addition to (2.8), the solution (ρ, ρu) satisfies √ √ 1,(4+2ν)/(4+ν) ρ ( ρu) ∈ L 2 (0, T ; W0 ()), i.e.,

√

(2.16)

√ ρ ( ρu) satisfies the Dirichlet boundary (2.6) in the sense of trace.

Remark 2.6. (1) Theorem 2.1 above holds for the compressible shallow water equation (1.2). (2) For the Dirichlet case (2.6), one of √ the √ available ways (available for the case α ∈ (1/2, 3/2) and γ ≥ 1) to obtain that ρ ρu ∈ L p (0, T ; W 1,q ()), for some p ≥ 1, q ≥ 1, is to choose the positive constant ν in Theorem 2.1 such that ⎧ ν ∈ (0, 2γ − α), for α ∈ ( 21 , 1], γ ≥ 1, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪  ⎨

1+α  −2) 2(2γ −α) 3 , ν ∈ 2(α+γ 3−α−γ 1+α−2γ , for α ∈ (1, 2 ), γ ∈ 1, 2 , ⎪ ⎪ ⎪ ⎪

 ⎪ ⎪ ⎩ ν ∈ 4(α−1) , ∞ , for α ∈ (1, 23 ), γ ≥ 1+α 3−2α 2 .

(2.17)

Next, we show that there is a global entropy weak solution (ρ, u) in the sense of Definition 2.5 for which the vacuum states and the structure of interface, if existing initially, can be preserved for a short time, so long as the initial data has additional regularity besides (2.4) and the fluids and the vacuum states in initial data are connected “smoothly”. In addition, the weak solution (ρ, u) is actually a unique regular solution for the short time. For simplicity, we consider the case of one point vacuum state contained at x = x0 ∈ (0, 1) in the initial data (ρ0 , m 0 ) = (ρ0 , ρ0 u 0 ) with additional regularity A0 |x − x0 |σ ≤ ρ0 (x) ≤ A1 |x − x0 |σ , for any x ∈ , ¯ u 0 ∈ C 1 (),

γ −1+1/2 j (ρ0 )x

∈ L 2 j (),

−1+1/2 j α ρ0 (ρ0 u 0x )x

(2.18)

∈ L 2 j (), j = 1, n, (2.19)

with n ≥ 2 an integer; and in the case of continuous vacuum state of one piece initially on 0 = [x0 , x1 ] ⊂ (0, 1) in the initial data, we require ⎧ σ σ ⎪ ⎨ A0 (x0 − x) ≤ ρ0 (x) ≤ A1 (x0 − x) , x ∈ [0, x0 ), (2.20) x ∈ [x0 , x1 ], ρ0 (x) = 0, m 0 (x) = ρ0 u 0 (x) = 0, ⎪ ⎩ B (x − x )σ ≤ ρ (x) ≤ B (x − x )σ , x ∈ (x , 1] 0 1 0 1 1 1

Vanishing of Vacuum States and Blow-up Phenomena

409

and ⎧ γ −1+1/2 j ¯ \ 0 ), ⎨ (ρ0 )x ∈ L 2 j (), j = 1, n, u 0 ∈ C 1 ( ⎩

−1+1/2 j

ρ0

(2.21)

(ρ0α u 0x )x ∈ L 2 j ( \ 0 ), j = 1, n,

with n ≥ 2 an integer. Here, σ , A0 , A1 , and B0 , B1 are positive constants, and the power σ ∈ (σ− , σ+ ) with positive constants σ± given in (2.33) later. We also require that the initial data (ρ0 , m 0 ) = (ρ0 , ρ0 u 0 ) given by (2.3) is consistent with boundary value for the Dirichlet boundary condition. We have the following results on short time structure of global entropy weak solutions. Theorem 2.2. (Short time structure of vacuum states). In addition to the assumptions of Theorem 2.1, assume further that α > 21 , γ > max{1, α}

(2.22)

and that there is either one point vacuum state in initial data (ρ0 , u 0 ) with (2.18)–(2.19) satisfied or a piece of continuous vacuum states in initial data (ρ0 , u 0 ) with (2.20)–(2.21) satisfied. Then, there exists a global entropy weak solution (ρ, u) to the IBVP for the compressible Navier-Stokes equations (2.1)–(2.2) with initial data (2.3) and boundary value (2.6) or (2.7) in the sense of Definition 2.5. Moreover, there is a short time T∗ > 0, so that the global entropy weak solution (ρ, u) is unique1 and regular on the domain  × [0, T∗ ], and the initial structure of vacuum states is maintained for t ∈ [0, T∗ ] in the following sense: For the case of the one point vacuum state initially, (2.18), the solution (ρ, u) is ¯ × [0, T∗ ], regular and unique on the domain  ¯ × [0, T∗ ]), u x ∈ L ∞ (0, T∗ ; C 0 ()), ¯ (ρ, u) ∈ C 0 ( u L ∞ (×[0,T + u x  L ∞ ([0,T∗ ];C 0 ()) ¯ ¯ ≤ C(T∗ ). ∗ ])

(2.23) (2.24)

The one point vacuum state propagates along the particle path, namely, there is one particle path x = X 0 (t) : [0, T∗ ] →  with X 0 (t) ∈ C([0, T∗ ]) defined by X˙ 0 (t) = u(X 0 (t), t), X 0 (0) = x0 ∈ (0, 1),

(2.25)

a− |x − X 0 (t)|σ ≤ ρ(x, t) ≤ a+ |x − X 0 (t)|σ

(2.26)

so that

for (x, t) ∈  × [0, T∗ ], where the two positive constants a± are independent of time T∗ . In the case of a piece of continuous vacuum states initially, (2.20), there are two particle paths x = X i (t) : [0, T∗ ] →  with X i (t) ∈ C([0, T∗ ]), i = 0, 1 defined by X˙ i (t) = u(X i (t), t), X i (0) = xi ∈ (0, 1), i = 0, 1, 1 Here the uniqueness is specified for density ρ and momentum ρu = states of one piece.

(2.27)

√ √ ρ ρu for continuous vacuum

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H.-L. Li, J. Li, Z. Xin

so that it holds for some positive constants a± , b± independent of the time T∗ , a− (X 0 (t) − x)σ ≤ ρ(x, t) ≤ a+ (X 0 (t) − x)σ ,

(2.28)

for (x, t) ∈ [0, X 0 (t)) × [0, T∗ ], and b− (x − X 1 (t))σ ≤ ρ(x, t) ≤ b+ (x − X 1 (t))σ

(2.29)

for (x, t) ∈ (X 1 (t), 1] × [0, T∗ ] respectively, and the interfaces separating the fluid and vacuum coincide with the particle paths ρ(x, t) = 0, ρu(x, t) = 0, (x, t) ∈ [X 0 (t), X 1 (t)] × [0, T∗ ].

(2.30)

The solution (ρ, u) is regular and unique up to the vacuum boundary ¯ × [0, T∗ ]), u ∈ C 0 ( ¯ × [0, T∗ ] \ 0T ), ρ ∈ C 0 ( ∗ u L ∞ (×[0,T + u x  L ∞ (×[0,T 0 0 ¯ ¯ ∗ ]\ ) ∗ ]\

T∗ )

T∗

≤ C(T∗ ),

(2.31) (2.32)

where 0T∗ = (X 0 (t), X 1 (t)) × [0, T∗ ]. Remark 2.7. (1) The constant exponents σ± are defined as σ± = β± /(1 − β± ) > 0 with β± determined by 1 β− = max{ 2α ,

1 1 γ (1 − 2n )},

β+ = min{1,

1 1 1 1 α (1 − 2n ), 1+3α (4 − n )},

(2.33)

while the positive constants a± are independent of the time T∗ . (2) The regularity assumptions (2.18)–(2.19) are satisfied for the following initial data ρ0 (x) = 21 (A0 + A1 )(|x − x0 |2 )(σ− +σ+ )/4 , u 0 (x) = 0, x ∈ , and the regularity assumptions (2.20)–(2.21) are satisfied for the initial data ⎧ 1 (σ +σ )/2 ⎪ ⎨ 2 (A0 + A1 )(x0 − x) − + , x ∈ [0, x0 ), ρ0 (x) = 0, x ∈ (x0 , x1 ), ⎪ ⎩ 1 (B + B )(x − x )(σ− +σ+ )/2 , x ∈ (x , 1]. 0 1 1 1 2

u 0 (x) = 0, x ∈ ,

Next, we prove that for any global entropy weak solution (ρ, u) to the IBVP (2.1)– (2.3) together with boundary condition (2.6) or (2.7) in the sense of Definition 2.5, even though in some cases the vacuum states may exist for some finite time, for instance, in the cases as shown by Theorem 2.2, any possible vacuum state has to vanish within finite time after which the density is always away from vacuum. Simultaneously, not only can the velocity field be defined in terms of the density and momentum, and the nonlinear diffusion is represented in terms of the density and velocity, but also the global entropy weak solution (ρ, u) is shown to be a unique and strong solution after the vanishing of vacuum states. We have the following result.

Vanishing of Vacuum States and Blow-up Phenomena

411

Theorem 2.3. (Vanishing of vacuum states). Assume that α>

1 , γ ≥ 1. 2

(2.34)

Let (ρ, u) be any global entropy weak solution to the IBVP (2.1)–(2.2) with initial data (2.3) and boundary value (2.6) or (2.7) in the sense of Definition 2.5. Then, there exist some time T0 > 0 (depending on initial data) and a constant ρ− so that inf ρ(x, t) ≥ ρ− > 0, t ≥ T0 ,

(2.35)

¯ x∈

and the global entropy weak solution (ρ, u) becomes a unique strong solution (ρ, u) for t ≥ T0 and satisfies
ρ ∈ L ∞ (T0 , t; H 1 ()), ρt ∈ L ∞ (T0 , t; L 2 ()), (2.36) u ∈ H 1 (T0 , t; L 2 ()) ∩ L 2 (T0 , t; H 2 ()), with velocity u and nonlinear diffusion term given by √ ρu u
√ , (ρ α u x )x = x , ρ respectively. In addition, for ⎧ ⎨0  us
1 ⎩ m0d x ρ¯0 

(2.37)

for the Dirichlet case, for the periodic case,

there exist two positive constants µ0 , c0 both depending on initial data (ρ0 , m 0 ) and ρ− , such that (ρ − ρ¯0 , u − u s )(·, t) L 2 () ≤ c0 e−µ0 (t−T0 ) , t > T0 ,

(2.38)

where (and in what follows) f denotes the average of f over the bounded domain , i.e.,   1 f = f (x)d x = f (x)d x. ||   Remark 2.8. (1) Theorem 2.3 shows that any possible vacuum states must vanish in finite time. This theory applies to the compressible shallow water equation (1.2). (2) It is easy to verify (see the proof of Proposition 5.1) that the phenomena of vacuum vanishing (2.35) in finite time actually happens for any global weak solution (ρ, u) to the IBVP (2.1)–(2.3) with boundary condition (2.6) or (2.7) in the sense of Definition 2.3 satisfying the following entropy inequality:  T 

√ 2 α−1/2 2 sup | ρu| + |(ρ )x | + π(ρ) (x, t) d x + (ρ (γ +α−1)/2 )x 2L 2 dt 0≤t≤T



≤ C0

 

0

|2

α−1/2

( |mρ00 + |(ρ0

with C0 independent of T.

)x |2 + π+ (ρ0 ))(x)d x

(2.39)

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Finally, for any global entropy weak solution (ρ, u) to the IBVP (2.1)-(2.3) together with boundary condition (2.6) or (2.7) in the sense of Definition 2.5, the density is continuous, i.e., ρ ∈ C( × [0, T ]) for any T > 0, due to (2.8) and (2.9). Thus, the continuity of ρ and Theorem 2.3 imply that if the density contains vacuum states at least at one point, then there exists some critical time T1 ∈ [0, T0 ) with T0 > 0 given by ¯ such that (2.35) and a nonempty subset 0 ⊂  ⎧ 0 ⎪ ⎨ρ(x, T1 ) = 0, ∀ x ∈  , ¯ \ 0 , (2.40) ρ(x, T1 ) > 0, ∀ x ∈  ⎪ ⎩ρ(x, t) > 0, ¯ × (T1 , T0 ]. ∀ (x, t) ∈  It follows from (2.36) easily that for any δ > 0, it holds 

T0

u x  L ∞ ds < ∞.

(2.41)

T1 +δ

Under the condition that vacuum states appear, we shall prove that the spatial derivative of velocity (if regular enough and definable) blows up in finite time as the vacuum states vanish, even if the solution is regular enough for short time so that the velocity field and its derivatives are bounded as shown by Theorem 2.2. Theorem 2.4. (Finite time blow-up). Let (ρ, u) be any global entropy weak solution, which contains vacuum states at least at one point for some finite time, to the IBVP for the compressible Navier-Stokes equations (2.1)–(2.3) with boundary condition (2.6) or (2.7) in the sense of Definition 2.5. Let T0 > 0 and T1 ∈ [0, T0 ) be the time such that (2.35) and (2.40) holds respectively. Then, the solution (ρ, u) blows up as vacuum states vanish. Namely, for T1 satisfying (2.40) and for any given fixed η > 0, it holds  lim+

t→T1

T1 +η

u x  L ∞ ds = ∞.

(2.42)

t

On the other hand, if there exists some T2 ∈ (0, T0 ) such that the weak solution (ρ, u) satisfies u L 1 (0,T2 ;W 1,∞ ()) < ∞, then, there is a time T3 ∈ [T2 , T0 ) so that the blowup phenomena happens for (ρ, u), i.e.,  lim

t→T3− 0

t

u x  L ∞ ds = ∞.

(2.43)

Remark 2.9. Theorem 2.4 implies that for any global entropy weak solution (ρ, u) to the IBVP for the compressible Navier-Stokes equations (2.1)–(2.2) with initial data (2.3) and boundary value (2.6) or (2.7) in the sense of Definition 2.5, which contains vacuum states at least at one point initially, the finite time blowup phenomena (2.42) happens for such a solution (ρ, u).

Vanishing of Vacuum States and Blow-up Phenomena

413

Remark 2.10. Theorems 2.1–2.4 provide a complete dynamical description on the vanishing of vacuum states and blow-up phenomena for the global entropy weak solutions to the compressible Navier-Stokes equations with density-dependent viscosity. That is, a global entropy weak solution exists for general large initial data with finite entropy. For short time, such weak solution is unique and regular with well-defined velocity field subject to additional initial regularity, and any existing vacuum state is maintained with the same interface structure as the initial. Then, within finite time the vacuum states vanish definitely and the velocity blows up (even if it is regular enough and definable along the interfaces). After the vanishing of vacuum states, the global entropy weak solution becomes a strong one and tends to the non-vacuum equilibrium state exponentially in time. This dynamical phenomena is quite similar to those well-known for the 3-D incompressible Navier-Stokes equations. However, before the time of vacuum-vanishing, the uniqueness of the global entropy weak solution to the compressible Navier-Stokes equations with density-dependent viscosity subject to the initial data is not known yet. Remark 2.11. All theories established in Theorems 2.1-2.4 fit the shallow water equation (1.2). We believe that such phenomena described by Theorems 2.1-2.4 are also observed for other compressible fluids with density-dependent viscosity, such as Navier-Stokes equations with capillarity and/or drag friction, Navier-Stokes–Poisson system, etc. Remark 2.12. It is interesting to investigate the (global) dynamics of (one-dimensional) interface connecting vacuum from initial time until the vanishing of vacuum in order to investigate the dynamics of interface and the vanishing of vacuum state and to verify the formation of singularity for the general case. Remark 2.13. (1) Another interesting problem is whether the phenomena of the vanishing of vacuum states and blow-up happens for a multi-dimensional compressible isentropic Navier-Stokes system with density-dependent viscosity, especially on spatial bounded domain with the Dirichlet boundary condition (2.6). (2) It is also interesting to study whether any vacuum states shall vanish within finite time and blow-up phenomena happen for the multi-dimensional full Navier-Stokes system with density-dependent viscosity. It is not obvious yet (although we expect) since it is not clearly understood yet how the dynamics of temperature and heat-conduction shall affect the global existence of weak solutions and the evolution of vacuum states, especially in the case of spherical symmetry under the Dirichlet boundary condition (2.6). This is under further investigation [26]. 3. Global Existence of Entropy Weak Solutions In this section, we will establish the existence of global entropy weak solutions to the IBVP for the compressible Navier-Stokes equations (2.1)–(2.3) together with (2.6) or (2.7). Since the compactness arguments are straightforward in the framework of MelletVasseur [38], we only need to construct a sequence of approximate solutions by using some ideas developed by Jiang-Xin-Zhang in [23] for one-dimensional compressible Navier-Stokes equations with free boundary, to establish some uniform a priori estimates, in particular, the lower and upper bounds of the density for the approximate solution sequence with the help of BD entropy, and to justify the boundary condition (2.16) for (limiting) weak solution for the Dirichlet case as follows. Outline of proof of Theorem 2.1. Step 1. Construction of smooth approximate solutions. Let us consider the following approximate compressible Navier-Stokes equations

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inspired by Jiang-Xin-Zhang in [23]: ρεt + (ρε u ε )x = 0, (ρε u ε )t + (ρε u 2ε

+ p(ρε ))x − (µε (ρε )u εx )x = 0, (ρε , ρε u ε )(x, 0) = (ρ0ε , m 0ε )(x),

(3.1) (3.2) (3.3)

with one of the following boundary conditions: u ε (0, t) = u ε (1, t) = 0,

(3.4)

ρε , u ε are periodic in x of period 1.

(3.5)

or The viscosity µε is given by µε (ρ) = ρ α + ερ θ , ε > 0, θ ∈ (0, 1/2),

(3.6)

where we remark (see Remark 3.1 below) that although the modified viscosities (3.6) are limited to the case θ ∈ (0, 1/2), the methods adapted here to construct approximate solutions also can be applied to the case θ ∈ (0, 1) and γ > θ after some modification, the interested reader can refer to [16,23] and references therein, we omit the details here. The initial data ρ0ε , m 0ε ∈ C ∞ () satisfies
α−1/2 α−1/2 → ρ0 in H 1 (), ρ0ε → ρ0 in L 1 (), ρ0ε (3.7) (m 0ε )2 (ρ0ε )−1 → m 20 ρ0−1 , |m 0ε |2+ν (ρ0ε )−1−ν → |m 0 |2+ν ρ0−1−ν in L 1 () as ε → 0+ , and ρ0ε ≥ c0 ε1/(2α−2θ)

(3.8)

for some c0 independent of ε. By (3.7) we can assume without the loss of generality that  ρ0ε d x = 1. (3.9) 

By the standard arguments (see [23] for reference), after applying the classical theory of parabolic and hyperbolic equations, one can obtain that there exists some T∗ > 0 such that the approximate problem (3.1)–(3.3) together with the boundary condition (3.4) or (3.5) has a unique smooth solution (ρε , u ε ) on [0, T∗ ] with the density away from vacuum, i.e., ρε (x, t) > 0, for all (x, t) ∈  × [0, T∗ ].

(3.10)

Step 2. The a-priori estimates. To extend the local solution globally in time, one needs to control the lower and upper bounds of the density and get some a-priori estimates. As mentioned in the introduction, this relies on the BD entropy inequality developed by Bresch and Desjardins (see [2,3,6,7] for instance). In the Eulerian coordinates, the BD entropy for the approximate solutions (ρε , u ε ) of the IBVP problem for the system (3.1)–(3.1) reads (see [2,3,6,7] for instance) as  θ √ √ ε )x |2 + | ε((ρ √ε ) )x |2 + π(ρε ))(x, t) d x (| ρε u ε |2 + | µ(ρ ρε ρε   t p  (ρε )µε (ρε ) + (µε (ρε )|u εx |2 + |ρεx |2 )(x, s) d xds ≤ C, t > 0 (3.11) ρε 0



Vanishing of Vacuum States and Blow-up Phenomena

415

with positive constant C independent of t and ε. Indeed, multiplying (3.2) by u ε , integrating by parts on [0, 1], and using (3.1), we get the classical entropy estimate  

√ (| ρε u ε |2 + π(ρε ))(x, t) d x +

 t 0



µε (ρε )|u εx |2 d xds ≤ C, t > 0 (3.12)

with positive constant C independent of t and ε. Multiplying the transport equation (3.1) by ϕε (ρε ) with ρϕε (ρ) = µε (ρ), i.e., ϕε (ρ) = α1 ρ α + θε ρ θ as in [8], we get ϕε (ρε )t + ϕε (ρε )x u ε + ϕε (ρε )ρε u εx = 0.

(3.13)

Differentiating the above equation with respect to x and denoting vε = ϕε (ρε )x /ρε , we have (ρε vε )t + (ρε vε u ε )x + (µε (ρε )u εx )x = 0.

(3.14)

Summing Eqs. (3.14) and (3.1) together, multiplying by u ε + vε and integrating by parts on [0, 1], we have after a straightforward computation that 

√ (| ρε u ε |2 + | ϕε√(ρρεε)x |2 + π(ρε ))(x, t) d x   t p  (ρε )ϕε (ρε ) + (µε (ρε )|u εx |2 + |ρεx |2 )(x, s) d xds ≤ C, t > 0 (3.15) ρε 0



with positive constant C independent of t and ε. This together with (3.12), (ρε2α−2 + ε2 ρε2θ−2 )(ρεx )2 ≤ (ϕε (ρε )x )2 = (ρε2α−2 + 2ερεα+θ−2 + ε2 ρε2θ−2 )(ρεx )2 ≤ 2(ρε2α−2 + ε2 ρε2θ−2 )(ρεx )2 , gives rise to the BD entropy (3.11). In order to derive the lower bound and upper bounds of density for the approximate solution (ρε , u ε ), it is convenient to make use of the Lagrangian coordinates (y, t) with 

x

y= 0

ρε (z, t)dz, t = t,

because the approximate solution (ρε , u ε ) is away from vacuum at least for short time due to (3.10) and fluid density transports along the particle path. In Lagrangian coordinates, we have the equivalent system for (ρε , u ε ) as follows: ρεt + ρε2 u εy = 0, u εt + ( p(ρε )) y − (ρε µε (ρε )u εy ) y = 0,

(3.16) (3.17)

−1 )(y), (ρε , u ε )(y, 0) = (ρ0ε , m 0ε ρ0ε

(3.18)

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together with the boundary conditions similar to (3.4) or (3.5), where (y, t) ∈  L × (0, T∗ ] and   L
(0, L ε ), with L ε = ρ0ε (x)d x = 1 

due to (3.9). Correspondingly, the BD entropy equality (3.11) becomes in Lagrangian coordinates as follows:  T∗  sup E ε (t) + ρε µε (ρε )(u εy )2 dydt 0≤t≤T∗



T∗

+ 0

L

0

 L



ρεγ +α−2 + ερεγ +θ−2 (ρεy )2 dydt ≤ C0 E ε (0)

(3.19)

with C0 independent of T∗ and ε > 0, and E ε (t) defined as 

u 2ε + ((ρεα ) y )2 + ε2 ((ρεθ ) y )2 + ρε−1 π(ρε ) dy. E ε (t)
L

Since

 L

ρε−1 dy

 =



d x = || = 1,

(3.20)

it follows from the continuity of ρε that there exists some y0 (t) ∈  L such that ρε (y0 (t), t) = | L |/|| = 1. Hence, it implies from (3.19) that ρεα (y, t) = ρεα (y0 (t), t) +



y y0 (t)

(ρεα ) y dy

≤ (| L |||−1 )α + C0 E ε (0) + | L |.

(3.21)

This yields the uniform upper bound (w.r.t. time t ∈ [0, T∗ ]) for the density. To obtain the lower bound for the density ρε , we employ the idea in [23]. That is, we consider the upper bound for vε = ρε−1 , which can be estimated by (3.19) and (3.20) as follows:   vε (y, t)dy + (vε )2 |ρεy |dy vε (y, t) ≤ L L  θ+1/2 ≤1 + C max (vε ) (ρεθ ) y  L 2 () ( vε (y, t)dy)1/2 L

y∈ L

1 ≤1 + max vε + C(ε−2 E ε (0))1/(1−2θ) . 2 y∈ L

(3.22)

This shows that for all (y, t) ∈  L × [0, T∗ ], ρε (y, t) ≥

ε2/(1−2θ) . 2ε2/(1−2θ) + C(E ε (0))1/(1−2θ)

(3.23)

Thus, we have obtained the uniform upper bound (3.21) and the lower bound (3.23) (w.r.t. time t ∈ [0, T∗ ]) for density ρε . Using these uniform bounds, one can extend

Vanishing of Vacuum States and Blow-up Phenomena

417

the local smooth solution globally in time by standard arguments (see [23] for details). Moreover, the total mass of the approximate solutions is conserved due to the boundary conditions (3.4),   ρε (x, t)d x = ρ0ε (x)d x, ∀ t > 0. (3.24) 



Remark 3.1. Although the estimates (3.22) and (3.23) are established for the approximate solutions in the case that the modified viscosities (3.6) are limited to the case θ ∈ (0, 1/2), the methods adapted here can be applied to establish the lower bounds like (3.23) in the case θ ∈ (0, 1) and γ > θ . Indeed, one can also establish the following estimates, similar to [23,16], for the approximate solutions (ρ, u ε ) as:  sup (ρ θ )2n x (x, t)d x ≤ C(T ) t∈[0,T ]

for any integer n ≥ 1 and C(T ) a constant independent of ε, θ . Thus, we can verify similar to (3.22) that   vε ≤ 1 + C vε1+θ |(ρ θ )x |d x ≤ 1 + C(T )( vεq(θ+1) dy)1/q ≤ C + C(T )vε(q(θ+1)−1)/q with q = 2n/(2n − 1). This together with Young’s inequality implies the positive lower bound of density for integer n sufficiently large but fixed so that θ < 2n−1 2n so long as θ ∈ (0, 1). η

In addition to (3.11), we show ρε u ε is actually bounded in L ∞ (0, T ; L 2+ν ()) for some η ≥ 1/(2 + ν) with ν > 0 given by
ν > 0 arbitrary if 2γ − α ≥ 1, (3.25) ν ∈ (0, 2(2γ − α)/(1 + α − 2γ )] if 2γ − α ∈ (0, 1). Note here that for α ∈ (1/2, 3/2) the constant ν defined by (2.17) actually satisfies (3.25). It follows from (3.11) that sup (ρεα−1/2 )x  L 2 () ≤ C,

(3.26)

t≥0

which together with (3.24) and the mean value theorem gives rise to α−1/2

sup ρε  L ∞ t≥0

≤ ρ0ε α−1/2 + (ρεα−1/2 )x  L 2 () ≤ C,

(3.27)

where C denotes some generic positive constant depending on E ε (0) but independent of T. With the help of (3.27), we can make use of the idea due to [38] to derive the following a-priori estimate:  sup ρε |u ε |2+ν d x ≤ C, (3.28) 0≤t≤T



which together with (3.27) implies for η(2 + ν) ≥ 1 that     η 2+ν η(2+ν) 2+ν ρε |u ε | dx = ρε |u ε | d x ≤ C sup ρε |u ε |2+ν d x ≤ C, 



0≤t≤T



(3.29)

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H.-L. Li, J. Li, Z. Xin

and, in particular, for η = 1/2 that  √ 2+ν ρε |u ε | dx ≤ C sup 0≤t≤T

(3.30)



with C one generic positive constant depending on both T and E ε (0). Thus, we can follow the compactness arguments in [38] to prove the convergence of the approximate solution (ρε , u ε ) to some expected weak solution (ρ, u) as ε → 0+. Namely, ρε −→ ρ

in C([0, T ] × ),

(3.31)

(ρεα−1/2 )x  (ρ α−1/2 )x weakly in L 2 ( × (0, T )), √ √ ρε u ε −→ ρu, ρεα u ε −→ ρ α u in L 2+ν/2 ( × (0, T )),

(3.32) (3.33)

holds and there exists some function ∈ L 2 ( × (0, T )) such that ρεα u εx 

weakly in L 2 ( × (0, T )), as ε → 0.

(3.34)

Step 3. Justification of the Dirichlet boundary condition (2.16) for α ∈ (1/2, 3/2). For α ∈ (1/2, 1], it follows from (3.11) and (3.29) that (ρε u ε )x  L 2 (0,T ;L (4+2ν)/(4+ν) ())    

   α−1/2 1/2  ρ ≤ C ρεα/2 u εx  2 + C ρ u  2  ε ε ε L (0,T ;L 2 ()) x L (0,T ;L (4+2ν)/(4+ν) ())   

  α−1/2   1/2  ≤ C + C  ρε · ρε u ε  ∞  ∞ 2+ν 2 ≤ C.

x L (0,T ;L ())

L (0,T ;L

())

(3.35)

For α ∈ (1, 3/2), some additional estimates are needed here. First, for γ ≥ (1+α)/2, it is easy to check that for ν defined by (2.17) that (3/2 − α)/(2 + ν) > 1. Hence, for β = (4 + 2ν)/(4 + ν) one deduces from (3.29) and (3.27) that   (ρε u ε )x  2 L (0,T ;L β ())    

 α/2   α−1/2 −α+3/2  ρ ≤ C ρε u εx  2 + C ρ u  2  ε ε ε L (0,T ;L 2 ()) x L (0,T ;L β ())   

  α−1/2   −α+3/2  ≤ C + C  ρε uε  ∞  ∞ ρε 2+ν 2 ≤ C.

x L (0,T ;L ())

L (0,T ;L

())

(3.36)

In the case that 1 ≤ γ < (1 + α)/2, (2.17) yields that (3 − α − γ )(2 + ν) ≥ 2. Hence, one derives from (3.29) and (3.11) that   (ρε u ε )x  2 L (0,T ;L β ())    

   (γ +α−1)/2 (3−α−γ )/2  ρ ≤ C ρεα/2 u εx  2 + C ρ u  2  ε ε ε L (0,T ;L 2 ()) x L (0,T ;L β ())   

   (3−α−γ )/2   ≤ C + C  ρε(γ +α−1)/2  uε  ∞ ρε 2+ν 2 2 ≤ C.

x L (0,T ;L ())

L (0,T ;L

())

(3.37)

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Thus, the estimates (3.4), (3.36) and (3.37) imply that (2.16) still holds for α ∈ (1, 3/2). The proof of Theorem 2.1 on the existence of global entropy weak solution (ρ, u) is completed. 4. Dynamics of Vacuum States for Short Time 4.1. Short time structure of vacuum states. We prove Theorem 2.2 in this subsection in order to study the short time structure of vacuum states for global entropy weak solutions. To this end, it is sufficient to show that there is a unique entropy weak solution in short time for the compressible Navier-Stokes equations (2.1)–(2.2) with initial data (2.3) and boundary value (2.6) or (2.7) under the assumptions of Theorem 2.2 as follows. Proposition 4.1. (Vacuum states for short time). Under the assumptions of Theorem 2.2, there is a short time T∗ > 0 so that the unique entropy weak solution (ρ, ˜ u) ˜ in the sense of Definition 2.5 of the IBVP problem for the compressible Navier-Stokes equations (2.1)–(2.2) with initial data (2.3) and boundary condition (2.6) or (2.7) exists on the domain  × [0, T∗ ]. The initial vacuum state (2.19) or (2.21) is also propagated for the short time t ∈ [0, T∗ ]; more precisely, the properties (2.23)–(2.26) or (2.27)–(2.32) hold respectively for (ρ, ˜ u). ˜ In addition, it holds that (ρ˜ α−1/2 )x  L ∞ (0,T∗ ;L 2 ()) + ((ρ˜ u) ˜ 2 /ρ, ˜ (ρ˜ u) ˜ 2+ν /ρ˜ 1+ν ) L ∞ (0,T∗ ;L 1 ()) ≤ C(T∗ ). (4.1) Proof. The proof of Proposition 4.1 will be completed in Subsects. 4.2–4.3 later. Proof of Theorem 2.2. This is a consequence of Proposition 4.1 and Theorem 2.1. In fact, Proposition 4.1 shows not only that there exists a unique entropy weak solution (ρ, ˜ u) ˜ on the domain ×(0, T∗ ) in the sense of Definition 2.5 to the IBVP problem for the compressible Navier-Stokes equations (2.1)–(2.2) with initial data (2.3) and boundary condition (2.6) or (2.7), but also that this short time entropy weak solution satisfies all the properties (2.23)–(2.32). Now, choose time T∗ = T∗ − δ with δ > 0 a constant small enough. One can verify that at time t = T∗ the density and momentum also satisfies the assumptions (2.4), and particularly ⎧ ⎪ ρ(x, ˜ T∗ ) ≥ 0 on , ρ˜ u(., ˜ T∗ ) = 0, on {x ∈  | ρ0 (x) = 0}, ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ρ(., ˜ T∗ ) ∈ L 1 (), (ρ˜ α−1/2 (., T∗ ))x ∈ L 2 (), (4.2) ⎪ ⎪ ⎪ ⎪ ⎪ 2 2+ν ⎪ ˜ ˜ ∗ )| ∗ )| ⎩ |m(.,T + |ρm(.,T ∈ L 1 (). ρ(x,T ˜ ˜ 1+ν (x,T ) ∗) ∗

ˆ Thus, it follows from Theorem 2.1 that there is a global entropy weak solution (ρ, ˆ u) for time t ≥ T∗ in the sense of Definition 2.5 to the IBVP problem for the compressible Navier-Stokes equations (2.1)–(2.2) with initial data (ρ, ρu)(x, T∗ ) = (ρ, ˜ ρ˜ u)(x, ˜ T∗ ), x ∈  and boundary condition (2.6) or (2.7). Define (ρ, u) as
(ρ, ˜ u), ˜ for (x, t) ∈  × [0, T∗ ], (ρ, u) = ˆ (ρ, ˆ u), for (x, t) ∈  × [0, T∗ ].

(4.3)

(4.4)

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It is easy to verify that (ρ, u) is a global entropy weak solution in the sense of Definition 2.5 to the compressible Navier-Stokes equations (2.1)–(2.2) with initial data (2.3) and boundary condition (2.6) or (2.7) under the assumptions of Theorem 2.2. This global entropy weak solution is actually unique and regular for short time t ∈ [0, T∗ ] and satisfies all the properties (2.23)–(2.32). The proof of Theorem 2.2 is complete.   4.2. Compressible Navier-Stokes in Lagrangian coordinates. In order to prove Proposition 4.1, we present an equivalent proposition for the compressible Navier-Stokes equations in the Lagrangian coordinates (y, t), instead of the Eulerian coordinates (x, t), through the coordinate transformation  x y= ρ(z, t)dz, x ∈ (0, 1), t ≥ 0 (4.5) 0

for the Dirichlet boundary condition, or  X 0 (t)+x y= ρ(z, t)dz, x ∈ (0, 1), t ≥ 0 X 0 (t)

(4.6)

for the periodic boundary condition, where x = X 0 (t) is a particle path. In addition, both the case of initial one point vacuum state and the case of a piece of initial continuous vacuum states will be studied with some additional initial regularities. We first describe the equivalent proposition for the IBVP problem for the compressible Navier-Stokes equations in the Lagrangian coordinates in the case of initial one point vacuum state. Thus, consider the compressible Navier-Stokes equations ρt + ρ 2 u y = 0, u t + p(ρ) y − (ρµ(ρ)u y ) y = 0,

y ∈ , t > 0

(4.7) (4.8)

with initial data (ρ, u)(y, 0) = (ρ0 (y), u 0 (y)),

y ∈ ,

(4.9)

and the Dirichlet boundary condition u(0, t) = u(1, t) = 0, t ≥ 0

(4.10)

or the periodic boundary condition (ρ, u) is periodic w.r.t. x of period one.

(4.11)

For the case of Dirichlet boundary, the initial data (ρ0 , u 0 ) is assumed to be consistent with the boundary values. Moreover, we assume that there is a one point vacuum state at y = y0 for some fixed point y0 ∈ (0, 1) and that the initial data is of additional regularity ⎧ β β ⎪ ⎨ A0 |y − y0 | ≤ ρ0 (y) ≤ A1 |y − y0 | , y ∈ , (4.12) ⎪ γ ⎩u ∈ C 1 (), 1+α 2 2n ¯ (ρ0 ) y , (ρ0 u 0y ) y ∈ L () ∩ L () 0 with an integer n ≥ 2, where A0 , A1 are positive constants, and the constant β ∈ (β− , β+ ) with β± is determined by Remark 4.3. Then the following result holds in the case of the initial point vacuum state.

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Proposition 4.2. (Point vacuum state for short time). Assume that (2.22) and (4.12) hold. Then, there is a time T∗ > 0 so that the unique regular weak solution (ρ, u) with point vacuum of the IBVP (4.7)–(4.9) with boundary condition (4.10) or (4.11) exists on the domain  × [0, T∗ ] and satisfies ¯ × [0, T∗ ]) ∩ C 1 ([0, T∗ ]; L 2 ()), ρ ∈ C 0 ( ¯ × [0, T∗ ]) ∩ C 1 ([0, T∗ ]; L 2 ()), u ∈ C 0 (

(4.14)

ρ uy ∈ L (ρ α ) y  L ∞ ([0,T∗ ];L 2 ()) + ρu y  L ∞ (×[0,T∗ ]) ≤ C(T∗ ).

(4.15) (4.16)

1+α

∞

( × [0, T∗ ]) ∩ C 1/2 ([0, T∗ ]; L 2 ()),

(4.13)

In addition, the initial point vacuum state is maintained for the short time β

ρ(y0 , t) = 0, t ∈ [0, T∗ ], β

a− |y − y0 | ≤ ρ(y, t) ≤ a+ |y − y0 | , (y, t) ∈ [0, 1] × [0, T∗ ].

(4.17) (4.18)

Here, a± are positive constants independent of T∗ . Proof. The proof of Proposition 4.1 will be given in Subsect. 4.3. Remark 4.3. The choice of β± > 0 is such that 1 , β− = max{ 2α

1 1 γ (1 − 2n )},

β+ = min{1,

1 1 1 1 α (1 − 2n ), 1+3α (4 − n )}

(4.19)

for integer n ≥ 2. It should be emphasized here that all the assumptions required here are satisfied for the shallow water equations (1.2), i.e., γ = 2, α = 1. Next, we describe the equivalent proposition for the IBVP problem for the compressible Navier-Stokes equations in the Lagrangian coordinates for the case of initial continuous vacuum of one piece. In this case, we consider the IBVP problem for the compressible Navier-Stokes equation (4.7)–(4.9) with one of following boundary conditions: (1) Mixed boundary condition
u(0, t) = ρ(1, t) = 0, t ≥ 0, A0 (1 − y)β ≤ ρ0 (y) ≤ A1 (1 − y)β , y ∈ , or (2) Mixed free boundary condition
ρ(0, t) = u(1, t) = 0, t ≥ 0, B0 y β ≤ ρ0 (y) ≤ B1 y β ,

y ∈ ,

or (3) Free boundary condition
ρ(0, t) = ρ(1, t) = 0, t ≥ 0, A0 (y(1 − y))β ≤ ρ0 (y) ≤ A1 (y(1 − y))β , y ∈ ,

(4.20)

(4.21)

(4.22)

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where, β ∈ (β− , β+ ), and A0 , A1 , B0 , B1 , and A0 , A1 are some given positive constants. The initial data is also assumed to be regular γ

¯ (ρ0 ) y , (ρ01+α u 0y ) y ∈ L 2 () ∩ L 2n () u 0 ∈ C 1 (),

(4.23)

with an integer n ≥ 2. Then the following results for free boundary value problems for the compressible Navier-Stokes equations hold. Proposition 4.4. (Continuous vacuum state for short time). Assume that (2.22) and (4.23) hold. Then, there is a time T∗ > 0 so that the unique regular weak solution (ρ, u) of the IBVP problem for the compressible Navier-Stokes equations (4.7)–(4.8) with initial data (4.9) and free boundary condition (4.20), or (4.21), or (4.22) exists on the domain  × [0, T∗ ] and satisfies ¯ × [0, T∗ ]) ∩ C 1 ([0, T∗ ]; L 2 ()), ρ ∈ C 0 ( ¯ × [0, T∗ ]) ∩ C 1 ([0, T∗ ]; L 2 ()), u ∈ C 0 (

ρ 1+α u y ∈ L ∞ ( × [0, T∗ ]) ∩ C 1/2 ([0, T∗ ]; L 2 ()), and (ρ α ) y  L ∞ ([0,T∗ ];L 2 ()) + ρu y  L ∞ (×[0,T∗ ]) ≤ C(T∗ ). In addition, the initial vacuum state is also maintained for short time, namely, it holds that a− (1 − y)β ≤ ρ(y, t) ≤ a+ (1 − y)β , (y, t) ∈ (0, 1) × [0, T∗ ], corresponding to the mixed free boundary conditions (4.20), or b− y β ≤ ρ(y, t) ≤ b+ y β , (y, t) ∈ (0, 1) × [0, T∗ ] corresponding to the mixed free boundary conditions (4.21), or c− (y(1 − y))β ≤ ρ(y, t) ≤ c+ (y(1 − y))β , (y, t) ∈ (0, 1) × [0, T∗ ] corresponding to the free boundary (4.22). Here, a± , b± and c± are positive constants independent of T∗ . Proof. The short time existence, uniqueness and regularity of weak solutions of the free boundary problems for the compressible Navier-Stokes equations are well-investigated by many authors (see [23,32,33,56,57]). Proposition 4.4 can be proved in a similar way as [23,32,33,56], so we omit the details. Remark 4.5. Based on Proposition 4.4 for the compressible Navier-Stokes equations (4.7)–(4.9) with either mixed free boundary condition (4.20) and (4.21) or free boundary (4.22) and the coordinates transformation from the Lagrangian coordinates to the Eulerian coordinates, one can study easily the IBVP problem for compressible Navier-Stokes equations (2.1)–(2.2) in the Eulerian coordinates with either the Dirichlet boundary condition (2.6) or the periodic boundary condition (2.7) in the case of initial continuous vacuum state of one piece (2.20). In fact, for the compressible Navier-Stokes equations (2.1)–(2.2) with Dirichlet boundary condition (2.6) in the Eulerian coordinates, the short time existence of a unique

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solution subject to the case of a piece of initial continuous vacuum state (2.20)–(2.21) in initial data can be constructed, in the Lagrangian coordinates, by combining two mixed free boundary value problems (4.20) and (4.21) together with one continuous vacuum state in-between as follows. Denote two particle paths x = X i (t) (assumed to be definable for short time) starting from the initial vacuum boundary x = x0 and x = x1 respectively as X˙ i (t) = u(X i (t), t),

X i (0) = xi , i = 0, 1,

along which the vacuum boundary moves in the Eulerian coordinates so that ⎧ ⎪ x ∈ [X 0 (t), X 1 (t)], t ≥ 0, ⎨(ρ, ρu)(x, t) = 0, ⎪ ⎩ρu(x, t) > 0,

(4.24) x ∈ [0, X 0 (t)) ∪ (X 1 (t), 1], t ≥ 0.

We first choose the coordinate transformation from the Eulerian coordinates to the Lagrangian coordinates as ⎧ x ⎪ ⎨ y = 0 ρ(z, t)dz, x ∈ [0, X 0 (t)], (4.25) ⎪ ⎩ y =  X 0 (t) ρ(z, t)dz =  x0 ρ (z)dz < 1, conservation of mass, 0 0 0 0 which gives y ∈ [0, y0 ] and the mixed free boundary conditions (4.20). The application of Proposition 4.4  ywith  replaced by (0, y0 ) implies, via the inverse coordinate transformation x = 0 ρ −1 (z, t)dz, y ∈ [0, y0 ], the existence of unique solution (ρl , u l ) of the compressible Navier-Stokes equations (2.1)–(2.2) on [0, X 0 (t)] × [0, T∗ ] in the Eulerian coordinates with the initial data (ρl , u l )(x, 0) = (ρ0 , u 0 )(x), x ∈ (0, x0 ), ρ0 (x0 ) = 0, and mixed free boundary conditions u(0, t) = 0, ρ(X 0 (t), t) = 0, t ∈ [0, T∗ ]. Next, we choose the coordinate transformation from the Eulerian coordinates to the Lagrangian coordinates as
1 y = 1 − x ρ(z, t)dz, x ∈ [X 1 (t), 1], 1 1 (4.26) y1 = X 1 (t) ρ(z, t)dz = x1 ρ0 (z)dz < 1 conservation of mass, which gives y ∈ [1 − y1 , 1] and the mixed free boundary conditions (4.21). Then 1 Proposition 4.4 implies, via the inverse coordinate transformation x = 1− y ρ −1 (z, t)dz, y ∈ [1− y1 , 1], the existence of unique (ρr , u r ) of the compressible Navier-Stokes equations (2.1)–(2.2) on [X 1 (t), 1] × [0, T∗ ] in the Eulerian coordinates with initial data (ρr , u r )(x, 0) = (ρ0 , u 0 )(x), x ∈ (x1 , 1), ρ0 (x1 ) = 0, and mixed free boundary conditions u(1, t) = 0, ρ(X 1 (t), t) = 0, t ∈ [0, T∗ ].

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Consequently, we can construct the short time unique solution (ρ, u) to the IBVP problem for the compressible Navier-Stokes equations (2.1)–(2.3) with the Dirichlet boundary condition (2.6) and a piece of continuous vacuum state (2.20) in the initial data as ⎧ ⎪ ⎨(ρl , ρl u l ), on [0, X 0 (t)] × [0, T∗ ], (ρ, ρu) = (0, 0), on (X 0 (t), X 1 (t)) × [0, T∗ ], ⎪ ⎩(ρ , ρ u ), on [X (t), 1] × [0, T ]. r r r 1 ∗ Similarly, we can obtain the short time existence of unique solution to the IBVP problem for the compressible Navier-Stokes equations (2.1)–(2.2) with periodic boundary condition (2.7) which can be viewed as a free boundary problem after the choice of the spatial reference point. The details will be omitted.

4.3. Proof of Propositions 4.1–4.2. We first prove Proposition 4.2 in this subsection with the help of the a-priori estimates for (regularized) solutions and the construction of approximate solutions by a finite difference scheme, due to the modification of the ideas used in [23,33,43,56]. Without the loss of generality, we only prove Proposition 4.2 in the case of the Dirichlet boundary conditions with one point vacuum state in the initial data. First, we can easily derive some identities for (regularized) solutions as in [23,56]. Lemma 4.6. Let T > 0 and assume that the solution (ρ, u) of the IBVP problem (4.7)– (4.10) exists for t ∈ [0, T ] with ρ(y0 , t) = 0. Then, under the assumptions of Proposition 4.2, it holds that  y  y0 1+α γ ρ u y (y, t) = u t (z, t)dz + ρ (y, t) = − u t (z, t)dz + ρ γ (y, t), (4.27) y0

ρ α (y, t) + α

y



t 0

ρ γ (y, s)ds =ρ0α (y) + α

 t

=ρ0α (y) − α

0

y0

 t 0

u t (z, s)dzds

y y

(4.28) u t (z, s)dzds

y0

for all y ∈  with y = y0 . We also have the following useful a-priori estimates, whose proofs are similar to those for (3.11) and (3.19) and thus will be omitted. Lemma 4.7. Let T > 0 and assume that the solution (ρ, u) of the IBVP problem (4.7)– (4.10) exists for t ∈ [0, T ]. Then, under the assumptions of Proposition 4.2, it holds that  t γ −1 u(t)2L 2 () + ρ(t) L γ −1 () + |(ρ (γ +α)/2 ) y |2 d xds ≤ C0 , t ∈ [0, T ] , 0

ρ(t) L ∞ () + (ρ α (t)) y  L 2 () ≤ C0 , t ∈ [0, T ], with C0 > 0 a constant.

(4.29)

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Lemma 4.8. Let T > 0 and assume that the solution (ρ, u) of the IBVP problem (4.7)– (4.10) exists for t ∈ [0, T ] with ρ(y0 , t) = 0. Then, under the assumptions of Proposition 4.2, there is a time T∗ ∈ (0, T ] (depending on initial data) so that it holds that   t 2j 2 j−2 1+α 2 u t (y, t)dy + ρ u yt d xds ≤ C1 (4.30) ut 0

[0, T∗ ]

uniformly for t ∈ with j = 1, n and C1 > 0 a positive constant, and that ¯ is uniformly bounded for any t ∈ [0, T∗ ], ρu y (t) ∈ C 0 () ρu y  L ∞ (×[0,T∗ ]) ≤ C2

(4.31)

with C2 > 0 a constant. Moreover, the solution (ρ, u) is continuous ¯ × [0, T∗ ]), (ρ, u) ∈ C 0 (

(4.32)

and the initial one point vacuum state is maintained a− |y − y0 |β ≤ ρ(y, t) ≤ a+ |y − y0 |β , (y, t) ∈  × [0, T∗ ]

(4.33)

with a+ > a− > 0 two constants independent of time T∗ . Proof. Let us first assume that the weak solution is regular enough so that we can differentiate it through the equations and the interface as in [56], and the density is of the form a∗ |y − y0 |β ≤ ρ(y, t) ≤ a ∗ |y − y0 |β , (y, t) ∈  × [0, T ]

(4.34)

with a∗ , a ∗ two positive constants to be determined later. It will be assumed further that the following a-priori estimate holds: ρu y  L ∞ (×[0,T ]) ≤ M0

(4.35)

for some positive M0 to be determined later. We will prove (4.30) only for j = n below since the other case can be treated similarly. Taking the inner product between (4.8)t and 2n(u t )2n−1 over  leads to    d 2n γ 2n−1 dy − 2n (ρ 1+α u y ) yt (u t )2n−1 dy = 0 |u t | dy + 2n (ρ ) yt (u t ) dt from which, one deduces after integration by parts and using Eq. (4.7) that   d |u t |2n dy + 2n(2n − 1) ρ 1+α u 2yt (u t )2n−2 dy dt  =2n(2n − 1)γ ρ γ −1 ρt u yt (u t )2n−1 dy  + 2n(2n − 1)(1 + α) ρ α ρt u y u yt (u t )2n−2 dy  = − 2n(2n − 1)γ ρ γ +1 u y u yt (u t )2n−2 dy  − 2n(2n − 1)(1 + α) ρ 2+α u 2y u yt (u t )2n−2 dy

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 ≤ n(2n − 1) ρ 1+α u 2yt (u t )2n−2 dy   2γ +1−α 2 2n−2 +C ρ u y (u t ) dy + C ρ 3+α u 4y (u t )2n−2 dy  2γ −1−α ≤ n(2n − 1) ρ 1+α u 2yt (u t )2n−2 dy + Cρ L ∞ ρu y 2L ∞ u t (t)2n−2 L 2n−2  + C ρ 3+α u 4y (u t )2n−2 dy  2γ −1−α 2 ≤ n(2n − 1) ρ 1+α u 2yt (u t )2n−2 dy + CC0 M0 (1 + u t (t)2n+2 ) L 2n−2  (4.36) + C ρ 3+α u 4y (u t )2n−2 dy, where one has used (2.22), (4.29), the a-priori assumption (4.35), and Hölder’s inequality. The last term on the right hand side of (4.36) can be estimated as follows. For the case α ≥ 1, it follows from (4.29) and (4.35) that  ρ 3+α u 4y (u t )2n−2 dy ≤ρ 3+α u 4y  L ∞ u t (t)2n−2 ≤ C0α−1 M04 u t (t)2n−2 L 2n−2 L 2n−2 ≤ C M04 (1 + u t (t)2n+2 ). L 2n

(4.37)

For the case α ∈ ( 21 , 1), since it holds by (4.27) that  y |ρ 1+α u y | ≤ | u t (z, t)dz| + ρ γ ≤ u t  L 2n · |y − y0 |(2n−1)/2n + ρ γ ,

(4.38)

y0

which together (4.34) implies |ρ (3+α)n u 4n y |

=|ρ

−(1+3α)n

(ρ

1+α

uy) | = ρ 4n

−(1+3α)n



y

u t (z, t)dz + ρ

γ

4n

y0

≤ Cρ −(1+3α)n (u t 4n · |y − y0 |2(2n−1) + ρ 4nγ ) L 2n · |y − y0 |−(1+3α)nβ+2(2n−1) + ρ 3n(γ −α)+n(γ −1) , ≤ Cu t 4n L 2n

(4.39)

the last term in (4.36) is estimated by  1/n  (3+α)n 4n · u dy | ρ 3+α u 4y (u t )2n−2 dy| ≤ u t 2n−2 ρ y L 2n ≤ Cu t 2n−2 L 2n

 1/n   4n −(1+3α)nβ+2(2n−1) 3n(γ −α)+n(γ −1) · u t  L 2n |y − y0 | dy + ρ dy

≤ C(u t 2n+2 + 1), L 2n

(4.40)

4n−1 where one has used the fact β < n(1+3α) due to (4.19). Substituting (4.37) and (4.40) into (4.36) shows   d 2n |u t | dy + n(2n − 1) ρ 1+α u 2yt (u t )2n−2 dy dt ≤ C M02 (1 + M02 )(1 + u t (t)2n+2 ) ≤ C(1 + M04 )(u t (t)2n+2 + 1). (4.41) L 2n−2 L 2n

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Set u t (0)2n2n

2 −1 Ta = min{ n2n C(1+M 4 )C , n

0

L

C(1+M04 )

1

, T }.

(4.42)

One can apply Grönwall’s Lemma to obtain (4.30) for t ∈ [0, Ta ] with C1 given by C1 =: 2u t (0)2n ≥ u t (0)2n + C Ta (1 + M04 ). L 2n L 2n

(4.43)

To prove (4.31) and ensure the a-priori assumption (4.35), we use the equality (4.27) to get that near y = y0 , it holds that  y −α ρu y (y, t) = ρ (y, t) u t (z, t)dz + ρ γ −α (y, t), y = y0 , (4.44) y0

  1 which, together with (2.22), (4.29), and the fact β ≤ α1 1 − 2n due to (4.19), implies  y γ −α ≤C0 + |ρ −α (y, t) u t (z, t)dz| L ∞ ρu y  L ∞ ×[0,t] ×[0,t] γ −α ≤ C0 γ −α ≤ C0

y0 −α + a∗ |y − y0 |1−1/2n−αβ u t (t) L 2n γ −α 1/2n + a∗−α C1 ≤ C0 + 2a∗−α u t (0) L 2n

= : C 2 ≤ M0 so long as the constant M0 is chosen as

γ −α

M0 = 1 + C 0

+ 2a∗−α u t (0) L 2n .

(4.45)

Next, we verify the a-priori assumption (4.34). Set Tb = min{

Aα0

1/2n

3αC1

, Ta }.

It follows from Eq. (4.28) for y = y0 and t ∈ [0, Tb ] that  t  t ρ α (y, t) + α ρ γ (y, s)ds = ρ0α (y) + α 0

0

(4.46)

y

u t (z, s)dzds

y0

≥ Aα0 |y − y0 |αβ − αTb |y − y0 |1−1/2n sup u t (t) L 2n 0≤t≤Tb

≥

Aα0 |y

αβ

1/2n − C1 αTb |y

− y0 | − y0 | (4.47) 2 α 1 1/2n = A0 |y − y0 |αβ + |y − y0 |αβ ( Aα0 − C1 αTb |y − y0 |1−1/2n−αβ ) 3 3 2 ≥ Aα0 |y − y0 |αβ , 3   1 due to (4.19). On the where we have used the facts |y − y0 | ≤ 1 and β ≤ α1 1 − 2n other hand, it follows from (4.28) that  t  t y ρ α (y, t) + α ρ γ (y, s)ds = ρ0α (y) + α u t (z, s)dzds ≤ ≤

1−1/2n

0 0 y0 α αβ 1−1/2n A1 |y − y0 | + αt|y − y0 | u t (t) L 2n 1/2n α αβ (A1 + C1 αt)|y − y0 | .

(4.48)

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Define Tc ∈ (0, Tb ] by Tc = min{

Aα0 3αC(Aα1 + C1

1/2n

αTb )γ /α

, Ta , Tb }.

(4.49)

Set 

t

Z (t) =

ρ γ (y, s)ds.

0

It follows from (4.48) that (Z  (t))α/γ ≤ (Aα1 + C1

1/2n

αt)|y − y0 |αβ ,

which implies for t ∈ [0, Tc ] that 

t 0

ρ γ (y, s)ds ≤ Tc (Aα1 + C1

1/2n

αTc )γ /α |y − y0 |(γ −α)β |y − y0 |αβ

≤ C Tc (Aα1 + C1

1/2n

αTb )γ /α |y − y0 |αβ .

(4.50)

As a consequence of (4.48), (4.50), and (4.49), one gets 2 ρ (y, t) ≥ Aα0 |y − y0 |αβ − α 3 α



t

ρ γ (y, s)ds ≥

0

1 α A |y − y0 |αβ . 3 0

(4.51)

From (4.48) and (4.51), we can verify the a-priori assumption (4.34) and justify the property (4.33) by simply choosing a∗ = a− = (Aα0 /3)1/α , a ∗ = a+ = (Aα1 + C1

1/2n

αTc )1/α

(4.52)

for any t ∈ [0, T∗ ] with time T∗ = Tc determined by (4.49). One then derives from Eq. (4.7), (4.29), and (4.31) that ρ α ∈ L ∞ (0, T∗ , H 1 ()), (ρ α )t ∈ L ∞ (0, T∗ , L 2 ()),

(4.53)

while (4.31), (4.33), boundary condition (4.10), and (4.30) for j = 1 imply that u ∈ L ∞ (0, T∗ , W0 ()), u t ∈ L ∞ (0, T∗ , L 2 ()) 1, p

(4.54)

for any p ∈ (1, β −1 ), where one has used the fact β −1 > β+−1 ≥ 1 so that sup u y  L p () = sup ρ − p  L 1 () · ρu y  L ∞ (×[0,T  ]) ≤ C. p

t∈[0,T∗ ]

t∈[0,T∗ ]

∗

(4.55)

Equations (4.53)–(4.55) imply the continuity (4.32) of the solution (ρ, u), and the continuity of ρu y follows from Eq. (4.27) and that of (ρ, u). The proof of the lemma is completed.   Using Lemmas 4.6–4.8 and a direct computation, we can obtain the following result:

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Lemma 4.9. Let T∗ > 0 be given in Lemma 4.8 and (ρ, u) be the solution of the IBVP problem (4.7)–(4.10). Then, under the assumptions of Proposition 4.2, ρ 1+α u y (t) ∈ ¯ is uniformly bounded for any t ∈ [0, T∗ ], C 0 ()  lim ρ 1+α u y (y, t) = 0, ρ 1+α u y  L ∞ (0,T∗ ;C 0 ()) ¯ ≤ C(T∗ ),

y→y0

and (ρ, u) satisfies for 0 ≤ s < t ≤ T∗ that ρ(t) − ρ(s) L 2 () + u(t) − u(s) L 2 () ≤ C(T∗ )|t − s|, ρ 1+α u y (t) − ρ 1+α u y (s) L 2 () ≤ C(T∗ )|t − s|1/2 . ¯ and lim y→y0 Proof. The conclusions that for any t ∈ [0, T∗ ], ρ 1+α u y (t) ∈ C 0 () 1+α ρ u y (y, t) = 0 are due to the continuity of right hand side terms of (4.44) and (4.34). By (4.31) and (4.33), one can check easily that ρ 1+α u y  L ∞ (×[0,T∗ ]) ≤ ραL ∞ (×[0,T  ]) · ρu y  L ∞ (×[0,T∗ ]) ≤ C(T∗ ). ∗

Making use of Eq. (4.7), (4.30) and (4.31), one can obtain  t  t ρ(t) − ρ(s) L 2 () ≤ ρt (τ ) L 2 () dτ =  ρ 2 u y (τ ) L 2 () dτ s

s

≤ C(t − s)ρ L ∞ (×[0,T∗ ]) · ρu y  L ∞ (×[0,T∗ ]) ≤ C(T∗ )|t − s|,  u(t) − u(s) L 2 () ≤ 

t s

 u t (τ )dτ  L 2 () ≤ C

s

t

u t  L 2 () ≤ C(T∗ )|t − s|,

and  t ρ 1+α u y (t) − ρ 1+α u y (s) L 2 () ≤  (ρ 1+α u y (t))t dτ  L 2 () s  t  t ≤ C( ρt ρ α u y (τ ) L 2 () +  ρ 1+α u yt (τ )dτ  L 2 () ) s s  t  t ρ 2+α u 2y (τ ) L 2 () +  ρ 1+α u yt (τ )dτ  L 2 () ) ≤ C( s s  2+α 2 ∞ ≤ C(T∗ )(|t − s|ρ u y  L (×[0,T∗ ]) ≤C(T∗ )|t − s|1/2 .

The proof is completed.

+ |t − s|1/2 ρ (1+α)/2 u yt (τ ) L 2 (×[0,T∗ ]) )

 

Proof of the Proposition 4.2. With the help of Lemmas 4.6–4.8, we are ready to prove Proposition 4.2. (1) Existence of weak solution for short time. We only deal with the case for the Dirichlet boundary condition and one point vacuum state in the initial data, the case of periodic boundary and one point vacuum state in initial data can be done in a similar way. Once the a-priori estimates are established as in Lemmas 4.6–4.9, we are able to prove Proposition 4.2. First of all, we construct a sequence of approximate solutions by

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H.-L. Li, J. Li, Z. Xin

modifying the finite difference scheme used in [33,40,43]. Without the loss of generality, we assume y0 =

1 , namely, ρ0 ( 21 ) = 0. 2

(4.56)

For any given positive integer N = 2k + 1 with k ≥ 0 an integer, let h = 1/N . Consider the system of 2N ordinary differential equations ⎧ h h d h ⎪ h 2 u 2n+2 − u 2n ⎪ + (ρ ) ρ = 0, ⎪ 2n+1 2n+1 ⎪ ⎪ h ⎪ dt ⎨ h h (ρ2n+1 )γ − (ρ2n−1 )γ d h (4.57) u + ⎪ ⎪ dt 2n h ⎪ ⎪ ⎪ ⎪ ⎩ = 1 {(ρ h )1+α (u h − u h )/ h − (ρ h )1+α (u h − u h )/ h}, 2n+2 2n 2n−1 2n 2n−2 h 2n+1 with the boundary condition and point vacuum h u 0h (t) = u 2N (t) = 0, ρ2k+1 (t) = 0,

(4.58)

and initial data h ρ2n+1 (0) = ρ0 ((2n + 1) h2 ),

h u 2n (0) = u 0 ((2n) h2 ),

(4.59)

where n = 1, 2, 3, ..., N . Here, we also assume h h h h h h ρ2N +1 (t) = ρ2N −1 (t), ρ1 (t) = ρ−1 (t), u 2N +2 (t) = u −2 (t) = 0

which is consistent with the boundary condition and the fact that density is continuous and non-zero at the left boundary. By applying the idea as in [32,43] and the similar arguments mentioned above, we can obtain the following uniform (w.r.t. h) a-priori estimates about the solutions (ρ2n+1 , u 2n ) (here and below we omit the symbol h for simplicity) of (4.57)–(4.59) similar to Lemmas 4.6–4.9. Details will be omitted (the reader can refer to [32,43] for similar arguments in detail). Lemma 4.10. Let (ρ2n+1 , u 2n ) be the solution of (4.57)–(4.59). Then, it holds  t N N  1+α u 2n+2 (s)−u 2n (s) 2 1 2 ( 2 u 2n (t) + π(ρ2n+1 (t)))h + ρ2n+1 ( ) hds h 0 n=1

n=0

=

N  ( 21 u 22n (0) + π(ρ2n+1 (0)))h, n=0 N −1  n=0

−1 ρ2n+1 (t)h =

N −1 

−1 ρ2n+1 (0)h.

n=0

It follows from Lemma 4.10 and the standard theory of ordinary differential equations that there exists a global solution (ρ2n+1 , u 2n ) to (4.57)–(4.59) for any fixed positive N and h. Furthermore, the following properties hold:

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431

Lemma 4.11. Let (ρ2n+1 , u 2n ) with n = 0, 2, 3, ..., N − 1 with N = 2k + 1 be the solution of (4.57)–(4.59). Then, it holds for n ≥ k + 1 that  t   t n d γ α α ρ2n+1 (t) = ρ2n+1 (0) − α u 2 j (s)hds − α ρ2n+1 (s)ds, dt 0 0 j=k+1

1+α ρ2n+1 (t)

n  d u 2n+2 (t) − u 2n (t) γ = u 2 j (t)h + ρ2n+1 , h dt j=k+1

and for n ≤ k that α ρ2n−1 (t)

=

α ρ2n−1 (0) − α

 t  t k d γ u 2 j (s)hds − α ρ2n−1 (s)ds, dt 0 0 j=n

 d u 2n (t) − u 2n−2 (t) γ =− u 2 j (t)h + ρ2n−1 . h dt k

1+α ρ2n−1 (t)

j=n

Lemma 4.12. Under the assumptions of Proposition 4.2, there is a short time T∗ > 0 so that it holds for n = 0, 1, 2, ..., N that ρ2n+1 (t) +

N 

(

n=1 1 a− | 2 (2n

α α ρ2n+1 (t)−ρ2n−1 (t) 2 ) h h

2n (t) + |ρ2n+1 (t) u 2n+2 (t)−u | ≤ C(T∗ ), h

+ 1)h − 21 |β ≤ ρ2n (t) ≤ a+ | 21 (2n + 1)h − 21 |β ,

and for m = 1 or n that  t N N  d d 2m 2m−2 dt u 2n (s)− dt u 2n−2 (s) 2 d d ( dt u 2 j (t)) h + ρ21+α ( ) )h ≤ C(T∗ ). j+1 (s)( dt u 2 j (s)) h 0 j=0

j=0

Here C(T∗ ) > 0 and a± > 0 are constants. Lemma 4.13. Under the assumptions of Proposition 4.2 and Lemma 4.12, it holds for t ∈ [0, T∗ ] that 1+α 2n (t) |ρ2n+1 (t) u 2n+2 (t)−u | ≤ C(T∗ ), h

|u 2n (t)| +

N 

|u 2n (t) − u 2n−2 (t)| ≤ C(T∗ ),

n=1 N 

|(ρ2n+1 )1+α (u 2n+2 − u 2n )/ h − (ρ2n−1 )1+α (u 2n − u 2n−2 )/ h| ≤ C(T∗ ),

n=1 N  n=1 N  n=1

|ρ2n (t) − ρ2n (s)|2 h +

N 

|u 2n−1 (t) − u 2n−3 |2 h ≤ C(T∗ )|t − s|2 ,

n=1

2n−1 (t) 2n−3 (s) 2 |(ρ2n )1+α (t) u 2n+1 (t)−u − (ρ2n−2 )1+α (s) u 2n−1 (s)−u | h ≤ C(T∗ )|t − s|. h h

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With the help of Lemmas 4.10–4.13, we can define the sequence of approximate solutions (ρh , u h ) on the domain  × [0, T∗ ] as
ρh (y, t) = ρ2n+1 (t), u h (y, t) = h1 [(y − (n − 21 )h)u 2n+1 (t) + ((n + 21 )h − y)u 2n−1 (t)] for y ∈ ((2n) 21 h, (2n + 2) 21 h). It can be verified that the following properties hold for the approximate solutions ∂ y u h (y, t) = and

u 2n+1 (t) − u 2n−1 (t) h


a− |y − y0 |β ≤ ρh (y, t) ≤ a+ |y − y0 |β , |u h (y, t)| ≤ C(T∗ ), |ρh (y, t)∂x u h (y, t)| ≤ C(T∗ ),

for (y, t) ∈  × [0, T∗ ]. Then, the existence of a weak solution for short time t ∈ [0, t∗ ] follows from Helly’s theorem, the diagonal process together with Lebesgue’s theorem, and the a-priori estimates (see, for instance, [33]). The details are omitted. (2) Uniqueness of weak solution for short time. We prove the uniqueness of weak solutions for α ∈ ( 21 , γ ). Without loss of generality, only the case for the Dirichlet boundary condition (4.10) will be studied. Let (ρ1 , u 1 ) and (ρ2 , u 2 ) be two weak solutions of Eq. (4.7)–(4.10) satisfying Lemma 4.6–4.9. Denote n = ρ1 − ρ2 , ψ = u 1 − u 2 , (x, t) ∈ (0, 1) × [0, T ]. Obviously, the new unknown (n, ψ) with ψ(0, t) = ψ(1, t) = n(y0 , t) = 0 satisfies   n + ψ y = 0, y = y0 , (4.60) ρ1 ρ2 t γ

γ

ψt + (ρ1 − ρ2 ) y − (ρ11+α ψ y ) y − ((ρ11+α − ρ21+α )u 2y ) y = 0,

(4.61)

for (y, t) ∈ (0, 1) × (0, T∗ ] with zero initial data (n(y, 0), ψ(y, 0)) = (0, 0),

y ∈ (0, 1).

(4.62)

Take the inner product between ρ1α ρ2−1 n and (4.60) over [0, y0 ) ∪ (y0 , 1] to obtain    d ρ1−1+α ρ2−2 n 2 dy = −(1 + α) u 1y ρ1α ρ2−2 n 2 dy − 2 ρ1α ρ2−1 nψ y dy dt   1 ≤ (Cρ1 u 1y  L ∞ + 4) ρ1−1+α ρ2−2 n 2 dy + ρ11+α ψ y2 dy 4   1 ≤ C(T∗ ) ρ1−1+α ρ2−2 n 2 dy + (4.63) ρ11+α ψ y2 dy, 4 1  y where f d x =: 0 0 f d x + y0 f d x. Taking the inner product between ψ and (4.61) over , and noting that ψ(0, t) = ψ(1, t) = n(y0 , t) = 0 and ρi1+α u i (y0 , t) = 0, i = 1, 2, we have

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433

   1 d γ γ ψ(t)2 + ρ11+α ψ y2 dy = (ρ1 − ρ2 )ψ y dy − (ρ11+α − ρ21+α )u 2y ψ y dy 2 dt  γ  1+α γ ρ1 − ρ2 ρ1 − ρ21+α = nψ y dy − nu 2y ψ y dy ρ1 − ρ2 ρ1 − ρ2    γ  γ 2 ρ1 − ρ2 1 ρ1−1−α n 2 dy + ≤C ρ11+α ψ y2 dy ρ1 − ρ2 8 2   1+α  ρ1 − ρ21+α 1 ρ1−1−α n 2 u 22y dy + +4 ρ11+α ψ y2 dy ρ1 − ρ2 8   1 2(γ −1) 2(γ −1) −2α 2 1+α 2 + ρ2 )ρ1 ρ2 ρ1−1+α ρ2−2 n 2 dy ≤ ρ1 ψ y dy + C sup (ρ1 4 y= y0  + C sup (ρ12α + ρ22α )ρ1−2α ρ2 u 2y 2L ∞ ρ1−1+α ρ2−2 n 2 dy ≤

1 4



y= y0

ρ11+α ψ y2 dy + C sup |y − y0 |[2(γ −1)+2−2α]β y= y0

+ C sup |y − y0 | ≤

1 4



[2α−2α]β

y= y0

ρ11+α ψ y2 dy + C







ρ1−1+α ρ2−2 n 2 dy

ρ1−1+α ρ2−2 n 2 dy

ρ1−1+α ρ2−2 n 2 dy.

Summing up the two differential inequalities (4.63) and (4.64) leads to   d d −1+α −2 2 2 ψ(t) + ρ2 n (t)d x + ρ11+α ψ y2 dy ρ1 dt dt  ≤C(T∗ ) ρ1−1+α ρ2−2 n 2 dy

(4.64)

(4.65)

which, together with the initial data (4.62) and the Grönwall’s lemma, gives rise to  2 ψ(t) + ρ1−1+α ρ2−2 n 2 (x, t)d x ≡ 0, t ∈ [0, T ]. (4.66) This together with ρ1 (y0 , t) = ρ2 (y0 , t) = 0 implies the uniqueness of weak solution (ρ1 , u 1 ) ≡ (ρ2 , u 2 ).

(4.67)

The proof of Proposition 4.2 is completed.   Proof of Proposition 4.1. With the help of Propositions 4.2–4.4 and the inverse transformation from the Lagrangian coordinates to the Eulerian coordinates, we are able to prove Proposition 4.1. Since the case of periodic boundary conditions with one point vacuum state in the initial data can be dealt with in a similar framework, we only deal with the IBVP problem for the compressible Navier-Stokes equations (2.1)–(2.3) for the case of the Dirichlet boundary condition (2.6) with one point vacuum in the regular initial data (2.18)–(2.19). First, one can check easily that the IBVP problem (2.1)–(2.3), (2.6), and (2.18)– (2.19) in the Eulerian coordinates is equivalent to the corresponding IBVP problem in

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H.-L. Li, J. Li, Z. Xin

the Lagrangian coordinates for Eqs. (4.7)–(4.9) with the Dirichlet boundary conditions (4.10) and (4.12) through the coordinate transformation  x y= ρ(z, t)dz, x ∈ (0, 1), t ≥ 0. (4.68) 0

Note that y ∈ [0, 1] due to the conservation of mass. More importantly, the case of one point vacuum state in the initial  x data (2.18)–(2.19) isσ reformulated into the corresponding case (4.12) with y0 = 0 0 ρ0 (z)dz and β = 1+σ . It is easy to verify that all the assumptions of Proposition 4.2 are satisfied. Then Proposition 4.2 gives the existence and uniqueness of a weak solution (ρ, ˜ u) ˜ satisfying (4.13)–(4.18) in the Lagrangian coordinate to the compressible Navier-Stokes equations (4.7)–(4.9) with the Dirichlet boundary conditions (4.10) and one point vacuum state in the initial data. In particular, ¯ ¯ ρ, ˜ u˜ ∈ C 0 ([0, T∗ ]; C 0 ()), ρ˜ u˜ y (t) ∈ L ∞ (0, T∗ ; C 0 ()), ρ(y ˜ 0 , t) = 0, ρ(y, t) > 0, y = y0 .

(4.69) (4.70)

These in turn imply, in terms of the inverse coordinate transformation of (4.68), i.e.,  y ρ˜ −1 (z, t)dz, y ∈ [0, 1], x= 0

the existence and uniqueness of a weak solution (ρ, ˜ u) ˜ on the domain (x, t) ∈ [0, 1] × [0, T∗ ] to the compressible Navier-Stokes equations (2.1)–(2.3) with the Dirichlet boundary conditions (2.6) and initial one point vacuum state (2.18)–(2.19). Moreover, one can check that there exists one particle path x = X 0 (t) defined by ˜ 0 (t), t), t > 0, X˙ 0 (t) = u(X

X 0 (0) = x0

satisfying 

X 0 (t)

y0 ≡

 ρ(z, ˜ t)dz =

d d y(X 0 (t), t) = dt dt

ρ0 (z)dz

0

0

due to the fact

x0



X 0 (t)

ρ(z, ˜ t)dz = 0, t ≥ 0.

0

This, together with (4.70), gives rise to ρ(X ˜ 0 (t), t) = 0, and ρ(x, ˜ t) > 0, x = X 0 (t), t ≥ 0. Moreover, it is easy to verify that the solution (ρ, ˜ u) ˜ satisfies all the properties (2.23)– (2.26), and particularly ⎧ ˜ T∗ ) ≥ 0 on , m(., ˜ T∗ ) = ρ˜ u(., ˜ T∗ ) = 0, on {x ∈  | ρ0 (x) = 0}, ⎪ ⎨ρ(x, ρ(., ˜ T∗ ) ∈ L 1 () ∩ L γ (), (ρ˜ α−1/2 (., T∗ ))x ∈ L 2 (), 2 2+ν ⎪ ˜ ˜ ⎩ |m(.,T ∗ )| ∗ )| + |m(.,T ∈ L 1 (). ρ(x,T ˜ ∗)

ρ˜ 1+ν (x,T∗ )

The case of periodic boundary can be treated in a similar way; we omit the details. The proof of Proposition 4.1 is complete.  

Vanishing of Vacuum States and Blow-up Phenomena

435

5. Vanishing of Vacuum States and Blow-up Phenomena We shall prove that for any global entropy weak solution (ρ, u) to the IBVP for compressible Navier-Stokes equations (2.1)–(2.3) together with (2.6) or (2.7), any possible vacuum state vanishes in finite time and the velocity (if definable and regular enough) blows up in finite time if a vacuum state appears, for example, the density contains vacuum initially as in Theorem 2.2. The weak solution becomes a strong one after the vanishing of vacuum states and tends time-asymptotically to the non-vacuum equilibrium state exponentially. 5.1. Vanishing of vacuum states in finite time. Proposition 5.1. Let (2.34) hold. For any global entropy weak solution (ρ, u) to the IBVP for compressible Navier-Stokes equations (2.1)–(2.2) with initial data -(2.3) and boundary condition (2.6) or (2.7) in the sense of Definition 2.5, there exists a time T0 > 0 such that inf ρ(x, t) > 0, for all t ≥ T0 .

(5.1)

x∈

Proof. To prove (5.1), we will employ an idea which has been used in [28] (see also [22,27]) to show the blow up behavior of both the global strong solutions to the IBVP for (2.1)–(2.2) with the constant viscosity and the global strong solutions to the Stokes approximation equations, with initial data containing vacuum states. Let T ∈ (0, ∞) be fixed. In this subsection, C denotes some generic positive constant independent of T . First, it is noted that mass is conserved for any t ∈, (0, T ],   ρ(x, t)d x = ρ0 (x)d x. (5.2) 



Based on the entropy inequality (2.13), it can be deduced from (5.2) and (2.39) that for a constant b ≥ max{α + γ − 1, 2α + 1, 1},    T  2   b   + sup ρ L ∞ +  ρ b  (5.3)  ρ  dt ≤ C. x L2

0≤t≤T

It will be shown below that  4

  g(t)
 ρ b − ρ b (·, t) 4

L ()

where ρ b (t)

1 = ||

x L2

0

→ 0 as t → ∞,

(5.4)

 

ρ b (x, t)d x.

Now, we assume that (5.4) holds, and continue the proof of Proposition 5.1. In fact, the inequality (5.3) and the Poincaré-Sobolev inequality implies that  

  b  ρ − ρ b (·, t) C() 2/3 

  1/3 ≤ C  ρ b − ρ b (·, t) 4 (ρ b )x (·, t) L 2 L () 2/3 

  ≤ C  ρ b − ρ b (·, t) 4 → 0, as t → ∞. (5.5) L ()

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H.-L. Li, J. Li, Z. Xin

This suffices to finish the proof of Proposition 5.1 due to the following simple fact: ρ b (t) ≥ ρ b (t) ≡ ρ0 b = 1, for any t ≥ 0. It remains to prove (5.4). First, it follows directly from (5.3) and the Poincaré-Sobolev inequality that  T  2  T   g(t)dt ≤ C sup ρ b − ρ b  ∞ (ρ b )x 2L 2 dt 0

L

0≤t≤T

0

≤ C. Next, we prove that

(5.6) 

T

|g  (t)|dt ≤ C.

(5.7)

0

Note that (2.9) as well as the boundary condition (2.6) or (2.7) imply that  

3

3   b b−1 b g (t) = 4b ρ − ρ ρb − ρb d x ρ , ρt − 4 ρb t  H 1 ×H −1   

3

3  √ √ ρ b − ρ b ρ b−1 = −4b ρb − ρb d x ρ ρud x − 4 ρ b 

t

x




I1 + I2 .

(5.8)

It follows from (5.3) and (2.13) that   T  T 

2   b b b−1/2 √ b  ρ −ρ |I1 |dt ≤ C (ρ )x ρ ρud x  dt   0 0   T 

3   b b−1 √ √ b  +C ρ −ρ (ρ )x ρ ρud x  dt   0   T 

2   b α−1/2 b−α √ b   dt ≤C ρ − ρ (ρ ) ρ ρud x x    0  T   2 √  α−1/2    ≤C )x  2  ρu L 2 ρ b − ρ b  ∞ dt (ρ 

T

≤C ≤ C.

L

0

0

L

(ρ b )x 2L 2 dt (5.9)

The uniform entropy estimate (2.13), together with (2.9), gives that     d    sup  ρ b (t) = b sup  ρ b−1 , ρt  dt 0≤t≤T 0≤t≤T  

√ √   b−1  = b sup  ρ ρ ρud x  0≤t≤T



x

 

  α−1/2 b−α √  ≤ C sup  ρ ρ ρud x  x 0≤t≤T  



√  α−1/2  ≤ C sup  ρ   ρu L 2 ρb−α ∞ L 0≤t≤T

≤ C.

x L2

(5.10)

Vanishing of Vacuum States and Blow-up Phenomena

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This together with the Poincaré inequality and (5.3) yields 

T 0

 |I2 |dt ≤ C 0

T

  3  b  b ρ − ρ  ∞ dt ≤ C L

0

T

(ρ b )x 2L 2 dt ≤ C.

(5.11)

The estimate (5.7) thus follows directly from (5.8)–(5.11). Hence the desired estimate (5.4) follows from (5.6) and (5.7). The proof of Proposition 5.1 is complete.  

5.2. Regularity and asymptotics of weak solutions for large time. It is usually difficult √ to get information about the velocity field for the global entropy weak solution (ρ, ρu), in the sense of Definition 2.5 to the IBVP for the Compressible Navier-Stokes equations (2.1)–(2.2) with initial data (2.3) and boundary values (2.6) or (2.7) in the appearance of vacuum states. After vacuum states vanish, however, it will be shown that the velocity field u can be defined with enough regularity and the nonlinear diffusion term is represented in terms of the velocity u and the density ρ. The √ momentum equation becomes √ a uniform parabolic equation, and the weak solution (ρ, ρu) = (ρ, ρ · u) becomes a strong solution. Proposition 5.1 implies that there is a time T0 > 0 after which the density of the global entropy weak solution (ρ, u) to the IBVP problem for (2.1)–(2.3) together with (2.6) or (2.7) is strictly positive and (ρ, u) satisfies the finite entropy estimate (2.13). Consider the IBVP problem (2.1)–(2.2) again for time t ≥ T0 with data given at time t = T0 by √ ρu(x,t) √ , ρ(x,t) t→T0

ρ(x, T0 ) = lim ρ(x, t), u(x, T0 ) = lim t→T0

(5.1)

and note here that away from vacuum the Dirichlet boundary condition (2.6) reduces to u(0, t) = u(1, t) = 0, t ≥ T0 .

(5.2)

We then have the regularity property of the solution for the compressible Navier-Stokes equations (2.1)–(2.3) with the Dirichlet boundary condition (2.6) or the periodic boundary condition (2.7) for large time. √ Proposition 5.2. Under the assumptions of Theorem 2.3, let (ρ, ρu) be the global entropy weak solution to the IBVP for the compressible Navier-Stokes equations (2.1)– (2.2) with initial data (2.3) and boundary value (2.6) or (2.7) in the sense of Definition 2.5. Let T0 > 0 so that the global weak solution (ρ, u) satisfies for two positive constants ρ± that ¯ × [T0 , ∞). 0 < ρ− ≤ ρ(x, t) ≤ ρ+ , ∀ (x, t) ∈ 

(5.3)

√ √ Then, (ρ, ρu) = (ρ, ρ · u) is the unique strong solution to the IBVP for the compressible Navier–Stokes equations (2.1)–(2.2) and (5.1) with the boundary condition (2.6)2 or (2.7) for t ≥ T0 . Moreover, the regularity (2.36) and the long time behavior (2.38) hold. 2 Note here that away from vacuum the Dirichlet boundary condition (2.6) reduces to the usual one u(0, t) = u(1, t) = 0, t ≥ T0 .

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Proof. We only prove Proposition 5.2 for the Dirichlet case below, the periodic case can be treated similarly. Step 1. Regularity. It follows easily from Proposition 5.1 that there exist some T0 and a constant ρ− > 0 such that for all t ≥ T0 , inf ρ(x, t) ≥ ρ− > 0

(5.4)

x∈

which, together with (2.13), implies ρ ∈ L ∞ (T0 , T ; H 1 ())

(5.5)

¯ × [0, ∞)) there exists some σ > 0 small for any T > T0 . By the continuity of ρ ∈ C( enough such that for any t ≥ T0 − σ, inf ρ(x, t) ≥

x∈

ρ− > 0. 2

This implies that one can define the velocity u for any global entropy weak solution in the sense of Definition 2.5 after the vanishing of vacuum states by √ ρu u =: √ , for t ≥ T0 − σ. ρ It then follows from the definition and (2.8) that u ∈ L ∞ (T0 − σ, T ; L 2 ())

(5.6)

for any T > T0 . Noting that (2.12) implies that for any ϕ(x) ∈ C0∞ (), ψ(t) ∈ C0∞ (T0 − σ, T ), 



T T0 −σ

ψ(t) 

=−

T T0 −σ



ρ −α ϕd xdt 

ψ(t)



√ ρ α−1/2 ρu(ρ −α ϕ)x d xdt

 2α √ ψ(t) (ρ α−1/2 )x ρuρ −α ϕd xdt 2α − 1 T0 −σ    T √ ψ(t) ρuρ −1/2 ϕx d xdt =− 

−

 =−

T0 −σ T T0 −σ

T



 ψ(t)



uϕx d xdt,

we can define the spatial derivative of velocity and, together with (2.13), its regularity as ux =

∈ L 2 ( × (T0 − σ, T )). ρα

(5.7)

Vanishing of Vacuum States and Blow-up Phenomena

439

In terms of (5.6), (5.7) and (2.9) we are also able to justify the Dirichlet boundary condition (2.6) for the velocity u u(0, t) = u(1, t) = 0, for any t ≥ T0 − σ.

(5.8)

Thus, (5.6), (5.7) and (5.8) show u ∈ L 2 (T0 − σ, T ; H01 ()) ∩ L ∞ (T0 − σ, T ; L 2 ())

(5.9)

for the case of the Dirichlet boundary conditions. Note here that u ∈ L 2 (T0 − σ, T ; 1 ()) ∩ L ∞ (T − σ, T ; L 2 ()) in the case of periodic boundary conditions. We thus Hper 0 obtain from (2.9) and (2.10) that the solution (ρ, u) satisfies ρt + (ρu)x = 0

a.e. in

 × (T0 − σ, T ),

(5.10)

and 

T T0 −σ



 

ρuϕt d xdt +

T T0 −σ



ρu 2 − ρ α u x + ρ γ ϕx d xdt = 0 

(5.11)

for any ϕ(x, t) ∈ C0∞ ( × (T0 − σ, T )) for the Dirichlet case. Equation (5.11) can be re-written in terms of (5.9), (5.10) and (2.13) as follows:  T   T 

α−1 α−2 (uϕt − ρ u x ϕx + ρ ρx − u u x ϕ)d xdt = γρ γ −2 ρx ϕ d xdt T0 −σ



T0 −σ



(5.12) for any ϕ(x, t) ∈ C0∞ ( × (T0 − σ, T )) for the Dirichlet case. Noticing that ρ α−2 ρx − u ∈ L ∞ (T0 − σ, T ; L 2 ()) due to (2.8), (5.4) and (5.6), and using standard regularity results for linear parabolic equations (see [25]), we get that u ∈ L 2 (T0 , T ; H 2 ())) ∩ H 1 (T0 , T ; L 2 ())

(5.13)

2 ()) ∩ H 1 (T , T ; L 2 ()) for the Dirichlet case. It is noted here that u ∈ L 2 (T0 , T ; Hper 0 for the periodic case. Step 2. Uniqueness. We shall show that if there exists another solution (η, v) to the compressible Navier-Stokes equations (2.1)–(2.2) with the following initial data and Dirichlet boundary conditions
(η, v)(x, T0 ) = (ρ, u)(x, T0 ), (5.14) v(0, t) = v(1, t) = 0,

such that
ρ− ≤ η ∈ L ∞ (T0 , T ; H 1 ()), v ∈ L ∞ (T0 , T ; H 1 ()) ∩ L 2 (T0 , T ; H 2 ()) ∩ H 1 (T0 , T ; L 2 ()),

(5.15)

then ρ = η, u = v

a.e. in

 × (T0 , T ).

(5.16)

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H.-L. Li, J. Li, Z. Xin

In fact, it follows from (5.10), (5.11) and (5.13) that 1 2

 ρu d x + 2



 t 

T0

ρ α u 2x d xds

−

 t T0

1 ρ u x d xds = 2  γ

 

ρu 2 (x, T0 )d x

(5.17)

for all t ∈ (T0 , T ), while (5.11) and (5.15) imply that  

ρuvd x +

 t T0



=





 t

ρ α u x vx d xds −

ρuv(x, T0 )d x +



T0



T0

 t

ρ γ vx d xds

ρu(vt + uvx )d xds

(5.18)

for all t ∈ (T0 , T ). To estimate the second term on the right hand side of (5.18), we use the decomposition ρvt + ρuvx = (ρ − η)(vt + vvx ) + ρ(u − v)vx + (ηα vx )x − (ηγ )x .

(5.19)

Multiplying (5.19) by u and integrating by parts give  t T0

=



ρu(vt + uvx )d xds

 t 

T0

−

(ρ − η)u(vt + vvx )d xds +

 t 

T0

ηα vx u x d xds +



ρu(u − v)vx d xds



T0

 t T0

 t

ηγ u x d xds.

(5.20)

Substituting (5.20) into (5.18) gives, for a.e. t ∈ (T0 , T ), that  

ρuvd x +

 t T0



=





ρ α u x vx d xds −

ρuv(x, T0 )d x +

T0

 t

+



T0

 t 

 t 

T0

(ρ γ vx + ηγ u x )d xds

(ρ − η)u(vt + vvx )d xds

ρu(u − v)vx d xds −

 t 

T0

ηα vx u x d xds.

(5.21)

Multiplying (5.19) by v and integrating the result over  × (T0 , t) lead to 1 2

 

ρv 2 d x =

1 2

 ρv 2 (x, T0 )d x +

  t

+ T0  t

+ T0



 t T0







(ρ − η)v(vt + vvx )d xds

ρv(u − v)vx d xds −

 t T0

ηγ vx d xds.



ηα vx2 d xds (5.22)

Vanishing of Vacuum States and Blow-up Phenomena

441

We obtain after adding up (5.17) and (5.22) and subtracting (5.21) that for all t ∈ (T0 , T ),  t  1 2 ρ(u − v) d x + ρ α (u − v)2x d xds 2  T0   t  t α α (ρ − η )(v − u)x vx d xds + (ρ − η)(vt + vvx )(v − u)d xds = T0

−



 t 

T0

ρ(u − v)2 vx d xds −

 t T0



T0



 γ  ρ − ηγ (v − u)x d xds

 t

ρ α − ηα 2L 2 vx 2L ∞ + ρ − η2L 2 vt + vvx 2L 2 + ρ γ − ηγ 2L 2 ds ≤ Cε T0



t

+C



 vx 

T0

L∞

ρ(u − v) d xds + Cε 2



t T0

u − v2H 1 ds.

(5.23)

Next, we estimate the term g(t)
ρ − η2L 2 + ρ α − ηα 2L 2 + ρ γ − ηγ 2L 2 . For any β > 0, (5.15), (5.13) and (2.1) imply that (ρ β − ηβ )t + v(ρ β − ηβ )x + (u − v)(ρ β )x + βρ β (u − v)x + β(ρ β − ηβ )vx = 0. (5.24) One can derive from this that



ρ β − ηβ 2L 2 ≤ ρ β − ηβ 2L 2 Cvx  L ∞ + Cε ρx 2L 2 + Cε t

+εu − v2L ∞ + ε(u − v)x 2L 2 .

(5.25)

Thus, since g(T0 ) = 0, we obtain from (5.25) with β = 1, α, γ respectively that  t  t

2 g(s) Cvx  L ∞ + Cε ρx  L 2 + Cε ds + Cε u − v2H 1 ds. (5.26) g(t) ≤ T0

T0

Now (5.16) is a consequence from (5.23), (5.26), (5.15), (5.5) and (5.4). The proof of large time convergence (2.38) follows directly from the standard arguments (see [44] for instance) with the help of entropy inequality (2.13). The proof of Proposition 5.2 is completed.   5.3. Finite time blow-up. In this subsection, we shall prove Theorem 2.4 about the finite time blow-up phenomena as an immediate consequence of Theorem 2.3, Propositions 5.1 and Proposition 5.2. Proof of Theorem 2.4. We will prove (2.42) only. The proof of (2.43) is similar. If (2.42) fails, then there exists a fixed constant η > 0, such that  T1 +η u x  L ∞ ds < ∞. (5.27) T1

442

H.-L. Li, J. Li, Z. Xin

For any (x, t) ∈  × (T1 , T1 + η], the particle path x(s) = X (s; t, x) through (x, t) is given by
∂ ∂s X (s; t, x) = u(X (s; t, x), s), T1 ≤ s < t ≤ T1 + η, (5.28) X (t; t, x) = x, T1 ≤ t ≤ T1 + η, x ∈ , which is well-defined due to (5.27) and (2.36). Consequently, one obtains via a standard argument from the transport equation (2.1) !  t u y (y, s)| y=X (s;t,x) ds (5.29) ρ(x, t) = ρ(X (T1 ; t, x), T1 ) exp − T1

for any (x, t) ∈  × (T1 , T1 + η]. On the other hand, it follows from (5.27) and (5.28) that for any x ∈  there exists a trajectory x = x(t) ∈  for t ∈ [T1 , T1 + η] so that X (T1 ; t, x(t)) = x. In particular, there exists a trajectory x = x1 (t) ∈  for t ∈ [T1 , T1 + η] so that X (T1 ; t, x(t)) = x1 with (x1 , T1 ) determined by (2.40), namely, ρ(x1 , T1 ) = 0. Thus, due to (5.29), we deduce from (5.27) that ρ(x1 (t), t) ≡ 0

for all

t ∈ (T1 , T1 + η],

which contradicts (2.40). Thus, the blowup phenomena (2.42) happens. The proof of Theorem 2.4 is complete.   Acknowledgements. The authors would like to thank the referee for informing them of the recently; published references [4,5,39] and the helpful comments on the manuscript. They also would like to thank Prof. Didier Bresch for his interest in this work and helpful discussions about the BD entropy estimates he and his collaborators introduced. The main part of this research was done when H.Li and J. Li were visiting the Institute of Mathematical Sciences (IMS) of The Chinese University of Hong Kong. Financial support from the IMS and the hospitality of the staff at the IMS are appreciated greatly. The research of H. Li is partially supported by Beijing Nova Program, NNSFC No.10431060, the NCET support of the Ministry of Education of China, the Institute of Mathematics and Interdisciplinary Science at CNU, the Huo Ying Dong Foundation 111003, and the Zheng Ge Ru Foundation. The research of J. Li is partially supported by the JSPS Research Fellowship for foreign researchers, NNSFC No.10601059. The research of J. Li and Xin are partially supported by Hong Kong RGC Earmarked Research Grants CUHK4028/04 and 4040/02, and the Zheng Ge Ru Foundation.

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