Mathematical Physics, Analysis and Geometry - Volume 13

Math Phys Anal Geom (2010) 13:1–18 DOI 10.1007/s11040-009-9063-1

Griffith–Kelly–Sherman Correlation Inequalities for Generalized Potts Model Nasir Ganikhodjaev · Fatimah Abdul Razak

Received: 22 April 2009 / Accepted: 8 July 2009 / Published online: 17 July 2009 © Springer Science + Business Media B.V. 2009

Abstract Inspired by the work of D.G.Kelly and S.Sherman on general Griffiths inequalities on correlations in Ising ferromagnets, we formulate and prove Griffith–Kelly–Sherman-type inequalities for the ferromagnetic Potts model with a general number q of local states. We take as local state space for the q-state Potts model the set F c = {−l, −l+1, · · · , l−1, l},where l = q−1 . The 2 important properties of F c for what follows are that |F c | = q and F c = −F c . Keywords Correlation inequalities · Potts model · Griffith–Kelly–Sherman inequalities · Gibbs measure Mathematics Subject Classifications (2000) 82B20 · 82B26 1 Introduction Statistical physics seeks to explain the macroscopic behavior of matter on the basis of its microscopic structure. This includes the analysis of simplified mathematical models [3]. The Potts model [12] was introduced as a generalization of the Ising model to more than two components (spins). Ising model considered

N. Ganikhodjaev (B) · F. A. Razak Department of Computational and Theoretical Sciences, Faculty of Science, IIUM, 25200 Kuantan, Malaysia e-mail: [email protected] F. A. Razak e-mail: [email protected] N. Ganikhodjaev Institute of Mathematics and Information Technology, 100125, Tashkent, Uzbekistan

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N. Ganikhodjaev, F.A. Razak

only up and down spins [8] whereas Potts model incorporates more possibilities of spins and their interactions. The Potts model describes an easily defined class of statistical mechanics models. At the same time, its rich structure is surprisingly capable of illustrating almost every conceivable nuance of the subject. The Potts model encompasses a number of problems in statistical physics (see, e.g. [15]). Correlation inequalities play an important role in many areas of statistical mechanics. In addition to describing microscopic structure they also provide information about macroscopic properties: for ferromagnetic spin systems they give monotonicity of the critical temperature, inequalities for critical exponents, etc [1]. In the recent work of N.Macris [9] it was shown that a correlation inequality of statistical mechanics can be applied to linear lowdensity parity-check codes. In this paper we prove Griffith–Kelly–Sherman (GKS) inequalities [8] for the generalized (likely a generalization of the classical Ising model in [8]) Potts model. At present there are a lot papers and books where the authors proved correlation inequalities for different models (see for example [1–11]) and it is possible that our results follow from some of them . To the best knowledge of the authors, formulated here Griffith–Kelly–Sherman inequalities for Potts model are new and proof of these inequalities one can consider as new alternate combinatoric proof. Finally note that after putting our manuscript to arxiv [17], we received from Prof.G.R.Grimmett letter [7], where he derived our inequalities using the FKG inequality for the random-cluster representation of the Potts model.

2 Potts Model with Long-Range Binary Interaction Let N denote the index set {1, 2, · · · , n}, consider the space of all spin configurations (σ1 , σ2 , .., σn ) where each σi is allowed the values from 1 to q (q  2). A general configuration is denoted by γ and (σi )γ is the number of values (1, · · · , q) which appears as the ith spin (component) in γ . Let  be the set of all possible configurations. For each pair (i, j) of distinct indices in N the extended real number Jij = J ji  0

(1)

is given. The requirement J ji  0 is that the system be ferromagnetic. The Hamiltonian of the one-dimensional Potts model is the real valued function on configurations, whose value at the configuration γ is  Jijδ(σi )γ (σ j )γ (2) Hγ = − 1i< jn

where δ is the Kronecker’s delta defined as  1 if (σi )γ = (σ j)γ , δ(σi )γ (σ j )γ = 0 otherwise.

Griffith–Kelly–Sherman Correlation Inequalities for Generalized Potts Model

3

The Potts model with long-range interaction was considered by Fortuin and Kasteleyn [2], where the model is defined on a lattice whose sites coincide the node points of the network and whose edges coincide the resistors. For q = 2 the model equivalent to the Ising model [5]. The Gibbs probability P on the space of configurations  is defined by P(γ ) = Z −1 exp (−β Hγ ), where Z =



exp (−β Hγ )

(3)

(4)

γ

and β = (kT)−1 > 0,

(5)

where k is the Boltzman’s constant and T is the absolute temperature. For brevity, β will be assumed to be 1 for the rest of the paper which gives the probability as P(γ ) = Z −1 exp (−Hγ ).

(6)

The expected value of a random variable X on this probability space (, P) is called its thermal average and is denoted by angular brackets:  X(γ )P(γ ). (7) X = E(X) = γ

3 Centered Random Variables Let σi denote the random variable whose value at γ is (σi )γ , that is, it’s range is the following set F = {1, 2, · · · , q}, then  (σi )γ P(γ ). (8) σi  = γ

We introduce centered random variable σi whose values are derived from σi such as σi = σi − σi .

(9)

Proposition 1 For any given q and arbitrary i ∈ N, range F c of the centered random variables σi is the following set: F c = {−l, −(l − 1), · · · , l − 1, l}; where l =

q−1 , 2

that is, l is an integer or half-integer.

Proof Let ( j)

Ai = {γ ∈  : (σi )γ = j},

(10)

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N. Ganikhodjaev, F.A. Razak

where i ∈ N and j ∈ F. So that, any σi  can be written as       (q) σi  = 1 · P Ai(1) + 2 · P Ai(2) + · · · + q · P Ai . Definition 1 For arbitrary permutation π ∈ Sq let us define transformation Tπ :  →  by the following way: for any γ = (σ1 , σ2 , · · · , σn ) assume (Tπ )γ = {π(σ1 ), π(σ2 ), · · · , π(σn )}. Remark 1 For any transformation   Tπ defined  above, P((Tπ )γ ) = P(γ ) for (π( j)) ( j) arbitrary γ ∈ , that is P Ai = P Ai for any permutation π ∈ Sq and any j ∈ F (also for j ∈ F c ). The remark above follows from the fact that Kronecker’s delta only takes into account the similarity of spins. Since Tπ is a one-to-one transformation, it is also measure preserving. It follows that for any i ∈ N       (q) P Ai(1) = P Ai(2) = · · · = P Ai and since P is a probabilistic measure, then for any i ∈ N and j ∈ F,   ( j) P Ai = 1/q. Therefore we have σi  = (1 + 2 + · · · + q)/q = (q + 1)/2,

(11)

consequently enabling us to find σγ for any q values of spins by rewriting (9) as σi = σi − (q + 1)/2,

(12)

which implies that F c = F − (q + 1)/2 hence the statements of Proposition 1 follows.   Taking into account that changing the value of the spins from F → F c does not affect the Hamiltonian as well as the Gibbs probability and also that F c = −F c , then for any i ∈ N, σi  = 1/q

 j∈F c

Thus for all k σk  = 0 and var(σk ) =

q2 −1 . 2

j = 0.

(13)

Griffith–Kelly–Sherman Correlation Inequalities for Generalized Potts Model

5

Griffiths [5] proved in Ising ferromagnets, i.e., q = 2, the following sets of inequalities: σk σm   0 for all k, m ∈ N.

(14)

σk σm σ p σr  − σk σm σ p σr   0 for all k, m, p, r ∈ N.

(15)

(Note that k, m, p, r need not be distinct.) The Griffiths inequalities hold for Potts model (2) with q > 2 also. They will appear as a consequence of the extension below.

4 Generalized Potts Model Let us consider the following generalization of Potts model. In addition to the binary interactions, assume that there can also be multisite interactions as well as external fields. Interest in systems with multisite interactions has been stimulated by the seminal work of [14]. Such systems display varied critical behaviour, and proliferate under the action of the real space renormalization group [13]. Let N denote as above the index set {1, 2, · · · , n} and the spin variable σi at . a lattice point i takes the values F c = {−l, −l + 1, · · · , l − 1, l},where l = q−1 2 For each A = {i1 , · · · , ik } ⊂ N where k  2, let the real number J A  0 be given, and define the Hamiltonian by   Hγ = − J A δ(σ A )γ − Ji δ(σi )γ ,−l (16) i

A⊂N

where σA =



σi

(σ ∅ ≡ 1)

i∈A

and the generalized Kronecker’s delta δ(σ A )γ is  1 if (σi1 )γ = · · · = (σik )γ δ(σ A )γ = 0 otherwise. The Hamiltonian (16) generalize the following Hamiltonian [15]    δσi ,0 − K δσi σ j − K3 δσi σ j σk − · · · , H = −L i

i, j

(17)

i, j,k

where σi = 0, 1, · · · , q − 1 specifies the spin states at the ith site and Kn , n  3, is the strength of the n-site interactions, and L is an external field applied to the spin state 0. Consideration of Potts models with multisite interactions has proved to be fruitful in many fields of physics, ranging from the determination of phase diagrams in metallic alloys and exhibition of new types of phase transition, to site percolation [16]. Next model has the same local state space as above.

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N. Ganikhodjaev, F.A. Razak

The ∇ 2 − model Let L be a simple cubic lattice. The Hamiltonian of the Blume-Capel model is    (sa − sb )2 − g sa2 − h sa (18) H = 1/2 a,b 

a

a

where the spin variable sa at a lattice point a takes the values 0, ±1, · · · , ±l, l is an integer or half-integer, and the first sum is over pairs of nearest-neighbour points of the lattice. For l = 1/2 the model is equivalent to the Ising model. For l > 1/2 this model exhibits much more interesting low-temperature behaviour than the Ising model [13]. However it is easy to see that for l > 1/2 the Blume-Capel model is not equivalent to a generalized Potts model. Abelian ferromagnetic models Abelian model is defined as classical lattice gas model where a set of particle types forms a finite abelian group [10] and the finite volume Hamiltonian has following form:  ˆ B; ˆ J( B) ˆ  0, J( B) (19) H = − ˆ χˆ  B∈

where the summation is over all characters of the product group of particle configurations χ in a finite volume . Defined above generalized ferromagnetic Potts model provides an example of abelian model with cyclic group Z q , where multisite interaction is given by δ(X(a1 )X(a2 ) · · · X(am )) =

q−1 m−1 

1 qm−1 ×

q−1 

exp{(2πik/q)[X(a j) − X(a j+1 )]} ×

j=1 k=0

exp{(2πik/q)[X(am ) − X(a1 )]},

k=0

where m  2. By discussing the arbitrary finite abelian group case one can enlarge the family of multisite ferromagnetic models [10].

5 First Griffith–Kelly–Sherman Inequality Below we consider following Hamiltonian  Hγ = − J A δ(σ A )γ

(20)

A⊂N

where the sum is taken over all A with |A|  2. Let R be a list of indices from N,where R may contain repeated indices. Then for any R define  σi (σ ∅ ≡ 1). σR = i∈R

Griffith–Kelly–Sherman Correlation Inequalities for Generalized Potts Model

7

Let R ⊂ N be the set of all elements in R. The difference between R and R is that R may contain repeated indices while R may not since it is a set. When there are no repeated indices in R we have R = R. Remark 2 In the case of Ising model and also if l = 1 we can consider only subset of N and it is not necessary to consider list of indices since corresponding random variables take only values 0, ±1. Let x A = exp(J A )  1. Define Z γ = exp(−Hγ ) =



xA

(21)

A⊂N

which enables us to express the Gibbsprobability on the space of configurations  as P(γ ) = Z γ /Z , where Z = γ Z γ . Thus the expected value of any (σ R )γ is given by    R  R (σ )γ P(γ ) = Z −1 σ = σ R γ Zγ . γ

(22)

γ

Theorem 1 In probability space (, P) defined by (20)–(22), we have σ R   0 for any list R of indices from N. Proof Notice that since we allow repeated indices in R, then R is comprised of odd and even groups of repeated indices. For example when R = [1, 2, 3, 3, 4, 4, 4], it has three odd groups of indices which are [1], [2] and [4,4,4]. It also has one even group of indices [3,3]. Define odd groups in R as θi when i ∈ R is repeated an odd number of times. Similarly, define even groups in R as i when i ∈ R is repeated an even number of times. So now, we can say that R = [1, 2, 3, 3, 4, 4, 4] is comprised of θ1 , θ2 , 3 and θ4 thus R = [θ1 , θ2 , 3 , θ4 ]. Define R+ ⊂  to be a set of configurations where multiplication of all spins in R+ gives a positive value and similarly R− ⊂  is a set of configurations where multiplication of all spins in R− gives a negative value. Also let R0 ⊂  be the set where the multiplication of spins are zero so that we have  = R+ ∪ R− ∪ R0 . Note that we only have R0 when q = 2l + 1 is odd and it does not appear in σ R  (since σ R = 0), so we only need to consider R+ and R− . When R is only comprised of even groups of repeated indices then (σ R )γ = i − R i∈R (σ )γ  0 thus R = ∅ and σ   0 since P(γ )  0. Otherwise, consider cases where at least one odd group of repeated indices, θi , exist in R (which is true for all instances when |R| is odd). In these cases, for each γ ∈ R+ there exists a corresponding γ  ∈ R− due to symmetrical properties of centered value variables in F c . Any element of either subsets can be transformed into a corresponding element of the other subset by choosing

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N. Ganikhodjaev, F.A. Razak

any i ∈ R and multiplying σi with −1 or simply multiplying any one of the θi , i ∈ R with −1. Consequently, for these cases, if q is even then |R+ | = |R− | = ||/2 = qn /2 and if q is odd, then |R+ | = |R− | = | − R0 |/2, so that      R σ = σ R γ P(γ ) + σ R γ  P(γ  ). γ ∈R+

γ  ∈R−



 Since σ R γ  = − σ R γ , we can write      R σ = σ R γ [P(γ ) − P(γ  )] = Z −1 σ R γ [Z γ − Z γ  ]. γ ∈R+

γ ∈R+

When |R| is odd, the one-to-one correspondence between R+ and R−

R can also be obtained simply by multiplying σ γ with −1. By this way the difference of spins are preserved hence the Gibbs measure is preserved, consequently Z γ = Z γ  for any γ . In other words for |R| odd, Tπ : R+ → R− , where Tπ is stated in Definition 1. Hence, since σ R γ = − σ R γ  and Z γ = 

  Z γ  for all odd |R|, σ R = Z −1 γ ∈R+ σ R γ [Z γ − Z γ  ] = 0 and Theorem 1 stands. Let A ⊂  be a set of configurations, and then define ζ (R, A) =



σR

γ ∈A

 γ

Zγ .

For brevity if A =  let ζ (R) = ζ (R, ). Thus we can write ζ (R) =



σR

γ ∈

 γ

 Zγ = Z · σ R ,

and when R+ = {γ ∈  : σ R > 0} and R− = {γ ∈  : σ R < 0}, we have ζ (R) = ζ (R, R+ ) + ζ (R, R− ) =

 γ ∈R+

=



(σ R )γ Z γ +



(σ R )γ  Z γ 

γ  ∈R−

(σ R )γ [Z γ − Z γ  ].

γ ∈R+

Let B ⊂ N, where B(1) = {γ ∈  : δ(σ B )γ = 1} and B(0) = {γ ∈  : δ(σ B )γ = 0}, then since  = B(1) ∪ B(0) , ζ (R) =

    σ R γ Zγ + σ R γ Z γ = ζ (R, B(1) ) + ζ (R, B(0) ).

γ ∈B(1)

γ ∈B(0)

Griffith–Kelly–Sherman Correlation Inequalities for Generalized Potts Model

9

Similarly, let R+ B(1) = {γ ∈  : σ R > 0 and δ(σ B )γ = 1}, R− B(1) = {γ ∈  : σ R < 0 and δ(σ B )γ = 1}, R+ B(0) = {γ ∈  : σ R > 0 and δ(σ B )γ = 0} and R− B(0) = {γ ∈  : σ R < 0 and δ(σ B )γ = 0} for any B ⊂ N, we can write 

ζ (R, ) =

(σ R )γ Z γ +

γ ∈R+ B(1)

+





(σ R )γ  Z γ  +

γ  ∈R− B(1)

(σ R )γ Z γ +

γ ∈R+ B(0)



(σ R )γ  Z γ 

γ  ∈R− B(0)

= ζ (R, R+ B(1) ) + ζ (R, R− B(1) ) + ζ (R, R+ B(0) ) + ζ (R, R− B(0) ). For cases where |R| is even and (σ R )γ = i∈R (σ i )γ (there exists θi , i ∈ R), we seek to prove by induction on s, the number of J A > 0. To prove σ R   0 we only need to prove that ζ (R)  0 since ζ (R) = Z · σ R  and Z > 0. For s = 0 we have Z γ = Z γ  = 1, thus     ζ (R) = σ R γ [Z γ − Z γ  ] = σ R γ [1 − 1] = 0 γ ∈R+

γ ∈R+

and Theorem 1 is satisfied. Note that P(γ ) =  1/Z for all γ hence we have uniform measure which renders σ R  = 0 since γ ∈R (σ R )γ = 0 due to centered value properties. Let ζs (R) be ζ (R) for any s number of nonzero existing interactions. Assume ζs (R)  0 for all s  k such that for any s we add J Bs > 0. Then for s = k + 1 let J Bk+1 > 0 be the additional interaction. Since we know that x Bk+1 will only (1) multiply all the terms in B(1) k+1 where Bk+1 = {γ ∈  : δ(σ Bk+1 )γ = 1}, the terms

(0) in B(0) k+1 (Bk+1 = {γ ∈  : δ(σ Bk+1 )γ = 0}) remains the same as it was in s = k. Thus we have





(0)  (1)  (0)  ζk+1 (R) = ζk+1 R, B(1) k+1 + ζk+1 R, Bk+1 = x Bk+1 · ζk R, Bk+1 + ζk R, Bk+1 .

By induction hypothesis we have ζk (R)  0, and if we have ζk (R, B(1) k+1 )  0 we shall be able to write (0) ζk+1 (R) = x Bk+1 · ζk (R, B(1) k+1 ) + ζk (R, Bk+1 ) (0) > ζk (R, B(1) k+1 ) + ζk (R, Bk+1 ) = ζk (R)  0.

since x Bk+1 > 1 and it is multiplied with a positive sum. Thus given R ζk (R, B(1) k+1 )  0, we have σ   0 for any n number of vertices, q number of spins and R in which any i ∈ R is also in N.   Lemma 1 Let ζsN (R) be ζ (R) where s is the number of nonzero J A , N = {1, · · · , n} is the set of n vertices and R is a list where i ∈ R implies i ∈ N. Given that ζsN (R)  0 for any n vertices and q number of spins then we have ζsN (R, B(1) )  0 where B ⊂ N and B(1) = {γ ∈  : δ(σ B )γ = 1}.

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N. Ganikhodjaev, F.A. Razak

Proof Let B = {b 1 , b 2 · · · b m } ⊂ N. For γ ∈ B(1) , the spins are always similar for all its vertices, such that σb 1 = σb 2 = · · · = σb m , thus we seek to treat B as a single vertex, say b 1 . Firstly get B ∩ A for all existing J A ’s, if B ∩ A = ∅, then x A is left as it is, but if B ∩ A = ∅ then we shall do some alterations. For A’s where B ∩ A = ∅ let C A = A − (B ∩ A). If there exist A’s with similar C A , then we seek to group it together. Let C = C A ∪ b 1 where b 1 is ∗ the first element of B then we set xC = x A for all A’s with similar C A , to represent them in a group as a single interaction. We can do this because they will always appear together in ζs (R, B(1) ). Thus if there exist similar C A ’s for different A’s, the number of existing interaction is reduced but the remaining interaction has a larger size which does not matter since J A can even be ∞. If C A = ∅ then the x∗b 1 group if comprised of all of A ⊂ B. This group will appear in every in term in ζ (R, B(1) ) due to the fact that all spins in B are similar for γ ∈ B(1) . ∗ After replacing all the terms where B ∩ A = ∅ with its corresponding xC , then we will see that we have ∗

ζkN (R, B(1) ) = x∗b 1 · ζsN (R∗ )

(23)

where s  k, N ∗ = (N − B) ∪ b 1 and R∗ is obtain by simply replacing any i ∈ R which is also in B with b 1 (if none of i ∈ R is in B then R∗ = R). ∗ (σ R )γ = (σ R )γ since the spins in B are all similar, we are simply renaming the vertices. A simple example is that initially we have B = {2, 3} thus b 1 = 2 and R = [1, 2, 3, 4] in N = {1, 2, 3, 4} then we can see that the γ ∈ B(1) is exactly similar to γ ∈  for cases where R∗ = [1, 2, 2, 4] in N ∗ = {1, 2, 4} which can be modified by renaming index 4 as index 3 and then it can be obtained just like in the case where R = [1, 2, 2, 3] in N = {1, 2, 3}. ∗ We can find ζsN (R∗ ) exactly the same way we obtain ζsN (R) where N = ∗ ∗ {1, · · · , n }, n = n − |B| + 1 and R∗ is transformed accordingly to a new R where |R∗ | = |R|, the difference is only that the vertices have different position, but because any interactions are accounted for, this does not really matter since the existence and size of interactions does not depends on the ∗ vertices being neighbours or not. Now that we know x∗b 1  1, ζsN (R∗ )  0 (since ζsN (R)  0 for any n including n∗ and any R where i ∈ R implies i ∈ N), then we have ζkN (R, B(1) )  0.   Example 1 In this example we seek to illustrate Lemma 1. Let N = {1, 2, 3}, q = 3, R = [1, 3] and B = {1, 2}. The only possible interactions are J12 , J13 , J23 , J123 . Assume only x12 = 1 hence s = 3. We also have B(1) = {(−1, −1, −1), (−1, −1, 1), (1, 1, −1), (1, 1, 1)} and ζ (R, B(1) ) = 2(x13 x23 x123 − 1). Since C13 = {3}, C23 = {3} and C123 = {3}, assign x∗13 = x13 x23 x123 . Replace it in ζ (R, B(1) ), we have ζ (R, B(1) ) = 2(x∗13 − 1).

Griffith–Kelly–Sherman Correlation Inequalities for Generalized Potts Model

11

Consequently we have only one existing interaction, x∗13 . Note that if x12  1 than it will be ζ (R, B(1) ) = 2x∗1 (x∗13 − 1) where x∗1 = x12 . Now let N ∗ = (N − B) ∪ b 1 = {1, 3} thus the only possible interaction here is x∗13 . R∗ = R since the only similar elements of R and B is b 1 . The set of possible configuration are {(−1, −1), (−1, 0), (−1, 1), (1, −1), (1, 0), (1, 1)} thus we have R∗+ = {(−1, −1), (1, 1)} and R∗− = {(−1, 1), (−1, 1)}, ζ (R∗ ) = 2(x∗13 − 1) thus we see that equation (27) is verified. Note that N ∗ = {1, 3} can be treated like N = {1, 2} with n = 2 if we change vertex 3 to vertex 2 (thus R∗ = [1, 3] becomes R = [1, 2]). For N = {1, 2} we will get ζ (R) = 2(x12 − 1) where x12 = x∗13 . Since we can just replace B in Lemma 1 by Bk+1 due to the fact that both are subsets of N, thus by Lemma 1 we have ζk (R, B(1) k+1 )  0 when ζk (R)  0, hence we have proven σ R   0 for any n number of vertices, q number of spins and R where i ∈ R implies i ∈ N.

6 Second Griffith–Kelly–Sherman Inequality Let R and S be two lists of indices from N and   σ RS = σ R σ S = σi σ j, i∈R

j∈S

where the list RS is the joint of the lists R and S. Theorem 2 In probability space (, P) defined by (20)–(22), we have σ R σ S  − σ R σ S   0 for arbitrary lists R, S of indices from N. Note that, when either |R| or |S| is odd then |RS| is also odd thus we have  RS  R  S σ − σ σ =0−0=0 which fulfills Theorem 2. If both |R| and |S| are odd then |RS| is even and we have  RS  R  S  RS  σ − σ σ = σ − 0 = σ RS  0 which also fulfills Theorem 2. To complete prove for Theorem 2 we only need to prove for cases where |R| and |S| is even thus |RS| is also even. We seek to prove by induction on s,

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N. Ganikhodjaev, F.A. Razak

   the number of J A > 0. To prove σ RS − σ R σ S  0 we only need to prove that Z · ζ (RS) − ζ (R) · ζ (S)  0 since Z > 0 and  R S  R  S  RS  R  S σ σ − σ σ = σ − σ σ =

Z · ζ (RS) − ζ (R) · ζ (S) ζ (RS) ζ (R) ζ (S) − · = Z Z Z Z2

due to the fact that ζ (R) = Z · σ R  . For s = 0 we have Z γ = Z γ  = 1 for any γ , thus as long as there is correspondence between R+ and R− we have σ R  = 0. But we only have the correspondence when there is at least one odd group of repeated indices, θi in R. This is also true for S and RS. If there exist an odd group of indices in R, S or both of them ,then we have σ RS  − σ R σ S  = σ RS  − 0 = σ RS   0. When R and S are comprised of only even group of indices we do not have the correspondence. Let iR be i given R and θiR be θi given R, for R any R where i ∈ R implies that i ∈ N. In such cases (σ R )γ = i∈R (σ i )γ  S RS 0, (σ S )γ = i∈S (σ i )γ  0 and (σ RS )γ = i∈RS (σ i )γ  0. Thus σ R   0, σ S   0 and σ RS  ≥ 0. Examples are R = [1, 1, 2, 2] and S = [3, 3] thus RS = [1, 1, 2, 2, 3, 3]. Since Z γ = 1 we have      R  R  ζ (R) = σR γ = σ i = q|N−R | · j|i | γ

γ

γ

i∈R

i∈R j∈F c

and similarly for S      S  S  σS γ = σ i ζ (S) = = q|N−S | · j|i | . γ

γ

γ

i∈S

i∈S j∈F c

Also due to the fact that Z γ = 1 we have Z = || = q|N| = qn and 

   RS    RS   = q|N−RS | · j|i | . ζ (RS) = σ RS γ = σ i γ

γ

γ

i∈RS

i∈RS j∈F c

Consequently we have 

Z · ζ (RS) − ζ (R) · ζ (S) = qn q|N−RS | ·



j|i

RS

i∈RS j∈F c

⎡

− ⎣q|N−R | · 

 i∈R

|

−

⎤⎡ |iR |

j

⎦ ⎣q|N−S | · 

j∈F c

 i∈S

⎤ |iS |

j

⎦.

j∈F c

Let R and S be the subsets of N corresponding to lists R and S respectively. If R ∩ S = ∅ then we have   RS  R  S    qn q|N−RS | = q|N−R | q|N−S | and j|i | = j|i | · j|i | i∈RS j∈F c

i∈R j∈F c

i∈S j∈F c

Griffith–Kelly–Sherman Correlation Inequalities for Generalized Potts Model

thus we can write 

Z · ζ (RS) = qn q|N−RS | ·



j|i

RS

|



i∈RS j∈F c





= q|N−R | ·



= q|N−R | q|N−S | · 

R

i∈R j∈F c

j|i | · R

i∈R j∈F c



j|i | · q|N−S | ·



13



j|i | S

i∈S j∈F c

j|i | = ζ (R) · ζ (S) S

i∈S j∈F c

so that Z · ζ (RS) − ζ (R) · ζ (S) = 0 thus σ RS  − σ R σ S  = 0.      But if R ∩ S = ∅ then qn q|N−RS | = q|N−R | q|N−S | q|R ∩S | and only for i ∈ (R ∩ S ) do we have ⎤ ⎡ ⎤ ⎡    RS   R  S j|i | = ⎣ j|i | ⎦ · ⎣ j|i | ⎦ i∈RS −(R ∩S ) j∈F c

i∈R −(R ∩S ) j∈F c

i∈S −(R ∩S ) j∈F c

and we can write Z · ζ (RS) − ζ (R) · ζ (S) 

= qn q|N−RS | ·

 i∈RS





⎡ |iRS |

j

− ⎣q|N−R | · 

j∈F c 



= q|N−R | q|N−S | q|R ∩S | ·

|iRS |

j

=q





j

j



⎦ ⎣q|N−S | · 



−q|N−R | q|N−S | ·



j|i | · R

i∈R j∈F c



⎣q|R ∩S |



j|i

RS

|



−



i∈S j∈F c

−

j

i∈(R ∩S )

So now we only need to prove that   RS    j|i | − q|R ∩S | · i∈(R ∩S )





|iR |



⎤ |iS |

j



R

i∈(R ∩S )



j|i |  0. S

i∈(R ∩S )

We can do that by proving that for each i ∈ (R ∩ S ) we have  RS  R  S q j|i | − j|i | j|i | j∈F c

since 



q|R ∩S | ·





j∈F c

j|i

i∈(R ∩S )

=

 i∈(R ∩S )

⎡ ⎣q

RS

|

j∈F c



−



i∈(R ∩S )

 j∈F c

⎤

RS j|i | ⎦ −

⎦

i∈(R ∩S )

j|i | ·

 i∈(R ∩S )

j|i | · R

⎡ ⎣





i∈(R ∩S )

 j∈F c

j|i

R

|

 j∈F c

j



i∈(R ∩S )

i∈R and i∈(R ∩S )

⎤ |iS |

j|i | S

⎤ j|i | ⎦ . S

⎦

j∈F c

i∈S



⎡

|iRS |

⎤⎡ |iR |

j∈F c

i∈R

 i∈RS j∈F c

|N−R |+|N−S |



j|i | S

14

N. Ganikhodjaev, F.A. Razak

  RS S R  Let ξq = q j∈F c j|i | − j∈F c j|i | j∈F c j|i | for any i ∈ (R ∩ S ) and q. Note that we have |iRS | = |iR + iS | = |iR | + |iS | for any R, S and RS. When q = 2, F c = {−1/2, 1/2} and   RS RS ξ2 = 2 · (−1/2)|i | + (1/2)|i | −     S S R R − (−1/2)|i | + (1/2)|i | · (−1/2)|i | + (1/2)|i | = 2 · 2(1/2)|i

RS

= 2 · 2(1/2)|i

R

|

− 2(1/2)|i | · 2(1/2)|i | S

R

+iS |

− 2(1/2)|i | · 2(1/2)|i | S

R

= 2 · 2(1/2)|i | (1/2)|i | − 2(1/2)|i | · 2(1/2)|i | = 0. S

R

S

R

If q = 3 then F c = {−1, 0, 1} and   RS RS RS ξ3 = 3 · (−1)|i | + 0|i | + 1|i | −     S S S R R R − (−1)|i | + 0|i | + 1|i | · (−1)|i | + 0|i | + 1|i |  RS   R  S = 3 · 2 1|i | − 2 1|i | · 2 1|i | = 3 · 2 − 2 · 2 = 2 > 0. Now to find a general formula for any ξq , by obtaining ξq+2 − ξq , using the fact that ⎛ ξq+2 = (q + 2) ⎝ ⎛ −⎝



j|i

RS

|

 +2

j∈F c



|iR |

j

j∈F c



q+1 +2 2

ξq = q

and

q+1 2

|iR |



|iRS |

⎞⎛ ⎠⎝

⎞ ⎠−



|iS |

j

j∈F c

j|i

RS

|

−

j∈F c



j|i

R

j∈F c

|



q+1 +2 2



|iS |

⎞ ⎠

j|i | . S

j∈F c

As a result we have ξq+2 − ξq = 2

 j∈F c

 −2

|iRS |

j



q+1 + 2(q + 2) 2

q+1 2

|iS |  j∈F c

j|i | − 4 R

|iRS |





q+1 −2 2

q+1 2

|iR | 

|iR | 

q+1 2

j∈F c

|iS |

.

j|i | − S

Griffith–Kelly–Sherman Correlation Inequalities for Generalized Potts Model

15

We know that |iRS | = |iR + iS |, then ξq+2 − ξq = 2





q+1 + 2(q + 2 − 2) 2

|iRS |

j

j∈F c



q+1 −2 2 ⎡ = 2 ⎣q



q+1 2



q+1 − 2 =2

 j∈F c

|iR | 



|iS |

j

j∈F c

|iRS |



q+1 −2 2

−

|iS | 

j|i

R

|

j∈F c

|iRS |   R q + 1 |i |  | S | − j i − 2 c j∈F

|iS | 

j|i | + R

j∈F c

q+1 2

|iR |



⎤

RS j|i | ⎦

j∈F c |iR |

 

−j

q+1 2

|iS |

 |iS |

−j

0

thus [( q+1 )|i | − j|i | ] > 0 and [( q+1 )|i | − j|i | ] > 0. since the largest j is q−1 2 2 2 We have established that ξ2 = 0, ξ3 = 2 and      S   q + 1 |iR | q + 1 |i | |iR | |iS | −j −j ξq+2 = ξq + 2 2 2 j∈F c R

R

S

S

so that ξq  0 for any q  2. Therefore when s = 0 we have Z · ζ (RS) − ζ (S) · ζ (S)  0 thus σ RS  − σ R σ S   0 for any R and S. Let Z · ζ (RS) − ζ (S) · ζ (S)  0 for any s  k, number of nonzero J A ’s. Consider the case when s = k + 1, and the  added interaction isJ Bk+1 . Let x = x Bk+1 for brevity. Then define Z (1) = γ ∈B(1) Z γ and Z (0) = γ ∈B(0) Z γ k+1 k+1 so that Z = Z (1) · x + Z (0) . We also have these equations:



(0)  ζ (RS) = ζ RS, B(1) k+1 · x + ζ RS, Bk+1 .



(0)  ζ (R) = ζ R, B(1) k+1 · x + ζ R, Bk+1 . 



(0)  ζ (S) = ζ S, B(1) k+1 · x + ζ S, Bk+1 . Due to the fact that

  



(0)  Z · ζ (RS) − ζ (R) · ζ (S) = Z (1) · x + Z (0) ζ RS, B(1) − · x + ζ RS, B k+1 k+1 

 

(0)  − ζ R, B(1) × k+1 · x + ζ R, Bk+1  



(0)  , × ζ S, B(1) k+1 · x + ζ S, Bk+1

16

N. Ganikhodjaev, F.A. Razak

we can write Z · ζ (RS) − ζ (R) · ζ (S) = U x2 + Vx + W where





(1)  (1)  U = Z (1) · ζ RS, B(1) k+1 − ζ R, Bk+1 · ζ S, Bk+1 ,



 (0) V = Z (1) · ζ RS, B(0) · ζ RS, B(1) k+1 + Z k+1 −





(1)  (1)  (0)  − ζ R, B(0) k+1 · ζ S, Bk+1 − ζ R, Bk+1 · ζ S, Bk+1 



(0)  (0)  and W = Z (0) · ζ RS, B(0) k+1 − ζ R, Bk+1 · ζ S, Bk+1 . (1) (1) ∗ ∗ ∗ ∗ Since Z (1) · ζ (RS, B(1) k+1 ) − ζ (R, Bk+1 )·ζ (S, Bk+1 ) = xb 1 [Z ·ζ (RS )−ζ (R )· ∗ ∗ ∗ ∗ ∗ ζ (S )] by Lemma 1 and Z · ζ (RS ) − ζ (R ) · ζ (S )  0 by induction hypothesis, we have U  0 so that U x2 + Vx + W is a quadratic function with a minimum value. By definition we know that

  



(1)  (1)  − ζ R, B · ζ S, B + 2U + V = 2 Z (1) · ζ RS, B(1) k+1 k+1 k+1



 (0) +Z (1) · ζ RS, B(0) · ζ RS, B(1) k+1 + Z k+1 − 





(1)  (1)  (0)  −ζ R, B(0) k+1 · ζ S, Bk+1 − ζ R, Bk+1 · ζ S, Bk+1 



(1)  = Z (1) · ζ (RS) + Z · ζ RS, B(1) k+1 − ζ (R) · ζ S, Bk+1 −

 −ζ R, B(1) k+1 · ζ (S). It is easy to verify that since we have U  0 and U + V + W  0 (by induction hypothesis) we also have 2U + V  0. By differentiating U x2 + Vx + W where x > 1 , we obtain d(U x2 + Vx + W) = 2U x + V dx Given that we have 2U + V  0 , we can conclude that U x2 + Vx + W is an increasing function for any x  1. When x = 1, we have U x2 + Vx + W = U + V + W  0 , so now we know that U x2 + Vx + W is always positive. Hence we have Z · ζ (RS) − ζ (R) · ζ (S)  0 for any x  1. Consequently σ R σ S  − σ R σ S   0 for any R and S hence Theorem 2 is proven.

7 Discussions The GKS inequalities in Ising ferromagnets [8] express the positivity of certain correlations or equivalently the monotonicity of first and second derivatives of the free energy. The first GKS inequality states that 1 ∂ ln Z N = σ R   0 β ∂ JR

(24)

Griffith–Kelly–Sherman Correlation Inequalities for Generalized Potts Model

17

for any R ⊂ N. Thus the pressure ln Z N is monotone increasing with respect to each J A , hence there exist thermodynamic limit of the pressure, and bounds on the surface pressure. The second GKS inequality states that 1 ∂ 2 ln Z = σ R σ S  − σ R σ S   0 β 2 ∂ J R∂ JS

(25)

for any R, S ⊂ N. The GKS inequality for random spin systems have been derived recently in [1] and [11]. Let us consider the generalized Potts model  Hγ = − J A δ(σ A )γ (26) A⊂N

where the sum is taken over all A with |A|  2 and let J = {J A , A ⊂ N} be a family of all multi-site interactions. Then the partition function (19) is  Z N (l, J) = exp(−β Hγ ). (27) γ

Assume also that the model (24) is a finite range interactions, that is, J A = 0 for |A| > r for some fixed positive integer r  2. Then one can prove that 1 ∂ ln Z  Cσ R  β ∂ JR

(28)

for any R ⊂ N, where C = C(r, l). Here the constant C independent of size N. For example, if r = 2 and l = 3/2, then C = 4/5. The inequality (28) yields various results usually obtained for ferromagnetic systems [1, 5, 8, 9, 11]. From first GKS inequality follows that for any J A ∂ ln Z N (l, J)  0. ∂ JA Thus the pressure ln Z N (l, J) is monotone increasing with respect to each J A , hence there exist thermodynamic limit of the pressure, and bounds on the surface pressure. In the case of generalized ferromagnetic Potts model with l  1 probably one cannot expect validity of equality or inequality type (25) connected with second GKS inequality. Finally some remarks on GKS inequalities for ∇ 2 − and abelian ferromagnetic models. In our proof we essentially use the invariance of Gibbs state with respect to transformations Tπ introduced in Definition 1. It is easy to see that a transformation Tπ don’t preserve a Gibbs measure for ∇ 2 − model. Note that for this model ⎧ ⎪ ⎨positive, if h > 0 sa  = 0, if h = 0 ⎪ ⎩ negative, if h < 0

18

N. Ganikhodjaev, F.A. Razak

for any site a ∈ L. If h = 0, then for small |L| and l the GKS inequalities are implemented. For h > 0 and |L| = 2 we have s1 s2 −s1 s2  = 4(1− e−4 )e2g + 4e−0.5 (1−2e−2 )(eh + e−h )eg − e−1 (eh − e−h )2 (29) It is easy to see that for any fixed h there is g∗ such that s1 s2  − s1 s2  < 0 for g < g∗ and s1 s2  − s1 s2  > 0 for g > g∗ . Note that g∗ < h. It is verisimilar that for the ∇ 2 − model with parameters h > 0 and g one can find a number g∗ = g∗ (h) such that GKS inequalities hold for all g > g∗ . If abelian ferromagnetic model (19) satisfies following conditions: (i) J(B) = J(B−1 ) and (ii) if B2 can be obtained from B1 by a translation then J(B1 ) = J(B2 ), then extending considered above approach one can expect that GKS inequalities hold for this model. Acknowledgement The work was partially supported by the International Islamic University Malaysia short term research grant RC 04.

References 1. Contucci, P., Lebowitz, J.: Correlation Inequalities for Spin Glasses (2007). arxiv:0612.371v1 2. Fortuin, C.M., Kasteleyn, P.W.: On the random-cluster model. I. Introduction and relation to other models. Physica 57, 536–564 (1972) 3. Georgii, H.O.: Gibbs Measures and Phase Transitions. Walter de Gruyter, Berlin (1988) 4. Ginibre, J.: General formulation of Griffiths’s inequalities. Commun. Math. Phys. 16, 310–328 (1970) 5. Griffiths, R.B.: Correlations in Ising ferromagnets I,II. J. Math. Phys. 8, 478–483, 484–489 (1967) 6. Grimmett, G.R.: The Random-Cluster Model. Springer, Berlin (2006) 7. Grimmett, G.R.: Correlation Inequalities of GKS type for the Potts Model (2009). arxiv:0901.1625v1 8. Kelly, D.G., Sherman, S.: General Griffiths’ inequalities on correlations in Ising ferromagnets. J. Math. Phys. 9, 466–472 (1967) 9. Macris, N.: Griffith-Kelly-Sherman correlation inequalities: a useful tool in the theory of error correcting codes. IEEE Trans. Inf. Theory 53, 664–683 (2007) 10. Miekisz, J.: The decomposition property and equilibrium states of ferromagnetic lattice systems. Commun. Math. Phys. 109, 353–367 (1987) 11. Morita, S., Nishimori, H., Contucci, P.: Griffiths inequality for the Gaussian spin glass. J. Phys. A 37, 203–209 (2004) 12. Potts, R.B.: Some generalized order-disorder transformations. Proc. Camb. Philos. Soc. 48, 106–109 (1952) 13. Slawny, J.: Low temperature properties of classical lattice systems: phase transition and phase diagrams. In: Domb, C., Lebowitz, J.L. (eds.) Phase Transitions and Critical Phenomena, vol. 11. Academic, London (1987) 14. Wegner, F.G.: Duality in generalized Ising models and phase transitions without local order parameters. J. Math. Phys. 12, 2259–2272 (1971) 15. Wu, F.Y.: The Potts model. Rev. Mod. Phys. 54, 235–268 (1982) 16. Wu, F.Y., Eugene Stanley H.: Universality of Potts models with two- and three-site interactions. Phys. Rev. B 26, 6326–6329 (1982) 17. Ganikhodjaev, N., Razak, F.A.: Correlation Inequalities for Generalized Potts Model: General Griffiths’ Inequalities (2007). arxiv:0707.3848

Math Phys Anal Geom (2010) 13:19–28 DOI 10.1007/s11040-009-9064-0

On the Discrete Spectrum of a Spatial Quantum Waveguide with a Disc Window Hatem Najar · Samir Ben Hariz · Mounir Ben Salah

Received: 22 February 2009 / Accepted: 8 July 2009 / Published online: 24 July 2009 © Springer Science + Business Media B.V. 2009

Abstract In this study we investigate the bound states of the Hamiltonian describing a quantum particle living on three dimensional straight strip of width d. We impose the Neumann boundary condition on a disc window of radius a and Dirichlet boundary conditions on the remained part of the boundary of the strip. We prove that such system exhibits discrete eigenvalues below the essential spectrum for any a > 0. We give also a numeric estimation of the number of discrete eigenvalue as a function of da . When a tends to the infinity, the asymptotic of the eigenvalue is given. Keywords Quantum waveguide · Shrödinger operator · Bound states · Dirichlet Laplcian Mathematics Subject Classifications (2000) 81Q10 · 47B80 · 81Q15

1 Introduction The study of quantum waves on quantum waveguide has gained much interest and has been intensively studied during the last years for their important physical consequences. The main reason is that they represent an interesting physical effect with important applications in nanophysical devices, but also

H. Najar (B) · M. B. Salah Département de Mathématiques, ISMAI. Kairouan, Bd Assed Ibn Elfourat, 3100 Kairouan, Tunisia e-mail: [email protected] S. B. Hariz Département de Mathématiques, Université du Maine, Le Mans, France

20

H. Najar et al.

in flat electromagnetic waveguide. See the monograph [10] and the references therein. Exner et al. have done seminal works in this field. They obtained results in different contexts, we quote [2, 6, 8, 9]. Also in [12–14] research has been conducted in this area; the first is about the discrete case and the two others for deals with the random quantum waveguide. It should be noticed that the spectral properties essentially depends on the geometry of the waveguide, in particular, the existence of a bound states induced by curvature [3, 4, 6–8] or by coupling of straight waveguides through windows [8, 10] were shown. The waveguide with Neumann boundary condition were also investigated in several papers [5, 15, 17]. A possible next generalization are waveguides with combined Dirichlet and Neumann boundary conditions on different parts of the boundary. The presence of different boundary conditions also gives rise to nontrivial properties like the existence of bound states. The rest of the paper is organized as follows, in Section 2, we define the model and recall some known results. In Section 3, we present the main result of this note followed by a discussion. Section 4 is devoted for numerical experiments. 2 The Model The system we are going to study is given in Fig. 1. We consider a Schrödinger particle whose motion is confined to a pair of parallel plans of width d. For simplicity, we assume that they are placed at z = 0 and z = d. We shall denote this configuration space by   = R2 × [0, d]. Let γ (a) be a disc of radius a, without loss of generality we assume that the center of γ (a) is the point (0, 0, 0); γ (a) = {(x, y, 0) ∈ R3 ; x2 + y2  a2 }.

(2.1)

We set  = ∂γ (a). We consider Dirichlet boundary condition on  and Neumann boundary condition in γ (a). 2.1 The Hamiltonian Let us define the self-adjoint operator on L2 () corresponding to the particle Hamiltonian H. This is will be done by the mean of quadratic forms. Precisely, let q0 be the quadratic form  q0 ( f, g) = ∇ f · ∇gd3 x, with domain Q(q0 ) = { f ∈ H 1 (); f  = 0}, 

(2.2) where H () = { f ∈ L ()|∇ f ∈ L ()} is the standard Sobolev space and we denote by f , the trace of the function f on . It follows that q0 is a densely defined, symmetric, positive and closed quadratic form. We denote the 1

2

2

Discrete Spectrum of a Spatial Quantum Waveguide

21

Dirichlet boundary condtion

Neumann boundary condition

d

2a

Fig. 1 The waveguide with a disc window and two different boundaries conditions

unique self-adjoint operator associated to q0 by H and its domain by D(). It is the hamiltonian describing our system. From [18] (page 276), we infer that the domain D() of H is   ∂f D() = f ∈ H 1 (); − f ∈ L2 (), f  = 0, γ (a) = 0 , ∂z and H f = − f,

∀ f ∈ D().

2.2 Some Known Facts Let us start this subsection by recalling that in the particular case when a = 0, we get H 0 , the Dirichlet Laplacian, and a = +∞ we get H ∞ , the DirichletNeumann Laplacian. Since H = (−R2 ) ⊗ I ⊕ I ⊗ (−[0,d] ), on L2 (R2 ) ⊗ L2 ([0, d]),  π 2  (see [18]) we get that the spectrum of H 0 is 2d , +∞ . Consequently, we have 

 

 π 2 π 2 , +∞ ⊂ σ (H) ⊂ , +∞ . d 2d Using the property that the essential spectra is preserved under compact perturbation, we deduce that the essential spectrum of H is 

 π 2 σess (H) = , +∞ . d  π 2  π 2 An immediate consequence is the discrete spectrum lies in 2d , d .

22

H. Najar et al.

2.3 Preliminary: Cylindrical Coordinates Let us notice that the system has a cylindrical symmetry, therefore, it is natural to consider the cylindrical coordinates system (r, θ, z). Indeed, we have that L2 (, dxdydz) = L2 (]0, +∞[×[0, 2π [×[0, d], rdrdθdz), ˙,r , the scaler product in L2 (, dxdydz) = L2 (]0, +∞[×[0, 2π [× We note by ˙ [0, d], rdrdθdz) given by 

f, gr = fgrdrdθdz. ]0,+∞[×[0,2π [×[0,d]

We denote the gradient in cylindrical coordinates by ∇r . While the Laplacian operator in cylindrical coordinates is given by r,θ,z =

 1 ∂ d2 ∂ 1 ∂2 r + 2 2 + 2. r ∂r ∂r r ∂θ dz

(2.3)

Therefore, the eigenvalue equation is given by −r,θ,z f (r, θ, z) = E f (r, θ, z).

(2.4)

Since the operator is positive, we set E = k2 . The Eq. 2.4 is solved by separating variables and considering f (r, θ, z) = ϕ(r) · ψ(θ)χ(z). Plugging the last expression in Eq. 2.4 and first separate χ by putting all the z dependence in one term so that χχ can only be constant. The constant is taken as −s2 for

convenience. Second, we separate the term ψψ which has all the θ dependance. Using the fact that the problem has an axial symmetry and the solution has to be 2π periodic and single value in θ, we obtain ψψ should be a constant −n2 for n ∈ Z. Finally, we get the following equation for ϕ ϕ (r) +

  1 n2 ϕ (r) + k2 − s2 − 2 ϕ(r) = 0. r r

(2.5)

We notice that the Eq. 2.5, is the Bessel equation and its solutions could be expressed in terms of Bessel functions. More explicit solutions could be given by considering boundary conditions.

3 The Result The main result of this note is the following Theorem. Theorem 3.1 The operator H has at least one isolated eigenvalue in  π 2  π 2 for any a > 0. , d 2d

Discrete Spectrum of a Spatial Quantum Waveguide

23

Moreover for a big enough, if λ(a) is an eigenvalue of H less then have.

 π 2 1 +o 2 . λ(a) = 2d a

π2 , d2

then we

(3.6)

Proof Let us start by proving the first claim of the Theorem. To do so, we define the quadratic form Q0 , Q0 ( f, g) = ∇ f, ∇gr  =

 1 ∂r f ∂r g + 2 ∂θ f ∂θ g + ∂z f ∂z g rdrdθ dz, r

]0,+∞[×[0,2π [×[0,d]

(3.7) with domain   D0 () = f ∈ L2 (, rdrdθdz); ∇r f ∈ L2 (, rdrdθ dz); f  = 0 . Consider the functional q defined by q[] = Q0 [] −

π 2 d

 2L2 (,rdrdθ dz) .

(3.8)

 2 Since the essential spectrum of H starts at πd , if we construct a trial function  ∈ D0 () such that q[] has a negative value then the task is achieved. Using the quadratic form domain,  must be continuous inside  but not necessarily smooth. Let χ be the first transverse mode, i.e.    2 sin πd z if z ∈ (0, d) d (3.9) χ(z) = 0 otherwise. For (r, θ, z) = ϕ(r)χ(z), we compute π 2 q[] = ∇r ϕχ, ∇r ϕχ −

ϕχ 2L2 (,rdrdθ dz) , d    = |χ(z)|2 |ϕ (r)|2 + |ϕ(r)||χ (z)|2 rdrdθdz ]0,+∞[×[0,2π [×[0,d]

−

π 2 d

ϕχ 2L2 (,rdrdθ) ,

= 2π ϕ 2L2 ([0,+∞[,rdr) . Now let us consider an interval J = [0, b ] for a positive b > a and a function ϕ ∈ S ([0, +∞[) such that ϕ(r) = 1 for r ∈ J. We also define a family {ϕτ : τ > 0} by  ϕ(r) if r ∈ (0, b ) (3.10) ϕτ (r) = ϕ(b + τ (ln r − ln b )) if r  b .

24

H. Najar et al.

Let us write

ϕτ L2 ([0,+∞),rdr) =

 

=  =

(0,∞)

|ϕτ (r)|2rdr,

(b ,+∞)

(b ,+∞)



=τ

τ 2 |ϕ (b + τ (ln r − ln b ))|2rdr, τ2 |ϕ (b + τ (ln r − ln b ))|2rdr, r2

(b ,+∞)

 =τ

(0,+∞)

τ |ϕ (b + τ (ln r − ln b ))|2 dr, r |ϕ (s)|2 ds = τ ϕ 2L2 ((0,+∞)) .

(3.11)

Let j be a localization function from C0∞ (0, a) and for τ, ε > 0 we define τ,ε (r, z) = ϕτ (r)[χ(z) + εj(r)2 ] = ϕτ (r)χ(z) + ϕτ εj2 (r) = 1,τ,ε (r, z) + 2,τ,ε (r). (3.12) q[] = q[1,τ,ε + 2,τ,ε ] = Q0 [1,τ,ε + 2,τ,ε ] − = Q0 [1,τ,ε ] −

π 2

π 2 d

1,τ,ε + 2,τ,ε 2L2 (,rdrdθ dz) .

1,τ,ε 2L2 (,rdrdθ dz) d π 2 + Q0 [2,τ,ε ] −

2,τ,ε 2L2 (,rdrdθ dz) d π 2 + 2 ∇r 1,τ,ε , ∇r 2,τ,ε r −

1,τ,ε , 2,τ,ε r . d

Using the properties of χ, noting that the supports of ϕ and j are disjoints and taking into account Eq. 3.11, we get q[] = 2π τ ϕ L2 (0,+∞) − 8π dε j 2 L22 (0,+∞) π 2   + 2ε2 π 2 jj 2(L2 (0,∞),rdr) −

j 2 2(L2 (0,∞),rdr) . d

(3.13)

Firstly, we notice that only the first term of the last equation depends on τ . Secondly, the linear term in ε is negative and could be chosen sufficiently small so that it dominates over the quadratic one. Fixing this ε and then choosing τ sufficiently small the right hand side of (3.13) is negative. This ends the proof of the first claim.

Discrete Spectrum of a Spatial Quantum Waveguide

25

The proof of the second claim is based on bracketing argument. Let us split L2 (, rdrdθdz) as follows, L2 (, rdrdθdz) = L2 (a− , rdrdθ dz) ⊕ L2 (a+ , rdrdθdz), with a− = {(r, θ, z) ∈ [0, a] × [0, 2π [×[0, d]}, a+ = \a− . Therefore Ha−,N ⊕ Ha+,N  H  Ha−,D ⊕ Ha+,D . Here we index by D and N depending on the boundary conditions considered on the surface r = a. The min-max principle leads to 

 π 2 +,N +,D σess (H) = σess (Ha ) = σess (Hr ) = , +∞ . d Hence if Hr−,D exhibits a discrete spectrum below πd2 , then H do as well. We mention that this is not a necessary condition. If we denote by λ j(Ha−,D ), λ j(Ha−,N ) and λ j(H), the j-th eigenvalue of Ha−,D , Ha−,N and H respectively then, again the minimax principle yields the following 2

λ j(Ha−,N )  λ j(H)  λ j(Ha−,D ),

(3.14)

λ j−1 (Ha−,D )  λ j(H)  λ j(Ha−,D ).

(3.15)

and for 2  j

has a sequence of eigenvalues [1, 19], given by

 (2k + 1)π 2 xn,l 2 + . λk,n,l = 2d a

Fig. 2 We represent  π 2  x(i) 2 + a where a  → 2d x(1), x(2), x(3) are the first three zeros of the bessel functions increasingly ordered

Area of existence of the first three eigenvalues of H for d=π

Threshold of appearance of eigenvalues 1.5

x(1) x(2) x(3)

1.2

x(1) x(2) x(3)

1.1

(π/(2d))2 + (x(i)/a)2

1 0.9

(π/(2d))2 + (x(i)/a)2

Ha−,D

0.8 0.7 0.6

1

0.5

0.5 0.4 0.3 0.2 5

10 a

15

0

2

4

6 a

8

10

26

H. Najar et al.

Fig. 3 The number of the eigenvalues of the operator H D function of λ ≡ a/d

Number of eigen values of the operator HD less or equal than π 2/d2

The number

400 300 200 100 0 0

10

20

30

40

50 λ

60

70

80

90

100

2

4

6

8

10 λ

12

14

16

18

20

The number

10 8 6 4 2 0

0

Where xn,l is the l-th positive zero of Bessel function of order n (see [1, 19]). The condition λk,n,l < yields that k = 0, so we get λ0,n,l =

π 2

π2 , d2

+

(3.16)

x 2 n,l

. 2d a This yields that the condition (3.16) to be fulfilled, will depends on the value  2 of xan,l . We recall that xn,l are the positive zeros of the Bessel function Jn . So, for any λ(a), eigenvalue of H, there exists, n, l, n , l ∈ N, such that x2n,l x2n ,l π2 π2 +  λ(a)  + . 4d2 a2 4d2 a2

(3.17)

Fig. 4 The number of the eigenvalues of the operator H D function of d and a

The number

200 150 100 0.4 50 0.6 0 5

6

0.8 7

8 a

9

10

1

d

Discrete Spectrum of a Spatial Quantum Waveguide

27

The proof of (3.6) is completed by observing by that xn,l and xn ,l are independent from a. In Fig. 2, the domain of existence of λ1 (H), λ2 (H) and λ3 (H) are represented.   4 Numerical Computations This section is devoted to some numerical computations. In [11] and [16], the 2 number of positive zeros of Bessel functions less than λ is estimated by πλ 2 which is based on the approximate formula for the roots of Bessel functions for large l is

 1 π xn,l ∼ n + 2l − . (4.18) 2 2 taking into account (3.14), we get that for, d and a positives such that 1,9276, H has a unique discrete eigenvalue (Figs. 3 and 4).

a2 d2

< λ∗ =

Acknowledgement It is a pleasure for the authors to thanks Professor Pavel Exner for useful discussions, valuable comments and remarks which significantly improve this work.

References 1. Abramowitz, M., Stegun, I.A.: Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Dover, New York (1972) 2. Borisov, D., Exner, P.: Exponential splitting of bound states in a waveguide with a pair of distant windows. J. Phys. A 37(10), 3411–3428 (2004) 3. Bulla, W., Gesztesy, F., Renger, W., Simon, B.: Weakly coupled bound states in quantum waveguides. Proc. Amer. Math. Soc. 125(5), 1487–1495 (1997) 4. Davies, E.B.: Trapped modes in acoustic waveguides. Quart. J. Mech. Appl. Math. 51, 477 (1998) 5. Dittrich, J., Kˇríž, J.: Bound states in straight quantum waveguides with combined boundary conditions. J. Math. Phys. 43(8), 3892–3915 (2002) 6. Duclos, P., Exner, P.: Curvature-induced bound states in quantum waveguides in two and three dimensions. Rev. Math. Phys. 37, 4867–4887 (1989) 7. Exner, P., Šeba, P.: Bound states and scattering in quantum waveguides coupled laterally through a boundary window. J. Math. Phys. 30(10), 2574 (1989) ˇ D.: Bound states and scattering in quantum waveguides 8. Exner, P., Šeba, P., Tater, M., Vanek, coupled laterally through a boundary window. J. Math. Phys. 37(10), 4867–4887 (1996) 9. Exner, P., Vugalter, S.A.: Asymptotic estimates for bound states in quantum waveguide coupled laterally through a boundary window. Ann. Inst. H. Poincaré A Phys. Théor. 65, 109–123 (1996) 10. Hurt, N.E.: Mathematical Physics of Quantum Wires and Devices. Mathematics and its Application, vol. 506. Kluwer, Dordrecht (2000) 11. Gloge, D.: Weakly guiding fibers. Appl. Optim. 10, 2442–2258 (1971) 12. Najar, H.: Lifshitz tails for acoustic waves in random quantum waveguide. J. Statist. Phys. 128(4), 1093–1112 (2007) 13. Klein, A., Lacroix, J., Speis, A.: Localization for the Anderson model on a strip with singular potentials. J. Funct. Anal. 94(1), 135–155 (1990) 14. Kleespies, F., Stollmann, P.: Lifshitz asymptotics and localization for random quantum waveguides. Rev. Math. Phys. 12, 1345–1365 (2000) 15. Krejcirik, D., Kriz, J.: On the spectrum of curved quantum waveguides. Publ. Res. Inst. Math. Sci. 41(3), 757–791 (2005)

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16. Marcuse, D.: Theory of Dielectric Optical Waveguides (Quantum Electronics-Principles and Applications Series). Academic, New York (1974) 17. Nazarov, S.A., Specovius-Neugebauer, M.: Selfadjoint extensions of the Neumann Laplacian in domains with cylindrical outlets. Comm. Math. Phys. 185, 689–707 (1997) 18. Reed, M., Simon, B.: Methods of Modern Mathematical Physics. Analysis of Operators, vol. IV. Academic, New York (1978) 19. Watson, G.N.: A Treatise on the Theory of Bessel Functions. Cambridge University Press, Cambridge (1995)

Math Phys Anal Geom (2010) 13:29–46 DOI 10.1007/s11040-009-9066-y

Inverse Problem for Sturm–Liouville Operators with Coulomb Potential which have Discontinuity Conditions Inside an Interval Nilifer Topsakal · Rauf Amirov

Received: 12 November 2008 / Accepted: 27 August 2009 / Published online: 9 December 2009 © Springer Science + Business Media B.V. 2009

Abstract In this study, properties of spectral characteristic are studied for Sturm–Liouville operators with Coulomb potential which have discontinuity conditions inside a finite interval. Also Weyl function for this problem under consideration has been defined and uniqueness theorems for solution of inverse problem according to this function have been proved. Keywords Inverse problem · Coulomb singularity · Weyl function Mathematics Subject Classifications (2000) Primary 34A55; Secondary 34B24 · 34L05 1 Introduction In spectral theory, the inverse problem is the usual name for any problem in which it is necessary to ascertain the spectral data that will determine a differential operator uniquely and a method of construction for this operator from the data. A problem of this kind was first formulated and investigated by Ambartsumyan in 1929 [3]. Since 1946, various forms of the inverse problem have been considered by numerous authors—Borg [11], Levinson [4], Levitan [5], etc. and now there exists an extensive literature on the question [6–10]. Later, the inverse problems having specified singularities were considered by a number of authors [17–20]. N. Topsakal (B) · R. Amirov Department of Mathematics, Faculty of Arts and Sciences, Cumhuriyet University, 58140 Sivas, Turkey e-mail: [email protected] R. Amirov e-mail: [email protected]

30

N. Topsakal, R. Amirov

Some integral representations have been given in [21, 22] which are different from the integral representations in our study. In particular, the integral representation for special functions which are given in those papers have been obtained by using Fourier transformations. Moreover, this kind of representation is useful for investigating the spectral functions of singular Dirac and Sturm–Liouville differential operators in half- and all-axis, respectively. Spectral functions are important for determining the operators, that is, for solving the inverse problem for differential operators. However, in finite intervals, the integral representations for the solution of differential equations which generate the operator with initial conditions are more useful for investigating the spectral properties of this operator. In case of q(x) ≡ 0 since this operator is the singular Sturm–Liouville operator with Coulomb potential, linearly independent solutions of this kind of differential equation could be given with hypergeometric functions and this integral representation is also a representation for hypergeometric functions. For this reason, obtaining this kind of integral representation is so important. Therefore, when obtained, these integral representations can be used for asymptotic behaviours of hypergeometric functions as x → +∞. We consider the boundary value problem L for the equation: (y) := −y +

C y + q(x)y = k2 y x

(1.1)

on the interval 0 < x < π with the boundary conditions U(y) := y(0) = 0,

V(y) := y(π ) = 0

(1.2)

and with the jump conditions 

y(d + 0) = αy(d − 0) y (d + 0) = α −1 y (d − 0)

(1.3)

  where k is spectral parameter; C, α ∈ R, α = 1, α > 0, d ∈ π2 , π , q(x) is a real valued bounded function and q(x) ∈ L2 (0, π ). Boundary value problems with discontinuities inside the interval often appear in mathematics, mechanics, physics, geophysics and other branches of natural proper-ties. The inverse problem of reconstructing the material properties of a medium from data collected outside of the medium is of central importance in disciplines ranging from engineering to the geo-sciences. For example, discontinuous inverse problems appear in electronics for constructing parameters of heterogeneous electronic lines with desirable technical characteristics [23, 24]. After reducing corresponding mathematical model we come to boundary value problem L where q (x) must be constructed from the given spectral information which describes desirable amplitude and phase characteristics. Spectral information can be used to reconstruct the permittivity and conductivity profiles of a one-dimensional discontinuous medium [25, 26].

Inverse Problem for Sturm–Liouville Operators

31

Boundary value problems with discontinuities in an interiorpoint also appear in geophysical models for oscillations of the Earth [27, 28]. Here, the main discontinuity is cased by reflection of the shear waves at the base of the crust. Further, it is known that inverse spectral problems play an important role for investigating some nonlinear evolution equations of mathematical physics. Discontinuous inverse problems help to study the blow-up behaviour of solutions for such nonlinear equations. We also note that inverse problem considered here appears in mathematics for investigating spectral properties of some classes of differential, integrodifferential and integral operators. Direct and inverse spectral problems for differential operators without discontinuities have been thoroughly studied (see [10–16] and the references therein). The presence of discontinuities produces essential qualitative modifications in the investigation of the operators. Some aspects of direct and inverse problems for discontinuous boundary value problems in various formulations have been considered in [2, 26, 29–34] and other   works. In particular, it has shown in [29] that if q(x) is known a priori on 0, π2 then q(x) is uniquely deter  mined on π2 , π by the eigenvalues. In [26] the discontinuous inverse problem is considered on the half-line. Boundary value problems with singularities have been studied in [35–37, 39] for further discussion see the references therein. A representation with transformation operator of problem (1.1)–(1.3) was obtained in [40], as in [39]. In this study, Weyl function for considering operator has been defined and the theorem which is related to uniqueness of solution of inverse problem, according to Weyl function has been proved. In the last part, has been proved that the system of the eigenfunctions of the Sturm–Liouville boundary value problem L is complete and forms an orthogonal basis in L2 (0, π ).

2 Representation for the Solution We define y1 (x) = y(x), y2 (x) = (y)(x) = y (x) − u(x)y(x), u(x) = C ln x and let’s write the expression of left hand side of the (1.1) as the follows   (y) = − (y)(x) − u(x)(y)(x) − u2 (x)y + q(x)y = k2 y

(2.1)

then the (1.1) reduces to the system; 

y1 − y2 = u (x) y1 y2 + k2 y1 = −u (x) y2 − u2 (x) y1 + q (x) y1

(2.2)

with the boundary conditions y1 (0) = 0,

y1 (π ) = 0

(2.3)

32

N. Topsakal, R. Amirov

and with the jump conditions  y1 (d + 0) = αy1 (d − 0)

(2.4)

y2 (d + 0) = α −1 y2 (d − 0) . Matrix form of system (2.2)    y1 u = y2 −k2 − u2 + q

1 −u



y1 y2

 (2.5)

   y1 u(x) 1 or y = Ay such that A = , . − u(x) y2 −k2 − u2 (x) + q(x) x = 0 is a regular-singular end point for equation (2.5) and Theorem 2 in [1] (see Remark 1–2, p. 56) extends to interval [0, π ]. For this reason, by [1], there exists only one solution of the system (2.2) which satisfies the initial conditions y1 (ξ ) = υ1 , y2 (ξ ) = υ2 for each ξ ∈ [0, π ], υ = (υ1 , υ2 )T ∈ C2 , especially the initial conditions y1 (0) = 1, y2 (0) = ik. 



Definition 1 The first component of the solution of the system (2.2) which satisfies the initial conditions y1 (ξ ) = υ1 , y2 (ξ ) = (y)(ξ ) = υ2 is called the solution of the equation (1.1) which satisfies these same initial conditions. It was obtained in [40] by the successive approximations method that (see [13]) the following theorem is true. Theorem 1 For each solution of system (2.2) which satisfying the initial condi y

1

1 tions (0) = and the jump conditions (2.4), the following expression y2 ik is true: ⎧ x ⎪ ⎪ ⎪ ikx ⎪ y1 = e + K11 (x, t) eikt dt ⎪ ⎪ ⎨ −x , x
−x

⎧ x ⎪ ⎪ + ikx − ik(2d−x) ⎪ y1 = α e + α e + K11 (x, t) eikt dt ⎪ ⎪ ⎪ ⎪ ⎪ −x ⎪  + ikx  ⎪ ⎪ − ik(2d−x) ⎪ ⎨ y2 = ik α e − α e   ⎪ + b (x) α + eikx + α − eik(2d−x) ⎪ ⎪ ⎪ ⎪ ⎪ x x ⎪ ⎪ ⎪ ⎪ ikt ⎪ + K21 (x, t) e dt + ik K22 (x, t) eikt dt ⎪ ⎩ −x

−x

,

x>d

Inverse Problem for Sturm–Liouville Operators

33

where 1 b (x) = − 2

x



x

 − 12 u(t)dt ds u (s) − q(s) e s 2

0

K11 (x, x) =

+

α u (x) 2

1 K21 (x, x) = b (x) − 2 

x



 1 u (s) − q(s) K11 (s, s)ds − 2

0

K22 (x, x) = −

x u(s)K21 (s, s)ds

2

0

 α  u (x) + 2b (x) 2 +

K11 (x, 2d−x+0)− K11 (x, 2d−x − 0) =

α− u(x) 2

  ∂ Kij(x, .) ∂ Kij(x, .) 1 1 ± , ∈ L2 (0, π ), i, j= 1, 2, α = α± . ∂x ∂t 2 α

3 Inverse Problem In the present section, we study the inverse problems recovering the boundary value problem L from the spectral data. We consider three statements of the inverse problem of the reconstruction of the boundary value problem L from the Weyl function, from the spectral data {kn , an }n0 and from two spectra {kn , μn }n0 . These inverse problems are generalizations of the well-known inverse problems for Sturm–Liouville operator [24, 41]. Let us denote problem L as L0 in the case of C = 0 and q(x) ≡ 0. It is easily shown that solution ϕ0 (x, k) satisfying the initial conditions ϕ0 (0, k) = 0, (ϕ0 )(0, k) = 1 and the jump conditions (2.4) is shown as ⎧ 1 ⎪ ⎪ , x d. k k Let 0 (k) be a characteristic function of problem L0 . Then characteristic equation of problem L0 is the form

0 (k) =

α− α+ sin kπ + sin k (2d − π ) = 0. k k

(3.1)

We define characteristic function, eigenvalues sequence and normalizing constans sequence by (k), {kn } and {αn }, respectively. Denote

(k) = ψ (x, k) , ϕ (x, k) ,

(3.2)

34

N. Topsakal, R. Amirov

where y (x) , z (x) := y1 (x) z2 (x) − y2 (x) z1 (x) . According to Liouville formula, ψ(x, k), ϕ(x, k) is not depend on x. We shall assume that ϕ(x, k), C(x, k) and ψ(x, k) are solutions of the (1.1) under the following initial conditions: ϕ1 (0, k) = 0,

ϕ2 (0, k) = 1,

ψ1 (π, k) = 0,

ψ2 (π, k) = 1,

C1 (0, k) = 1,

C2 (0, k) = 0

(3.3)

Clearly, for each x, functions ψ(x, k), ϕ(x, k) are entire in k and

(k) = V (ϕ) = U (ψ) = ϕ1 (π, k) = ψ1 (0, k) .

(3.4)

By using representation of the function y(x, k) for the solution ϕ1 (x, k) : 1 ϕ1 (x, k) = ϕ10 (x, k) + k

x

11 (x, t) sin ktdt K

0

is obtained. Therefore, the characteristic function of the problem L is obtained as 1

(k) = 0 (k) + k

π

11 (π, t) sin ktdt K

(3.5)

0

11 (x, t) = K11 (x, t) − K11 (x, −t) . where K It is clear that Lemma 6 in [40] will be expressed for initial conditions ϕ1 (0, k) = 0, ϕ2 (0, k) = 1 as follows: Lemma 1 (Lemma 6, [40]) The eigenvalues kn of problem L have the following asymptotic behaviour: dn δn kn = k0n +  2 +  2 , k0n k0n

(3.6)

where δn ∈ 2 and dn =

α − cos k0n (2d − π ) + α + cos k0n π u (π ) .   2 k0n

is a bounded sequence. 



1 (x, k) be solution of (2.2) under the conditions U( ) =

2 (x, k)

1 (0, k) = 1 and V( ) = 1 (π, k) = 0 and under the jump conditions (2.4). Let (x, k) =

Inverse Problem for Sturm–Liouville Operators

35

We set M(k) := 2 (0, k). The functions (x, k) and M(k) are respectively called the Weyl solution and the Weyl function for the boundary value problem L. Denote δ (k) M (k) = (3.7)

(k) where δ (k) = ψ2 (0, k) . Clearly,

(x, k) = M (k) ϕ (x, k) + C (x, k) .

(3.8)

Weyl solution and the Weyl function are meromorphic functions of a parameter k with poles on the spectrum of the problem L. It follows from (3.7) and (3.8) that

(x, k) =

ψ (x, k) ψ2 (0, k) and 2 (0, k) = = M (k)

(k)

(k)

(3.9)

where ψ (x, k) = ψ2 (0, k) ϕ (x, k) + (k) C (x, k) .

(3.10)

Note that, by virtue of equalities C(x, k), ϕ(x, k) ≡ 1, (3.8) and (3.9) we have,  (x, k) , ϕ (x, k) ≡ 1,

ψ (x, k) , ϕ (x, k) ≡ (k)

(3.11)

Theorem 2 The following representation holds;   ∞  1 1 1  + + M (k) = α0 (k − k0 ) n=−∞ αn (k − kn ) αn0 k0n

(3.12)

Proof Let’s write a representation solution of ψ(x, k) as representation solution of ϕ(x, k) : for x > d ⎧ π −x  ⎪ ⎪ 1 1 ⎪ 11 (x, t) sin ktdt ⎪ ψ1 (x, k) = − sin k (π − x) + N ⎪ ⎪ k k ⎪ ⎪ ⎪ 0 ⎪ ⎪ ⎪ ⎨ 1 ψ2 (x, k) = cos k (π − x) − b (x) sin k (π − x) ⎪ k ⎪ ⎪ ⎪ ⎪ π −x π −x ⎪   ⎪ ⎪ 1 ⎪ ⎪ 21 (x, t) sin ktdt + 22 (x, t) cos ktdt ⎪ + N N ⎪ ⎩ k 0

0

36

N. Topsakal, R. Amirov

for x < d ⎧ 1 1 ⎪ ⎪ ψ1 (x, k) = − α + sin k (π − x) + α − sin k (π − 2d + x) ⎪ ⎪ k k ⎪ ⎪ ⎪ ⎪ π −x ⎪  ⎪ ⎪ 1 ⎪ 11 (x, t) sin ktdt ⎪ + N ⎪ ⎪ k ⎪ ⎪ ⎪ 0 ⎪ ⎪ ⎨ ψ2 (x, k) = α + cos k (π − x) + α − cos k (π − 2d + x) ⎪ ⎪   ⎪ ⎪ 1 + 1 − ⎪ ⎪ + b (x) − α sin k − x) + α sin k − 2d + x) (π (π ⎪ ⎪ ⎪ k k ⎪ ⎪ ⎪ ⎪ π −x π −x ⎪   ⎪ ⎪ 1 ⎪ ⎪ 21 (x, t) sin ktdt + 22 (x, t) cos ktdt ⎪ + N N ⎪ ⎩ k 0

0

ij(x, t) = Nij(x, t) − Nij(x, −t), i, j = 1, 2. In the case of C = 0 and where N q(x) ≡ 0 we write solution with ψ01 (x, k) and ψ02 (x, k), so we have 

ψ1 (x, k) = 01 (x, k) + f1 ψ2 (x, k) = 02 (x, k) + f2

where 1 f1 = k

π −x  11 (x, t) sin ktdt N 0



 1 + 1 − f2 = b (x) − α sin k (π − x) + α sin k (π − 2d + x) k k 1 + k

π −x π −x   21 (x, t) sin ktdt + 22 (x, t) cos ktdt. N N 0

0

On the other hand, we can write M (k) − M0 (k) =

ψ2 (0, k) ψ02 (0, k) f2 f1 − = − M0 (k) . ψ1 (0, k) ψ01 (0, k)

(k) (k)

Since lim e−| Im k|π | fi (k)| = 0 and (k) > Cδ e| Im k|π for k ∈ Gδ , we get that |k|→∞

lim sup |M (k) − M0 (k)| = 0.

|k|→∞ k∈Gδ

(3.13)

Inverse Problem for Sturm–Liouville Operators

37

Weyl function M(k) is meromorphic with respect to poles kn . Using (3.5) and Lemma 2 in [40], we calculate that ⎧ 1 1 ψ2 (0, kn ) ⎪ ⎪ = . = Re s M (k) = . ⎪ ⎪ k=k α ⎨ n n

(kn )

(kn ) ϕ2 (π, kn ) . (3.14)   0 ⎪ 1 ⎪ ⎪ Re s M0 (k) = ψ02. 0, kn = .   1  ⎪ = 0 ⎩ k=k0 αn n

k0n

0 k0n ϕ02 π, k0n Consider the contour integral  1 M (μ) − M0 (μ) In (k) = dμ , 2πi k−μ

k ∈ intn .

n

By virtue of (3.13), we have lim In (k) = 0. On the other hand, the residue n→∞

theorem and (3.14) yield In (k) = −M (k) + M0 (k) +

 kn ∈intn

 1 1   − 0 αn (k − kn ) αn k0n − k 0 kn ∈intn



and theorem is proved.

Let us formulate a theorem on the uniqueness of a solution of the inverse problem with the use of the Weyl function. For this purpose, parallel with L, we consider the boundary-value problem  L of the same form but with different coefficients  q(x). It is assumed in what follows that if a certain symbol α denotes an object related to the problem L, then  α denotes the corresponding object related to the problem  L.  Theorem 3 If M(k) = M(k) then L =  L Thus the specification of the Weyl function uniquely determines the operator. Proof Since



ψ (υ) (x, k) = O |k|υ−1 exp (|Im k| (π − x)) | (k)|  Cδ |k| exp (|Im k| π ) ,

δ , k∈G

then it follows from (3.15) and (3.9) that  (υ)   (x, k)  Cδ |k|υ−1 exp (− |Im k| π ) ,

υ = 0, 1

k ∈ Gδ .

Let us define the matrix P(x, k) = [P jk (x, k)] j,k=1,2 by the formula     1  ϕ1 ϕ1 1 P (x, k) = . 2 ϕ2 2  ϕ2

(3.15)

(3.16)

(3.17)

38

N. Topsakal, R. Amirov

Using (3.17) and (3.10) we calculate ⎧ 2 (x, k) − 1 (x, k)  P11 (x, k) = ϕ1 (x, k) ϕ2 (x, k) ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ P12 (x, k) = 1 (x, k)  1 (x, k) ϕ1 (x, k) − ϕ1 (x, k)

(3.18)

⎪ 2 (x, k) − 2 (x, k)  ϕ2 (x, k) P21 (x, k) = ϕ2 (x, k) ⎪ ⎪ ⎪ ⎪ ⎩ 1 (x, k) P22 (x, k) = 2 (x, k)  ϕ1 (x, k) − ϕ2 (x, k) and ⎧ ϕ1 (x, k) = P11 (x, k)  ϕ1 (x, k) + P12 (x, k)  ϕ2 (x, k) ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ϕ2 (x, k) = P21 (x, k)  ϕ1 (x, k) + P22 (x, k)  ϕ2 (x, k) ⎪ 1 (x, k) + P12 (x, k) 2 (x, k)

1 (x, k) = P11 (x, k) ⎪ ⎪ ⎪ ⎪ ⎩ 1 (x, k) + P22 (x, k) 2 (x, k)

2 (x, k) = P21 (x, k)

.

(3.19)

It follows from (3.18), (3.8) and (3.11)

P11 (x, k) = 1 +

1  1 (x, k)( ϕ2 (x, k) − ϕ2 (x, k))

(k)   2 (x, k) − 2 (x, k) , − ϕ1 (x, k) 

 1  1 (x, k) − 1 (x, k)  ϕ1 (x, k)  ϕ1 (x, k) ,

(k)  1  2 (x, k) , 2 (x, k)  ϕ2 (x, k) − ϕ1 (x, k)  P21 (x, k) =

(k)   1  1 (x, k) −  2 (x, k) P22 (x, k) = 1 + ϕ2 (x, k) 

(k)  − 2 (x, k)( ϕ1 (x, k) −  ϕ2 (x, k)) . P12 (x, k) =

According to (3.18) and (3.8), for each fixed x, the functions P jk (x, k) are δ . meromorphic in k with poles in the points kn and  kn . Denote G0δ = Gδ ∩ G (υ) υ By virtue of ϕ (x, k) = O(|k| exp(| Im k|x)), (3.15) and (3.17) this yields |P11 (x, k)−1| 

Cδ , |k|

|P12 (x, k)| 

Cδ , |k|

k ∈ G0δ ,

|k|  k∗ , (3.20)

|P22 (x, k)−1| 

Cδ , |k|

|P21 (x, k)| 

Cδ , |k|

k ∈ G0δ ,

|k|  k∗ (3.21)

Inverse Problem for Sturm–Liouville Operators

39

According to (3.8), (3.10) and (3.16) we have 2 (x, k) − C1 (x, k)  P11 (x, k) = ϕ1 (x, k) C ϕ2 (x, k)    (k) − M (k) ϕ1 (x, k)  + M ϕ2 (x, k) , 2 (x, k) − C 1 (x, k) ϕ1 (x, k) P12 (x, k) =  ϕ1 (x, k) C    (k) ϕ1 (x, k)  ϕ1 (x, k) , + M (k) − M 2 (x, k) − C2 (x, k)  ϕ2 (x, k) P21 (x, k) = ϕ2 (x, k) C    (k) − M (k) ϕ2 (x, k)  + M ϕ2 (x, k) , 1 (x, k) ϕ2 (x, k) ϕ1 (x, k) C2 (x, k) − C P22 (x, k) =     (k) ϕ2 (x, k)  ϕ1 (x, k) . + M (k) − M  Thus if M(k) = M(k) then the functions P jk (x, k) are entire in k for each fixed x. Together with (3.20) and (3.21) this yields, P11 (x, k) ≡ 1,

P12 (x, k) ≡ 0,

P21 (x, k) ≡ 0,

P22 (x, k) ≡ 1.

Substituting into (3.19) we get ϕ1 (x, k) ≡  ϕ1 (x, k) ,

ϕ2 (x, k) ≡  ϕ2 (x, k) ,

1 (x, k) , ≡

2 (x, k)

2 (x, k) ≡

1 (x, k)

for all x and k. Consequently L =  L.



kn , αn =  αn , n  0 then L =  L. Thus, the specification of Theorem 4 If kn =  the spectral data {kn , αn }n0 uniquely determines the operator. Proof

 ∞   1 1 1 + + 0 0 α0 (k − k0 ) n=−∞ αn (k − kn ) αn kn   ∞  1 1 1  (k) = M  +  + k0  α0  k − k0  αn  k − kn  α 0 M (k) =

n=−∞

(3.22)

n n

Under the hypothesis of the theorem and in view of (3.22), we get that M(k) =  M(k) and consequently by Theorem 2, L =  L. 

kn , μn =  μn ,n  0, then L =  L. Theorem 5 : If kn = 

40

N. Topsakal, R. Amirov

 (k), it is clear that Proof In view of properties of functions (k) and

(k)   lim

 (k) = 1. Under the hypothesis kn = kn and (k) and (k) functions are k→∞

 (k). entire we get that (k) =

n ϕ(x,  n  (x,  kn ) = β ϕ (x, kn ) and From Lemma 2 in [40], we have ψ kn ) = β     ϕ (x, kn ). It follows βn = βn and so αn =  αn . Conse (x, kn ) =  (x, kn ) = βn  quently by Theorem 3, L =  L. 

4 Properties of Eigenfunctions In this section, properties of eigenfunctions of problem L will be learned. We have representation of eigenfunctions as follows: 1 1 ϕ (x, kn ) = sin kn x + kn kn

x

11 (x, t) sin kn tdt, K

x < d,

0

ϕ (x, kn ) =

α+ α− sin kn x + sin kn (2d − x) kn kn 1 + kn

x

11 (x, t) sin kn tdt, K

x > d.

0

Theorem 6 (i) The system of eigenfunctions {ϕ(x, kn )}n0 of the boundary value problem L is complete in L2 (0, π ). (ii) Let f (x), x ∈ [0, d) ∪ (d, π ] be an absolutely continuous function and satisfy the jump conditions:  f (d + 0) = α f (d − 0) f  (d + 0) = α −1 f  (d − 0) . Then ∞  f (x) = an ϕ (x, kn ) , n=0

1 an = αn

π ϕ (t, kn ) f (t) dt

(4.1)

0

and the series converges uniformly on [0, d) ∪ (d, π ]. Proof (i) From Theorem 3, Theorem 1 and Theorem 8 in [38] the system of eigenfunctions {ϕ(x, kn )}n0 of the problem L is a Riesz Bazis in L2 (0, π ). Thus (i) is proved.

Inverse Problem for Sturm–Liouville Operators

(ii) Denote



1 G (x, t, k) = −

(k)

41

ϕ (x, k) ψ (t, k) , ϕ (t, k) ψ (x, k) ,

xt xt

and consider the function π Y (x, k) = G (x, t, k) f (t) dt 0

1 =− ψ (x, k)

(k)

x ϕ (t, k) f (t) dt 0

⎧ d ⎨

1 ψ (t, k) f (t) dt + − ϕ (x, k) ⎩

(k) x

where

ϕ (x, k) =

π ψ (t, k) f (t) dt

⎫ ⎬ ⎭

d

⎧ x ⎪ ⎪ 1 1  ⎪ ⎪ K11 (x, t) sin ktdt, sin kx + ⎪ ⎪ k ⎨k

x
0

x ⎪ ⎪ α+ α− 1  ⎪ ⎪ K11 (x, t) sin ktdt, sin kx + sin k (2d − x) + ⎪ ⎪ ⎩ k k k

x>d

0

(4.2) and

ψ (x, k) =

⎧ x ⎪ ⎪ 1 1  ⎪ ⎪ cos kx+ K11 (x, t) cos ktdt, ⎪ ⎪ k ⎨k

x
0

x ⎪ + − ⎪ α 1 α ⎪ 11 (x, t) cos ktdt, ⎪ cos kx+ cos k (2d−x) + K ⎪ ⎪ ⎩ k k k

x > d.

0

(4.3) The function G(x, t, k) is called Green’s function for L. G(x, t, k) is the kernel of the inverse operator for the Sturm–Liouville operator, i.e. Y(x, k) is the solution of the boundary value problem Y − k2 Y = f (x) U (Y) = 0, V (Y) = 0 and satisfies the jump condition:  Y (d + 0) = αY (d − 0) Y  (d + 0) = α −1 Y  (d − 0) this easily verified by differentiation.

(4.4)

(4.5)

42

N. Topsakal, R. Amirov

Let now f, x ∈ [0, d) ∪ (d, π ] be an arbitrary absolutely continuous function. Since ϕ(x, k) and ψ(x, k) are solutions of (1.1), we transform Y(x, k) as follows ⎧ x    1 ⎨ Y (x, k) = − 2 ψ (x, k) − (ϕ) (t, k) − u (t) (ϕ) (t, k) k (k) ⎩ 0   − u2 (t) − q (t) ϕ (t, k) f (t) dt + ϕ (x, k)

d  x

+ ϕ (x, k)

π  d

  − (ψ) (t, k) − u (t) (ψ) (t, k)   − u2 (t) − q (t) ψ (t, k) f (t) dt   − (ψ) (t, k) − u (t) (ψ) (t, k) ⎫ ⎬  2  − u (t) − q (t) ψ (t, k) f (t) dt . (4.6) ⎭

Integrating of the terms containing second derivates by parts yields in view of (3.2), Y (x, k) =

" f (x) 1 ! Z 1 (x, k) + Z 2 (x, k) , − 2 k k

(4.7)

where ⎧ x d 1 ⎨ Z 1 (x, k) = ψ (x, k) (ϕ) (t, k) g (t) dt + ϕ(x, k) (ψ) (t, k) g(t)dt k (k) ⎩ x 0 ⎫ π ⎬ + ϕ (x, k) (ψ) (t, k) g (t) dt ⎭ d

 1   + f (0) ψ (x, k) + ψ (π, k) f  (π ) ϕ (x, k) , g (t) := f  (t) , k (k) x !   " 1 ψ (x, k) Z 2 (x, k) = u (t) (ϕ) (t, k) − u2 (t) − q (t) ϕ (t, k) f (t) dt k (k) 0

d   1 + ϕ (x, k) u (t) (ψ) (t, k) − u2 (t) − q (t) ψ (t, k) f (t) dt k (k) x

+

1 k (k)

π   ϕ (x, k) u (t) (ψ) (t, k) − u2 (t) − q (t) ψ (t, k) f (t) dt. d

Inverse Problem for Sturm–Liouville Operators

43

Using (3.14), we get for a fixed δ > 0 and sufficiently large k∗ > 0 : max |Z 2 (x, k)| 

0  x π

C , |k|

|k|  k∗ .

k ∈ Gδ ,

(4.8)

Let us show that lim max |Z 1 (x, k)| = 0.

(4.9)

|k|→∞0xπ k∈Gδ

First we assume that g(x) is absolutely continuous on [0, d) ∪ (d, π ]. In this case another integration by parts yields of Z 1 (x, k) we infer max |Z 1 (x, k)| 

0  x π

C , |k|

|k|  k∗ .

k ∈ Gδ ,

Let now g (t) ∈ L [0, π ] . Fix ε > 0 and choose an absolutely continuous function gε (t) such that π |g (t) − gε (t)| dt <

ε , 2C+

0

where

⎧ ⎨

x

1 |ψ (x, k)| C = max sup 0xπ k∈Gδ | (k)| ⎩ +

⎛ ×⎝

|(ϕ) (t, k)| dt + |ϕ (x, k)| 0

d

π |(ψ) (t, k)| dt +

x

⎞⎫ ⎬ |(ψ) (t, k)| dt⎠ . ⎭

d

Then, for k ∈ Gδ , |k|  k∗ , we have max |Z 1 (x, k)|  max |Z 1 (x, k; gε )| + max |Z 1 (x, k; g−gε )|

0  x π

0  x π



0  x π

ε C (ε) . + |k| 2

Hence, there exists k0 such that max |Z 1 (x, k)|  ε for |k| > k0 . By virtue of 0  x π

the arbitrariness ε > 0, we arrive at (4.9). Consider the contour integral  1 I N (x) = Y (x, k) dk, 2πi N

where

   σ n = k : |k| = k0n  + , n = 0, ± 1, ± 2, ... . 2

44

N. Topsakal, R. Amirov

It follows from (4.7)–(4.9) that I N (x) = f (x) + ε N (x) ,

lim

max |ε N (x)| = 0.

N→∞ 0xπ

(4.10)

On the other hand, we can calculate I N (x), with the help of the residue theorem. Using Lemma 2 and Lemma 4 in [40], we calculate Re sY (x, k) = − k=kn

−

=−

.

1

(kn )

.

1

(kn ) .

βn

(kn )

x ϕ (t, kn ) f (t) dt

ψ (x, kn ) 0

ϕ (x, kn )

⎧ d ⎨ ⎩

π ψ (t, kn ) f (t) dt +

x

ψ (t, kn ) f (t) dt

⎫ ⎬ ⎭

d

π ϕ (x, kn )

ϕ (t, kn ) f (t) dt 0

.

and by virtue of αn βn = − (kn ), 1 ϕ (x, kn ) Re sY (x, k) = k=kn αn

π ϕ (t, kn ) f (t) dt.

(4.11)

0

Since (4.6), we get N  I N (x) = an ϕ (x, kn ) , n=0

1 an = αn

π ϕ (t, kn ) f (t) dt. 0

Comparing this with (4.10) we arrive at (4.1), where the series converges uniformly on [0, d) ∪ (d, π ], i.e. (ii) is proved. 

References 1. Naimark, M.A.: Linear Differential Operators. Moscow, Nauka (1967) (in Russian) 2. Amirov, Kh.R., Yurko, V.A.: On differential operators with singularity and discontinuity conditions inside an interval. Ukrainian Math. J. 53(11), 1443–1458 (2001) 3. Ambartsumyan, V.A.: Über eine frage der eigenwerttheorie. Z. Phys. 53, 690–695 (1929) 4. Levinson, N.: The inverse Sturm–Liouville problem. Math. Tidsskr. B. 1949, 25–30 (1949) 5. Levitan, B.M.: On the determination of the Sturm–Liouville operator from one and two spectra. Math. USSR Izv. 12, 179–193 (1978) 6. Chadan, K., Colton, D., Paivarinta, L., Rundell, W.: An Introduction to Inverse Scattering and Inverse Spectral Problems. SIAM, Philadelphia (1997) 7. Gesztesy, F., Simon, B.: Inverse spectral analysis with partial information on the potential II, the case of discrete spectrum. Trans. Amer. Math. Soc. 352, 2765–2789 (2000) 8. Gesztesy, F., Kirsch, A.: One dimensional Schrödinger operators with interactions singular on a discrete set. J. Reine Angew. Math. 362, 28–50 (1985)

Inverse Problem for Sturm–Liouville Operators

45

9. Gilbert, R.P.: A method of ascent for solving boundary value problem. Bull. Am. Math. Soc. 75, 1286–1289 (1969) 10. Pöschel, J., Trubowitz, E.: Inverse Spectral Theory. Orlando Academic, Orlando (1987) 11. Borg, G.: Eine umkehrung der Sturm–Liouvillesehen eigenwertaufgabe. Acta Math. 78, 1–96 (1945) 12. Levitan, B.M., Sargsyan, I.S.: Introduction to Spectral Theory. Am. Math. Soc. Transl. Math. Monogr, vol. 39. American Mathematical Society, Providence (1975) 13. Marchenko, V.A.: Sturm–Liouville Operators and their Applications. Naukova Dumka, Kiev, English transl. Birkhauser, Basel (1986) 14. Levitan, B.M.: Inverse Sturm-Louville Problems. Nauka, Moscow, 1984, English transl. VNU Sci. Press, Utrecht (1987) 15. Yurko, V.A.: Inverse Spectral Problems fo Differential Operators and their Applications. Gordon and Breach, New York (2000) 16. McLaughlin, J.R.: Analytical methods for recovering coefficients in differential equations from spectral data. SIAM Rev. 28, 53–72 (1986) 17. Carlson, R.: A Borg-Levinson theorem for bessel operators. Pac. J. Math. 177(1), 1–26 (1997) 18. Hochstadt, H.: The Functions of Mathematical Physics. Wiley Interscience, New York (1971) 19. Koyunbakan, H., Panakhov, E.S.: Transformation operator for singular Sturm–Liouville equations. Int. J. Appl. Math. 14(2), 135–143 (2003) 20. Marchenko, V.A.: Certain problems of the theory of one dimensional linear differential operators of the second order. In: Trudy, I. (ed.) Moskovskogo Matematicheskogo Obshchestva, vol. 1, pp. 327–340 (1952) 21. Srivastava, H.M., Tuan, V.K., Yakubovich, S.B.: The Cherry transform and its relationship with a singular Sturm–Liouville problem. Q. J. Math. Oxf. Ser. 51(2), 371–383 (2000) 22. Zayed, A.I., Tuan, V.K.: Paley-Wiener-type theorem for a class of integral transforms arising from a singular Dirac system. Z. Anal. ihre Anwend. 19(3), 695–712 (2000) 23. Meschanov, V.P., Feldstein, A.L.: Automatic Design of Directional Couplers. Sviaz, Moscow (1980) 24. Litvinenko, O.N., Soshnikov, V.I.: The Theory of Heterogeneous Lines and their Applications in Radio Engineering. Radio, Moscow (1964) (in Russian) 25. Krueger, R.J.: Inverse problems for nonabsorbing media with discontinuous material properties. J. Math. Phys. 23(3), 396–404 (1982) 26. Shepelsky, D.G.: The inverse problem of reconstruction of te medium’s conductivity in a class of discontinuous and increasing functions. Adv. Sov. Math. 19, 209–231 (1994) 27. Anderssen, R.S.: The effect of discontinuities in density and shear velocity on the asypmtotic overtone sturcture of toritonal eigenfrequencies of the Earth. Geophys. J. R. Astron. Soc. 50, 303–309 (1997) 28. Lapwood, F.R., Usami, T.: Free Oscillations of the Earth. Cambridge Univ. Press, Cambridge (1981) 29. Hald, O.H.: Discontinuous inverse eigenvalue problems. Comm. Pure Appl. Math. 37, 59–577 (1984) 30. McNabb, A., Andersson, R.S., Lapwood, E.R.: Asypmtotic behaviour of the eigenvalues of a Strum-Liouville system with discontinious coefficients. J. Math. Anal. Appl. 54, 741–751 (1976) 31. Symes, W.W.: Impedence profile inversion via the first transport equation. J. Math. Anal. Appl. 94, 435–453 (1983) 32. Aktosun, T., Klaus, M., Mee, C.: Inverse wave scattering with discontinious wave speed. J. Math. Phys. 36(6), 2880–2928 (1995) 33. Eberhard, W., Freiling, G.G., Schneider, A.A.: On the distribution of the eigenvalues of a class of indefinite eigenvalue problem. Differ. Integral Equ. 3(6), 1167–1179 (1990) 34. Carlson, R.: An inverse spectral problem for Sturm–Liouville operators with discontinuous coefficients. Proc. Amer. Math. Soc. 120(2), 475–484 (1994) 35. Yurko, V.A.: On higher-order differential operators with a singular point. Inverse Problems 9, 495–502 (1993) 36. Yurko, V.A.: On higher-order differential operators with a regular singularity. Mat. Sb. 186(6), 133–160 (1995) 37. Yurko, V.A.: Integral transforms connected with differential operators having singularities inside the interval. Integral Transform. Spec. Funct. 5(3–4), 309–322 (1997)

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38. He, X., Volkmer, H.: Riesz bazes of solutions of Sturm–Liouville equations. Fourier Anal. Appl. 7, 297–307 (2001) 39. Amirov, Kh.R., Topsakal, N.: A representation for solutions of Sturm–Liouville equations with Coulomb potential inside finite interval. Journal of Cumhuriyet University Natural Sciences 28(2), 11–38 (2007) 40. Amirov, Kh.R., Topsakal, N.: On Sturm–Liouville operators with Coulomb potential which have discontinuity conditions inside an interval. Integral Transform. Spec. Funct. 19(12), 923–937 (2008) 41. Levitan, B.M.: Inverse Sturm–Liouville Problems. Nauka, Moskov, 1984, English transl. VNU Science, Utrecht (1987)

Math Phys Anal Geom (2010) 13:47–67 DOI 10.1007/s11040-009-9067-x

Wegner Estimates for Some Random Operators with Anderson-type Surface Potentials Yoshihiko Kitagaki

Received: 16 July 2008 / Accepted: 9 September 2009 / Published online: 20 November 2009 © Springer Science + Business Media B.V. 2009

Abstract For Schrödinger operator with random potentials concentrated near a surface, Wegner-type estimates are proven by using the spectral averaging method of Combes, Hislop and Klopp. These estimates allow us to show the local regularity of the integrated density of surface states at the gap of the background periodic operator. Acoustic operator with random surface potentials is treated similarly. Keywords Random Schrödinger operators · Density of states · Wegner estimates · Surface states Mathematics Subject Classifications (2000) 35P20 · 46N50 · 47B80

1 Introduction Recently, Combes, Hislop and Klopp [10] proved an optimal Wegner-type estimate at all energies. In this article, we consider Schrödinger operators with the random surface potential Hω = H0 + Vs,ω on L2 (Rd ) for d  2 and apply this idea to prove the regularity of the integrated density of surface states Ns (E) (IDSS as an acronym) outside of the spectrum of the background periodic Schrödinger operators H0 = − + V0 . For x = (x1 , x2 ) ∈ Rd1 × Rd−d1 , an Anderson-type random surface potential constructed from the nonzero

Y. Kitagaki (B) Graduate School of Human and Environmental Studies, Kyoto University, Kyoto 606-8501, Japan e-mail: [email protected]

48

Y. Kitagaki

single site potential u  0 and the bounded random variables {ωγ }γ ∈Zd1 is defined as  Vs,ω (x) = ωγ u(x1 − γ , x2 ). (1.1) γ ∈Zd1

This potential is concentrated near a surface Rd1 × {0} for 1  d1 < d. By Wegner-type estimate, we prove that if the conditional probability measure of a single site random variable is Hölder continuous of order 0 < α  1, then the IDSS is Hölder continuous of order α at the gap of H0 . In discrete case, if the probability measure has a bounded density with compact support, it is known to be true that the IDSS is locally Lipchitz continuous outside of the spectrum of the discrete Laplacian [8, 21]. This corresponds exactly to our case of α = 1. Indeed, we study the corresponding discrete Schrödinger operator with random surface potentials on 2 (Zd ). As a simple application, we also study the following random acoustic operators formally written by  1 1 ∇=− ∂x j ∂x ρs,ω ρs,ω j j=1 d

Aω = −∇

(1.2)

on L2 (Rd ) for d  2, where ρs,ω = ρ0 (1 + Vs,ω ) is a random media. ρ0 is a positive background periodic potential and Vs,ω is an Anderson-type random surface perturbation defined as (1.1). We can also show that the IDSS for these models have the same property. The Wegner estimate [40] was originally used to show a bound of the density of states, later it was applied as a probabilistic key estimate for the proof of Anderson localization (the pure point spectrum and the exponential decaying eigenfunctions) via the multiscale analysis. For the moment, we assume that the periodic potential V0 = 0 and that the random coupling constants {ωγ }γ ∈Zd1 ω are independent and identically distributed. Let H be the Dirichlet restriction of Hω = − + Vs,ω to the cube  = (−L/2, L/2)d . Let us assume that the bottom of the spectrum of Hω is strictly negative, that is, σ (Hω ) = [E0 , ∞) almost surely and E0 < 0. Set Iε = [E − ε, E − ε] ⊂ σ (Hω ) \ σ (−) = [E0 , 0). Kirsch and Veseli´c [28] proved a Wegner-type estimate for some random sparse potentials, which include random surface potentials. By applying the theory of spectral shift function (SSF for the short form) in [11], they showed ω E Tr E (Iε )  Cg∞ εα Ld1 .

(1.3)

ω ω for any 0 < α < 1. E (Iε ) is the spectral projection of H to the interval Iε . ω This trace is the number of eigenvalues of H in Iε . Hence, this estimate allow us to show the α-Hölder continuity of IDSS at negative energies. However, they assumed that the single site measure had a bounded density g with compact support.

Wegner Estimates for Operators with Anderson-type Potentials

49

Boutet de Monvel and Stollmann [5] proved Anderson localization (Exponential localization and Dynamical localization) for random surface models near the bottom of the spectrum at negative energies, i.e. 1. (Exponential localization) There is E1 ∈ (E0 , 0) such that almost surely Hω has pure point spectrum in I EL = [E0 , E1 ] with exponentially decaying eigenfunctions in the L2 sense, i.e. I EL ∩ σ pp (Hω ) = ∅ ∃ C,

and

I EL ∩ σc (Hω ) = ∅ P a.s.

γ > 0 s.t. χx φ2  Ce−γ |x|

where φ is the corresponding eigenfunction with the eigenvalue in [E0 , E1 ],  · 2 is L2 (Rd ) norm, χx is the characteristic function of the unit cube centered at x and |x| := max{|x1 |, |x2 |, · · · , |xd |}. 2. (Dynamical localization) There exists an E1 ∈ (E0 , 0) such that in I DL = [E0 , E1 ], we have the dynamical localization in the sense that:   n −itHω ω 2 E sup  |X| e E (I DL )χ K 2 < ∞ t>0

for any compact set K ⊂ Rd and for some n > 0, where Eω (I) is the spectral projection of Hω associated with I,  · 2 is Hilbert-Schmidt class norm and |X| is the multiplication operator of |x|. Anderson localization can be interpreted from either the spectral or the dynamical view of point as above. Sometimes, as a weaker version of the exponential localization, for the random operator Hω , the following absence of continuous spectrum in an energy interval I SL is only proven, I SL ∩ σ pp (Hω ) = ∅,

I SL ∩ σc (Hω ) = ∅

(1.4)

for P almost all ω. This is called the spectral localization. We also remark that the dynamical localization implies the exponential localization, that is, for each energy region I DL , I EL and I SL , the relation I DL ⊂ I EL ⊂ I SL holds. See e.g. [20, 30]. The proof of Andeson localization was given by the standard argument called the multiscale analysis, which was started by Fröhlich and Spencer [19]. This technique was further developed and simplified in order to apply the various models, see [7, 20, 25, 30, 38]. In this proof, the key Wegner-type estimate was given as    ω   P dist σ H , E  ε  Cεα L2d1 . (1.5) They assumed α-Hölder continuity of the single site probability measure. This Wegner-type estimate is sufficient to prove Anderson localization via the

50

Y. Kitagaki

multiscale analysis. However, one can not derive a property of IDSS from this estimate because of the volume factor L2d1 . Therefore, we will improve with respect to both the volume term in (1.5) and the term of the single site measure in (1.3). Another method to prove Anderson localization is to use the fractional moment method initiated by Aizenman and Molchanov in [1]. In [4], the localization properties for random surface potentials were shown by applying this method. Without using Wegner-type estimates, Kostrykin and Schrader [33] proved some regularity of the IDSS at all energies by analyzing SSF for − + V and −, where the local random potential V is defined by V (x) =



ωγ u(x1 − γ , x2 ).

(1.6)

γ ∈Zd1 ∩ p

Using the argument in [11], they obtained that Ns (E) ∈ Lloc (R) for 1  p < ∞, in other words, the IDSS is a measurable locally integrable function. Our results generalize this regularity property only at the energies exterior to the spectrum of the background operator. Moreover, both continuous and discrete Anderson-type random surface models are studied extensively, especially about the behavior of IDSS at the edge of the spectrum (Lifshitz asymptotic) and the existence of the absolutely continuous spectrum. See e.g. [6, 15, 16, 23, 24, 26, 29], Klopp and Tchebotareva (unpublished manuscript) and references therein. This paper is organized as follows. In Section 2, we present the main hypotheses and results. The proof of Wegner-type estimate is given in Section 3. In Section 4, we treat the corresponding discrete model. In Section 5, we explain an application for the random surface acoustic operator.

2 The Main Results We study the random self-adjoint Schrödinger operators Hω = − + V0 + Vs,ω acting on the Hilbert space L2 (Rd ) of complex-valued square integrable functions on Rd (d  2), where = dj=1 ∂ 2 /∂ x2j is the free Laplacian, V0 is a bounded, Zd -periodic potential such that V0 (x) = V0 (x + γ ),

∀γ ∈ Zd ,

∀x ∈ Rd

We define H0 = − + V0 . Vs,ω is an Anderson-type random surface potential defined by (1.1). The single site potential u(x) ∈ L∞ 0 is a nonzero and nonnegative function with |u|∞  1 and its support is included in a unit cube 1 = (1/2, 1/2)d . The non constant random coupling constants {ωγ }γ ∈Zd1 take values in the finite interval [m0 , M0 ] and form a real-valued, bounded process on Zd1 with probability space (P, ). In order to describe the dependence

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51

on the probability measure P, we let μ j denote the conditional probability measure for the random variables ω j at the site j ∈ Zd1 , conditioned on all the random variables {ωk }k = j; that is,   μ j([E, E + ε]) = P ω j ∈ [E, E + ε]|{ωk }k = j . The Wegner-type estimate and the regularity for the IDSS are expressed in terms of the quantity

 s(ε) = sup E sup μ j([E, E + ε]) . j∈Zd1

E∈R

The measure μ j([E, E + ε]) depends on all the other values {ωk }k = j of the random configuration. As a remark, in applications to continuity of the IDSS or Anderson localization, the rate of vanishing of s(ε) as ε 0 is essential. Clearly, if we assume that {ωγ } is i.i.d. with a common distribution μ, one has simply μ j = μ for any j ∈ Zd1 and s(ε) = sup E∈R μ([E, E + ε]). ω The main result in this paper is the following. Let H be the the Dirichlet, periodic or Neumann restriction of Hω to the cube  = (−L/2, L/2)d . Let ω ω E (I) be the spectral projection of H to an interval I. Theorem 2.1 Fix an interval I ⊂ R \ σ (H0 ) with |I| small enough. Then, there exists a finite constant C > 0 such that for any compact interval Iε = [E − ε, E + ε] ⊂ I, one has      ω   ω P dist σ H ([E − ε, E + ε]) , E  ε  E Tr E  Cs(2ε)Ld1 .

(2.1)

Remark 2.1 We always assume σ (Hω ) \ σ (H0 ) is not empty. This assumption is always satisfied for Schrödinger operators of the type H0 + Vs,ω if m0 < 0. Indeed, since the periodic operator H0 is bounded from below, the set σ (Hω ) ∩ (−∞, inf σ (H0 )) is non-empty in this case. ω Of course, it is important and interesting to study E Tr E (I) for any interval I ⊂ R. However, we do not obtain same Wegner-type estimate on the spectrum of the unperturbed operator H0 . Moreover, we do not know how to prove the following estimate; there is a constant C > 0 such that  ω  0   C| supp f |Ld1 E Tr f H − f H (2.2)

for sufficient smooth compact supported function f on R, provided some assumptions for ωγ . If we proceed along the same lines of [10], it seems to need some estimate like Theorem 2.1 in [10] for the surface potential. But then, if we only consider the spectral gaps of unperturbed operator, especially the energies below inf σ (H0 ), our proof of Wegner estimate may be applicable for not only surface-type random models but also another random models. In this article, for the simplicity, we will take the random potentials with the

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Y. Kitagaki

lattice structure Zd1 , but this is not essential in our proof, hence we expect to generalize the results to cover the sparse random potentials described in [3, 28]. In order to state the corollary, we first recall the integrated density of states N(E) (IDS as an acronym) for the standard Anderson model Hω = − + V0 + Vω (d1 = d and the random variables {ωγ }γ ∈Zd are i.i.d). This is ω defined as follows: we describe by H the Dirichlet, periodic or Neumann restriction of Hω to the cube  = (−L/2, L/2)d . As Hω is elliptic, the resolvent ω ω of H is compact and, consequently, the spectrum of H is discrete and is made of isolated eigenvalues of finite multiplicity. We define the eigenvalue ω counting function N Lω (λ), which is the number of eigenvalues of H smaller d than λ. Let || = L be the volume of  in the sense of Lebesgue measure. It is known that the limit of N Lω (E)/|| when  tends to Rd exists almost surely and is independent of the boundary condition. It is called the IDS of Hω . There are many literatures on the IDS of random Schrödinger operators, for example, we refer to the recent review [27] and the books [7, 36, 39]. Next, let us consider the surface Anderson model Hω = − + V0 + Vs,ω . We assume that the random variables {ωγ }γ ∈Zd are i.i.d. Similarly, we let N Lω (E) ω smaller than E and N L0 (E) be also the be the number of eigenvalues of H 0 eigenvalue counting function of H . Since one has to get rid of the bulk spectrum to recover properties of the surface potential, we treat the limit of  1  ω N L (E) − N L0 (E) d L1 when L → ∞. This limit is taken in a distribution sense, that is,    1 ω 0 f (λ)dN L (λ) − f (λ)dN L (λ) lim L→∞ Ld1

(2.3)

(2.4)

for sufficiently smooth functions f with compact support. In other words, we take the limit of the following trace ν Ls,1 ( f ) =

 ω  0  1 Tr f H − f H . d 1 L

(2.5)

One can also consider the following ν Ls,2 ( f ) =

1 Tr[χ ( f (Hω ) − f (H0 ))], Ld1

(2.6)

where χ is the characteristic function of a cube  = (−L/2, L/2)d . Moreover, one can also defines the functional, ν Ls,3 ( f ) =

   1 Tr f H0 + Vω − f (H0 ) , d 1 L

(2.7)

where Vω is the local random potential as (1.6). The existence of these limits as L → ∞ is not clear. However, it is known that each limit of (2.5), (2.6) and (2.7) exists almost surely and is non random. Moreover, (2.5), (2.6) and

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53

(2.7) have same limit. For example, under the discrete setting, [8] proved the existence of the limit for all f ∈ C2 (R) such that (1 + |x|)2 f ( j) ∈ L2 (R), j = 1, 2. For the continuous models, [16] showed the existence for all f ∈ C3 (R) with f (x) = O(e−α|x| ) for some α > 0 and [32] proved the existence for all f ∈ C02 (R). We write this limit as ν s ( f ). So, taking the expectation, one has ν s ( f ) = lim

L→∞

 ω  0  1 E Tr f H − f H = d L1

 f (λ)dνs (λ).

(2.8)

The measure νs is defined by

  νs (A) = E Tr χs0 (χ A (Hω ) − χ A (H0 ))χs0

(2.9)

for A a Borel set in R, where s0 = [1/2, 1/2] × Rd−d1 and χs0 is the characteristic function of s0 . This is called the density of surface states measure. And this distribution function Ns of νs , defined by Ns (E) = νs ((−∞, E])

(2.10)

is known as the integrated density of surface states. See [3, 8, 15, 16, 29], Klopp and Tchebotareva (unpublished manuscript). In [32], with respect to the spectral shift function (SSF), Kostrykin and Schrader studied the limit of (2.7). In fact, if we use the SSF ξ(λ; H0 + Vω , H0 ) for the pair H0 + Vω and H0 (see e.g.[11, 32, 33]), by the famous trace formula, (2.7) can be written as    1 f  (λ)ξ λ; H0 + Vω , H0 dλ. (2.11) d 1 L They showed that the limit of ν Ls,3 ( f ) exists almost surely and is non random. This limit functional ν s ( f ) defines a signed Borel measure dns . So, taking the expectation, one also has 

 ω  0  1 (2.12) − f H = f  (λ)dns (λ). lim d E Tr f H L→∞ L 1 Moreover, in [33], they proved that the measure dns is Lebesgue absolutely continuous, i.e. dns (λ) = Ns (λ)dλ. We will call also this density Ns (λ) the IDSS. So one also gets  

 ω  0  1  lim E Tr f H − f H = f (λ)Ns (λ)dλ = f (λ)Ns (λ)dλ. L→∞ Ld1 (2.13) By definition, the IDS N(E) is a monotone nondecreasing function on R. But the IDSS is not monotone decreasing on R. Now, let us consider the restricted energies R \ σ (H0 ). Then the function N Lω (E) − N L0 (E) is a monotone nondecreasing function for E ∈ R \ σ (H0 ). Because the value of N L0 (E) is constant on R \ σ (H0 ). So the Ns (E) is also the monotone nondecreasing on 0 R \ σ (H0 ). In other words, if supp f ⊂ R \ σ (H0 ), since the trace Tr f (H ) vanishes, consequently, (2.8) implies that the restriction of the measure dνs

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Y. Kitagaki

on R \ σ (H0 ) is a positive measure and that the distribution function Ns (E) for E ∈ R \ σ (H0 ) is the monotone nondecreasing function. In this case, the argument for the IDSS on R \ σ (H0 ) is essentially similar to the IDS. Hence, by Theorem 2.1, we can state a corollary to finish this section. Corollary 2.1 Assume that the density of surface states measure νs and the IDSS Ns exist. Let I0 ⊂ R \ σ (− + V0 ) be small enough. Then, there exists a finite constant C > 0 such that for any I ⊂ I0 , we have νs (I)  Cs(|I|).

(2.14)

Assume that the single site measure is α-Hölder continuous, that is, there exists a finite constant c > 0 such that s(ε)  cεα for 0 < α  1. Then, there exists a finite constant C > 0 such that for any E1 < E2 ∈ I we have 0  Ns (E2 ) − Ns (E1 )  C|E2 − E1 |α .

(2.15)

In particular, when α = 1, the IDSS Ns (E) is locally Lipschitz continuous on R \ σ (H0 ) and the density of surface states ρs (E) = dNs (E)/dE exists as a locally bounded function. Remark 2.2 For (2.14), we assume that (2.8) holds for any f ∈ C0∞ (R). This is satisfied for i.i.d. case. Moreover, for (2.15), assume that for almost all E ∈ R \ σ (H0 ) the IDSS defined by Ns (E) = lim

L→∞

 1  ω E N L (E) − N L0 (E) d 1 L

(2.16)

exists. It is known that if E < inf σ (H0 ) and the random variables {ωγ }γ ∈Zd1 are i.i.d, this assumption (2.16) is satisfied for E < inf σ (H0 ) where the limit function Ns (E) is continuous. See Theorem 1.3. in [29]. By N L0 (E) = 0 for E < inf σ (H0 ), the similar method for IDS is applicable for the IDSS. As we said above, because Ns (E) is a monotone nondecreasing for E < inf σ (H0 ), it has at most a countable discontinuities. We define it to be right continuous. Hence, by Theorem 2.1, one has for E1 < E2 < inf σ (H0 ) in a dense set of energies S ⊂ (−∞, inf σ (H0 )), and it follows (2.15). The monotonicity of the IDSS below inf σ (H0 ) implies its Hölder continuity. For further discussion about the IDSS, we remark that the following estimate is known, lim

E↓E0

d1 log | log Ns (E)| =− , log(E − E0 ) 2

where E0 < inf σ (H0 ) is the bottom of the spectrum of Hω . This is shown by Kirsch and Warzel (Theorem 1.4. in [29]). This behavior for Ns (E) is known as Lifshitz tails (see, e.g. [7, 36]). Kirsch and Klopp in [26] studied the discrete

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55

models. For the internal Lifshitz tails, we refer to Klopp and Tchebotareva (unpublished manuscript). We note that this upper bound of Ns (E), i.e. −d1 /2

Ns (E)  c1 e−c2 (E−E0 )

is used for the initial estimate, which gives the first step in the multiscale analysis for the proof of Anderson localization, see Proposition 5.2 in [29].

3 Proof of the Main Theorem Before the proof, we will prepare some lemmas. We first note an essential d estimate in [10]. We let u ∈ L∞ 0 (R ) be a bounded function with compact support appeared in our model as a single site potential. We write u j(x) =  = Zd1 ∩ . u(x1 − j, x2 ) for j ∈  Lemma 3.1 For any φ ∈ L2 (Rd ) and any Iε = [E − ε, E + ε] ⊂ R, one has   ω E φ, u j E (Iε )u jφ  4|u|∞ (1 + |u|∞ )s(2ε)φ2 .

(3.1)

To show this lemma, we quote the general result from ([10], Theorem 3.1). Lemma 3.2 Let A and B be two self-adjoint operators on a separable Hilbert space H , and suppose that B is bounded and nonnegative. Then, for any φ ∈ H , we have the bound 

  sup Bφ, ((A + (n + y)B)2 + 1)−1 Bφ  π B(1 + B)φ2 .

(3.2)

n∈Z y∈[0,1]

Proof (Lemma 3.1) Because of the structure of our random operator, we can ω ω = H (ω j = 0) + write our model as the one-parameter family of operators H ˜ ω ju j for j ∈ . Hence, we can apply new spectral averaging method by Combes, Hislop and Klopp for general probability measure. By using the inequality given in ([10], Eq. 3.1), one has ω E (Iε )

4  π

 Iε

 ω −1 dE H − E − i2ε .

(3.3)

Proceeding as in the proof of Proposition 3.2 in [10], one has 

ω φ, u j H

− E − i2ε

−1

−1    ω j 2 1 u jφ = Bφ , (3.4) Bφ, A + B +1 2ε 2ε

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Y. Kitagaki

ω where A = 2ε1 (H (ω j = 0) − E) and B = u j. Hence, taking the expectation of (3.4), dividing the integration and changing the variables, we then obtain

 E

R

 −1   ω dμ j(ω j) φ, u j H − E − i2ε u jφ

  −1   ω j 2 1  2(n+1)ε B +1 dμ j(ω j) Bφ, A + Bφ 2ε n∈Z 2nε 2ε  1  E sup μ j ([2mε, 2(m + 1)ε]) 2ε m∈Z   2 −1 sup Bφ, ((A + (n + y)B) + 1) Bφ . × =E

(3.5)

n∈Z y∈[0,1]

From (3.2) with B  |u|∞ and (3.5), we have 

 −1   ω π dμ j(ω j) φ, u j H − E − i2ε u jφ  s(2ε)|u|∞ (1 + |u|∞ )φ2 . 2ε R (3.6) To obtain (3.1), we combine (3.3) with (3.6). E

Here, we quote a variant of Combes-Thomas estimate. This gives the exponential estimate for p-th powers of the resolvent in q-th von NeumannSchatten class Iq . Let A be the operator norm and Aq be the q-th von Neumann-Schatten class norm (see, e.g., [37]) Let χ0 be the characteristic function of the cube of center 0 and side length 1 in Rd . Let χα be its translated by the vector α, i.e. χα (x) = χ0 (x − α). Lemma 3.3 Let p, q be integers such that pq > d/2. Then, there exists c1 , c2 > 0 such that for any V ∈ L∞ (Rd ), for any z ∈ σ (− + V) and for any (α, β) ∈ Zd × Zd , the operator χα (− + V − z)− p χβ is q-th Schatten class and one has χα (− + V − z)− p χβ q  c1 e−c2 |α−β| .

(3.7)

For example, the proof is found in [31]. We note that this Combes-Thomas estimate is also true for the restriction of the operator − + V in a cube  with Dirichlet, periodic, or Neumann boundary condition. Originally, CombesThomas inequality was proved in [13]. This estimate concerns the exponential decay estimate in the operator norm of the resolvent operator. Afterward, the estimate in the trace-class norm was improved in [2]. This decay estimate is very useful tool. In [10], the decay estimate for the second power of the resolvent in trace class norm has been also applied. Now, let us start to prove the main theorem. First, we give the following lemma.

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57

Lemma 3.4 Fix I ⊂ R \ σ (H0 ). Let |I| be small enough. Then for m = 0, 1, 2..., there exists a constant Cm > 0 such that for any Iε = [E − ε, E + ε] ⊂ I, ω ω ω Tr Km E (Iε )(K∗ )m  Tr Km+1 E (Iε )(K∗ )m+1 + Cm ε Tr E (Iε ).

(3.8)

−1   0  −E Here K = H − Vω .  0 −1 Proof (Lemma 3.4) We write R0 = H −E and Vω = γ ∈  ωγ uγ . By  ω  ω ω ω (Iε ) = KE (Iε ) + R0 H − E0 E (Iε ) (3.9) E and its adjoint version ω ω ω ω E (Iε ) = E (Iε )K∗ + E (Iε )(H − E0 )R0 ,

(3.10)

 ω  ω  ω 2   ω ω E (Iε ) = E (Iε ) = KE (Iε ) + R0 H − E0 E (Iε )  ω    ω ω · E (Iε )K∗ + E (Iε ) H − E0 R0 .

(3.11)

We get

Now, in (3.11), operate Km from the left and (K∗ )m from the right. Then one has ω ω K m E (Iε )(K∗ )m = Km+1 E (Iε )(K∗ )m+1   ω ω +Km R0 H − E0 E (Iε )(K∗ )m+1  ω  ω +Km+1 E (Iε ) H − E0 R0 (K∗ )m  ω  ω   ω +Km R0 H − E0 E (Iε ) H − E0 R0 (K∗ )m . (3.12)

We will estimate this trace except for the first term. First, the trace for the second term of (3.12) is dominated by  ω  ω  ω ω Km R0   H − E0 E (Iε ) E (Iε )1 (K∗ )m+1   Cε Tr E (Iε )

(3.13)

Here we used the fact that our local random potential Vω is bounded. By R0   1/δ, δ = dist(σ (H0 ), I), we see that the constant C > 0 in (3.13) is uniformly bounded in E and ω. Next, the estimate for the third term of (3.12) is same and the trace for the forth term of (3.12) is also dominated similarly by  ω  ω  ω  ω  ω  Km R0   H − E0 E (Iε ) E (Iε )1  E (Iε ) H − E0  R0 (K∗ )m  ω  Cε2 Tr E (Iε ).

So, this finish the proof. Now, we combine the previous lemmas in order to finish the proof.

(3.14)  

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Y. Kitagaki

Proof (Theorem 2.1) Choose an integer m > d/4. By (3.8) in Lemma 3.4, we have ω ω ω (Iε )  Tr KE (Iε )K∗ + C0 ε Tr E (Iε ) Tr E ω ω  Tr K2 E (Iε )(K∗ )2 + (C0 + C1 )ε Tr E (Iε ) m−1   ω ω .....  Tr Km E (Iε )(K∗ )m + Ck ε Tr E (Iε ).

(3.15)

k=0

  m−1 ε is strictly smaller C Hence, if we choose |I| small enough so that k k=0 than 1, we can find some constant M > 0 such that ω ω Tr E (Iε )  M Tr Km E (Iε )(K∗ )m .

(3.16)

This constant M is independent of ω, hence we get ω ω E Tr E (Iε )  ME Tr Km E (Iε )(K∗ )m .

(3.17)

Expanding Vω and using the bound of random coupling constant ωγ , we can estimate this trace as  ω (3.17)  C E Tr R0 uγ2 R0 · · · uγm R20 uγm+1 · · · R0 uγ2m E (Iε )uγ1 . )2m γ1 ,··· ,γ2m ∈(

(3.18) Here, we used Tr AB = Tr B A. Taking a characteristic function χγ1 , χγ2m such that uγ1 = uγ1 χγ1 , uγ2m = uγ2m χγ2m , we get  ω ω E Tr E (Iε )  C E Tr T(γ ) uγ2m E (Iε )uγ1 . (3.19) )2m γ1 ,γ2 ,·,γ2m ∈(

Here, we defined T(γ ) = χγ1 R0 uγ2 R0 · · · uγm R20 uγm+1 · · · R0 χγ2m . It is not hard to show that T(γ ) is trace class for any (γ ) = (γ1 , γ2 , ·, γ2m ). In fact, using Hölder inequality and the fact that χγ R0 ∈ I2m for m > d/4 (see e.g., [37]), one has T(γ ) 1  χγ1 R0 2m uγ2 R0 2m · · · uγm R0 2m R0 uγm+1 2m · · · R0 χγ2m 2m

< ∞.

(3.20)

Since T(γ ) is a trace-class operator, and hence a compact operator, we have the singular value decomposition for T(γ ) . So, for the index (γ ), there exist a pair  (γ )   (γ )   (γ )  of orthonormal basis, φ j j, ψ j j and the singular values λ j j such that T(γ ) =

∞  j=1

 (γ ) (γ )  (γ ) λj ψj , · φj .

(3.21)

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59

Inserting the representation (3.21) into (3.19) and expanding the trace in  (γ )  φ j , we have E



ω | Tr T(γ ) uγ2m E (Iε )uγ1 |

(γ )

=E

 (γ )



j

1  2

(γ )

(γ )  (γ ) (γ )  ω λ j ψ j , uγ2m E (Iε )uγ1 φ j

 (γ ) (γ )  (γ ) (γ )  (γ )  ω ω λ j E ψ j , uγ2m E (Iε )uγ1 ψ j + φ j , uγ2m E (Iε )uγ1 φ j .

j

In order to estimate the above expectations by some factor independent of (γ ) (γ ) (γ ) and m, we apply Lemma 3.1. By (3.1), ψm  = φm  = 1 and |u|∞  1, we get  ω ω E Tr E (Iε )  C E Tr T(γ ) uγ1 E (Iε )uγ2 (γ )

 Cs(2ε)



T(γ ) 1

(3.22)

(γ )

Finally, we use the Combes-Thomas bound in order to estimate the sum of the trace norm of T(γ ) 1 . Using Hölder inequality and taking a characteristic function χγ such that uγ = uγ χγ , one has     χγ R0 uγ R0 · · · uγ R2 uγ · · · R0 χγ  T(γ ) 1 = 1 2 m 2m 1 0 m+1 )2m γ1 ,··· ,γ2m ∈(

(γ )





χγ1 R0 uγ2 2m χγ2 R0 uγ3 2m · · ·

)2m ,γ ∈∩Zd γ1 ,··· ,γ2m ∈(

· · · χγm R0 uγ 2m χγ R0 uγm+1 2m · · · χγ2m−1 R0 χγ2m 2m . We used R20 = γ ∈∩Zd R0 χγ R0 . Since we took 2m > d/2, we can apply the exponential decay estimate (3.7) in Lemma 3.3 for each term of (3.23). So, we obtain  ˜ T(γ ) 1  C||. (3.23) (γ )

Combine (3.22) with (3.23) to finish the proof.

 

4 Discrete Model In this section, we treat the corresponding discrete model. Consider a random self-adjoint discrete Schödinger operator Hω = + Vs,ω on the Hilbert space

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Y. Kitagaki

2 (Zd ) of the complex-valued square summable functions for d  2, where is the discrete Laplacian defined as 

u(x) =

u(x + y)

|y|=1

for u(x) ∈ 2 (Zd ). Vs,ω is an Anderson-type random surface potential described as  Vs,ω = ωγ γ , γ ∈Zd1

where γ = |δγ δγ | is the projector onto the standard basis vector δγ ∈ 2 (Zd ). {ωγ }γ ∈Zd1 are the non constant random coupling constants, which take values in the finite interval [m0 , M0 ]. When {ωγ } are i.i.d. and the single site measure has a compactly supported bounded density g with |g|∞ < ∞, Chahrour [8] gave an upper bound of the density of surface states ns (E) on R \ [−2d, 2d] as follows. Fix E0 > 2d, then ns (E) =

dNs (E) 2K|g|∞ ,  dE E0 − 2d

(4.1)

for all E ∈ R \ [−E0 , E0 ]. If we use the density of surface states measure νs , for any compact interval I ⊂ R \ [−E0 , E0 ], it follows that νs (I) 

2K|g|∞ |I|. E0 − 2d

(4.2)

Here, the constant K is given by K = max{|m0 |, |M0 |}. Remark that we know σ ( ) = [−2d, 2d] easily by Fourier transform. Moreover, Kostrykin and Schrader show that |Ns (E)|  1 for Lebesgue almost every E ∈ R. See Theorem 2 in [33]. In this section, following the analysis in Section 3, we prove a Wegner-type estimate for this discrete model, and consequently we can improve (4.2) with respect to the single site measure. ω Let H be the the Dirichlet, periodic or Neumann restriction of Hω to the ω cube  = [−L, L]d ⊂ Zd for L ∈ N. Simply, we may take H on () defined ω by the matrix elements H (n, m) = δn , Hω δm  whenever n and m belong to . ω ω Let E (I) be the spectral projection of H to an interval I. Theorem 4.1 Fix I ⊂ R \ σ ( ) with |I| small enough. Then, there exists a finite constant C > 0 such that for any compact interval Iε = [E − ε, E + ε] ⊂ I, we have    ω     ω P dist σ H , E  ε  E Tr E ([E − ε, E + ε])  Cs(2ε)Ld1 .

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61

The essential estimate is the following. ˜ =  ∩ Zd1 . For any δγ Lemma 4.1 Take γ ∈  E + ε] ⊂ R,   ω E δγ , E  (Iε )δγ  8s(2ε).

and any Iε = [E − ε, (4.3)

This is the discrete analogue for Lemma 3.1 in Section 3. The proof is similar to the continuous case. For the reader, we refer the appendix in [9] and the work of Krishna [34]. One can also find the spectral averaging results for the rank one perturbation under the general setting. Proof (Lemma 4.1) We can also write our discrete model as the one-parameter ω ω ˜ So, proceeding as family of operators H = H (ωγ = 0) + ωγ γ for γ ∈ . the continuous case, one has  −1     ω −1  ω j 2 1 Bδγ ,  δγ , H − E − i2ε δγ = Bδγ , A + B +1 2ε 2ε

(4.4)

ω where A = 2ε1 (H (ωγ = 0) − E) and B = γ . Hence, taking the expectation of (4.4) like (3.5), one gets    ω −1  E dμ j(ω j) δγ , H − E − i2ε δγ R

     1 2 −1  E sup μ j([2mε, 2(m+1)ε]) sup Bδγ , ((A+(n+ y)B) + 1) Bδγ . 2ε m∈Z n∈Z y∈[0,1] (4.5) Using (3.2) in Lemma 3.2, one has 

  sup Bδγ , ((A + (n + y)B)2 + 1)−1 Bδγ  π B(1 + B)δγ 2 .

n∈Z y∈[0,1]

(4.6) From (4.5), (4.6), δγ  = 1 and B  1, we obtain    ω −1  π E dμ j(ω j) δγ , H − E − i2ε δγ  s(2ε). ε R Combine (4.7) with (3.3) to obtain (4.3). Proof (Theorem 4.1) Set Vω =



˜ γ ∈

(4.7)  

ωγ γ and R0 = (  − E)−1 . By

  ω  ω  ω ω E (Iε ) = R0 − Vω E (Iε ) + R0 H − E E (Iε ),

(4.8)

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Y. Kitagaki

we have ω Tr E (Iε )

  ω   ω  ω ω ω E (Iε ) + R0   H  R0  Tr − V − E E (Iε ) E (Iε )]1  ω ω  1/δ |ωγ | Tr γ E (Iε ) + ε/δ Tr E (Iε ), (4.9) ˜ γ ∈

where we took δ = dist(σ ( ), I) > 0. Since the coupling constants {ωγ }γ ∈˜ are bounded and ε is small enough, there is a finite constant C > 0 such that    ω ω Tr γ Eω (Iε ) = C δγ , E  (4.10) Tr E (Iε )  C (Iε )δγ .  ˜ γ ∈

˜ γ ∈

Taking the expectation of (4.10) and using (4.3) in Lemma 4.1, we obtain ω ˜ E Tr E (Iε )  Cs(2ε)||.

(4.11)  

5 Acoustic Operator We finally consider random surface acoustic operators. This operator is formally described by (1.2) acting on the Hilbert space L2 (Rd ) for d  2. ρ0 is a positive bounded Zd -periodic potential, that is, we assume that ρ0 (x) = ρ0 (x + γ ),

∀γ ∈ Zd ,

∀x ∈ Rd

and 0 < ρ0−  ρ0 (x)  ρ0+ < ∞. ρs,ω is a positive random surface potential defined as    ωγ u(x1 − γ , x2 ) ρs,ω (x) = ρ0 (x) 1 + γ ∈Zd1

for x = (x1 , x2 ) ∈ Rd1 × Rd−d1 , where 1  d1 < d. The single site potential u(x) ∈ L∞ 0 is a nonzero and nonnegative function with |u|∞  1 and its support is included in a unit cube. The non constant random coupling constants {ωγ }γ ∈Zd1 take values in the finite interval [0, M0 ]. Let A (ρs,ω ) be the quadratic form defined as follows : for u ∈ H 1 (Rd ),  1 A (ρs,ω )[u, u] = ∇u(x)∇u(x)dx. Rd ρs,ω (x) This is a symmetrical, closed and positive quadratic form. Aω is defined to be the self-adjoint operator associated to A (ρs,ω ). Similarly, we define the background periodic operator A0 = −∇

1 ∇ ρ0

Wegner Estimates for Operators with Anderson-type Potentials

63

as the self-adjoint operator associated to the quadratic form A (ρ0 ). For the standard case d1 = d, there are some works about Anderson localization and Lifshitz tails. See e.g. [12, 17, 35]. Now, we state a Wegner-type estimate for our random surface acoustic operator. Let Aω be the the Dirichlet, periodic or Neumann restriction of Aω ω to the cube  = (−L/2, L/2)d . Let E (I) be the spectral projection of Aω to an interval I. Theorem 5.1 Fix I ⊂ [0, ∞) \ σ (A0 ) with |I| small enough. Then, there exists a finite constant C, c > 0 such that for any compact interval Iε = [E−ε, E+ε] ⊂ I, we have         ω P dist σ Aω , E  ε  E Tr E ([E − ε, E + ε])  Cs(2cε)Ld1 .

(5.1)

Remark 5.1 We also assume that [0, ∞) \ σ (A0 ) is not empty. This is non trivial assumption, because 0 belongs to the spectrum of Aω independently of the choice of ρs,ω . Figotin and Kuchment in [18] studied the possibility of the existence of open spectral gaps in the spectrum of the periodic acoustic operators. Indeed, the fact that the operator A0 has a spectral gap for certain periodic potential ρ0 is known. In this case, we can choose M0 > 0 large enough so that σ (Aω ) \ σ (A0 ) is not empty. See [12, 17]. Before the proof of Theorem 3, we quote an useful lemma. See e.g. [14]. Let σ p (T) be the set of eigenvalues of the operator T. Lemma 5.1 Let H1 and H2 be two separable Hilbert spaces. Let T be a densely defined closed operator from H1 to H2 . Then, T ∗ T on H1 and TT ∗ on H2 are the nonnegative self-adjoint operators. Then, one has σ p (T ∗ T) \ {0} = σ p (TT ∗ ) \ {0}.

(5.2)

Moreover, for any λ ∈ σ p (T ∗ T) \ {0}, one has dim ker(T ∗ T − λ) = dim ker(TT ∗ − λ).

(5.3)

By this lemma, we first reduce our acoustic operator to a Schrödinger-type operator. Let χ I (T) be the spectral projection of the self-adjoint operator T to a compact interval I. We have the following reduction formula. Proposition 5.1 Assume that 0 ∈ / [E − ε, E + ε]. There exists a finite constant c > 0 such that    ω  E Tr χ[E−ε,E+ε] Aω  E Tr χ[−cε,cε] H (E) , (5.4) where Hω (E) = − − Eρs,ω .

64

Y. Kitagaki

Proof (Proposition 5.1) By Lemma 5.1, if 0 ∈ / Iε = [E − ε, E + ε], as for the number of eigenvalues in an interval Iε , we know

So we obtain

Tr χ Iε (TT ∗ ) = Tr χ Iε (TT ∗ ).

(5.5)

   ω Tr χ Iε Aω = Tr χ Iε  ,

(5.6)

1 1 ω = √ (−) √ . ρs,ω ρs,ω

(5.7)

where

ω Hence, it is enough to estimate the number of eigenvalues of the operator  in the interval Iε . Now, let us define the auxiliary Schrödinger operator H(λ) = − − λρs,ω and let μn (λ) be the n-th eigenvalues of H (λ). Now, we will use a variant of the Birman-Schwinger principle (see e.g. [37]). Suppose that φλ is an ω ω eigenfunction of  satisfying  φλ = λφλ . If we take

1 ψ=√ φλ , ρs,ω

(5.8)

this ψ satisfies H(λ)ψ = 0. So we see

 ω . 0 ∈ σ p (H (λ)) ⇔ λ ∈ σ p 

(5.9)

Consider the number of eigenvalues μn (λ) for some λ ∈ Iε . By the assumption of ρ0 (x) and ρs,ω (x), there exists c > 0 such that H(λ) − cε  H(E)  H(λ) + cε.

(5.10)

By using the min-max principle, we have μn (λ) − cε  μn (E)  μn (λ) + cε.

(5.11)

Now, take λn ∈ [E − ε, E + ε] satisfying μn (λn ) = 0. Since the function μn (λ) is decreasing in λ, then, the n-th eigenvalue of H (E) satisfies −cε  μn (E)  cε by (5.11). Hence, the number of eigenvalues μn (λ) for some λ ∈ Iε is dominated by

This implies

#{n | μn (E) ∈ [−cε, cε]}.

(5.12)

    ω (E) . Tr χ[E−ε,E+ε] Aω  Tr χ[−cε,cε] H

(5.13)

Take the expectation of (5.13) to finish the proof.

 

In virtue of this reduction inequality, the Wegner-type estimate for Aω can be proven easily. In other word, this proposition states that if we proved the regularity of the IDSS for Hω (E) near 0, then we obtain the regularity of the IDSS for Aω near E. This idea is already used in order to study the IDS for Schrödinger operators with Anderson-type sign-indefinite random

Wegner Estimates for Operators with Anderson-type Potentials

65

potential. For example, Hislop and Klopp [22] studied the IDS for the BirmanSchwinger-type random operator. Consider H = − + V, where V is a standard Anderson-type random potential and the sign of single site potential is indefinite. Set E < 0. They used the inequality as follows, Tr χ[E−ε,E+ε] (H )  Tr χ[−1−cε,−1+cε] ( ), where c is some constant independent of , ε and ω. The Birman-Schwingertype operator  is defined as  = (− − E)−1/2 V (− − E)−1/2 , and thus they estimated the expectation of the number of eigenvalues of  near −1 with the theory of the SSF in [11]. We believe that this method is also applicable for our random surface models with Anderson-type sign-indefinite random potential. Proof (Theorem 5.1) By (5.4), it is enough to estimate  ω  E Tr χ[−cε,cε] H (E) , where Hω (E) = − − Eρs,ω . We remark that 0 is not in the spectrum of the background periodic operator H0 (E) = − − Eρ0 when E is not in the spectrum of σ (A0 ), that is,  0    (E) ⇔ E ∈ σ A0 . 0 ∈ σ H (5.14) Since the interval [−cε, cε] in the resolvent set of H0 (E) for ε small enough, consequently, we can apply the same method in Section 3 for the auxiliary ω Schrödinger operator H (E).   Acknowledgements The author would like to thank especially Professor F. Klopp for his hospitality at the Université Paris-Nord. This work was done under his direction and supported by Collège Doctoral Franco-Japonais.

References 1. Aizenman, M., Molchanov, S.A.: Localization at large disorder and at extreme energies: an elementary derivation. Commun. Math. Phys. 157, 245–278 (1993) 2. Barbaroux, J.M., Combes, J.M., Hislop, P.D.: Localization near band edges for random Schrödinger operators. Helv. Phys. Acta 70, 16–43 (1997) 3. Böcker, S., Kirsch, W., Stollmann, P.: Spectral theory for nonstationary random models. Interacting Stochastic Systems, pp. 103–117. Springer, Berlin (2005) 4. Boutet de Monvel, A., Naboko, S., Stollmann, P., Stoltz, G.: Localization near fluctuation boundaries via fractional moment and applications. J. Anal. Math. 100, 83–116 (2006) 5. Boutet de Monvel, A., Stollmann, P.: Dynamical localization for continuum random surface models. Arch. Math. 561(1), 87–97 (2003) 6. Boutet de Monvel, A., Stollmann, P., Stolz, G.: Absence of continuous spectral types for certain non-stationary random Schrödinger operators. Ann. Henri Poincaré 6, 309–326 (2005) 7. Carmona, R., Lacroix, J.: Spectral Theory of Random Schrödinger Operators. Birkhäuser, Boston (1990)

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8. Chahrour, A.: Densité inégrée d’états surfaciques et fonction généralisée de déplacement spectral pour un opérateur de Schrödinger surfacique ergodique. Helv. Phys. Acta 72, 93–122 (1999) 9. Combes, J.M., Germinet, F. and Klein, A.: Generalized eigenvalue-counting estimates for the Anderson model. J. Stat. Phys. 135, 201–216 (2009) 10. Combes, J.M., Hislop, P.D., Klopp, F.: An optimal Wegner estimate and its application to the global continuity of the integrated density of states for random Schrödinger operators. Duke Math. J. 140, 469–498 (2007) 11. Combes, J.M., Hislop, P.D., Nakamura, S.: The L p -theory of the spectral shift function, the Wegner estimate, and the integrated density of states for some random operators. Commun. Math. Phys. 218, 113–130 (2001) 12. Combes, J.M., Hislop, P.D., Tip, A.: Band edge localization and the density of states for acoustic and electromagnetic waves in random media. Ann. I.H.P. sec.A 70, 381–428 (1999) 13. Combes, J.M., Thomas, L.: Asymptotic behaviour of eigenfunctions for multiparticle Schrödinger operators. Commun. Math. Phys. 34, 251–270 (1973) 14. Deift, P.: Applications of a commutation formula. Duke Math. J. 45, 267–310 (1978) 15. Englisch, H., Kirsch, W., Schröder, M., Simon, B.: Density of surface states in discrete models, De. Phys. Rev. Lett. 61(11), 1261–1262 (1988) 16. Englisch, H., Kirsch, W., Schröder, M., Simon, B.: Random Hamiltonians Ergodic in all but one direction. Commun. Math. Phys. 128, 613–625 (1990) 17. Figotin, A., Klein, A.: Localization of classical waves I: acoustic waves. Commun. Math. Phys. 180, 439–482 (1996) 18. Figotin, A., Kuchment, P.: Band-gap structure of spectra of periodic dielectric and acoustic media. I. Scalar model. SIAM J. Appl. Math. 56, 68–88 (1996) 19. Fröhlich, J., Spencer, T.: Absence of diffusion in the Anderson tight binding model for large disorder or low energy. Commun. Math. Phys. 88, 151–184 (1983) 20. Germinet, F., Klein, A.: Bootstrap multiscale analysis and localization in random media. Commun. Math. Phys. 222, 415–448 (2001) 21. Grinshpun, V.: Localization for random potentials supported on a subspace. Lett. Math. Phys. 34, 103–117 (1995) 22. Hislop, P.D., Klopp, F.: The integrated density of states for some random operators with nonsign definite potentials. J. Funct. Anal. 195, 12–47 (2002) 23. Hundertmark, D., Kirsch, W.: Spectral theory of sparse potentials. Stochastic Process, Physics and Geometry: New Interplays, I (Leipzig, 1999), pp. 213–238. American Mathematical Society, Providence (2000) 24. Jakˇsi´c, V., Molchanov, S.: Localization of surface spectra. Commun. Math. Phys. 208, 153–172 (1999) 25. Kirch, W.: An invitation to random operators. Panoramas. Syn. 25, 1–119 (2008) 26. Kirsch, W., Klopp, F.: The band-edge behavior of the density of surfacic states. Math. Phys. Anal. Geom. 8, 315–360 (2005) 27. Kirsch, W., Metzeger, B.: The integrated density of states for random Schrödinger operators. Proc. Sympos. Pure Math. 76, 649–696 (2007) 28. Kirsch, W., Veseli´c, I.: Wegner estimate for sparse and other generalized alloy type potentials. Proc. Ind. Acad. Sci. 112, 131–146 (2002) 29. Kirsch, W., Warzel, S.: Anderson localization and Lifshitz tails for random surface potentials. J. Funct. Anal. 230, 222–250 (2006) 30. Klein, A.: Multiscale analysis and localization of random operators. Panoramas. Syn. 25, 121–159 (2008) 31. Klopp, F.: An asymptotic expansion for the density ofstates of a random Schrödinger operator with Bernoulli disorder. Random Oper. Stoch. Equ. 3(4), 315–331 (1995) 32. Kostrykin, V., Schrader, R.: The density of states and the spectral shift density of random Schrödinger operators. Rev. Math. Phys. 12, 807–847 (2000) 33. Kostrykin, V., Schrader, R.: Regularity of the surface density of states. J. Funct. Anal. 187, 227–246 (2001) 34. Krishna, M.: Continuity of integrated density of states-independent randomness. Proc. Indian Acad. Sci. 117, 401–410 (2007)

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35. Najar, H.: 2-dimensional localization of acoustic waves in random perturbation of periodic media. J. Math. Anal. Appl. 322, 1–17 (2006) 36. Pastur, L.A., Figotin, A.: Spectra of Random and Almost-Periodic Operators. Springer, Berlin (1992) 37. Simon, B.: Trace Ideals and their Applications. London Math. Soc. Lecture Note Ser. (35). Cambridge University Press, Cambridge (1979) 38. Stollmann, P.: Caught by Disorder, Bound States in Random Media. PMP(20). Birkhäuser, Boston (2001) 39. Veseli´c, I.: Existence and Regularity Properties of the Integrated Density of States of Random Schrödinger Operators. LNM(1917). Springer, New York (2008) 40. Wegner, F.: Bounds on the density of states in disordered systems. Z. Phys. B. 44, 9–15 (1981)

Math Phys Anal Geom (2010) 13:69–82 DOI 10.1007/s11040-009-9069-8

Perturbations of Functions of Operators in a Banach Space Michael I. Gil’

Received: 4 February 2009 / Accepted: 23 September 2009 / Published online: 6 October 2009 © Springer Science + Business Media B.V. 2009

Abstract We consider an analytic function f of bounded operators A and A˜ represented by infinite matrices in a Banach space with a Schauder basis. ˜ are established. Applications Sharp inequalities for the norm of f (A) − f ( A) to differential equations are also discussed. Keywords Operator valued functions · Perturbations · Infinite matrices · Hille-Tamarkin matrices · Operators with Hilbert-Schmidt Hermitian components · Differential equations in a Banach space · Green’s functions Mathematics Subject Classifications (2000) 47A56 · 47A55 · 47A60

1 Introduction and Statement of the Main Result This paper is devoted to perturbations of analytic operator valued functions of operators in a Banach space. The theory of perturbations of functions of operators in a Banach space still do not attract much attention of mathematicians although it is very important for various applications. Mainly perturbations of concrete functions are considered, such as the exponential function (semigroup) [5, 11], sine and cosine operator functions [15]. The paper [19] should be mentioned; it deals with a trace class perturbation of a normal

This research was supported by the Kamea Fund of Israel. M. I. Gil’ (B) Department of Mathematics, Ben Gurion University of the Negev, P.0. Box 653, Beer-Sheva 84105, Israel e-mail: [email protected]

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operator with the spectrum on a smooth curve. The results of that paper can be applied to perturbation theory, scattering theory, functional models, and others. The interesting inequality ˜  const f (A − A) ˜  f (A) − f ( A) was derived in [3]. Here f is a holomorphic function admitting certain integral representation. In paper [9] perturbations of entire Banach valued functions are investigated. Of course we cannot survey the whole subject here and refer the reader to the above pointed papers and references given therein. In this paper we consider perturbations of analytic functions of bounded operators in a Banach space, represented by infinite matrices. Let X be a complex Banach space with a Schauder basis {dk } and a norm .. The unit operator in X is denoted by I. Let σ (A) be the spectrum of a bounded linear operator A and Rz (A) = (A − zI)−1 (z ∈ σ (A)) be the resolvent of A; rs (A) denotes the spectral radius of A. It is assumed that A is represented in basis {dk } by a matrix (a jk )∞ j,k=1 . We denote that matrix also by A. So A = D + V where D = diag [a11 , a22 , ... ] and V := A − D is the off diagonal matrix. That is, the entries v jk of V are v jk = a jk ( j = k) and v jj = 0 ( j, k = 1, 2, ...). Simultaneously we consider another matrix ˜ ˜ A˜ = (˜a jk )∞ j,k=1 = D + V ˜ respectively. We ˜ V˜ are the diagonal and off-diagonal entries of A, where D, , i.e. |A| is the matrix whose entries are absolute values put |A| = (|a jk |)∞ j,l=1 of A in basis {dk }. We also write C  0 if all the entries of a matrix C are nonnegative. If C and B are two matrices, then we write C  B if C − B  0. The same sense have the symbols |h|, h  0 and h  g for vectors h, g ∈ X. Put   ˜ := max rs (D) + rs (|V|), rs ( D) ˜ + rs (|V|) ˜ τ (A, A) and (r) := {z ∈ C : |z|  r} for an r > 0. In the sequel f (λ) is a function ˜ The matrix valued function holomorphic on a neighborhood of (τ (A, A)). f (A) is defined by  1 f (A) = − f (λ)Rλ (A)dλ, (1.1) 2πi  ˜ is a closed contour surrounding σ (A). where  ⊂ (τ (A, A)) ˜ the closed convex hull of the diagonal entries a11 , a22 , ... Denote by co(D, D) and a˜ 11 , a˜ 22 , .... Now we are in a position to formulate the main result of the paper. ˜ Theorem 1.1 Let f (λ) be holomorphic on a neighborhood of (τ (A, A)). Then with the notation η j,k :=

| f (k+ j+1) (z)| ( j, k = 0, 1, 2, ...), ˜ (k + j + 1)! z∈co (D, D) sup

Perturbations of Functions of Operators in a Banach Space

71

the inequality ˜  | f (A) − f ( A)|

∞ 

˜ V| ˜ k η j,k |V| j|A − A||

(1.2)

j,k=0

is valid. Below we check that the series in (1.2) really strongly converges. The proof of this theorem is presented in the next section. Theorem 1.1 and its corollary supplements the very interesting recent investigations of infinite matrices and their applications [4, 7, 13, 14, 22].

2 Proof of Theorem 1.1 Lemma 2.1 Under the hypothesis of Theorem 1.1, let A and A˜ have ndimensional ranges (n < ∞). Then inequality (1.2) is valid. Proof We have by (1.1),

 ˜ ˜ =− 1 f (A) − f ( A) f (λ)(Rλ (A) − Rλ ( A))dλ 2πi |λ|=ˆr  1 ˜ = f (λ)Rλ ( A)ER λ (A)dλ, 2πi |λ|=ˆr

(2.1)

˜ But Rλ (A) = (D + V − Iλ)−1 = (I + Rλ (D)V)Rλ (D). where E = A − A. Consequently, Rλ (A) =

∞  (−1)k (Rλ (D)V)k Rλ (D), k=0

provided the spectral radius rs (Rλ (D)V) of the matrix Rλ (D)V is less than one. The entries of this matrix are a jk (λ = a jj, j = k) a jj − λ and the diagonal entries are zero. Clearly, |Rλ (D)V| 

|V| . mink |akk − λ|

But |akk − λ|  |λ| − |akk |  |λ| − rs (D), provided |λ| > rs (D). So in this case |V| . |λ| − rs (D)

|Rλ (D)V|  Therefore, if |λ| > rs (|V|) + rs (D), then rs (Rλ (D)V) 

rs (|V|) <1 |λ| − rs (D)

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M.I. Gil’

and the series ∞ 

(Rλ (D)V)k (−1)k

k=0

converges. Similarly, ˜ = Rλ ( A)

∞  ˜ V) ˜ k Rλ ( D), ˜ (−1)k (Rλ ( D) k=0

˜ + rs ( D). ˜ So by (2.1) we have provided |λ| > rs (|V|) ∞ 

˜ = f (A) − f ( A)

Cmk

(2.2)

m,k=0

where Cmk = (−1)

k+m

1 2πi

 |λ|=ˆr

˜ V) ˜ k Rλ ( D)dλ. ˜ f (λ)(Rλ (D)V)m Rλ (D)E(Rλ ( D)

˜ are diagonal matrices with respect to basis {dk }, we can write out Since D, D Rλ (D) =

n  j=1

Qj , a jj − λ

˜ = Rλ ( D)

n  j=1

Qj , a˜ jj − λ

where Qk h = hk . Consequently, Cmk =

n 

Qi1 V

i1 =1

. . . V˜

n  i2 =1

n 

Qi2 V . . . V

n 

Qim+1 E

im+1 =1

n 

Q j1 V˜

j1 =1

n 

Q j2 V˜

j2 =1

Q jk+1 Ii1 ,i2 ,...,im+1 , j1 j2 ... jk+1

jk+1 =1

Here Ii1 ,i2 ,...,im+1 , j1 j2 ... jk+1  (−1)k+m f (λ)dλ . = 2πi (a − λ) . . . (a − λ)(˜a j1 j1 − λ) . . . (˜a jk+1 jk+1 − λ) i i i i |λ|=ˆr 1 1 m+1 m+1 By Lemma 1.5.1 [8], |Ii1 ,i2 ,...,im+1 , j1 j2 ... jk+1 |  ηm,k . Hence, |Cmk |  ηmk

n  j1 =1

Q j1 |V|

n  j2 =1

Q j2 |V| . . . |V|

n  jk =1

Q jm |E|

n 

˜ . . . |V| ˜ Qll |V|

l1 =1

˜ k . Now (2.2) implies the required result. Thus |Cmk |  ηmk |V|m |E||V|

n 

Qlm .

lk =1

 

Perturbations of Functions of Operators in a Banach Space

73

Proof of Theorem 1.1 Passing to the limit as n → ∞ in the previous lemma we get required result due to the Banach-Schteihaus theorem.  

3 Perturbations in Terms of Lattice Norms In the sequel the norm . in X is a lattice norm. That is,  f   h whenever | f |  |h|. In addition, A := supx∈X Ax/x and A  |A|. Theorem 2.1 implies Corollary 3.1 Under the hypothesis of Theorem 1.1 we have ˜  | f (A) − f ( A)| ˜ ˜  f (A) − f ( A)  |A − A|

∞ 

˜ k . η j,k |V| j|V|

j,k=0

(3.1) If A and A˜ are diagonal: V = V˜ = 0, inequality (3.1) takes the form ˜  max |akk − a˜ kk |  f (A) − f ( A) k

sup ˜ z∈co (D, D)

| f (z)|.

Furthermore, since |V|  |A|, from (3.1) it follows ˜ ˜ | f (A) − f ( A)|  |A − A|

∞ 

˜ k η j,k |V| j|V|

j,k=0

˜  |A − A|

∞ 

˜ k. η j,k |A| j| A|

(3.2)

j,k=0

For instance, let X = l p for some integer p  2, and V and V˜ be HilleTamarkin matrices, cf. [16]. Namely, ⎛ N p (V) := ⎝

∞  j=1

⎡ ⎣

∞ 

⎤ p/q ⎞1/ p |a jk |q ⎦

⎠

<∞

˜ < ∞ (3.3) N p (V)

and

k=1, k= j

with 1/ p + 1/q = 1. So under (3.3), A = (˜a jk ) and A˜ = (˜a jk ) represent ˜ are bounded. As it is wellbounded linear operators in l p , provided D and D known |V|  N p (|V|) = N p (V), cf. [16]. Besides   ˜  max N p (V) + max |akk |, N p (V) ˜ + max |˜akk | . τ (A, A) k

k

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Now (3.2) yields Corollary 3.2 Under the hypothesis of Theorem 1.1, let X = l p , 2  p < ∞ and condition (3.3) hold. Then ∞ 

˜  | f (A) − f ( A)| ˜ ˜  f (A) − f ( A)  |A − A|

˜ η j,k N pj (V)N kp (V),

j,k=0

provided the series converges.

4 Green’s Functions of Differential Equations In this section we consider the scalar function Gk (λ, t) (λ ∈ C, t  0), for an integer k  1, satisfying the equation ∂ k Gk (λ, t) = ξ λk , t>0 ∂tk with a real constant ξ and the initial conditions ∂ j Gk (λ, 0) = 0 ( j = 0, ..., k − 2), ∂t j

f (k−1) (0) = 1.

(4.1)

(4.2)

For example, if k = 1, ξ ≡ 1, then G1 (λ, t) = eλt . If k = 2, ξ ≡ −1, then G2 (λ, t) = λ−1 sin (λt), etc. The function Gk satisfying (4.1), (4.2) is the Green function to equation (4.1). For any continuous function h, a solution x(t) of the equation dk x(t) − ξ λk x(t) = h(t) dtk with the initial conditions x( j) (0) = 0 ( j = 0, ..., k − 1), can be represented as



t

x(t) =

Gk (λ, t − s)h(s)ds.

(4.3)

0

Thus under (4.1), (4.2), we can write out,   dk Gk (λ, t) dk Gk (μ, t) − = ξ λk Gk (λ, t) − μk Gk (μ, t) (λ, μ ∈ C; t  0). dtk dtk So   dk (Gk (λ, t) − Gk (μ, t)) − ξ λk (Gk (λ, t) − Gk (μ, t)) = ξ λk − μk Gk (μ, t). k dt According to (4.3), this gives  t   Gk (λ, t) − Gk (μ, t) = ξ Gk (λ, t − s) λk − μk Gk (μ, s)ds. 0

Perturbations of Functions of Operators in a Banach Space

75

Hence we get the following result. Lemma 4.1 Let the Green function Gk (λ, t) to (4.1) be regular in λ on σ (A) for each t  0. Then dk Gk (A, t) = Ak Gk (A, t), dtk

t>0

(4.4)

f (k−1) (0) = I,

(4.5)

and ∂ j Gk (A, 0) = 0 ( j = 0, ..., k − 2), ∂t j

˜ then where I is the unit operator. If, in addition, Gk (λ, t) is regular in λ on σ ( A), ˜ t) = ξ Gk (A, t) − Gk ( A,



t

  ˜ s)ds. Gk (A, t − s) Ak − A˜ k Gk ( A,

(4.6)

0

In particular,



˜

t

e At − e At =

˜ ˜ As e A(t−s) [A − A]e ds

0

and ˜ = A sin (At) − A˜ −1 sin ( At) −1



t

  ˜ A−1 sin (A(t − s)) A2 − A˜ 2 A˜ −1 sin ( As)ds.

0

From (4.6) it follows ˜ t)|  |ξ | |Gk (A, t) − Gk ( A,



t

˜ s)|ds. |Gk (A, t − s)||Ak − A˜ k ||Gk ( A,

0

and ˜ t)  |ξ | Gk (A, t) − Gk ( A,



t

˜ s)ds. Gk (A, t − s)|Ak − A˜ k |Gk ( A,

0

Denote by co (D) the closed convex hull of the diagonal numbers a11 , a22 , .... Put τ A = rs (D) + rs (|V|). We need the following result from [10]. Theorem 4.2 Let V be the off-diagonal part of an matrix A. Let f (λ) be holomorphic on a neighborhood of the circle (τ A ). Then with the notation γˆ j :=

| f ( j) (z)| ( j = 0, 1, 2, ...), j! z∈co (D) sup

we have | f (A)| 

∞  j=0

γˆ j|V| j.

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M.I. Gil’

The series in the right-hand part of the latter formula strongly converges. This result and (4.6) imply ˜ Corollary 4.3 Let the Green function Gk (λ, t) to (4.1) be regular on τ (A, A). Then with the notation γ j(A, t) :=

1 ∂ j Gk (z, t) sup j! z∈co (D) ∂t j

the inequality ˜ t)|  |Gk (A, t) − Gk ( A,

∞ 

˜ t)|V|m |Ak − A˜ k ||V| ˜ j γm (A, t)γ j( A,

m, j=0

is valid. This corollary gives us the inequality ∞ 

˜ t)  |Ak − A˜ k | Gk (A, t) − Gk ( A,

˜ t)|V|m |V| ˜ j. γm (A, t)γ j( A,

m, j=0

In particular, we have ˜ ˜ e At − e At   |A − A|



t

˜

eα(D)(t−s)+α( D)s

0

˜ = |A − A|



t

∞  ˜ j (t − s)m |V|m s j|V| ds m! j! j,m=0

  ˜ + |V|) ˜ exp (t − s)(α(D) + |V|) + s(α( D) ds,

0

where α(D) = maxk Re akk . Thus ˜

e At − e At  

˜  (α(D)+|V|)t  |A − A| ˜ ˜ e − e(α( D)+|V|)t . ˜ − |V| ˜ α(D) + |V| − α( D)

In particular, let X = l p , V and V˜ be Hille-Tamarkin operators, then we obtain ˜ t)  |Ak − A˜ k | Gk (A, t) − Gk ( A,

∞ 

˜ t)N m (V)N j (V), ˜ γm (A, t)γ j( A, p p

m, j=0

provided the series converges. Hence, ˜

e At − e At  

˜  (α(D)+N p (V))t |A − A| ˜ ˜  e − e(α( D)+N p (V))t . ˜ − N p (V) ˜ α(D) + N p (V) − α( D)

5 Functions of Operators with Hilbert-Schmidt Hermitian Components In this section we improve Theorem 1.1 in the case of operators with HilbertSchmidt Hermitian components acting in a separable Hilbert space H with a √ scalar product (., .) and the norm . = (., .). Everywhere below A and A˜

Perturbations of Functions of Operators in a Banach Space

77

are linear bounded operators in H, λk (A) are the eigenvalues of A taken with their multiplicities, σ (A) denotes the spectrum of A, A∗ is the adjoint to A and A I = (A − A∗ )/2i. Let N2 (B) be the Hilbert-Schmidt norm of a HilbertSchmidt operator B: N22 (B) = Trace BB∗ . It is assumed that N2 (A I ) < ∞

˜ < ∞. N2 (A − A)

and

(5.1)

The following quantity plays an essential role hereafter 1/2  ∞  2 2 |Im λk (A)| . u(A) := 2N2 (A I ) − 2 k=1

√ Obviously, u(A)  2N2 (A I ). It is not hard to show that, if A is a normal operator: A A∗ = A∗ A, then u(A) = 0; if A is a Hilbert-Schmidt operator, then u(A) = g(A) where

 g(A) :=

N22 (A)

−

∞ 

(5.2) 1/2

|λk (A)|

2

.

k=1

For details see [8, Section 7.7]. Since ∞  ∞      2 2 |λk (A)|   λk (A) = |Trace A2 |,   k=1

k=1

one can write g2 (A)  N22 (A) − |Trace A2 |. ˜ the closed convex hull of σ (A) ∪ σ ( A). ˜ Denote by co(A, A) Theorem 5.1 Let conditions (5.1) hold and f (λ) be holomorphic on a neigh˜ Then with the notation borhood of co(A, A). ψ j,k :=

| f (k+ j+1) (z)|  ( j, k = 0, 1, 2, ...), k! j!(k + j + 1)! ˜ z∈co (A, A) sup

the inequality ˜  N2 (A − A) ˜ N2 ( f (A) − f ( A))

∞ 

˜ ψ j,k u j(A)uk ( A)

j,k=0

is valid. The proof of this theorem is presented in the next section. The perturbation theory of operator functions in a Hilbert space is rather rich. I would like to mention the fundamental papers on Double Operator Integrals by Birman and Solomyak, which are reflected in the survey [2], and

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also the papers by Ando [1], Matsaev [12] and Peller [17] which contributed to the topic most substantially. The remarkable Birman-Solomyak results allow ˜ in the case when A and A˜ us to establish bounds for the norm of f (A) − f ( A) are selfadjoint and A − A˜ belongs to some ‘nice‘ ideal. Besides, A and A˜ may be unbounded. At the same time we do not assume that A and A˜ are selfadjoint. Let A and A˜ be normal operators and f (λ) be holomorphic on a neigh˜ Then due to equalities u(A) = u( A) ˜ = 0, Theorem 5.1 borhood of co(A, A). implies the inequality. ˜  N2 (A − A) ˜ N2 ( f (A) − f ( A))

sup ˜ z∈co (A, A)

| f (z)|.

(5.3)

Actually this result and even slightly better one can be directly proved by the spectral theorem for normal operators. To consider the sharpness of Theorem 5.1 assume that A and A˜ are commuting Hermitian and f is real on the real axis. Then ˜ = N22 ( f (A) − f ( A))

n  ˜ 2. ( f (λk (A)) − f (λk ( A))) k=1

Clearly, ˜  M|λk (A) − λk ( A)| ˜ | f (λk (A)) − f (λk ( A))|

(5.4)

where M = sup | f (x)|. x∈[a,b ]

˜ That is, Here [a, b ] is the smallest segment containing the spectra of A and A. ˜ in the considered case [a, b ] = co (A, A). Thus ˜  MN2 (A − A). ˜ N2 ( f (A) − f ( A)) Comparing this inequality with (5.3), we can assert that if the matrices A and A˜ are commuting Hermitian and f is real, then the sharpness of Theorem 5.1 depends on the sharpness of inequality (5.4). √ Furthermore, thanks to Theorem 5.1 and the inequality u(A)  2N2 (A I ), we have ˜  N2 (A − A) ˜ N2 ( f (A) − f ( A))

∞ 

√ j ψ j,k 2k+ j N2 (A I )N2k ( A˜ I ).

j,k=0

Theorem 5.1 supplements the interesting recent investigations of operators in Schatten-von Neumann ideals [6, 18, 20, 21].

Perturbations of Functions of Operators in a Banach Space

79

6 Proof of Theorem 5.1 Lemma 6.1 Let A and A˜ have n-dimensional ranges (n < ∞) and f (λ) be ˜ Then holomorphic on a neighborhood of co(A, A). ˜  N2 (A − A) ˜ N2 ( f (A) − f ( A))

n−1 

˜ ψ j,k u j(A)uk ( A).

j,k=0

Proof By (1.1),

 1 ˜ ˜ f (A) − f ( A) = − f (λ)(Rλ (A) − Rλ ( A))dλ 2πi L  1 ˜ = f (λ)Rλ ( A)ER λ (A)dλ 2πi L

(6.1)

˜ By the triangular (Schur) where E = A˜ − A and L surrounds σ (A) ∪ σ ( A). representation A = D + V (σ (A) = σ (D))

(6.2)

where D is a normal and V is a nilpotent operators having the same invariant subspaces. Similarly, ˜ = σ ( D)) ˜ + V˜ (σ ( A) ˜ A˜ = D

(6.3)

˜ is a normal and V˜ is a nilpotent operators having the same invariant where D subspaces. But Rλ (A) = (D + V − Iλ)−1 = (I + Rλ (D)V)Rλ (D). Consequently, Rλ (A) =

n−1 

(−1)k (Rλ (D)V)k Rλ (D).

k=0

Similarly, ˜ = Rλ ( A)

n−1 

˜ V) ˜ k Rλ ( D). ˜ (−1)k (Rλ ( D)

k=0

So we have ˜ = f (A) − f ( A)

n−1 

Cmk

(6.4)

m,k=0

where Cmk = (−1)

k+m

1 2πi



˜ V) ˜ k Rλ ( D)dλ. ˜ f (λ)(Rλ (D)V)m Rλ (D)E(Rλ ( D) L

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M.I. Gil’

Since D is a diagonal matrix in the orthonormal basis of the triangular ˜ is a diagonal matrix in the representations of A (the Schur basis) {ek }, and D ˜ we can write out Schur basis {˜ek } of A, Rλ (D) =

n  j=1

Qj , λj − λ

˜ = Rλ ( D)

n 

˜j Q

j=1

λ˜ j − λ

,

˜ Qk = (., ek )ek , Q ˜ k = (., e˜k )˜ek . Consequently, where λ j = λ j(A), λ˜ j = λ j( A), Cmk =

n 

Qi1 V

i1 =1

n 

Qi2 V . . . V

i2 =1 n 

. . . V˜

n 

Qim+1 E

n 

im+1 =1

˜ j1 V˜ Q

j1 =1

n 

˜ j2 V˜ Q

j2 =1

˜ jk+1 Ji1 ,i2 ,...,im+1 , j1 j2 ... jk+1 . Q

(6.5)

jk+1 =1

Here Ji1 ,i2 ,...,im+1 , j1 j2 ... jk+1  (−1)k+m f (λ)dλ = . ˜ j1 − λ) . . . (λ˜ jk+1 − λ) 2πi L (λi1 − λ) . . . (λim+1 − λ)(λ Below the symbol |V| means the operator whose entries are absolute values ˜ means the operator whose entries are absolute of V in the basis {ek } and |V| values of V˜ in the basis {˜ek }. Furthermore, put Ekj = (Ee˜ j, ek ). Then E=

n 

Ekj(., e˜ j)ek

Ee˜ j =

and

j,k=1

n 

Ekjek .

k=1

Put |E| =

n 

|Ekj|(., e˜ j)ek .

j,k=1

By Lemma 1.5.1 [8], |Ji1 ,i2 ,...,im+1 , j1 j2 ... jk+1 |  ψm,k :=

| f (k+m+1) (z)| . ˜ (m + j + 1)! z∈co (A, A) sup

Now (6.5) implies |Cmk |  ψm,k

n 

Qi1 |V|

i1 =1

˜ . . . |V|

n 

n  i2 =1

Qi2 |V| . . . |V|

n  im+1 =1

Q jm+1 |E|

n 

˜ j2 |V| ˜ Q

j1 =1

˜ jk+1 . Q

jk+1 =1

Thus ˜ k. |Cmk |  ψm,k |V|m |E||V|

(6.6)

Perturbations of Functions of Operators in a Banach Space

81

Note that N22 (|E|) =

n 

|E|˜ek 2 =

n  n 

|E jk |2 = N22 (E).

k=1 j=1

k=1

Hence (6.6) yields the inequality ˜ k . N2 (Cmk )  ψm,k |V|m N2 (E)V| By Theorem 2.5.1 from [8] we have N2m (V) |V|m   √ . m!

(6.7)

In addition by Lemma 2.3.2 from [8], N2 (V) = u(A). So N2 (Cmk )  ψm,k Now (6.4) implies the required result.

˜ um (A)uk ( A) . √ √ m! k!  

Proof of Theorem 5.1 We get the required result by passing to the limit n → ∞ in the previous lemma.   Acknowledgement

I am very grateful for the referee of this paper for his really helpful remarks.

References 1. Ando, T., Szymaiski, W.: Order structure and Lebesgue decomposition of positive definite operator functions. Indiana Univ. Math. J. 35(1), 157–173 (1986) 2. Birman, M., Solomyak, M.: Double operator integrals in a Hilbert space. Integr. Equ. Oper. Theory 47(2), 131–168 (2003) 3. Boyadzhiev, K.N.: Some inequalities for generalized commutators. Publ. RIMS, Kyoto University 26(3), 521–527 (1990) 4. Candan, M., Solak, I.: On some difference sequence spaces generated by infinite matrices. Int. J. Pure Appl. Math. 25(1), 79–85 (2005) 5. Engel, K.-J., Nagel, R.: One-parameter Semigroups for Linear Operators. Springer, New York (2000) 6. Gheorghe, L.G.: Hankel operators in Schatten ideals. Ann. Math. Pures Appl. IV Ser. 180(2), 203–210 (2001) 7. Gil’, M.I.: Spectrum localization of infinite matrices. Math. Phys. Anal. Geom. 4(4), 379–394 (2001) 8. Gil’, M.I.: Operator functions and localization of spectra. In: Lectures Notes in Mathematics, vol. 1830. Springer, Berlin (2003) 9. Gil’, M.I.: Inequalities of the Carleman type for Schatten-von Neumann operators. AsianEuropean J. Math. 1(2), 203–212 (2008) 10. Gil’, M.I.: Estimates for entries of matrix valued functions of infinite matrices. Math. Phys. Anal. Geom. 11(2), 175–186 (2008) 11. Hasse, M.: The Functional Calculus for Sectorial Operators. Birkháuser Verlag, Boston (2006) 12. Matsaev, V.: Volterra operators obtained from self-adjoint operators by perturbation. Dokl. Akad. Nauk SSSR 139, 810–813 (1961) (Russian) 13. Mittal, M.L., Rhoades, B.E., Mishra, V.N., Singh, U.: Using infinite matrices to approximate functions of class Lip(α, p) using trigonometric polynomials. J. Math. Anal. Appl. 326(1), 667–676 (2007)

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14. Nagar, D.K., Tamayo-Acevedo, A.C.: Integrals involving functions of Hermitian matrices. Far East J. Appl. Math. 27(3), 461–471 (2007) 15. Palencia, C., Piskarev, S.: On multiplicative perturbations of C0 -groups and C0 -cosine operator functions. Semigroup Forum 63(2), 127–152 (2001) 16. Pietsch, A.: Eigenvalues and s-numbers. Cambridge University Press, Cambridge (1987) 17. Peller, V.: Hankel operators in perturbation theory of unbounded self-adjoint operators. In: Analysis and Partial Differential Equations. Lecture Notes in Pure and Appl. Math., vol. 122, pp. 529–544. Dekker, New York (1990) 18. Sigg, M.: A Minkowski-type inequality for the Schatten norm. J. Inequal. Pure Appl. Math. 6(3), paper no. 87, 7 pp. (2005) (electronic only) 19. Tikhonov, A.: Boundary values of operator-valued functions and trace class perturbations. Rev. Roum. Math. Pures. Appl. 47(5–6), 761–767 (2002) 20. Wong, M.W.: Schatten-von Neumann norms of localization operators. Arch. Inequal. Appl. 2(4), 391–396 (2004) 21. Xia, J.: On the Schatten class membership of Hankel operators on the unit ball. Ill. J. Math. 46(3), 913–928 (2002) 22. Zhao, X., Wang, T.: The algebraic properties of a type of infinite lower triangular matrices related to derivatives. J. Math. Res. Expo. 22(4), 549–554 (2002)

Math Phys Anal Geom (2010) 13:83–103 DOI 10.1007/s11040-009-9070-2

Time-dependent Delta-interactions for 1D Schrödinger Hamiltonians Toufik Hmidi · Andrea Mantile · Francis Nier

Received: 31 March 2009 / Accepted: 18 November 2009 / Published online: 3 December 2009 © Springer Science + Business Media B.V. 2009

2 Abstract The non autonomous Cauchy problem i∂t u = −∂xx u + α(t)δ0 u with 2 ut=0 = u0 is considered in L (R) . The regularity assumptions for α are accurately analyzed and show that the general results for non autonomous linear evolution equations in Banach spaces are far from being optimal. In the mean time, this article shows an unexpected application of paraproduct techniques, initiated by J.M. Bony for nonlinear partial differential equations, to a classical linear problem.

Keywords Point interactions · Solvable models in Quantum Mechanics · Non-autonomous Cauchy problems Mathematics Subject Classifications (2000) 37B55 · 35B65 · 35B30 · 35Q45 1 Introduction This work is concerned with the dynamics generated by the particular class d2 of non-autonomous quantum Hamiltonians: Hα(t) = − dx 2 + α(t)δ, defining the time-dependent delta shaped perturbations of the 1D Laplacian. Quantum Hamiltonians with point interactions were first introduced by physicists as a

T. Hmidi · A. Mantile · F. Nier (B) IRMAR, UMR-CNRS 6625, Université Rennes 1, Campus de Beaulieu, 35042 Rennes Cedex, France e-mail: [email protected] T. Hmidi e-mail: [email protected] A. Mantile e-mail: [email protected]

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computational tool to study the scattering of quantum particles with small range forces. Since then, the subject has been widely developed both in the theoretical framework as well as in the applications (we refer to [2] for an extensive presentation). For real values of the coupling parameter α, the d2 rigorous definition of: Hα = − dx 2 + αδ arises from the Krein’s theory of selfadjoint extensions. In particular, Hα identifies with the selfadjoint extension of d2 ∞ the symmetric operator: H0 = − dx 2 , D(H0 ) = C0 (R\ {0}) defined through the boundary conditions   + ψ (0 ) − ψ  (0− ) = αψ(0) (1.1) ψ(0+ ) − ψ(0− ) = 0 ψ(0± ) denoting the right and left limit values of ψ(x) as x → 0 [2]. Explicitly, one has    D(Hα ) = ψ ∈ H 2 (R\ {0}) ∩ H 1 (R)  ψ  (0+ ) − ψ  (0− ) = αψ(0) (1.2) Hα ψ = −

d2 ψ dx2

in R\ {0} .

(1.3)

When α(t) is assigned as a real valued function of time, the domain D(Hα(t) ) changes in time with the boundary condition (1.1), while the form domain is 1 given by H (R). The quantum evolution associated to the family of operators Hα(t) is defined by the solutions to the equation ⎧ ⎨i d u = H u α(t) dt (1.4) ⎩ u|t=0 = u0 . The mild solutions are the solutions to the associated integral equation t it u(t) = e u0 − i ei(t−s) q(s)δ0 ds

(1.5)

0

with q(s) = α(s)u(s, 0) . The questions are about: • •

the regularity assumptions on t → α(t) for which (1.4) defines a unitary strongly continuous dynamical system on L2 (R) the meaning of the differential equation (1.4), according to the regularity of t → α(t) .

General conditions for the solution of this class of problems have been long time investigated. In the framework of evolution equations in Banach spaces, Kato was the first who obtained a result for the Cauchy problem ⎧ ⎨ d u = A(t)u dt , (1.6) ⎩ ut=0 = u0 when t → A(t) is an unbounded operator valued function [11]. This result, which applies to the quantum dynamical case for A(t) = −iH(t), requires the

Time-dependent Delta-interactions for 1D Schrödinger Hamiltonians

85

strong differentiability of the map t → A(t) and the time independence of the domain D(A(t)). Afterwards, a huge literature was devoted to this problem in the main attempt of relaxing the above conditions (e.g. in [18] and [16]; a rather large bibliography and an extensive presentation of the subject are also given in [9]). In particular, the time-dependent domain case was explicitly treated in 2 ´ [13] by Kisynski using coercivity assumptions and Cloc -regularity of t → A(t). The regularity conditions in time were substantially relaxed into a later work of Kato [12] who proves weak and strong existence results, for the solutions to (1.6), when {A(t)} forms a stable (a notion defined in [12]-Definition 2.1 expressing uniform bounds for the norms of resolvents) family of generators of contraction semigroups leaving invariant a dense set Y of the Banach space X, and the map t → A(t) is norm-continuous in L(Y, X) . When A(t) = −iH(t) and H(t) defines a non autonomous selfadjoint Hamiltonian acting on the Hilbert space H, a different approach to the problem (1.6) consider the evolution equation d ϕ(σ ) = −ih ϕ(σ ) dσ

(1.7)

where σ is the new time variable, while t and −i dtd are conjugate variables of the Floquet operator h = −i dtd + H(t) acting on L2 (R; H). In this setting, one can use the Stone’s theorem to express the solution of (1.7) through the action of ˆ ). If the a strongly continuous one parameter unitary group on L2 (R; H), U(σ equation ⎧ ⎨ −i d u = H(t)u dt (1.8) ⎩ ut=0 = u0 ˆ ) by admits a unitary propagator, U(t, s), this is related to U(σ

 ˆ )ϕ (t, ·) = U(t, t − σ )ϕ(t − σ, ·) U(σ

(1.9)

where ϕ(t, ·) ∈ L2 (R; H). On the other hand, it can be shown that  U(t, s) is ˆ )Tσ ϕ (t, ·), whenuniquely determined from (1.9) as: U(t, t − σ )ϕ(t, ·) = U(σ ˆ )Tσ is a multiplication by an operator valued function and Tσ ϕ(t, ·) = ever U(σ ϕ(t + σ, ·). Therefore, one can solve the non-autonomous problem (1.8) by discussing the propagator’s properties for the larger, autonomous, quantum system defined by (1.7). As remarked in [16], this method, due to J. Howland, does not requires the strong differentiability of H −1 (t), neither a constant domain condition. Then it can be adopted in those situations where one does not expect strong differentiability and the Yoshida’s theorem does not apply (we refer to [16] or to [15] and references therein; applications of the Howland’s method to time periodic Hamiltonians and ionizations problem are accounted in [19]). Time-dependent point interaction Hamiltonians have been largely used to build up solvable models for a number of different physical situations (between

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the others, the ionization problem [5–7], concentrated quantum nonlinearities [1], control problems [14]). Due to their particular structure, these operators allow rather explicitly energy and resolvent estimates, so that most of the techniques employed in the analysis of non-autonomous Hamiltonians can be used to deal with the (1.4), provided that α(t) is regular enough. At this concern, the Yafaev’s works [20, 21] and [17] (with M. Sayapova) on the scattering problems for a time-dependent delta interaction in the 3D setting 2 are to be recalled: There the condition α ∈ Cloc (t0 , +∞) is used to ensure the existence of a strongly differentiable time propagator for the quantum evolution. Such a condition, however, could be considerably relaxed. In our case, for instance, a first approach consists in adapting the strategy of [13] ∗ by constructing a family of unitary maps Vt,t0 such that: Vt,t0 Hα(t) Vt,t has a 0 constant domain; then, it is possible to solve the evolution problem for the deformed operator by using results from [12]. To fix the idea, let yt be the timedependent vector field defined by y˙ t = g(yt , t) (1.10) yt0 = x

 with g(·, t) ∈ C0 0, T; C0∞ (R) , supp g(·, t) ⊂ (−1, 1), and g(0, t) = 0 for each t. Under these assumptions, the (1.10) has a unique solution continuously depending on time and Cauchy data {t0 , x}. Using the notation: yt = F(t, t0 , x), one has t

∂x F(t, t0 , x) = e

t0

∂1 g(ys ,s)ds

>0

∀x ∈ R ,

(1.11)

∂1 g(·, s) denoting the derivative w.r.t. the first variable. This condition allows to consider the map of x → F(t, t0 , x) as a time-dependent local dilation and one can construct the family of time dependent unitary transformations associated to it ⎧ ⎨ V u (x) = ∂ F(t, t , x) 12 u F(t, t , x) t0 ,t x 0 0 ⎩ V = V −1 t,t0

t0 ,t

Under the action of Vt,t0 , the (1.4) reads as ⎧   ⎪ ⎨ i d v = Vt,t Hα(t) Vt ,t v − i 1 [∂ y g](y, t) + g(y, t)∂ y v 0 0 dt 2 ⎪ ⎩ vt=0 = u0 with

(1.12)

 Vt,t0 Hα(t) Vt0 ,t = −∂ y b 2 ∂ y + a2 − ∂ y ab + b 0 α(t)δ t

b (y, t) = e a(y, t) =

t0

∂1 g(F(s,t,y),s) ds

1 12 tt ∂1 g(F(s,t,y),s) ds e 0 2

t0

t

;

b 0 = b (0, t)

  t ∂ g F(s ,t,y),s ) ds ∂12 g F(s, t, y), s e t0 1 ( ds

Time-dependent Delta-interactions for 1D Schrödinger Hamiltonians

87

 and Vt,t0 u = v. Set: A(t) = iVt,t0 Hα(t) Vt0 ,t + 12 ∂ y g(y, t) + g(y, t)∂ y , the domain D(A(t)) is the subspace of H 2 (R\ {0}) ∩ H 1 (R) identified by the boundary condition

 b (0, t) u (0+ ) − u (0− ) = α(t)u(0) For α ∈ C1 (0, T) and α(t)  = 0 for all t, one can determine (infinitely many) g(·, t) ∈ C0 0, T; C0∞ (R) such that α(t) ˙ = ∂1 g(0, t) α(t) In particular, this implies: b (0, t) = α(t), and the domain’s boundary condition 

 + u (0 ) − u (0− ) = u(0) becomes independent of the time. With this choice of g(·, t), the operator domain is constant, D(A(t)) = Y, and one can show that the corresponding A(t) define a stable family of skew-adjoint operators t-continuous in L(Y, L2 (R))-operator norm. Thus, the Theorem 5.2 and Remark 5.3 in [12] apply and provide strongly solutions for the related evolution problem. This is summarized in the following Proposition. Proposition 1.1 Let α ∈ C1 (0, T), sign(α(t)) = const. and u0 ∈ D(Hα(0) ). There exists a unique solution ut of the problem (1.4), with: ut ∈ D(Hα(t) ) for each t and ut ∈ C0 (0, T; L2 (R)). An analogous problem for infinitely many time-dependent quantum point interactions, have been recently considered by H. Neidhardt and V. Zagrebnov in [15]; following a combination of the Howland’s method and the evolution semigroup approach, they prove the existence of H 1 -weak solutions provided that the coupling constants are nonnegative and locally Lipschitz continuous functions of time. In spite of those improved results with already known general tools, our aim is to prove that they are far from being optimal. An additional structure, indeed, can lead to the same conclusions with weaker regularity assumptions. This is the question that we propose to explore with one dimensional δinteractions which allow direct and explicit computations. The main result of this paper is the following. Theorem 1.2 1

4 1) Assume α ∈ Hloc (R), then for any u0 ∈ H 1 (R) the integral equation (1.5) admits a unique solution u ∈ C(R; H 1 (R)) with i∂t u − α(t)u(t, 0)δ0 ∈ C(R; H −1 (R)) and (1.4) is weakly well-posed. 2) With the same assumption, (1.5) defines a unitary strongly continuous dynamical system U(t, s) on L2 (R) .

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3) If additionally α ∈ Hloc (R), then for any u0 ∈ D(Hα(0) ) the solution u of (1.5) belongs to the space C1 (R; L2 (R)), with u(t) ∈ D(Hα(t) ) for every t ∈ R. According to (1.5), the essential informations about the dynamics of our system are coded by the finite dimensional auxiliary variable, q(t), usually denoted as the charge of the quantum state. A straightforward computation allows to establish a simple relation between the spatial Fourier transform of the solution, u and the time Fourier transform of χq, where χ is a sharp cutoff function (see relation (2.6) below). Then, using product laws and standard techniques of functional analysis, the regularity estimates for u, stated in Theorem 1.2, are obtained in terms of the fractional Sobolev norm of the charge. It is worthwhile to notice that the possibility of mixing the time and the space Fourier variables as in (2.6) as well as the related product laws used below, are specific of the one dimensional model. The extension of this analysis to the three dimensional case does not appear immediate. With this concern, let us also recall that the 3-D heat equation with time-dependent delta-shaped potentials have been considered in [8]. In this work the authors prove the existence of strong solutions when the time profile of the interaction is an Hölder continuous function α ∈ Cη (0, T) with η > 14 . It is interesting to notice that this result is rather ’near’ to the one stated by Theorem 1.2 for the 3/4 corresponding 1-D quantum system (notice that the condition α ∈ Hloc implies 1 α ∈ C 4 − ). Nevertheless, the passage to the 3-D quantum case, may require a stronger regularity for α. The problem of defining the quantum evolution for 1D time-dependent delta interactions has also been considered in a nonlinear setting. In [1] α is assigned as a function of the particle’s state, in our notation: α(t) = γ |ut (0)|2σ , with γ ∈ R, σ ∈ R+ . In this framework, the authors prove that solutions to the nonlinear evolution problem exists locally in time for u0 ∈ H ρ with ρ > 12 and: ρ

+1

ν 2 4 as a function of time. This corresponds to: α ∈ Hloc with U(t, 0)u0 (0) ∈ Hloc 1 ν > 2 . Since the result of Theorem 1.2, for the linear model, improves this condition, it would be interesting to check if our technique can be adapted to improve the prescription u0 ∈ H ρ , ρ > 12 , for this particular nonlinear system. In [10], the author consider the non linear Schrödinger equation on the half line with a boundary condition in the origin assigned through a function of time f (t). In particular, he proves that a solution u ∈ L2 (R+ ) of this problem 1 exists locally in time provided that f ∈ H 4 : It is interesting to notice that this condition, whereas obtained in a rather different situation, corresponds to the minimal regularity request on the coupling term α presented in Theorem 1.2.

Notation In what follows, Dx denotes 1i ∂x = F −1 ◦ (ξ ×) ◦ F , where F is the Fourier transform in position (or in time) normalized according to F ϕ(ξ ) =

R

e−iξ x ϕ(x) dx .

Time-dependent Delta-interactions for 1D Schrödinger Hamiltonians

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The Sobolev spaces are denoted by H s (R), s ∈ R, their local version s by Hloc (R). The notation u ∈ H s+0 (R) means that there exists ε > 0 such that u ∈ H s+ε (R) (inductive limit) and its local version u ∈ s+0 Hloc (R) allows ε R > 0 to depend on R > 0 while considering the interval [−R, R]. More generally the Besov spaces are defined through dyadic decomposition: For ( p, r) ∈ [1, +∞]2 and s ∈ R, the space Bsp,r is the set of tempered distributions u such that   u Bsp,r := 2qs q u L p r < +∞ , 

−q

where q = ϕ(2 D), q ∈ N, is a cut-off in the Fourier variable supported in C−1 2q ≤ |ξ | ≤ C2q . Details are given in Appendix A. Finally, the notation ’’, appearing in many of the following proofs, denotes the inequality: ’≤ C’, being C a suitable positive constant.

2 Proof of Theorem 1.2 This theorem is a consequence of simple remarks, explicit calculations and standard applications of paraproduct estimates in 1D Sobolev spaces. Let us start with some elementary rewriting of the Cauchy problem (1.4). 2.1 Preliminary remarks •

•

First of all (1.4) or its integral version (1.5) are local problems in time so that t0 = 0, t ∈ [−T, T] for some T > 0 and even supp α ⊂ [−T/2, T/2] can be assumed after replacing α with αT (s) = α(s)χ( Ts ) for some fixed χ ∈ C0∞ ((−1/2, 1/2)) and χ ≡ 1 near s = 0. The dependence of H s -norms of αT with respect to T will be discussed when necessary. The (1.4) or its integral version (1.5) makes sense in S  (Rx ) as soon as u(t, 0) is well defined for almost all t ∈ [−T, T] and q(t) = αT (t)u(t, 0) is locally integrable. Then it can be written after applying the Fourier transform as a local problem in ξ ∈ R  i∂t u(t, ξ ) = |ξ |2 u(t, ξ ) + q(t) (2.1)  u(0, ξ ) =  u0 (ξ ) with q(t) = αT (t)u(t, 0) = αT (t)  u(t, ξ ) dξ . (2.2) R

This is equivalent to the integral form  u0 (ξ ) − i u(t, ξ ) = e−it|ξ |  2

with q(t) = q0 (t) − iαT (t)



t 0

t

e−i(t−s)|ξ | q(s) ds 2

(2.3)

0

e−i(t−s)|ξ | q(s) ds 2

R

dξ 2π

(2.4)

90

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by setting q0 (t) = αT (t)[eit u0 ](0) . The assumption u0 ∈ H s (R), s > 1/2, (resp. u0 ∈ L1 (R)) ensures that [eit u0 ](0) ∈ C0 ([−T, T]) (resp. t1/2 [eit u0 ](0) ∈ C0 ([−T, T])). Such an assumption as well as looking for u(t) ∈ H 1 (R) ensures that the quantities q0 (t) and q(t) make sense for almost all t ∈ [−T, T]. With the support assumption supp αT ⊂ [−T/2, T/2], the convolution equation (2.3) can be written t dξ 2 q(t) = q0 (t) − iαT (t) 1[−T,T] (t − s)e−i(t−s)|ξ | 1[−T,T] (s)q(s) ds 2π 0 R in D (R) := q0 (t) − iαT (t)Lq(t) := q0 (t) + Lα q(t) .

•

(2.5)

Once q is known after solving (2.5), (2.3) with t ∈ R reads simply

  2 2  u(t, ξ ) = e−it|ξ |  u0 (ξ ) − ie−it|ξ | F q1[0,t] −|ξ |2 .

•

(2.6)

When u0 and q are regular enough the time-derivative of the quantity (2.3) gives i∂t (∂t u)(t, ξ ) = |ξ |2 ∂t u(t, ξ ) + q (t). By Duhamel formula, this implies −it|ξ |2

u(t, ξ ) = e ∂t 



t

∂t  u(0, ξ ) − i

e−i(t−s)|ξ | q (s)ds , 2

0

while (2.1) says for t = 0 ∂t u(0, ξ ) = −i|ξ |2 u0 (ξ ) − iq(0) . Therefore we obtain for t ∈ R

  2

2 u(t, ξ ) = e−it|ξ | |ξ |2 u0 (ξ ) + q(0) + e−it|ξ | F (∂s q)1[0,t] (−|ξ |2 ) . (2.7) i∂t 2.2 Reduced scalar equation for q Let us now study the (2.5) written: q = q0 + Lα q with Lα q := −iαT (t)Lq = −iαT (t)

t 0

R

1[−T,T] (t − s)e−i(t−s)|ξ | 1[−T,T] (s)q(s) ds 2

Solving this fixed point equation relies on the next result.

dξ . 2π

Time-dependent Delta-interactions for 1D Schrödinger Hamiltonians

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Proposition 2.1 The estimate   1 1 Lq Hs  T 2 1[−1,1] (Dt )q L2 + T 2 −θ q 1[0,T] Hs−θ + q 1[−T,0] Hs−θ holds for every s ∈ R and θ ∈ [0, 12 ] . Proof Owing to



e±iλ|ξ |

2

R

dξ 2π

π

e±i 4 √ 4π λ

=

1 π √ Lq(t) = e−i 4 π

for λ > 0, Lq writes as

R π

1[0,T] (s)q(s)

1[0,T] (t − s) 1

(t − s) 2



+ ei 4

R

1[−T,0] (s)q(s)

ds

1[−T,0] (t − s) 1

(s − t) 2

ds

Passing to the Fourier transform, we get 1  π q(τ ) = e−i 4 √ L π

 e−itτ √ dt F (1[0,T] q)(τ ) t 0  T itτ  e √ dt F (1[−T,0] q)(τ ) t 0



π

+ ei 4

T

One easily checks    

T 0

 √ e±itτ  √ dt ≤ 2 T t

and

   

T 0

 e±itτ  1 √ dt  |τ |− 2 . t

This yields for every θ ∈ [0, 12 ], τ ∈ R    

T 0

 e±itτ  1 1 √ dt  T 2 1[−1,1] (τ ) + T 2 −θ |τ |−θ 1{R\[−1,1]} (τ ). t

Thus we get for every s ∈ R, θ ∈ [0, 12 ],   1 1 Lq Hs  T 2 1[−1,1] (Dt )q L2 + T 2 −θ 1[0,T] q Hs−θ + 1[−T,0] q Hs−θ .   Proposition 2.2 1

4 1. Let u0 ∈ H s (Rx ) with s > 1/2 and let α ∈ Hloc (Rt ) . Then the (2.5) has a 1

4 unique solution q ∈ Hloc (Rt ). Moreover, for a fixed u0 ∈ H s (Rx ) with s > 1 1 1/2, the map α → q is locally Lipschitzian from H 4 (Rt ) to H 4 (Rt ) .

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4 2. Let u0 ∈ H 1 (Rx ) and let α ∈ Hloc (Rt ) . Then the (2.5) has a unique solution 3

4 (Rt ). Moreover, for a fixed u0 ∈ H 1 (Rx ) the map α → q is locally q ∈ Hloc 3 3 Lipschitzian from H 4 (Rt ) to H 4 (Rt ). 3

+ε

4 3. Let u0 ∈ H 1+2ε (Rx ) and let α ∈ Hloc (Rt ) , for some ε > 0. Then the (2.5) 3

+ε

4 has a unique solution q ∈ Hloc (Rt ).

Proof 1) Let us first prove that q0 ∈ H 4 when u0 ∈ H 2 +ε . Write first 1

(eit u0 )(0) =

1

−1

e−it|ξ | u0 (ξ ) 2

1

dξ + 2π



+∞

√ √  e−itτ u0 ( τ ) + u0 (− τ )

1

dτ √ 2π τ

= I(t) + I I(t) . The first term I(t) defines a C∞ function with I Bs∞,∞  u0 L2 , for every s ∈ R . On the other hand, the Fourier transform of I I equals

√ √  1 II(−τ ) = 1[1,∞[ (τ ) u0 ( τ ) + u0 (− τ ) √ . τ The Sobolev regularity of the second term, I I, is given by: I I 2Hν

+∞



τ

2ν

1

 √ 2  u0 ( τ )   √  dτ  τ 

 u0 2H2ν−1/2 ,

(2.8)

for any ν ∈ R . Now, write q0 (t) = αT (t)I(t) + αT (t)I I(t). Lemma A.2-b) applied to the first term, implies αT I

1

H4

 αT

H4

1

I Bs+ε ∞,∞

 αT

H4

1

u0 L2 .

For the second term, use Lemma A.2-a), Sobolev embeddings and (2.8) αT I I

1

H4

 αT

H4

1

I I

 αT

H4

1

I I

 αT

H4

1

u0

1

2 B2,∞ ∩L∞ 1

ε

1

.

H2+2

H 2 +ε

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93

By combining these estimates, we get q0

 αT

1

H4

1

H4

u0

1

H 2 +ε

.

It remains to estimate αT . Let χ  ∈ D(R) with χ  ≡ 1 in [−1, 1] and set  α (t) = χ (t)α(t) . By using again Lemma A.2-a), we get for 0 ≤ T ≤ 1   α 14 χ(T −1 ·) 12 . αT 14   ∞ H

H

B2,∞ ∩L

 A change of variable in the Fourier transform F χ(T −1 .) (τ ) = T χ (Tτ ) leads to     χ(T −1 .) μ ≤ T 12 −μ χ Hμ and χ(T −1 .) μ  T 12 −μ χ Bμ , H B 2,∞ 2,∞

(2.9) for μ ≥ 0 and T ≤ 1. Hence we get αT

1

H4

  α

1

H4

.

and q0

1

H4

  α

1

H4

u0

(2.10)

1

H 2 +ε

In order to estimate the the operator L, use Lemma A.2-a) Lα q

1

H4

 αT   α

1

H4 1

H4

Lq

Lq

1

2 B2,∞ ∩L∞ 1

H 2 +ε

,

while Proposition 2.1 says Lq

1 H 2 +ε

 1 1  T 2 q L2 + T 4 −ε 1[0,T] q

+ 1[−T,0] q

1 H4

 1 H4

.

Hence we get for 0 ≤ T ≤ 1 and by Lemma A.2-a) Lq

 T 4 −ε q 1

1

H 2 +ε

1

H4

This yields Lα q

1

H4

  α

T 4 −ε q 1

1

H4

1

H4

.

1

(2.11)

This proves that L is a contracting map in H 4 for sufficiently small time T. The time T depends only on  α 14 and then we can construct globally a 1

H

4 unique solution q ∈ Hloc (R) for the linear problem (2.5).

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It remains to prove the continuity dependence of q with respect to α. Let 1

4 and q, q¯ the corresponding solutions then we have α, α¯ ∈ Hloc

¯ ¯ = αT (t)Lq(t) − α¯ T (t)Lq(t), q(t) − q(t)

α¯ T (t) = α(t)χ(t/T). ¯

with

Since L is linear on q then ¯ ¯ = αT (t)L(q − q)(t) ¯ q(t) − q(t) + (αT − α¯ T )(t)Lq(t) ¯ ¯ = Lα (q − q)(t) + Lα−α¯ q(t). To estimate the terms of the r.h.s we use (2.11) ¯ 14 ,   α 14 T 4 −ε q − q H H   1 ¯ 14 .   α − α¯  14 T 4 −ε q 1

¯ Lα (q − q)

H4

¯ Lα−α¯ q

H4

1

1

H

H

With the choice of T done above we get ¯ q − q

  α − α¯   

1 H4

1

H4

¯ q

1

H4

.

This achieves the proof of the continuity. 2) Write again q0 (t) = αT (t)I(t) + αT (t)I I(t). Lemma A.2-b) implies αT I

3

H4

 αT

H4

3

I

 αT

H4

3

u0 L2 .

3 +ε

4 B∞,∞

3

Since H 4 is an algebra the inequality αT

3

H4

 χ(T −1 ·)  T − 4  α

3

H4

 α

3

H4

1

3

H4

holds for T ∈ [0, 1], owing to (2.9). It follows αT I

 T − 4  α 1

3

H4

3

H4

u0 L2 .

The second term is estimated with (2.8): αT I I

3

H4

 αT

3

H4

I I

 T − 4  α 1

3

H4

3

H4

u0 H1 .

Time-dependent Delta-interactions for 1D Schrödinger Hamiltonians

95

Finally we get for T ∈ [0, 1] q0

 T − 4  α 1

3

H4

u0 H1 .

3

H4

Using Lemma A.2-a)–d), Sobolev embeddings and Proposition 2.1 (with 5 and θ = 16 ), gives θ = 12 Lα q

3

H4

 αT L∞ Lq

+ αT

3

H4

1   α L∞ T 12 q1[0,T]

  α

T 12 q

Lq L∞

+ q1[−T,0]

+ T − 4  α

1

3

H4

1

H3

3

H4

1

1

H3

 1

H3

+ T − 4  α 1

1

3

H4

T 3 q

1

H 3 +ε

3

H4

Lq

.

1

H 2 +ε

(2.12)

Thus we get for 0 ≤ T ≤ 1, Lα q

  α

3

H4

1

3

H4

T 12 q

3

H4

.

3

This proves that L is a contracting map in H 4 for sufficiently small time T. The time T depends only on  α 34 and then we can construct globally a H

3

4 (R) for the linear problem. unique solution q ∈ Hloc For the locally Lipschitz dependence with respect to α, the proof is left to 1 the reader: it can be easily done like for the case α ∈ H 4 . 3) Like in the proof of the second point 2) we get

q0

 αT

3

H 4 +ε

3

H 4 +ε

u0 L2 + I I

 3

H 4 +ε

 T − 4 −ε u0 H1+2ε . 1

Reproducing the same computation as (2.12) leads to Lα q

3

H 4 +ε

 αT L∞ Lq

+ αT

3

H 4 +ε

1   α L∞ T 12 q1[0,T]

+ T − 4 −ε  α 1

3

H 4 +ε

1

  α L∞ T 12 q 1

 T 12  α

3

H 4 +ε

Lq L∞

+ q1[−T,0]

1

H 3 +ε

Lq

3

H 4 +ε

1

H 2 +ε

+ T − 4 −ε  α 1

1

H 3 +ε

q

3

H 4 +ε

 1

H 3 +ε

T 3 +ε q 1

3

H 4 +ε

1

H 3 +2ε

.

With the fixed point argument we can conclude the proof.

 

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2.3 Regularity of u We start with the following result. Lemma 2.3 For s ∈ R, let HTs be the closed subset of H s (Rt )   HTs = u ∈ H s (Rt ), supp u ⊂ [−T, T] , endowed with the norm Hs . For any T > 0 and any s ∈ R, there is a constant CT,s such that 2s−1

∀ f ∈ HT 4 ,

 −1

 F F f (−|ξ |2 )  Hs ≤ CT,s f

Proof It suffices to compute  2 (1 + |ξ |2 )s   f (−|ξ |2 ) dξ =

H

2s−1 4

.

2 (1 + τ )s   f (−τ ) dτ 1/2 2τ 0 ∞  2  2    f (−τ ) dτ ≤ max f (−τ ) + (1 + τ )s−1/2  

R

+∞

τ ∈[0,1]

1

 2 ≤ max   f (−τ ) + f 2

s

1

H2−4

τ ∈[0,1]

,

(x) , f  with χ  ∈ D(R) with value 1 in [−T, T]. By duality where  f (τ ) = eiτ x χ we have for ν ∈ R sup |  f (τ )| ≤ f Hν sup eiτ · χ (·) H−ν

0≤τ ≤1

0≤τ ≤1

≤

C1T,ν

f Hν .  

The main result of this section is the following. Proposition 2.4 1

4 (Rt ), then the (1.5) has a unique solution u ∈ 1. Let u0 ∈ H 1 (Rx ), α ∈ Hloc 1 C(R; H (R)) . 3

4 2. Let u0 ∈ D(Hα(0) ), α ∈ Hloc (Rt ), then the (1.5) has a unique solution u 1 belonging to the space C (R; L2 (R)), with u(t) ∈ D(Hα(t) ) for all t ∈ R .

Proof 1) The solution of (1.5) is obtained via the (2.6)

 2 2  u(t, ξ ) = e−it|ξ |  u0 (ξ ) − ie−it|ξ | F q1[0,t] (−|ξ |2 ) .

Time-dependent Delta-interactions for 1D Schrödinger Hamiltonians

97

Let us check that we have the required regularity for u. Applying Lemma 2.3 to (2.6) implies u(t) H1 ≤ u0 H1 + C q1[0,t]

1

H4

.

(2.13)

Lemma A.2-a) and Lemma B.1 yield q1[0,t]

1

H4

 q

1

H4

≤ Ct q

1[0,t] 1

H4

1

2 B2,∞ ∩L∞

.

∞ This proves that u ∈ Lloc (R; H 1 ). It remains to prove the continuity in time of u. First notice that we need for this purpose to prove only the continuity in time of v(t) := u(t) − eit u0 . This will be done in two steps. In the first 3

4 one we deal with the case α ∈ Hloc . In the second one, we go back to the 1

4 . case α ∈ Hloc

•

3

4 Case α ∈ Hloc . Remark that according to Proposition 2.2-2) we can 3

4 construct a unique solution q ∈ Hloc for the problem (2.5). An easy  computation gives for t, t ∈ R    

2   2 2 t−t 2 |ξ | (1 + |ξ |2 ) F q1[0,t] (−|ξ |2 ) dξ v(t) − v(t ) H1  sin 2 R

+ q(1[0,t] − 1[0,t ] ) 2 1 . H4

Using the fact sin x ≤ |x|ε , ∀ε ∈ [0, 1], and Lemma 2.3 gives   

2  t − t 2 sin2 |ξ | (1 + |ξ |2 ) F q1[0,t] (−|ξ |2 ) dξ 2 R   ε  t − t  q1[0,t] 14 + 2ε . 2

H

It suffices now to use Lemma A.2-a)    

2  2 t−t 2 sin |ξ | (1 + |ξ |2 ) F q1[0,t] (−|ξ |2 ) dξ 2 R  ε   t − t  q 14 + 2ε . H

For the second term we use again Lemma A.2-c) combined with the proof of Lemma B.1 q(1[0,t] − 1[0,t ] )

1

H4

 q

H 4 +ε

1

1[0,t] − 1[0,t ]

 q

H 4 +ε

1

|t − tε

1

H 2 −ε

3

4 This concludes the proof of the time continuity of u when α ∈ Hloc .

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•

1

4 Case α ∈ Hloc . We smooth out the function α leading to a sequence 1

4 . To each of smooth functions αn that converges strongly to α in Hloc αn we associate the unique solutions qn and un . From the first step un belongs to C(R; H 1 ). Similarly to (2.13) we get for n, m ∈ N 1 ≤ C T qn − qm un − um L∞ [−T,T] H

1

H4

. 1

By Proposition 2.2-a), {qn } is a Cauchy sequence in H 4 and 1 thus {un } converges uniformly to u in L∞ T H . This gives that u ∈ C([−T, T], H 1 ), for every T > 0. 2) Recall from (2.7) that

 2 2 u(t, ξ ) = e−it|ξ | F (Hα(0) u0 )(ξ ) + e−it|ξ | F (∂s q)1[0,t] (−|ξ |2 ) . i∂t Since u0 ∈ D(Hα(0) ) then the first term of the r.h.s belongs to C(R; L2 ). On 3 the other hand we have D(Hα(0 ) ⊂ H 2 −d for any d > 0. It follows from 3 Proposition 2.4-1) that we can construct a unique solution q ∈ H 4 . Now, it let w(t) := i∂t u − e Hα(0) u0 . Then Lemma 2.3 yields w(t) L2  q 1[0,t]

.

1

H− 4

Lemma A.2-a) and Lemma B.1 imply q 1[0,t]

1

H− 4

 q

1

H− 4

≤ CT q

1[0,t] 3

H4

1

2 B2,∞ ∩L∞

.

Thus we get for every t ∈ [−T, T] w(t) L2 ≤ CT q

3

H4

.

(2.14)

∞ It follows that w ∈ Lloc (R; L2 ). To prove the continuity in time of w we use the same argument as for the first point of this proposition. We 3

+ε

4 start with a smooth function α, that is α ∈ Hloc . This gives according 3 +ε to Proposition 2.2-3) a unique solution q ∈ H 4 . We have the following 3

4 discussed above, estimate, obtained similarly to case α ∈ Hloc

w(t) − w(t ) L2  |t − t |ε q 1[0,t] Using Lemma A.2-a) with s = − 41 + q 1[0,t]

1

1

ε

H− 4 + 2

ε 2

gives

ε

 q

H− 4 + 2

+ q (1[0,t] − 1[0,t ] )

3

H 4 +ε

.

1

H− 4

.

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For the second term of the r.h.s we use Lemma A.2-c- with s = − 41 + ε, s = 12 − ε and Lemma B.1 q (1[0,t] − 1[0,t ] )

1

H− 4

 |t − t |ε q

3

H 4 +ε

. 3

+ε

4 This achieves the proof of the continuity of w in time for α ∈ Hloc . Now 3 4

for α ∈ Hloc we do like the first point of the proposition: we smooth out α and we use the continuity dependence of q with respect to α stated in Proposition 2.2-2) combined with the estimate (2.14). By writing i∂t u = Hα(t) u(t) we get that for every t ∈ R, u(t) ∈ D(Hα(t) ).  

Appendix A: Paraproducts and product laws The aim of this section is to prove some product laws used in the proof of the main results. For this purpose we first recall some basic ingredients of the paradifferential calculus. Start with the dyadic partition of the unity: there exists two radial positive functions χ ∈ D(R) and ϕ ∈ D(R\{0}) such that  χ(ξ ) + ϕ(2−q ξ ) = 1, ∀ξ ∈ R. q≥0

For every tempered distribution v ∈ S  , set −1 v = χ(D)v ; ∀q ∈ N, q v = ϕ(2

−q

and Sq =

D)v

q−1 

 j.

j=−1

For more details see for instance [3, 4]. Then Bony’s decomposition of the product uv is given by uv = Tu v + Tv u + R(u, v), with Tu v =



Sq−1 uq v

and

R(u, v) =



q uq v .

|q −q|≤1

q

Let us now recall the definition of Besov spaces through dyadic decomposition. For ( p, r) ∈ [1, +∞]2 and s ∈ R, the space Bsp,r is the set of tempered distribution u such that   u Bsp,r := 2qs q u L p r < +∞. 

This definition does not depend on the choice of the dyadic decomposition. One can further remark that the Sobolev space H s coincides with Bs2,2 . Below is the Bernstein lemma that will be used for the proof of product laws and which is a straightforward application of convolution estimates and Fourier localization.

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Lemma A.1 There exists a constant C such that for q, k ∈ N, 1 ≤ a ≤ b and for f ∈ La (R), sup ∂ α Sq f Lb ≤ Ck 2q(k+ a − b ) Sq f La , 1

1

|α|=k

C−k 2qk q f La ≤ sup ∂ α q f La ≤ Ck 2qk q f La . |α|=k

The following product laws have been used intensively in the proof of our main result. Lemma A.2 In dimension d = 1 the product (u, v) → uv is bilinear continuous 1

a) b) c) d)

2 from H s × (B2,∞ ∩ L∞ ) to H s as soon as |s| < 12 ; s s+ε from H × B∞,∞ to H s as soon as s ≥ 0 and ε > 0.  1  from H s × H s to H s+s − 2 as soon as s, s < 12 and s + s > 0. For s ≥ 0, H s ∩ L∞ is an algebra. For s > 12 , H s is an algebra.

Proof a) Using the definition and Bernstein lemma we obtain Tu v 2Hs 



22qs Sq−1 u 2L∞ q v 2L2

q



 v 2

1 2 B2,∞

1

q

 

 v 2

1 2 B2,∞

q

1 2 B2,∞

1

2 B2,∞

p

2q(s− 2 ) 2 2  p u L2 1

2

p≤q−1

 

 v 2  v 2

22q(s− 2 ) Sq−1 u 2L∞

q

2 1 2(q− p)(s− 2 ) (2 ps  p u L2 )

p≤q−1

u 2Hs .

We have used in the last line the convolution law 1  2 → 2 . For the second term Tv u we use the fact that Sq−1 maps L∞ to itself uniformly with respect to q. Tv u 2Hs 



22qs Sq−1 v 2L∞ q u 2L2

q

 v 2L∞ u 2Hs .

Time-dependent Delta-interactions for 1D Schrödinger Hamiltonians

101

To estimate the remainder term we use the fact  q R(u, v) = q ( ju j v). j≥q−4 | j− j |≤1

According to Bernstein lemma one gets  1  ju L2  j v L2 2qs q (R(u, v)) L2  2q(s+ 2 ) 



j≥q−4 | j− j |≤1

j≥q−4 | j− j |≤1

 v

1

2(q− j)(s+ 2 ) 2 js  ju L2 2 j 2  j v L2

1 2 B2,∞

1



2(q− j)(s+ 2 ) 2 js  ju L2 . 1

j≥q−4

It suffices now to apply the convolution inequalities. b) First remark that the case s = 0 is obvious: L2 × L∞ → L2 and Bε∞,∞ → L∞ . Hereafter we consider s > 0 . To estimate the first paras product, use the embedding Bs+ε ∞,∞ → B∞,2 , for ε > 0.  Tu v 2Hs  22qs Sq−1 u 2L2 q v 2L∞ q

 u L2 v Bs∞,2  u L2 v Bs+ ∞,∞

For the second term we use the result obtained in the part a):  Tv u 2Hs  22qs Sq−1 v 2L∞ q u 2L2 q

 v 2L∞ u 2Hs  v 2Bs+ u 2Hs ∞,∞

To estimate the remainder term we write   ju L2  j v L∞ 2qs q (R(u, v)) L2  2qs j≥q−4 | j− j |≤1

 v L∞



2(q− j)s 2 js  ju L2 .

j≥q−4 | j− j |≤1

Since s > 0 then we obtain by using the convolution inequalities R(u, v) Hs ≤ v L∞ u Hs . c), d) These results are standard, see for example [4].

 

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Appendix B: Sobolev and Besov regularity of cut-offs Lemma B.1 1. For any ν < 1/2, t → 1[0,t] (s) belongs to C(R+ ; H ν (R)) . More precisely we have for |t − t | ≤ 1 1[0,t] − 1[0,t ] Hν  |t − t | 2 −ν . 1

1

2 2. The map t → 1[0,t] (s) belongs to L∞ (R+ ; B2,∞ (R)).

Proof −iτ t

1) The Fourier transform of 1[0,t] (s) equals F (1[0,t] )(τ ) = e −iτ−1 . One gets for ν ∈ [0, 1/2)  2 (1 + τ 2 )ν F (1[0,t1 ] ) − F (1[0,t2 ] ) (τ ) dτ R



=4

R

(1 + τ 2 )ν

Let λ > 1 then

|sin(τ |t2 − t1 |/2)|2 dτ := I. τ2



λ

I  |t2 − t1 |2

τ 2ν dτ +

λ

0

 |t2 − t1 | λ

2 2ν+1

+λ

2ν−1

+∞

τ 2ν−2 dτ

.

Choosing judiciously λ then we obtain for |t2 − t1 | ≤ 1   1[0,t ] − 1[0,t ]  ν ≤ Cν |t2 − t1 |1/2−ν . 1 2 H 2) We set ft (s) := 1[0,t] (s), then have ft 2

1

2 B2,∞



≤ f 2L2 + max 2q q∈N



 |t| + max 2q q∈N

2q ≤|τ |≤2q+1

2q ≤|τ |≤2q+1

| ft (τ )|2 dτ

|τ |−2 dτ

 |t| + 1.

 

References 1. Adami, R., Teta, A.: A class of nonlinear Schrödinger equations with concentrated nonlinearitie. J. Funct. Anal. 180, 148–175 (2001) 2. Albeverio, S., Gesztesy, F., Högh-Krohn, R., Holden, H.: Solvable Models in Quantum Mechanics, 2nd edn, with an appendix by P. Exner. AMS, Providence (2005) 3. Bony, J.-M.: Calcul symbolique et propagation des singularités pour les équations aux dérivées partielles non linéaires. Ann. Sci. Ec. Norm. Super. (4), 14(2), 209–246 (1981)

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4. Chemin, J.-Y.: Perfect Incompressible Fluids. Translated from the 1995 French original by Isabelle Gallagher and Dragos Iftimie. Oxford Lecture Series in Mathematics and its Applications, 14. Clarendon, Oxford University Press, New York (1998) 5. Costin, O., Costin, R.D., Lebowitz, J.L., Rokhlenko, A.: Evolution of a model quantum system under time periodic forcing: conditions for complete ionization. Commun. Math. Phys. 221(1), 1–26 (2001) 6. Correggi, M., Dell’Antonio, G.F., Figari, R., Mantile, A.: Ionization for three dimensional time-dependent point interactions. Commun. Math. Phys. 257, 169–192 (2005) 7. Correggi, M., Dell’Antonio, G.F.: Decay of a bound state under a time-periodic perturbation: a toy case. J. Phys., A, Math. Gen. 38, 4769–4781 (2005) 8. Dell’Antonio, G.F., Figari, R., Teta, A.: A limit evolution problem for time-dependent point interactions. J. Funct. Anal. 142, 249–275 (1996) 9. Fattorini, H.O., Kerber, A.: The Cauchy Problem. Cambridge University Press, Cambridge (1984) 10. Holmer, J.: The initial boundary value problem for the 1D nonlinear Schrödinger equation onthe half line. Differ. Integral Equ. 18(6), 647–668 (2005) 11. Kato, T.: Integration of the equation of evolution in a Banach spac. J. Math. Soc. Jpn. 5, 208– 234 (1953) 12. Kato, T.: Linear evolution equations of ‘hyperbolic’ type. J. Fac. Sci., Univ. Tokyo, Sect. 1A, Math. 17, 241–258 (1970) ´ 13. Kisynski, J.: Sur les opérateurs de Green des problèmes de Cauchy abstraits. Stud. Math. 23, 285–328 (1964) 14. Mantile, A.: Point interaction controls for the energy transfer in 3D quantum systems. Int. J. Control 81(5), 752–763 (2008) 15. Neidhardt, H., Zagrebnov, V.: Linear non-autonomous Cauchy problems and evolution semigroups. Adv. Differ. Equ. 14(3–4), 289–340 (2009) 16. Reed, M., Simon, B.: Methods of Modern Mathematical Physics, vol. 2. Academic, New York (1975) 17. Sayapova, M., Yafaev, D.: The evolution operator for time-dependent potentials of zero radius. Trudy Mat. Inst. Steklov. 159, 167–174 (1983) 18. Simon, B.: Quantum Machanics for Hamiltonians Defined as Quadratic Forms. Princeton University Press, Princeton (1971) 19. Soffer, A., Weinstein, M.I.: Nonautonomous Hamiltonians. J. Stat. Phys. 93(1–2), 359–391 (1998) 20. Yafaev, D.: On “eigenfunctions” of a time-dependent Schrödinger equation (Russian). Teor. Mat. Fiz. 43(2), 228–239 (1980) 21. Yafaev, D.: Scattering theory for time-dependent zero-range potentials. Ann. IHP, Phys. Théor. 40 (1984)

Math Phys Anal Geom (2010) 13:105–143 DOI 10.1007/s11040-009-9071-1

Pictorial Representation for Antisymmetric Eigenfunctions of PS−3 Integral Equations Andrei Borisovich Bogatyrev

Received: 27 March 2008 / Accepted: 15 December 2009 / Published online: 16 January 2010 © Springer Science+Business Media B.V. 2009

Abstract Eigenvalue problem for Poincare-Steklov-3 integral equation is reduced to the solution of three transcendential equations for three unknown numbers, moduli of pants. The complete list of antisymmetric eigenfunctions of integral equation in terms of Kleinian membranes is given. Keywords Spectral parameter · Riemann surface · Pair of pants · Branched complex projective structure · Dessin d’Enfants Mathematics Subject Classifications (2000) 30Fxx · 14H15 · 30C20 · 45C05 · 33E30

1 Introduction Traditionally, integral equations are the subject of functional analysis and operator theory. In the contrast we show that methods of complex geometry and combinatorics are efficient for the study of the following singular integral Poincare-Steklov (briefly, PS) equations  λ V. p. I

u(t) dt − V. p. t−x

 I

u(t) dR(t) = const, R(t) − R(x)

x ∈ I := (−1, 1), (1)

Supported by RFBR grant 09-01-12160 and RAS Program “Contemporary problems of theoretical mathematics”. A. B. Bogatyrev (B) Institute for Numerical Mathematics, Russian Academy of Sciences, ul. Gubkina 8, 119991 Moscow GSP-1, Russia e-mail: [email protected]

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where λ is the spectral parameter; u(t) is the unknown function; const is independent of x. The functional parameter R(t) of the equation is a given smooth nondegenerate change of variable on the interval I:   d   0 <  R(t) < ∞, when t ∈ [−1, 1]. (2) dt Under the assumption that R(t) =: R3 (t) is a degree three rational function with separate real critical values different from the endpoints of the interval I, we give the constructive representation for the eigenvalues λ and eigenfunctions u(x) of (1). First, we say a few words about the origin of PS integral equations and the related background. Spectral Boundary Value Problem Let a domain in the plane be subdivided into two simply connected domains 1 and 2 by a smooth simple arc . We are looking for the values of the spectral parameter λ when the following problem has nonzero solution: Find a harmonic function U s in the domain s , s = 1, 2, vanishing on the outer portion of the boundary: ∂s \ . On the interface  the functions U 1 and U 2 coincide while their normal derivatives dif fer by the factor of −λ: −λ

∂U 2 ∂U 1 = . ∂n ∂n

(3)

Applications Boundary value problems for the Laplace equation with spectral parameter in the boundary conditions were first considered by H.Poincare (1896) and V.A.Steklov (1901). The problems of this kind arise e.g. in the analysis of diffraction, (thermo-) conductivity of composite materials and the motion of two-phase liquids in porous medium. This particular problem (3) arises in justification and optimization of domain decomposition method for the solution of boundary values problems for elliptic PDE. The eigenvalues λ of the spectral problem and the traces of eigenfunctions U 1 = U 2 on the interface  are respectively the critical values and critical points of the following functional, the ratio of two Dirichlet integrals   F(U) =  2 1

|∇U 2 |2 d2 |∇U 1 |2 d1

,

1/2 U ∈ Hoo (),

(4)

where U s is the harmonic continuation of the function U from interface  to the domain s , s = 1, 2, vanishing at the outer boundary of the domain. Integral Equation The reduction of the stated above boundary value problem to the interface brings to the (1). Let Vs be the harmonic function conjugate to U s , s = 1, 2. From Cauchy-Riemann equations and (3) it follows that tangent to the interface derivatives of V1 and V2 differ by the same factor −λ. Integrating along  we get λV1 (y) + V2 (y) = const,

y ∈ .

(5)

Antisymmetric Eigenfunctions of PS − 3 Integral Equations

107

For the half-plane the boundary values of conjugate harmonic functions are related via Hilbert transform. To take advantage of this transformation we consider a conformal mapping ωs (y) from s to the open upper halfplane H with normalization ωs () = I, s = 1, 2. Now (5) may be rewritten as

−

λ V. p. π

 I

U 1 (ω1−1 (t)) 1 dt − V. p. t − ω1 (y) π

 I

U 2 (ω2−1 (t ))  dt = const, t − ω2 (y)

y ∈ .

Introducing new notation x := ω1 (y) ∈ I; R := ω2 ◦ ω1−1 : I →  → I; u(t) := U 1 (ω1−1 (t)) and the change of variable t = R(t) in the second integral, we arrive at the Poincare-Steklov (1). Note that here R(t) is the decreasing function on I. Operator analysis of two equivalent spectral problems, boundary value problem (3) and Poincare-Steklov (1), may be found e.g. in [1, 2]. Here we only mention that the spectrum is discreet if (2) holds, the eigen values are positive and converge to λ = 1. Philosophy of the Research The aim of our study is to give explicit expressions for the eigen pairs (λ, u) of the PS integral equation. For the rational degree two functions R(x) = R2 (x) the eigen pairs were expressed in terms of elliptic functions [3]. Next natural step is to consider degree three rational functions. Here the notion of explicit solution should be specified. Usually this term means an answer in terms of elementary function of parameters or a quadrature of it or an application of other permissible operations in the spirit of Umemura classical functions. The history of mathematics however knows many disappointing results when the solution of the prescribed form does not exist. The nature always forces us to introduce new types of transcendent objects to enlarge the scope of search. Cf.: “Mais cette étude intime de la nature des fonctions integrales ne peut se faire que par l’introduction de transcendantes nouvelles” [4]. Take for instance the algebraic equations. The ancients were able to solve quadratic equations. But after the invention of the formulas for cubic and quartic equations in the 16th century no progress was made until the 19th century when it became clear that no formula including arithmetic operations and radicals only can solve equations of higher orders. After that Ch.Hermite and L.Kronecker suggested formulae involving elliptic modular function to find the roots of degree five equations. The ideas of C.Jordan elaborated by H.Umemura resulted in a formula (involving hyperelliptic integrals and theta constants) for the roots of arbitrary degree polynomial. In modern mathematical physics it is very often that the problems are “explicitly” solved in terms of suitable transcendential functions, say solutions of Painleve equations. From the philosophical point of view our goal is to study the nature of the solutions of integral (1) and the means for their constructive representation.

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Brief Description of the Result Given rational degree three function R3 (t), we explicitly associate it to a pair of pants in Section 2.3. On the other hand, given spectral parameter λ and two auxiliary real parameters, we explicitly construct in Section 4 another pair of pants which additionally depend on one or two integers. When the above two pants are conformally equivalent, λ is the eigenvalue of the PS integral equation with parameter R3 (x). Essentially, this means that to find the spectrum of the given integral (1) one has to solve three transcendential equations involving three moduli of pants. Whether this representation of the solutions may be considered as a constructive or not is a matter of a discussion. Two arguments in favour: today it is possible to numerically evaluate the conformal structure of surfaces (e.g. via circle packing). Also, this representation allows us to conceive the valuable features of the solution: to find the number of zeroes of eigenfunction u(t), to localize the spectrum and to show the discrete mechanism of generating the eigenvalues.

2 Space of PS-3 Equations In what follows we consider integral equations (1) with rational degree three real functional parameter R(x) = R3 (x) and call such equations PS-3. We restrict ourselves to the case when R3 (x) has four distinct real critical values dif ferent from ±1. Other possible configurations are discussed in [11]. The details of our further constructions depend on the topological properties of functional parameter of the integral equation. One may encounter one of five described in Section 2.2 typical situations A, B 1, B 21, B 22, B 23 corresponding to the components in the space of admissible functions R3 (x). 2.1 Topology of the Branched Covering Degree three rational function R3 (x) defines the three- sheeted branched covering of a Riemann sphere by another Riemann sphere. The RiemannHurwitz formula suggests that R3 (x) typically has four separate branch points as , s = 1, 4. This means that every value as is covered by a critical point bs , and an ordinary point cs . We have assumed that all four points as are distinct, real and differ from ±1. ˆ := R ∪ {∞} belongs to Every point y = as of the extended real axis R exactly one of two types. For the type (3:0) the pre-image R−1 3 (y) consists of three distinct real points. For the type (1:2) the pre-image consists of a real and two complex conjugate points. The type of the point is locally constant on the extended real axis and changes when we step over the branch point. Let ˆ so that the branch points as be enumerated in the natural cyclic order of R the intervals (a1 , a2 ) and (a3 , a4 ) are filled with the points of the type (1:2). We specify the way to exclude the relabeling a1 ↔ a3 , a2 ↔ a4 of branch points in Section 2.2.

Antisymmetric Eigenfunctions of PS − 3 Integral Equations

109

Fig. 1 The topology of the covering R3 with real branch points

ˆ The total pre-image R−1 3 (R) consists of the extended real axis and two pairs ˆ at points b 1 , b 2 , b 3 , b 4 as shown of complex conjugate arcs intersecting R at the left side of Fig. 1. The compliment of this pre-image on the Riemann sphere has six components, each of them is mapped 1-1 onto upper or lower half plane. Note that the points b 1 , c4 , c3 , b 2 ... on the left picture of Fig. 1 may follow in inverse order as well. 2.2 Classification of Parameters R3 The functional parameter R3 (x) is a nondegenerate change of variable on the segment [−1, 1]. This in particular means that no critical point bs belongs to this segment. So exactly one of two cases is realized:1 Case A : [−1, 1] ⊂ [b 2 , b 3 ], Case B : [−1, 1] ⊂ [b 3 , b 4 ].

(6)

Remaining possibilities (like [−1, 1] ⊂ [b 1 , b 2 ]) are reduced either to A or B by the clever choice of labeling the branch points as —see Section 2.1. For the case B it is important whether [−1, 1] intersects [c2 , c1 ] or not. So we consider two subcases: Case B 1 : [−1, 1] ∩ [c2 , c1 ] = ∅, Case B 2 : [−1, 1] ∩ [c2 , c1 ] = ∅.

(7)

The case B 2 in turn is subdivided into three subcases: Case B 21 : [−1, 1] ⊂ [c2 , c1 ], Case B 22 : [−1, 1] ⊃ [c2 , c1 ], Case B 23 : all the rest.

(8)

2.3 Pair of Pants Associated to R3 For the obvious reason, a pair of pants is the name for the Riemann sphere ˆ with three with three holes in it. Pair of pants may be conformally mapped to C nonintersecting real slots. This mapping is unique up to real linear-fractional mappings. The conformal class of pants with labelled boundary components depends on three real parameters varying in a cell. 1 Two points on a circle (extended real axis) define two segments. It should be clear which segment

we mean: e.g. b 1 , b 2  ∈ [b 3 , b 4 ]; b 1 , b 4  ∈ [b 2 , b 3 ], etc.

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Definition To every PS-3 equation we associate the pair of pants:   ˆ \ {([−1, 1][a1 , a2 ]) ∪ [a3 , a4 ]} P (R3 ) := Cl C

(9)

where  is the symmetric difference (union of two sets minus their intersection). Closure here and everywhere below is taken with respect to the intrinsic spherical metric when every slot acquires two banks. Boundary components of pants are colored in accordance with the palette: [−1, 1] \ [a1 , a2 ] [a1 , a2 ] \ [−1, 1] [a3 , a4 ]

– red – blue – green

Thus obtained pair of pants will have boundary ovals of all three colors, but in cases B 21 (green and two blue ovals) and B 22 (green and two red ovals). Note that in case A the red, green and blue slots always follow in the natural cyclic order of the extended real axis. 2.4 Gauge Transformations There is a two-parametric transformation of the functional parameter R(x) which essentially does not affect the spectral characteristics of integral equation (1). Let us recall that R(x) is not uniquely determined by two domains 1 and 2 . Pre- and post- composition with linear-fractional transformations preserving the segment [−1, 1] is admissible. The general appearance of such a mapping is L± α (t) := ±

t+α , αt + 1

α ∈ (−1, 1).

(10)

Lemma 1 1. The gauge transformation R → L± α ◦ R does not change neither eigenvalues λ nor the eigenfunctions u(t) of any PS integral equation. 2. The gauge transformation R → R ◦ L± α does not change the eigenvalues λ and slightly changes the eigenfunctions: u(t) → u(L± α (t)). Proof To simplify the notations we put L(t) := L± α (t). 1. The gauge transformation just adds a constant term to the right hand side of equation.  1  1 u(t)dL(R(t)) u(t)L (R(t))dR(t) =   1/2 (R(t) − R(x)) −1 L(R(t)) − L(R(x)) −1 [L (R(t))L (R(x))]  1  1 u(t)dR(t) u(t)dR(t) = − . −1 −1 R(t) − R(x) −1 R(t) − L (∞)

Antisymmetric Eigenfunctions of PS − 3 Integral Equations

111

2. We define the new variable x∗ := L(x) and the new function u∗ (x∗ ) := u(x).  1  1 u(t)dR(L(t)) u∗ (t∗ )dR(t∗ ) =± , R(L(t)) − R(L(x)) R(t ∗ ) − R(x∗ ) −1 −1  1  1  1  1 u(t)dt u∗ (t∗ )dL−1 (t∗ ) u∗ (t∗ )dt∗ u∗ (t)dt ∓ =± = ± . −1 −1 −1 t − x −1 L (t∗ ) − L (x∗ ) −1 t∗ − x∗ −1 t − L(∞)   We see that essentially the space of PS − 3 equations has real dimension 3, the same as the dimension of the moduli space of pants. It is easy to check the following: • • •

Any gauge transformation of the parameter R3 (x) does not change the type (A, B 1, . . . ) of integral equation. The transformation R3 → R3 ◦ L± α does not change the associated pants (9) and preserves the colors of the boundary ovals. ± Associated to functional parameter L± α ◦ R3 are the pants Lα P (R3 ). The ± colors of the boundary ovals are transferred by Lα , but in one case. When the type of integral equation is A, the transformation L− α interchanges blue and green colours on the boundaries.

2.5 Reconstruction of R3 (x) from the Pants The parameter R3 (x) of integral equation may be reconstructed, given the pants P (R3 ) and the type A, B 1 . . . of the equation. One has to follow the routine described below. Restore the Labeling of the Branch Points In case B 2 we temporarily paint the real segment separating two non-green slots in blue. The (extended) blue segment is set to be [a1 , a2 ]; the green segment is [a3 , a4 ]. The relabeling a1 ↔ a2 and a3 ↔ a4 is eliminated by the natural cyclic order of the points a1 , a2 , a3 , a4 on the extended real axis. Normalized Covering Let La be the unique linear-fractional map sending the points a1 , a2 , a3 , a4 to respectively 0, 1, a > 1, ∞. The conformal motion Lb of the covering Riemann sphere sends the critical points b 1 , b 2 , b 3 , b 4 of R3 (x) (unknown at the moment) to respectively 0, 1, b > 1, ∞. The function La ◦ R3 ◦ L−1 b with the normalized critical points and critical values takes a simple form:  R3 (x) = x2 L(x), with real linear fractional function L(x) satisfying the restrictions: L(1) = 1, L(b ) = a/b 2 ,

L (1) = −2, L (b ) = −2a/b 3 .

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We got four equations for three parameters of L(x) and the unknown b . The first two equations suggest the following expression for the linear-fractional function: (c − 1)(x − 1) . L(x) = 1 + 2 x−c The other two equations are solved parametrically in terms of c: b (c) = c

3c − 2 ; 2c − 1

a(c) = c

(3c − 2)3 . 2c − 1

Given a > 1, there are exactly two real solutions of the equation a(c) = a. One of the solutions c lies in the segment (1/3, 1/2), the other lies in (1, ∞). In the case c ∈ (1/3, 1/2) the segments La (a1 , a2 ) = (0, 1) and La (a3 , a3 ) = (a, ∞) are filled with the points of the type (1:2), which corresponds to our choice of labelling the branchpoints in Section 2.1. The solution c ∈ (1, ∞) is a fake as the same segments bear the points of the type (3:0). Both functions b (c) and a(c) increase from 1 to ∞ when the argument c runs from 1/3 to 1/2. Reconstruction of the Mapping Lb In case A the red, green and blue slots follow in the natural cyclic order. Hence, the segment La [−1, 1] is a subset of −1 the interval (1, a). We choose the unique component of the pre-image  R3 of the segment La [−1, 1] belonging to the interval (1, b ) – see Fig. 2. For the case B the segment La [−1, 1] is a subset of the ray (−∞, a) and we choose the pre-image of this segment which lie in (b , ∞). The requirement: Lb maps [−1, 1] to the chosen segment determines this map up to a pre-composition with the function (10). We see that given the pants P (R3 ), the functional parameter is recovered up to a gauge transformation R3 → R3 ◦ L± α . It is not difficult to check, that the described above procedure applied to the pair of pants L± β P (R3 ) (in case A and the mapping L− reversing the orientation of real axis we additionally have β Fig. 2 The graph of R˜ 3 (x) when c ∈ ( 31 , 12 )

a 1 c

1

b

x

Antisymmetric Eigenfunctions of PS − 3 Integral Equations

113

to exchange the blue and green colors of the slots) returns the covering map ± L± β ◦ R3 ◦ Lα . Roughly speaking, the classes of gauge transformation of R3 (x) correspond to the conformal classes of pants with suitably colored boundary components and each conformal class of pants corresponds to a class of certain functional parameter R3 (x).

3 Reduction to Projective Structures PS integral equations possess rich geometrical content [10, 11] which we disclose in this section. We describe a three-step reduction of the integral equation to a certain problem [11] for projective structures on a riemann surface which has essentially combinatorial solution. 3.1 Step 1: Functional Equation Let us expand the kernel of the second integral in (1) into a sum of elementary fractions:  R3 (t) 1 d Q = log(R3 (t) − R3 (x)) = − (t), R3 (t) − R3 (x) dt t − zk (x) Q 3

(11)

k=1

where Q(t) is the denominator in noncancellable representation of R3 (t) as the ratio of two polynomials; z1 (x) = x, z2 (x), z3 (x)—are all solutions (including multiple and infinite) of the equation R3 (z) = R3 (x). This expansion suggests to rewrite the original equation (1) as certain relationship for the Cauchy-type integral 

(x) := I

u(t) dt + const∗ , t−x

ˆ \ [−1, 1]. x∈C

(12)

Known (x), the solution u(t) may be recovered by the Sokhotskii-Plemelj formula: u(t) = (2πi)−1 [ (t + i0) − (t − i0)] ,

t ∈ I.

The constant term const∗ in (12), which we assume to be ⎡ ⎤   1 u(t)Q (t) ⎣ const∗ := dt − const⎦ λ−3 Q(t)

(13)

(14)

I

is introduced to cancel the constant terms arising after substitution of expression (12) to the equation (1). In this way the following result was proven [10]: Lemma 2 For λ = 1, 3 the transformations (12) and (13) bring about a one-toone correspondence between the Hölder’s eigenfunctions u(t) of PS-3 integral

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equation and the holomorphic on the Riemann sphere outside the slot [−1, 1] nontrivial solutions (x) of the functional equation



(x + i0) + (x − i0) = δ (z2 (x)) + (z3 (x)) ,

x ∈ I,

δ = 2/(λ − 1),

(15)

(16)

with Hölder boundary values (x ± i0) at the banks of the slot [−1, 1]. 3.2 Step 2: Riemann Monodromy Problem In this section we reduce our functional (and therefore integral) equation to the Riemann monodromy problem in the following form. Find a holomorphic vector W(y) = (W1 , W2 , W3 )t on the slit sphere P (R3 ) \ [−1, 1] whose boundary values on the opposite sides of every slot are related by the constant matrix specif ied for each slot. 3.2.1 Monodromy Generators To formulate the Riemann monodromy problem we introduce 3 × 3 permutation matrices 100 D1 := 0 0 1 ; 010

001 D2 := 0 1 0 ; 100

010 D3 := 1 0 0 001

(17)

and a matrix depending on the spectral parameter λ: D :=

−1 δ δ 0 10 , 0 01

δ = 2/(λ − 1).

(18)

Lemma 3 Matrices D1 , D2 , D3 , D, D1 D = DD1 have order two as GL3 group elements. 3.2.2 Separating Branches of R−1 3 Let domain O be the compliment to the segments [a1 , a2 ] and [a3 , a4 ] on the Riemann sphere. The pre-image R−1 3 O consists of three components O j , j = 1, 2, 3, mapped one-one to O—see Fig. 1. Two of the components are (topological) discs with a slot and the third is an annulus. The enumeration of domains O j is determined by the following rule: the segment [−1, 1] lies in the closure of O1 , the segment [c3 , c4 ] lies on the border of O2 .

Antisymmetric Eigenfunctions of PS − 3 Integral Equations

115

3.2.3 Let u(x) be the solution of integral equation (1) in the case A. We consider the vector W(y) = ( (x1 ), (x2 ), (x3 ))t ,

y ∈ O \ [−1, 1],

(19)

where (x) is from (12) and xs is the unique solution of the equation R3 (xs ) = y in Os . Vector W(y) will be holomorphic and bounded in the domain O \ [−1, 1] as all three points xs , s = 1, 2, 3, remain in the holomorphy domain of the function (x). We claim that W(y + i0) = DW(y − i0), W(y + i0) = D3 W(y − i0), W(y + i0) = D2 W(y − i0),

when y ∈ [−1, 1], when y ∈ [a1 , a2 ], when y ∈ [a3 , a4 ].

Indeed, let y+ := y + i0 and y− := y − i0 be two points on the opposite − ± ∓ banks of [a1 , a2 ]. Their inverse images x+ 3 = x3 , x1 = x2 lie outside the cut [−1, 1]. Hence W(y+ ) = D3 W(y− ). For two points y± lying on the opposite − banks of the slot [a3 , a4 ], their inverse images satisfy the relations x+ 2 = x2 , ± ∓ x1 = x3 , which means W(y+ ) = D2 W(y− ). Finally, let y± lie on the banks of − + − [−1, 1]. Now two points x+ 2 = x2 and x3 = x3 lie in the holomorphy domain of + −

(x) while x1 and x1 appear on the opposite sides of the cut [−1, 1]. According to the functional equation (15), − − −

(x+ 1 ) = − (x1 ) + δ( (x2 ) + (x3 )),

therefore W(y+ ) = DW(y− ) holds on the slot [−1, 1]. 3.2.4 Conversely, let W(y) = (W1 , W2 , W3 )t be the bounded solution of the Riemann monodromy problem stated above. We define a piecewise holomorphic function on the Riemann sphere:

(x) := Ws (R3 (x)),

when x ∈ Os \ R−1 3 [−1, 1],

s = 1, 2, 3.

(20)

From the boundary relations for the vector W(y) it immediately follows that the function (x) has no jumps on the lifted cuts [a1 , a2 ], [a3 , a4 ], [−1, 1] apart from the cut [−1, 1] from the upper sphere. Say, if the two points y± lie on the opposite sides of the cut [a1 , a2 ], then W3 (y+ ) = W3 (y− ) and W1 (y± ) = W2 (y∓ ) which means that the function (x) has no jump on the components of R−1 3 [a1 , a2 ]. From the boundary relation on the cut [−1, 1] it follows that (x) is the solution for the functional equation (15). Therefore it gives a solution of Poincare- Steklov integral equation with parameter R3 (x). Combining formulae (13) with (20) we get the reconstruction rule

 −1 u(x) = (2πi) W1 (R3 (x) + i0) − W1 (R3 (x) − i0) , x ∈ [−1, 1]. (21)

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3.2.5 We have just proved for the case A the following Theorem 1 [11] If λ = 1, 3 then two formulas (19) and (21) implement the one-to-one correspondence between the solutions u(x) of the integral equation (1) and the bounded solutions W(y) of the Riemann monodromy problem in the ˆ \ {[a1 , a2 ] ∪ [a3 , a4 ] ∪ [−1, 1]} with the following matrices assigned slit sphere C to the slots: Case A : Case B 1 : Case B 2 :

[−1, 1] \ [a1 , a2 ] [a1 , a2 ] \ [−1, 1] [a3 , a4 ] [−1, 1] ∩ [a1 , a2 ] D D3 D2 D D1 D2 D D1 D2 D1 D = DD1

(22)

3.2.6 Monodromy Invariant It may be checked that the matrices D, D1 , D2 , D3 generating the monodromy group for the solution W(y) are pseudo-orthogonal, that is preserve the same quadratic form J(W) :=

3 

Wk2 − δ

3 

W j Ws .

(23)

j<s

k=1

This form is not degenerate unless −2 = δ = 1, or equivalently 0 = λ = 3. Since the solution W(y) of our monodromy problem is bounded near the cuts, the value of the form J(W) is independent of the variable y. Therefore the 3 solution ranges either in the smooth quadric {W ∈ C : J(W) = J0 = 0}, or 3 the cone {W ∈ C : J(W) = 0}. 3.2.7 Geometry of Quadric Surface The nondegenerate projective quadric {J(W) = J0 } contains two families of line elements which for convenience we denote by the signs  + and  − . Two different lines from the same family are disjoint while two lines from different families intersect. The intersection of those lines with the ’infinitely distant’ secant plane gives points on the conic {(W1 : W2 : W3 )t ∈ CP : 2

J(W) = 0}

(24)

which by means of stereographic projection p may be identified with the Riemann sphere. Therefore we have introduced two global coordinates p± (W) on the quadric, ’infinite part’ of which (i.e. conic (24)) corresponds to coinciding coordinates: p+ = p− (see Fig. 3).

Antisymmetric Eigenfunctions of PS − 3 Integral Equations

117

Fig. 3 Global coordinates p+ and p− on quadric

To obtain explicit expressions for the coordinate change W ↔ p± on quadric we bring the quadratic form J(W) to the simpler form J• (V) := V1 V3 − V22 by means of the linear coordinate change W = KV

(25)

where −1/2

K := (3δ + 6)

11 1 0 μ−1 0 2 1ε ε · 00 1 , 1 ε ε2 10 0 

ε := exp(2πi/3),

μ :=

δ−1 = δ+2



(26)

3−λ . 2λ

Translating the first paragraph of the current section into the language of formulae we get √ V2 ± i J0 V3 ± = (27) p (W) := √ ; V1 V2 ∓ i J0 and inverting this dependence, ⎛ ⎞ √ 1 J 2i 0 W( p+ , p− ) = + K ⎝ ( p+ + p− )/2 ⎠ . p − p− p+ p−

(28)

The point W( p+ , p− ) with coordinate p+ (resp. p− ) being fixed moves along the straight line with the directing vector K(1 : p+ : ( p+ )2 ) (resp. K(1 : p− : ( p− )2 )) belonging to the conic (24).

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Lemma 4 There exists a (spinor) representation χ : O3 (J) → PSL2 (C) such that: 1) The restriction of χ(·) to SO3 (J) is an isomorphism to PSL2 (C). 2) For coordinates p± on the quadric the following transformation rule holds: p± (TW) = χ(T) p± (W), p± (TW) = χ(T) p∓ (W),

T ∈ SO3 (J), T ∈ O3 (J) \ SO3 (J).

(29)

3) The linear-fractional mapping χ p := (ap + b )/(cp + d) is the image of the matrix: d2 2cd c2 1 T := K b d ad + b c ac K−1 ∈ SO3 (J). ad − b c b 2 2ab a2

(30)

4) The generators of the monodromy group are mapped to the following elements: χ(Ds ) p = ε1−s / p, μp − 1 χ(D) p = . p−μ

s = 1, 2, 3; (31)

Proof We define the action of matrix A ∈ SL2 (C) on the vector V ∈ C by the formula: 3

A :=

ab : cd

V3 V2 V2 V1

−→

A

V3 V2 At . V2 V1

(32)

It is easy to check that (32) gives the faithful representation of connected 3dimensional group PSL2 (C) := SL2 (C)/{±1} into SO3 (J• ) and therefore, an isomorphism. Let us denote χ• the inverse isomorphism SO3 (J• ) → PSL2 (C) and let χ(±T) := χ• (K−1 TK) for T ∈ SO3 (J). The obtained homomorphism χ : O3 (J) → PSL2 (C) will satisfy statement 1) of the lemma. To prove 2) we replace vector V components in the right-hand side of (32) with their representation in terms of the stereographic coordinates p± : √   i J0 A ( p+ , 1)t · ( p− , 1) + ( p− , 1)t · ( p+ , 1) At + − p −p  (cp+ + d)(cp− + d)   + t − − t + = i J0 (χ p , 1) · (χ p , 1) + (χ p , 1) · (χ p , 1) p+ − p− =

=

√   i J0 (χ p+ , 1)t · (χ p− , 1) + (χ p− , 1)t · (χ p+ , 1) + − χp − χp V3 (χ p+ , χ p− ) V2 (χ p+ , χ p− ) , V2 (χ p+ , χ p− ) V1 (χ p+ , χ p− )

Antisymmetric Eigenfunctions of PS − 3 Integral Equations

119

where we set χ p := (ap + b )/(cp + d). Now (29) follows immediately for T ∈ SO3 (J). It remains to check the transformation rule for any matrix T from the other component of the group O3 (J), say T = −1. Writing the action (32) component-wise we arrive at conclusion 3) of the lemma. An finally, expressions 4) for the generators of monodromy group may be obtained either from analyzing formula (30) or finding the eigenvectors of the matrices Ds , D which correspond to the fixed points of linear- fractional transformations.   3.3 Step 3: Projective Structures Speaking informally, a (branched) complex projective structure [5–9] on the  of Riemann surface M is a meromorphic function p on the universal cover M the surface that transforms linear fractionally under the deck transformations. The appropriate representation π1 (M) → PSL2 (C) is called the monodromy of the structure p. The projective structure is called branched when p has critical points. The set of all critical points of p(t) with their multiplicities . The projection of this set to survives under the cover transformations of M the Riemann surface M is known as the branching divisor D( p) of projective structure and the branching number of the structure p(t) is deg D( p). The classical examples of unbranched projective structures arise in Fuchsian or Schottky uniformization of Riemann surfaces. 3.3.1 Projective Structures Generated by Eigenfunction Stereographic coordinates p± (y) := p± (W(y)) for the solution of the Riemann monodromy problem (22) will give two nowhere coinciding meromorphic functions in the sphere with three possibly overlapping slots. As it follows from the transformation rules (29), the boundary values of two functions p± (y) on the slot D∗ , one of [a1 , a2 ] \ [−1, 1], [−1, 1] \ [a1 , a2 ], [a1 , a2 ] ∩ [−1, 1] or [a3 , a4 ], are related by the formulas y ∈ D∗ = [a1 , a2 ] ∩ [−1, 1], p± (y + i0) = χ(D∗ ) p∓ (y − i0), p± (y + i0) = χ(DD1 ) p± (y − i0), y ∈ [a1 , a2 ] ∩ [−1, 1],

(33)

where D∗ is the matrix assigned to the slot D∗ in (22). Relations (33) allow us to analytically continue both functions p+ (y) and − p (y) through any slot to locally single valued functions on the genus 2 Riemann surface   4  2 2 (34) M := w = (y − 1) (y − as ) , s=1

since all matrices D∗ are involutive—see Lemma 3. Further continuation ˜ Traveling gives single valued functions p± (·) on the universal covering M. of the argument y along the handle of the surface M may result in the

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linear- fractional transformation of the value p± (y). Say, the continuations of p+ (y) from the pants through the red and green slots will give two different functions on the second sheet related by the linear-fractional mapping χ(DD2 ). 3.3.2 Branching of Structures p± The way we have carried out the continuation of functions p± (y) suggests that the branching divisors of the arising projective structures are related via formula: D( p+ ) = HD( p− )

(35)

where H(y, w) := (y, −w) is the hyperelliptic involution of the surface M. We determine the branching numbers of the structures in the proof of Theorem 2 [11] When λ ∈ {0, 1, 3} the solutions u(x) of the PS-3 integral equation are in one-one correspondence with the couples of meromorphic in ˆ \ {[a1 , a2 ] ∪ [a3 , a4 ] ∪ [−1, 1]} functions p± (y) with boundary the slit sphere C values satisfying (33) and either non or two critical points in common. The correspondence u(x) → p± (y) is implemented by the sequence of formulae (12), (19) and (27). The inverse dependence is given up to proportionality by the formula  (y) u(x) = ( p+ (y) p− (y) − μ( p+ (y) + p− (y)) + 1), (36) dp+ (y)dp− (y) (dy)2 is the w 2 (y) holomorphic quadratic dif ferential on the Riemann surface M with zeroes at the critical points of the (possibly coinciding) functions p+ and p− , or with double zeroes y1 = y2 (otherwise arbitrary) when p+ = p− is unbranched.

where x ∈ [−1, 1] and y := R3 (x) + i0, (y) = (y − y1 )(y − y2 )

Remark The number of the critical points of the structures in the slit sphere is counted with their weight and multiplicity: 1) the branching number of p± (y) at the branch point a ∈ {±1, a1 , . . . , a4 } of the surface M is computed √ with respect to the local parameter z = y − a, 2) every critical point on the boundary should be considered as a half-point. Proof 1.

Let u(x) be an eigenfunction of integral equation PS-3. The stereographic coordinates p± (y) of the solution W(y) of the associated Riemann monodromy problem are nowhere equal meromorphic functions when the invariant J0 = 0, or two identically equal functions when J0 = 0. In any case they inherit the boundary relationship (33).

Antisymmetric Eigenfunctions of PS − 3 Integral Equations

121

What remains is to find the branching numbers of the entangled structures p± (y). To this end we consider the O3 (J)-invariant quadratic differential form J(dW) = J• (dV) transferred to the slit sphere. In the general case J0 = 0 we get (up to proportionality) the Kleinian quadratic differential: (y) =

dp+ (y)dp− (y) , ( p+ (y) − p− (y))2

ˆ. y∈C

(37)

This expression is the infinitesimal form of the cross ratio, hence it remains unchanged after the same linear-fractional transformations of the functions p+ and p− . Therefore, (37) is a well defined quadratic differential on the entire sphere. Lifting (y) to the surface M we get a holomorphic differential. Indeed, p+ = p− everywhere and applying suitable linear-fractional transformation we assume that p+ = 1 + zm+ + {terms of higher order} and p− = czm− + ... in terms of local parameter z of the surface, m±  1, c = 0. Then  = cm+ m− zm+ +m− −2 (dz)2 + {terms of higher order}. Therefore D( p+ ) + D( p− ) = (). Any holomorphic quadratic differential on genus 2 surface has 4 zeroes. The curve M consists of two copies of the slit sphere interchanged by the hyperelliptic involution H. Taking into account the symmetry (35) of branching divisors, we see that the structures p± together have two critical points in the slit sphere. In the special case J0 = 0 two structures merge: p± (y) =: p(y) and the same quadratic differential J(dW) = J• (dV) on the curve M has the appearance: (y) = [V1 (y)dp(y)]2 ,

(38)

here V1 (y) is the first component in the vector V(y) defined by formula (25). The analysis of this representation in local coordinates suggests that 2D( p) + 2(W) = (),

(39)

where (W) is the divisor of zeroes of the locally holomorphic (but globally multivalued) on M vector W(y). To characterize (W) we need the following lemma, which we prove at the end of the current section. Lemma 5 The vector W(y) cannot have simple zeroes at the f ixed points of the hyperelliptic involution of M when J0 = 0 and λ = 0, 3. The divisor (W) is obviously invariant under the hyperelliptic involution H. From this Lemma it follows that either (W) = 0 (therefore deg D( p) = 2) or (W) consists of two points interchanged by H (therefore the structure p is unbranched). In other words, p(y) has the branching number 0 or 1 on the slit sphere and the quadratic differential  is a square of a holomorphic linear differential. 2.

Conversely, let p+ (y) and p− (y) be two not identically equal meromorphic functions on the slit sphere, with boundary conditions (33) and the

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total branching number either zero or two (see remark above). For the meromorphic quadratic differential (37) on the Riemann surface M we establish (using local coordinate on the surface) the inequality: D( p+ ) + D( p− )  ()

(40)

where the deviation from equality means that there is a point where p+ = p− . But the degree of the divisor on the left of (40) is zero or four and deg() = 4. Therefore, p+ = p− everywhere (and the total branching of this pair of functions in the slit sphere is two). The holomorphic vector W( p+ (y), p− (y)) in the slit sphere solves the Riemann monodromy problem specified in Theorem 1. We already know how to convert the latter vector to the eigenfunction of integral equation PS-3. Careful computation gives the restoration formula  (δ + 2)J0 p+ (y) p− (y) − μ( p+ (y) + p− (y)) + 1 , (41) 2π u(x) = 3 p+ (y) − p− (y) where x ∈ [−1, 1] and y := R3 (x) + i0. Formula (36) appears after the substitution of (37) to the latter formula. Finally, suppose that two functions p± (y) with necessary branching and boundary behaviour are identical. For the solution on the cone, V = V1 (1, p, p2 )t and the first component V1 may be taken from (38). Therefore we consider the vector on the slit sphere: W(y) :=

(y − y1 ) K(1, p(y), p2 (y))t , w(y) p (y)

(42)

where y1 is the critical point of p(y) or arbitrary point when p(y) is unbranched. One immediately checks that it is holomorphic and solves the Riemann monodromy problem specified in Theorem 1. Now to find the corresponding eigenfunction is a routine task.   Proof of Lemma 5 Let the hyperelliptic involution H changes the sign of the local coordinate z defined in the vicinity of the fixed point z = 0 of the involution. Boundary relationship of the vector W on the slots implies the symmetry: W(−z) = D∗ W(z)

(43)

where D∗ is one of the matrices D1 , D2 , D3 or D. Suppose that W has a simple zero at the fixed point: W(z) = W − z + {terms of higher order}. We immediately see that W − = 0 is the eigenvector of D∗ corresponding to eigenvalue −1 of this matrix. Another obvious property of this vector: J(W − ) = 0. The matrix D∗ has the invariant (complex) plane corresponding to the double eigenvalue +1. The nullset of the quadratic form J(·) on this plane is nontrivial, in other words there is an eigenvector W + = D∗ W + = 0 lying in the cone: J(W + ) = 0. The chain of equalities is valid: J(W + , W − ) = J(D∗ W + , D∗ W − ) = −J(W + , W − ) = 0,

Antisymmetric Eigenfunctions of PS − 3 Integral Equations

123

where J(·, ·) is the bilinear form polar to the quadratic form J(·). Now we see that the cone contains the entire plane generated by the vectors W + and W − . Therefore, the quadratic form J(·) is degenerate which only happens when λ = 0, 3.   3.4 Mirror Symmetry of Solution In what follows we are looking for real solutions u(x) of the integral equation (1). There is no loss of the generality. Indeed, the restrictions on the monodromy of projective structures [11] imply that the spectrum of any PS3 integral equation belongs to the segment [0, 3]. Now both real and imaginary parts of any complex eigenfunction u(x) are the solutions of the integral equation. Real solutions u(x) of the integral equation correspond to the solutions of Riemann monodromy problem with mirror symmetry: W( y¯ ) = W(y). This symmetry for the vector V(y) := K−1 W(y) takes the form V( y¯ ) = (V3 (y) sign(δ + 2), (V2 (y) sign(δ − 1), (V1 (y) sign(δ + 2)). The values δ + 2 and δ − 1 have the same sign as it follows from the range of spectral parameter λ ∈ [0, 3]. Therefore, real solutions are split into two classes depending on the sign of (δ + 2)J0 : Symmetric ((δ+2)J0 > 0), Antisymmetric ((δ+2)J0  0),

p± ( y¯ ) = 1/ p± (y) , p± ( y¯ ) = 1/ p∓ (y)

y ∈ P (R3 ) \ [−1, 1],

In the remaining part of the article we give explicit parametrization of all antisymmetric solutions for the integral PS-3 equations of the considered type—when six points ±1, a1 , . . . , a4 are real and pairwise distinct. Restricting ourselves to the search of antisymmetric solutions we have to find only one function in the pants, say p(y) := p+ (y) while the remaining function may be recovered from the mirror antisymmetry: p− (y) = 1/ p+ ( y¯ ).

(44)

On the boundary components of the slit sphere this function obeys the rule: (33)

(44)

p+ (y ± i0) = χ(D∗ ) p− (y ∓ i0) = χ(D∗ D1 ) p+ (y ± i0), y ∈ D∗ = [−1,1]∩[a1 , a2 ]. Therefore ˆ, p∈R

when D∗ = D1 ;

ˆ, p ∈ εR

when D∗ = D2 ;

ˆ, p ∈ ε2 R

when D∗ = D3 ;

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and finally when D∗ = D, the value of p lies on the circle C := { p ∈ C :

−1 2

−2

|p − μ | = μ

− 1 },

μ :=



3−λ 2λ

(45)

As an immediate consequence of this observation we give a universal restriction for the spectrum of our integral equation. Lemma 6 Antisymmetric eigenfunctions correspond to eigenvalues λ ∈ [1, 3]. Proof For the cases A, B 1, B 22, B 23 the slot [−1, 1] \ [a1 , a2 ] is not empty and the boundary value of p(y) on this slot belongs to the circle C. This circle is an empty set for μ > 1, or equivalently λ ∈ (0, 1). The proof for the remaining case B 21 requires special machinery and will be given in Section 7.   The critical points of two functions p+ (y) and p− (y) in the considered antisymmetric case are complex conjugate as it follows from (44). Taking this fact into account we reformulate Theorem 2 for the antisymmetric solutions: Theorem 3 When λ ∈ {0, 1, 3}, the antisymmetric solutions u(x) of integral equation PS-3 are in one-two correspondence with the meromorphic in the ˆ \ {[a1 , a2 ] ∪ [a3 , a4 ] ∪ [−1, 1]} functions p(y) that have either none slit sphere C or one critical point in the domain and the following values on the boundary components: y∈

[−1, 1] \ [a1 , a2 ] (red) [a1 , a2 ] \ [−1, 1] (blue) [a3 , a4 ] (green) ˆ (Case A) ε2 R ˆ p(y ± i0) ∈ C εR ˆ (Case B ) R In case B 2 the function p(y) has the jump on the remaining part of the boundary: p(y + i0) = χ(DD1 ) p(y − i0),

y ∈ [−1, 1] ∩ [a1 , a2 ].

(46)

Remark By ’one-two’ correspondence we mean the following: given any function p(y) satisfying the conditions of this theorem, it’s easy to check that its antisymmetrization 1/ p( y¯ ) also satisfies all the conditions. Therefore, we have a correspondence of an eigenfunction u(x) to a couple: function p(y) and its antisymmetrization, only one of them being independent.

4 Statement of the Main Result From the Section 3.4 it follows that every antisymmetric eigenfunction u(x) of PS-3 integral equation induces a mapping p(y) of the pants P (R3 ) to a multivalent domain spread possibly with branching over the Riemann sphere. Such surface is known as Kleinian membrane or Überlagerungsfläche and the complete list of them is given in this section.

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4.1 Tailoring the Pants We define pants PQ(λ, h1 , h2 |m1 , . . . ) of several fashions Q which parametrically depend on spectral parameter λ, two other reals h1 , h2 and one or two integers m1 , . . . . Each boundary oval of our pair of pants covers a circle and acquires its color in the following way: C ˆ ˆ or χ(DD1 )εR εR 2 ˆ or ε R ˆ R

– red, – green, – blue.

Any constructed pair of pants may be obtained from the “basic” pants PQ(λ, h1 , h2 | . . . ) with lowest possible integer parameters by a surgery procedure known as “grafting” and introduced independently by B.Maskit, D.Hejhal and D.Sullivan–W.Thurston in 1969-1983. 4.1.1 Cases A, B 1 For real λ ∈ (1, 2) we consider (depending on λ) open annulus α bounded by ˆ ˆ . Another annulus bounded by C and ε2 R two circles: C defined in (45) and εR we denote α. ¯ Note that for the considered values of λ the circle C does not intersect the lines ε±1 R. The m− sheeted unbranched covering of the annuli, m = 1, 2, . . . , we denote as m · α or m · α¯ correspondingly. The annuli we have just introduced may be sewn together in the way specified in Table 1 to get the pants of four fashions PA1 , PA2 , PA3 , PB1. Sign ’+’ in the definitions of Table 1 means certain surgery explained below. Instructions on sewing annular patches together 1. PA1 (λ, h1 , h2 | m1 , m2 ). Take two annuli m1 · α and m2 · α. Cut the top sheet of each annulus along the same segment (dashed red line in the Fig. 4) starting at the point h := h1 + ih2 and ending at the circle C. Now sew the left bank of one cut on the right bank of the other. The emerging surface is the pair of pants.

Table 1 Three-parametric families of pairs of pants for the cases A, B1; parameter 1 < λ < 2 Fashion of pants

Range of h1 , h2 and m1 , m2

Definition

PA1 (λ, h1 , h2 | m1 , m2 )

h := h1 + ih2 ∈ α ∩ α, ¯ |h|  1; m1 , m2 = 1, 2, . . . 0 < h1 < h2 , h1 h2  1; m1 = 1, 2, . . . , m2 = 0, 1, 2, . . . 0 < h1 < h2 , h1 h2  1; m1 =0, 1, 2, . . . , m2 = 1, 2, 3, . . . μ−1 + μ−2 − 1 < h1 < h2 ; m = 1, 2, 3, . . .

Cl{(m1 · α) \ [μ−1 , h]}+ Cl{(m2 · α) ¯ \ [μ−1 , h]} Cl{(m1 · α) \ −ε 2 [h1 , h2 ]}+ Cl{m2 · α} ¯ Cl{(m2 · α) ¯ \ −ε[h1 , h2 ]}+ Cl{m1 · α} Cl{(m · α) \ [h1 , h2 ]}

PA2 (λ, h1 , h2 | m1 , m2 ) PA3 (λ, h1 , h2 | m1 , m2 ) PB1(λ, h1 , h2 | m)

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C

C +

0

"red" "green" "blue"

0

h

h

Fig. 4 The pair of pants PA1 (λ, h1 , h2 | m1 , m2 ) is sewn down of two annuli

2. PA2 (λ, h1 , h2 | m1 , m2 ). The annulus m1 · α with the segment −ε2 [h1 , h2 ] removed from the top sheet is a pair of pants PA2 (λ, h1 , h2 |m1 , 0). Cut the obtained pair of pants along the segment joining the circle C to the slot (dashed blue line in the Fig. 5). Also cut top sheet of the annulus m2 · α along the same segment and sew the left bank of one cut on the right bank of the other. The arising surface is the pair of pants. 3. PA3 (λ, h1 , h2 | m1 , m2 ). The annulus m2 · α¯ with the segment −ε[h1 , h2 ] removed from the top sheet, is a pair of pants PA3 (λ, h1 , h2 |0, m2 ). As in the previous passage, we may sew in the annulus m1 · α to the obtained pants to get the result. 4. PB1(λ, h1 , h2 | m) is just the annulus m · α with the segment [h1 , h2 ] removed from its top sheet. The limit case of the pants PA1 , when the branch point h1 + ih2 tends to ε±1 R, coincides with the limit cases of pants PA2 or PA3 , when h1 = h2 > 0. The corresponding unstable two-parametric families of pants PA12 and PA13 are defined in Table 2.

– ε 2h2 – ε 2 h1

0

C

C +

0

"red" "green" "blue"

Fig. 5 The pair of pants PA2 (λ, h1 , h2 | m1 , m2 ) is sewn of simpler pants and the annulus

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Table 2 Unstable two-parametric families of pants Fashion of pants

Definition

PA12 (λ, h| m1 , m2 )

PA1 (λ, −Re(ε 2 h), −Im(ε 2 h)|m1 , m2 ) = PA2 (λ, h, h|m1 , m2 ) PA1 (λ, −Re(εh), −Im(εh)|m1 , m2 ) = PA3 (λ, h, h|m1 , m2 )

PA13 (λ, h| m1 , m2 )

The range of parameters: 1 < λ < 2, h > 0, m1 and m2 = 1, 2, 3, . . .

4.1.2 Case B 2 ˆ and χ(DD1 )εR ˆ do not intersect when λ ∈ (1, 3). They bound Two circles: εR the open annulus β depending on λ. The m− sheeted unbranched covering of the annulus we denote as m · β, m = 1, 2, 3, . . . The points of the annulus m · β may be described in the form p = μ−1 + ρ exp(iφ), where ρ > 0 and the argument φ ∈ R mod 2π m. The action of χ(DD1 ) on the ˆ ) lifts to the involution of sphere (i.e. consecutive reflections in circles C and R the multi-sheeted annulus m · β in the following way:  : μ−1 + ρ exp(iφ) → μ−1 + where r :=

r2 exp(−iφ) ρ

(47)

 μ−2 − 1 is the radius of the circle C.

Definition We introduce three pairs of pants PB21, PB22 and PB23, each of them depend on three reals λ, h1 , h2 and an integer m: PB 2s(λ, h1 , h2 | m) := Cl{(m · β) \ (Es1 (h1 ) ∪ Es2 (h2 ))}/,

s = 1, 2, 3,

Table 3 Slots for the subcases of B2, parameter 1 < λ < 3 Definition of slots

Range of h1 , h2

E11 (h1 ) := μ−1 + r exp[−h1 , h1 ],

h1  h2 > 0,

E12 (h2 ) := μ−1 + r exp[−h2 , h2 ] exp(iπm)

when m is even; (μ−1 + r exp h1 )· (μ−1 − r exp h2 )  1, when m is odd.

E21 (h1 ) := μ−1 + r exp[−ih1 , ih1 ],

h1  h2 , when m is even;

E22 (h2 )

:=

μ−1

+ r exp[−ih2 , ih2 ] exp(iπm)

E31 (h1 ) := μ−1 + r exp[−h1 , h1 ], E32 (h2 ) := μ−1 + r exp[−ih2 , ih2 ] exp(iπm)

Arg(exp(ih1 ) + μr)  Arg(exp(ih2 ) − μr) when m is odd; h1 + h2 < mπ, h2 > 0, for any m. h1 > 0, mπ > h2 > 0

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where the slots Es1 , Es2 are defined in Table 3. The slots are invariant with respect to the involution  and do not intersect each other as well as the boundary of the annulus m · β. To understand the interrelation of introduced constructions it is very useful to imagine how the pants PB1 are transformed to the pants of fashion Q = B 23 and the latter—to the pair of pants PB21 or PB22. 4.2 The Main Theorem Theorem 4 When λ = 1, 3 the Hölder’s antisymmetric eigenfunctions of PS3 integral equation for the case Q = A, B 1, B 21, B 22, B 23 are in one to one correspondence with the pairs of pants2 PQ(λ, h1 , h2 |m1 ..) conformally equivalent to the pants (9) associated to the functional parameter of integral equation. Let the function p(y) conformally maps the pair of pants P (R3 ) to the pants PQ(λ, h1 , h2 |m1 ..) and respects the colours of the boundary ovals, then up to proportionality ⎧ ⎪ (y − y1 )(y − y2 ) p(y+ ) − p(y− ) ⎪ ⎪ ⎨ , x ∈ [−1, 1] \ [a1 , a2 ], p (y+ ) p (y− ) w(y) (48) u(x) = + ⎪  Im p(y ) ⎪ ⎪ , x ∈ [−1, 1] ∩ [a1 , a2 ]. ⎩ (y − y1 )(y − y2 ) w(y)| p (y+ )| Here y := R3 (x), y± := y ± i0. For the fashion Q = A1 , y1 = y2 is the inner critical point of the function p(y); for other fashions Q real y1 and y2 are boundary critical points of the function p(y). The proof of the main theorem for the cases A, B 1 is given in Section 6 and for the case B 2—in Section 7. 4.3 Corollaries The representation of eigenfunction (48) cannot be called explicit in the usual sense, since it comprises a transcendent function p(y). We show that nevertheless the representation allows us to understand the following properties of the solutions. 1. The “antisymmetric” part of the spectrum is always a subset of [1, 3]; for the equations of types A, B 1 this part of the spectrum always lies in [1, 2] ∪ {3}. 2. Every λ ∈ (1, 3) is the eigenvalue for inf initely many equations PS-3.

2 For

the case A there are three stable and two unstable pants fashions PA∗ (. . . )

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Proof Any of the constructed pants may be conformally mapped to the standard form: the sphere with three real slots. Now we can apply the procedure of the Section 2.5 and get the functional parameter R3 (x) such that the associated pair of pants is conformally equivalent to the pants we started from. 3. Eigenfunction u(x) related to the pants PQ(. . . |m1 , m2 ) has exactly m1 + m2 + 1 zeroes on the segment [−1, 1] when Q = A, B 1. Proof According to the formula (48), the number of zeroes of eigenfunction u(x) is equal to the number of points y ∈ [−1, 1] where p(y+ ) = p(y− ). This number in turn is equal to the number of solutions of the inclusion S(y) := Arg[ p(y− ) − μ−1 ] − Arg[ p(y+ ) − μ−1 ]

∈ 2π Z,

y ∈ [−1, 1]. (49) Let the point p(y) goes m times around the circle C when its argument y travels along the banks of [−1, 1]. Integer m is naturally related to the integer parameters of pants PQ(. . . ). The function S(y) strictly increases from 0 to 2π m on the segment [−1, 1], therefore the inclusion (49) has exactly m + 1 solutions on the mentioned segment.   4. The mechanism for arising the discrete spectrum of the integral equation is explained. Sewing annuli down to the pants PQ(λ, h1 , h2 | . . . ) one changes the conformal structure of the latter. To return to the conformal structure specified by P (R3 ) we have to change the real parameters of the pants, one of them is the spectral parameter λ. In a sense, the eigenvalue problem (1) is reduced to the solution of three equations for three unknown numbers. These equations relate moduli of given pants P (R3 ) to the moduli of membrane with real parameters λ, h1 , h2 and extra discreet parameters.

5 Auxiliary Constructions The combinatorial analysis of the arising projective structure p(y) is based on two constructions we describe below. Let p(y) be a holomorphic map from a Riemann surface M with a boundary to the sphere and the selected boundary component (∂ M)∗ is mapped to a circle. The reflection principle allows us to holomorphically continue p(y) through this selected component to the double of M. Therefore we can talk of the critical points of p(y) on (∂ M)∗ . When the argument y passes through a simple critical point, the value p(y) reverses the direction of its movement on the circle. So there should be even number of critical points (counted with multiplicities) on the selected boundary component.

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5.1 Construction 1 (No Boundary Critical Points) Suppose that p(y) has no critical points on the selected component of ∂ M which is mapped to the boundary of the unitary disc

U := { p ∈ C :

| p|  1}.

(50)

We define the mapping from a disjoint union M  U to a sphere: ˜ p(y) :=

p(y), y ∈ M, y−d , y ∈ U,

(51)

where integer d = 0 is the degree of the mapping p : (∂ M)∗ → ∂ U, d is positive (negative) if a small annular vicinity of the selected boundary component is mapped to the interior (exterior) of the unit disc. We also call d the winding number of p(y) on the boundary oval (∂ M)∗ Now we fill in the hole in M by the unit disc, identifying the points of (∂ M)∗ and the points of ∂ U with the same value of p˜ (there are |d| ways to ˜ do so). The holomorphic mapping p(y) of the new Riemann surface M∪U to the sphere will have exactly one additional critical point of multiplicity |d| − 1 at the center of the glued disc. 5.2 Construction 2 (Two Boundary Critical Points) Let again p(y) be a holomorphic mapping of a bounded Riemann surface M to the sphere with selected boundary component (∂ M)∗ being mapped to the boundary of the unit disc U. Now the mapping p(y) has two simple critical points on the selected boundary component (the case of coinciding critical values is not excluded). Those two points (Fig. 6) divide the oval (∂ M)∗ into two segments. We are going to modify the Riemann surface M, sewing down one segment of (∂ M)∗ to the other and filling the remaining hole (if any) with the patch U.

*

*

Fig. 6 Mapping of the boundary component (∂M)∗ with two simple critical points ∗ on it and the degree d = −1

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Again, we define the mapping from the disjoint union M  U to the sphere by the formula (51) where integer d is the degree of the mapping p : (∂ M)∗ → ∂ U. In other words, 2π d is the increment of arg p(y) when the point y goes around the selected boundary oval in the positive direction. We glue one segment of (∂ M)∗ to a part of the other, identifying the points of the boundary oval equidistant in the metric |dp| from (any) chosen boundary critical point. If d = 0, the hole disappears. Otherwise we identify the remnant ˜ of (∂ M)∗ with the boundary of U, gluing points with the same value of p(y) as shown in the Fig. 7a. ˜ The holomorphic mapping p(y) of the modified Riemann surface to the sphere will have no additional critical points when d = 0. When d = 0 one or two additional critical point arise: one of multiplicity |d| − 1 at the center of the artificially attached disc and a simple critical point the place of the old boundary critical point other than chosen in the previous paragraph. Remark The application of Constructions 1 and 2 to a given function p(y) mapping boundary component of the surface M to the circle contains an arbitrary element—linear fractional function mapping the given circle to the standard one, ∂ U. Changing this element we can (a) arbitrary move the additional critical value of p˜ within appropriate disc and (b) change the sign of d. The choice of the additional critical value will simplify the arising combinatorial analysis and we always assume w.l.o.g. that d  0.

6 Proof for the Cases A, B1 6.1 Eigenfunction Gives Pair of Pants We already know that every antisymmetric eigenfunction of integral equation PS-3 generates the mapping p(y) from the pants P (R3 ) to the sphere. The boundary ovals of the pants are mapped to three circles specified in Theorem 3 and the function p(y) may have either (a) no critical points, (b) one simple critical point inside the pants, (c) two boundary simple critical points or (d)

Fig. 7 (a) Filling in the hole bounded by (∂M)∗ (b) Splitting of the mapping p(y)

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one double critical point on the boundary. The first two possibilities will be considered in Section 6.1.1 and the other two—in the Section 6.1.2. 6.1.1 No Critical Points on the Boundary of Pants Branched Covering of a Sphere Suppose that the point p(y) winds around the corresponding circle dr , dg and db times when the argument y runs around the ‘red’,‘green’ and ‘blue’ boundary component of P (R3 ) respectively. We can apply the construction of Section 5.1 and glue three discs, Ur , Ug , Ub to the holes of the pants. Essentially, we have split our mapping p(y)—see the commutative diagram on the Fig. 7b. The holomorphic mapping p˜ has three or four ramification points, three of them are in the artificially glued discs and the fourth (if any) is inherited from the projective structure. Applying the Riemann–Hurwitz formula we get: dr + dg + db = 2N, dr + dg + db = 2N + 1,

p is branched, p is unbranched,

˜ N := deg p.

(52)

Intersection of Circles Lemma 7 In case A the required projective structure p(y) with a critical point inside the pants may exist only if the spectral parameter 1 < λ < 2 (i.e. when ˆ ). The structure without the circle C does not intersect two other circles ε±1 R branching does not exist for any λ. Proof ˆ and ε2 R ˆ. 1. We know that the point 0 lies in the intersection of two circles: εR −1 The total number { p˜ (0)} of the pre-images of this points (counting the multiplicities) is N and cannot be less than db + dg —the number of preimages on the blue and green boundary oval of the pants. Comparing this to (52) we get dr  N which is only possible when dr = dg + db = N.

(53)

ˆ we repeat Assuming that the circle C intersects any of the circles ε±1 R the above argument for the intersection point and arrive at the conclusion db = dr + dg = N or dg = dr + db = N incompatible with already established (53). 2. For the unbranched structure p(y) the established inequalities db + dg  N and dr  N contradict the Riemann-Hurwitz formula (52).   The above arguments may be applied to the case B 1 as well. Taking into ˆ always intersect we arrive at account that the circles C and R

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Lemma 8 In case B 1 the mapping p(y) (if any) will have a boundary critical point. Image of Pants Let us investigate where the artificially glued discs in case A are mapped to. Suppose for instance that the disc Ur is mapped to the exterior of the circle C. The point 0 will be covered then at least dr + dg + db = 2N times which is impossible. The discs Ug and Ub are mapped to the left of the lines εR and ε2 R respectively, otherwise points from the interior of the circle C will be covered more that N times. The image of the pair of pants P (R3 ) is shown on the Fig. 8. We use the ambiguity in the construction of gluing the discs to the pants and require that the critical values of p˜ in the discs Ug , Ub coincide. Now the branched covering p˜ has only three different branch points shown as •, ◦ and ∗ on the Fig. 8. The branching type at • is the cycle of length N; at the point ◦ there are cycles of lengths dg and db ; and the branch point ∗ is simple. The coverings with three branch points are called Belyi maps and are described by certain graphs known as Grothendieck’s “Dessins d’Enfants”. In our case the dessin is the lifting of the segment connecting white and black branch points:  := p˜ −1 [•, ◦]. Combinatorial Analysis of Dessins There is exactly one critical point of p˜ over the branch point ∗. Hence, the compliment to the graph  on the upper sphere of the diagram on the Fig. 7b contains exactly one cell mapped 2 − 1 to the lower sphere. All the rest components of the compliment are mapped 1 − 1. Two types of cells are shown in the Fig. 9a and b, the lifting of the red circle is not shown to simplify the pictures. The branch point ∗ =: h1 + ih2 should lie in the intersection of two annuli α and α otherwise the discs Ug , Ub glued to different boundary components of our pants will intersect: the hypothetical case when the branch point of p(y) belongs to one annulus but does not belong to the other is shown in the Fig. 9c.

Fig. 8 Shaded area is the image of pants in case A, index db  = 0

C

*

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(a)

*

*

(b)

(c)

Fig. 9 (a) Simple cell (N − 2 copies) (b) Double cell (1 copy) (c) Impossible double cell

The cells from the Fig. 9a, b may be assembled in a unique way shown in the Fig. 10. The pants are colored in white, three artificially sewn discs are shaded. Essentially this picture shows us how to sew together the patches bounded by ˆ to get the pants conformally equivalent to P (R3 ). our three circles C, ε±1 R As a result of the surgery procedure we get the pants PA1 (λ, h1 , h2 |dg , db ). Changing the superscript of the projective structure p± (y) gives us the change of sign for the eigenfunction u(x) and the reflection of the pants PA1 (. . . ) in the unit circle ∂ U. This is why we consider only the pants with |h1 + ih2 |  1. 6.1.2 Boundary Critical Points First of all we consider the stable case of two simple critical points on the boundary oval. At the moment we do not know the color of this oval and we use the ‘nicknames’ {1, 2, 3} for the set of colours {r, g, b } so that the critical points will be on the oval 3. Branched Covering of the Sphere The usage of both constructions from Section 5 allows us to include the pants P (R3 ) to the sphere attaching two discs U1 and U2 to the first two ovals, collapsing the boundary of the third oval and

"red" "green" "blue"

Fig. 10 Dessin for dg = 3, db = 2; the pre-image of the branch point ∗ is at infinity

Antisymmetric Eigenfunctions of PS − 3 Integral Equations

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sewing in the third disc U3 if necessary. Nonnegative winding numbers arising in those constructions we denote as d1 , d2 , d3 respectively. We arrive at the branched covering p˜ of the diagram on the Fig. 7b. This mapping has either two, three or four critical points. Two of them are in the centers of the discs U1 , U2 ; another one or two points arise only when d3 = 0— the center of U3 and one of the boundary critical points of the mapping p(y). The multiplicities of those critical points are respectively d1 − 1, d2 − 1, d3 − 1, 1. Riemann-Hurwitz formula for this covering reads d1 + d2 + d3 = 2N,

˜ N := deg p.

(54)

Lemma 9 The images of the ovals 1 and 2 do not intersect. Proof Suppose the inverse is true and a point Pt lies in the intersection of images of the first two ovals. Then N   p˜ −1 (Pt)  d1 + d2 . Any of the critical points from the third oval has at least d3 + 1  N pre-images counting multiplicities. The last two inequalities contradict (54).   Corollaries 1. In case A the critical points of p(y) lie either on the blue or on the green ˆ intersect) boundary of pants. (Two circles ε±1 R 2. In case B 1 the critical points of p(y) lie on the blue boundary of pants. (Two ˆ intersect) circles C and R 3. In both cases the required function may only exist when μ ∈ ( 12 , 1), or ˆ intersect) equivalently λ ∈ (1, 2). (Otherwise the circles C and ε±1 R

To save space, further proof will be given for the case A only when both critical points lie on the blue oval. The omitted cases require no extra technique. Now the notations Ur , Ug , Ub , dr , dg , db have the obvious meaning.

Image of Pants Lemma 10 The image p(P (R3 )) of the pants is the union α ∪ α¯ when db = 0 or the annulus α when db = 0. Proof Setting Pt = 0 in the proof of Lemma 9 we establish the equalities dg + db = dr = N. Now, repeating the arguments of the same title paragraph of the Section 6.1.1 ˜ Ur ) fills the interior of C, the disc Ug is mapped to we conclude that the disc p( the left of the line εR and the disc Ub (if any) is mapped to the left of ε2 R. So  arg p  4π } is covered dg + db = N times by the artificially the sector { 2π 3 3 inserted discs. The disc Ug covers the half-plane to the left of the line εR exactly dg times, the latter number is N when db = 0.  

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Fig. 11 (a) Simple cell (N − 2 copies) (b) Double cell (1 copy)

*

(a)

(b)

Corollary Both critical values of p(y) lie on the ray −ε2 (0, ∞). Dessin d’Enfants Again, we put the critical values of p˜ in the discs Ug , Ub to the same point ◦ (see Fig. 8). The only difference from the Section 6.1.1: now the inherited from the pants branch point ∗ (if db = 0) lies on the ray −ε2 (0, ∞). We introduce the Grothendieck’s Dessin as the lifting of the segment connecting white and black branch points:  := p˜ −1 [•, ◦]. The compliment to  is composed of cells shown in the Fig. 11. In the assembly the double cell may be used only once and only when db = 0. Given the winding numbers dg , db , the cells from the Fig. 11a, b may be attached to each other in a unique way. When db = 0 the Dessin  has the same combinatorial structure as in Fig. 10. Of course, one has to replace the old cells by those shown in Fig. 11. Shown in the Fig. 12 is the assembly of cells for db = 0, dg = 5. The pants are colored in white, two artificially inserted discs are shaded. As a result of the surgery procedure we get the pants PA2 (λ, h1 , h2 |dg , db ) with positive reals h1 , h2 determined by the critical values of p(y). To discern the pair of pants PA2 (. . . ) from its reflection in the unit circle we consider the restriction h1 h2  1. Junction of critical points To study the remaining case when the boundary critical points of projective structure merge, one has to apply the limit case of

Fig. 12 Degenerate Dessin for dg = 5, db = 0; the pre-image of the branch point ◦ is at infinity

"red" "green" "blue"

Antisymmetric Eigenfunctions of PS − 3 Integral Equations

137

the Construction 2. In this way one arrives at the unstable membranes PA12 and PA13 . To save space we omit the details. 6.2 Pair of Pants Corresponds to Eigenfunction Let the pair of pants (9) associated to the functional parameter R3 (x) of the type Q = A, B1 integral equation be conformally equivalent to the pants PQ(λ, . . . ). This exactly means that there exists a conformal mapping p(y) from P (R3 ) to PQ(λ, . . . ) respecting the colors of the boundary ovals. This mapping is unique since the conformal self-mapping of pants keeping all boundary ovals invariant is trivial. The mapping p(y) has one simple critical point inside the pants (for the membrane PA1 (. . . )) or two simple boundary points (for PA2 (. . . ), PA3 (. . . ), PB1(. . . )) or a double boundary critical point (for PA12 (. . . ), PA13 (. . . )). Moreover, p(y) maps the boundary components of P (R3 ) to the circles specified by Theorem 3. Hence, given p(y) one can consecutively restore: two projective structures p± (y), the solution W(y) of Riemann monodromy problem and the eigenfunction u(x). Combining the formulae (44), (36) we obtain the top of the reconstruction formulae in (48).

7 Proof for the Case B2 7.1 Eigenfunction Gives Pair of Pants Any antisymmetric eigenfunction of the integral equation PS-3 generates the mapping p(y) from the pants P (R3 ) to the sphere. The principal difference of this case from the one considered in Section 6 lies in the two-valuedness of the function p(y) in the pants. To reflect this phenomenon we consider the two sheeted unbranched cover P2 → P (R3 ) with trivial monodromy around the green boundary oval. This new surface is a sphere with four holes, each boundary inherits the color of the oval it covers—see Fig. 13a. The mapping

*

*

* *

*

* (a)

* *

(b)

Fig. 13 (a) The surface P2 is the double cover of pants P; (b) P4 is the reflection of P2 in the blue boundary oval

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p(y) is lifted to the single-valued mapping p2 : P2 → C P1 satisfying the equivariance condition: p2  = χ(DD1 ) p2 ,

(55)

where  is the cover transformation (change of sheets) of P2 , represented as the rotation by 180◦ around the horizontal axis on Fig. 13a. Now we can complete the proof of the Lemma 6. Suppose that there exists the required function p2 (y) in the case B 21 and μ > 1. We show that all possible locations of the critical points of this function lead to the contradiction: (a) p2 has no boundary critical points; (b) all critical points lie on the green ovals of P2 ; (c) p2 has at least one critical point on a blue oval. (a) We use the Construction 1 and attach four discs to the surface P2 . For the arising ramified covering p˜ 2 the Riemann-Hurwitz formula reads 2dg + db + db = 2N, 2dg + db + db = 2N + 2,

p is branched, p is unbranched,

N := deg p˜ 2 ,

where dg is the winding number (degree) of p2 (y) on each of the green ovals; db and db are the winding numbers for two blue ovals of the surface ˆ ∩ εR ˆ is covered at least dg + db + d  N times. This P2 . The point 0 ∈ R b

agrees the previous formula only if dg  N. But now db = db = 0 which is impossible. (b) Now each of two green ovals has two boundary critical points of p2 (y). We use both Constructions and eliminate all holes in P2 attaching two discs to the blue ovals and possibly two more discs to the green ovals of the surface. The Riemann-Hurwitz formula for the arising ramified covering p˜ 2 reads 2dg + db + db = 2N,

N := deg p˜ 2 ,

where dg  0 is the degree of p(y) for each of the green ovals of P2 . Further argument is exactly as in the previous paragraph. (c) We claim that in this case there are exactly four critical points of p2 on a blue oval of the surface P2 . Indeed, given a critical point Pt, Pt is also a critical point because of the equivariance (55). When μ > 1, the mapping χ(DD1 ) conserves the orientation of the real axis. This means that those two critical points are of the same type (say, local maxima of the real value p2 ). Hence, Pt and Pt are separated by the critical points of the opposite type (local minima in our case). There cannot be more that four boundary critical points of the function p2 on the double cover of pants P (R3 ), so we have listed them all. Let us consider the double of the surface P2 and cut it along all boundary ovals of P2 , but the blue oval containing all critical point of p2 (y). This new surface, P4 , is a sphere with six holes shown in the Fig. 13b, four boundary ovals are green and two are blue. The reflection principle allows to continue

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analytically p2 (y) to the mapping p4 (y) of the entire surface P4 to the sphere. This continuation has four inner critical points and no boundary critical points. ˆ and four green ovals to the circles ε±1 R ˆ, It maps both blue ovals to R ±1 ˆ χ(DD1 ))ε R. The usage of Construction 1 allows to fill in all the holes of P4 . The Riemann-Hurwitz formula for the arising ramified covering p˜ 4 reads 2dg + db = N,

N := deg p˜ 4 ,

where the numbers dg , db have the obvious meaning. The point 0 is covered at least 2dg + 2db > N times which is impossible. The location of the critical points of the mapping p2 (y) is given by the following lemma. Lemma 11 The mapping p2 (y) has exactly two boundary critical points on each of the non-green ovals of the surface P2 . ˆ, Proof The mapping χ(DD1 ) changes the orientation of the circles C and R when μ ∈ (0, 1). When the point y runs along the blue or red oval of P2 , the value p2 (y) changes the orientation of its motion at least twice due to (55). This means that the argument y comes through at least two boundary critical points. Since the mapping p(y) has at most two boundary critical points in pants, the lifted mapping p2 (y) has at most four in the double cover of pants.   Branched covering of the sphere The mapping p2 (y) from P2 to the sphere has equal winding numbers d = dg on both green ovals. The degree of p2 (y) on each of non-green ovals equals zero. Both statements are simple consequences of the equivariance condition (55). Applying Constructions 1 and 2 to the mapping p2 defined on P2 , we get a ramified covering p˜ 2 with two critical points, both of multiplicity dg − 1. Image of the Surface The Riemann-Hurwitz formula for the ramified covering p˜ 2 reads dg = N := deg p˜ 2 . It is easily seen that two discs attached to the green ovals of P2 are mapped to the left of the line ε R and to the interior of ˆ . Therefore, the surface P2 is conformally equivalent to the circle χ(DD1 ) εR the closure of the annulus dg · β with two slots in it. The involution  of P2 (the interchange of sheets) induces the involution of the multisheeted annulus. The latter involution is the lifting of χ(DD1 ) to dg · β and is given by the formula (47). The slots of dg · β are invariant with respect to  and therefore pass through the fixed points μ−1 + r and μ−1 + r exp iπ m of the involution. The red slots are projected to the circle C, the blue slots are projected to the real line. Given in Table 3 inequalities for the parameters h1 , h2 specifying the endpoints of the slots allow us to relate any given antisymmetric eigenfunction to exactly one picture. A by-product of the explicit description of the image of the pants is the following

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Lemma 12 In antisymmetric case B 2 two structures p± (y) are dif ferent when λ = 3. Proof Suppose the opposite is true, that is p(y) p( y¯ ) ≡ 1

(56)

for meromorphic function p(y) satisfying the conditions of Theorem 3. In case B 21 the value p(a) ∈R when a ∈ {a1 , a2 } is the endpoint of the blue slot. From (56) it immediately follows that p(a) = ±1. But the image of pants pP (R3 ) = β avoids both points ±1. ¯ when a ∈ {a1 , a2 } is In case B 22 the value p(a ± i0) ∈ C = { p = χ(DD1 ) p} the endpoint of the red slot. From (56) and the jump relationship (46) on [a1 , a2 ] it follows that p(a ± i0) = ±1. Again, the image of pants p(P ) avoids both points ±1. In case B 23 any of the above two arguments is applicable.   Corollary In case B 2 any eigenvalue corresponds to no more than one antisymmetric eigenfunction. Proof Suppose, there are two linearly independent antisymmetric eigenfunctions of the integral equation (1) with common eigenvalue. They generate two couples of meromorphic functions in the slit sphere P (R3 ) \ [−1, 1], say p± , p± 0 . The analytic continuation of those four functions gives four projective structures on the riemann surface M with common monodromy determined by the eigenvalue. From Lemma 12 it follows that no two of the four structures are identical and therefore (see the second part of the proof of the Theorem 2) all four values p± (y), p± 0 (y), are different at any point y. We consider the following differential form on the Riemann surface M:

 1 1 ω := dp+ − . p+ − p+ p+ − p− 0 0 This form ω is the infinitesimal form of the cross ratio and it is invariant + under the same linear-fractional transformations of three functions p± 0, p . Therefore ω is well defined on the entire Riemann surface M. Using local coordinates on M, it’s easy to check that the form is holomorphic and (ω) = D( p+ ). Any holomorphic differential on the genus 2 surface has two zeroes which are interchanged by the hyperelliptic involution of M. According to Lemma 11, the branching divisor of p1 (y) is different as it has a branchpoint   on each of non-green ovals of the pants P . 7.2 Pair of Pants Corresponds to Eigenfunction Let the pair of pants PB2s(λ, h1 , h2 |m) be conformally equivalent to the pair of pants P (R3 ) associated to type B 2s, s = 1, 2, 3, integral equation. This

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exactly means that there exists respecting the colors of the boundary ovals equivariant conformal mapping p2 (y) from the double cover P2 (R3 ) of pants to the closure of the multisheeted annulus m · β with two slots Es1 (h1 ) and Es2 (h2 ) in it. We represent the double cover P2 (R3 ) as two copies of pants P (R3 ) cut along the segment [−1, 1] ∩ [a1 , a2 ] and attached one to the other. The restriction of p2 (y) to one of such copies gives the function p(y) satisfying all the assumptions of the Theorem 3. The antisymmetric eigenfunction of the integral equation now may be reconstructed via the known procedure which gives the formulae (48). Moreover, from Lemma 12 we have learned that the invariant J0 = 0 for antisymmetric solutions in the case B 2. So we can use the alternative formula (41) to reconstruct the eigenfunction u(x) when x ∈ [−1, 1] \ [a1 , a2 ]. On the remaining part of the segment [−1, 1] we can use along with (48) the other formula: u(x) =

Im p(y+ ) , | p(y+ )−μ|2 +1−μ2

y+ := R3 (x + i0),

x ∈ [−1, 1] ∩ [a1 , a2 ].

The only nuisance here consists in possible non-uniqueness of the mapping p(y). Indeed, when two of the boundary ovals of pants have the same color (blue in case B 21 or red in case B 22), the pants may admit conformal involution interchanging the ovals of the same color. Such pants fill in a codimension one manifold in the corresponding moduli space. The Corollary to Lemma 12 nevertheless guarantees the uniqueness of the antisymmetric eigenfunction for the given membrane PB2s: the composition of p(y) with the conformal automorphism of pants coincides with either p(y) or its antsymmetrization 1/ p( y¯ ).

8 Conclusion The geometric and combinatorial analysis of the spectral problem for the Poincare-Steklov integral equation is given in this paper. Another powerful approach to the study of integral equations is the theory of operators. It is always useful to compare two different viewpoints on the same subject. The study of the spectral problem (1) based on the theory of singular integral operators was carried out in [1, 2]. The following results were obtained for general smooth coordinate changes R(x) provided the non-degeneracy condition (2) holds: (i) The spectrum is discrete; the eigenvalues are positive ! and converge to λ = 1; (ii) |λ − 1|2 < ∞ (a constructive estimate in λ∈Sp terms of R(x) may be given); (iii) The eigenfunctions u(x) make up an orthogonal (with respect to a special scalar product) basis in the Sobolev space 1/2 Hoo (I). The geometric approach was first applied to the PS − 2 integral equations with R(x) = x + (2C)−1 (x2 − 1), the parameter C > 1. The complete set of

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eigen-values and functions from the above Sobolev space has been explicitly found [3]: ⎡ ⎤ (C+x)/(C−1)  nπ un (x) = sin ⎣  (s2 − 1)−1/2 (1 − k2 s2 )−1/2 ds⎦ , (57) K 1

λn = 1 + 1/ cosh 2π τ n,

n = 1, 2 . . . ,

where τ = K/K is the ratio of the complete elliptic integrals of modulus k = (C − 1)/(C + 1). All Poincare-Steklov equations with the eigenfunctions expressed in terms of elliptic functions were listed in [10]. The next natural step was to study equations (1) with a rational degree 3 functions R(x) [11]. In this paper we show that all the eigenfunctions of PS-3 equations are split into two classes (symmetric/antisymmetric) with respect to the symmetries of the underlying geometric structures and we give the representation for all antisymmetric eigenfunctions (further analysis shows that this class is larger than the remaining one). The representation is given by an explicit formula (48) which contains a transcendental function p(y) conformally mapping some explicitly constructed pair of pants to the standard form. Of course, this answer is less explicit than e.g. formula (57). The immediate consequences of this formula listed in Section 4.3 include: the exact locus of the spectrum for the family of equations, the oscillatory behaviour of the higher eigenfunctions; the mechanism for the emergence of the separate eigenvalues. Further study of the eigenvalues and eigenfunctions is now reduced to the problems of geometric function theory. Say, the explicit asymptotical formulae for the solutions may be written in the case when the conformal mapping of the pants to the standard three-slit domain is known approximately (short slots, small holes etc.). It seems that the results of the mentioned two approaches are complementary. Operator theory may be applied in a rather general situation and it gives rather general answers. The usage of geometrical analysis is more restricted (e.g. may be applied to rational parameters R(x)) and more difficult. But if we are lucky, this analysis may give us the best of the answers—the explicit formulae for the solutions. The author hopes that some of the techniques used in this paper may be helpful for the study of other integral equations with rational low degree kernels.

References 1. Bogatyrev, A.B.: The discrete spectrum of the problem with a pair of Poincare-Steklov operators. Doklady RAS 358(3) (1998) 2. Bogatyrev, A.B.: The spectral properties of Poincare-Steklov operators. PhD Diss., INM RAS, Moscow (1996) 3. Bogatyrev, A.B.: A geometric method for solving a series of integral PS equations. Math. Notes 63(3), 302–310 (1998) 4. Poincare, H.: Analyse des travaux scientifiques de Henri Poincare. Acta Math. 38, 3–135 (1921)

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5. Gunning, R.C.: Special coordinate coverings of Riemann surfaces. Math. Ann. 170, 67–86 (1967) 6. Tyurin, A.N.: On the periods of quadratic differentials. Russ. Math. Surv. 33(6) (1978) 7. Gallo, D., Kapovich, M., Marden, A.: The monodromy groups of Schwarzian equations on closed Riemann surfaces. Ann. Math. (2) 151(2), 625–704 (2000) See also arXiv, math.CV/9511213 8. Hejhal, D.A.: Monodromy groups and linerly polymorphic functions. Acta Math. 135, 1–55 (1975) 9. Mandelbaum, R.: Branched structures and affine and projective bundles on Riemann surfaces. Trans. AMS 183, 37–58 (1973) 10. Bogatyrev, A.B.: Poincare-Steklov integral equations and the Riemann monodromy problem. Funct. Anal. Appl. 34(2), 9–22 (2000) 11. Bogatyrev, A.B.: PS-3 integral equations and projective structures on Riemann surfaces. Sb. Math. 192(4), 479–514 (2001)

Math Phys Anal Geom (2010) 13:145–157 DOI 10.1007/s11040-010-9072-0

From Uncertainty Principles to Wegner Estimates Peter Stollmann

Received: 29 August 2009 / Accepted: 20 January 2010 / Published online: 6 February 2010 © Springer Science+Business Media B.V. 2010

Abstract We give a shortcut from simple uncertainty principles to Wegner estimates with correct volume term. Keywords Wegner estimates · Uncertainty principle Mathematics Subject Classifications (2000) 81Q10 · 47B80 · 60H25 · 35P05 1 Introduction The path described in the title has been laid by Combes, Hislop and Klopp, [5] to prove Wegner estimates with the correct volume factor for Anderson type models without a covering condition. Subsequently we showed in [3] how to use this approach to deduce localization for a great number of models that couldn’t be treated before. Actually, for low energies we derived an uncertainty principle by an extremely simple method. However, for the link from these uncertainty principles to Wegner estimates with the correct volume factor we still had to rely upon the analysis of [5] which is technically quite involved. Here, we will use the fact that the uncertainty principles in [3] apply to the spectral projections of the random operators and not just to to the spectral projections of the underlying periodic background, as was the case in [5]. This allows the substantial shortcut we present in this paper. To keep the paper selfcontained, we also present the necessary general spectral averaging result. Although this is basically wellknown, our discussion

Dedicated to the memory of Pierre Duclos. P. Stollmann (B) Fakultät für Mathematik, Technische Universität, 09107 Chemnitz, Germany e-mail: [email protected]

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here seems to be somewhat more streamlined than what is available in the literature so far. Before stating the heart of the matter, we introduce and analyze the type of uncertainty principles that we referred to above. See [14] for Wegner’s original work and [13] for references to the recent literature.

2 An Uncertainty Principle Here we just briefly review the uncertainty principles that came up in connection with random Schrödinger operators and mention relations to more classical uncertainty principles from Harmonic Analysis. Roughly speaking, the classical uncertainty principle says that a function cannot be localized in position and momentum space at the same time. In what we consider here, momentum space does not necessarily refer to the usual Fourier transform. Actually, we start in an abstract setting: H denotes a Hilbert space, H a selfadjoint operator in H, bounded from below and 0 ≤ W a bounded operator on H. By I ⊂ R we always denote an interval, and by P I (H) = 1 I (H) the corresponding spectral projection. The uncertainty principle we need is an estimate of the form P I (H)W P I (H) ≥ κ P I (H),

()

with some κ > 0. For the applications we have in mind, think of H as a Schrödinger operator in L2 and of W as a function, that has to be spread out in a certain sense. If W itself is bounded below by a strictly positive constant, () holds trivially. Let us record some easy facts in the following: Proposition 2.1 (1) Denote the range of P I (H) by H I . Then () is equivalent to 1

W 2 f 2 ≥ κ f 2 for all f ∈ H I . (2) If () holds and J ⊂ I is a smaller interval, then () holds for J in place of I. (3) Abbreviate P I := P I (H), Q I := 1 − P I . If 1

Q I f 2 + W 2 f 2 ≥ η f 2 for all f ∈ H,

()

then () holds with κ = η. Conversely, if () holds for some κ > 0, then () holds for some η > 0. In [3] we showed: Theorem 2.2 ([3], Theorem 1.1) Def ine λ(t) := inf σ (H + tW) for t ≥ 0. If λ(t0 ) > E0 := λ(0) and E1 < λ(t0 ), then () holds for I = [E0 , E1 ] with κ=

λ(t0 ) − E1 . t0

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We now turn to the more specific situation that is relevant to the applications we have in mind: H = L2 (Rd ), H is a Schrödinger operator. Without too much restriction, we could even assume that W = 1 E , where E ⊂ Rd is a measurable set. Let us first consider the case H = −. Then       H I = f ∈ L2 Rd | supp fˆ ⊂  I , where  I = ξ ∈ Rd | |ξ |2 ∈ I is an annulus or a ball. In virtue of Proposition 2.1 (1), the uncertainty principle () is equivalent to  f 2L2 (E) ≥ κ f 2 if supp fˆ ⊂  I . In the language of uncertainty principles of Harmonic Analysis this latter inequality means that E and  I form a strongly annihilating pair, cf [6]. Following [10] we say that E is thick, provided there exist a1 , ..., ad > 0 and α > 0 such that for C := [0, a1 ] × ... × [0, ad ]: |E ∩ (C + y)| ≥ α for all y ∈ Rd , where | · | denotes the Lebesgue measure. The following generalization of the Logvinenko-Sereda Theorem can be found in [10]: Theorem 2.3 ([10], Theorem 4) For any bounded interval I and any thick subset E: P I (−)1 E P I (−) ≥ κ P I (−). In fact, the statement in [10] is much more precise, the constant can be estimated from below and the L2 estimate has an L p counterpart. Combined with the analysis from [5], the latter estimate can be used to prove continuity of the IDS and Wegner estimates at all energies for Anderson models with quite irregular geometry of the set of impurities. We do not pursue this issue here. Evidently, the latter theorem was not known to Combes, Hislop and Klopp who proved a similar but somewhat different result with H0 = − + V0 , where V0 and W are periodic over the same lattice in [4]: Theorem 2.4 Let H0 = − + V0 , with V0 and W periodic over the same lattice and W = 0 on some open set. Then P I (H0 )W P I (H0 ) ≥ κ P I (H0 ). Let us now comment on how the above three uncertainty principles differ. Remark 2.5 The above three theorems are mutually incomparable: as shown in [3], Theorem 2.2 applies in situations where H = H(ω) and W may even be concentrated near a subspace of Rd , so that W is not “thick” at all; the decisive input is the mobility of the ground state energy. However, we can only treat intervals I close to the ground state energy.

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In Theorem 2.3, W maybe fairly general, I is just supposed to be bounded, but H = − is essential. In Theorem 2.4, H0 = − + V0 with a periodic V0 , I is just supposed to be bounded, but W has to be periodic.

3 Spectral Averaging for General Measures The main result here is, in fact, basically known although the formulation is sometimes slightly more complicated or restricted to a less general setting. We start with a real analysis lemma. Denote, for a probability measure μ on R the number s(μ, ε) := sup{μ([a, a + ε]) | a ∈ R}, that is sometimes called the Lévy concentration function. Lemma 3.1 For any probability measure μ on R, 0 < λ ≤ 1, a ∈ R and ε > 0:  λε2 1 dμ(x) ≤ 4 s(μ, ε). 2 2 2 λ R (x − a) + λ ε Proof First observe that it suffices to treat a = 0 and that λε2 1 ε2 ≤ . x 2 + λ2 ε 2 λ x2 + ε 2 Now we divide R=



ε ε kε − , kε + , 2 2

k∈Z

so that

 R

ε2 dμ(x) ≤ 2 2 x +ε k∈Z

 [kε− 2ε ,kε+ 2ε ]

≤ s(μ, ε) 1 + 2

ε2 dμ(x) x2 + ε 2

k∈N



k−

1  1 2 2

 +1

,

where we used the monotonicity of the integrand to estimate the integrals by the maximum of the integrand times the integral over [kε − 2ε , kε + 2ε ], which gives a factor of s(μ, ε). We can now compare the sum with the integral of (x2 + 1)−1 over [1, ∞) plus an extra term to account for the first term in the sum. The best estimate we get in this fashion is ( 95 + π2 )s(μ, ε) which proves the claim. 

The following result can be seen as a streamlined version of Theorem 3.1 in [5]. We fix a Hilbert space H and denote its inner product by (· | ·).

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Theorem 3.2 Let μ be probability measure on R, A a selfadjoint operator in H and 0 ≤ B a bounded operator on H. For any interval I of length |I| we get    1 1 B 2 P I (A + tB)B 2  |  dμ(t) ≤ 6B2 s(μ, |I|), R

for any  ∈ H. Proof By functional calculus P I (A + tB) ≤

4 π



  dE Im(A + tB − E − iε)−1 , I

where we set |I| =: ε. Therefore, 

 4 1 1 B 2 P I (A + tB)B 2  |  ≤ π



 1  1 dE B 2 Im(A + tB − E − iε)−1 B 2  |  . I

As we can approximate A, B we can assume without restriction that A is bounded and B is invertible. Thus we do not have any domain problems and we can write the integrand as 

  1 1 −1 Im B− 2 (A + tB − E − iε)B− 2 | . Next, we want to realize the inverse as the resolvent of some maximally accretive operator in order to be able to make use of the spectral theorem. (In [1] operators with strictly positive imaginary part are called dissipative; we are facing operators with strictly negative imaginary part so that accretive seems to be an appropriate term; note, however that these notions are not standardized at all.) 1 else. In any case, we To do so, choose λ = 1 if B ≤ β < 1 and 0 < λ < B get 

  −1   −1 − 12 − 12  |  = Im A E,ε,λ + t − iλε | , Im B (A + tB − E − iε)B where A E,ε,λ = B− 2 (A − E − iε(1 − λB))B− 2 1

1

is maximally accretive. As discussed in [1], Appendix B and in [5], Section 3, the resolvent of a maximally accretive operator can always be written as the resolvent of a selfadjoint dilation. Therefore, we get an auxiliary Hilbert space H E,ε,λ that contains H and a selfadjoint operator L E,ε,λ in H E,ε,λ such that       −1 −1 Im A E,ε,λ + t − iλε  |  = Im L E,ε,λ + t − iλε |

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 which can be written in terms of the spectral measure ρ E,ε,λ of L E,ε,λ in the state  as  λε  ... = dρ E,ε,λ (x). 2 2 2 R (x + t) + λ ε

We are now almost ready to use the previous Lemma. Putting the above estimates together we get    1 1 B 2 P I (A + tB)B 2  |  dμ(t) R

4 ≤ π

  R

 dE I

R

λε dρ  (x)dμ(t) (x + t)2 + λ2 ε2 E,ε,λ

We can now use Fubini and the previous Lemma, giving   1  4 1 ... ≤ dE 4 s(μ, ε) dρ E,ε,λ (x) π I ε R λ  4 1 1 ≤4 dE s(μ, ε) 2 π I λ ε  since the total mass of ρ E,ε,λ is just 2 . Now the length of I cancels the that

... ≤ 4 Since

1 λ

1 ε

so

41 s(μ, ε)2 . πλ

can be chosen arbitrarily close to B we get the assertion.



As the reader can easily check, two special cases of the previous result stand out as particularly simple: for B = 1 there is no need to pass to selfadjoint dilations; for B a rank one operator, the Birman-Schwinger principle gives a scalar expression immediately so that one does not need to refer to spectral measures. See [11] for a discussion of the latter case and for uniformly αHölder continuous measures. The following reformulation of Theorem 3.2 is particularly neat. Recall that  we write ρ H for the spectral measure of the selfadjoint operator H in the state . Theorem 3.3 Let μ be probability measure on R, A a selfadjoint operator in H and 0 ≤ B a bounded operator on H. Then the measure  1 B2  ρ= ρ A+tB dμ(t) R

satisf ies s(ρ, ε) ≤ 6B2 s(μ, ε).

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Note that the continuity of the averaged spectral measure does not depend on A and its dependence on B is very explicit and easy to control. The above mentioned special cases lead to: Corollary 3.4 (1) In the situation of the preceding theorem assume that, furthermore, B = 1. Then    s ρ A+tB dμ(t), ε ≤ 62 s(μ, ε) R

(2) In the situation of the preceding theorem assume that, furthermore, B = P is the orthogonal projection onto the onedimensional space spanned by the normed . Then   s ρ dμ(t), ε ≤ 6s(μ, ε) A+tB R

4 Wegner Estimates Here we show that an uncertainty principle for random Hamiltonians implies Wegner estimates. We use some clever tricks borrowed from [5, 9] to show how to reduce the estimation of the trace of eigenprojectors to an application of Theorem 3.2 above. Note, however, that both the input, uncertainty principles for random operators, and the statement of the following result are new. To state our Wegner estimate we have chosen the following setup (inspired by [2, 7]) which can be regarded as a compromise between a general abstract theorem and an application to a specific type of random model. In fact, since the Wegner bound is dealing with a Hamiltonian on one particular cube, we fix an open cube in Rd   L L d  =  L (0) = − , , 2 2 with L ∈ 2N + 1. Observe that =



1 (k).

k∈∩Zd

Here is the Setup: (S1) V0 ∈ L p (), where p = 2 if d ≤ 3 and p > d if d > 3 and H0 := − + V0 with Dirichlet boundary conditions. (S2) I is a finite index set, CU ≥ 0 and U α ∈ L∞ (),

0 ≤ U α ≤ CU

for all α ∈ I . Denote Ik := {α ∈ I | U α χ1 (k) = 0}.

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 (S3) For the probability measure P on  = α∈I [0, ηmax ] we use regular conditional probabilities to define the following uniform bound   s(P, ε) = sup ess sup ess sup P ωα ∈ [E, E + ε] | (ωβ )β=α . α

E∈R

(ωβ )β=α

(See [8] for definition and existence of regular conditional probabilities.) (S4) We set Vω := ωα U α for ω = (ωα )α∈I α∈I

and H := H(ω) := H0 + Vω . This setup typically arises when Anderson type random Schrödinger operators are restricted to cubes. Here is our main result: Theorem 4.1 Let H be as in the setup, W := R. Assume that there is κ > 0 such that P-a.s.

 α∈I

U α and I be some interval in

P I (H)W P I (H) ≥ κ P I (H). Then, for a constant C(V0 ) that only depends on V0 ,   2 E{tr[P I (H)]} ≤ C(V0 ) · CU · ( max |Ik |) κ −2 emax I ||s(P, |I|). k∈∩Zd

()

(4.1)

Before proceeding to the proof let us comment on the estimate in the inequality (4.1) above. The term in square brackets is a structural constant that, typically, does not depend on . As we showed in [3], we can find an uncertainty estimate of the form () with κ independent of  for very general Anderson type random models. The remaining terms give the right volume dependence as well as the continuity of the single site random variables. We now single out several wellknown facts that will be used in the proof of the above theorem. Remarks 4.2 (1) For operators on a general Hilbert space we record (a) Cyclicity of the trace: tr[AB] = tr[B A], provided A and B are Hilbert Schmidt operators. (b) For A a selfadjoint Hilbert Schmidt operator, 0 ≤ B ≤ C bounded: tr[AB A] ≤ tr[AC A].

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(2) By monotonicity of the functional calculus, 0 ≤ P I (H) ≤ emax I e−H for any semibounded selfadjoint H and any bounded interval I in R. (3) Denote  N H0N := (− + V0 ) , 1 (k) k∈∩Zd

the direct sum of − + V0 restricted to the cube 1 (k) with Neumann boundary conditions. Then H0N ≤ H0 ≤ H(ω) for all ω ∈ . (4) The operator H0N has compact resolvent; denote, for k ∈  ∩ Zd , by k, j, j ∈ N, an orthonormal basis of eigenfunctions with corresponding eigenvalues Ek, j. Then N e−Ek, j = tr[e−H0 ] ≤ C(V0 ) · ||. k, j

For V0 ≡ 0 this can be read off explicit foemulæ for the eigenvalues, cf. [12], p. 266. Under assumption (S1), V0 is relatively bounded with respect to − and so we get an estimate of the above type. Proof of Theorem 1 We start by using Remarks 4.2(1) and (2): tr[P I (H)] ≤ emax I tr[P I (H)e−H P I (H)] = emax I tr[P I (H)e−H ]; expanding into an orthonormal basis (ϕk (ω))k∈N of H(ω) with eigenvalues μk (ω) we obtain ... = emax I e−μk (ω) μk (ω)∈I

= emax I



e−(H(ω)ϕk (ω)|ϕk (ω))

μk (ω)∈I

≤ emax I



μk (ω)∈I

≤ emax I

e−(H0 ϕk (ω)|ϕk (ω))



N

 N e−H0 ϕk (ω) | ϕk (ω) ,

μk (ω)∈I

where we used Remark 4.2(3) in the next to last and Jensen’s inequality in the last step. This important trick is borrowed from [9]. We go on with

 N ... = emax I tr P I (H)e−H0

1 N  1 N = emax I tr e− 2 H0 P I (H)e− 2 H0 . (4.2)

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We now estimate the latter expression, using the abbreviation P I := P I (H) and the main condition, ():

1 N  

1 N 1 1 N N tr e− 2 H0 P I e− 2 H0 ≤ κ −1 tr e− 2 H0 P I W P I e− 2 H0 ;

(4.3)

Now comes a nice idea from [5]. We denote by  ·  HS the Hilbert-Schmidt norm and by (· | ·) HS the corresponding inner product, given by (A | B) HS = tr[B∗ A] so that the Cauchy Schwarz inequality yields

1 N   1 N N κ −1 tr W P I e−H0 = κ −1 tr e− 2 H0 P I P I W P I e− 2 H0 ≤ κ −1 P I W P I e− 2 H0  HS P I e− 2 H0  HS   1 c 1 1 N N ≤ κ −1 P I W P I e− 2 H0 2HS + P I e− 2 H0 2HS 2 2c 



 1 c N N tr P I e−H0 + tr P I W P I W P I e−H0 = κ −1 2 2c 1

1

N

N

(4.4) which holds for any c > 0. We pick c = κ, insert (4.4) into (4.3) and move the first term in (4.4) to the left:

 

N N tr P I e−H0 ≤ κ −2 tr P I W P I W P I e−H0 

N = κ −2 tr P I W P I e−H0 P I W P I 

N ≤ κ −2 tr P I We−H0 W P I

1 N  1 N ≤ κ −2 tr e− 2 H0 W P I We− 2 H0 .

(4.5)

We now pick an eigenbasis as described in Remark 4.2(4) and get ... = κ −2



W P I We− 2 H0 k, j | e− 2 H0 k, j 1

N

1

N



k, j

= κ −2



  e−Ek, j W P I Wk, j | k, j

k, j

= κ −2

k

j

e−Ek, j

  U α P I U α k, j | k, j .

(4.6)

α,α  ∈Ik

In the last step we used that U β k, j = 0 for β ∈ Ik . Now we want to control the inner sum. Note that   L∞ × L∞ → R, (U, V) → U P I Vk, j | k, j

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is a nonnegative bilinear form (on real-valued functions). The Cauchy-Schwarz inequality gives that    1  1 | U P I Vk, j | k, j | ≤ U P I Uk, j | k, j 2 · V P I Vk, j | k, j 2 ≤

   1  U P I Uk, j | k, j + V P I Vk, j | k, j , 2

so that     U α P I U α k, j | k, j ≤ 2 U α P I U α k, j | k, j . α,α  ∈Ik

α∈Ik

Inserting into (4.6) and interchanging summation and expectation we get 

 N E{tr[P I (H)]} ≤ emax I E tr W P I e−H0   ≤ 2emax I κ −2 e−Ek, j E U α P I U α k, j | k, j , k

α∈Ik

j

1

where we used the estimates (4.3), (4.4) and (4.5). Since U α ≤ CU U α2 for all α, we get E{tr[P I (H)]} ≤ 2CU emax I κ −2

k

j

e−Ek, j



E

 1  1 U α2 P I U α2 k, j | k, j ,

α∈Ik

(4.7) which makes the expectation accessible to an application of Theorem 3.2. So consider, E

 1 1  U α2 P I U α2 k, j | k, j  ηmax    1 1 =E U α2 P I (H(ω))U α2 k, j | k, j d P(ωα | (ωβ )β=α ) . 0

The inner integral can be written as ⎛ ⎞  ηmax 1 1 ⎝U α2 P I (H0 + U β + ωα U α )U α2 k, j | k, j⎠ d P(ωα | (ωβ )β=α ). 0

β=α

 With A = H0 + β=α U β , B = U α , the substitution t = ωα and μ = P(· | (ωβ )β=α ) the latter integral is 

 R

 1 1 B 2 P I (A + tB)B 2 k, j | k, j dμ(t) ≤ 6CU s(μ, |I|),

(4.8)

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since the k, j are normed. By definition, s(μ, |I|) ≤ s(P, |I|), which gives an estimate independent of k, j. Inserting (4.8) into (4.8) we get 2 max I −2 E{tr[P I (H)]} ≤ 12 · CU e κ e−Ek, j s(P, |I|) (4.9) k 2 max I −2 ≤ 12 · CU e κ

k

α∈Ik

j

j

e−Ek, j



 max |Ik | s(P, |I|)

k∈∩Zd

   2 · max |Ik | κ −2 emax I ||s(P, |I|), (4.10) ≤ C(V0 ) · CU k∈∩Zd

after using Remark 4.2(4), and incorporating 12 in the constant C(V0 ).



An immediate consequence is the usual form of Wegner estimates:  Corollary 4.1 Let H be as in the setup, W := α∈I U α and I be some interval in R. Assume that there is κ > 0 such that P-a.s. P I (H)W P I (H) ≥ κ P I (H).

()

Then, for a structural constant C, P{σ (H(ω)) ∩ I = ∅} ≤ Cemax I ||s(P, |I|).

For the proof, put the explicit constants into C and note that 1{ω|σ (H(ω))∩I=∅} ≤ tr[P I (H(ω))], since P I (H(ω)) is a projection. In [3] we showed that we have an uncertainty principle () with κ independent of . Moreover, for the models considered there, the other constants are uniformly bounded. Consequently, we get a fairly simple proof of Wegner estimates with a linear volume term. Note that Theorem 4.1 can also be used to obtain improvements (better volume factors) of some of the results in [7], notably of Theorem 8 and Corollary 9. Acknowledgement acknowledged.

Fruitful discussions with Daniel Lenz and Ivan Veseli´c are gratefully

References 1. Aizenman, M., Elgart, A., Naboko, S., Schenker, J., Stolz, G.: Moment analysis for localization in random Schrödinger operators. Invent. Math. 163, 343–413 (2006) 2. Boutet de Monvel, A., Naboko, S., Stollmann, P., Stolz, G.: Localization near fluctuation boundaries via fractional moments and applications. J. Anal. Math. 100, 83–116 (2006) 3. Boutet de Monvel, A., Lenz, D., Stollmann, P.: An uncertainty principle, Wegner estimates and localization near fluctuation boundaries. Preprint arXiv:0905.2845 4. Combes, J.-M., Hislop, P.D., Klopp, F.: Hölder continuity of the integrated density of states for some random operators at all energies. Int. Math. Res. Not. 2003(4), 179–209 (2003)

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5. Combes, J.-M., Hislop, P.D., Klopp, F.: An optimal Wegner estimate and its application to the global continuity of the integrated density of states for random Schrödinger operators. Duke Math. J. 140(3), 469–498 (2007) 6. Havin, V.P., Jöricke, B.: The Uncertainty Principle in Harmonic Analysis. Springer-Verlag, Berlin Heidelberg (1994) 7. Kirsch, W., Veseli´c. I.: Wegner estimates for sparse and other generalized alloy type potentials. Proc. Indian Acad. Sci. Math. Sci. 112(1), 131–146 (2001) 8. Klenke, A.: Probability theory. Universitext. Springer-Verlag London Ltd., London (2008). A comprehensive course, Translated from the 2006 German original. 9. Klopp, F., Zenk, H.: The integrated density of states for an interacting multiparticle homogenous model and applications to the Anderson model. Adv. Math. Phys. 2009(Article ID 679827), 15 (2009). doi:10.1155/2009/679827 10. Kovrijkine, O.: Some results related to the Logvinenko-Sereda theorem. Proc. Am. Math. Soc. 129, 3037–3047 (2001) 11. Krishna, M.: Continuity of integrated density of states—independent randomness. Proc. Indian Acad. Sci. Math. Sci. 117(3), 401–410 (2007) 12. Reed, M., Simon, B.: Methods of Modern Mathematical Physics. IV: Analysis of Operators. Academic Press, New York (1978) 13. Veseli´c, I.: Existence and regularity properties of the integrated density of states of random Schrödinger operators. Lecture Notes in Mathematics, vol. 1917. Springer-Verlag, Berlin (2008) 14. Wegner, F.: Bounds on the density of states in disordered systems. Z. Phys. B 44(1–2), 9–15 (1981)

Math Phys Anal Geom (2010) 13:159–189 DOI 10.1007/s11040-010-9073-z

The Escape Rate of a Molecule Andreas Knauf · Markus Krapf

Received: 9 May 2008 / Accepted: 1 February 2010 / Published online: 12 February 2010 © Springer Science+Business Media B.V. 2010

Abstract We show existence and give an implicit formula for the escape rate of the n-centre problem of celestial mechanics for high energies. Furthermore we give precise computable estimates of this rate. This exponential decay rate plays an important role especially in semiclassical scattering theory of n-atomic molecules. Our result shows that the diameter of a molecule is measurable in a (classical) high-energy scattering experiment. Keywords Uniformly hyperbolic systems · Thermodynamic formalism · Scattering theory Mathematics Subject Classifications (2000) 37D20 · 37D35 · 70F05 · 78A45

1 Introduction and Statement of Results The n-centre problem in three dimensions is given by n nuclei with charges Z 1 , . . . , Z n ∈ R \ {0} fixed at positions q1 , . . . , qn ∈ R3 . We assume that the nuclei are in general position, which means that no three qk lie on one line.

A. Knauf (B) · M. Krapf Department Mathematik, Universität Erlangen-Nürnberg, Bismarckstr. 1 12 , 91054 Erlangen, Germany e-mail: [email protected]

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The n-atomic molecule generates a Coulombic potential on the conf iguration ˆ := R3 \ {q1 , . . . , qn } : space M ˆ → R is called Coulombic if Definition 1.1 A smooth potential V : M  −Z k ˆ with W : R3 → R 1. V has the form V(q) = nk=1 q−q + W(q) (q ∈ M) k smooth. 2. The potential vanishes at infinity, i.e. limq→∞ V(q) = 0, and its difference to a Coulomb potential is of short range. I.e. there exists Z ∞ ∈ R, called the asymptotic charge, ε ∈ (0, 1] and Rmin > 2 max(q1 , . . . , qn ) such that for some C1 > 0     ∇V(q) − Z ∞ q  < C1 Rmin  3 q  q2+ε q1 −q2  and ∇V(q1 ) − ∇V(q2 ) < C1 min(q 2+ε 1 ,q2 )

(q  Rmin ) (q1 , q2   Rmin ).

Remark 1.2 1. In [7] the high energy dynamics in Coulombic potentials was analyzed, using symbolic dynamics. The results are used here to calculate the escape rate, a quantity measurable in scattering experiments. For semiclassical aspects of the model, see [3], for topological methods also applicable to non-singular potentials, see [8]. ´ Derezinski and Gérard [4] is used as a general reference for scattering theory. Narnhofer analyzed time delay for short range potentials in [12]. 2. In the context of celestial mechanics, V is the sum of singular Kepler potentials, with Z i > 0interpreted as masses, and W = 0. Note that for n Z i. W = 0 we have Z ∞ = i=1 For electrostatic potentials the charges may be positive and negative, i.e. the force can be attractive as well as repulsive. The scattering of a classical electron by a molecule can be well modeled in this setting by positive charges of the nuclei, i.e. Z i > 0, and an additional smooth shielding (electronic) potential W, say of Thomas–Fermi type, such that, up to a Coulombic term Z ∞ /q given by the net charge of the molecule, the resulting potential is of short range, see [3]. Both for W = 0 and in the Thomas–Fermi case one may take ε = 1 in Definition 1.1. ˆ → R on the phase space T ∗ M ˆ  R3 × M ˆ is The Hamilton function Hˆ : T ∗ M given by ˆ p, q) := 1  p 2 + V(q). H( 2

(1.1)

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For n = 1 centres this generalizes the Kepler problem. For large energies no bounded orbits exist, so that the time delay is bounded. Thus we assume n  2 in the following. Due to collision orbits for nuclei with positive charge (respectively mass, depending on the interpretation) the Hamiltonian flow generated by (1.1) is incomplete. As we are interested in time-related quantities like time delay and escape rate, we use a regularization method which—unlike the so-called Kustaanheimo–Stiefel transform—does not involve a time reparametrization. In Section 5 of [7] such a regularization is done by phase space extension ˆ of T ∗ M. We generally denote by k (N) the space of k-forms on a manifold N. After regularization we get a smooth Hamiltonian system (P, ω, H) with a sixˆ H ∈ C ∞ (P, R) with H  ∗ ˆ = Hˆ and dimensional smooth manifold P ⊃ T ∗ M, T M 3 dqi ∧ a symplectic two-form ω ∈ 2 (P) such that ωT ∗ Mˆ = ω0 , with ω0 = i=1 ˆ dpi the canonical symplectic form on T ∗ M. This Hamiltonian system generates a smooth complete Hamilton flow  : R × P → P. Although for collisions the momentum p diverges, for simplicity we write the flow in the forms   p(t, x), q(t, x) := t (x) := (t, x) (x ∈ P, t ∈ R). 1.1 Classification of States in Phase Space We are dealing with Hamiltonian dynamics for which the energy (i.e. the value of the Hamilton function) is conserved. So for a given energy E ∈ H(P) the dynamics is confined on the energy surface  E := H −1 (E) ⊂ P. For large E this is a smooth manifold of dimension five. We now classify the points in the (extended) phase space P by their asymptotic behaviour: •

states b ± := {x ∈ P : lim sup q(t, x) < ∞}, bounded in the future respect→±∞

• • •

tively past, and the bounded states b := b + ∩ b − states s± := P \ b ± , scattered in the future/past and the scattered states s := s+ ∩ s− states t± := s∓ \ s± = s∓ ∩ b ± trapped in the future/past and the trapped states t := s+ s− = (b + ∩ s− ) ∪ (b − ∩ s+ ) = t+ ∪ t− . with a subscript E we denote the corresponding sets restricted to the energy surface  E , i.e. b E := b ∩  E .

The assumption that V is Coulombic gives rise to the virial inequality       d q(t), p(t) = 2 E − V q(t) + q(t), ∇V q(t)  E > Eth  0. (1.2) dt This is valid for H(x) = E and the part of a trajectory t → q(t) := q(t, x) outside an interaction zone defined by a virial radius Rvir

 Zone(E) := q ∈ R3 : q   Rvir (E) (E > 0). (1.3)

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So a trajectory leaving the interaction zone at some time t0 will move away from the origin for all future times t > t0 , and for any scattered state x ∈ s±E also limt→±∞ q(t, x) = ∞. The function E → Rvir (E) can be chosen to be continuous and nonincreasing in E, and as we consider high energies, i.e. energies above an energy threshold Eth > 0, we can assume a energy-independent virial radius Rvir := Rvir (Eth ) and energy-independent interaction zone Zone := Zone(Eth ), see [7], p. 11 for details. In the course of the article several other lower bounds on the constant Eth will arise. We consider the asymptotic behaviour of the flow t : P → P, x → ( p(t, x), q(t, x)) ∈ P by defining for E > 0 and a point x ∈ s±E the asymptotic velocities, directions and impact parameters p± (x) := lim p(t, x)

,

t→±∞

q⊥± (x) := lim

t→±∞

p ± (x) ∈ S2 pˆ ± (x) := √ 2E

(1.4)

  q(t, x) − q(t, x), pˆ ± (x) · pˆ ± (x) .

These are t -invariant and depend continuously on the point x, see Theorem 6.5 of [7]. Noting that pˆ ± (x) ⊥ q⊥± (x), we define the continuous asymptotic maps   A±E : s±E → T ∗ S2 , x → pˆ ± (x), q± (1.5) ⊥ (x) . We denote the canonical symplectic two-form on the cotangent bundle T ∗ S2 of the sphere by ω0 ∈ 2 (T ∗ S2 ), and use the volume four-forms1  E := −E ω0 ∧ ω0 ∈ 4 (T ∗ S2 ) D±E

A±E (s E )

(E > Eth ).

(1.6)

∗ 2

The sets := ⊂ T S represent the possible asymptotic data in the corresponding time direction, for a given energy E. The asymptotic scattering map of energy E AS E : D−E → D+E maps the initial asymptotic data ( pˆ + , q⊥+ ) ∈ D+E .

,

AS E := A+E ◦ (A−E )−1

( pˆ − , q⊥− )

∈

D−E

(1.7)

to the final asymptotic data

Remark 1.3 (Asymptotic completeness) Note that A±E (s±E ) = T ∗ S2 but T ∗ S2 \ D±E = ∅ if the sets s±E \ s E = t∓ E of past/future trapped orbits are not empty. The asymptotic completeness of the n-centre problem implies that D±E is of full

1 Choice of the volume form: The canonical symplectic volume form on a 2d-dimensional cotangent d/2

bundle T ∗ M with symplectic two-form ω0 equals −1d! ω0d ∈ 2d (T ∗ M). In (1.4) we normalize the √ momentum by a factor 1/ 2E. Thus in order to obtain symplectomorphisms between asymptotic data and Poincaré surfaces in phase space (see (2.10)), we multiply that canonical symplectic volume form on T ∗ S2 by 2E.

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measure with respect to the canonical volume form  E on T ∗ S2 , see Corollary 6.4 in [7]. Since the asymptotic maps A±E are t -invariant on their domains s±E , the asymptotic scattering map AS E carries no information about time-related quantities. 1.2 Møller Transformation Far from the origin in configuration space, the n-centre problem is well approximated by the Kepler problem, given by the phase space Pˆ ∞ := T ∗ (R3 \ {0}) and Hamilton function Hˆ ∞ : Pˆ ∞ → R

Z∞ Hˆ ∞ ( p, q) := 12  p 2 − . q

,

Being for Z ∞ > 0 a special case of (1.1), it can always be regularized to yield a smooth complete flow   ∞ (1.8) t : P∞ → P∞ , x  → p∞ (t, x), q∞ (t, x) of the Kepler problem, with the extended Hamilton function H∞ ∈ C ∞ (P∞ , R). The scattering states of the flow ∞ with a non-vanishing asymptotic momentum form the set P∞,+ := {x ∈ P∞ : H∞ (x) > 0}, ∞

consisting of  -orbits projecting to Kepler hyperbolae (resp. straight lines for Z ∞ = 0) in configuration space. Scattering theory in general deals with the comparison of two dynamics, in this case the dynamics of (P, t ) and (P∞ , ∞ t ). This is done by the Møller transformations ± : P∞,+ → s±

,

± := lim −t ◦ Id ◦ ∞ t . t→±∞

(1.9)

Here Id identifies P∞ with P outside a region near the singularities. In [7], Sect. 6 it was shown that the Møller transformations ± exist point-wisely and are measure-preserving homeomorphisms, and if the partial derivatives of V decay at infinity like

    Z ∞ q→∞ β β ∈ N30 (1.10) ∂q V(q) + = O q−|β|−1−ε q for some 0 < ε  1, then the Møller transformations are C ∞ symplectomorphisms. Similar statements hold for the asymptotic scattering map, defined in (1.7). Like in Remark 1.2, both for W = 0 and in the Thomas–Fermi case one may take ε = 1. ± ± We denote by ± ∗ : s → P∞,+ the inverse of  .

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From this follows in particular that ± (P∞,+ ) = s± , i.e. given a Kepler hyperbola there exists a unique scattered orbit of the n-centre problem which is asymptotic to this hyperbola in the time direction described by the sign. Conversely any one-sided scattered orbit is one-sided asymptotic to a unique Kepler hyperbola. The choice of the Kepler problem as the “comparison dynamics” for defining the Møller transformation is justified by the existence of the limit in (1.9). 1.3 Time Delay and Escape Rate Next we define the time delay τ (x) for a point x belonging to a scattered orbit by comparison with the Kepler dynamics ( denoting the Heaviside step function):       τ (x) := lim R − ||q (t, x)|| − 12 R − ||q∞ (t, + ∗ (x))|| R→∞ R

 + (R − ||q∞ (t, − (x))|| dt. ∗

(1.11)

So τ is the difference of dwell time of the trajectory inside a large ball and the mean dwell time of its asymptotic Kepler hyperbolae. As τ is -invariant, the time delay τ (x) only depends on the asymptotic data ∗ 2 ∗ 2 ˆ − , q− ( pˆ + , q+ ⊥ ) ∈ T S (respectively ( p ⊥ ) ∈ T S ) of x. So for E > 0 we define AsTim E := τ  E and the asymptotic time delay   −1 AsTim E ◦ A+E (x) if x ∈ D+E ∪ {+∞} , +∞ else (1.12) (remember that D+E is the set A+E (s E ) of asymptotic data of the scattering states).  + AsVol E (t) := 1l{AsTim+E (x)t}  E (t > 0) (1.13) AsTim+E : T ∗ S2 → R

AsTim+E (x) :=

T ∗ S2

denotes the  E -volume of the set of asymptotic data with a time delay greater than or equal a given time delay t > 0. Clearly AsVol+E (t) is monotone decreasing in t. As the asymptotic scattering map (1.7) is measure-preserving, this quantity would not change if one would use A−E instead of A+E in Def. (1.12). Remark 1.4 The passage from the time delay to the asymptotic time delay is motivated by measure theoretical properties of these maps: because of invariance of τ , for any time interval I ⊂ R the Liouville measure of the set {x ∈ s E : AsTim E (x) ∈ I} of scattered states is either zero or infinite, whereas (1.13) turns out to be finite for t > 0.

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Although our main interest lies in the analysis of orbits with large time delay, we also show 1. that the time delay is bounded below on the energy surface, 2. that the volume of orbits with time delay greater than or equal a given positive time is bounded, 3. and that complicated dynamics of scattering orbits occur for large time delay. More precisely we have the following results: Proposition 1.5 There exist constants C1 , C2 , C3 > 0 such that for all energies E > Eth 1. AsTim E > −C2 /E1/2  2. with ε ∈ (0, 1] from Def inition 1.1 we have AsVol+E (t) < C1 t−2/ε E1−3/ε t ∈  (0, C3 /E3/2 ) 3. the orbit through x intersects the interaction zone if AsTim E (x) ∈   C3 /E3/2 , ∞ . The proof of Proposition 1.5 is in the Appendix. The escape rate β E is defined as the exponential decay rate of AsVol+E (t) for large t, i.e. β E := − lim

t→∞

 1  ln AsVol+E (t) t

(E > Eth )

(1.14)

in the case of existence. Now we are ready to state our main result. To ease the notation, for real valued functions f, g we write f  g (or sloppily f (t)  g(t)) if there exist constants C1  1, C2 > 0 such that C1−1 g(t)  f (t)  C1 g(t) for all t  C2 . 1.4 A Matrix Perron–Frobenius Problem We now set up a finite matrix problem which we show to approximate the escape rate (1.14) with optimal precision (see Remark 1.6). This then allows to compute the escape rate very precisely, using only a few parameters of the model. Symbolic dynamics will be based on the alphabet A := {(i, j) : i, j = 1, . . . , n with i = j},

(1.15)

since we will erect Poincaré surfaces between pairs of nuclei. Thus only those symbol  sequences will be relevant, whose successive pairs (k0 , k1 ) = (i, j), (k, l) ∈ A × A have j = k. Then we define the charge Z k0 ,k1 := Z j and, for distance di, j := qi − q j, the mean distance dk0 ,k1 := 12 (di, j + d j,l ). We set     2di, j cos2 12 α(i, j, l) f k0 , k1 := , (1.16) −Z j

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α(i, j, l) denoting the angle between the vectors qi − q j and ql − q j. With

2E| f (k0 , k1 )| dk0 ,k1 Z k0 ,k1 ln(E)   and F E (k0 , k1 ) := 2 ln T E (k0 , k1 ) := √ − (2E)3/2 dk0 2E (1.17) playing the role of approximate Poincaré time respectively unstable Jacobian, we define the weighted transfer matrix (consult Baladi [1] for the subject of transfer operators)    E (k0 , k1 ) + β T E (k0 , k1 ) if j = k    exp − F M E β k0 ,k1 := . (1.18) 0 else Note that the definition of M E only involves the energy E, the positions qi and the charges Z i of the n nuclei and is independent of the potential W. The Perron–Frobenius eigenvalue λPF > 0 of M E now depends on E and β, and will be shown to have a unique solution β˜E of λPF (β, E) = 1

(E > Eth ).

By dmax we denote the maximal mutual distance of the nuclei, that is the diameter of the molecule. Main Theorem: Let V be a Coulombic n-centre potential, n  2, and the energy E > Eth . Then for the escape rate β E (defined in Eq. 1.14) it holds: (i) (ii)

β E exists, even more AsVol+E (t)  exp(−β E t). β E is given implicitly by a Perron–Frobenius problem. √ 2 2E ln E The escape rate β E is asymptotic to dmax . It is approximated by β˜E , with relative error of order O(1/E).

Remark 1.6 1. As a W-independent estimate, the O(1/E) estimate of (ii) is optimal. This can be seen by adding to V a cut off function W ∈ Cc∞ (R3 ) which equals a constant C in the interaction zone Zone, see (1.3). Thus the dynamics in  E over Zone equals the one without W for energy E − C. 2. We base our proof on the application of renewal theory, worked out by Lalley in [9], precisely determining the E-dependence of all quantities. This is possible by applying a simple symbolic dynamics from [7], see Fig. 1.

2 Proof of the Main Theorem While our estimates are optimal in their dependence on the energy E, we will be somewhat vague in denoting most energy-independent constants by C, without tracing back their mutual dependence. Generally speaking we suppress the dependence on E whenever this is feasible. We hope that this

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Fig. 1 Left: Scattering orbit for n = 3 centres, with symbol sequence 1, 2, 1, 2, 3. Right: Poincaré surfaces projected to the plane in configuration space, containing the three centres

makes the following part more readable. The interested reader, however, may consult [7] to find in many cases more explicit estimates. 2.1 Proof of Part (i) Although the escape rate is defined in the realm of scattering theory, the key of showing its existence and determining its value is to study the bounded states. As the escape rate is a limit of large time delay, it is natural that the trapped states play an important role. In our case for E > Eth the omega-limit set of the trapped states t E even equals the set b E of bounded states. 2.1.1 Symbolic Dynamics The set of non-wandering points of the flow  on the energy surface  E equals b E , the subset of bounded states. Moreover, for high enough energies E > Eth the bounded states b E form a hyperbolic set so that the flow t  E satisfies Axiom A (see [7], Thm. 12.8). This allows, by using Poincaré sections, to model a time-discretized version of the dynamics t b E with symbolic dynamics, given by a two-sided shift space (X, σ ) of finite type, see (2.5) below. Therewith the left shift σ on X is conjugated to the Poincaré map P E , restricted to the bounded states on the Poincaré surfaces. The continuous f low on b E is modelled by the suspension flow technique as follows. Definition 2.1 Given a function r ∈ C0 (X, R+ ) (called roof function), we set X r := X × R/ ∼

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where ∼ is the equivalence relation defined by       k, t + r(k) ∼ σ (k), t (k, t) ∈ X × R . Then the r-suspension f low is given by σtr : X r → X r

,

[(k, s)] → [(k, s + t)]



 t∈R .

(2.1)

In the present case of Poincaré time r = T E we abbreviate X E := X T E and σtE := σtT E . Lemma 2.2 (Theorem 12.8 of [7]) For energies E > Eth the f low t b E of the n-centre problem is conjugated to a suspended f low (X E , σtE ) by a Hölder continuous homeomorphism X E : X E → b E . Remark 2.3 (Hölder continuity) In this lemma, Hölder continuity relates to the choice of a metric on the shift space X, see (2.6) below (and an arbitrary choice of Riemannian metric on the Poincaré sections, denoted by dI E ). The roof function of the suspended flow (X E , σtE ) is the pull-back T E : X → + R of the Poincaré time, see (2.4) below. In particular it is Hölder continuous on the shift space X. From Theorem 19.1.6 and its Corollary 19.1.13 of Katok and Hasselblatt [5] it follows that the logarithm of the unstable Jacobian, denoted by F E (see (2.18) below) is also a Hölder continuous function on the intersection of the Poincaré surfaces with the bounded states. The conjugacy X E of Lemma 2.2 is established by introducing Poincaré surfaces labelled by the alphabet A from (1.15). For C > 0 the hypersurfaces in the energy shell  E , labelled by (i, j) ∈ A  C di, j i, j , I E := ( p, q) ∈  E : q − mi, j, qˆ i, j = 0, |q − mi, j| < 2E  p      p, qˆ i, j > 0,  √ × qˆ i, j < 2 C E−1 , 2E

(2.2)

are located in configuration space near the midpoint mi, j := 12 (qi + q j) between the centre i and j, and are perpendicular to the direction qˆ i, j := (q j − qi )/di, j (with di, j = qi − q j), see Fig. 1. These are Poincaré surfaces for all E > Eth , i.e. they are transversal to the flow t  E . We denote the disjoint union of these  i, j inner Poincaré surfaces by I E := (i, j)∈A I E . Remark 2.4 Although these surfaces become small for large E, for a large enough constant C > 0 in (2.2) every bounded orbit meets these surfaces between its successive near-collisions with the nuclei. The same applies to the scattering and the trapped orbits.

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Thus our strategy is to decompose every scattering orbit into three parts: • •

•

The part outside the interaction zone; either this is the whole orbit or it consists of two components. The segment between entry of the interaction zone and exit or the first encounter with an inner Poincaré surface, and similarly the segment between the last encounter with an inner Poincaré surface and exit from the interaction zone. The segment between the first and last encounter with an inner Poincaré surface. Only this segment is relevant for the value of the escape rate. However, the other two have to be controlled, too in order to guarantee its existence.

To realize the above strategy, we add further Poincaré surfaces.

 D E := x = ( p, q) ∈  E : q  Rvir denotes the part of the energy surface lying over the interaction zone (1.3). The two submanifolds of the boundary

 O± (2.3) E := ( p, q) ∈  E : q  = Rvir , ±q, p > 0 ⊂ ∂ D E , consisting of states leaving resp. entering the interaction zone, are transversal to the flow, too and called outer Poincaré surfaces. We use their disjoint unions − O E := O+ E ∪ OE

± H± E := I E ∪ O E

and H E := I E ∪ O E .   + The Poincaré map P E : H− E → H E , P E (x) :=  T E (x), x is defined using Poincaré time  

for x ∈ H− inf t > 0 : t (x) ∈ H+ E E . T E : H E → [0, ∞) , T E (x) := 0 for x ∈ O+ E (2.4) The inner Poincaré surfaces are labelled by the indices of their respective pair of nuclei. So the Poincaré map P E , restricted to I E ∩ b E , gives rise to the twosided shift space (X, σ )

 X := k = (. . . , k−1 , k0 , k1 , . . .) ∈ AZ : ∀i ∈ Z : (ki+1 )1 = (ki )2 . (2.5) ,

As usual, the shift σ : X → X acts by σ (k)i := ki+1 . Using similar definitions, with X + ⊂ AN0 we denote the one-sided shift space and with X ∗ the set of (unindexed) words in X + resp. in X. With the metric

 (2.6) d(k, l) := 2− sup i∈N0 : ∀| j|i: k j =l j the sets X resp. X + become metric spaces. According to Lemma 12.2 of [7] we have symbolic dynamics in the following sense: there exists a Hölder continuous homeomorphism FE : X → b ∩ IE,

(2.7)

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conjugating the shift on X with the Poincaré map on the bounded orbits. This allows us to consider functions on b ∩ I E like Poincaré time T E as functions on X. For a word k = (k0 , . . . , km ) ∈ X ∗ of length m + 1 we denote the cylinder over k by [k] := {l ∈ X : li = ki ∀i = 0, . . . , m}. This corresponds to the open submanifold on the inner Poincaré surface   I E (k) := x ∈ I E : P i (x) ∈ I Eki , i = 0, . . . , m . (2.8) The canonical symplectic form ω on P, restricted to H E makes this fourdimensional Poincaré surface a symplectic manifold. Next we define the total inner Poincaré time InTim E :  E → [0, ∞] := [0, ∞) ∪ {+∞}, the time spent between first and last encounter of the set I E of inner Poincaré surfaces, by InTim E (x)  sup{t ∈ R : t (x) ∈ I E } − inf{t ∈ R : t (x) ∈ I E } := 0

if I E ∩ (R, x) = ∅ . else

The t -invariance of InTim E permits us to define the asymptotic total inner Poincaré time   −1 InTim E ◦ A+E (x) if x ∈ D+E + + ∗ 2 . InTim E : T S → [0, ∞] , InTim E (x) := ∞ else We use the four-form  := − 12 ω ∧ ω ∈ 4 (P) (and will compare this with the volume form  E on T ∗ S2 defined in (1.6)). We consider the volumes 

 InVol E (t) := 1l{InTim E t}  of the sets VI E (t) := x ∈ O− E : InTim E (x)  t , O± E

(2.9) as well as InVol+E (t)

 :=

T ∗ S2

1l{InTim+E t}  E .

The motivation for studying the function InVol E is that it is controllable by symbolic dynamics, and at the same time it is asymptotically near to the function AsVol+E (defined in (1.12)), which gives the escape rate: Lemma 2.5 InVol+E (t) = InVol E (t) for all times t > C3 /E3/2 , and AsVol+E  InVol+E . Proof • By Remark 1.3 the set of points x ∈ T ∗ S2 with InTim+E (x) = ∞ has measure zero. So they do not contribute to InVol+E . We thus disregard the trapped orbits and only consider the scattering orbits.

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According to Part 3 of Proposition 1.5 all scattering orbits with time delay larger than C3 /E3/2 enter the interaction zone. In case of sufficient smoothness of the potential (e.g. assuming (1.10) to be valid for all |β|  2) the asymptotic maps A±E (see (1.5) map the outer Poincaré surfaces O± C1 -diffeomorphically to their images A±E (O± E E) ⊂ ∗ 2 T S . By using the cotangential lift of the polar diffeomorphism R3 \ {0} → [0, ∞) × S2 , see Section 6.3 of Marsden and Ratiu [11], one can compute that for the pullback with A±E of the volume form  E on T ∗ S2 holds (A±E )∗  E  A±E (O±E ) = O±E .

(2.10)

Thus for the -volume InVol E (t) of the set it holds for t > C3 /E3/2    + ± ∗ E = InVol E (t) = (A E )  E =  = InVol E (t).   V (t) V (t) A± E VI E (t)

IE

IE

But this relation of measures holds true even under the slightly milder conditions on the potential, given in Definition 1.1. • By Proposition 9.2 of [7] and Theorem 10.6 of [6] (adapted to three dimensions) it follows that there exists an energy dependent constant C(E) > 0 such that (x ∈ s E ). |AsTim E (x) − InTim E (x)|  C(E) So (setting ∞ − ∞ = 0) we have the uniform estimate    AsTim+ − InTim+  < C(E) E E on T ∗ S2 . Thus, assuming the estimate InVol E (t)  e−β E t which will be shown below,   + + AsVol E (t) = 1l{AsTim E (x)t}  1l{InTim+E t}  E = InVol+E (t). (2.11) T ∗ S2

T ∗ S2

With Eq. 2.11 it follows finally that AsVol+E  InVol E .

 

In the following we show the estimate InVol E (t)  e−β E t for the volume of the sets VI E (t) ⊂ O− E . As a first step for a given time t > 0 we partition X into cylinders which give rise to Poincaré times approximately equal to t. More precisely, we define the sets of best f itting words X t,E := {(k0 , . . . , km ) ∈ X ∗ : ∀ l ∈ [k] : Sm T E (l)  t

and ∃ l ∈ [k] : Sm−1 T E (k) < t}

and similarly X t,E := {(k0 , . . . , km ) ∈ X ∗ : ∃ l ∈ [k] : Sm T E (l)  t

and ∀ l ∈ [k] : Sm−1 T E (l) < t}

as subsets of words in X ∗ , with the summatory function of f : X → C S0 f := 0

,

Sm f :=

m−1  i=0

f ◦ σi

(m ∈ N).

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Lemma 2.6 The sets of cylinders [X t,E ] := {[k] : k ∈ X t,E } respectively [X t,E ] := {[k] : k ∈ X t,E } constitute partitions of X. Proof • To show that [X t,E ] covers X, take some k ∈ X. According to Lemma 9.3, Eq. 9.21 in [7] the Poincaré times have inf(T E ) > 0. Thus there is a minimal m ∈ N0 such that Sm T E (l)  t for all l ∈ [(k0 , . . . , km )]. Then (k0 , . . . , km ) ∈ X t,E . • To show that [X t,E ] is a partition of X, suppose that k = (k1 , . . . , km(k) ) ∈ X t,E and l = (l1 , . . . , lm(l) ) ∈ X t,E with [k] ∩ [l] = ∅. Then w.l.o.g. [k] ⊂ [l], i.e. m(k)  m(l). For all x ∈ [l] it holds Sm(l) T E (x)  t, and there exists an x ∈ [k] ⊂ [l] with Sm(k)−1 T E (x) < t. Together this implies that m(l)  m(k). So m(l) = m(k) and k = l. • The proof for [X t,E ] is analogous.   With the aid of the ‘best fitting words’ we approximate VI E (t), defined in (2.9), in the following manner: Proposition 2.7 There exists a constant Cτ (E) > 0 such that for any t > 0 the inclusions         ˙ ˙ − −1 −1 O− ∩ P ( I ⊂ V (t) ⊂ O ∩ P ( I (2.12) k) k) E I E E E E k∈X t+Cτ (E)

k∈X t−Cτ (E)

hold for the iterated inner Poincaré surfaces I E (k), def ined in (2.8). Proof • By the hyperbolicity of P on I E ∩ b , see [7], and the Lipschitz-continuity of T E on I E it follows that there exists a positive constant Cτ = Cτ (E) such that m−1     i  m−1  i   T E P (x) − T E P (y)  < Cτ    i=0

i=0

for all words k ∈ X ∗ and any x, y ∈ I E (k), m + 1 being the lengthof k. • To show the first inclusion in (2.12), take some x ∈ P −1 I E (k) ∩ O− E for m−1  some k = (k0 , . . . , km(k) ) ∈ X t+Cτ ,E . From T E (P i (y))  t + Cτ for all y ∈ i=0

  I E (k) ∩ b it follows that T E P i (y)  t for all y ∈ I E (k). P (x) ∈ I E (k) i=0   together with InTim E (x) = InTim E P (x) imply that x ∈ VI E (t). This shows the first inclusion. • To show the second inclusion in (2.12), that t > 0 and let x ∈ m−1 suppose T E (P i (P (x)))  t and a word k = VI E (t). Then there exists m ∈ N with i=0 (k0 , . . . , km ) ∈ X ∗ such that P i+1 (x) ∈ I E (ki ) for i = 0, . . . , m. Since P (x) ∈ m−1 

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I (k0 , . . . , km ), it follows that there exists y ∈ I E (k0 , . . . , km ) ∩ b such that Em−1 i i=0 T E (P (y))  t − Cτ . So after eventually shortening the finite sequence (k0 , . . . , km ) ∈ X ∗ to length m#  m there exists k# = (k0 , . . . , km# ) ∈ X t−Cτ ,E # such that P (x) ∈ I E (k) ⊂ I E (k# ) and so x ∈ P −1 (I E (k# )) ∩ O− E for some k ∈ X t−Cτ ,E showing the second inclusion. • The fact that these unions are disjoint follows from Lemma 2.6 and completes the proof.  

2.1.2 Measure-Theoretical Estimates In order to estimate VI E (t) using (2.12), we now approximate the volume of −1 ∗ O− E ∩ P (I E (k)) for the words k ∈ X . In [7] local coordinates (y, z) = (y1 , y2 , z1 , z2 ) on the inner Poincaré suri, j faces I E were introduced, with z affine in the position q, y affine in the momentum p, and the volume form 2di,2 j

I i, j . (2.13) E E By using the Euclidean metric on R4 , these coordinates serve also for defining a metric dI E on I E . In these coordinates the Poincaré surfaces defined in (2.2) i, j take the form I E = B y × Bz , B y and Bz being two-dimensional disks whose radii are proportional to 1/E. With f from (1.16), the linearized Poincaré map equals dy1 ∧ dy2 ∧ dz1 ∧ dz2 =

Tx P = f (k0 , k1 ) E

 1l 1l  1l 1l

+ O(E0 )



 x ∈ I E (k0 , k1 ) ,

(2.14)

see Prop. 11.2 of [7]. In order to use symbolic dynamics, we compare these maps with the ones along the bounded orbits. Since I E ∩ b is a hyperbolic set for P , the tangent space has the T P -invariant splitting Tx I E = Txu ⊕ Txs

(x ∈ I E ∩ b )

(2.15)

into the unstable and stable Lagrangian subspace. In (y, z)-coordinates, the cone f ield Cone+E ≡ Cone E on I E is defined by   C Cone E (x) := (δy, δz) ∈ T(y,z) I E : |δy − δz|  |δy + δz| , E

(2.16)

and Cone−E denotes the image of Cone+E under time reversal. So the aperture of these four-dimensional cones is of order O(1/E). Furthermore, by estimate (2.14), for an appropriately chosen constant C > 0, for all E > Eth the cone field (2.16) is strictly P -invariant on I E ∩ P −1 (I E ).  −1 The logarithmic Jacobian of a subbundle U of T I E ∩ P (I E ) is generally defined by F E,U : I E ∩ P −1 (I E ) → R

,

   F E,U (x) = ln det Tx PU x ,

(2.17)

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with the determinant depending on the choice of the Riemannian metric dI E . For the two-dimensional Lagrangian unstable bundle in (2.15) the logarithmic unstable Jacobian is given by 

 +

F E : I E ∩ b → R+

,

F E := F E,T u .

(2.18)

Let L− , L be a pair of transversal Lagrangian subbundles on I E . Then any tangent vector w ∈ Tx I E has a unique decomposition w = w − + w+ with w± ∈ L± (x). This decomposition gives rise to the quadratic form Q E (w) := 2ω(w− , w + ) and defines the sector field 

 Sect E := w ∈ Tx I E : Q E (w)  0 , x∈I E



 both depending on the pair L− , L+ .   Lemma 2.8 The pair L− , L+ of transversal Lagrangian subbundles with  1 ∓ C/E  L± (x) := (δy, δz) ∈ T(y,z) I E : δy = δz (x ∈ I E ) 1 ± C/E spans the cone: Cone E = Sect E . Proof Transversality follows from C/E ∈ (0, 1), the Lagrangian property from    #     #    δz   δz#  1∓C/E δy δy δy ± , , = 0 , ω δy = ω ∈ L (x) . # δz δz δz δz 1±C/E δz# δz#   E (C/E)2 |δy + δz|2 − |δy − δz|2 for The quadratic form equals Q E (w) = 2C  δy  w = δz . This implies equality of the cone and the sector.   Lemma 2.9 For any E > Eth the tangent map of the Poincaré map P E is strictly monotone with respect to the sector Sect E , i.e.     x ∈ I E ∩ P −1 (I E ) . Tx P Sect E,x \ {0} ⊂ int Sect E,P (x) Proof Let v ∈ Sect E,x \ {0} for some point x ∈ I E ∩ P −1 (I E ). Since T yu I E ⊂ Cone E,y = Sect E,y and the logarithmic unstable Jacobian of T y P is of order E for y ∈ I E ∩ b the claim follows by a compactness argument.   Following Liverani and Wojtkowski [10], we denote for x ∈ I E by Lagr E,x := {Lagrangian subspace U ⊂ Tx I E : ∀u ∈ U \ {0} : Q E,x (u) > 0} the set of positive Lagrangian subspaces. With Lagr E we denote the set of positive Lagrangian subbundles of T I E . For two subspaces U, V ∈ Lagr E,x the distance      ωx (u, v)    Dist E,x (U, V) := sup Asinh   Q E,x (v) Q E,x (w)  u∈U\{0}, v∈V\{0} 

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gives a complete metric on Lagr E,x , see [10]. By Theorem 1 of [10]and the strict monotonicity of the Poincaré map P E it follows that Q E Tx P (v) > Q E (v) for any v ∈ Cone E,x , v = 0 and thus γ E :=

sup

Q E (v)   <1 Q E Tx P (v)

sup

x∈I E ∩P −1 (I E ) v∈Sect E,x : vI E =1

by a compactness argument. Thus for any u, v ∈ int Sect E,x it holds (note that P E is a symplectomorphism) that  ⎛ ⎞     

     ω Tx P m (u), Tx P m (v) Asinh ⎝ ⎠  Asinh √ ω(u, v) γ Em  m→∞         −→ 0 Q(u)Q(v)  Q E Tx P m (u) Q E Tx P m (v)    implying limm→∞ Dist E,P m (x) Tx P m (U), Tx P m (V) = 0 for any positive Lagrangian subspaces U, V ∈ Lagr E,x ⊂ Tx I E and x ∈ I E ∩ b . Since the set of positive Lagrangian subspaces lying in the cone Cone E,x is compact, it follows that there exists a constant Cs,E > 0 such that     m x ∈ ∩i=0 Dist E,x Tx P m (U), Tx P m (V)  Cs,E · γ Em P −i (I E ) . (2.19) Given a two-dimensional subspace U x ⊂ Tx I E we denote with F E,U x (x) the logarithm of the Jacobian of Tx P restricted to U x with respect to the metric dI E . For a two-dimensional subbundle U of T I E and m  1 we have Sm F E,U (x) :=

m−1 

  F E,Tx P i (U x ) P i (x)



 m−1 −i x ∈ ∩i=0 P (I E ) .

(2.20)

i=0

Note that Sm F E,U (x) is the logarithm of the Jacobian of P m restricted to U x ⊂ m Tx I E , x ∈ ∩i=0 P −i (I E ). In order to estimate the volumes in (2.12), in the proof of Proposition 2.11 we control Sm F E,U for a concrete positive Lagrangian subbundle U ∈ Lagr E . The following lemma relates this to Sm F E , defined on the much smaller set IE ∩ b . Lemma 2.10 There exists a constant C F E such that for any x, y ∈ I E (k), x ∈ b with k = (k0 , . . . , km ) ∈ X ∗ and any smooth positive Lagrangian subbundle V ∈ Lagr E ⊂ Cone E it holds:    Sm F E,V (y) − Sm F E (x) < C F . E Proof Let x, y ∈ I E (k), x ∈ b with k = (k0 , . . . , km ) ∈ X ∗ . Then we have        Sm F E (x)− Sm F E,V (y)   Sm F E (x)− Sm F E,V (x) +  Sm F E,V (x)− Sm F E,V (y) . (2.21) Since for any x ∈ I E ∩ P −1 (I E ) the set (Lagr E,x , Dist E ) of positive Lagrangian subspaces in the cone Cone E,x is a compact metric space and since the topology defined by this metric coincides with the standard topology, see

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Corollary on p. 8 of [10], the map Lagr E,x → R, U → F E,U (x) is continuously differentiable. Thus it holds by a compactness argument that this map is Lipschitz continuous, i.e. |F E,U (x) − F E,V (x)|  C · Dist E (U x , Vx ) for an appropriate constant C > 0. Hence by Eq. 2.19 |Sm F E (x) − Sm F E,V (x)| 

m−1 

      F E,T P i (T u I ) P i (x) − F E,T P i (V) P i (x)  x E

i=0

C

m−1  i=0

m−1    C · Cs,E Dist E Tx P i (Vx ), Tx P i (Txu I E )  C · Cs,E γ Ei  , 1 − γE i=0

showing that the first summand in Eq. 2.21 is bounded uniformly in m. The second sum in Eq. 2.21 is bounded by a constant by the fact that the function x → F E,U x is smooth on I E ∩ P −1 (I E ), and thus also Lipschitzcontinuous on I E ∩ P −1 (I E ) by a compactness argument.   m−1 F E (P i (x)) is Note that by the chain rule of differentials Sm F E (x) = i=0 m the logarithm of the unstable Jacobian of P (x) for x ∈ I E ∩ b . Proposition 2.11 With F E : I E ∩ b → R+ the logarithm of the unstable Jacobian it holds: There exists a constant C > 1 such that for all E > Eth  exp(−Sm F E (x)) −1 exp(−Sm F E (x)) C   C (2.22) 3 E E3 I E (k) uniformly for any k = (k0 , . . . , km ) ∈ X ∗ and x ∈ I E (k) ∩ b . Proof • For the word k ∈ X ∗ and  = 0, . . . , m we denote by (y , z ) the local (y, z)coordinates on I E (k ). By (2.13) the canonical volume form  appearing in (2.22) and the standard coordinate area forms  y := dy1 ∧ dy2 on the disk B y , z := dz1 ∧ dz2 on Bz are related by I E (k0 ) = 2dE2  y0 ∧ z0 . k0

For all points z0 ∈ Bz the set B y0 (z0 , k) := {x = (y, z) ∈ I E (k) : z = z0 } is a two-disk, and is identified with its image in B y . Then     E = 2  y0  z0 . (2.23) 2dk0 Bz I E (k) B y0 (z0 ,k) We restrict the iterated Poincaré maps P m to the two-disks B y0 (z0 , k) ⊂  I E (k0 ) (z0 ∈ Bz0 ), and denote by πz0 : P m B y0 (z0 , k) → Bzm the projections of their images to the zm -coordinate plane. By Prop. 11.5 (1) of [7] the composition of these maps gives rise to the diffeomorphisms Zz0 ,k := πz0 ◦ P Em  B y0 (z0 ,k) : B y0 (z0 , k) → Bzm

(z0 ∈ Bz0 ).

(2.24)

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177

In the inner integral on the right hand side of (2.23) we apply the transformation rule of integration     y0 =  y0 = Zz0 ,k ∗  y0 (2.25) B y0 (z0 ,k)

Zz−1,k (Bzm ) 0

B zm

• We denote by ver E the vertical bundle whose form in (y, z)-coordinates is   ver E,x := {(δy, δz) ∈ Tx I E : δz = 0} x ∈ I E ∩ P −1 (I E ) . Using the notation (2.20), the push-forward of the two-form  y0 with the diffeomorphism (2.24) equals for x ∈ B y0 (z0 , k)       −1  Zz0 ,k ∗  y0 Zz0 ,k (x) = exp − Sm F E,ver E (x) Jπz0 P Em (x) zm Zz0 ,k (x) , (2.26) with Jπz0 the Jacobian of the projection πz0 , of the surface S := P Em (B y0 (z0 , k)). We insert (2.26) in (2.25), estimating its constituents. • That Jacobian in (2.26) is estimated uniformly in the parameter z0 ∈ Bz0 by J π z0 =

1 2

+ O(1/E).

(2.27)

To show this we choose an orthonormal basis w (1) , w(2) of Ta S at a := P Em (x). Then the tangent vectors are contained in the cone at a. Writing   local  w(1) = (δy(i) , δz(i) ), this implies that |δy(i) |2 − 12  = |δz(i) |2 − 12   C/E, with C from (2.16). On the other hand   (1) δy , δy(2) = − δz(1) , δz(2)   by orthogonality of w (1) and w(2) . With (2.16) this implies  δz(1) , δz(2)  =  2 O(1/E). As Jπz0 = |δz(1) |2 |δz(2) |2 − δz(1) , δz(2) , we have proven (2.27). • Note that both the unstable bundle T u (I E ∩ b ) and the iterates of the vertical bundle ver E are contained in the cone Cone E . Furthermore the smoothness of the map x → F E,ver E (x) − F E,V for V ∈ Lagr E a positive Lagrangian subbundle together with a compactness argument and Lemma 2.10 show the existence of a constant C such that the first factor on the right hand side of (2.26) is estimated by       C−1 exp − Sm F E (x0 )  exp − Sm F E,ver E (x)  C exp − Sm F E (x0 ) (2.28) uniformly in E > Eth and for any x ∈ I E (k), x0 ∈ I E (k) ∩ b . • Insertion of (2.27) and (2.28) in (2.26) completes the proof, taking into regard the fact that  y0 (B y )  1/E2 and zm (Bz )  1/E2 .   Propositions 2.7 respectively 2.11 concern subsets of the outer resp. inner Poincaré surfaces. To combine them note that the sets appearing in Proposition 2.7 can be written as     −1 O− I E (k) = P −1 I E (k) \ I E . (2.29) E ∩P # We henceforth abbreviate A  by (A).

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Since  is P E -invariant, (2.29) trivially implies the upper bound in        −1 1  I E (k)   O− I E (k)   I E (k) (E > Eth , k ∈ X ∗ ). E ∩P 2 (2.30)   −1     The volume of the r.h.s. in (2.30) equals  I E (k) −  P I E (k) ∩ I E . The explicit formula (2.14) for the differential T P shows that the unstable Jacobian diverges (like E2 ) as E → ∞ and so does its logarithm F E . So the lower bound in (2.30) (even with any constant smaller than one instead of 1/2) also follows for large enough threshold energy Eth from (2.29) and the estimate of Proposition 2.11 in terms of scaling factors F E . Thus for an appropriate constant C > 1 we obtain the estimate       C−1 exp − Sm(k) F E (xk )  InVol E (t)  C exp − Sm(k) F E (xk ) k∈X t+Cτ ,E

k∈X t−Cτ ,E

(2.31) valid for all E > Eth and arbitrary representatives xk ∈ [k] ⊂ X. The hyperbolicity of b E assures that also the logarithm of the unstable Jacobian F E is Hölder continuous on I E ∩ b resp. on X, see Thm. 19.1.6 and its Corollary 19.1.13 in [5]. Like for T E from (2.4) we will, using the homeomorphism (2.7), consider F E : b ∩ I E → R+ (see (2.18)) also as an element of Fα (X + , R). Since the following approach depends on the cohomology classes of the functions T E and F E only, we can assume that T E and F E are Hölder continuous functions on X only depending on the future, and by a natural identification that T E and F E are Hölder continuous functions on the onesided shift X + . See Bowen [2], Sect. 6 for more details. Next we define, using the one-sided shift (X + , σ ), for x ∈ X + the function InVol E : R → R+ x

x

, InVol E :=

∞ 



m=0 y∈σ −m (x)

  exp −Sm F E (y) 1l−∞,S

.

m T E (y)

(2.32) This quantity models InVol E , defined in (2.9), but is only based on data of bounded orbits. We start with a rough upper estimate, needed later on for renewal theory. Lemma 2.12 For a suitable energy threshold Eth > 0 and C > 0 for all energies x E > Eth the sum InVol E (t) converges for any t ∈ R and any choice x ∈ X + . Furthermore   x 0 < InVol E (t)  C exp − ω− (E) max(t, 0) with 0 < ω− (E) :=

inf(F E ) − htop (X + , σ ) sup(F E ) − htop (X + , σ )  ω+ (E) := , sup(T E ) inf(T E ) (2.33)

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htop (X + , σ )  0 being the (E-independent) topological entropy of the shift space (X + , σ ). Proof • For all n  2 there exists a constant Ch  1 such that Ch−1 · enhtop (X

+

,σ )

 |σ −m (x)|  Ch · emhtop (X

+

,σ )

(n ∈ N, x ∈ X + ).

For n = 2 this follows since the topological entropy htop (X + , σ ) = 0 and |σ −m (x)| = 1. For n  3 the estimate follows since then the shift is topologically mixing. • We already remarked that the logarithmic unstable Jacobian F E diverges as E → ∞. Thus for Eth  1 large enough we have F E − htop (X + , σ )  ln(2)

(E > Eth ),

(2.34)

which in particular vindicates the first inequality in (2.33). $ • With the% constant C > 0 from Proposition 2.11 and for m0 := max(t, 0)/ sup(T E ) x

InVol E (t) 

∞    C Ch  exp m htop (X + , σ ) − inf(F E ) 3 E m=m 0

  ˜  C(E) exp −ω− (E) max(t, 0) ˜ is finite, with C(E) := 2 CEC3h e F E −htop (X the geometric series.

+

,σ )

= O( E1 ), using (2.34) in estimating  

The following lemma shows the asymptotic equivalence of the functions x InVol E and InVol E : Lemma 2.13 For any energy E > Eth it holds uniformly in x ∈ X + that x

InVol E (t)  InVol E (t).

(2.35)

Proof We show the existence of a constant C > 1 such that for all E > Eth , and x ∈ X + C−1 InVol E (t)  E3 InVol E (t)  C InVol E (t) x

x

(t > 0).

(2.36)

We start with the first inequality in (2.36). Since by Lemma 2.6 the cylinders over the set X t+CAsTim E ,E ⊂ [X + ] of best fitting words constitute a partition of the shift space X + , it holds: x

InVol E (t) =

 ∞

 k∈X t+CAsTim

E

,E



m=0 y∈σ −m (x)

e−Sm F E (y) 1l−∞,S

 (t) m T E (y)

· 1l[k] (y). (2.37)

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By Eq. 2.22 and the fact that Sm(k) T E [k] < t it follows that for an appropriate C1 > 1 and any best fitting word k ∈ X t+CAsTim E ,E the corresponding term in (2.37) is dominated by ∞ 



m=0

y∈σ −m (x)

=

∞ 

e−Sm F E (y) 1l 

 (t) −∞,Sm T E (y)

∞ 



m=m(k)

y∈σ −m (x)

· 1l[k] (y) 

e−Sm F E (y) 1l[k] (y)

   exp − Sm(k) F E (y) − Sm# F E σ m(k) (y) 1l[k] (y)

m# =0 y∈σ −(m# +m(k)) (x) ∞     C1 E3  I E (k)



e−m

#

inf(F E )

.

(2.38)

m# =0 y∈σ −m# (x)

Note that |σ −m (x)|  ehtop (X,σ )m uniformly in x ∈ X + . Thus since the unstable Jacobian diverges as E → ∞, the double sum in (2.38) converges (if Eth is chosen large enough) for all E > Eth and x, with upper bound 2. So the first estimate in (2.36) follows by using (2.31). With analogous arguments we get ∞   x InVol E (t)  C1−1 E3  I E (k)



e−m

#

max{F E }

,

m# =0 y∈σ −m# (x)

showing (after an adaptation on the constant C if necessary) the second estimate in (2.36).   x

The key-feature of the function t → InVol E (t) which allows for a precise study of its asymptotic behaviour is the following: x

Lemma 2.14 The function InVol E : R → R satisf ies the renewal equation   z x InVol E (t) = 1l{t0} + e−F E (z) InVol E t − T E (z) . (2.39) z∈σ −1 (x)

Proof Noting that the sums S0 T E = 0, we decompose (2.32) into x

InVol E (t) = 1l{t0} +

∞  

  exp − S F E (y) · 1l{S T E (y)t}

=1 y∈σ − (x)

= 1l{t0} +



−F E (z)

e





  exp −Sm F E (y) · 1l{Sm+1 T E (y)t}

m=0 y∈σ −m (z)

z∈σ −1 (x)

= 1l{t0} +

∞ 

 z e−F E (z) InVol E t − T E (z) ,

z∈σ −1 (x)

that is, the renewal equation.

 

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For any E > Eth and α ∈ C the function α T E − F E is Hölder continuous. The associated Ruelle transfer operator  Lα,E : C (X + , R) → C (X + , R) , Lα,E f (k) := eαT E (l)−F E (l) f (l) (2.40) l∈σ −1 (k)

is a Perron–Frobenius (PF) operator if α ∈ R. We denote with λα,E , hα,E and να,E its PF eigenvalue, its normalized positive PF eigenfunction and its adjoint Borel PF probability measure respectively, i.e. (omitting the index E)  Lα hα = λα hα , L∗α να = λα να and hα dνα = 1. (2.41) X+

The solution α0 (E) of the implicit equation λα,E = 1 turns out to be the escape rate. Lemma 2.15 For all E > Eth there exists a unique solution α0 (E) ∈ R+ of the equation λα,E = 1, and α0 (E) ∈ [ω− (E), ω+ (E)] with α ± from (2.33). Proof • The fact that α → λα is continuously differentiable with   dλα d Lα hα dνα = λα T E hα dνα > 0, = dα dα X X+

(2.42)

using the normalizations (2.41), shows uniqueness of the solution. • For existence and localization first notice that for α  ω− (E) = inf(F E )−htop (X + ,σ ) and independent of k ∈ X + sup(T E ) λα,E = lim

1/m Lm α hα (k) hα (k)

m→∞

 lim

m→∞

  ⎞1/m exp Sm (αT E − F E )(l) hα (l) ⎜ l∈(σ m )−1 (k) ⎟ = lim ⎝ ⎠ m→∞ hα (k) ⎛



   1/m  Ch sup(hα ) exp m α − ω+ (E) sup(T E )  1, inf(hα )

since for E > Eth we have htop (X + , σ ) < F E , see (2.33). • Similarly for α  ω+ (E) = ⎛ ⎜ λα,E = lim ⎝ m→∞

 l∈σ −m (k)

sup(F E )−htop (X + ,σ ) inf(T E )

independent of k ∈ X +

  ⎞1/m exp Sm (α T E − F E )(l) hα (l) ⎟ ⎠ hα (k)

 1/m    inf(hα ) Ch−1  1, exp m α − ω+ (E) inf(T E ) m→∞ sup(hα )

 lim

together showing existence of a solution α0 (E) ∈ [ω− (E), ω+ (E)].

 

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Proposition 2.16 With InVol E from (2.32) and α0 (E) from Lemma 2.15 there exists a constant C  1 such that   x C−1 K E (x) exp − α0 (E)t  InVol E (t)    CK E (x) exp − α0 (E)t (t ∈ R+ , x ∈ X + ), K E ∈ C (X + , R+ )

K E (x) :=

,

hα0 (x) # α0 (1 − e−α0 ) T E hα0 dνα0 X+

being def ined with the help of (2.41). Proof • We first suppose that T E is integer-valued, but the image T E (X + ) is not contained in a proper subgroup nZ, n > 1 of Z. Then the piecewise constant x map t → InVol E (t) has jumps only at t ∈ Z. We claim that even InVol E ∼ K E (x)e−α0 t x

(t → ∞),

(2.43)

and it is sufficient to check (2.43) for t ∈ N. By Lemma 2.12 the FourierLaplace transform  x x  E (α) := InVol InVol E (t)eαt (x ∈ X + ) t∈Z

  converges absolutely in a strip Re(α) ∈ 0, ω− (E) of the complex of plane, with ω− (E) > 0 defined in (2.33). Namely for these α one has, with C > 0 from Lemma 2.12, x  InVol E (0) x InVol E (t)eαt = 1 − e−α t∈Z\N x InVol E

and

  InVolx (t)eαt   E t∈N

C  . 1 − exp Re(α) − ω− (E)

The Fourier-Laplace transform of the renewal equation (2.39) is given by    x z 1  E (α) =  E (α) InVol + exp αT E (z)− F E (z) InVol (x ∈ X + ). −α 1−e −1 z∈σ

(x)

In terms of the Ruelle transfer operator (2.40) this leads to the formula x

 E (α) = (1 − e−α )−1 (1l − Lα )−1 1l(x) InVol

(x ∈ X + ).

(2.44)

The right hand side of Eq. 2.44 is real-analytic in the extended strip Re(α) ∈ (0, α0 ), with λα0 = 1 from Lemma 2.15. Similar as in Prop. 7.2 of Lalley [9] one decomposes the PF-operator in the form Lα = λα να (·)hα + L##α

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such that (with (2.41)) L##α maps C (X + , C) to the subspace {g ∈ C (X + , C) : νk (g) = 0}. Thus ∞  ∞ ∞    m m Lα = λα να (·)hα + (L##α )m m=0

m=0

m=0

and we obtain in some punctured neighbourhood of α0 (1l − Lα )−1 = (1 − λα )−1 να (·)hα + (1l − L##α )−1 ,

(2.45)

the second term of the right hand side being holomorphic. Combining Eqs. x 2.44, 2.45 and 2.42 and using λα0 = 1 we see that the residue of α → κˆ I E (α) at α = α0 equals

−1 hα0 (x) dλα −α0 −1 (α0 ) hα0 (x) = −(1 − e−α0 )−1 = −K E (x). (1 − e ) − dα να0 (T E hα0 ) (2.46) To show (2.43) we introduce, similar to the proof of Thm. 2 of [9], the function F(z, x) :=

∞ 

  x zm emα0 InVol E (m) − K E (x)

(x ∈ X + ).

m=0

By Lemma 2.12 the function z → F(z, x) is holomorphic in an open disk around 0 of radius exp(ω− (E) − α0 )  1 (see Lemma 2.15). For z in the open − annulus e−α0 < |z| < eω (E)−α0 we rewrite ∞  x x  E (ln z + α0 ) + K E (x) − F(z, x) = InVol z− e−α0 InVol E (−). z−1 =1

By Lemma 2.12 the last term is analytic in z for |z| > e−α0 , whereas the residue of the sum of the first and second term vanishes by (2.46). Thus one can analytically extend as in [9] the function z → F(z, x) to an open disk |z| < 1 + 2ε(E) for some ε(E) > 0. Then Cauchy’s integral formula gives  x F(z, x)z−m−1 dz emα0 InVol E (m) − K E (x) = (2πi)−1 

|z|=1+μ(E) ˜

= O (1 + ε(E))−m x



and thus InVol E (m) ∼ e−mα0 K E (x) for non-negative integer m. This shows the claim for the case when T E is integer-valued and not contained in a proper subgroup of Z. • The general lattice case, i.e. the image T E (X + ) generates the discrete subgroup cZ, c > 0 of R is treated by a multiplication of T E by the factor 1/c. • The non-lattice case follows by appropriate modification of the proof of Thm. 1 in [9], as the above lattice case followed by modifying the proof of Thm. 2 in [9].  

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The Lemmata 2.5 and 2.13, together with this proposition imply Part (i) of the Main Theorem:   x AsVol+E (t)  InVol E (t)  InVol E (t)  exp − α0 (E)t

(E > Eth ).

2.2 Proof of Part (ii) The escape rate β E from (1.14) has been shown to equal the solution α0 (E) of the implicit eigenvalue formula λPF (Lα0 ,E ) = 1 of the transfer operator Lα,E defined in (2.40). To obtain a finite-dimensional approximation of Lα,E , we approximate the functions F E , T E , originally defined in (2.4) and (2.18) and later considered as elements of Fα (X + , R), using the homeomorphism (2.7), by the matrices E and F E from (1.17), depending only on two symbols (k0 , k1 ), but also T considered as (locally constant) functions in Fα (X + , R). It follows from (2.14) and Lemma 10.6 of [7] that there exists a C > 0, so that with δT (E) := CE−3/2 and δ F (E) := CE−1 for all E > Eth E − δT (E)  T E  T + − := T E + δT (E) =: T T E E

and

(2.47)

− := F E + 2 ln(1 − δ F (E))  F E  F E + 2 ln(1 + δ F (E)) =: F + . (2.48) F E E Next we define for δ = (δ1 , δ2 ) ∈ R2 and M E from (1.18) the weighted transfer matrices M E (β, δ) ∈ RA×A

,

M E (β, δ)k0 ,k1 := exp(βδ1 + δ2 ) M E (β)k0 ,k1 .

Note that M2E (β, δ) has strictly positive entries, i.e. M E (β, δ) is a Perron– Frobenius matrix. With λPF (M E (β, δ)) we denote the Perron–Frobenius eigenvalue of the matrix M E (β, δ). The approximate escape rate β˜E and its bounds β˜E± are defined implicitly by    λPF M E β˜E = 1

and

   λPF M E β˜E± , (±δ F (E), ∓δT (E)) = 1.

(2.49)

The following lemma tells that β˜E and β˜E± are well defined for large enough energies E:   Lemma 2.17 For all δ ∈ R2 the map β → λPF M E (β, δ) is continuous. If δ2 > E ), it is strictly monotone increasing. − inf(T Proof It is well known that λPF is related to the topological pressure by      E + δ1 + β(T E + δ2 ) . ln λPF M E (β, δ) = Ptop X, σ, − F

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  Continuity of β → ln λPF (M E (β, δ)) follows from Theorem 9.7 of [13]. By using the variational principle for the topological pressure and the fact that E + δ2 > 0 one gets T     ln λPF (M E (β2 , δ)) − ln λPF (M E (β1 , δ)) E ) + δ2  inf(T (β2 > β1 ). β2 − β1   2.2.1 High Energy-Limit of the Approximate Escape Rate With dmax being the maximum of the distances of the n centres, we denote the approximation to the escape rate, appearing in Part (ii) of the Main Theorem, √ ln E by β E∞ := 2 2E . dmax Lemma 2.18 β˜E from (2.49) satisf ies lim E→∞

β˜E β E∞

= 1.

Proof     M E β˜E  λPF M E (β˜E ) = i, j   1 implies that not all entries of the PF-matrix M E β˜E can tend to zero for E → ∞. This, together with the asymptotic formula    M E β˜E k0 ,k1 ∼ f −2 (k0 , k1 ) exp 

 Z k0 ,k1 ln E dk0 ,k1 + (k0 , k1 ∈ A) × −2 ln E + β˜E √ (2E)3/2 2E

• First note that the estimate maxi∈A





j∈A

  for the non-zero entries of M E β˜E , implies lim inf E→∞

β˜E  1. β E∞

• Next assume that β˜E ∞ >1 E→∞ β E   such that at least one entry of M E β˜E is unbounded for E → ∞. The symmetry lim sup

(M E )(i, j),( j,k) = (M E )(k, j),( j,i) implies that for two centres i0 , j0 ∈ {1, . . . , n} of maximal distance di0 , j0 = dmax the entry (M E )k0 ,k#0 = (M E )k#0 ,k0 → ∞ for E → ∞, k0 = (i0 , j0 ) ∈ A

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   2 and k#0 = ( j0 , i0 ) ∈ A. Then the k0 -th diagonal element of M E β˜E is unbounded as E → ∞, since  (M E )k0 , (M E ),k0  (M E )k0 ,k#0 (M E )k#0 ,k0 = (M E )2k0 ,k# . (M2E )k0 ,k0 = 0

∈A

This is a contradiction to the fact that for the matrix M2E the diagonal    2 elements has to be bounded above by one, as M E β˜E is a PF-matrix whose PF-eigenvalue equals one. β˜E ∞ E→∞ β E

Thus it follows lim

= 1.

 

2.2.2 Quality of the Approximation We will now estimate the quality of the approximation of the escape rate β E by β˜E . Recall that β E was implicitly defined by λPF (L−F E +β E T E ) = 1. The ± and the phase ± resp. F inequalities (2.47) and (2.48) between the constants T E E space functions F E resp. T E , together with the monotonicity of topological pressure, show that β˜E−  β E  β˜E+ . The next lemma together with Lemma 2.18 prove Part (ii) of the Main Theorem.   Lemma 2.19 β˜E± = β˜E 1 + O(1/E) . Proof We are now going to express the bounds β˜E± for the escape rate β E in terms of the approximate escape rate β˜E . For this consider the Taylor expansion   β(δ) = β(0) dδ β(0), δ + O δ 2      of β(δ), implicitly defined by 1 = λPF M E β(δ), δ =: λPF β(δ ), δ . This implicit definition of β gives   . 0 = dδ λPF (β(δ ), δ ) = ∂β λPF (β, δ ) dβ(δ ) + ∂δ λPF (β(δ), δ ) By taking the derivative of the formula     λPF M E (β, δ ) = vlPF (β, δ ), M E (β, δ) vrPF (β, δ) with vlPF (β, δ), vrPF (β, δ) = 1 it follows

 l vPF (β, δ) , ∂δi M E (β, δ) vrPF (β, δ) −∂δi λPF (β, δ) =− l ∂δi β(δ1 , δ2 ) = ∂β λPF (β, δ) vPF (β, δ ) , ∂β M E (β, δ) vrPF (β, δ)

(i = 1, 2).

E from (1.17) we get, ∗ denoting the pointwise With the A × A matrix T product,    E ∗ M E β(0) , ∂β M E (β, δ )(β(0),0) = T

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    ∂δ1 M(β, δ)(β(0),0) = M E (β(0)) and ∂δ2 M(β, δ)(β(0),0) = β(0)M E β(0) . (     )  E ∗ M E (β(0) vr β(0), 0 one gets Setting τ E := vlPF β(0), 0 , T PF ∂δ1 β(δ )|(0) = −1/τ E

∂δ2 β(δ )|(0) = −β(0)/τ E   and finally β(δ ) = β(0) − (δ1 + β(0)δ2 ) + O δ2 . As the non-zero en√ E are bounded below by C/ E, we get τ −1 ∈ O(E1/2 ). From tries of T E δ F (E) ∈√O(E−1 ) and δT (E) ∈ O(E−3/2 ) (see (2.47) and (2.48)) and β(0) = β˜E  2 2E ln(E)/dmax we get the approximation β˜E± = β˜E 1 + O(1/E) .   ,

1 τE

Appendix: Proof of Proposition 1.5 We first prove an upper bound for the modulus of time delay for orbits not entering the interaction zone. We use the following estimate from [7], Thm. 6.5, valid for the energies E > Eth : If we have outgoing initial conditions, that is  (±p0 , q±0) ≡ x0 ±∈  E with r0 := q0   Rvir and ± q0 , p0   0, then for x± := p∞ , q∞ :=  ( p0 , q0 ) ∞

1  −ε −1  −1−ε − 2 E , q± p± . (2.50) ∞ − p0 = O r0 ∞ − q0 = O r 0 E for the phase space Here the symbol O means existence of a bound valid √ region indicated above. From (2.50) and p± ∞ = O ( E) we conclude that in the pericentric case q0 , p0  = 0

1  ± ±  ± ±  ± −ε − 2 ± . (2.51) q∞ , p∞ = q∞ − q0 , p∞ + q∞ , p∞ − p0 = O r0 E For the Kepler flow (1.8), on the other hand we use the Lagrange–Jacobi equation, followed by an inequality (which is valid since potential energy is bounded above by E): d Z∞ q∞ (t), p∞ (t) = 2E +  E > Eth  0. dt q∞ (t)

(2.52)

The time t0 needed to reach the pericentre of a Kepler hyperbola is thus uniquely defined by the implicit equation q∞ (t0 ), p∞ (t0 ) = 0. Together with (2.51) this shows for the Kepler hyperbolae asymptotic to our orbit that their times t0± are estimated by  ±   | q± ∞ , p∞ | ± (2.53) |t0 |  = O r0−ε E−3/2 . E For a suitable constant C > 0 this implies the estimate |AsTim E (x)|  C r0−ε E−3/2 .

(2.54)

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We now show the three assertions of the proposition: As.1: Equation √ 2.53 implies Assertion 1 (that is, the lower bound AsTim E > −C2 / E on time delay): If the -scattering orbit has a pericentre whose distance from the origin is larger than Rvir , then by the virial inequality (1.2) that pericentre is unique, and the total time delay is smaller than 2|t0 |, with t0 from (2.53). Otherwise it enters the interaction zone at a unique time, which we assume to equal zero w.l.o.g.. At that moment r0 = Rvir and q0 , p0   0 and so by (2.50)   −1  2Rvir q∞   Rvir + O R−ε vir E for Eth large enough. With (2.52) we get that the time spent √ by the Kepler orbit inside the interaction zone is smaller than 2Rvir / E. −ε As.2: We want to prove for  C3 := C Rvir and the small but strictly positive 3/2 that the measure of orbits with time delay larger times t ∈ 0, C3 /E than t is bounded by AsVol+E (t) < C1 t−2/ε E1−3/ε .   If AsTim E (x)  t for t ∈ 0, C3 /E3/2 , then by (2.54) the -orbit through x cannot have a pericentre ( p0 , q0 ) with distance from the origin r0 >  −1/ε . r˜0 (t) := tE3/2 /C So every such scattering orbit enters the Poincaré surface 

( p, q) ∈  E : q = r˜0 (t), q, p  < 0 . = Rvir As r˜0 (t)  r˜0 (C3 /E3/2 ) √  , points ( p, q) on that surface have momenta of order p = O E . So the symplectic volume of the surface is of order r˜0 (t)2 E. This implies Assertion 2. As.3: Proving by contradiction, we assume that the orbit through x with large time delay (AsTim E (x)  C3 /E3/2 ) does not enter the interaction zone. By our choice of C3 = C R−ε vir and (2.54) we get −3/2  AsTim E (x)  C r0−ε E−3/2 C R−ε vir E

so that its minimal distance from the origin r0  Rvir .

References 1. Baladi, V.: Positive Transfer Operators and Decay of Correlations. Advanced Series in Nonlinear Dynamics, vol. 16. World Scientific, Singapore (2000) 2. Bowen, R.: Equilibrium states and the ergodic theory of Anosov diffeomorphisms. Lecture Notes in Mathematics, vol. 470. Springer, Berlin (1975) 3. Castella, F., Jecko, Th., Knauf, A.: Semiclassical resolvent estimates for Schrödinger operators with Coulomb singularities, math-ph/0702009. Ann. Henri Poincaré 9, 775–815 (2008) ´ 4. Derezinski, J., Gérard, C.: Scattering Theory of Classical and Quantum N-Particle Systems. Texts and Monographs in Physics. Springer, Berlin (1997) 5. Katok, A., Hasselblatt, B.: Introduction to the Modern Theory of Dynamical Systems. Cambridge University Press, Cambridge (1998)

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6. Klein, M., Knauf, A.: Classical planar scattering by Coulombic potentials. Lecture Notes in Physics, vol. m13. Springer, Berlin (1992) 7. Knauf, A.: The n-centre problem of celestial mechanics for large energies. J. Eur. Math. Soc. 4, 1–114 (2002) 8. Knauf, A., Krapf, M.: The non-trapping degree of scattering, math-ph/0706.3124. Nonlinearity 21, 2023–2041 (2008) 9. Lalley, S.P.: Renewal theorems in symbolic dynamics, with applications to geodesic flows, noneuclidean tessellations and their fractal limits. Acta Math. 163, 1–55 (1989) 10. Liverani, C., Wojtkowski, M.P.: Generalization of the Hilbert metric to the space of positive definite matrices. Pac. J. Math. 166(2), 339–355 (1994) 11. Marsden, J.E., Ratiu, T.S.: Introduction to Mechanics and Symmetry. Springer, Berlin (1999) 12. Narnhofer, H.: Another definition for time delay. Phys. Rev. D 22, 2387–2390 (1980) 13. Walters, P.: An Introduction to Ergodic Theory, Graduate Texts in Mathematics 79. Springer, Berlin (1982)

Math Phys Anal Geom (2010) 13:191–204 DOI 10.1007/s11040-010-9074-y

Alternative Evaluation of a ln tan Integral Arising in Quantum Field Theory Mark W. Coffey

Received: 27 February 2009 / Accepted: 16 April 2010 / Published online: 7 May 2010 © Springer Science+Business Media B.V. 2010

Abstract A certain dilogarithmic integral I7 turns up in a number of contexts including Feynman diagram calculations, volumes of tetrahedra in hyperbolic geometry, knot theory, and conjectured relations in analytic number theory. We provide an alternative explicit evaluation of a parameterized family of integrals containing this particular case. By invoking the Bloch–Wigner form of the dilogarithm function, we produce an equivalent result, giving a third evaluation of I7 . We also alternatively formulate some conjectures which we pose in terms of values of the specific Clausen function Cl2 . Keywords Clausen function · Dilogarithm function · Hurwitz zeta function · Functional equation · Duplication formula · Triplication formula Mathematics Subject Classifications (2010) 33B30 · 11M35 · 11M06

1 Introduction The particular integral 24 I7 ≡ √ 7 7



π/2 π/3

   tan t + √7    ln  √  dt,  tan t − 7 

(1)

occurs in a number of contexts and has received significant attention in the last several years [3–5, 7]. This and related integrals arise in hyperbolic geometry,

M. W. Coffey (B) Department of Physics, Colorado School of Mines, Golden, CO 80401, USA e-mail: [email protected]

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M.W. Coffey

knot theory, and quantum field theory [7–9]. The integral (1) originates from reducing a higher dimensional Feynman diagram integral. Very recently [10] we obtained an explicit evaluation of (1) in terms of the specific Clausen function Cl2 . However, some questions remained open. For instance, there is the conjectured relation between a Dirichlet L series and I7 [7], ?

I7 = L−7 (2)  ∞   1 1 1 1 1 1 = + − + − − . (7n+1)2 (7n+2)2 (7n+3)2 (7n+4)2 (7n+5)2 (7n+6)2 n=0 (2) The ? here indicates that numerical verification to high precision has been performed but that no proof apparently existed, the approximate numerical value of I7 being I7  1.15192547054449104710169. The statement (2) is equivalent √ to the conjecture, with θ7 ≡ 2 tan−1 7,  ? 1 1 3Cl2 (θ7 ) − 3Cl2 (2θ7 ) + Cl2 (3θ7 ) = Z Q(√−7) 2 4 



 7 2π 4π 6π = Cl2 +Cl2 −Cl2 , 4 7 7 7 (3) relating triples of Clausen function values. Herein, we alternatively evaluate I7 directly in terms of the left side of (3). In addition, we present another evaluation of I7 , based upon a property of the Bloch–Wigner form of the dilogarithm function. We also present (Appendix 2) new series and an integral representation for the function Cl2 . That (3) had in fact been proved some time before [6, 21] we discuss below. We recall that the L series L−7 (s) has occurred in several places before, including hyperbolic geometry [20] and Dedekind sums of analytic number theory [2]. Let ζ Q(√− p) denote the Dedekind zeta function of an imaginary √ quadratic field Q( − p). Then indeed we have [2, 20, 22] ζ Q(√−7) (s) =

1 2

 m,n∈Z

1 (m2 + mn + 2n2 )s

(4)

6    ν ν ζ s, , 7 7 ν=1

(5)

(m,n)=(0,0)

= ζ (s)L−7 (s) = ζ (s)7

−s

 where ν7 is a Legendre symbol, ζ (s, a) is the Hurwitz zeta function, and ζ (s) = ζ (s, 1) is the Riemann zeta function.

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The series L−7 (s) is an example of a Dirichlet L function corresponding to a real character χk [here, modulo 7] with χk (k − 1) = −1. Such L functions, extendable to the whole complex plane, satisfy the functional equation [22]  sπ 1 (1 − s)L−k (1 − s). L−k (s) = (2π )s k−s+1/2 cos (6) π 2 Owing to the relation (1 − s)(s) = π/ sin(π s), this functional equation may also be written in the form πs (s)L−k (s). L−k (1 − s) = 2(2π )−s ks−1/2 sin (7) 2 Integral representations are known for these L-functions [11, 22]. From the functional equation (6) we find   k3/2 ∂ = (8) L−k (s) L−k (2). ∂s 4π s=−1 In turn, we have  √ ζ Q( (−1) = − −k)

k3/2 L−k (2), 48π

(9)

where we used ζ (−1) = −1/12 and L−k (−1) = 0. Concerning the much publicized integral (1), we very recently found that relation (3) was effectively proved in [21]. There, Zagier introduces the functions

 x 1 4 dt, (10) A(x) = ln 2 1 + t2 0 1+t and ∞

(θ) =

1  sin 2nθ =− 2 n=1 n2



θ

ln |2 sin t|dt,

(11)

0

to evaluate ζ Q(−√7) (2). Given the relations (θ ) = 12 Cl2 (2θ), A(x) = √ √ 2 (cot−1 x) = Cl2 (2 cot−1 x), tan θ7 = − 7/3, and tan(3θ7 /2) = 7/5, (5) and (6) of [21] give the equivalent of (3). Zagier’s method was to use the interpretation of the value ζ K (2) as the volume of a hyperbolic manifold. We have Proposition 1 We have  4  I7 = √ 3Cl2 (θ7 ) − 3Cl2 (2θ7 ) + Cl2 (3θ7 ) . 7 7 In fact, we treat integrals  I(a) ≡

π/2 π/3

   tan t + a    dt, ln  tan t − a 

(12)

(13)

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and more general ones with varying limits. For (13), we assume that π/3 < ϕ = tan−1 a < π/2. These other integrals permit us to explicitly write other conjectures directly in terms of linear combinations of specific Clausen function values. The Clausen function Cl2 can be defined by (e.g., [16, 18]) 

 θ   1  t  x sin θ dx  ln 2 sin  dt = tan−1 Cl2 (θ) ≡ − (14) 2 1 − x cos θ x 0 0 

1

= − sin θ 0

∞

 sin(nθ) ln x dx = . 2 x − 2x cos θ + 1 n2 n=1

(15)

When θ is a rational multiple of π it is known that Cl2 (θ ) may be written in terms of the trigamma and sine functions [12, 15]. This Clausen function is odd and periodic, Cl2 (θ) = −Cl2 (−θ), and Cl2 (θ) = Cl2 (θ + 2π ). It also satisfies the duplication

triplication

1 Cl2 (2θ) = Cl2 (θ) − Cl2 (π − θ), 2

(16)





2π 4π 1 Cl2 (3θ) = Cl2 (θ) + Cl2 θ + + Cl2 θ + , 3 3 3

(17)

and quadriplication

 1 π 3π Cl2 (4θ) = Cl2 (θ) + Cl2 θ + + Cl2 (θ + π ) + Cl2 θ + , 4 2 2

(18)

formulas, as well as a more general multiplication formula [16]. We recall the specific relation

6  2π j = 0, (19) Cl2 7 j=1 that arises as a special case of [16] (pp. 95, 253)

n−1  2π Cl2 j = 0. n j=1

(20)

In (19), pairwise cancellation occurs, as Cl2 (θ) = −Cl2 (2π − θ). Further information on the special functions that we employ may readily be found elsewhere [11, 17–19]. In particular, with Li2 (z) =

∞  zk , k2

|z| ≤ 1,

(21)

ln(1 − t) dt, t

(22)

k=1

or

 Li2 (z) = − 0

z

Alternative Evaluation of a ln tan Integral Arising

195

the dilogarithm function, we have the relation Li2 (eiθ ) =

π2 1 − θ(2π − θ) + iCl2 (θ ), 6 4

0 ≤ θ ≤ 2π.

(23)

We omit discussion of further relations between the Clausen function Cl2 and the dilogarithm function. Proof of Proposition 1 For the proof of Proposition 1 we repeatedly rely on [16] (pp. 227, 272)  θ 1 1 1 ln(tan θ +tan ϕ)dθ = −θ ln(cos ϕ)− Cl2 (2θ +2ϕ)+ Cl2 (2ϕ)− Cl2 (π −2θ). 2 2 2 0 (24) We may split the integral in (13), writing



 ϕ  π/2 a + tan t tan t + a ln ln I(a) = dt + dt a − tan t tan t − a π/3 ϕ  =

π/2

π/3

 ln(a + tan t)dt −

ϕ π/3

 ln(a − tan t)dt −

π/2 ϕ

ln(tan t − a)dt.

(25)

By the use of (24) we obtain for the first integral on the right side of (25) 

  π/2 1 π 2π Cl2 + 2ϕ − Cl2 (π + 2ϕ) ln(a + tan t)dt = − ln cos ϕ + 6 2 3 π/3  π 1 + Cl2 , (26a) 2 3 the second integral,     ϕ  π π 1 ln(a − tan t)dt = − ϕ − ln cos ϕ − Cl2 2 ϕ − 3 2 3 π/3    1 π + Cl2 − Cl2 (π − 2ϕ) , 2 3 and the third integral,  π/2 π ln(tan t − a)dt = − − ϕ ln cos ϕ = − cot−1 a ln cos ϕ. 2 ϕ

(26b)

(26c)

This latter integral is readily obtained from (24) by taking a → −a so that simply tan ϕ → − tan ϕ. Then per (25) we have 

 1 2π 2π I(a) = Cl2 2ϕ + + Cl2 2ϕ − − Cl2 (π + 2ϕ). (27) 2 3 3

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Then we apply both the duplication formula (16) and the triplication formula (17) wherein Cl2 (θ + 4π/3) = Cl2 (θ − 2π/3) as Cl2 (θ ) = Cl2 (θ − 2π ) by the 2π -periodicity of Cl2 . We find I(a) = When ϕ = tan−1

 1 Cl2 (6ϕ) − 3Cl2 (4ϕ) + 3Cl2 (2ϕ) . 6

(28)

√ 7, the case (12) follows.



2 Discussion and Other Results We next present some reference integrals. We then apply them to write expressions for combinations of the integrals    (n+1)π/24  tan t + √7    In ≡ (29) ln  √  dt,  tan t − 7  nπ/24 where n ≥ 0 is an integer. We supplement (24) with  y  1 ln(a − tan t)dt = −(y − x) ln cos ϕ + Cl2 [2(ϕ − y)] − Cl2 [2(ϕ − x)] 2 x −

 1 Cl2 (π − 2y) − Cl2 (π − 2x) , 2

(30)

where a = tan ϕ. We also have   y   tan t + a     dt = 1 Cl2 (2x + 2ϕ) − Cl2 (2x − 2ϕ) + Cl2 (2y − 2ϕ) ln   tan t − a 2 x  − Cl2 (2y + 2ϕ) , (31) with x < ϕ = tan−1 a < y. We now write expressions for the linear combinations ?

C1 ≡ −2(I2 + I3 + I4 + I5 ) + I8 + I9 − (I10 + I11 ) = 0,

(32)

and ?

C2 ≡ I2 + 3(I3 + I4 + I5 ) + 2(I6 + I7 ) − 3I8 − I9 = 0.

(33)

These relations have been detected with further PSLQ computations [5]. A similar conjecture for integrals In with increments nπ/60 has also been written [4] (p. 508). The latter linear combination may also be expressed in terms of Cl2 values, but here we concentrate on (32) and (33). We decompose the left side of (32) as indicated, and find for these contributions π π π 2(I2 + I3 + I4 + I5 ) = Cl2 + 2ϕ − Cl2 − 2ϕ + Cl2 − 2ϕ 6 6 2 π −Cl2 + 2ϕ , (34) 2

Alternative Evaluation of a ln tan Integral Arising

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√ where ϕ = tan−1 7,





2π 3π 2π 3π 2I8 = Cl2 +2ϕ −Cl2 +2ϕ +Cl2 2ϕ− −Cl2 2ϕ − , (35) 3 4 3 4 2I9 =Cl2 and I10 + I11







3π 3π 5π 5π +2ϕ −Cl2 −2ϕ +Cl2 −2ϕ −Cl2 +2ϕ , (36) 4 4 6 6





 1 5π 5π = −Cl2 (π + 2ϕ) + Cl2 + 2ϕ − Cl2 − 2ϕ . 2 6 6

(37)

Therefore, we obtain π π π π C1 = −Cl2 + 2ϕ + Cl2 − 2ϕ − Cl2 − 2ϕ + Cl2 + 2ϕ 6 6 2 2 



2π 2π 5π 1 + 2ϕ + Cl2 2ϕ − + Cl2 − 2ϕ × Cl2 2 3 3 6

5π + 2ϕ + Cl2 (π + 2ϕ). −Cl2 (38) 6 For the combination C2 we have π π π π 2I2 = Cl2 + 2ϕ − Cl2 − 2ϕ + Cl2 − 2ϕ − Cl2 + 2ϕ , (39) 6 6 4 4

− 2(I6 + I7 ) = Cl2



π π 2π 2π +2ϕ −Cl2 −2ϕ +Cl2 −2ϕ −Cl2 +2ϕ , 3 3 2 2 (40)

and 2(I3 + I4 + I5 ) = Cl2

π π π +2ϕ −Cl2 −2ϕ +Cl2 −2ϕ −Cl2 +2ϕ . 4 4 2 2 (41)

π

Therefore, we find π π π π 2C2 = Cl2 + 2ϕ − Cl2 − 2ϕ + Cl2 − 2ϕ − Cl2 + 2ϕ 6 6 2 2







2π 2π 5π 5π +2ϕ −5Cl2 2ϕ − −Cl2 −2ϕ +Cl2 +2ϕ −5Cl2 3 3 6 6

  π π 3π 3π +2 Cl2 +2ϕ −Cl2 −2ϕ +Cl2 +2ϕ +Cl2 2ϕ − . 4 4 4 4 (42)

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By a combination of the quadriplication formula (18) and the duplication formula (16) we may write   1 π π 1 Cl2 (4θ) = Cl2 θ + + Cl2 θ − + Cl2 (2θ). 4 2 2 2

(43)

This enables other expressions for C1 and C2 . Similarly, one may use the 6- and 12-fold multiplication formulas. In regard to the combination on the right side of (3), we comment on an observation given previously [10]. We have that ± sin(2π/7), ± sin(4π/7), and ± sin(6π/7) are the nonzero roots of the Chebyshev polynomial T7 (x). Indeed, if we write the cubic polynomials √ √





4π 6π 2π 7 2 7 3 p1 (x) = x − sin x − sin x + sin =x − x + , (44a) 7 7 7 2 8 and

6π p2 (x) = x − sin 7



2π x + sin 7



4π x + sin 7



√ √ 7 2 7 =x + x − , (44b) 2 8 3

we then have the factorization p1 (x) p2 (x) = T7 (x)/64x. This invites questions as to whether scaled versions of these or other Chebyshev polynomials could be useful in developing identities underlying (3), (32), (33), or the like. Given the close relation of the Clausen function Cl2 and the dilogarithm function, one wonders if a set of ladder relations for the latter may be carried over to explain (3) and relations amongst the integrals In . In developing ladder relations, cyclotomic equations for the base have proven very useful. It would be of interest to see if Cl2 relations with θ7 could be discovered in this way. We remark on using Kummer’s relation [16] (pp. 107, 254) to rewrite the right side of (3) in terms of the dilogarithm of complex argument. We have  1 7 Z Q(√−7) = Im Li2 (Reiφ ) − b ln R , 4 2

(45)

where R=

tan b . sin φ + tan b cos φ

(46)

Here, we may take φ = π/7 and b = 2π/7, or vice versa. Then by Proposition 2 of [10] we have the integral representation √





2π 4π 6π 7 ? I7 = Cl2 + Cl2 − Cl2 2 7 7 7 π  ∞ ln y dy , (47) = 2 sin 2 7 d y − 2y cos(π/7) + 1 where d = [2 cos(π/7) − 1]−1 .

Alternative Evaluation of a ln tan Integral Arising

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Finally, we use relations from [18] (Appendix 1) and [20] to write a third evaluation of the integral I7 . For this we introduce the angle θ75 ≡ √ 2 tan−1 ( 7/5) and the Bloch–Wigner dilogarithm1 D(z) = Im[Li2 (z)] + arg(1 − z) ln |z|,

(48)

for which we have [18] (p. 246) 1 (49) [Cl2 (2θ) + Cl2 (2ω) − Cl2 (2θ + 2ω)], 2 where θ = arg z and ω = arg (1 − z¯ ). We note the interpretation that for z ∈ C, the volume of the asymptotic simplex with vertices 0, 1, z, and ∞ in 3dimensional hyperbolic space is given by |D(z)| [18] (p. 271). We then rewrite the expression ([18], p. 384 or [20], p. 246)    √  √  4π 2 −1 + i 7 1+i 7 √ ζ Q( −7) (2) = ζ (2)L−7 (2) = √ 2D +D . (50) 2 4 21 7 D(z) =

We apply (49), giving

   √  √  8 −1 + i 7 1+i 7 I7 = L−7 (2) = √ 2D +D 2 4 7 7 ?

 4  (51) = √ 4Cl2 (π − θ7 ) − Cl2 (θ7 ) + Cl2 (θ75 ) + Cl2 (θ7 − θ75 ) . 7 7 √ In the case of D[(1 + i 7)/2] we used the duplication formula√(16). In contrast to (51), the expression in [10] for I7 involves θ+ ≡ tan−1 ( 7/3). With the various analytic evaluations now known for I7 or L−7 (2), we have enlarged the set of possible relations amongst Cl2 values. From (12), (16), and (51) we obtain the conjecture ?

Cl2 (3θ7 ) − Cl2 (2θ7 ) = Cl2 (θ75 ) + Cl2 (θ7 − θ75 ).

(52)

In fact, we have θ7 − θ75 = 2θ+ , and we conclude by proving (52), and thereby (51). We quickly show that both Cl2 (3θ7 ) = Cl2 (θ75 )

(53)

Cl2 (2θ7 ) = −Cl2 (θ7 − θ75 ),

(54)

and for we have 3θ7 − 2π = θ75 and θ7 − π = −θ+ . The latter relations require nothing more than the identity tan(x/2) = sin x/(1 + cos x). We have similarly found many other angular pairs (θ1 , θ2 ) satisfying 3θ1 − 2π = ±θ2 , immediately giving Cl2 (3θ1 ) = ±Cl2 (θ2 ). As these may be useful

1 Apparently in (11.22) in [18], the lower limit of the integral for

with our (22)

D(z) is intended to be 0, consistent

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M.W. Coffey

elsewhere [7, 18], we record several of them in Appendix 1. We also relegate to this Appendix a possibly new log trigonometric integral in terms of Cl2 . In Appendix 2, we develop new series and integral representations of the Clausen function. Acknowledgement

I thank J. Zhao for useful discussion.

Appendix 1 We let θk ≡ 2 tan−1

√ √ k, and θk, j ≡ 2 tan−1 k/j. We find the relations 3θ2 − 2π = −θ2,5 ,

3θ11 − 2π = −θ11,4 , and

3θ5 − 2π = θ5,7 ,

−1

3θ13 − 2π = −2 tan

(55)

 √  5 13 , 19

(56)

 √   √  8 91 2 91 , 3θ91,5 − 2π = 2 tan−1 , 3θ91,3 − 2π = −2 tan−1 99 155

3θ91,7 − 2π = −θ91,28 . (57) √ √ √ With θ32 √ ≡ 2 tan−1 3/2, θ53 ≡ 2 tan−1 5/3, θ133 ≡ 2 tan−1 13/3, θ73 ≡ 2 tan−1 7/3, we have       3 5 −1 3 −1 1 , 3θ53 − 2π = −2 tan , (58) 3θ32 − 2π = −2 tan 7 2 3 3 and 3θ133 − 2π = 2 tan

−1

   1 13 , 9 3

−1

3θ73 − 2π = −2 tan

   1 7 . 9 3

(59)

Based upon the trigonometric identity 3 + 4 cos θ + cos 2θ = 2(1 + cos θ)2 , we have found the integral  x ln(3 + 4 cos θ + cos 2θ)dθ = −x ln 2 + 4Cl2 (π − x), 0 ≤ x ≤ π. (60) 0

This provides an integral expression for the Catalan constant G =  obviously k (−1) /(2k + 1)2 = Cl2 (π/2) when x = π/2. k≥0 The function Cl2 (t) for t ∈ (0, π ) has its only maximum at t = π/3, when Cl2 (π/3)  1.014941606409653625021. We mention that near this value Cl2 has a fixed point, Cl2 (y) = y for y  1.01447193895251725798414.

Alternative Evaluation of a ln tan Integral Arising

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Related to the equality of expressions (12) and (51) for I7 we have the relation    √  √  1 − 3i 7 1 + 3i 7 6 Li2 + Li2 = 3(π − 2θ+ )2 − π 2 8 8 √ = 3(θ7 − θ+ )2 − π 2 = 3[π − tan−1 (3 7)]2 − π 2 . (61) Such relations follow readily from (21) as we have 6[Li2 (eiθ ) + Li2 (e−iθ )] = 2π 2 + 3θ 2 ,

0 ≤ θ ≤ 2π.

(62)

Appendix 2 We have Proposition 2 We have for θ < π/n and n ≥ 1 an integer    ∞ 1 1 ζ (2 j) (n2 j − 1) 2 j+1 − θ ln n. Cl2 (2nθ) − Cl2 (2θ) = θ 2 n jπ 2 j (2 j + 1) j=1

(63)

This result gives several Corollaries, including Corollary 1

 n−1 ∞ 1 2π ζ (2 j) (n2 j − 1) 2 j+1 Cl2 2θ + − θ ln n, k = θ 2 n jπ 2 j (2 j + 1) j=1

(64)

k=1

Corollary 2    2θ ∞ 1 dx 1 1 Cl2 (2nθ) − Cl2(2θ) = −θ ln n + [sinh(nx) − n sinh x] π x/θ , 2 2 n n 0 x (e − 1) (65) Corollary 3   1 1 Cl2 (2nθ) − Cl2 (2θ) 2 n



  2 θ θ π − n2 θ 2 −1 nθ −1 = −θ ln n + 2nθ tanh − 2θ tanh + π ln 2π π π π2 − θ2

  ∞ θ nθ 1 + 2π tanh−1 − tanh−1 P1 (x)dx. (66) π x n π x 1 In the last equation, P1 (x) = x − [x] − 1/2 is the f irst periodized Bernoulli polynomial.

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Proof The Proposition is based upon the relation [1] (p. 75)

 ∞ n sin x ζ (2 j) 2 j ln = (n − 1)x2 j, |x| < π/n, 2j sin nx jπ j=1

(67)

wherein we have used the relation between ζ (2 j) and the Bernoulli numbers B2 j. (For more details, see the end of this Appendix.) With the series of (67) being boundedly convergent, we may integrate term-by-term over any finite interval avoiding the singularity at x = π/n. Doing so, integrating over [0, θ ], and using the first integral representation for Cl2 on the right side of (14) gives (63). Corollary 1 follows from the multiplication formula for Cl2 [16] (pp. 94, 253). Corollary 2 uses a standard integral representation of the Riemann zeta function. With the interchange of summation and integration, with the integral being absolutely convergent, the Corollary follows. Corollary 3 uses the representation for Re s > −1,  ∞ 1 1 P1 (x) ζ (s) = + −s dx. (68) 2 s−1 xs+1 1 Again, the interchange of summation and integration is employed.



Remark In connection with (67) we may note the relation with the Chebyshev polynomials U n of the second kind [13, 14] (p. 1032), U n−1 (cos φ) =

sin nφ . sin φ

(69)

For θ = π/4 and other values, Proposition 2 gives many relations involving the Catalan constant G. More generally, for θ a rational multiple of π , the results are expressible in terms of ψ  , the trigamma function [12, 15]. If we let r(θ, n) ≡

∞  ζ (2 j) (n2 j − 1) j=1

jπ 2 j (2 j + 1)

θ 2 j+1 ,

θ ≤ π/n,

we may write several simple examples:   π 1 √ ψ  (1/3) ,2 = π ( 3π + 6 ln 2) − √ , r 3 18 4 3π r

  ψ  (1/3) 1 √ ,3 = π ( 3π + 9 ln 3) − √ , 3 27 6 3π

π

r

r

π

(70)

(71a)

(71b)

1 π , 2 = − G + ln 2, 4 2 4

(72a)

2 π , 3 = − G + ln 3, 4 3 4

(72b)

π

Alternative Evaluation of a ln tan Integral Arising

r and

203

π

1 π , 4 = − G + ln 2, 4 2 2

(72c)

 ψ  (1/3) 1 √ r ,2 = π (2 3π + 9 ln 2) − √ , 6 54 6 3π π







 1 √ ψ  (1/3) ,3 = π ( 3π + 3 ln 3) − √ , r 6 18 4 3π

π

 7ψ  (1/3) 1 √ r ,4 = π (7 3π + 36 ln 2) − , √ 6 108 24 3π r

π



π



6



,5 = π

(73b)



1 √ (2 3π + 5 ln 5) − 30

 √  3ψ (1/3) , 10π

 ψ  (1/3) 1 √ . r ,6 = π ( 3π + 3 ln 2 + 3 ln 3) − √ 6 18 4 3π π

(73a)

(73c)

(73d)



(73e)

Finally, we supply a derivation of (67). We have

d n sin x ln = cot x − n cot nx dx sin nx =

∞  22k |B2k | k=1

(2k)!

(n2k − 1)x2k−1 ,

n|x| < 1, π

(74)

where we used a series representation for cot [13, 14] (p. 35). Since 2ζ (2k) 22k |B2k | , = (2k)! π 2k

(75)

we have cot x − n cot nx = 2

∞  ζ (2k) k=1

π 2k

(n2k − 1)x2k−1 .

(76)

Integrating both sides of this relation gives (67).

References 1. Abramowitz, M., Stegun, I.A.: Handbook of Mathematical Functions. National Bureau of Standards (1972) 2. Almkvist, G.: Asymptotic formulas and generalized Dedekind sums. Exp. Math. 7, 343–359 (1994) 3. Bailey, D.H. et al.: Experimental Mathematics in Action. A. K. Peters, Wellesley (2007) 4. Bailey, D.H., Borwein, J.M.: Experimental mathematics: examples, methods and implications. Not. Am. Math. Soc. 52, 502–514 (2005)

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5. Bailey, D.H., Borwein, J.M.: Computer-assisted discovery and proof. In: Amdeberhan, T., Moll, V. (eds.) Tapas in Experimental Mathematics, Contemp. Math., Am. Math. Soc., vol. 457, pp. 21–52. http://crd.lbl.gov/∼dhbailey/dhbpapers/comp-disc-proof.pdf (2008) 6. Bailey, D.H., Borwein, J.M., Broadhurst, D., Zudilin, W.: Experimental mathematics and mathematical physics. In: Gems in Experimental Mathematics, Contemp. Math., Am. Math. Soc., vol. 517, pp. 41–58. http://www.carma.newcastle.edu.au/∼jb616/bbbz09.pdf (2009) 7. Borwein, J.M., Broadhurst, D.J.: Determination of rational Dedekind-zeta invariants of hyperbolic manifolds and Feynman knots and links. arxiv:hep-th/9811173 (1998) 8. Broadhurst, D.J.: Massive 3-loop Feynman diagrams reducible to SC∗ primitives of algebras of the sixth root of unity. Eur. Phys. J. C 8, 311–333 (1999) 9. Broadhurst, D.J.: Solving differential equations for 3-loop diagrams: relation to hyperbolic geometry and knot theory. arxiv/hep-th/9806174v2 (1998) 10. Coffey, M.W.: Evaluation of a ln tan integral arising in quantum field theory. J. Math. Phys. 49, 093508-1-15 (2008) 11. Coffey, M.W.: On a 3-dimensional symmetric Ising tetrahedron and contributions to the theory of the dilogarithm and Clausen functions. J. Math. Phys. 49, 043510-1-32 (2008) 12. de Doelder, P.J.: On the Clausen integral Cl2 (θ) and a related integral. J. Comput. Appl. Math. 11, 325–330 (1984) 13. Gradshteyn, I.S., Ryzhik, I.M.: Table of Integrals, Series, and Products, Academic Press, New York (1980) 14. Gradshteyn, I.S., Ryzhik, I.M.: Table of Integrals, Series, and Products, 7th edn. In: Jeffrey, A., Zwillinger, D. (eds.) Academic, New York (2007) 15. Grosjean, C.C.: Formulae concerning the computation of the Clausen integral Cl2 (θ ). J. Comput. Appl. Math. 11, 331–342 (1984) 16. Lewin, L.: Dilogarithms and Associated Functions. Macdonald, London (1958) 17. Lewin, L.: Polylogarithms and Associated Functions. North Holland, Amsterdam (1981) 18. Lewin, L. (ed.): Structural Properties of Polylogarithms. American Mathematical Society (1991) 19. Srivastava, H.M., Choi, J.: Series Associated with the Zeta and Related Functions. Kluwer, Boston (2001) 20. Zagier, D.: The dilogarithm function in geometry and number theory. In: Number Theory and Related Topics, Bombay Tata Institute of Fundamental Research, pp. 231–249 (1988) 21. Zagier, D.: Hyperbolic manifolds and special values of Dedekind zeta-functions. Inv. Math. 83, 285–301 (1986) 22. Zucker, I.J., Robertson, M.M.: Some properties of Dirichlet L-series. J. Phys. A 9, 1207–1214 (1976)

Math Phys Anal Geom (2010) 13:205–217 DOI 10.1007/s11040-010-9075-x

Wegner-type Bounds for a Two-particle Lattice Model with a Generic “Rough” Quasi-periodic Potential Martin Gaume

Received: 24 June 2009 / Accepted: 29 April 2010 / Published online: 19 June 2010 © Springer Science+Business Media B.V. 2010

Abstract In this paper, we consider a class of two-particle tight-binding Hamiltonians, describing pairs of interacting quantum particles on the lattice Zd , d ≥ 1, subject to a common external potential V(x) which we assume quasiperiodic and depending on auxiliary parameters. Such parametric families of ergodic deterministic potentials (“grands ensembles”) have been introduced earlier in Chulaevsky (2007), in the framework of single-particle lattice systems, where it was proved that a non-uniform analog of the Wegner bound holds true for a class of quasi-periodic grands ensembles. Using the approach proposed in Chulaevsky and Suhov (Commun Math Phys 283(2):479–489, 2008), we establish volume-dependent Wegner-type bounds for a class of quasi-periodic two-particle lattice systems with a non-random short-range interaction. Keywords Schrödinger operator · Wegner bounds · Two-particle · Quasi-periodic Mathematics Subject Classifications (2010) 82B44 · 82B20 · 81Q10 · 47B80 · 47N50

1 Introduction Let Zd be the standard d-dimensional cubic lattice. If x = (x1 , . . . , xd ) ∈ Zd , we denote |x| = max{|x1 |, . . . , |xd |}. Let  ≡ Tν = Rν /Zν be the ν-dimensional

M. Gaume (B) Institut de Mathématiques de Jussieu, Université Paris Diderot, Site Chevaleret, case 7012, 75205 Paris Cedex 13, France e-mail: [email protected]

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unit torus equipped with the Haar measure P. Let also  be some auxiliary probability space. In this paper we consider a random operator H(ω; θ) acting in 2 (Z2d ), of the form H(ω; θ) =

2  

  j + V(x j; ω; θ) + U(x1 , x2 ),

(1.1)

j=1

where xl ∈ Zd , l = 1, 2, ω ∈ , θ ∈ , (1 f )(x1 , x2 ) =



f (x, x2 ),

x∈Zd |x−x1 |=1

(2 f )(x1 , x2 ) =



f (x1 , x),

x∈Zd |x−x2 |=1

and V(xl ; ω; θ), U(x1 , x2 ) are multiplication operators. The function V : Zd × Tν ×  → R is defined as follows. For every n ≥ 1, we consider of the unit cir the partition  i −n cle T1 = R1 /Z1 into the 2n intervals In,i = i−1 of size 2 , , for i = 1, . . . , 2n . n n 2 2   The family Cn = Cn,k 1≤k≤2νn of all Cartesian products In,i1 × · · · × In,iν , is a partition of the torus Tν into a family of cubes of sidelength 2−n . Further, let ϕn,k = 1Cn,k be the indicator function of the cube Cn,k . Next, we introduce a parametric family of functions on the torus v(ω; θ) =

∞  n=1

an

 Kn

θn,k ϕn,k (ω) ,

(1.2)

k=1

where the θn,k ’s, for n ≥ 1 and 1 ≤ k ≤ Kn < ∞, are i.i.d. random variables on the auxiliary probability space , with uniform distribution in [0, 1]. We identify  with the set of all samples {θn,k }n≥1, 1≤k≤Kn . Assumptions Let {T x }x∈Zd be an action of the additive group Zd on the torus Tν , which we assume to satisfy a Diophantine condition of the form (for all x ∈ Zd \ {0}, ω ∈ Tν ) dist(ω, T x ω) ≥

C1 , x B

(1.3)

for some constants 0 < B < ∞ and 0 < C1 < ∞. Concerning the rate of decay of the coefficients an in the expansion (1.2), we assume that for all n ≥ 1, c1 c2 ≤ |an | ≤ κ nM n

(1.4)

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for some constants c1 , c2 ∈ (0, +∞) and 1 < κ ≤ M < ∞. The upper bound on |an | guarantees the convergence of the above expansion, while the lower bound is required for our method. Finally, we set V(x; ω; θ) = v(T x ω; θ)

(1.5)

for x ∈ Zd , ω ∈  ≡ Tν , and θ ∈ . Following [8], we call such a parametric family of functions on  a “grand ensemble” of “randelette type”. Expansions of the form (1.2) will be called “randelette expansions”.

2 Wegner-type Bounds. Main Results 2.1 The One-particle Case In the spectral theory of random operators, e.g., Anderson tight-binding Hamiltonians of the form H(ω) =  + V(x; ω),

x ∈ Zd ,

(2.1)

with an i.i.d. random potential V(x; ω), an important role is played by eigenvalue concentration bounds. The first fairly general result of such kind was obtained by F. Wegner [23], so they are usually called Wegner-type bounds. Namely, consider a lattice Hamiltonian of the form (2.1), where the random variables V(x; · ) are identically distributed with a common probability cumulative distribution function (CDF) F(s) = FV (s), having a bounded probability density p(s) = pV (s):  pV ∞ < ∞.

(2.2)

Given a finite lattice cube  L (u) ⊂ Zd of arbitrary center u ∈ Zd and of sidelength 2L + 1, L ≥ 0, we consider the restriction H (ω) of the Hamiltonian (2.1) on  ≡  L (u) with Dirichlet boundary conditions. Further, let (H (ω)) = {Ej , j = 1, . . . , || = (2L + 1)d } be the (random) spectrum of the finite-volume Hamiltonian H (ω), i.e., the set of its (random) eigenvalues counted with multiplicities. Then for all E ∈ R and all ε ∈ (0, 1), we have       P dist (H (ω)), E ≤ ε = P there exists Ej ∈ [E − ε, E + ε] ≤ || ·  pV ∞ · ε.

(2.3)

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The presence of the factor || ≡ card  at the RHS of (2.3) allows to prove the absolute continuity of the so-called (limiting) density of states

 card j : Ej ≤ E N(E) = lim , L→∞ | L (0)| where the limit exists with probability one and is non-random (cf. [6, 13, 19] and references therein). In other words, the absolute continuity of the marginal cumulative distribution function FV (s) implies that N(E) also admits a density, called “density of states”, ρ(E), so that E ρ(E ) dE . N(E) = −∞

Many generalizations of the Wegner bound (2.3) are well-known by now. In particular, analogs of the Wegner bound can be given in cases where the marginal distribution of the random potential field V(x; ω) does not admit a density, but the CDF FV (s) is Hölder-continuous; see, e.g., [21, 22]. On the other hand, in the case of sufficiently regular marginal distributions optimal eigenvalue concentration bounds are known; see [10]. The main difficulty to extend the Wegner bound to deterministic (e.g., quasi-periodic) potentials is that all known “probabilistic” methods applicable to the eigenvalue concentration problem require a greater freedom in varying individual values of the potential. For example, the usual Wegner bound requires the random variables V(x; · ) to be independent. It is wellunderstood by now that the independence requirement can be replaced by that of asymptotic decay of dependence. In [9], even a greater degree of correlation was allowed. Still, an ensemble of quasi-periodic potentials (ω ∈ T1 = R1 /Z1 , α ∈ R \ Q) V(x; ω) = v(ω + xα), is too “rigid” and does not allow a direct application of probabilistic methods to the eigenvalue concentration problem. A reader interested in eigenvalue concentration bounds and, in particular, in the analysis of regularity of the density of states, can find an extensive bibliography in references [1, 5, 10–12, 20], as well in the above mentioned monographs [6, 13, 19, 22]. In this connection, we mention an alternative powerful approach developed earlier by J. Bourgain, M. Goldstein and W. Schlag (see, e.g., [2–4]). Unfortunately, their approach requires the function v : Tν → R to be analytic, which greatly limits the applications. 2.2 The Two-particle Case This is why we use an approach proposed in [8]. Instead of an individual ergodic ensemble of quasi-periodic potentials {V( · ; ω), ω ∈ Tν }, a parametric family {V( · ; ω; θ)}ω∈Tν , θ ∈ labelled by points of a specially constructed

Wegner-type Bounds for a Two-particle Lattice Model

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parameter space  is considered. While such an ensemble, called “grand ensemble” in [8], is not ergodic, it turns out that some analog of Wegner-type bounds can be established for generic parameter values θ ∈ . A drawback of this method is that the obtained eigenvalue concentration bounds are non-uniform in size, unlike the conventional Wegner-type bounds. Before stating the theorems, let us give some definitions: Definition 1 Let u = (u1 , u2 ) ∈ Zd × Zd . We denote by  L (u) =  L (u1 ) ×  L (u2 ) ⊂ Z2d a “two-particle box”, or “two-particle lattice cube”. We define its “total shadow” by Π L (u) :=  L (u1 ) ∪  L (u2 ) ⊂ Zd . Definition 2 Let  L (u) be a two-particle box. We denote by HL (u) (ω; θ) the “finite-volume operator” obtained by restriction of H(ω; θ) on  L (u) with Dirichlet boundary conditions. Definition 3 The spectrum of the finite-volume operator HL (u) (ω; θ) is   (2.4) u,L (ω; θ) = Ej L (u) 1≤ j≤|L (u)| , where the Ej L (u) ’s are the eigenvalues counted with multiplicities. The main results of the present paper are the following two statements. Theorem 1 Consider a two-particle box  L (u) ⊂ Z2d . Let (v, r) ∈ Zd × R+ such that Π L (u) ⊂  Lr (v). Given a positive number b ∈ (0, +∞), there exists (i) a constant C, which only depends on C1 , B, c1 , and d, (ii) a subset  L ⊂  of measure μ( L ) ≥ 1 − L−b , such that for any θ ∈  L the following inequality holds: P{ω ∈ Tν : dist[ u,L (ω; θ)), E] ≤ ε} ≤ C L M+b +3d+r ε.

Theorem 2 Consider two two-particle lattice cubes  L (u ),  L (u

) ⊂ Z2d and (v, r) ∈ Zd × R+ , r > 1 such that u − u  ≤ Lr , min{u − u , u − S(u )} > 8L, and Π L (u) ∪ Π L (u ) ⊂  Lr (v).

(2.5)

Here S : Z → Z is the symmetry (u1 , u2 ) → (u2 , u1 ) exchanging the coordinates of two particles. 2d

2d

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Given a positive number b ∈ (0, +∞), there exists (i) a constant C , which only depends on C1 , B, c1 and d, (ii) a subset  L ⊂  of measure μ( L ) ≥ 1 − L−b , such that for any θ ∈  L the following inequality holds:   P ω ∈ Tν : dist[ u,L (ω; θ), u ,L (ω; θ)] ≤ ε ≤ C L M+b +r+5d ε. For the proofs of Theorems 1 and 2, see Subsections 4.4 and 4.5, respectively. Remark 1 To explain the role of condition (2.5), recall that the potential energy of a two-particle system is invariant under the symmetry S : (u1 , u2 ) → (u2 , u1 ). Obviously, Π L (u1 , u2 ) = Π L (u2 , u1 ), i.e., Π L (u) = Π L (S(u)), so that the samples {V(x; ω), x ∈ Π L (u)} and {V(x; ω), x ∈ Π L (S(u))} are identical. As a consequence, the two Hamiltonians H( L (u)) and H( L (S(u))) have identical spectra. Further discussion on this condition can be found in [9]. Naturally, the above upper bounds are useful only for sufficiently small ε, when the RHS is smaller than 1.

3 Diagonally Monotone Operator Families Here we briefly recall some notions and results from [21, 22] and their extensions to two-particle systems proposed in [9]. Let J be a finite set of cardinality |J| = m ≥ 1. Consider the Euclidean space RJ ∼ = Rm with standard basis (e1 , . . . , em ), and its positive orthant R+J = {q ∈ R J : q j ≥ 0, j = 1, 2, . . . , m}.

For any measure μ on R, we will denote by μm the product measure μ ⊗ · · · ⊗ μ on R J . Furthermore, for any probability measure μ and for any ε > 0, define the following quantity: s(μ, ε) = sup μ([a, a + ε]). a∈R

Furthermore, let μ on q = (q2 , . . . , qm ).

m−1

be the marginal probability distribution induced by μm

Definition 4 Let J be a finite set with |J| = m. A function  : R J → R is called “diagonally monotone” (DM) if it satisfies the following two conditions: (i) For any r ∈ R+J and any q ∈ R J , (q + r) ≥ (q).

Wegner-type Bounds for a Two-particle Lattice Model

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(ii) Moreover, for e = e1 + · · · + em ∈ R J , for any q ∈ R J and for any t > 0 (q + t · e) − (q) ≥ t. It is convenient to introduce the notion of DM operators considered as quadratic forms. In the following definition, we use the same notations as above. Definition 5 Let H be a Hilbert space and J a finite set. A family of self-adjoint operators B(q) : H → H,

q ∈ RJ,

is called “diagonally monotone” (DM) if the following two conditions are satisfied: (i) For any q ∈ R J and any r ∈ R+J , we have B(q + r) ≥ B(q), in the sense of quadratic forms. (ii) For any vector f ∈ H with  f  = 1, the function  f : R J → R defined by  f (q) = (B(q) f, f ) is diagonally monotone. Remark 2 By virtue of the variational principle for self-adjoint operators, if an operator family H(q) in a finite-dimensional Hilbert space H is DM, then H(q) each eigenvalue Ek of H(q) is a DM function. In addition, it is readily seen that if H(q), q ∈ R J , is a DM operator family in some Hilbert space H, and H0 : H → H is an arbitrary self-adjoint operator, then the family H0 + H(q) is also DM. Lemma 1 (Stollmann [21]) Let J be a f inite index set, μ a probability measure on R, and μ J the product measure μ ⊗ · · · ⊗ μ on R J . If the function  : R J → R is diagonally monotone, then for any open interval I ⊂ R we have μ J {q : (q) ∈ I} ≤ |J| · s(μ, |I|). Remark 3 It is not difficult to see that the identical distribution of the random variable V is not essential for the Stollmann’s bound. In a more general case, when the measure μ J = ⊗ j∈J μ j is the direct product of measures {μ j} j∈J , the above bound can be replaced by P{q : (q) ∈ I} ≤ |J| · max s(μ j, I). j

In the particular case where the μ j’s admit bounded probability densities p j(t), the Stollmann’s bound takes the form P{q : (q) ∈ [a, a + ε]} ≤ |J| · max  p j · ε. j

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The proof of Stollmann’s lemma can be found in [21, Section 1, p. 309], so we omit it. As was shown in [9], an ensemble of two-particle tight-binding Hamiltonians H = H0, + U + (V(x1 ; ω) + V(x2 ; ω)) in a finite cube  = (1) × (2) ⊂ Z2d with external potential V(x; ω) taking independent values is a diagonally monotone family, with respect to the index set J = Π := (1) ∪ (2) . This allows a fairly straightforward application of Stollmann’s bound to two-particle systems. In our case, we achieve a similar goal, using a particular conditioning in the “grand ensemble” of quasi-periodic potentials V(x; ω; θ); see Subsection 4.2.

4 Proof of the Main Results 4.1 Spacing Along Finite-volume Trajectories Owing to the Diophantine condition (1.3), for any cube  L (u) ⊂ Zd , L ≥ 1, and any point ω ∈ Tν , we have   dist(T x ω, T y ω) ≥ C1 L−B . (4.1) δ( L (u), ω) := min x,y∈ L (u) x= y

The usual distance on the torus being shift-invariant, δ(ω) does not depend on ω, hence we drop the argument ω in δ( L (u)) = δ( L (u), ω). For the same reason, δ( L (u)) does not depend upon u, and we will use a simpler notation δ L for the quantity δ( L (u)). Given a positive integer n, consider again the partition Cn of the torus Tν νn into  i−1 cubes  Cn,k , k = 1, . . . , 2 , of the form Cn,k = In,i1 × · · · × In,iν , with In,i = i , , as defined in Section 1. It is convenient for our purposes to use the 2n 2n max-norm in Rν , ω∞ := max |ωi |, 1≤i≤ν

ω ∈ Rν ,

and the distance induced by it on Tν = Rν /Zν . Below we always use this distance on the torus, unless otherwise specified. Then diam Cn,k = 2−nν , and we see that points of any finite-volume trajectory T (ω,  L (u)) = {T x ω, x ∈  L (u)}

are separated by elements of any partition Cn with n≥

ln δ L B ln C1 ≥ ln L + . ln 2 ln 2 ln 2

Let

n0 (L) =

 ln C1 + B ln L + 1, ln 2

(4.2)

Wegner-type Bounds for a Two-particle Lattice Model

213

where  ·  denotes the integer part. For any L ≥ 2, we have the estimate n0 (L) ≤

ln C1 + B + ln 2 ln L = C2 ln L. ln 2

As a consequence, given any point ω ∈ Tν , any positive integer n ≥ n0 (L) and any lattice cube  L (u), the finite family of random variables   θn,k : supp ϕn,k ∩ T (ω,  L (u)) = ∅ (4.3) (with n fixed) is a family of independent random variables. Recall also that each of the above random variables θn,k is uniformly distributed in [0, 1]. Hence, its probability density is bounded by 1. Moreover, for any lattice cube  L (u) ⊂ Z2d , so that Π L (u) ⊂  N (v), we see that the points of any finite-volume trajectory T (ω, Π L (u)) = {T x ω, x ∈ Π L (u)}

are separated by elements of any partition Cn with n ≥ n0 (N). 4.2 Conditional Independence of the Potential Values Now we analyze the values of the function v : Tν ×  → R along the points of a given finite trajectory T (ω, Π L (u)). Relative to the product measure dω × μ on the product probability space Tν × , these values are not independent. (Here, dω is the normalized Haar measure on the torus.) However, they are conditionally independent given the sigma-algebra B ( N (v)) generated by the RVs ωi , 1 ≤ i ≤ ν and by all RVs {θn,k : n < n0 (N)}. Indeed, we can re-write the expansion (1.2) (“randelette” expansion) as follows: v(ω; θ) =

∞ 

Kn 

an

n=1

=

θn,k ϕn,k (ω)

k=1



an

n
Kn  k=1

θn,k ϕn,k (ω) +

 n≥n0 (N)

an

Kn 

θn,k ϕn,k (ω)

(4.4)

k=1

It is straightforward now that the first sum at the RHS becomes constant, given the sigma-algebra B ( N (v)). Fix two points ω , ω

∈ Tν with dist(ω , ω

) ≥ δ N . For any n ≥ n0 (L), they are separated by the elements of partition Cn . Observe that, actually, an

Kn 

θn,k ϕn,k (ω ) = an θn,k(ω ) ϕn,k(ω ) (ω )

k=1

where k(ω ) is uniquely defined by the condition ω ∈ supp ϕn0 ,k(ω )

(4.5)

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Further, with k(ω

) defined in a similar way for the point ω

, k(ω ) = k(ω

), since ω and ω

are separated by elements of Cn . This yields the representations ∞ 

v(ω ; θ) = ξ +

an θn,k(ω ) ϕn,k(ω ) (ω ) = ξ + η ,

n≥n0 (N)





v(ω ; θ) = ξ +

∞ 

an θn,k(ω

) ϕn,k(ω

) (ω

) = ξ

+ η

,

(4.6)

n≥n0 (N)

where ξ , ξ

are B ( N (v))-measurable and η , η

are conditionally independent, given B ( N (v)). Actually, the entire family of the RVs v(T x ω; θ), x ∈ Π L (u), admits the decomposition v(T x ω; θ) = ξx +

∞ 

an θn,k(T x ω) ϕn,k(T x ω) (T x ω)

n≥n0 (N)

= ξx + ηx ,

(4.7)

where all ξx are B ( N (v))-measurable and the family of RVs {ηx , x ∈  L (u)} is independent. Respectively, the family {ξx + ηx , x ∈ Π L (u)} is conditionally independent, given B ( N (v)), and so are the values {v(T x ω, θ), x ∈ Π L (u)}. Next, we rewrite (4.7) as follows: v(T x ω; θ) = ξx + θn 0 (N) + ηx ,

(4.8)

with θn 0 (N) = an0 (N) θn0 (N),k(T x ω) ϕn0 (N),k(T x ω) (T x ω) and ηx =

∞ 

an θn,k(T x ω) ϕn,k(T x ω) (T x ω).

n>n0 (N)

The random variable θn 0 (N) is uniformly distributed in [−an0 (N) , an0 (N) ], so its probability density pθn (N) exists and is bounded by (2an0 (N) )−1 . Therefore, by 0 (1.4) and (4.2), we have −1 M a−1 n0 (N) ≤ c1 n0 (N) ≤

C2M M ln N. c1

The random variable ηx admits some probability density pηx (as a sum of a convergent series of RVs with uniform distributions). Since θn 0 (N) and ηx are independent, their sum ζx ≡ θn 0 (N) + ηx admits a probability density pζx given by the convolution pθn (N) ∗ pηx , which is bounded by the L∞ -norm of any of 0 them. Hence,  pζx ∞ ≤ (2an0 (N) )−1 ≤

C2M M ln N. 2c1

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4.3 Two-particle Wegner-type Bound for Independent Potentials Here is a slightly adapted version of the main results of [9]. In this article, Y. Suhov and V. Chulaevsky deal with a finite-volume two-particle Hamiltonians with an external random potential which is supposed i.i.d. Their proofs are based on the observation that those operators can be represented as diagonally monotone operator families, so that Stollmann’s method applies to such ensembles. As in Stollmann’s lemma, the identical distribution of the random variables is not essential. The proof requires only that the random variables are independent. Let Fx = FV,x be the marginal cumulative distribution function (CDF) of the external random potential {V(x; ω)}. Proposition 1 ([9], Thm. 1) Consider a two-particle Hamiltonian H = H0 + (V(x1 ; ω) + V(x2 ; ω)) + U(x1 , x2 ), where {V(x; ω), x ∈ Π L (u)} is an independent random f ield relative to a probability space (, F , P), with a marginal CDF Fx (t). Set a+ε sx (ε) = sup dFx (t), 0 < ε ≤ 1. (4.9) a∈R

a

and s(ε) =

sup

x∈Π  L (u)

sx (ε)

Then for all E ∈ R, L ≥ 1, u ∈ Zd × Zd and ε > 0,     P dist u,L (ω; θ), E ≤ ε ≤ | L (u)|3/2 · s(2ε).

(4.10)

(4.11)

Proposition 2 ([9], Thm. 2) Under the same assumptions as in Proposition 1, consider a pair of two-particle cubes  L (u),  L (u ), L ≥ L ≥ 1. Let S : (u1 , u2 ) → (u2 , u1 ) be the symmetry in Z2d exchanging the coordinates of two particles and suppose that min{u − u , u − S(u )} > 8L.

(4.12)

Set s(ε) =

sup

x∈Π  L (u)∪Π  L (u )

sx (ε).

(4.13)

Then for all ε > 0,     P dist u,L (ω; θ), u ,L (ω; θ) ≤ ε ≤ | L (u)|3/2 · | L (u )| · s(2ε). (4.14) 4.4 One-volume Wegner-type Bound for Quasi-periodic Potentials Now we return to the analysis of our two-particle Hamiltonian HL (u) with quasi-periodic potential V(x; ω; θ) in a finite volume  L (u) ⊂ Z2d , where L ≥ 1, u = (u1 , u2 ) ∈ Z2d ,  L (u) =  L (u1 ) ×  L (u2 ), and Π L (u) ⊂  N (v).

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Fix a real number E. We have to estimate the probability       P dist HL (u) , E ≤ ε .

(4.15)

First, we can apply the identity       P dist HL (u) , E ≤ ε         = E P dist HL (u) , E ≤ ε  B ( N (v)) ,

(4.16)

so that it suffices to bound the inner, conditional probability. It is clear that the results of Subsection 4.3 apply to the two-particle Hamiltonian HL (u) under conditioning by the sigma-algebra B ( N (v)). Indeed, this conditioning makes the values of the external potential conditionally independent. Moreover, the conditional probability density for each value of the external potential V(x; ω; θ), x ∈  L (u1 ) ∪  L (u2 ), admits a density bounded by 2a−1 n0 (N) with n0 (N) ≤ C2 ln N. By assumption (1.4), such a density is bounded by C2M

C2M 2c1

ln M N.

Therefore, owing to (1.4), and for C = 23d 2c1 the following inequality holds true:        P dist HL (u) , E ≤ ε  B ( N (v)) ≤ C ln M N · L3d · ε. Finally, we see that, by (4.11),       P dist HL (u) , E ≤ ε ≤ C ln M N · L3d · ε. This concludes the proof of Theorem 1.

 

4.5 Two-volume Wegner-type Bound for Quasi-periodic Potentials We can argue as in the one-volume case, but apply now Proposition 2 instead of Proposition 1. We deal here with a pair of two-particle cubes  L (u),  L (u ), L ≥ L ≥ 1. Suppose that Π( L (u) ∪  L (u )) ⊂  N (v) ⊂ Zd for some N ∈ N, v ∈ Zd and condition (4.12) holds. We have to estimate the probability     P dist u,L (ω; θ), u ,L (ω; θ) ≤ ε . First, we notice that     P dist u,L (ω; θ), u ,L (ω; θ) ≤ ε       = E P dist u,L (ω; θ), u ,L (ω; θ) ≤ ε B ( N (v)) . Therefore, one can apply the Proposition 2 of Subsection 4.3 to bound the conditional probability. Under this conditioning, the values of the external potential are independent, and the marginal conditional probability density for each value of the external potential V(x; ω; θ), x ∈ Π( L (u) ∪  L (u )) is bounded by a−1 n0 (N) ≤

C2M M ln N. 2c1

Wegner-type Bounds for a Two-particle Lattice Model

217

CM

Therefore, for C = 25d 2c21 , the following inequality holds true:      P dist u,L (ω; θ), u ,L (ω; θ) ≤ ε B ( N (v)) ≤ C L5d ln M N · ε. This concludes the proof of Theorem 2.

 

References 1. Bovier, A., Campanino, M., Klein, A., Perez, F.: Smoothness of the density of states in the Anderson model at high disorder. Commun. Math. Phys. 114(3), 439–461 (1988) 2. Bourgain, J., Goldstein, M.: On nonperturbative localization with quasiperiodic potentials. Ann. Math. 152(3), 835–879 (2000) 3. Bourgain, J., Goldstein, M., Schlag, W.: Anderson localization for Schrödinger operators on Z with potential generated by the skew-shift. Commun. Math. Phys. 220(3), 583–621 (2001) 4. Bourgain, J., Schlag, W.: Anderson localization for Schrödinger operators on Z with strongly mixing potentials. Commun. Math. Phys. 215(1), 143–175 (2000) 5. Campanino, M., Klein, A.: A supersymmetric transfer matrix and differentiability of the density of states in the one-dimensional Anderson model. Commun. Math. Phys. 104(2), 227– 241 (1986) 6. Carmona, R., Lacroix, J.: Spectral Theory of Random Schrödinger Operators. Birkhäuser, Boston (1990) 7. Chan, J.: Method of variations of potential of quasi-periodic Schrödinger equations. Geom. Funct. Anal. 17(5), 1416–1478 (2008) 8. Chulaevsky, V.: Wegner–Stollmann estimates for some quantum lattice systems. In: Contemp. Math., vol. 447, pp. 17–28. Amer. Math. Soc., Providence (2007) 9. Chulaevsky, V., Suhov, Y.: Wegner bounds for a two-particle tight binding model. Commun. Math. Phys. 283(2), 479–489 (2008) 10. Combes, J.-M., Hislop, P.D., Klopp, F.: An optimal Wegner estimate and its application to the global continuity of the integrated density of states for random Schrödinger operators. Duke Math. J. 140(3), 469–498 (2007) 11. Constantinescu, F., Fröhlich, J., Spencer, T.: Analyticity of the density of states and replica method for random Schrödinger operators on a lattice. J. Stat. Phys. 34(3–4), 571–596 (1984) 12. Craig, W., Simon, B.: Log Hölder continuity of the integrated density of states for stochastic Jacobi matrices. Commun. Math. Phys. 90(2), 207–218 (1983) 13. Cycon, H.L., Froese, R.G., Kirsch, W., Simon, B.: Schrödinger Operators. Springer, Berlin (1987) 14. Delyon, F., Souillard, B.: Remark on the continuity of the density of states of ergodic finite difference operators. Commun. Math. Phys. 94(2), 289–291 (1984) 15. von Dreifus, H., Klein, A.: A new proof of localization in the Anderson tight binding model. Commun. Math. Phys. 124(2), 285–299 (1989) 16. Fröhlich, J., Martinelli, F., Scoppola, E., Spencer, T.: Constructive proof of localization in Anderson tight binding model. Commun. Math. Phys. 101(1), 21–46 (1985) 17. Fröhlich, J., Spencer, T.: Absence of diffusion in the Anderson tight binding model for large disorder or low energy. Commun. Math. Phys. 88(2), 151–184 (1983) 18. Fröhlich, J., Spencer, T., Wittwer, P.: Localization for a class of one dimensional quasi-periodic Schrödinger operators. Commun. Math. Phys. 132(1), 5–25 (1990) 19. Pastur, L.A., Figotin, A.L.: Spectra of Random and Almost Periodic Operators. Springer, Berlin (1992) 20. Simon, B., Taylor, M.: Harmonic analysis on SL(2, R) and smoothness of the density of states in the one-dimensional Anderson model. Commun. Math. Phys. 101(1), 1–19 (1985) 21. Stollmann, P.: Wegner estimates and localization for continuous Anderson models with some singular distributions. Arch. Math. (Basel) 75(4), 307–311 (2000) 22. Stollmann, P.: Caught by Disorder. A Course on Bound States in Random Media. Birkhäuser, Boston (2001) 23. Wegner, F.: Bounds on the density of states in disordered systems. Z. Phys., B. Condens. Matter 44(1–2), 9–15 (1981)

Math Phys Anal Geom (2010) 13:219–233 DOI 10.1007/s11040-010-9076-9

On the AC Spectrum of One-dimensional Random Schrödinger Operators with Matrix-valued Potentials Richard Froese · David Hasler · Wolfgang Spitzer

Received: 2 December 2009 / Accepted: 4 May 2010 / Published online: 10 June 2010 © Springer Science+Business Media B.V. 2010

Abstract We consider discrete one-dimensional random Schrödinger operators with decaying matrix-valued, independent potentials. We show that if the 2 -norm of this potential has finite expectation value with respect to the product measure then almost surely the Schrödinger operator has an interval of purely absolutely continuous (ac) spectrum. We apply this result to Schrödinger operators on a strip. This work provides a new proof and generalizes a result obtained by Delyon et al. (Ann. Inst. H. Poincaré Phys. Théor. 42(3):283–309, 1985). Keywords Random Schrödinger operators · Spectral theory 1 Model and Statement of Results In this paper we are interested in the absolutely continuous (ac) spectrum of quasi one-dimensional random Schrödinger operators with decaying potentials. To this end, it is convenient to formulate the problem in terms of matrixvalued potentials on the one-dimensional lattice, Z.

R. Froese Department of Mathematics, University of British Columbia, Vancouver, British Columbia, Canada e-mail: [email protected] D. Hasler (B) Department of Mathematics, College of William & Mary, Williamsburg, Virginia, USA e-mail: [email protected] W. Spitzer Fakultät für Mathematik und Informatik, FernUniversität Hagen, Hagen, Germany e-mail: [email protected]

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Let us first introduce some standard notation that is used throughout this paper. If H is an operator on some Hilbert space H, then we denote by ρ(H), σ (H), σac (H), σess (H) its resolvent set, spectrum, ac spectrum, respectively its essential spectrum. By H we denote the operator norm of H. For some m ∈ N, let Sym(m) denote the set of real symmetric m × m matrices. Let D ∈ Sym(m) be some fixed matrix and let q = (qn )n∈Z be a family of independent Sym(m)-valued random variables. We assume here that (i) the mean of each random variable qn is zero and (ii) there is a compact set K ⊂ Sym(m) so that the support of each qn is contained in K. By νn we denote the probability measure of qn . The probability measure for q is then the product measure ν = ⊗n∈Z νn . We use the notation E to denote the expectation value with respect to this product measure, ν. On the Hilbert space 2 (Z; Cm ) (of Cm -valued functions on Z equipped with the usual Euclidean norm) we consider the operator H :=  + D + q ,

(1)

which is defined as (Hϕ)(n) := −ϕ(n − 1) − ϕ(n + 1) + D ϕ(n) +qn ϕ(n) ,

ϕ ∈ 2 (Z; Cm ) , n ∈ Z .

(2)

To state the first result of this paper we introduce the following set which depends on the (eigenvalues of the) constant “potential” D,  I D := [λ − 2, λ + 2] . (3) λ∈σ (D)

 Theorem 1 Let E[ n∈Z qn 2 ] < ∞. Then almost surely σac (H) ⊇ I D and the spectrum of H is purely absolutely continuous in the interior of I D . Theorem 1 will be used to prove the second result of this paper. Theorem 2 Let E[

 n∈Z

qn 2 ] < ∞. Then almost surely σac (H) ⊇ σ ( + D).

Remarks The case of a random potential with m = 1 has been analyzed in great detail by Delyon et al. [8]. For m = 1 they not only prove Theorem 1 (even under weaker conditions on the measures νn ) but also that the rate of decay of qn is necessary in order to have absolutely continuous spectrum. A deterministic version of a result in the direction of Theorem 2 has been announced by Molchanov and Vainberg [20] but, to the best of our knowledge, has not yet been published. Other previous work by Kirsch, Krishna, Obermeit and Sinha on decaying potentials can be found in [14, 17] and [18]. We would also like to mention also the work of Kotani and Simon [16] and SchulzBaldes [22] on random Schrödinger operators on the strip. Example The most important application of Theorem 2 is to Schrödinger operators on a strip. More generally, let C := {1, 2, . . . , L}d denote the discrete

On the AC Spectrum of Random 1-D Matrix Schrödinger Operators

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d-dimensional cube with side length L. Then 2 (C ) ∼ = Cm with m = Ld and 2 2 2 ∼  (Z;  (C )) =  (Z × C ). We introduce the multi-index n := (n1 , n2 , . . . , nd ) ∈ Zd with |n| := |n1 | + |n2 | + · · · + |nd |. Let D be the Dirichlet Laplacian on C , i.e., for all n ∈ C ,  ψ(m) , ψ ∈ 2 (C ) . (D ψ)(n) := − m∈C :|m−n|=1

Note that  + D is equivalent to the (nearest neighbor) Dirichlet Laplace operator on 2 (Z × C ). The eigenvalues of D are indexed by n ∈ C and are d given by −2 i=1 cos(π ni /(L + 1)). Observe that    I D = λ ∈ R : |λ| ≤ 2 1 − d cos(π/(L + 1)) . If d ≥ 2 this set is empty unless L = 1. If d = 1, then I D is non-empty but its length converges to 0 as L tends to infinity. By Theorem 2, σac (H) ⊇ σ ( + D) = [−2 − 2d cos(π/(L + 1)), 2 + 2d cos(π/(L + 1))]. By formally setting L to infinity the last interval becomes [−2(d + 1), 2(d + 1)]. Remarks On the full two-dimensional lattice Z2 , Bourgain [3] proved σac ( + q) ⊇ σ () for Bernoulli and Gaussian distributed, independent random potentials whose variances decay faster than |n|−1/2 . In [4], Bourgain improves this result to the weaker |n|−1/3 decay rate. For a deterministic potential, q, on Zd , Simon [23] conjectured that if (qn / 1 + |n|d−1 )n∈Zd ∈ 2 (Zd ), then σac ( + q) = σ (). In dimension one this was proved by Deift and Killip [7]. A recent improvement of this result has been obtained by Denisov [11]. In the analogous continuous setting, progress has been made towards this L2 conjecture e.g. by Denisov [9] and Laptev et al. [19]. For additional references see [5, 24].

2 Proofs of Theorem 1 and 2 In order to prove the two main theorems in this paper we will study the Green’s functions defined by Gn := Pn (H − λ)−1 Pn ,

n ∈ Z.

(4)

Here, Pn denotes the orthogonal projections of H = 2 (Z; Cm ) onto the subm space 2 ({n}; Cm ) ∼ , and λ denotes the spectral parameter. Let Pn+ := = C  − k≥n Pk and Pn := k≤n Pk be the orthogonal projections of H onto the subspaces 2 ({n, n + 1, . . .}; Cm ) and 2 ({. . . , n − 1, n}; Cm ), respectively. Let

± −1 G± Pn , n ∈ Z, (5) n := Pn Hn − λ be the so-called forward and backward Green’s functions, where Hn± := Pn± H Pn±  RanPn± . Then we have the recursion relation − −1 Gn = −(G+ , n+1 + Gn−1 + λ − D − qn )

n ∈ Z,

(6)

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which follows by using the decomposition H = RanPn ⊕ RanPn ⊥ and a resolvent identity. If Imλ > 0, it is elementary to see that for each n ∈ Z Gn , G± n ∈ SHm := {Z = X + iY : X, Y ∈ Sym(m), Y > 0} .

(7)

Henceforth we will assume that Imλ > 0. We equip the vector space SHm with the metric

1 d(Z , W) := cosh−1 1 + cd(Z , W) , Z , W ∈ SHm , 2 where we have introduced

  cd(Z , W) := tr (ImZ )−1 (Z − W)∗ (ImW)−1 (Z − W) .

The space (SHm , d) is called Siegel half space and is a generalization of the usual Poincaré upper half plane. + By symmetry it will suffice to study G+ 0 . Using the decomposition RanP0 = + RanP0 ⊕ RanP1 , we see that + G+ 0 = q0 (G1 ) ,

(8)

with the following mapping on SHm δ (Z ) := −(Z + λ − D − δ)−1 ,

δ ∈ Sym(m) , Z ∈ SHm .

Iterating Eq. 8 we arrive at + G+ 0 = q0 ◦ q1 · · · ◦ qn (Gn+1 ) ,

n ∈ N0 .

(9)

We will use the following theorem, which is a special case of a theorem obtained in [12]. Theorem 3 Let Imλ > 0 and ( n )n∈N0 ⊂ SHm be any sequence. Then G+ 0 = lim q0 ◦ · · · ◦ qn ( n ) . n→∞

We present a direct proof of this theorem in Appendix A, which in this case is simpler than the proof given in [12] for more general graphs. The next theorem measures the distance of G+ 0 from the free forward Green’s function, which is determined by the following fixed point relation in SHm : Z λ = 0 (Z λ ) . Solving for Z λ yields

(10)



D−λ D−λ 2 Zλ = +i 1− . 2 2

(11)

Note that for real λ, we have ImZ λ > 0 if and only if λ is in the interior of I D . To formulate the next theorem we define cdλ (Z ) := cd(Z λ , Z ) ,

Z ∈ SHm .

(12)

On the AC Spectrum of Random 1-D Matrix Schrödinger Operators

Theorem 4 Suppose E[ rior of I D . Then

 n∈Z

223

qn 2 ] < ∞. Let J be a closed subset of the inte-

sup

λ∈J+i(0,1]

  E cd2λ (G± n) < ∞.

(13)

Proof By symmetry it suffices to consider without loss of generality G+ 0. We assume λ ∈ J + i(0, 1] is fixed. By Theorem 7, we know that G+ 0 = limn→∞ Z 0,n , where Z 0,n = q0 ◦ q1 ◦ · · · ◦ qn (Z λ ). Moreover, by Lemma 7 from Appendix B, we know that there exists a hyperbolic ball B ⊂ SHm such that Z 0,n ∈ B for all n ≥ 2 and potentials q with qk ∈ K. By continuity of the function Z → cd2λ (Z ), we have lim cd2λ (Z 0,n ) = cd2λ (G+ 0 ).

n→∞

Since cd2λ (Z ) is bounded on the ball B, it follows from dominated convergence that   2   E cd2λ (G+ 0 ) = lim E cdλ (Z 0,n ) . n→∞

It remains to show that the right-hand side is bounded uniformly in λ ∈ J + i(0, 1]. To this end we set Z ,n := q ◦ q+1 ◦ · · · ◦ qn (Z λ ). Note that Z ,n = q (Z +1,n ). Using the inequality of Lemma 5 below, we find   E cd2λ (Z 0,n ) + 1 

2  = cdλ (Z 0,n ) + 1 dν0 (q0 ) · · · dνn (qn )  = 

Kn+1

cd2λ [q0 (Z 1,n )] + 1

cd2λ (Z 1,n )

Kn+1

+1

 cd2λ (Z 1,n ) + 1 dν0 (q0 ) · · · dνn (qn )

(1+ A(Z 1,n , q0 )+C0 q0 2 ) dν0 (q0 )

≤ K



×

Kn

 cd2λ (Z 1,n ) + 1 dν1 (q1 ) · · · dνn (qn ) 

= (1 + C0 E[q0  ])



2

Kn

 cd2λ (Z 1,n ) + 1 dν1 (q2 ) · · · dνn (qn )

.. . ≤

n 

i=0



1 + C0 E[qi 2 ]



≤ exp C0

∞  i=0

 E[qi  ] 2

,

 where we have used A(z, q) dνi (q) = 0, which follows from the assumption   that qi is a random variable with mean zero.

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Lemma 5 Suppose K is a compact subset of SHm . Let J be a closed interval contained in the interior of I D . Then there exists a constant C0 and a linear functional A(Z , ·) : Sym(m) → R, depending continuously on Z ∈ SHm , such that for all λ ∈ J + i(0, 1] and Z ∈ SHm , cd2λ (δ (Z )) + 1 cd2λ (Z ) + 1

≤ 1 + A(Z , δ) + C0 δ2 ,

∀δ ∈ K .

(14)

Proof Using that 0 is a hyperbolic contraction, we see cdλ (δ (Z )) = cd(δ (Z ), Z λ ) ≤ cd(Z − δ, Z λ ) = cdλ (Z − δ). By the definition of the distance function we have cd(Z − δ, Z λ ) = cd(Z , Z λ ) + a(Z , δ) + b (Z , δ) , where (with Z λ = Xλ + iYλ and Z = X + iY),  −1/2  −1/2 −1/2  −1/2  − tr Yλ (Z − Z λ )∗ Y −1 δYλ , a(Z , δ) := −tr Yλ δY −1 (Z − Z λ )Yλ

−1/2 −1 −1/2  b (Z , δ) := tr Yλ δY δYλ . Using the Cauchy-Schwarz inequality it follows that L. H. S. of (14) ≤ 1 + A(Z , δ) + C(Z , δ) , with A(Z , δ) := C(Z , δ) :=

2 cdλ (Z ) a(Z , δ) cd2λ (Z ) + 1

,

2 a(Z , δ)2 + 2 cdλ (Z ) b (Z , δ) + 2 b (Z , δ)2 cd2λ (Z ) + 1

.

It remains to show that C(Z , δ) ≤ C0 δ2 for some C0 . Let us use the bounds, a(Z , δ)2 ≤ 4 cdλ (Z ) b (Z , δ) ,

1/2 1/2  b (Z , δ) ≤ Yλ−1 2 δ2 tr Yλ Y −1 Yλ .

(15) (16)

Equation 15 follows from the Cauchy-Schwarz inequality. The trace in the −1/2 −1/2 function b can be written as tr(EF E) with E := Yλ δYλ and F := 1/2 −1 1/2 2 Yλ Y Yλ . This trace is estimated from above by E  trF. Then use −1/2 −1/2 E2  ≤ Yλ 2 Yλ−1  δ2 . Since Yλ is self-adjoint Yλ 2 = Yλ−1 , and (16) follows.

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The next estimate allows us to bound the right-hand side of (16) in terms of cdλ (Z ).

1/2 1/2  tr Yλ Y −1 Yλ



1/2 1/2  ≤ tr Yλ Y −1 Yλ + tr Y 1/2 Yλ−1 Y 1/2  −1/2 −1/2  + 2m = tr Yλ (Y − Yλ )Y −1 (Y − Yλ )Yλ  −1/2  −1/2 ≤ tr Yλ (Y − Yλ )Y −1 (Y − Yλ )Yλ  −1/2 −1/2  + tr Yλ (X − Xλ )Y −1 (X − Xλ )Yλ + 2m = cdλ (Z ) + 2m .

(17)

The claim now follows by inserting the above estimates and using that Yλ  and Yλ−1  are uniformly bounded for λ ∈ J + i(0, 1] and that δ is contained in a bounded set.   Proof of Theorem 1 Step 1 Almost surely σ (H) ⊇ σ ( + D).   2 2 The condition E n∈Z qn  implies that almost all potentials are in  and thus decay at infinity. H is thus a compact perturbation of  + D and hence σ ( + D) = σess ( + D) = σess (H) ⊆ σ (H) by Weyl’s Theorem. Step 2 Let J be any closed interval contained in the interior of I D . Let Wλ := − (2Z λ + λ − D)−1 . Then   sup E cd2 (Gn , Wλ ) < ∞ . λ∈J+i(0,1]

If we use the recursion relation (6), the fact that Z → −Z −1 is a hyperbolic isometry, and the inequalities of Lemma 8 (given in the Appendix B) we find that

 − cd(Gn , Wλ ) ≤ cd G+ n+1 + Gn−1 + qn , 2Z λ



−  ≤ cd G+ n+1 + qn /2, Z λ + cd Gn−1 + qn /2, Z λ 



−  2 ≤ C 1 + cd G+ n+1 , Z λ + cd Gn−1 , Z λ (1 + qn  ) . Then,

 





−  2 E cd2 (Gn , Wλ ) ≤ C 1 + cd2λ G+ n+1 + cdλ Gn+1 ,

and Step 2 follows from Theorem 4. Step 3 Almost surely H has purely ac spectrum in the interior of I D . For x ∈ Z × {1, ..., m} let μx denote the spectral measure of H for the indicator function at x, 1x ∈ H ∼ = 2 (Z × {1, ..., m}). Step 2 implies that almost surely μx is absolutely continuous on any closed subset of the interior of I D . This can be seen for example by applying Lemma 1 in [13] and noting

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that for any closed subset J contained in the interior of I D there exists a constant C such that (see Lemma 8) tr (ImZ ) ≤ C (cdλ (Z ) + 1) for all λ ∈ J of closed and Z ∈ SHm ; see also [15, Theorem 4.1]. Now choosing a sequence  subsets (Jn )n∈N of the interior of I D , such that Jn ⊂ Jn+1 and ∞ J = I D , and n n=1 using that countable unions of sets of measure zero have again measure zero, we find almost surely that for all x ∈ Z × {1, ..., m} the spectral measure μx is absolutely continuous on the interior of I D . The theorem now follows by combining Steps 1 and 3.   To prove Theorem 2 we will use Theorem 1 in combination with Theorem 6 below. Theorem 6 is an extension of a theorem by Denisov [10, Theorem 1.2]. A proof can also be found in Albeverio and Konstantinov [1]. We give a proof in Appendix C following arguments given in [2, 10]. Theorem 6 (Denisov) Let H1 and H2 be two bounded self-adjoint operators on the Hilbert spaces H1 and H2 , respectively. Assume that for a < b , [a, b ] ⊆ σac (H1 ) and σess (H2 ) ⊆ (−∞, a] ∪ [b , ∞). Let V : H2 → H1 be a Hilbert∗ ∗ Schmidt operator (i.e.,  V V and  VV are trace class operators on H2 respectively H1 V H1 ) and let HV := . Then, [a, b ] ⊆ σac (HV ). V ∗ H2 Proof of Theorem 2 Let {μ1 , . . . , μm } be the eigenvalues of D and let λ ∈ R. Then the eigenvalues of Z λ are given by zλ,k = (μk − λ)/2 + i 1 − ((μk − λ)/2)2 . If λ ∈ [μk − 2, μk + 2] then zλ,k lies on the unit semicircle above the real axis. Otherwise zλ,k lies on the real axis outside the unit circle (see diagram)

Since Z λ is related to the Green’s function for  + D, a point λ lies in σ ( + D) if and only if at least one of the zλ,k lies on the semicircle, and thus has positive imaginary part. Let m(λ) denote the number of zλ,k on the semicircle. As we vary λ, the function m(λ) is locally constant, with jumps when one of the zλ,k moves in or out of the semicircle. Pick a λ0 and let I be the largest interval containing λ0 on which m(λ) is constant. Notice that σ ( + D) is a finite disjoint union of such intervals. The collection of zλ,k that remain in the semicircle for λ ∈ I corresponds to a subset of eigenvalues of D, and thus to a spectral projection P I on Cm . We identify the range of P I with Cm(λ) . We use the same notation for the projection on H = 2 (Z; Cm ) where P I acts as a (constant) multiplication operator. Introducing P I := 1 − P I we have the decomposition   P I ( + D + q)P I P I qP I H = + D+q = . P I qP I P I ( + D + q)P I

On the AC Spectrum of Random 1-D Matrix Schrödinger Operators

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Note that  I := P I  P I is just the Laplace operator (2) on 2 (Z; Cm(λ) ). Furthermore, let D I be the restriction of D onto Cm(λ) . By Theorem 1, P I ( + D + q)P I =  I + D I + P I qP I has almost surely ac spectrum on I, since I ⊂ I DI , see (3). Since almost surely q is in 2 and thus decays at infinity the essential spectrum of P I ( + D + q)P I is contained in the complement of the interior of I. Since P I qP I is Hilbert-Schmidt almost surely, we can apply Theorem 6 and hence conclude that almost surely I ⊆ σac (H + D + q). Repeating the above arguments for the remaining intervals of non-zero length in the decomposition of the spectrum of  + D yields the claim.   Acknowledgements W.S. wants to thank the University of British Columbia and the University of Erlangen-Nürnberg for hospitality and financial support. D. H. wants to acknowledge the summer research grant awarded by the College of William & Mary.

Appendix A: Proof of Theorem 3 Let us start with the following lemma. Lemma 7 Suppose that |λ|, δ1 , δ2  ≤ C and Imλ ≥ 1/C. Then there exists a compact set B ⊂ SHm (depending on C) such that δ1 ◦ δ2 (SHm ) ⊆ B. Proof Applying δ once yields an upper bound on the norm, which can be seen from the basic inequality (Z + λ − D − δ)−1  ≤ (Imλ)−1 ,

Z ∈ SHm .

Applying δ a second time yields a lower bound on the imaginary part, which can be seen by the following estimate   Im −(Z +λ− D−δ)−1 = (Z ∗ + λ∗ − D − δ)−1 Im(Z + λ) (Z + λ − D − δ)−1 ≥

Imλ . Z + λ − D − δ2  

Proof of Theorem 3 Step 1 Let B be a compact subset of SHm as in the previous Lemma 7. Then there exists a γ < 1, such that for all Z , W ∈ B, d(δ (Z ), δ (W)) ≤ γ d(Z , W) . Using that the maps Z → −Z −1 and Z → Z − D − δ are hyperbolic isometries on SHm , we find that d(δ (Z ), δ (W)) = d(Z + λ, W + λ) .

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In order to estimate the last expression we use that [Im(W +λ)]−1 = [Im(W)]−1/2 [Im(W)]1/2 [Im(W +λ)]−1 [Im(W)]1/2 [Im(W)]−1/2    √ ≤ γ

≤

√ γ [Im(W)]−1

for some real number γ < 1 since W is in a bounded set. If we apply this estimate also to Z we obtain that d(δ (Z ), δ (W)) = d(Z + λ, W + λ) ≤ γ d(Z , W) . Step 2 The sequence (q0 ◦ · · · ◦ qn ( n ))n∈N0 converges to a limit independent of the choice of ( n )n∈N0 . n )n∈N0 is a different sequence. Then Suppose ( n )) ≤ γ n−2 C → 0 , d(q0 ◦ · · · ◦ qn ( n ), q0 ◦ · · · ◦ qn (

(n → ∞), (18)

with C := sup(Z ,W)∈B2 d(Z , W). We conclude that if the limit exists it must be independent of the sequence ( n )n∈N0 . On the other hand the sequence (q0 ◦ n := · · · ◦ qn ( n ))n∈N0 is a Cauchy sequence, which can be seen by inserting qn+1 ◦ · · · ◦ qn+m ( n+m ) for m ∈ N into (18). The theorem now follows from Step 2 and Eq. 9.  

Appendix B: Some Inequalities Lemma 8 Let Z i ∈ SHm , i ∈ {0, 1, 2} and δ ∈ Sym(m). Then   (a) cd(2Z 0 , Z 1 + Z 2 ) ≤ 12 cd(Z 0 , Z 1 ) + cd(Z 0 , Z 2 ) .   (b) cd(Z 0 , δ + Z 1 ) ≤ C (1 + δ2 ) cd(Z 0 , Z 1 ) + 1 for some constant C that depends on the (norm of the) imaginary part of Z 0 . (c) For λ ∈ I D (see def inition (3)) there is a constant C (depending on λ and m) so that tr (ImZ 0 ) ≤ C (cdλ (Z 0 ) + 1). Proof (a) Let us define 

U1 A := U2



 ,

B := −1/2

Y1 (Y1 + Y2 )−1/2 1/2 Y2 (Y1 + Y2 )−1/2 1/2

−1/2



(Z 0 − Z i )Y0 . Then with Yi := Im(Z i ) and U i = Yi  1  1/2 1/2 1/2 1/2 cd(2Z 0 , Z 1 + Z 2 ) = tr (U 1∗ Y1 +U 2∗ Y2 )(Y1 +Y2 )−1 (Y1 U 1 +Y2 U 2 ) 2  1  = tr A∗ BB∗ A . 2

On the AC Spectrum of Random 1-D Matrix Schrödinger Operators

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∗ ∗ Since B = 1, BB∗ is a projection ≤ 1. There B  ∗ and  hence  ∗ BB  ∗ ∗ ∗ fore, tr A BB A ≤ tr A A = tr U 1 U 1 + tr U 2 U 2 = cd(Z 0 , Z 1 ) + cd(Z 0 , Z 2 ). (b) By expanding the product in the trace and using tr[A∗ B] ≤ (tr|A|2 )1/2 (tr|B|2 )1/2 ≤ 12 tr|A|2 + 12 tr|B|2 we obtain −1/2

cd(Z 0 , δ + Z 1 ) ≤ 2 cd(Z 0 , Z 1 ) + 2 tr(Y0

−1/2

≤ 2 cd(Z 0 , Z 1 ) + 2 Y0 −1/2

−1/2

−1/2

δY1−1 δY0 −1/2 2

δY0

)

 tr(Y0 Y1−1 Y0 ) . 1/2

1/2

−1/2

Now we use Y0 δY0  ≤ δ Y0 2 and inequality (17), i.e., 1/2 −1/2 tr(Y0 Y −1 Y0 ) ≤ cd(Z 0 , Z 1 ) + 2m, which, all put together, proves the claimed inequality.  −1/2 −1/2  ≥ (c) We have cdλ (Z 0 ) ≥ tr Yλ (Y0 − Yλ )Y0−1 (Y0 − Yλ )Yλ  −1/2 −1/2    tr Yλ Y0 Yλ − 2m ≥ C tr(Y0 ) − 2m, where the constant C depends on λ throught the estimate on Yλ−1 . Finally, we choose the constant C := 2m/C and the stated inequality follows.  

Appendix C: Proof of Theorem 6 To prove Theorem 6 we will use the following theorem by Albeverio et al. [2]. For the convenience of the reader we also present a proof that follows closely the one given in [2] but uses analytic perturbation theory to obtain the graph subspaces. Theorem 9 (Albeverio-Makarov-Motovilov) Let H1 , H2 be two bounded selfadjoint operators on the Hilbert spaces H1 and H2 , respectively. Assume that σ (H1 ) ⊂ (a, b ) ⊆ ρ(H2 ) and V : H2 → H1 is a Hilbert-Schmidt operator. Let     0 V H1 0 , W := , HV := H0 + W . H0 := (19) V∗ 0 0 H2 Then σac (HV ) = σac (H0 ). Proof Since finite rank perturbations F do not change the ac spectrum and such operators are norm-dense in the space of Hilbert-Schmidt operators we can replace V by V + F and achieve that its norm is small. Henceforth we assume that V is small. Let H := H1 ⊕ H2 , and for i = 1, 2 we denote by pi the orthogonal projection of H onto Hi . Step 1 For V sufficiently small, there exist orthogonal projections P1 and P2 such that P1 + P2 = 1, Pi H = H Pi , and pi Pi pi : Hi → Hi is bijective, for i ∈ {1, 2}. Furthermore, Pi − pi is Hilbert-Schmidt and its norm can be made arbitrarily small if we choose V small enough.

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Let 1 [2 ] be a counter-clockwise contour in C around the spectrum of H1 [H2 ] and contained in the resolvent set of H2 [H1 ]. Then we define the spectral projections  dz 1 Pi := , i ∈ {1, 2} . (20) 2πi i z − HV The first two properties follow from this representation. In order to verify the other statements we use a similar representation for the projections pi and the resolvent identity so that  1 Pi − pi = (z − H0 )−1 W(z − HV )−1 dz . 2πi i Hence, Pi − pi is Hilbert-Schmidt. If V is small then  pi Pi pi − pi  is small, too, and therefore pi Pi pi can be inverted on Hi . Step 2 For V sufficiently small, there exist operators Q1 : H1 → H2 and Q2 : H2 → H1 such that RanP1 = {(x, Q1 x)|x ∈ H1 } and RanP2 = {(Q2 x, x)|x ∈ H2 }. Moreover Q2 = −Q∗1 and Qi , for i ∈ {1, 2}, is Hilbert-Schmidt and its norm can be chosen arbitrarily small for V sufficiently small. Let i = j. First observe that the operator p j Pi = p j(Pi − pi ) as well as its adjoint are Hilbert-Schmidt and can be made arbitrarily small by Step 1. Using the identity P1 p1 + P2 p2 = 1 − P1 p2 − P2 p1 and noting that the r.h.s. can be made arbitrarily close to one, we see that H = RanP1 p1 + RanP2 p2 . This implies that RanPi = RanPi pi . Define Qi := p j Pi ( pi Pi pi )−1 . If we set x := pi (Pi z) for z ∈ Hi , then p j(Pi z) = Qi x. Hence, the range of Pi pi equals the graph of Qi . The statement Q2 = −Q∗1 follows by orthogonality. Step 3 For V sufficiently small HV is unitary equivalent to   H1 + T1 0 , 0 H2 + T2 where Ti are trace class operators. Since by Step 2, HV leaves the graphs of the operators Q1 and Q2 invariant, there exist operators Ai ∈ Hi such that HV (1 + Q) = (1 + Q)A , with

 Q :=

0 Q2 Q1 0



 and

A :=

A1 0 0 A2

(21)  .

To construct these operators, let x ∈ H1 . Then by Step 2, z := (x, Q1 x) ∈ RanP1 . By Step 1, HV z ∈ RanP1 , and again by Step 2, HV z = (y, Q1 y) for some uniquely determined y ∈ H1 . This defines the operator A1 : H1 → H1 by setting A1 x := y. A similar construction gives the operator A2 . As a result we obtain (21). Expanding the product in (21) we see more concretely that A1 =

On the AC Spectrum of Random 1-D Matrix Schrödinger Operators

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H1 + V Q1 and A2 = H2 + V ∗ Q2 . Since Q has purely imaginary spectrum the operator 1 + Q is bijective. Using the polar decomposition 1 + Q = U|1 + Q|, with U unitary, we find U ∗ HV U = |1 + Q| A |1 + Q|−1 . Note that



(1 + Q∗1 Q1 )1/2 0 |1 + Q| = 0 (1 + Q∗2 Q2 )1/2

(22)  .

Using 0 ≤ (1 + Q∗1 Q1 )1/2 − 1 ≤ Q∗1 Q1 , the operator (1 + Q∗1 Q1 )1/2 (H1 + V Q1 ) (1 + Q∗1 Q1 )−1/2 − H1 is trace class. A similar statement holds for the second diagonal operator on the right-hand side of (22). Step 4 σac (H) = σac (H0 ). This follows from the result of Step 3 and the fact the trace class perturbations preserve ac spectrum.   Our proof of Theorem 6 follows closely the one given by Denisov [10], but uses almost analytic functional calculus (cf. [6]) to control the function of an operator. Proof of Theorem 6 Fix ε > 0. We will show that [a + ε, b − ε] ⊂ σac (HV ). Since finite-rank perturbations do not change the ac and the essential spectrum and σess (H2 ) ⊂ (−∞, a] ∪ [b , ∞), we can assume w.l.o.g. that σ (H2 ) ⊂ (−∞, a + ε/2] ∪ [b − ε/2, ∞). Step 1 Define H1 := H1 χ[a+ε,b −ε] (H1 ). Then [a + ε, b − ε] ⊆ σac ( HV ), where H1 V . HV := V ∗ H2 This follows directly from Theorem 9 by noting that σ ( H1 ) = [a + ε, b − ε] ⊂ ρ(H2 ). Step 2 Let f ∈ C0∞ ([a + ε, b − ε]). Then f (HV ) − f ( HV ) is trace class. To show Step 2 we use almost analytic functional calculus. Let  f ∈ C0∞ (C)   be an almost analytic extension of f , satisfying !f |R = f! , f (x + iy) = 0 if x∈ / supp f , and that for some constant C we have !∂z  f (z)! ≤ C|Imz|3 for all 1 z ∈ C. [Of course, ∂z := 2 (∂x + i∂ y ) for z = x + iy ∈ C and z := x − iy.] Setting R(A, z) := (z − A)−1 for a bounded self-adjoint operator A we recall the Helffer–Sjöstrand formula (see [6, (5)])  1 f (z) R(A, z) dxdy . (23) f (A) := − ∂z  π C

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0 V Setting W := V∗ 0 tity twice, we find



 and H0 :=

 H1 0 , and applying the resolvent iden0 H2

f (HV ) − f ( HV )    1 f (z) R(H0 , z) − R( H0 , z) dxdy ∂z  =− π C    1 − f (z) R(H0 , z)W R(H0 , z) − R( H0 , z)W R( H0 , z) dxdy ∂z  π C  1 f (z) [R(H0 , z)W R(H0 , z)W R(H, z) ∂z  − π C  −R( H0 , z)W R( H0 , z)W R( H, z) dxdy . The first term on the right-hand side equals f (H0 ) − f ( H0 ) and thus vanishes. The third term on the right-hand side is a trace class operator. The second term also vanishes, as we now show. It is a combination of terms of the following form,  1 f (z)R(H1 , z)V R(H2 , z) dxdy − ∂z  π C   1 1 f (z)R(H1 , z)V dP H2 (t) dxdy =− ∂z  π C z − t σ (H2 )  1 f (H1 ) = V dP H2 (t) . H1 − t σ (H2 ) In the first line we have applied the Spectral Theorem for R(H2 , z) with the spectral projections P H2 of H2 . In the last equality we have used that f˜(z)(z − t)−1 is an almost analytic extension of f (x)(x − t)−1 since ∂z (z − t)−1 = 0 for z = t, and the Helffer–Sjöstrand formula (23). By inspection, the right-hand side of the last displayed formula does not change if we replace H1 by H1 . Step 3 [a + ε, b − ε] ∈ σac (HV ). By the statement of Step 2 and the Kato-Rosenblum Theorem [21, Theorem XI.8] we know that σac ( f (HV )) = σac ( f ( HV )) for all f ∈ C0∞ ([a + ε, b − ε]). By the Spectral Theorem we conclude that σac (HV ) ∩ [a + ε, b − ε] = σac ( HV ) ∩ [a + ε, b − ε] = [a + ε, b − ε], where the second equality follows from the result of Step 1.  

References 1. Albeverio, S., Konstantinov, A.: On the absolutely continuous spectrum of block operator matrices. Math. Nachr. 281(8), 1079–1087 (2008) 2. Albeverio, S., Makarov, K., Motovilov, A.: Graph subspaces and the spectral shift function. Can. J. Math. 55(3), 449–503 (2003)

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3. Bourgain, J.: On random Schrödinger operators on Z2 . Discrete Contin. Dyn. Syst. 8(1), 1–15 (2002) 4. Bourgain, J.: Random lattice Schrödinger operators with decaying potential: some higher dimensional phenomena. In: Milman, V.D., Schechtman, G. (eds.) LNM, vol. 1807, pp. 70– 98 (2003) 5. Cycon, H., Froese, R., Kirsch, W., Simon, B.: Schrödinger Operators with Application to Quantum Mechanics and Global Geometry. Springer-Verlag, Berlin (1987) 6. Davies, E.B.: The functional calculus. J. Lond. Math. Soc. 52(2), 166–176 (1995) 7. Deift, P., Killip, R.: On the absolutely continuous spectrum of one-dimensional Schrödinger operators with square summable potentials. Commun. Math. Phys. 203(2), 341–347 (1999) 8. Delyon, F., Simon, B., Souillard, B.: From power pure point to continuous spectrum in disordered systems. Ann. Inst. H. Poincaré Phys. Théor. 42(3), 283–309 (1985) 9. Denisov, S.A.: Absolutely continuous spectrum of multidimensional Schrödinger operator. Int. Math. Res. Not. 2004(74), 3963–3982 (2004) 10. Denisov, S.: On the preservation of absolutely continuous spectrum for Schrödinger operators. J. Funct. Anal. 231(1), 143–156 (2006) 11. Denisov, S.: On a conjecture by Y. Last. arXiv:0908.3681 12. Froese, R., Hasler, D., Spitzer, W.: Transfer matrices, hyperbolic geometry and absolutely continuous spectrum for some discrete Schrödinger operators on graphs. J. Funct. Anal. 230(1), 184–221 (2006) 13. Froese, R., Hasler, D., Spitzer, W.: Absolutely continuous spectrum for a random potential on a tree with strong transverse correlations and large weighted loops. Rev. Math. Phys. 21, 1–25 (2009) 14. Kirsch, W., Krisha, M., Obermeit, J.: Anderson Model with decaying randomness-mobility edge. Math. Z. 235(3), 421–433 (2000) 15. Klein, A.: Extended states in the Anderson model on the Bethe lattice. Adv. Math. 133(1), 163–184 (1998) 16. Kotani, S., Simon, B.: Stochastic Schrödinger operators and Jacobi matrices on the strip. Commun. Math. Phys. 119(3), 403–429 (1988) 17. Krishna, M.: Anderson model with decaying randomness-extended states. Proc. Indian Acad. Sci., Math. Sci. 100(4), 285–294 (1990) 18. Krishna, M., Sinha, K.B.: Spectral properties of Anderson Type operators with decaying randomness. Proc. Indian Acad. Sci., Math. Sci. 111(2), 179–201 (2001) 19. Laptev, A., Naboko, S., Safronov, O.: A Szego˝ condition for a multidimensional Schrödinger operator. J. Funct. Anal. 219(2), 285–305 (2005) 20. Molchanov, S., Vainberg, B.: Schrödinger operators with matrix potentials. Transition from the absolutely continuous to the singular spectrum. J. Funct. Anal. 215(1), 111–129 (2004) 21. Reed, M., Simon, B.: Methods of Modern Mathematical Physics III: Scattering Theory. Academic Press (1979) 22. Schulz-Baldes, H.: Perturbation theory for Lyapunov exponents of an Anderson model on a strip. Geom. Funct. Anal. 14, 1089–1117 (2004) 23. Simon, B.: Schrödinger operators in the twenty-first century. In: Fokas, A., Grigoryan, A., Kibble, T., Zegarlinski, B. (eds.) Mathematical Physics 2000, pp. 283–288. Imperial College Press, London. 24. Stollmann, P.: Caught by Disorder. Bound States in Random Media. Progress in Mathematical Physics. Birkhäuser, Boston (2001)

Math Phys Anal Geom (2010) 13:235–253 DOI 10.1007/s11040-010-9077-8

Comparison Theorems for Eigenvalues of Elliptic Operators and the Generalized Pólya Conjecture Qiaoling Wang · Changyu Xia

Received: 31 October 2009 / Accepted: 30 June 2010 / Published online: 16 July 2010 © Springer Science+Business Media B.V. 2010

Abstract We establish comparison theorems for eigenvalues between higher order elliptic equations on compact manifolds with boundary. As an application, it follows that if the Pólya conjecture is true then so is the generalized Pólya conjecture proposed by Ku et al. (J Differ Equ 97:127–139, 1992). We also obtain new lower bound for the eigenvalues of higher order elliptic equations on bounded domains in a Euclidean space. Keywords Eigenvalues · Lower bounds · Elliptic operator · Pólya conjecture · Riemannian manifolds · Euclidean space Mathematics Subject Classifications (2010) 35P15 · 53C20 · 53C42 · 58G25

1 Introduction Let  be a bounded domain in an n( 2)-dimensional Euclidean space Rn with smooth boundary ∂. Let  be the Laplacian of Rn and consider the Dirichlet eigenvalue problem  u = −λu, in , (1.1) u|∂ = 0.

Qiaoling Wang and Changyu Xia were partially supported by CNPq, CAPES/PROCAD. Q. Wang · C. Xia (B) Departamento de Matemática, UnB, 70910-900, Brasília-DF, Brazil e-mail: [email protected] Q. Wang e-mail: [email protected]

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Let 0 < λ1 < λ2  · · · → ∞ denote the eigenvalues (repeated with multiplicity) of the problem (1.1). The Weyl’s asymptotic formula [15] asserts that   k 2/n as k → ∞ (1.2) λk ∼ C(n) || −2/n

where || is the volume of  and C(n) = (2π )2 ωn with ωn being the volume of the unit ball in Rn . In 1960, Pólya [14] showed that for any “plane covering domain”  in R2 (those that tile R2 ) the above asymptotic relation is in fact a one-sided inequality (his proof also works for Rn -covering domains) and conjectured, for any domain  ⊂ Rn , the inequality   k 2/n for all k. (1.3) λk  C(n) || In 1982, Li-Yau [12] showed the lower bound k  i=1

nkC(n) λi  n+2



k ||

2/n (1.4)

which yields an individual lower bound on λk in the form   nC(n) k 2/n λk  . n + 2 ||

(1.5)

Similar bounds for eigenvalues with Neumann boundary conditions have been proved in [7, 8] and [10]. It was pointed out in [11] that (1.4) also follows from an earlier result by Berezin [2] by the Legendre transformation. In 2002, Melas [13] proved the following improvement of (1.4): k  i=1

nkC(n) λi  n+2



k ||

2/n + dn k

|| I()

(1.6)

where the constant dn depends only on the dimension and I() = mina∈Rn  |x − a|2 dx is the “moment of inertia” of . Further developments on Berezin-Li-Yau’s inequality (1.4) have been made in [4–6] and [16]. It would be interesting to obtain Melas’ result and its generalizations for domains of finite Lebesgue measure, not necessary bounded. Consider now the the elliptic operator of order 2t defined by Lu =

t 

am−r (−)m u, u ∈ C∞ ()

m=r+1

where r  0 is an integer, am−r ’s are constants with am−r  0, r + 1  m  t, at−r = 1, t a fixed positive integer. The following eigenvalue problem about

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L is important in the study of various branches of mathematics, such as differential equations, differential geometry and mathematical physics:  Lu = λ(−)r u, u ∈ C∞ (), (1.7) (∂/∂ν) ju|∂ = 0, j = 0, 1, 2, · · · , t − 1, where ν is the outward unit normal vector field of ∂. Let 0 < λ1,r  λ2,r  · · ·  λk,r  · · · → ∞.

(1.8)

be the eigenvalues of the problem (1.7). In [9], the following conjecture was proposed : Generalized Pólya Conjecture The eigenvalues λk,r , k = 1, 2, · · · , of the eigenvalue problem (1.7) of the operator L satisfies the inequality λk,r 

t−r 

 am C

m

m=1

k ||

2m/n ,

(1.9)

where C = C(n) is defined as before. With respect to the above generalized Pólya conjecture, Ku-Ku-Tang showed [9] that if r is even, then λk,r

t−r 

nam  Cm n + 2m m=1



k ||

2m/n , k = 1, 2, · · · .

(1.10)

In this paper, we prove comparison theorems between the k-th eigenvalues of the problem (1.1) and that of the problem (1.7) which shows that if the Pólya conjecture (1.3) is true then so is the generalized Pólya conjecture (1.9). In fact, the comparison theorem holds for compact manifolds with boundary. Theorem 1.1 Let (M, , ) be an n( 2)-dimensional compact Riemannian manifold with boundary. Denote by the Laplacian operator on M and let L be the elliptic operator given by Lu = tm=r+1 am−r (−)m u, u ∈ C∞ (M), where r  0 is an integer, am−r ’s are constants with am−r  0, r + 1  m  t, at−r = 1, t a f ixed positive integer. Denote by ν the unit outward normal vector f ield on ∂ M. Consider the following eigenvalue problems:  Lu = λ(−)r u, u ∈ C∞ (M), (1.11) j (∂/∂ν) u|∂ M = 0, j = 0, 1, 2, · · · , t − 1, 

(−)r+1 u = (−)r u, (∂/∂ν) ju|∂ M = 0, 

−u = λu in u|∂ M = 0.

u ∈ C∞ (M), j = 0, 1, 2, · · · , r, M,

(1.12)

(1.13)

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Denote by 0 < λ1,r  λ2,r  · · · → ∞

(1.14)

0 < 1,r  2,r  · · · → ∞

(1.15)

0 < λ1 < λ2  · · · → ∞

(1.16)

and

the successive eigenvalues for (1.11), (1.12) and (1.13), respectively. Here each eigenvalue is repeated according to its multiplicity. Then for any k = 1, 2, · · · , we have + t−r λk,r  a1 k,r + a2 2k,r + · · · + at−(r+1) t−(r+1) k,r k,r

(1.17)

k,r  λk .

(1.18)

and

We then prove the following eigenvalue comparison theorem. Theorem 1.2 Let (M, , ) be an n( 2)-dimensional compact Riemannian manifold with boundary. Let l, r, s be positive integers with l > r + s. Consider the following eigenvalue problems ⎧ l u = (−)r u in M, (−) ⎨  ∂u  ∂ l−1 u  (1.19) = · · · = = 0. ⎩ u|∂ M =   l−1 ∂ν ∂ M ∂ν ∂M ⎧ ⎨ ⎩ u|∂ M

l r+s (−)  u = (−)l−1 u in M,  ∂u  ∂ u  = = ··· = = 0.  ∂ν ∂ M ∂ν l−1 ∂ M

(1.20)

Denote by 0 < 1  2  3  · · · and 0 < 1  2  3  · · · the successive eigenvalues for (1.19) and (1.20), respectively. Then we have (l−r−s)/(l−r)

k  k

, ∀ k = 1, 2, · · · .

(1.21)

Remark 1.1 If M is a bounded domain in Rn , we know from the inequality λk,r  a1 λk + a2 λ2k + · · · + at−(r+1) λt−(r+1) + λt−r k k

(1.22)

which is a combination of (1.17) and (1.18) that if the Pólya conjecture (1.3) is true then so is the generalized Pólya conjecture (1.9). Also despite of r being

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239

even or odd, the inequality (1.22) gives a lower bound on λk,r by using Li-Yau’s inequality (1.4) or more generally the Melas’ inequality (1.6). Namely, we have

λk,r 

t−r 

ai

i=1

nC(n) n+2



k ||

2/n

|| + dn I()

i .

(1.23)

In the special case that the a m s are zero, we have stronger estimate on the eigenvalues of the operator L. Theorem 1.3 Let  be a connected bounded domain in an Rn and let r  0 be an integer and let l be a positive integer with l  r + 1. Let λi be the i-th eigenvalue of the problem: ⎧ r ⎨ (−)l u = λ(−)  u in , l−1   ∂u  ∂ u  = ··· = = 0. ⎩ u|∂ =  ∂ν ∂ ∂ν l−1 ∂

(1.24)

and let μ1 , · · · , μn be the f irst n nonzero eigenvalues of the Neumann problem ⎧ ⎨ −v  = μv in , ∂v  = 0. ⎩ ∂ν ∂

(1.25)

i) If r is even, then for each k = 1, 2, · · · , we have k 

λj 

j=1

n n + 2m



2π

2m

1/n ωn



n  1 + b n,m k μi i=1

k1+ n ||− 2m

2m n

(1.26)

−m ;

ii) If r is odd, then we have for k = 1, 2, · · · ,

n λk  n + 2p



2π 1/n

ωn

2 p

2p n

k ||

− 2np

+ b n, p

n  1 μi i=1

− p p−1 p . (1.27)

Here m = l − r, p = l + 1 − r, b n,q is a positive constant depending only on n and q.

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2 Proof of the Results k be a set of orthonormal eigenfunctions of the Proof of Theorem 1.1 Let {ui }i=1 k problem (1.11) corresponding to {λi }i=1 , that is, ⎧ r ⎪ ⎪ Lui = λ(−)  ui , in M, t−1  ⎨  ∂ui  ∂ ui  = ··· = = 0, ui |∂ M =  ⎪ ∂ν ∂ M ∂ν t−1 ∂ M ⎪ ⎩  r M ui (−) u j = δij , i, j = 1, · · · , k. k Similarly, let {vi }i=1 be a set of orthonormal eigenfunctions of the problem k (1.12) corresponding to { i }i=1 , that is, ⎧ (−)r+1 vi = λi (−)r vi in M, ⎪  ⎪ ⎨ ∂vi  ∂ r vi  vi |∂ M = = ··· = = 0, ⎪ ∂ν ∂ M ∂ν r ∂ M ⎪ ⎩  r M vi (−) v j = δij , i, j = 1, · · · , k.  Let w = kj=1 α ju j = 0 be such that  w(−)r v j = 0, ∀ j = 1, · · · , k − 1. (2.1) M

Such an element w exists because {α j|1  j  k} is a non-trivial solution of a system of (k − 1)-linear equations  k  αj u j(−)r vi = 0, 1  i  k − 1, (2.2) j=1

M

in k unknowns. Notice that if u ∈ C∞ (M) satisfies   ∂u  ∂ t−1 u  = · · · = = 0, u|∂ M = ∂ν ∂ M ∂ν t−1 ∂ M then u|∂ M = ∇u|∂ M = u|∂ M = ∇(u)|∂ M = · · · =   =  p−1 u∂ M = ∇( p−1 u)∂ M = 0, when t = 2 p and

 u|∂ M = ∇u|∂ M = u|∂ M = ∇(u)|∂ M = · · · = k−1 u∂ M   = ∇( p−1 u)∂ M =  p u∂ M = 0, when t = 2 p + 1.  Let us show that M w(−)r w = 0. In fact, from divergence theorem, we have    r/2 2 , if r is even, r M ( w)  2  w(−) w = (r−1)/2   ∇( w) , if r is odd. M M  Thus, if M w(−)r w = 0, then r/2 w = 0 when r is even and (r−1)/2 w = 0 when r is odd. It then follows from the maximum principle about harmonic

Eigenvalues of Elliptic Operators and the Generalized Pólya Conjecture

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functions that r/2−1 w = 0 when r is even and (r−1)/2−1 w = 0 when r is odd. Continuing this process, we conclude that w = 0. This is a contradiction. Thus  r w(−) w = 0. We can assume without loss of generality that M  w(−)r w = 1. (2.3) M

Hence we infer from the Rayleigh-Ritz inequality that  k,r  w(−)r+1 w

(2.4)

M

We claim that for any j = 1, · · · , t − r, 

 j+1



w(−)r+ jw

j w(−)r+ j+1 w



M

.

(2.5)

M

Let us first prove that (2.5) holds when j = 1. In fact, if r = 2h is even, then one deduces from the divergence theorem and the Hölder’s inequality that 2

 w(−)

r+1

w



2

=

M

(−) w(−) h



M

h+1

w

 



((−)h w)2



((−)h+1 w)2

M

 =

M

w(−)r+2 w. M

On the other hand, if r = 2h + 1 is odd, then  2 r+1 w(−) w M



=

((−) 

M

= 

h+1

2 ∇((−)h+1 w)∇((−)h w)

M

  |∇((−)





2 w)(−)((−) w) h

h+1

w|

  (−) w(−)

M

h+1

 

M

 =

w(−)r+2 w. M



w

(−) M

w(−)r w

=

2

M

h



|∇((−) w)|

M

=

 h

2

w(−)2h+3 w M

h+1



w(−)

h+2

w

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Thus (2.5) holds when j = 1. Suppose now that (2.5) holds for j − 1, that is   j   j−1 w(−)r+ j−1 w  w(−)r+ jw . (2.6) M

M

When r + j is even, we have  w(−)r+ jw M



  (r+ j)/2−1 w (r+ j)/2 w

= M



    ∇ (r+ j)/2−1 ui ∇ (r+ j)/2 ui

=− 

M

  (r+ j)/2−1 2 ∇  w 



1/2 

M

  (r+ j)/2 2 ∇  w 

1/2

M

  1/2   1/2 (r+ j)/2−1 (r+ j)/2 (r+ j)/2 (r+ j)/2+1 = −  w w  w w − M

 =

M

1/2 

w(−)r+ j−1 w

1/2

w(−)r+ j+1 w

M

,

(2.7)

M

On the other hand, when r + j is odd,  w(−)r+ jw M



(−)(r+ j−1)/2 w(−)(r+ j+1)/2 w

= M









(−)(r+ j−1)/2 w

M

=

2

1/2 

(−)(r+ j+1)/2 w

M

1/2  w(−)r+ j−1 w

M



2

1/2

1/2

w(−)r+ j+1 w

.

(2.8)

M

Thus we always have  1/2  1/2  r+ j r+ j−1 r+ j+1 w(−) w  (−) w w(−) w . M

M

(2.9)

M

Substituting (2.6) into (2.9), we know that (2.5) is true for j. Using (2.5) repeatedly, we get



1/s



 w(−)r+1 w 

w(−)r+s w

, s = 1, · · · , t − (r + 1).

M

which, combining with (2.4) gives  sk,r  w(−)r+s w, s = 1, 2, · · · , t − (r + 1). M

(2.10)

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Thus we have a1 k,r + a2 2k,r + · · · + at−(r+1) t−(r+1) + t−r k,r k,r   w(a1 (−)r+1 + a2 (−)r+2 + · · · + at−(r+1) (−)t−1 + (−)t )w M

 =

wLw = M

=

k 



k 

αi α j

ui Lu j M

i, j=1

 αi α j M

i, j=1

k 

ui λ j(−)r u j =

αi α jλ jδij =

k 

i, j=1

αi2 λi  λk ,

i=1

where in the last equality, we have used the fact that k 

 αi2 =

w(−)r w = 1.

i=1

This proves (1.17). In order to prove (1.18), let us take a set of orthonormal eigenfunctions k k of the problem (1.13) corresponding to {λi }i=1 , that is, {zi }i=1 zi = −λi zi in M, 

zi |∂ M = 0, zi z j = δij, i, j = 1, · · · , k. M

Let ξ =

k j=1

β jv j be such that   ξ 2 = 1 and ξ z j = 0, ∀ j = 1, · · · , k − 1. M

(2.11)

M

It follows from the Rayleigh-Ritz inequality that  ξ(−ξ ) λk 

(2.12)

M

Using the same arguments as in the proof of (2.5), we have 

 j+1 ξ(−) jξ



j ξ(−) j+1 ξ



M

,

j = 1, · · · , r.

(2.13)

M

Thus we have  λk 

ξ(−)r ξ M

 1r

⎛ =⎝

k  j=1

⎞ 1r β 2j ⎠

(2.14)

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On the other hand, taking j = r in (2.13), we get 



r+1 ξ(−) ξ

r ξ(−)



r

r+1

M

ξ

,

(2.15)

M

which implies that k 

 β 2j 

ξ(−)r+1 ξ

r  r+1

M

j=1

⎛ =⎝



k 

r ⎞ r+1

βi β jvi (−)r+1 v j⎠

M i, j=1

⎛ =⎝



k 

r ⎞ r+1

βi β jvi j,r (−)r v j⎠

M i, j=1

⎛ =⎝

k 

r ⎞ r+1

j,r β 2j ⎠

j=1

⎛ r r+1

 k,r ⎝

k 

r ⎞ r+1

β 2j ⎠

.

(2.16)

j=1

Thus we have k 

β 2j  rk,r .

(2.17)

j=1

Combining (2.17) and (2.14), one gets (1.18). This completes the proof of Theorem 1.1.

 k Proof of Theorem 1.2 Let {ui }i=1 be a set of orthonormal eigenfunctions of the k , that is, problem (1.20) corresponding to { i }i=1

(−)l ui = i (−)r+s ui in M,   ∂ui  ∂ l−1 ui  ui |∂ M = = · · · = = 0, ∂ν ∂ M ∂ν l−1 ∂ M



ui (−)r+s u j = δij, i, j = 1, · · · , k. M

Eigenvalues of Elliptic Operators and the Generalized Pólya Conjecture

245

k Similarly, let {vi }i=1 be a set of orthonormal eigenfunctions of the problem k (1.19) corresponding to { i }i=1 , that is,

(−)l vi = i (−)r vi in M,   ∂vi  ∂ l−1 vi  vi |∂ M = = · · · = = 0, ∂ν ∂ M ∂ν l−1 ∂ M



vi (−)r v j = δij, i, j = 1, · · · , k. M

Let w =

k j=1

α ju j be such that



 w(−) w = 1 and

w(−)r v j = 0, ∀ j = 1, · · · , k − 1.

r

M

(2.18)

M

It follows from the Rayleigh-Ritz inequality that 

 M

=

k 

w(−) w =

k 

l

αi α jui λ j(−)r+s u j

M i, j=1

  k M i=1

αi2 λi  λk

k 

αi2 .

(2.19)

i=1

As in the proof of Theorem 1.1, we have for any j = 1, · · · , l − r that 

 j+1 w(−)r+ jw



j w(−)r+ j+1 w



M

.

(2.20)

M

Using (2.20) repeatedly, we get k  j=1

s/(l−r)



 α 2j

=

w(−)

r+s

w

M

w(−) w l

M

⎞s/(l−r) ⎛ ⎞s/(l−r) ⎛   k k  =⎝ α 2j λ j⎠ ⎝ α 2j λk ⎠ , M j=1

(2.21)

j=1

which implies that k 

s/(l−r−s)

α 2j  λk

(2.22)

j=1

Combining (2.19) with (2.22), one gets (1.21). Theorem 1.2 follows.



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Proof of Theorem 1.3 Let us consider first the case that r = 2h is even. We k will use the method in [13]. Let {ui }i=1 be a set of orthonormal eigenfunctions k corresponding to {λi }i=1 , that is,



(−)l ui = λi (−)r ui in ,   ∂ui  ∂ l−1 ui  = ··· = = 0, ui |∂ = ∂ν ∂ ∂ν l−1 ∂ 

ui r u j = δij, i, j = 1, · · · , k.

Let v1 , · · · , vn be n orthonormal eigenfunctions corresponding to μ1 , · · · , μn , that is, − vi = μi vi in ,  ∂vi  = 0, ∂ν ∂  vi v j = δij, i, j = 1, · · · n. 

By a translation of the origin and a suitable rotation of axes, we can assume that (Cf. [1])  

xi dx = 0, ∀ i = 1, · · · , n

and  

x jvi dx = 0, for j = 2, · · · , n and i = 1, · · · , j − 1.

It then follows from [1] that n  1  μi i=1

 

|x|2 dx . ||

(2.23)

Also, by a simple rearrangement argument, we have 

  || 2/n n || |x| dx  . n+2 ωn  2

(2.24)

Eigenvalues of Elliptic Operators and the Generalized Pólya Conjecture

247

Let us extend each ui to Rn by letting ui (x) = 0 for x ∈ Rn \  and set f j(x) ≡ h u j(x). For a function g on Rn , we will denote by F(g) the Fourier transformation of g. By definition, we have  F( f j)(z) = (2π )−n/2 e−i<x,z> f jdx. (2.25) 

Since { f j}kj=1 is an orthonormal set in L2 (), we have from Bessel’s inequality that  k  |F( f j)(z)|2  (2π )−n |e−i<x,z> |2 dx = (2π )−n ||. (2.26) 

j=1

For each q = 1, · · · , n, we have



zq F( f j)(z) = −iF

∂ fj ∂zq

 (z).

(2.27)

Set m = l − r; then by using (2.27) repeatedly, we have F((−)s f j)(z) = |z|2s F( f j)(z), when m = 2s is even,

(2.28)

 2 n   s     F ∂( f j) (z)   ∂z q

q=1

= |z|4s+2 |F( f j)(z)|2 , when m = 2s + 1 is odd. It then follows from the Plancherel formula that  |z|2m |F( f j)(z)|2 dz = λ j.

(2.29)

(2.30)

Rn

Also we have k 

−n

|∇ F( f j)(z)|  (2π ) 2

j=1

 

|x|2 dx.

(2.31)

Set G(z) =

k 

|F( f j)(z)|2 ;

(2.32)

j=1

then 0  G(z)  (2π )−n ||, ⎞1/2 ⎛ ⎞1/2 ⎛ k k   |∇G(z)|  2 ⎝ |F( f j)(z)|2 ⎠ ⎝ |∇ F( f j)(z)|2 ⎠ j=1

j=1



 2(2π )−n ||

1/2

 

|x|2 dx

(2.33)

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Q. Wang, C. Xia

for every z ∈ Rn and   G(z)dz = k, Rn

Rn

|z|2m G(z)dz =

k 

λ j.

(2.34)

j=1

Let G∗ (z) = g(|z|) be the decreasing spherical rearrangement of G. By approximation, we may assume that g : [0, +∞) → [0, (2π )−n ||] is absolutely continuous. Setting α(t) = |{G∗ > t}| = |{G > t}|, we have α(g(s)) = ωn sn

(2.35)

which implies that nωn sn−1 = α (g(s))g (s) for almost every s. Also the co-area formula [3] tells us that  1 α (t) = − (2.36) dAt . |∇G| {G=t}  1/2  Set η = 2(2π )−n ||  |x|2 dx ; then one gets from (2.33) and the isoperimetric inequality that − α (g(s))  η−1 area({G = g(s)})  η−1 nωn sn−1

(2.37)

− η  g (s)  0

(2.38)

and so

for almost every s. It follows from (2.34) that    ∗ k= G(z)dz = G (z)dz = nωn Rn

Rn

and k 

∞

sn−1 g(s)ds

(2.39)

0

 λj =

j=1

Rn

|z|2m G(z)dz

 

Rn

|z|2m G∗ (z)dz 

∞

= nωn

sn+2m−1 g(s)ds

(2.40)

0

since z → |z|2m is radial and increasing. Before we can finish the proof of (1.26), we will need the following lemma. 

Lemma 2.1 Let n  2, η, A > 0 and h : [0, +∞) → [0, +∞) be decreasing (and absolutely continuous) such that − η  h (s)  0 and



∞ 0

sn−1 h(s)ds = A.

(2.41)

(2.42)

Eigenvalues of Elliptic Operators and the Generalized Pólya Conjecture

249

Then for a f ixed positive integer m, we have  ∞ n+2m (nA) n 2m n−1+2m h(0)− n s h(s)ds  n + 2m 0 +

mAh(0)2m . (n + 2m)(2m + 1)22m−1 η2m

(2.43)

Proof of Lemma 2.1 We can suppose without loss of generality that η = 1 and h(0) = 1 since otherwise we can consider the function αh(βt) for appropriate α, β > 0 to obtain the desired inequality. Let us assume without loss of ∞ generality that B ≡ 0 sn−1+2m h(s)ds < ∞ and so lim dn−1+2m h(d j) = 0 j

(2.44)

j→∞

for some sequence (d j) with d j → ∞ as j → ∞. Set v(s) = −h (s) for s  0. Then 0  v(s)  1 and  ∞ v(s)ds = h(0) = 1.

(2.45)

0

Also we have  ∞

sn+2m v(s)ds

0

 = lim

T→∞

 −T

n+2m

h(T) + (n + 2m)

T n−1+2m

s

 h(s)ds

0

 (n + 2m)B.

∞

(2.46)

It follows from (2.45), (2.46) and Hölder’s inequality that 0 sn v(s)ds is finite. Observing (2.44), one deduces from integration by parts that    ∞  T n n n−1 s v(s)ds = lim −T h(T) + n s h(s)ds T→∞

0



∞

=n

0

sn−1 h(s)ds = nA.

(2.47)

0

Consider the function w : [0, ∞) → [0, ∞) given by  n (s − 1)(v(s) − 1) on [0, 1], w(s) = (sn − 1)v(s) on (1, ∞). Integrating w on [0, ∞), we obtain  ∞  n s v(s)ds  0

sn ds

(2.48)

0

Thus we can find an a  0 such that  a+1  n s ds = a

1

∞ 0

sn v(s)ds = nA.

(2.49)

250

Q. Wang, C. Xia

Let us choose λ, μ ∈ R such that the function p(s) = sn+2m − λsn + μ

(2.50)

satisfies p(a) = p(a + 1) = 0. Since p has at most one zero in (0, ∞) we know that p|(a,a+1) < 0, p|[0,∞)\(a,a+1) > 0 and that λ, μ  0 and consequently, we have p(s)(ξ(s) − v(s))  0, ∀s ∈ [0, ∞),

(2.51)

where ξ(s) denotes the characteristic function of the interval [a, a + 1]. Integrating (2.51) on [0, ∞), taking into account the choice of a and using (2.45), (2.46) and (2.49), we have  ∞  a+1 (n + 2m)B  sn+2m v(s)ds  sn+2m ds. (2.52) 0

a

We claim that for any s  0 and τ > 0 nsn+2m − (n + 2m)τ 2m sn + 2mτ n+2m  2mτ n (s − τ )2m .

(2.53)

Dividing the above inequality by τ n+2m , we know that our claim is equivalent to the following inequality nxn+r − (n + r)xn + r  r(1 − x)r , ∀ x  0,

(2.54)

where r is any fixed positive even integer. It is trivial that (2.54) holds when x = 0, 1. Let us separate two cases to prove (2.54). 

Case 1 0 < x < 1. In this case, since nxn+r − (n + r)xn + r = nxn (xr − 1) + r(1 − xn ), the inequality (2.54) is equivalent to     r 1 + x + · · · + xn−1 − nxn 1 + x + · · · + xr−1  r(1 − x)r−1 . From 0 < x < 1, it is easy to see that     r 1 + x + · · · + xn−1 − nxn 1 + x + · · · + xr−1  r(1 − x)  r(1 − x)r−1 .

Case 2 x > 1. In this case, (2.54) is equivalent to     nxn 1 + x + · · · + xr−1 − r 1 + x + · · · + xn−1  r(x − 1)r−1 . Observe that r is an integer no less than 2. One concludes easily that xr−1 − 1  (x − 1)r−1 + (r − 2)(x − 1)r−2 .

(2.55)

Eigenvalues of Elliptic Operators and the Generalized Pólya Conjecture

251

Thus we infer from n  2 and x > 1 that     nxn 1 + x + · · · + xr−1 − r 1 + x + · · · + xn−1     = nxn+r−1 + nxn 1 + x + · · · + xr−2 − r 1 + x + · · · + xn−1  nxn+r−1 + nxn (r − 1) − rnxn−1  nxn (xr−1 − 1)  2x2 (xr−1 − 1)     2 (x − 1)2 + 2(x − 1) + 1 (x − 1)r−1 + (r − 2)(x − 1)r−2  2(2r − 3)(x − 1)r−1  r(x − 1)r−1 .

(2.56)

Thus our claim is true. Integrating (2.53) over [a, a + 1], we get n(n + 2m)B − (n + 2m)τ 2m nA + 2mτ n+2m  a+1 n  2mτ (s − τ )2m ds a

  2mτ n

1/2 −1/2

t2m dt =

mτ n . 22m−1 (2m + 1)

(2.57)

Taking τ = (nA)1/n we infer n+2m

B

(nA) n mA . + n + 2m (n + 2m)(2m + 1)22m−1

(2.58)

This completes the proof of Lemma 2.1. Let us continue on the proof of Theorem 1.3. Applying Lemma 2.1 to the   1/2 function g with A = (nωn )−1 k, η = 2(2π )−n ||  |x|2 dx and using (2.40), we infer k 

λj 

j=1



mkg(0)2m n 2m 2m − 2m ωn n k1+ n g(0)− n + n + 2m (n + 2m)(2m + 1)22m−1 η2m n ckg(0)2m 2m 2m − 2m ωn n k1+ n g(0)− n + n + 2m (n + 2m)η2m

where c is any constant with 0 < c   η  2(2π )−n

(2.59)

m . From (2.24) we know that (2m + 1)22m−1 n n+2 −2 ωn n || n +1 . n+2

(2.60)

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Let us choose the constant c = c(n, m) by   m  16π 2 n m , c(n, m) = min . (2m + 1)22m−1 (n + 2)ωn4/n With the above choice of c, it is easy to see that the function θ(t) =

n ckt2m 2m 2m − 2m ωn n k1+ n t− n + n + 2m (n + 2m)η2m

satisfies θ ((2π )−n ||)  0 and so θ is decreasing on [0, (2π )−n ||]. Hence k 

λ j  θ((2π )−n ||)

j=1

n ck((2π )−n ||)2m 2m 2m − 2m ωn n k1+ n ((2π )−n ||)− n + n + 2m (n + 2m)η2m  m  2m n || 2π 1+ 2m − 2m n n  = k || + dn,m k , 2 n + 2m ωn1/n  |x| dx =

(2.61)

c . Substituting (2.23) into (2.61), one gets (1.26). + 2m) Let us consider now the case that r is odd. In this case, we deduce from item i) that the eigenvalues 1 , 2 , · · · , of the problem ⎧ r−1 u in ,  ⎨ (−)l u = (−)   ∂u  ∂ l−1 u  (2.62) = ··· = = 0. ⎩ u|∂ =  ∂ν ∂ ∂ν l−1 ∂ where d(n, m) =

22m (n

satisfy n k  n + 2p



2π 1/n

ωn

2 p

2p n

k ||

− 2np

+ b n, p

n  1 μ i i=1

− p .

(2.63)

Here p = l + 1 − r. One then obtains (1.27) by using Theorem 1.2. This completes the proof of Theorem 1.3. Acknowledgements The authors are very grateful to the referee for the careful reading of the manuscript and the valuable suggestions.

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Math Phys Anal Geom (2010) 13:255–273 DOI 10.1007/s11040-010-9078-7

Lagrangian Curves on Spectral Curves of Monopoles Brendan Guilfoyle · Madeeha Khalid · José J. Ramón Marí

Received: 21 May 2008 / Accepted: 1 July 2010 / Published online: 16 July 2010 © Springer Science+Business Media B.V. 2010

Abstract We study Lagrangian points on smooth holomorphic curves in TP1 equipped with a natural neutral Kähler structure, and prove that they must form real curves. By virtue of the identification of TP1 with the space L(E3 ) of oriented affine lines in Euclidean 3-space E3 , these Lagrangian curves give rise to ruled surfaces in E3 , which we prove have zero Gauss curvature. Each ruled surface is shown to be the tangent lines to a curve in E3 , called the edge of regression of the ruled surface. We give an alternative characterization of these curves as the points in E3 where the number of oriented lines in the complex curve  that pass through the point is less than the degree of . We then apply these results to the spectral curves of certain monopoles and construct the ruled surfaces and edges of regression generated by the Lagrangian curves. Keywords Neutral Kaehler structure · Monopoles · Lagrangian curves Mathematics Subject Classifications (2010) Primary—53A25; Secondary—81T13

B. Guilfoyle (B) · M. Khalid Department of Mathematics, Institute of Technology, Tralee, Clash, Tralee, Co. Kerry, Ireland e-mail: [email protected] M. Khalid e-mail: [email protected] J. J. Ramón Marí IT Tralee Research Institute, Institute of Technology, Tralee, Clash, Tralee, Co. Kerry, Ireland e-mail: [email protected]

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The space L(E3 ) of oriented affine lines in Euclidean 3-space E3 can be identified with the total space TS2 of the tangent bundle to the 2-sphere. The standard complex structure on P1 lifts to a complex structure J on TP1 , and hence L(E3 ). This is well-known and has found a variety of uses, most notably in the construction of monopoles in E3 [6]. What is less well-known is the canonical symplectic structure on L(E3 ) which is compatible with J and enjoys many remarkable geometric properties [3, 4, 11]. Together the complex and symplectic structures form a Kähler structure with the property that the associated metric is of signature (+ + −−). In this paper we consider Lagrangian points on complex curves in TP1 equipped with this neutral Kähler structure. At such points the metric induced on the surface vanishes. The only complex curves that are Lagrangian at every point are the oriented normals to planes and spheres in E3 , and we exclude these curves. Our first main result is: Main Theorem 1 Let  be a smooth compact complex curve in T P1 . (i) The branch points of the composition  → TP1 → P1 are Lagrangian, (ii) there do not exist any isolated Lagrangian points on , (iii) if C ⊂  is a Lagrangian curve, then the associated ruled surface in E3 has zero Gauss curvature. We show that a ruled surface in part (iii) is tangent to a curve in E3 , called the edge of regression of the ruled surface. We can characterize this curve in E3 another way. Given a holomorphic curve  consider the number of oriented lines in  that pass through a given point in E3 . Generically, we show that this number is the degree of  when it is considered as a curve in P3 . Moreover, we prove: Main Theorem 2 Let  be a smooth complex curve in T P1 . Then the genus of  is (m − 1)2 for m = 1, 2, 3... and a generic point in E3 has 2m distinct oriented lines of  passing through it. This is the maximum number of distinct oriented lines of  that can pass through a point (the minimum number being one). The points in E3 lying on less than 2m distinct oriented lines of  form the edges of regression of the ruled surfaces generated by the Lagrangian curves on . In fact,  embeds in P3 as a smooth complex curve of degree 2m and the edge of regression exactly parameterizes the family of real sections of T P1 which intersect  non-transversally. We apply this to the spectral curves of the charge 2 and charge 3 monopoles [8], paying particular attention to the ruled surfaces and the edges of regression. We show that the Lagrangian curves in the charge 2 case consists of 4

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disjoint curves, 2 passing through the branch points and 2 which do not. These curves generate two ruled surfaces in E3 which have edges of regression an ellipse in the x1 x2 −plane, and a hyperbola in the x1 x3 −plane. The eccentricity of the ellipse is k, while that of the hyperbola is with 1/k, where k is the spectral parameter of the monopole. For the charge 3 monopole we show that the set of Lagrangian points consists of ten (intersecting) simple closed curves. These fall naturally into two classes: six of the curves are rulings of planes, in particular, six planes that form the edges of a tetrahedron in E3 . The remaining four curves are rulings of flat, non-planar surfaces forming the faces of the tetrahedron. Finally, in the last section, we discuss the results and place them in the more general setting of null curves on neutral Kähler surfaces.

1 Lagrangian Curves on Holomorphic Curves The total space TS2 of the tangent bundle to the 2-sphere is a 4-manifold with a natural complex structure defined as follows. Let ξ be the standard holomorphic coordinate on P1 given by stereographic projection from the south pole. Let η be the corresponding complex coordinate in the fibre of the bundle π : TS2 → S2 obtained by identifying the pair of complex numbers (ξ, η) with the tangent vector η

∂ ∂ ∈ Tξ S2 . + η¯ ∂ξ ∂ ξ¯

This yields holomorphic coordinates on TS2 − π −1 {south pole}, which can be supplemented by an analogous coordinate system on TS2 − π −1 {south pole}. The transition functions on the overlap are holomorphic and so we obtain a complex structure J on TS2 . For short we write TP1 for TS2 with this complex structure. The space L(E3 ) of oriented affine lines in Euclidean 3-space E3 can be identified with the total space of TP1 , and thus inherits the above complex structure. In fact, this complex structure is natural in the sense that it is invariant under the action of the Euclidean group acting on L(E3 ) [3]. Geometric data can be transferred between L(E3 ) and E3 by use of a correspondence space. Definition 1 The map  : L(E3 ) × R → E3 is defined to take (γ , r) ∈ L(E3 ) × R to the point in E3 on the oriented line γ that lies a distance r from the point on the line closest to the origin. The double fibration below gives us the correspondence between the points in L(E3 ) and oriented lines in E3 : we identify a point γ in L(E3 ) with  ◦ π1−1 (γ ) ⊂ E3 , which is an oriented line. Similarly, a point p in E3 is identified

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with the 2-sphere π1 ◦ −1 ( p) ⊂ L(E3 ), which consists of all of the oriented lines through the point p. L(E3 ) × R

π1

@

? L(E3 )

@ @ R @

E3

If ((ξ, η), r) = (z(ξ, η, r), t(ξ, η, r)), then it has the following coordinate expression [2]: z=

2(η − ηξ ¯ 2 ) + 2ξ(1 + ξ ξ¯ )r (1 + ξ ξ¯ )2

t=

−2(ηξ¯ + ηξ ¯ ) + (1 − ξ 2 ξ¯ 2 )r , (1.1) (1 + ξ ξ¯ )2

where z = x1 + ix2 , t = x3 and (x1 , x2 , x3 ) are Euclidean coordinates in E3 . The symplectic structure on TP1 can be motivated in a number of ways—for our purposes we will simply use its coordinate expression (more details can be found in [3]):   2 2(ξ η¯ − ξ¯ η) ¯ ¯ dξ ∧ dξ . (1.2)

= dη ∧ dξ + dη¯ ∧ dξ + (1 + ξ ξ¯ )2 1 + ξ ξ¯ This is clearly a closed non-degenerate 2-form on TP1 which is compatible with J

(J(X), J(Y)) = (X, Y)

for all

X, Y ∈ Tγ TP1 .

The symplectic 2-form is also invariant under the action of the Euclidean group [3]. The neutral Kähler metric on TP1 is defined by G(·, ·) = (J·, ·) and has local coordinate expression   2i 2(ξ η¯ − ξ¯ η) ¯ ¯ dξ dξ . G= dηdξ − dηdξ ¯ + (1 + ξ ξ¯ )2 1 + ξ ξ¯ We now consider a real 2-dimensional surface  ⊂ TP1 . Definition 2 A point γ ∈  is a complex point if J :Tγ P1 →Tγ P1 . A surface  is called a complex curve (or holomorphic curve) if every point of  is a complex point. A point γ ∈  is said to be a Lagrangian point if the symplectic 2-form

pulled back to Tγ  is zero. A surface  is called Lagrangian if every point of  is a Lagrangian point. The only real surfaces in TP1 that are both complex and Lagrangian at every point are the oriented normal lines to planes and spheres in E3 . In what follows we exclude this case.

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We now prove: Main Theorem 1 Let  be a smooth compact complex curve in T P1 . (i) The branch points of the composition  → T P1 → P1 are Lagrangian, (ii) there do not exist any isolated Lagrangian points on , (iii) if C ⊂  is a Lagrangian curve, then the associated ruled surface in E3 has zero Gauss curvature. Proof Let  be a (real) surface in TP1 . About any point γ ∈  there is a local ¯ η(ν, ν¯ )), where we assume, without parameterization C → C2 :ν → (ξ(ν, ν), loss of generality, that γ does not lie in the fibre over the south pole. The real surface  is holomorphic iff about each point of  we have σ = 0 where [2] ¯ − ∂ξ ¯ ∂η, σ = ∂ξ ∂η

(1.3)

∂ being differentiation with respect to the parameter ν. On the other hand, by pulling back the 2-form , we see that a real surface is Lagrangian at a point γ ∈  iff we have λ = Im ρ = 0 at γ , where ¯   ¯ ξ¯ − 2ξ η ∂ξ ∂¯ ξ¯ − ∂ξ ¯ ∂ ξ¯ . ρ = ∂η∂¯ ξ¯ − ∂η∂ ¯ 1 + ξξ

(1.4)

It is also clear that  is locally the graph of a section of the bundle π : TP1 → P iff 1

¯ ∂ ξ¯ = 0. ∂ξ ∂¯ ξ¯ − ∂ξ We now turn to the proofs of statements (i) to (iii). Let  be a holomorphic curve in TP1 so that σ = 0. Proof of (i) Suppose that γ ∈  is a branch point. Then, as it is smooth, the curve osculates the fibre of the bundle π : TP1 → P1 at γ and so ¯ ∂ ξ¯ = 0, ∂ξ ∂¯ ξ¯ − ∂ξ at γ . A short calculation shows that ¯ η)(∂ξ ¯ ∂ ξ¯ ), ρ ρ¯ − σ σ¯ = (∂η∂¯ η¯ − ∂η∂ ¯ ∂¯ ξ¯ − ∂ξ which therefore vanishes at γ . However, σ = 0 and so we conclude that at a branch point ρ = 0. In particular, λ = Im ρ = 0 at γ , and so the point is Lagrangian, as claimed. Proof of (ii) We argue by contradiction. Let γ ∈  be an isolated Lagrangian point, which we assume, without loss of generality, does not lie on π −1 {south pole}. Thus there exists an open neighbourhood U⊂  containing γ such that λ(γ ) = 0

λ (U − {γ }) = 0.

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First suppose that the Lagrangian point γ is a branch point of the mapping  → TP1 → P1 . Then, since the projection restricted to  is not of maximal rank,  cannot be locally parameterized by a section of this bundle. However, as  is smooth, we can use the fibre coordinate as a local parameter about ¯ = 0 and by (1.4) γ : η → (ξ(η, η), ¯ η). Since σ = 0, by (1.3) we have that ∂ξ λ = 2i (∂ξ − ∂¯ ξ¯ ). Thus λ is the imaginary part of a holomorphic function and therefore its zeros cannot be isolated. This proves the claim when γ is a branch point. Now suppose that the Lagrangian point γ is not a branch point. Then we can parameterize a neighbourhood U of γ on  by a local section of the ¯ = 0 and bundle: ξ → (ξ, η(ξ, ξ¯ )). Since  is holomorphic, by (1.3) we have ∂η by (1.4)   η λ = Im (1 + ξ ξ¯ )2 ∂ . (1 + ξ ξ¯ )2 A short computation shows then that λ satisfies the second order equation ¯ + ∂ ∂λ

2λ = 0. 1 + ξ ξ¯

The strong maximum and minimum principles (cf. Theorem 2.2 of [1]) imply that zeros of λ cannot be isolated. For, suppose that λ  0 on U. Then λ is ¯  0, while λ(γ ) = infU λ. By the strong minimum superharmonic on U: ∂ ∂λ principle, λ is constant, in fact zero, on U, which is a contradiction (as we have ruled out the oriented normals to planes and spheres). ¯  0, while λ(γ ) = supU λ. If λ  0 on U, then λ is subharmonic on U: ∂ ∂λ By the strong maximum principle, λ is constant, in fact zero, on U, which again is a contradiction. We conclude that none of the Lagrangian points on the holomorphic curve can be isolated. Proof of (iii) Classically, a ruled surface is a 1-parameter family of oriented lines in E3 . From our point of view, a ruled surface is a real curve C in TP1 . Suppose that this curve is given locally by s → (ξ(s), η(s)). Then, by (1.1), the ruled surface is z(r, s) =

t(r, s) =

2 ] + 2ξ(s)[1 + ξ(s)ξ¯ (s)]r 2[η(s) − η(s)ξ(s) ¯ , [1 + ξ(s)ξ¯ (s)]2

−2[η(s)ξ¯ (s) + η(s)ξ(s)] ¯ + [1 − ξ(s)2 ξ¯ (s)2 ]r , [1 + ξ(s)ξ¯ (s)]2

where, as before, z = x1 + ix2 , t = x3 and (x1 , x2 , x3 ) are Euclidean coordinates in E3 , and r is an affine parameter along the lines of the ruling.

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By a straightforward, if lengthy, curvature calculation, the Gauss curvature of such a ruled surface is found to be:    2 (1 + ξ ξ¯ )2 Im (1 + ξ ξ¯ )η˙ ξ˙¯ + 2ξ η¯ ξ˙ ξ˙¯ K=−

, (1.5)

(1 + ξ ξ¯ )η˙ − 2ξ¯ ηξ˙ + (1 + ξ ξ¯ )rξ˙ 4 where a dot represents differentiation with respect to s. If the real curve C lies on a holomorphic curve we have η˙ = ∂ηξ˙ and the Gauss curvature simplifies to K=−

λ2 , (λ + (r + ψ)2 )2

where ρ = ψ + iλ as in (1.4). Along a Lagrangian curve λ = 0, and so the Gauss curvature of the ruled surface vanishes for such a curve. This completes the proof of the theorem.

Flat ruled surfaces (referred to as developable ruled surfaces) were studied in classical surface theory. Aside from rulings of a plane, other examples of flat ruled surfaces include generalized cones and cylinders. In fact: Theorem 1 The ruled surface generated by a Lagrangian curve on a holomorphic curve is the tangent lines to an oriented curve in E3 . Proof A well-known result of classical surface theory states that every developable surface can be subdivided into portions of a cylinder, a cone or the tangent line to a curve in E3 (for example, see Thm. 58.3 of [9]). For the situation stated in the theorem we eliminate the first two possibilities: generalised cylinder and cone, as follows. The generalised cylinder is obtained by translating an oriented line along a curve in the plane orthogonal to the oriented line. Clearly the direction of the lines in this ruling do not change and such a curve in TP1 must therefore lie in a fibre of the projection π : TP1 → P1 . This is not the case for Lagrangian curves on holomorphic curves (as we have ruled out the case of the oriented normals to planes) and so the ruled surface cannot be a generalised cylinder. The generalised cone is a 1-parameter family of oriented lines passing through a fixed point p in E3 . Such a curve in TP1 lies on the holomorphic sphere of all oriented lines through p. Thus, were a Lagrangian curve C on a holomorphic curve to form a generalized cone, it would lie on the intersection of two holomorphic curves, an impossibility for a 1-dimensional set.

Thus every Lagrangian curve on a holomorphic curve gives rise to a curve in E3 : the edge of regression of the ruled surface. This subset of E3 can be defined in a different way, as we show in the next section.

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2 A Different Characterization In this section we give an alternative characterization of the set of Lagrangian points on a smooth complex curve in TP1 . In particular, we prove: Main Theorem 2 Let  be a smooth complex curve in TP1 . Then the genus of  is (m − 1)2 for m = 1, 2, 3... and a generic point in E3 has 2m distinct oriented lines of  passing through it. This is the maximum number of distinct oriented lines of  that can pass through a point (the minimum number being one). The points in E3 lying on less than 2m distinct oriented lines of  form the edges of regression of the ruled surfaces generated by the Lagrangian curves on . Proof The set of oriented lines through a point in E3 forms a global holomorphic section of the complex vector bundle TP1 → P1 . In particular, an oriented line (ξ, η) passes through a fixed point (z, t) ∈ C ⊕ R = E3 iff  1 (2.1) z − 2tξ − z¯ ξ 2 . 2 Let F2 = P(O ⊕ O(2)) be the Hirzebruch surface obtained by the quotient of C2 − {0} × C2 − {0} by the equivalence relation η=

(x0 , x1 , y0 , y1 ) ∼ (ax0 , ax1 , b y0 , b a2 y1 ), for a, b ∈ C∗ . Projecting onto the first factor, we see that F2 is, in fact, a P1 bundle over P1 . TP1 is isomorphic to F2 − E∞ , where E∞ = {(x0 , x1 , 0, y1 )}/ ∼ is the infinity section. That is, F2 is isomorphic to TP1 after the one point compactification of each of the fibres of the canonical projection TP1 → P1 . Now, the Picard group of F2 is Pic(F2 ) = Z[h] ⊕ Z[ f ] where [h] and [ f ] are the divisor classes of the holomorphic sections and fibres, respectively, of the bundle F2 → P1 . These classes have intersection pairing [h] · [h] = 2,

[ f ] · [ f ] = 0,

[h] · [ f ] = 1,

and in this basis [E∞ ] = [h] − 2[ f ].

(2.2)

To see this, note that [E∞ ] = k[h] + l[ f ] for some k, l ∈ Z, and taking the intersection with [h] and [ f ] we find that, since [E∞ ] · [h] = 0 and [E∞ ] · [ f ] = 1, 2k + l = 0 and k = 1, which yield (2.2). Thus, if  is a compact complex curve in TP1 = F2 − E∞ , we have 0 = [E∞ ] · [] = [h] · [] − 2[ f ] · []. We conclude that [h] · [] = 2[ f ] · [] = 2m for some m ∈ N, and therefore  intersects a generic holomorphic section of TP1 in 2m distinct points. This extends to generic holomorphic sections of the form (2.1) and so we conclude that a generic point in E3 has 2m distinct oriented lines of  passing through it.

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To determine the genus g of , we use the adjunction formula  

K F2 + []  = K , where K F2 and K are the canonical bundles of F2 and , respectively. This implies that [] · [] + K F2 · [] = 2g − 2.

(2.3)

By Lemma V 2.10 of [5] and (2.2) K F2 = −2[E∞ ] − 4[ f ] = −2[h], and a routine calculation shows that [] = m[h] for some m ∈ N. Thus [] · [] + K F2 · [] = 2m2 − 4m, and by (2.3) we find that g = (m − 1)2 as claimed. Away from the branch points,  is given locally by a holomorphic section η = F(ξ ). Let p∈ E3 and consider the oriented lines in  that pass through p. If p has coordinates (z, t) as above, then we are seeking to find the roots of  1 z − 2tξ − z¯ ξ 2 . 2 An oriented line (ξ0 , F(ξ0 )) is a multiple root of G iff G(ξ0 ) = 0 and ∂G(ξ0 ) = 0. These are equivalent to G=F−

F(ξ0 ) =

(1 + ξ ξ¯ )2 ∂



 1 z − 2tξ0 − z¯ ξ02 , 2



¯ ¯

F

= − zξ0 + z¯ ξ0 + t(1 − ξ0 ξ0 ) . (1 + ξ ξ¯ )2 ξ0 1 + ξ0 ξ¯0

(2.4)

(2.5)

Suppose now that (ξ0 , F(ξ0 )) ∈  is a multiple root of G and so the above equations hold for some (z, t). Then clearly   

F

= 0, Im (1 + ξ ξ¯ )2 ∂

2 ¯ (1 + ξ ξ ) ξ0 and so the point is Lagrangian. Conversely, suppose that (ξ0 , F(ξ0 )) ∈  is Lagrangian, and define  

F

. r0 = −(1 + ξ ξ¯ )2 ∂ (1 + ξ ξ¯ )2 ξ0

(2.6)

By the Lagrangian condition this is a real number. Now consider the following point (z, t) ∈ E3 : z=

2[F(ξ0 ) − F(ξ0 )ξ02 ] + 2ξ0 (1 + ξ0 ξ¯0 )r0 , (1 + ξ0 ξ¯0 )2

t=

−2[F(ξ0 )ξ¯0 + F(ξ0 )ξ0 ] + (1 − ξ02 ξ¯02 )r0 , (1 + ξ0 ξ¯0 )2

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A calculation shows that (2.4) and (2.5) hold, so that less than 2m distinct oriented lines pass through the point (z, t), and this point lies on the oriented

line (ξ0 , F(ξ0 )) ∈ . Note 1 F2 is a resolution of the quadric cone Q in P3 , where E∞ is the exceptional divisor. The intersection number 2m above is the degree of  when considered as a curve in P3 . This can be described in local coordinates as follows. Let [z0 : z1 : z2 : z3 ] be homogenous coordinates on P3 . As described above, TP1 can be identified with an open subset of F2 , and F2 can be mapped to P3 by (x0 , x1 , y0 , y1 ) → [x20 y0 : x0 x1 y0 : x21 y0 : y1 ] Clearly, F2 maps to the quadric cone Q given by z0 z2 = z21 , and the infinity section E∞ maps to the vertex p = [0 : 0 : 0 : 1] of the cone. The result is an identification of TP1 with Q − p which can be written locally (ξ, η) ←→ [1 : ξ : ξ 2 : η].

(2.7)

A curve  in TP1 maps to a curve on Q − p ⊂ P3 . But for any complex curve in P3 there is a well-defined degree d = deg  = #( ∩ P2 ) where P2 is a generic complex plane in P3 . Now, a generic P2 is given by a0 z0 + a1 z1 + a2 z2 + a3 z3 = 0, so that, using the identification (2.7), Q ∩ P2 is a0 + a1 ξ + a2 ξ 2 + a3 η = 0. For a4 = 0 this defines a global holomorphic section of TP1 → P1 and so d is exactly the total number of points of intersection of  and a generic holomorphic section, that is, d = 2m. Note 2 Main Theorem 2 restricts the genus of a smooth holomorphic curve in TP1 . In contrast, taking the oriented normal lines to a smooth surface of genus g in E3 we obtain a smooth Lagrangian surface of genus g in TP1 , and so there exist smooth Lagrangian surfaces of any genus in TP1 . 3 Lagrangian Curves on Spectral Curves The complex structure on TP1 plays a crucial role in Hitchin’s approach to BPS monopoles for SU(2) Yang-Mills-Higgs theory in E3 [6]. Each such monopole can be constructed from its spectral curve: a compact holomorphic curve in TP1 which is given in our coordinates by ηm + α1 (ξ )ηm−1 + ... + αm (ξ ) = 0,

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where each α j is a complex polynomial of degree less than or equal to 2 j satisfying the reality condition   1 α j(ξ ) = (−1) jξ 2 jα j − . ξ¯ Here m ∈ N is the charge of the monopole, and the moduli space of gauge inequivalent charge m monopoles is a (4m-1)-dimensional manifold. We now consider the Lagrangian curves on the spectral curves of the charge 2 and the tetrahedrally symmetric charge 3 monopole. 3.1 The Charge 2 Monopole The moduli space of gauge inequivalent charge 2 monopoles is a 7-dimensional manifold. However, six of these degrees of freedom can be removed by the action of the Euclidean group, so that there is only a 1-parameter family of charge 2 monopoles. The spectral curve  can be described in local coordinates on TP1 by [10]

 η2 = α k2 (1 + ξ 4 ) − 2(2 − k2 )ξ 2 , (3.1) where the spectral parameter k satisfies 0  k < 1 and   π 2 1 2 dθ  α= . 2 0 1 − k2 sin2 θ The parameter k measures the separation of the 2 monopoles. As k → 1, the separation of the monopoles goes to infinity. The case k = 0, which corresponds to the 2 monopoles coinciding at the origin, leads to a singular spectral curve and will be excluded in what follows. For k = 0 the spectral curve is a smooth compact complex curve of genus 1 in TP1 which double covers P1 with four branch points γ1 , γ2 , γ3 , γ4 . These points lie at the four real points  √ 2 − k2 ± 2 1 − k2 ξ =± . k2 We start by finding where the symplectic form on TP1 pulled back to  vanishes:      η η¯ 4λi ¯

| = 2 ∂ −∂ dξ ∧ dξ¯ = dξ ∧ dξ¯ = 0. 2 2 ¯ ¯ (1 + ξ ξ ) (1 + ξ ξ ) (1 + ξ ξ¯ )2 Differentiating (3.1) we compute that  2α 2 3 k ξ − (2 − k2 )ξ , ∂η = η and so ∂η −

2  2ξ¯ η 2α = −k ξ¯ − (2 − k2 )ξ + (2 − k2 )ξ 2 ξ¯ + k2 ξ 3 . ¯ ¯ 1 + ξξ η(1 + ξ ξ )

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Upon squaring, the Lagrangian condition λ = 0 becomes

2 2 −k ξ¯ − (2 − k2 )ξ + (2 − k2 )ξ 2 ξ¯ + k2 ξ 3 k2 (1 + ξ 4 ) − 2(2 − k2 )ξ 2 2

2 −k ξ − (2 − k2 )ξ¯ + (2 − k2 )ξ ξ¯ 2 + k2 ξ¯ 3 = . k2 (1 + ξ¯ 4 ) − 2(2 − k2 )ξ¯ 2 This can be simplified and factorized to k2 (1 − k2 )(1 − ξ ξ¯ )(1 + ξ ξ¯ )3 (ξ − ξ¯ )(ξ + ξ¯ ) = 0. C1

γ

1

γ

γ

2

γ

3

4

C3

C2 |ξ|=1

Aside from the limits k = 0 and k = 1, we therefore have |ξ | = 1, ξ = ξ¯ or ξ = −ξ¯ . The last of these does not lift to a Lagrangian curve on , being an artefact of the squaring of the Lagrangian condition. Thus the Lagrangian points on  project down to two curves on P1 : the equator and a line of longitude, which we now consider in detail. Firstly, parameterize the equator C1 : |ξ | = 1 by ξ = eiθ . Then, we compute 

η2 = α k2 (1 + e4iθ ) − 2(2 − k2 )e2iθ = −4α(1 − k2 cos2 θ)e2iθ . √ √ The curve C1 lifts to two curves η = ±2i α 1 − k2 cos2 θ eiθ on . Substituting this in (1.1) we find the ruled surface in E3 is √ √ x1 = ∓2 α 1 − k2 cos2 θ sin θ + r cos θ √ √ x2 = ±2 α 1 − k2 cos2 θ cos θ + r sin θ, x3 = 0. The other Lagrangian points are given by ξ = ξ¯ . Suppose ξ = tan(θ/2) for −π  θ  π then √  2 α(k2 − sin2 θ) α k − sin2 θ 2 or η = ± . η = cos4 (θ/2) cos2 (θ/2)

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To be Lagrangian, clearly we must have η ∈ R, so sin2 θ  k2 . This implies that θ must lie in the following domains: −π  θ  −π + sin−1 k

− sin−1 k  θ  sin−1 k

π − sin−1 k  θ  π.

The first and last of these domains are connected: they form the curve C3 when projected onto P1 and contain the branch points γ1 and γ4 . The middle domain forms the curve C2 which passes through the branch points γ2 and γ3 . Of course each interval lifts to two copies in TP1 joined at the branch points, forming circles in the total space. The ruled surfaces in E3 , obtained by inserting our parameterized curves in (1.1), are √  x1 = ∓2 α k2 − sin2 θ cos θ + r sin θ x2 = 0,

√  x3 = ±2 α k2 − sin2 θ sin θ + r cos θ. We thus have shown that the Lagrangian curves on the charge 2 spectral curve yields rulings of the x1 x2 − and x1 x3 −planes. We can find the edge of regression of such a ruling by using the procedure given by (2.6). In particular, the edge of regression of these lines is obtained by substituting r = −∂η +

2ξ¯ η , 1 + ξ ξ¯

in (1.1). The result for the lift of curve C1 is √ √ 2 α sin θ 2 α(1 − k2 ) cos θ x2 = ± √ x1 = ∓ √ 1 − k2 cos2 θ 1 − k2 cos2 θ

x3 = 0,

This is an ellipse of eccentricity k, as can be seen by noting that the parameterized curve satisfies: (x2 )2 (x1 )2 + = 1. 4α 4α(1 − k2 ) For the lift of curves C2 and C3 we obtain the edge of regression √ √ 2 αk2 cos θ 2 α(k2 − 1) sin θ x3 = ±  x1 = ∓  x2 = 0 , k2 − sin2 θ k2 − sin2 θ which is a hyperbola with eccentricity 1/k. To see this, note that (x1 )2 (x2 )2 = 1. − 4αk2 4α(1 − k2 ) In the diagram below we show the ruled surface generated by the Lagrangian curves on the charge 2 monopole with k = 0.8—the edges of regression (a hyperbola and an ellipse) are clearly identifiable. Note that the

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4 branch points of the spectral curve are the two asymptotes of the hyperbola (counted once with each orientation).

2 1 0 -1 -2 2

0

-2

-4

-6

-4

-2

0

2

4

6

3.2 The Charge 3 Monopole The spectral curve  of the tetrahedrally symmetric charge 3 monopole is [8]:  9 √  13 η = √ (1 − 5 2ξ 3 − ξ 6 ). 48 6π 3 3

(3.2)

We start by pulling back the symplectic 2-form to  and finding that the Lagrangian curves are given by the equation:    √ √ Im η2 2ξ + 5 2ξ¯ 2 − 5 2ξ ξ¯ 3 + 2ξ¯ 5 = 0. Substitute (3.2) in this and cube the resulting equation to get Im



2  3 √ √ √ 1 − 5 2ξ 3 − ξ 6 2ξ + 5 2ξ¯ 2 − 5 2ξ ξ¯ 3 + 2ξ¯ 5 = 0.

We factorize this to (ξ − ξ¯ )(1 + ξ ξ¯ )3 (ξ 2 + ξ ξ¯ + ξ¯ 2 )Re( f (ξ, ξ¯ )) = 0,

(3.3)

where √ √ √ f (ξ, ξ¯ ) = − 4ξ 9 ξ¯ 9 − 165 2ξ 9 ξ¯ 6 − 42ξ 9 ξ¯ 3 + 10 2ξ 9 + 162ξ 8 ξ¯ 8 + 810 2ξ 8 ξ¯ 5 √ √ − 162ξ 8 ξ¯ 2 − 162ξ 7 ξ¯ 7 − 810 2ξ 7 ξ¯ 4 + 162ξ 7 ξ¯ − 2988ξ 6 ξ¯ 6 + 360 2ξ 6 ξ¯ 3 √ √ + 42ξ 6 + 8100ξ 5 ξ¯ 5 − 810 2ξ 5 ξ¯ 2 − 8100ξ 4 ξ¯ 4 + 810 2ξ 4 ξ¯ √ + 2988ξ 3 ξ¯ 3 − 165 2ξ 3 + 162ξ 2 ξ¯ 2 − 162ξ ξ¯ + 4.

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The (3.3) defines the set of points on P1 obtained by the projection of the Lagrangian curves on . The set can be plotted numerically, which we do below (after stereographic projection onto the plane) on two scales:

8

2

6 4

1

2 0

0

-2 -4

-1

-6 -2

-1

0

1

2

-2

-8

-6

-4

-2

0

2

4

6

8

-8

It is clear from this plot, that the zero set of the symplectic 2-form on the spectral curve projects to a union of 10 simple closed curves on P1 . More formally, we can exploit the tetrahedral symmetry to factorize the polynomial f (ξ, ξ¯ ) of (3.3). In our coordinates, the tetrahedral group T is the subgroup of SU(2) generated by the 12 fractional linear transformations ξ → gk (ξ )ev ji for j = 1, 2, 3 and k = 0, 1, 2, 3, where g0 (ξ ) = ξ,

g j(ξ ) =

α jξ − β¯ j , β jξ + α¯ j

and √ 3+i α= √ , 2 3

√ 2i β j = √ e−v ji , 3

for v1 = 0, v2 = 2π and v3 = 4π . Note that the spectral curve we are studying 3 3 is invariant under this group action, as can be seen by transforming the defining (3.2). Now, one factor of (3.3) is ξ − ξ¯ and so one of the curves is ξ = ξ¯ . Thus one of the Lagrangian curves projects to a great circle on P1 , and, acting by the tetrahedral group, we pick out five more great circles over which lie Lagrangian points. In fact, the explicit description of the tetrahedral group above allows us to extract these six circles directly, and we find the following factor of f : f (ξ, ξ¯ ) = (4ξ 3 ξ¯ 3 +

√ 3 √ 2ξ − 18ξ 2 ξ¯ 2 + 18ξ ξ¯ + 2ξ¯ 3 − 4)g(ξ, ξ¯ ).

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Thus the projection of the Lagrangian curves splits naturally into two classes: six great circles and the solutions of the remainder of f , which is Re(g(ξ, ξ¯ )) = 0 where √ √ √ g(ξ, ξ¯ ) = ξ 6 ξ¯ 6 + 41 2ξ 6 ξ¯ 3 − 10ξ 6 − 36ξ 5 ξ¯ 5 − 9 2ξ 5 ξ¯ 2 − 126ξ 4 ξ¯ 4 + 9 2ξ 4 ξ¯ √ (3.4) + 302ξ 3 ξ¯ 3 − 41 2ξ 3 − 126ξ 2 ξ¯ 2 − 36ξ ξ¯ + 1. These two classes of curves on P1 are shown below, the left hand being the great circles, while the right hand shows the solution set of Re g(ξ, ξ¯ ) = 0. 8 6 4 2 0 -2 -4 -6 -8 -8

-6

-4

-2

0

2

4

8 -8

6

-6

-4

-2

0

2

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6

8

Let us now look at the ruled surfaces in E3 generated by the great circles. First, these curves contain the six branch points. Taking the circle ξ = ξ¯ = s, and using (1.1) we find the ruled surface to be √ √ 1 ( 13 )3 (1 − s2 )(1 − 5 2s3 − s6 ) 3 + 2 6π s(1 + s2 )r 1 x = , x2 = 0, √ 6π(1 + s2 )2 √ √ 1 −2( 13 )3 s(1 − 5 2s3 − s6 ) 3 + 6π(1 − s4 )r 3 . x = √ 6π(1 + s2 )2 The edge of regression of this ruling turns out to be ( 1 )3 (1 − s2 + s4 ) x =√ 3 , √ 2 6π(1 − 5 2s3 − s6 ) 3 1

x = 0, 2

√ ( 13 )3 s(2 − 2s2 − 5 2s) x =− √ . √ 2 2 6π(1 − 5 2s3 − s6 ) 3 3

Together with the other five ruled planes, we get the six planes that pass through the edges and centroid of a tetrahedron. We turn now to the remaining Lagrangian curves, which project to Re(g(ξ, ξ¯ )) = 0 with g given by (3.4). While the zero set is made up of four simple closed curves, it turns out that g is not factorizable over R. We argue this as follows. If g factorized into four components, then the tetrahedral action will either leave a component invariant, or move it to another component. From the diagram, it is clear that under rotation about the origin through 2π/3 and 4π/3

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(which is the stereographic projection of a cyclic subgroup of T ) three of the components change place and one is left invariant. Thus if g factorizes into four components, at least one of the components must be invariant under the cyclic group C3 . Consider now the points where our curve crosses the real axis. If ξ = u + iv, then Re(g(u, u)) is factorizable over R, in fact, it is equal to: √ √ √ √ √ (u2 − 2 2u− 1)2 (u4 + 5 2u3 − 3u2 − 5 2u + 1)(u4 − 2u3 + 3u2 + 2u + 1). Each of these factors (and their products) contains a term that is linear in u, and therefore cannot come from the restriction of a C3 -invariant factor to ξ = u. Thus, no such factorization exists. We can still numerically plot the associated (non-planar) ruled surface in E3 , although we cannot write down the edge of regression parametrically. Combining the information we have obtained on the Lagrangian curves of the charge three monopole, we show the ten edges of regression in the figure below.

Note that, once again, the branch points of the spectral curve are the asymptotes of the edges of regression.

4 Discussion The broader geometric context of the preceding is that of Kähler surfaces. Thus we consider a real 4-manifold M endowed with a complex structure J, compatible symplectic structure and metric G. In the case we have considered, the metric G is of neutral signature (2,2), and so exhibits a rich interplay between holomorphic and symplectic structures that is absent for Hermitian metrics. To appreciate this distinction, consider the following

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calibration identity for Kähler surfaces. Let p ∈ M and v1 , v2 ∈ T p M span a plane. Then [4]

(v1 , v2 )2 + ς 2 (v1 , v2 ) = det G(vi , v j), where ς 2 (v1 , v2 )  0 with equality iff {v1 , v2 } span a complex plane. Here,  = 1 for G Hermitian, while  = −1 for G neutral. Thus, in the former case the above yields a version of the Wirtinger inequality: det G(vi , v j)  (v1 , v2 )2 , while in the latter case we have the metric as a balancing between holomorphic and symplectic structures. In particular, at a point in a neutral Kähler surface a plane can be both holomorphic and Lagrangian—a situation that cannot arise in the Hermitian case. It is natural then to study complex points on Lagrangian surfaces and Lagrangian points on complex curves in neutral Kähler surfaces. As shown in [2], the complex points on Lagrangian surfaces in TP1 correspond precisely to umbilic points on surfaces in E3 . In this paper we considered Lagrangian points on holomorphic curves in TP1 . In fact, there is a direct connection between these two situations implicit in the proof of part (iii) of Main Theorem 1. We rephrase our result more generally: Theorem 2 Let S be a ruled surface in E3 associated with a real curve C’ in TP1 . Then S is f lat if f C’ is null.  Proof The result follows by noting that G(γ˙ , γ˙ ) = Im (1 + ξ ξ¯ )η˙ ξ˙¯ + 2ξ η¯ ξ˙ ξ˙¯ , where γ˙ is the tangent vector to the curve C’, and recalling (1.5).



On the one hand, the metric at Lagrangian points on holomorphic curves is zero and so we obtain part (iii) of our Main Theorem. On the other hand, the normals along the lines of curvature of a surface S in L(E3 ) are also null curves in TP1 [2]. Thus we have proven the result in classical surface theory [9]: Theorem 3 Consider a surface S in E3 and let C be a curve on S. The ruled surface generated by the 1-parameter family of normals to S along C is f lat if f C is a line of curvature of S. Spectral curves of monopoles have been considered from a number of perspectives. While in principle it is possible to reconstruct the Higgs and Yang-Mills fields in E3 from the spectral curve via the Nahm data, this procedure is difficult and has only been carried out numerically for a small number of symmetric cases (see [10] and references therein). One alternative approach has been to consider the minimal surface in E3 generated by the Weierstrass representation applied to the spectral curve [6]. This has been carried out in detail for the charge two case, where the

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resulting geometry has been found to be rich and complicated [12]. The set of points in E3 where the charge two spectral curve lines are orthogonal has also been considered in [7], although no explicit calculations were given. Neither approaches have been extended to higher charge monopoles. Our techniques, however can be applied to monopoles of any charge. An additional advantage is that it works for all holomorphic curves and hence avoids the difficulty of the transcendental constraints [8]. Moreover, the symplectic form (and hence the edges of regression) are natural in the following sense. The Euclidean group O(3)  R3 acting on E3 sends oriented lines to oriented lines and hence acts on L(E3 ). The symplectic structure , along with J, and hence G, is invariant under this action. In fact, up to addition of the round metric on S2 , G is the unique metric on L(E3 ) which is invariant under any subgroup of the Euclidean group [11]. Finally, our techniques can be extended to holomorphic curves in the space of oriented geodesics of any 3-dimensional space of constant curvature—in particular to the spectral curves for monopoles in hyperbolic 3-space. What is obviously missing is the relationship between the edges of regression and the physical fields (gauge and Higgs field). Should this relationship be established, the techniques may yield a new way of localising these fields in E3 and beyond. Acknowledgements The first author would like to thank Nick Manton for suggesting the application to monopoles and Wilhelm Klingenberg for many helpful discussions, and the second author would like to thank Bernd Kreussler. The second author was supported by the IRCSET Embark Initiative Postdoctoral Fellowship Scheme.

References 1. Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order. Springer, Berlin (2001) 2. Guilfoyle, B., Klingenberg, W.: Generalised surfaces in R3 . Math. Proc. R. Ir. Acad. 104A, 199–209 (2004) 3. Guilfoyle, B., Klingenberg, W.: An indefinite Kähler metric on the space of oriented lines. J. Lond. Math. Soc. 72, 497–509 (2005) 4. Guilfoyle, B., Klingenberg, W.: On Weingarten surfaces in Euclidean and Lorentzian 3-space. Differential Geom. Appl. bf 28, 454–468 (2010) 5. Hartshorne, R.: Algebraic Geometry, Graduate Texts in Mathematics. Springer, New York (1977) 6. Hitchin, N.J.: Monopoles and geodesics. Commun. Math. Phys. 83, 579–602 (1982) 7. Hitchin, N.J.: Monopoles, minimal surfaces and algebraic curves. NATO Advanced Study Institute, University of Montreal Press, Quebec (1987) 8. Houghton, C.J., Manton, N.S., Ramão, N.M.: On the constraints defining BPS monopoles. Commun. Math. Phys. 212, 219–243 (2000) 9. Kreyszig, E.: Differential Geometry. Dover Publications, New York (1991) 10. Manton, N.S., Sutcliffe, P.: Topological Solitons. Cambridge University Press, Cambridge (2004) 11. Salvai, M.: On the geometry of the space of oriented lines in Euclidean space. Manuscr. Math. 118, 181–189 (2005) 12. Small, A.: On algebraic minimal surfaces in R3 deriving from charge 2 monopole spectral curves. Int. J. Math. 16, 173–180 (2005)

Math Phys Anal Geom (2010) 13:275–286 DOI 10.1007/s11040-010-9079-6

On Models with Uncountable Set of Spin Values on a Cayley Tree: Integral Equations Utkir A. Rozikov · Yusup Kh. Eshkobilov

Received: 11 October 2009 / Accepted: 16 July 2010 / Published online: 27 July 2010 © Springer Science+Business Media B.V. 2010

Abstract We consider models with nearest-neighbor interactions and with the set [0, 1] of spin values, on a Cayley tree of order k  1. We reduce the problem of describing the “splitting Gibbs measures” of the model to the description of the solutions of some nonlinear integral equation. For k = 1 we show that the integral equation has a unique solution. In case k  2 some models (with the set [0, 1] of spin values) which have a unique splitting Gibbs measure are constructed. Also for the Potts model with uncountable set of spin values it is proven that there is unique splitting Gibbs measure. Keywords Cayley tree · Configuration · Gibbs measures · Potts model Mathematics Subject Classifications (2010) Primary 82B05 · 82B20; Secondary 60K35

1 Introduction One of the central problems in the theory of Gibbs measures is to describe infinite-volume (or limiting) Gibbs measures corresponding to a given Hamiltonian. The existence of such measures for a wide class of Hamiltonians was established in the ground-breaking work of Dobrushin (see, e.g. [12]).

U. A. Rozikov (B) Institute of Mathematics and Information Technologies, Tashkent, Uzbekistan e-mail: [email protected] Yu. Kh. Eshkobilov National University of Uzbekistan, Tashkent, Uzbekistan e-mail: [email protected]

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However, a complete analysis of the set of limiting Gibbs measures for a specific Hamiltonian is often a difficult problem. In this paper we consider models with a nearest neighbor interaction and uncountably many spin values on a Cayley tree. A Cayley tree  k = (V, L) of order k  1 is an infinite homogeneous tree (see [1]), i.e., a graph without cycles, with exactly k + 1 edges incident to each vertices. Here V is the set of vertices and L that of edges (arcs). We consider models where the spin takes values in the set [0, 1], and is assigned to the vertexes of the tree. For A ⊂ V a configuration σ A on A is an arbitrary function σ A : A → [0, 1]. Denote  A = [0, 1] A the set of all configurations on A. A configuration σ on V is then defined as a function x ∈ V → σ (x) ∈ [0, 1]; the set of all configurations is [0, 1]V . The (formal) Hamiltonian of the model is :  ξσ (x)σ (y) , (1) H(σ ) = −J x,y∈L

where J ∈ R \ {0} and ξ : (u, v) ∈ [0, 1]2 → ξuv ∈ R is a given bounded, measurable function. As usually, x, y stands for nearest neighbor vertices. Let λ be the Lebesgue measure on [0, 1]. On the set of all configurations on A the a priori measure λ A is introduced as the |A|fold product of the measure λ. Here and further on |A| denotes the cardinality of A. We consider a standard sigma-algebra B of subsets of  = [0, 1]V generated by the measurable cylinder subsets. A probability measure μ on (, B ) is called a Gibbs measure (with Hamiltonian H) if it satisfies the DLR equation, namely for any n = 1, 2, . . . and σn ∈ Vn :     Vn  μ σ ∈  : σ Vn = σn = μ(dω)νω| (σn ), W 

Vn where νω| W

n+1

n+1

is the conditional Gibbs density

Vn νω| W

n+1

(σn ) =

    1    exp −β H σn || ωW , n+1 Z n ωWn+1

and β = T1 , T > 0 is temperature. Here and below, Wl stands for a ‘sphere’ and Vl for a ‘ball’ on the tree, of radius l = 1, 2, . . ., centered at a fixed vertex x0 (an origin): Wl = {x ∈ V : d(x, x0 ) = l}, Vl = {x ∈ V : d(x, x0 )  l}; and Ln = {x, y ∈ L : x, y ∈ Vn }; distance d(x, y), x, y ∈ V, is the length of (i.e. the number of edges in) the shortest path connecting x with y. Vn is the set of configurations in Vn  (and Wn that in Wn ; see below). Furthermore, σ Vn and ωWn+1 denote the restrictions of configurations σ, ω ∈  to Vn and Wn+1 , respectively. Next,

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   σn : x ∈ Vn → σn (x) is a configuration in Vn and H σn || ωWn+1 is defined    as the sum H (σn ) + U σn , ωWn+1 where  ξσn (x)σn (y) , H (σn ) = −J x,y∈Ln

   U σn , ωWn+1 = −J



ξσn (x)ω(y) .

x,y: x∈Vn ,y∈Wn+1

   Finally, Z n ωWn+1 stands for the partition function in Vn , with the boundary  condition ω : Wn+1

     Z n ω Wn+1 =

Vn

    exp −β H σn || ωWn+1 λVn (d σn ).

Due to the nearest-neighbor character of the interaction, the Gibbs measure possesses a natural Markov property: for given a configuration ωn on Wn , random configurations in Vn−1 (i.e., ‘inside’ Wn ) and in V \ Vn+1 (i.e., ‘outside’ Wn ) are conditionally independent. We use a standard definition of a translation-invariant measure (see, e.g., [12]). The main object of study in this paper are translation-invariant Gibbs measures for the model (1) on Cayley tree. We reduce the problem of description of such measures to the description of the solutions of a nonlinear integral equation. For finite and countable sets of spin values this argument is well known (see, e.g. [2–6, 11, 13, 14, 16]).

2 An Integral Equation Write x < y if the path from x0 to y goes through x. Call vertex y a direct successor of x if y > x and x, y are nearest neighbors. Denote by S(x) the set of direct successors of x. Observe that any vertex x = x0 has k direct successors and x0 has k + 1. Let h : x ∈ V → hx = (ht,x , t ∈ [0, 1]) ∈ R[0,1] be mapping of x ∈ V \ {x0 } with |ht,x | < C where C is a constant which does not depend on t. Given n = 1, 2, . . ., consider the probability distribution μ(n) on Vn defined by ⎛ ⎞  μ(n) (σn ) = Z n−1 exp ⎝−β H(σn ) + hσ (x),x ⎠ , (2) x∈Wn

Here, as before, σn : x ∈ Vn → σ (x) and Z n is the corresponding partition function: ⎞ ⎛   Zn = exp ⎝−β H( σn ) + h σ (x),x ⎠ λVn (d σn ). (3) Vn

x∈Wn

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Remark Note  that Z n is finite, since λ is a probability measure and exp(−β H( σn )+ x∈Wn h σ (x),x ) is bounded on Vn . We say that the probability distributions μ(n) are compatible if for any n  1 and σn−1 ∈ Vn−1 :  Wn

μ(n) (σn−1 ∨ ωn )λWn (d(ωn )) = μ(n−1) (σn−1 ).

(4)

Here σn−1 ∨ ωn ∈ Vn is the concatenation of σn−1 and ωn . In this case there a  unique measure μ on V such that, for any n and σn ∈ Vn ,  exists   = μ(n) (σn ). μ σ  = σn Vn

Definition 1 The measure μ is called splitting Gibbs measure corresponding to Hamiltonian (1) and function x → hx , x = x0 . The following statement describes conditions on hx guaranteeing compatibility of the corresponding distributions μ(n) (σn ). Proposition 1 The probability distributions μ(n) (σn ), n = 1, 2, . . ., in (2) are compatible if f for any x ∈ V \ {x0 } the following equation holds: 1



f (t, x) =

0

exp(Jβξtu ) f (u, y)du

0

exp(Jβξ0u ) f (u, y)du

1

y∈S(x)

.

(5)

Here, and below f (t, x) = exp(ht,x − h0,x ), t ∈ [0, 1] and du = λ(du) is the Lebesgue measure. Proof Necessity Suppose that (4) holds; we want to prove (5). Substituting (2) in (4), obtain that for any configurations σn−1 : x ∈ Vn−1 → σn−1 (x) ∈ [0, 1]: Z n−1 Zn

⎛





exp ⎝ Wn

⎛ = exp ⎝



⎞ (Jβξσn−1 (x)ωn (y) + hωn (y),y )⎠ λWn (dωn )

x∈Wn−1 y∈S(x)



⎞

hσn−1 (x),x ⎠ ,

(6)

x∈Wn−1

where ωn : x ∈ Wn → ωn (x).



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From (6) we get: Z n−1 Zn =







Wn x∈W n−1 y∈S(x)



exp (Jβξσn−1 (x)ωn (y) + hωn (y),y )d(ωn (y))

exp (hσn−1 (x),x ).

x∈Wn−1

Consequently, for any t ∈ [0, 1], 1



0

exp (Jβξtu + hu,y )du

0

exp (Jβξ0u + hu,y )du

1

y∈S(x)

= exp (ht,x − h0,x ),

which implies (5). Suf f iciency Suppose that (5) holds. It is equivalent to the representations  

1

exp (Jβξtu + hu,y )du = a(x) exp (ht,x ), t ∈ [0, 1]

(7)

y∈S(x) 0

for some function a(x) > 0, x ∈ V. We have LHS of (4) =

1 exp(−β H(σn−1 ))λVn−1 (d(σn )) Zn    1 × exp (Jβξσn−1 (x)u + hu,y )du.

(8)

x∈Wn−1 y∈S(x) 0

Substituting (7) into (8) and denoting An (x) = RHS of (8) =

 x∈Wn−1

a(x), we get

 An−1 exp(−β H(σn−1 ))λVn−1 (dσ ) hσn−1 (x),x . Zn x∈W

(9)

n−1

Since μ(n) , n  1 is a probability, we should have   λVn−1 (dσn−1 ) λWn (dωn )μ(n) (σn−1 , ωn ) = 1 Vn−1

Wn

Hence from (9) we get Z n−1 An−1 = Z n , and (4) holds. From Proposition 1 it follows that for any h = {hx ∈ R[0,1] , x ∈ V} satisfying (5) there exists a unique Gibbs measure μ and vice versa. However, the analysis of solutions to (5) is not easy. This difficulty depends on the given function ξ . In the next sections we will consider several examples of such functions and give some solutions of corresponding integral equations.

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3 The Potts Model Note that if ξtu = δtu where δ is the Kronecker’s symbol then model (1) becomes the Potts model with uncountable set of spin values. It is easy to see that  1  1 exp (Jβδtu ) f (u, y)du = exp (Jβδ0u ) f (u, y)du 0

0

for any t ∈ [0, 1], y ∈ V. Consequently the equation (5) has the unique solution f (t, x) = 1, t ∈ [0, 1], x ∈ V for any k  1, J ∈ R, and any β > 0. Thus we have Theorem 1 The Potts model with uncountable set of spin values on Cayley tree of order k  1 has unique splitting Gibbs measure for any J ∈ R and β > 0. The following remarks give a comparison of the result with known results about ordinary Potts model. Remark 1. It is known (see, for example [7]) that the Potts model with q  2 spin values on Z d , d  2 undergoes a first-order phase transition at a certain transition temperature Tcr = Tcr (q), provided q is large enough. Namely, the model (on Z d ) has q different Gibbs measures for temperatures T < Tcr , q + 1 measures at T = Tcr and one measure for T > Tcr . 2. Note that (see [4, 5, 8]) for the ferromagnetic Potts model with q spin values on Cayley tree for any q  2 (even for q = 2 i.e. for the Ising model (see [2, 3])) there are q + 1 distinct translation-invariant Gibbs measures. Namely, there are two critical temperatures 0 < Tc < Tc such that (1) for T ∈ (0, Tc ] there are q + 1 Gibbs measures. Among them only one, say μ0 , (with μ0 (σ (x) = i) = q1 , i = 1, .., q) is not extreme and called unordered Gibbs measure; (2) for T ∈ (Tc , Tc ] the q + 1 Gibbs measures still exist and all of them are extreme. (3) for T > Tc there is one Gibbs measure. 3. In [6] it was proven that the Gibbs measure is unique if q → ∞ i.e. when the set of spin values is a countable set. Theorem 1 shows that the uniqueness is also true for an uncountable set of spin values.

4 Translational—Invariant Solutions of (5) In this section we consider ξtu as a continuous function and we are going to solve (5) in the class of translational-invariant functions f (t, x) i.e f (t, x) = f (t), for any x ∈ V. For such functions (5) can be written as 1 k 0 K(t, u) f (u)du f (t) =  1 , (10) K(0, u) f (u)du 0 where K(t, u) = exp(Jβξtu ) > 0, f (t) > 0, t, u ∈ [0, 1].

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We shall find positive continuous solutions to (10) i.e. such that f ∈ C+ [0, 1] = { f ∈ C[0, 1] : f (x) > 0}. Note that (10) is not linear for any k  1. Define the linear operator W : C[0, 1] → C[0, 1] by  1 K(t, u) f (u)du (W f )(t) =

(11)

0

and defined the linear functional ω : C[0, 1] → R by  1 ω( f ) ≡ (W f )(0) = K(0, u) f (u)du. 0

Then (10) can be written as  f (t) = (Ak f )(t) =

(W f )(t) (W f )(0)

k

, f ∈ C+ [0, 1], k  1.

(12)

4.1 Case k = 1 In this subsection we consider k = 1 and assume K(·, ·) ∈ C+ [0, 1]2 and f (·) ∈ C+ [0, 1]. Proposition 2 If f ∈ C+ [0, 1] is a solution to (10) then f (t) 

κ min , for any t ∈ [0, 1], κ0max

where κ min = inft,u∈[0,1] K(t, u), κ0max = supu∈[0,1] K(0, u). Proof Straightforward. Denote   κ min C0+ = h ∈ C+ [0, 1] : h(t)  max κ0 The following Lemma is also obvious Lemma 1 1. The set C0+ is a closed and convex subset of the space C[0, 1]. 2. The set C0+ is invariant w.r.t. operator A1 i.e. A1 (C0+ ) ⊂ C0+ . Lemma 2 Operator A1 is continuous on C0+ .



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Proof Let f ∈ C0+ is an arbitrary element and { fn } ⊂ C0+ such that limn→∞ fn = f . We shall prove that A1 fn − A1 f  → 0 as n → ∞. We have |A1 fn − A1 f | 

W fn |ω( fn ) − ω( f )| + ω( fn )|W fn − W f | . ω( f )ω( fn )

(13)

Since the functional ω(·) and the operator W(·) are continuous on C[0, 1], for any small ε > 0 there exits n0 = n0 (ε) ∈ N such that |ω( fn ) − ω( f )| < ε, W fn − W f  < ε, ∀n > n0 . Consequently A1 fn − A1 f  <

W fn  + ω( fn ) · ε. (ω( f ) − ε)ω( f )

(14)

There are Mi , i = 0, 1, 2 such that ω( f )  M0 , for all f ∈ C0+ and W( fn )  M1 , ω( fn )  M2 , n ∈ N. Thus from (14) we get A1 fn − A1 f  <

M1 + M2 · ε, n > n0 . (M0 − ε)M0 

This completes the proof. Lemma 3 The set A1 (C0+ ) is relatively compact in C[0, 1].

Proof By Arzelá–Askoli’s theorem (see [15], ch.III, §3) it suffices to prove that all functions of A1 (C0+ ) are uniformly continuous and there exists M > 0 such that |h(t)|  M, ∀t ∈ [0, 1] and ∀h ∈ A1 (C0+ ). Let h ∈ A1 (C0+ ) be an arbitrary function, then for a function f ∈ C0+ we have h = A1 f . Consequently |h(t)| 

κ max , ∀t ∈ [0, 1]. κ0min

Now we shall prove that any h ∈ A1 (C0+ ) is uniformly continuous. For arbitrary t, t ∈ [0, 1] we have (h = A1 f )  1 1 |h(t) − h(t )|  |K(t, u) − K(t , u)| f (u)du. (15) ω( f ) 0 Since the kernel K(t, u) is uniformly continuous on [0, 1]2 we conclude that h also is a uniformly continuous function. This completes the proof. 

By Lemmas 1–3 and Schauder’s theorem (see [9], p.20) one obtains Proposition 3 The equation A1 f = f has at least one solution in C+ [0, 1].

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Now we shall prove that A1 f = f has a unique solution in C+ [0, 1]. Since the equation A1 f = f is equivalent to (W f )(t) = ω( f ) · f (t), f ∈ C+ [0, 1],

(16)

we shall study eigenvalues of the operator W f . Lemma 4 If ϕ0 ∈ C+ [0, 1] is an eigenfunction of the operator W i.e. Wϕ0 = λ0 ϕ0 , λ0 > 0 then there are a1 > 0 and b 1 > 0 such that a1 ω1 ( f )ϕ0 (t)  (W f )(t)  b 1 ω1 ( f )ϕ0 (t), ∀t ∈ [0, 1], ∀ f ∈ C+ [0, 1], 1 where ω1 ( f ) = 0 f (u)du.

(17)

Proof Note that aω1 ( f )  W f  b ω1 ( f ), f ∈ C[0, 1]

(18)

where a = mint,u∈[0,1] K(t, u) and b = maxt,u∈[0,1] K(t, u). We have aω1 (ϕ0 )  Wϕ0 = λ0 ϕ0  b ω1 (ϕ0 ). Hence λ0 ϕ0 (t) λ0 ϕ0 (t) 1 , ∀t ∈ [0, 1]. b ω1 (ϕ0 ) aω1 (ϕ0 )

(19)

Using (18) and (19) we get (17) with a1 =

aλ0 b λ0 > 0, b 1 = > 0. b ω1 (ϕ0 ) aω1 (ϕ0 )

Theorem 2 If λ0 > 0 is an eigenvalue of W then W f = λ0 f has a unique solution f ∈ C+ [0, 1]. Proof Assume that there are two solutions f0 ∈ C+ [0, 1] and f1 ∈ C+ [0, 1] i.e W fi = λ0 fi , i = 0, 1. Denote δ0 = sup{δ ∈ [0, ∞) : f0 (t) − δ f1 (t) ∈ C+ [0, 1]}. We have W( f0 − δ0 f1 ) = W( f0 ) − δ0 W( f1 ) = λ0 ( f0 − δ0 f1 ) > 0. 

By Lemma 4 we get W( f0 − δ0 f1 )  a2 f0 (t) > a2 δ0 f1 (t) with some a2 > 0, where we used a2 ( f0 (t) − δ0 f1 (t)) > 0. Consequently λ0 ( f0 − δ0 f1 ) > a2 δ0 f1 (t)

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i.e.

  a2 f1 (t) > 0 for any t ∈ [0, 1]. f0 (t) − δ0 1 + λ0

This contradicts the maximality of δ0 . Theorem 3 The equation A1 f = f has a unique solution f ∈ C+ [0, 1]. Proof By Proposition 3 the equation has at least one solution. We shall prove its uniqueness. Assume that A1 f = f has two solutions f0 and f1 , then there are λ0 = λ0 ( f0 ) and λ1 = λ1 ( f1 ) such that W fi = λi fi , i = 0, 1. By Theorem 2 we have λ0 = λ1 . Assume λ0 < λ1 (the case λ0 > λ1 is similar). Consider hδ (t) = f0 (t) − δ f1 (t), δ ∈ [0, ∞) and δ0 = sup{δ ∈ [0, ∞) : hδ (t) ∈ C+ [0, 1]} We have W(hδ0 )(t) = λ0 ( f0 (t) − δ0

λ1 f1 (t)) > 0, ∀t ∈ [0, 1]. λ0

Since δ0 is maximal we get λλ10  1 i.e. λ0  λ1 , this contradicts our assumption λ0 < λ1 . The theorem is proved. Example 1 If K(t, u) = α(t) + α(u) where α is a given function, then one can easily check that  1 2 α(t) + 0 α (u)du  f (t) = 1 2 α(0) + 0 α (u)du is the unique solution of the equation A1 f = f . As a corollary of Theorem 3 and Proposition 1 we get Theorem 4 For model (1) with an arbitrary continuous function ξtu on [0, 1]2 , ∀J ∈ R and for any β > 0 on the Cayley tree of order 1 there exists a unique splitting Gibbs measure. 4.2 Case k  2 The analysis of solutions to (10) for k  2 is not easy. In this subsection for k  2 we shall consider several examples of K(t, u) > 0 which are easily solvable. Proposition 4 The function f (t) ≡ 1 is a solution to the equation (10) if f  1 (exp(Jβξtu ) − exp(Jβξ0u )) du = 0, t ∈ [0, 1]. 0

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1 Proof Denoting At = 0 exp(Jβξtu ) f (u)du, t ∈ [0, 1] one can see that the equation (10) is equivalent to   + ... + Ak−1 f (t) − 1 = (At − A0 )A−k Ak−1 , t 0 0 

this completes the proof.

Example 2 For any k  1 we shall consider one simple case: let ξtu = a(t) + b (u), where a(t) and b (u) are arbitrary given functions. Very simple calculations show that (10) (even (5)) has unique solution f (t, x) = f (t) = exp(kJβ(a(t) − a(0))). Thus for the model (1) with ξtu = a(t) + b (u) there is unique splitting Gibbs measure. 1 ln(α(t) + α(u)), where Example 3 Consider K(t, u) = α(t) + α(u) i.e. ξtu = Jβ α is a given positive function on [0, 1]. Then the unknown function f can be written as     α(t)X + Y k α(t)x + 1 k f (t) = = , α(0)X + Y α(0)x + 1 1 1 where X = 0 f (u)du and Y = 0 α(u) f (u)du, x = YX . It is easy to see that x satisfies the equation k aj j j=0 j!(k− j)! x , x>0 (20) x = k a j+1 j j=0 j!(k− j)! x 1 where ai = 0 α i (t)dt, i = 0, 1, ..., k + 1.

From (20) we get γ (x) = ak+1 xk+1 + b k xk + b k−1 xk−1 + ... + b 1 x − 1 = 0,

(21)

where bj =

k!(2 j − k − 1) a j, j = 1, ..., k. j!(k − j + 1)!

It is well known (see [10], p.28) that the number of positive roots of the polynomial (21) does not exceed the number of sign changes of the sequence: ak+1 , b k , b k−1 , ..., b 1 , −1.

(22)

It is obvious that ak+1 > 0, bj > 0 if j > k+1 and bj < 0 if j < k+1 . Thus the 2 2 number of positive roots of the polynomial (21) is at most one. Since γ (0) = −1 and γ (+∞) = +∞ we get that (21) has a unique positive root. Consider the Hamiltonian 1  ln (α(σ (x)) + α(σ (y))) , (23) H(σ ) = − β x,y∈L where α is a given positive, integrable function.

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Thus we have proved the following Theorem 5 For any k  1 the model (23) has unique splitting Gibbs measure. Remark Is there a kernel K(t, u) > 0 of the equation (10) when the equation has more than one solutions? This is an open problem. Acknowledgements A part of this work was done in the Abdus Salam International Center for Theoretical Physics (ICTP), Trieste, Italy and UAR thanks ICTP for providing financial support and all facilities. He also thanks S. Albeverio for support of his visit to Bonn University. Authors thank referees for their useful suggestions.

References 1. Baxter, R.J.: Exactly Solved Models in Statistical Mechanics. Academic, London (1982) 2. Bleher, P.M., Ganikhodjaev N.N.: On pure phases of the Ising model on the Bethe lattice. Theor. Probab. Appl. 35, 216–227 (1990) 3. Bleher, P.M., Ruiz, J., Zagrebnov V.A.: On the purity of the limiting Gibbs state for the Ising model on the Bethe lattice. J. Stat. Phys. 79, 473–482 (1995) 4. Ganikhodjaev, N.N.: On pure phases of the ferromagnet Potts with three states on the Bethe lattice of order two. Theor. Math. Phys. 85, 163–175 (1990) 5. Ganikhodjaev, N.N., Rozikov, U.A.: On disordered phase in the ferromagnetic Potts model on the Bethe lattice. Osaka J. Math. 37, 373–383 (2000) 6. Ganikhodjaev, N.N., Rozikov, U.A.: The Potts model with countable set of spin values on a Cayley Tree. Lett. Math. Phys. 75, 99–109 (2006) 7. Kotecky, R., Shlosman, S.B.: First-order phase transition in large entropy lattice models. Commun. Math. Phys. 83, 493–515 (1982) 8. Mossel, E.: Survey: information flow on trees. In: Graphs, Morphisms and Statistical Physics. DIMACS Ser. Discrete Math. Theor. Comput. Sci., vol. 63, pp. 155–170. AMS Providence, RI (2004) 9. Nirenberg, L.: Topics in Nonlinear Functional Analysis. Courant Lec. Notes in Math, 6. AMS, NY (2001) 10. Prasolov, V.V.: Polynomials. Springer, Berlin (2004) 11. Preston, C.: Gibbs States on Countable Sets. Cambridge University Press, London (1974) 12. Sinai, Ya.G.: Theory of Phase Transitions: Rigorous Results. Pergamon, Oxford (1982) 13. Spitzer, F.: Markov random fields on an infinite tree. Ann. Probab. 3, 387–398 (1975) 14. Suhov, Y.M., Rozikov, U.A.: A hard-core model on a Cayley tree: an example of a loss network. Queueing Syst. 46, 197–212 (2004) 15. Yosida, K.: Functional Analysis. Springer, Berlin (1965) 16. Zachary, S.: Countable state space Markov random fields and Markov chains on trees. Ann. Probab. 11, 894–903 (1983)

Math Phys Anal Geom (2010) 13:287–297 DOI 10.1007/s11040-010-9080-0

The Representation of Isometric Operators on C(1) (X) Jingke Li

Received: 4 December 2008 / Accepted: 29 July 2010 / Published online: 21 August 2010 © Springer Science+Business Media B.V. 2010

Abstract In this paper,we introduce a new norm on C(1) (X), which is induced by a hexagon on R2 , and prove that every isometric operator on C(1) (X) can be induced by a homeomorphism of X, where X is a connected subset of R. Keywords Hexagon · Isometry · Extreme point Mathematics Subject Classification (2010) 46B04

1 Introduction Let X be a connected compact subset of R, C(1) (X) denotes the set consisting of all continually differential function f on X which satisfies: sup {| f (x)|} < ∞ x∈X



and sup {| f (x)|} < ∞, for f ,g ∈ C(1) (X), define x∈X

( f + g)(x) = f (x) + g(x),

(α f )(x) = α f (x),

∀α ∈ R ∀x ∈ X. 

Define a norm on C(1) (X) by  f = sup { ( f (x), f (x))  H }, ∀ f ∈ C(1) (X), x∈X

where  (·, ·)  H is a norm on R2 . We use BY to denote the unit ball of Y, and extBC(1) (X)∗ denotes all extreme points of BC(1) (X)∗ . The classical Banach–Stone theorem states that any isometry from C(X) onto C(Y) is induced by a homeomorphism of Y and X,where X and Y are compact spaces. This result has been extended to various other Banach spaces by many other authors.

J. Li (B) School of Mathematical Sciences, Nankai University, Tianjin 300071, China e-mail: [email protected]

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Define L-norm by          f = sup  f (x), f (x) l1 = sup | f (x) | + | f (x) | x∈X

x∈X

Where X is compact Hausdorff space. In 1965, Cambern first considered the representations of surjective linear isometries on C(1) [0, 1] with the L-norm (complex case only). In 1981, Cambern and Pathak studied the representations of surjective linear isometries between C0 (1) (X) and C0 (1) (Y) with the L-norm (complex case only), for X, Y any compact subsets of the real line, without isolated points. In the same year, Pathak generalized the conditions and considered the representations of surjective linear isometries on C0 (n) [0, 1] with the L-norm, for n  1 (complex case only). In 1994, Risheng Wang generalized the conditions of Cambern and Pathak, and studied the representations of surjective linear isometries between  | f (r) (x)| , for n  1, X, Y any C0 (1) (X) and C0 (n) (Y) with the norm  f = sup r! x∈X r∈

locally compact Hausdorff spaces, without isolated points. More important this conclusion holds in both complex case and real case. In 1996, Risheng Wang considered the most general case, and discussed the representations of surjective linear isometries between C0 (n1 ) (X) and  | f (r) (x)| C0 (n2 ) (Y) with the norm  f = sup , for m1 , m2 , n1 ,n2  1 and X ⊆ r! x∈X r∈

Rm1 , Y ⊆ Rm2 are locally compact Hausdorff spaces, without isolated points. This conclusion also holds in both complex case and real case [1]. In this paper, we introduce a new norm  (·, ·)  H which is different from other authors’ mentioned above. With this norm, the unit ball of normed linear space Y = (R2 ,  (·, ·)  H ) is a regular hexagon. The proof follows the standard route established by M. Cambern and V.D. Pathak. We hope that the idea this paper used is helpful to the norm induced by a regular octagon, even by a regular polygon having n sides.

2 Lemma √

√

On R2 , Let e1 = (1, 0), e2 = ( 12 , 23 ), e3 = (− 12 , 23 ), e4 = −e1 , e5 = −e2 , e6 = −e3 , the regular hexagon formed by the closure convex of these six points is denoted by H. {ei : i = 1, 2, ..., 6} divide the hexagon into six parts, noted by H1 ,H2 ,H3 ,H4 ,H5 ,H6 respectively, as shown in Fig. 1. We use Y = (R2 ,  (·, ·)  H ) to denote the normed linear space with H as its unit ball. 2 ∈ SY = {y ∈ Y : y  H = 1}, by Hahn–Banach theorem, there Since e1 +e 2 ∗ 2 exists f1 ∈ SY ∗ , such that f1 ∗ ( e1 +e ) = 1. As f1 ∗ (e1 ), f1 ∗ (e2 ) ≤ 1, we have 2 ∗ ∗ f1 (e1 ) = f1 (e2 ) = 1.

The Representation of Isometric Operators on C(1) (X) Fig. 1 We use H to denote the regular hexagon formed by the closure convex of these six points. The vectors {ei : i = 1, 2, ..., 6} divide the hexagon into six parts, noted by H1 ,H2 ,H3 ,H4 ,H5 ,H6 respectively

289

e3 e4

e2 H2

H3

o H 1 e1

H4

H6 H5

e5

1 a 2

e6

For√ (s, t) ∈ Y, let f1 ∗ (s, t) = as + √b t, then 1 = f1 ∗ (e1 ) = a,√1 = f1 ∗ (e2 ) = + 23 b , it follows that a = 1, b = 33 , so we have f1 ∗ = (1, 33 ). Similarly, √

3 there exists f2 ∗ ∈ SY ∗ , such that f2 ∗ ( e2 +e ) = 1, then f2 ∗ = (0, 2 3 3 ). 2 ∗ Generally, there exists fi ∈ SY ∗ (i = 1, 2, ..., 6) such that fi ∗ ( ei +e2 i+1 ) = 1 (e7 = e1 ), then fi ∗ (ei ) = fi ∗ (ei+1 ) = 1. So we have



f1 ∗

  √  √  √  2 3 3 3 = 1, = − f4 ∗ , f2 ∗ = 0, = − f5 ∗ , f3 ∗ = −1, = − f6 ∗ . 3 3 3

For y ∈ Y, we show that  y  H = max



1i6



fi ∗ (x) = max | fi ∗ (x) | . 1i3

Indeed, on the one hand, for y ∈ Y, it is easy to see that max fi ∗ (y)  y  H . 1i6

On the other hand, if y = sei + tei+1 (s, t  0, i ∈ {1, 2, ..., 6}), it follows that fi ∗ (y) = s + t =| s | + | t | y  H . So for any y ∈ Y,  y  H = max

1i6





fi ∗ (y) = max | fi ∗ (y) | . 1i3

Let fi ∗ = (ai , b i ), i = 1, 2, ..., 6. Then  y  H = max {ai s + b i t} = max {| ai s + 1i6

1i3

b i t |}. It is easy to prove that  (s, t)  H is a norm on R2 . For f ∈ C(1) (X), define  f  = sup 



      f (x), f (x)  H = sup max fi ∗ f (x), f (x)

x∈X

x∈X 1i6

   = sup max ai f (x) + b i f (x) , x∈X 1i6

where fi ∗ = (ai , b i ), i = 1, 2, ..., 6.

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Proposition 2.1 C(1) (X) is a Banach space. Define W = X × { f1 ∗ , ..., f6 ∗ }. Then W is a compact Hausdorff space. We use C(W) to denote the Banach space consisting of all continuous function f : W → R with the sup norm ( f ∞ = sup {| f (ω) |} < ∞). For f ∈ ω∈W

 C(1) (X), define continuous function  f on W by  f (x, fi ∗ ) = ai f (x) + b i f (x), ∗ where fi = (ai , b i ), i = 1, 2, ..., 6.

Lemma 2.2 The map f →  f is a surjective linear isometry between C(1) (X) and the closed subspace S of C(W). (Then we can look C(1) (X) and S as the same space.) Proof This follows from the definition of the function  f. Since the set of extreme points of BC(W)∗ is {αδω : α = ±1, ω ∈ W}, it follows that any extreme point f ∗ ∈ extBC(1) (X)∗ is the restriction on S of some g∗ ∈ extBC(W)∗ , i.e f ∗ = g∗ | S , where g∗ = αδω (α = ±1, ω ∈ W). thus, we have    f ∗ ( f ) = g∗ (  f ) = αδω (  f) = α  f (ω) = α  f (x, fi ∗ ) = α ai f (x) + b i f (x) 

= a j f (x) + b j f (x). where α fi ∗ = α(ai , b i ) = (a j, b j), some i, j ∈ {1, 2, ..., 6}.



For the inverse, we have the following proposition: Proposition 2.3 For ω ∈ W, def ine the linear function δω on C(1) (X) by δω ( f ) = ai f (x) + b i f  (x), where ω = (x, fi ∗ ), fi ∗ = (ai , b i ) i = 1, 2, ..., 6.. Then δω ∈ extBC(1) (X)∗ , and from the symmetry of { fi ∗ : i = 1, 2, ..., 6} we have that {δω : ω ∈ W} are all the extreme points of BC(1) (X)∗ . Before the proof of this proposition, let’s study the following lemmas: Definition 2.4 Let W be any compact Hausdorff space and S be a closed subspace of C(W) with the sup norm. For ω ∈ W, the function p ∈ S is called a peak function on ω with respect to the subspace S, if p(ω) = p  and | p(z) | p , ∀z ∈ W, z = ω. The equality holding if and only if for those z satisfying h(z) = h(ω) or h(z) = −h(ω)(∀h ∈ S). The following lemma can be found in [3]. Lemma 2.5 Let W be a compact Hausdorf f space and S be a closed linear subspace of C(W) with the sup norm. For ω ∈ W, if there exists a peak function on ω with respect to the subspace S,then the linear function δω : h → h(ω),(∀h ∈ S) is a extreme point of B S∗ .

The Representation of Isometric Operators on C(1) (X)

291

Lemma 2.6 For any ω0 ∈ W, there exists  f ∈ S such that  f is a peak funtion on ω0 with respect to the subspace S. Proof f ∈ S,  f (ω0 ) = f (x0 ) + 1) If ω0 = (x0 , f1 ∗ ), x0 ∈ X, then ∀ 

√

3 3



f (x0 ).

Take ⎧ ⎪ ⎪ 0, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨1 + 2(x − x0 )

g (x) =

1 x  x0 − , 2 1 x0 − < x  x0 , 2 1 x0 < x  x0 + , 2 1 x0 + < x, 2

⎪ ⎪ 1 − 2(x − x0 ), ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ 0,

(2.1)

let

 g(x) =

x x0

g (t) dt =

⎧ 1 ⎪ ⎪ − , ⎪ ⎪ ⎪ 4 ⎪ ⎪ ⎪ ⎨(x − x0 )(1 + x − x0 ), ⎪ ⎪ (x − x0 )(1 − x + x0 ), ⎪ ⎪ ⎪ ⎪ ⎪ 1 ⎪ ⎩ , 4

1 x  x0 − , 2 1 x0 − < x  x0 , 2 1 x0 < x  x0 + , 2 1 x0 + < x. 2

√

(2.2)

So we have g(x0 ) = 0. Take f (x) = g | X + 2 3 3 . It is easy to see that f ∈ C(1) (X) √ √ √ √  f (ω0 ) = f (x0 ) + 33 f (x0 ) = 2 3 3 + 33 = 3. and f  (x) = g (x) (x ∈ X). Then  For any ω ∈ W, ω = ω0 , Case A If ω = (x, fi ∗ ),i ∈ {2, 5}, then we have √ √ 2 3  | f (ω) |= | f  (x) |< 3, 3

∀x ∈ R.

Case B If ω = (x, fi ∗ ),i ∈ {1, 3, 4, 6}, x = x0 , then When x  x0 − 12 , |  f (ω) |=| f (x) |=| − 14 +

√ 2 3 3

|<

√ 3.

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When x0 −

1 2

< x < x0 ,

√ 3  | f  (x) | | f (ω) |  | f (x) | + 3

√ √ 3 2 3  | x − x0 + (x − x0 ) | + | 1 + 2(x − x0 ) | + 3 3 √ √   3 2 3 2 [1 + 2(x − x0 )] + = − x − x0 + (x − x0 ) + 3 3 √ √ 2 3 (x − x0 ) + 3 = −(x − x0 )2 + 3 √ < 3. 2

When x0 < x  x0 + 12 ,

√ √ 2 3 3 2  | f (ω) |  | x − x0 − (x − x0 ) | + | 1 − 2(x − x0 ) | + 3 3 √ √   2 3 = x − x0 − (x − x0 )2 − (x − x0 ) + 3 3 √ < 3. √ √ When x > x0 + 12 , |  f (ω) | 14 + 2 3 3 < 3. Case C If ω = (x0 , fi ∗ ), i ∈ {3, 4, 6}, then √ √ When i ∈ {3, 6}, |  f (ω) |= 33 < 3. √ When i√= 4, we have |  f (ω) |= 3 =  f (ω0 ), and for any  g ∈ S, g(ω) = −g(x0 ) − 33 g (x0 ) = − g(ω0 ). Thus,  f is a peak function on ω0 with respect to the subspace S. 

1)

√

If ω0 = (x0 , f4 ∗ ), take f (x) = −g(x) | X − 2 3 3 . √

2) If ω0 = (x0 , f3 ∗ ), take f (x) = g(x) | X − 2 3 3 . 

2)

√

If ω0 = (x0 , f6 ∗ ), take f (x) = −g(x) | X + 2 3 3 .

Where g(x) is the function we take in 1). 3) If ω0 = (x0 , f2 ∗ ), take ⎧ ⎪ ⎪ 0, ⎪ ⎪ ⎪ √ ⎪ ⎪ ⎪ 3 ⎪ ⎨2(x − x0 ) + ,  2 √ f (x) = ⎪ 3 ⎪ ⎪ −2(x − x0 ) + , ⎪ ⎪ 2 ⎪ ⎪ ⎪ ⎪ ⎩ 0,

√ 3 x  x0 − √ 4 3 x0 − < x  x0 4 √ 3 x0 < x  x0 + 4 √ 3 x0 + <x 4

(2.3)

The Representation of Isometric Operators on C(1) (X)

let

 f (x) =

293

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨

x

−∞

0,  √ 2 3 x − x0 + , 4    f (t) dt = √ ⎪ 3 3 ⎪ ⎪ ⎪ (x − x0 ) − (x − x0 ) + , ⎪ ⎪ 2 16 ⎪ ⎪ ⎪ ⎪ ⎪ 3 ⎪ ⎩ , 8 √

√ 3 x  x0 − 4 √ 3 < x  x0 x0 − 4 √ 3 x0 < x  x0 + 4 √ 3 x0 + <x 4 (2.4)

So we have  f (ω0 ) = 2 3 3 f  (x) = 1. For ω ∈ W, ω = ω0 , there are three cases to discuss. √ If ω = (x, fi ∗ ) and i ∈ {1, 3, 4, 6}, then |  f (ω) || f (x) | + 33 | f  (x) | √

3 3

×

√

3 2

< 1.

3 8

+

√

If ω = (x, fi ∗ ), i ∈ {2, 5}, and x = x0 , then |  f (ω) |= 2 3 3 | f  (x) |< 1. If ω = (x0 , f5 ∗ ), we have |  f (ω) |= 1 =  f (ω0 ), and ∀ g ∈ S,  g(ω) = √ 2 3  g(ω0 ). − 3 g (x0 ) = − Thus,  f is a peak function on ω0 with respect to the subspace S. 

3)

If ω = (x, f5 ∗ ), take f (x) = − f (x), where f (x) is the function in 3). Similarly, we can prove that f is a peak function on ω0 with respect to the subspace S. By Lemmas 2.5 and 2.6, it is easy to prove Proposition 2.3.



Lemma 2.7 If T is a surjective linear isometry on C(1) (X), the image by T of the constant function 1 of C(1) (X) is a non-zero constant function.(T(1) = 1 or T(1) = −1) Proof First, for f ∈ C(1) (X), f is a constant function on X if and only if the set E f = {| ( f ) |:  ∈ extBC(1) (X)∗ } contains at most two points. Indeed, if f is a constant function on X, by Proposition 2.3, it is easy to see that E f has at most two points. Conversely, If E f contains at most two points, √ √ √ 3  3  by Proposition 2.3, we have E f = {| f (x) − 3 f (x) |, | f (x) + 3 f (x) |, 2 3 3 | f  (x) | x ∈ R}. Since f  (x) is a continuous function on X, we get that f  (x) is a constant function. Case 1 If f  (x) ≡ 0, ∀x ∈ X, then f (x) is a constant function. Case 2 If f  (x) = 0, ∀x ∈ X, without loss of generality, we can assume √ that f  (x) = A > 0. Then ∀x ∈ X, | f (x) − 33 f  (x) | and | f (x) + √

3 3

f  (x) | are equal, or one of them is equal to

If | f (x) −

√

3 3

f  (x) |=| f (x) +

√

3 3

√ 2 3 3

A.

f  (x) |, it is easy to prove f (x) = 0.

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√

√

√

√

If | f (x) − 33 f  (x) |= 2 3 3 A, i.e. | f (x) − 33 A |= 2 3 3 A, then f (x) = − 33 A √ or f (x) = 3A. √ √ √ √ If | f (x) + 33 f  (x) |= 2 3 3 A, then f (x) = 33 A or f (x) = − 3A. Since f (x) is a continuous function, we get that f (x) is a constant function on X. Since T ∗ carries the extreme points of BC(1) (X)∗ onto themselves, by ∗ T δω (1) = δω (T(1)), (ω ∈ W) and the property of E f we get above, it is easy to see that T(1) is a non-zero constant function on X. 

Lemma 2.8 If T is a surjective linear isometry on C(1) (X), then T ∗ δ(x, fi ∗ ) = δ(y, f j ∗ ) ,

i, j ∈ {1, 3, 4, 6},

T ∗ δ(x, fk ∗ ) = δ(y, fl ∗ ) ,

k, l ∈ {2, 5},

x, y ∈ X. Proof If not, suppose T ∗ δ(x, fi ∗ ) = δ(y, fl ∗ ) , i ∈ {1, 3, 4, 6}, l ∈ {2, 5}, x, y ∈ X. Then T ∗ δ(x, fi ∗ ) (1) = δ(y, fl ∗ ) (1), i.e. ai T(1)(x) + b i T(1) (x) = 0, where (ai , b i ) = fi ∗ . From Lemma 2.7, we get ai T(1)(x) = 0. This yields a contradiction with Lemma 2.7. Thus T ∗ δ(x, fi ∗ ) = δ(y, f j ∗ ) , i, j ∈ {1, 3, 4, 6}. Similarly it is easy to prove T ∗ δ(x, fk ∗ ) = δ(y, fl ∗ ) , k, l ∈ {2, 5}, x, y ∈ X.  Lemma 2.9 Let T be a surjective linear isometry on C(1) (X) and P : W → X be the natural projection. Def ine ρ : ω → δω . Let PT = Pρ −1 T ∗ . Then, {PT δ(x0 , fi ∗ ) | i = 1, 2, ...6} is a singleton for the f ixed point x0 ∈ X. Proof First, the map ρ : ω → δω is a homeomorphism of W onto (extBC(1) (X)∗ , weak∗ ), Thus PT makes sense. If there exists i1 , i2 ∈ {1, 2, ..., 6},(i1 = i2 ), such that PT δ(x0 , fi1 ∗ ) = y1 and PT δ(x0 , fi2 ∗ ) = y2 ,(y1 = y2 ), then there exists g ∈ C(1) (R) (then g ∈ C(1) (X))  and the neighborhood V y1 ⊆ R, V y2 ⊆ R of y1 and y2 respectively (V y1 V y2 = ∅), such that g(y) = 1,

∀y ∈ V y1 ,

g(y) = 0,

∀y ∈ V y2 .

Since PT is continuous, it follows that for the fixed point i1 , i2 , there exists a neighborhood U 1 , U 2 of x0 , such that PT δ(x, fi1 ∗ ) ⊆ V y1 ,

∀x ∈ U 1 ,

PT δ(x, fi2 ∗ ) ⊆ V y2 ,

∀x ∈ U 2 .

(**)

If i1 , i2 ∈ {1, 4} or i1 , i2 ∈ {3, 6}, by Lemma 2.8 and the property of g(x), this yields a contradiction. If i1 ∈ {1, 4} and i2 ∈ {3, 6}, it is easy to prove that T(g)(x) ≡ 12 or T(g)(x) ≡ √ √  − 12 , meanwhile T(g) (x) ≡ 23 or T(g) (x) ≡ − 23 , ∀x ∈ U 1 U 2 . This obviously a contradiction.

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If i1 ∈ {1, 4} and i2 ∈ {2, 5}, then T(g)(x)  ≡ 1 or T(g)(x) ≡ −1, meanwhile T(g) (x) ≡ −1 or T(g) (x) ≡ 1, ∀x ∈ U 1 U 2 . This obviously a contradiction. Similarly,it is easy to see that the case i1 ∈ {3, 6} and i2 ∈ {2, 5} also yields contradictions. Furthermore, i1 , i2 ∈ {2, 5} doesn’t hold also. Then {PT δ(x0 , fi ∗ ) | i = 1, 2, ...6} is a singleton. 

Lemma 2.10 Let T be a surjective linear isometry on C(1) (X) and T ∗ δ(x, fi ∗ ) = δ(y, f j ∗ ) , i, j ∈ {1, 2, ..., 6}, x,y ∈ X. Then the map τ (x) = y is a homeomorphism on X. Proof By Lemma 2.9 and the fact that T ∗ is a surjective linear isometry ∗ ∗ on C(1) (X) (T ∗ maps the extreme points of the unit ball of C(1) (X) onto themselves), it is easy to prove. 

Lemma 2.11 Let T be a surjective linear isometry on C(1) (X) and fi ∗ = (ai , b i ),i ∈ {1, 3, 4, 6}. If T ∗ δ(x0 , fk ∗ ) = δ(y0 , f1 ∗ ) , T ∗ δ(x0 , fl ∗ ) = δ(y0 , f3 ∗ ) , k.l ∈ {1, 3, 4, 6}, then ak = al , b k = −b l . Proof By T ∗ δ(x0 , fk ∗ ) (1) = δ(y0 , f1 ∗ ) (1), T ∗ δ(x0 , fl ∗ ) (1) = δ(y0 , f3 ∗ ) (1), we have ak T(1) = 1 and al T(1) = 1. By Lemma 2.7, we get that ak = al , and by 

the property of fi ∗ (i ∈ {1, 3, 4, 6}) we have b k = −b l . Lemma 2.12 Let T be a surjective linear isometry on C(1) (X). For x ∈ X and f ∈ C(1) (X), if f (x) = 0, then T −1 ( f )(τ (x)) = 0. Proof Let ak , al , b k , b l as supposed in Lemma 2.11. Since   δ(τ (x), f1 ∗ ) T −1 ( f ) = (T ∗ )−1 δ(τ (x), f1 ∗ ) ( f ), by Lemma 2.11, that is, √ 3  −1  T f (τ (x)) = ak f (x) + b k f  (x) = b k f  (x). T −1 ( f )(τ (x)) + 3

(2.5)

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Similarly, δ(τ (x), f3 ∗ ) (T −1 ( f )) = (T ∗ )−1 δ(τ (x), f3 ∗ ) ( f ), that is, √ 3  −1  −1 T f (τ (x)) = al f (x) − b l f  (x) = −b l f  (x). T ( f )(τ (x)) − 3 If f (x) = 0, by (2.5) + (2.6), we have T −1 ( f )(τ (x)) = 0.

(2.6) 

3 Theorem Theorem 3.1 T is a surjective linear isometry on C(1) (X) if and only if T is of the following form: T( f ) = f ◦ τ

or

T( f ) = − f ◦ τ,

∀ f ∈ C(1) (X),

where τ is a homeomorphism on X. More over, if X = [a, b ], then τ is one of the two functions F and a + b − F, where F is the identity on X. Proof Since T(1) = 1 or T(1) = −1, thus assume without loss of generality T(1) = 1. Let T be a surjective linear isometry on C(1) (X) and g(y) = f (y) − f (x), for any f ∈ C(1) (X), x, y ∈ X. Then g ∈ C(1) (X) and g(x) = 0. By Lemma 2.12, we have 0 = T −1 (g)(τ (x)) = T −1 ( f )(τ (x)) − f (x) · T −1 (1)(τ (x)). Since T is a surjective linear isometry on C(1) (X), so is T −1 . By Lemma 2.7, we get that T −1 (1) is a constant function on X. Moreover, by Lemma 2.8, we have that T −1 (1) = ±1. Assume without loss of generality T −1 (1) = 1. Thus, f (x) = T −1 ( f )(τ (x)). Take T( f ) in place of f , we have T( f ) = f ◦ τ, ∀ f ∈ C(1) (X).(If T −1 (1) = 1, similarly, we can prove that T( f ) = − f ◦ τ, ∀ f ∈ C(1) (X).) If X = [a, b ], then T(F)(x) = F(τ (x)) = τ (x). Thus we have τ (x) ∈ C(1) (X), and √   3  ∗ τ (x). δ x, f1 (T(F)) = τ (x) + 3 By Lemma 2.8, we get ∗



T δ x, f1

∗



√ 3 (F) = τ (x) + . 3

then τ  (x) = ±1. Since τ is a homeomorphism on X, it is easy to see that τ is one of the two functions F and a + b − F. 

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297

Acknowledgement The author is grateful to Professor Wang Risheng for his encouraging discussion and kind help.

References 1. Wang, R.: Linear isometric operators on the C0 (n) (X) type spaces. Kodai Math. J. 19, 259–281 (1996) 2. Pathak, V.D.: Isometrics of Cn [0, 1]. Pac. J. Math. 94, 211–222 (1981) 3. De Leeuw, K.: Banach spaces of Lipschitz functions. Stud. Math. 21, 55–66 (1961) 4. Jarosz, K., Pathak, V.D.: Isometries between function spaces. Trans. Am. Math. Soc. 305(1), 193–206 (1988)

Math Phys Anal Geom (2010) 13:299–313 DOI 10.1007/s11040-010-9081-z

Wegner Estimates for Sign-Changing Single Site Potentials Ivan Veseli´c

Received: 26 July 2009 / Accepted: 2 August 2010 / Published online: 19 August 2010 © Springer Science+Business Media B.V. 2010

Abstract We study Anderson and alloy-type random Schrödinger operators on 2 (Zd ) and L2 (Rd ). Wegner estimates are bounds on the average number of eigenvalues in an energy interval of finite box restrictions of these types of operators. For a certain class of models we prove a Wegner estimate which is linear in the volume of the box and the length of the considered energy interval. The single site potential of the Anderson/alloy-type model does not need to have fixed sign, but it needs be of a generalised step function form. The result implies the Lipschitz continuity of the integrated density of states. Keywords Random Schrödinger operators · Alloy-type model · Integrated density of states · Wegner estimate · Single site potential · Non-monotone Mathematics Subject Classifications (2010) 82B44 · 60H25 · 35J10

1 Model and Results We study spectral properties of Schrödinger operators which are given as the sum H = − + V of the negative Laplacian  and a multiplication operator

This work has been partially supported by the DFG within the Emmy-Noether-Project “Spectral properties of random Schrödinger operators and random operators on manifolds and graphs”. I. Veseli´c (B) Fakultät für Mathematik, 09107 TU-Chemnitz, Germany URL: www.tu-chemnitz.de/mathematik/stochastik/

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V. The operators can be considered in d-dimensional Euclidean space Rd or on the lattice Zd . To be able to treat both cases simultaneously let us use the symbol X d for either Rd or Zd . On the continuum the Laplace operator is the d ∂ 2 sum of second derivatives i=1 and V is a bounded function Rd → R. Thus ∂ x2 i

H is selfadjoint on the usual Sobolev space W 2,2 (Rd ). In the discrete case the d Laplacian is given by the rule φ(k) = i=1 φ(k + ei ) + φ(k − ei ), where φ is a sequence in 2 (Zd ) and (e1 , . . . , ed ) is an orthonormal basis which defines the lattice Zd as a subset of Rd . The potential is given by a bounded function V : Zd → R, and thus H is a bounded selfadjoint operator. The operators we are considering are random. More precisely, the potential V = Vper + Vω decomposes into a part Vper which is translation invariant with respect to some sub-lattice nZd , n ∈ N, i.e. Vper (x + k) = Vper (x) for all x ∈ X d and all k ∈ nZd , and a part Vω which  is random. The random part of the potential is a stochastic field Vω (x) := k∈Zd ωk u(x − k), x ∈ X d , of alloy or Anderson type. Here u : X d → R is a bounded, compactly supported function, which we call single site potential. The coupling constants ωk , k ∈ Zd form a sequence of independent, identically distributed real random variables. We assume that the random variables are bounded and distributed according to a density f of bounded variation. In the discrete case the random operator Hω = − + Vper + Vω is called Anderson model, and in the continuum case Hω is called alloy-type model. There is a well defined spectral distribution function N : R → R of the family (Hω )ω which is closely related to eigenvalue counting functions on finite cubes. To explain this precisely, we need some more notation. Denote by χ the characteristic function of the set [−1/2, 1/2]d ∩ X d . Thus in the continuum case this set is a unit cube, and in the discrete case it is a single lattice point. Also, for k ∈ Zd , let χk (x) := χ(x − k) be the translate of χ. The cube [−l − 12 , l + 12 ]d ∩ X d will be abbreviated by l , its intersection with Zd by Ql , the restriction of Hω to l with selfadjoint boundary conditions (e.g. Dirichlet, Neumann or periodic ones) by Hωl , the spectral projection associated to Hω (respectively to Hωl ) and an interval I by Pω (I) (respectively by Plω (I)), and the number of eigenvalues of Hωl in ] − ∞, E] by Nωl (E) := Tr[Pωl ( ] − ∞, E])]. With this notation we can define the integrated density of states, which is the spectral distribution of the family (Hω )ω , by N(E) := E {Tr[χ Pω (] − ∞, E])]} . Here χ is understood as a multiplication operator. The function N has the following self-averaging property: for all E where N is continuous (that’s a set with countable complement) the relation liml→∞ (2l + 1)−d Nωl (E) = N(E) holds almost surely. This implies that if E1 and E2 are two continuity points of N, we have   lim (2l + 1)−d E Nωl (E2 ) − Nωl (E1 ) = N (E2 ) − N (E1 ) .

l→∞

(1)

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Thus if one is able to show that there is a function CW : R → R and an exponent β ∈ ]0, 1], such that for all E1 , E2  E and for all l ∈ N the so-called Wegner bound (named after the paper [29])   E Nωl (E2 ) − Nωl (E1 )  CW (E) (2l + 1)d |E2 − E1 |β

(2)

holds, it follows that the integrated density of states is (locally uniformly) Hölder-continuous with exponent β. Note that this shows a posteriori that there are no points of discontinuity of N and thus the convergence in (1) holds actually for all E1 , E2 ∈ R. This is only one of the reasons why one is interested in bounds on the averaged quantity E {Tr[Pωl (]E1 , E2 ])]}. It plays also a crucial role in arguments leading to the proof of localisation, i.e. the phenomenon that there is a subset ∅ = Iloc ⊂ R such that Iloc ∩ σpp (Hω ) = Iloc and Iloc ∩ (σac (Hω ) ∪ σsc (Hω )) = ∅ almost surely. In fact, usually localisation goes along with quite explicit bounds on the decay of eigenfunctions and on the non-spreading of electron wavepackets (see for instance the monograph [23] or the characterisation established in [12]). For recent surveys on the integrated density of states see [18, 27]. Now we specialise to a specific class of single site potentials. The important point is that the resulting Anderson/alloy-type model allows for random potentials where the potential values at different points in space are negatively correlated: Let κ > 0, v : X d → R a function satisfying v  κχ, and α : Zd → R a function with support such that its Fourier transform αˆ : [0, 2π [d →  compact −ik·θ C, α(θ) ˆ := k∈Zd αk e does not vanish on [0, 2π [d . Then we call u(x) :=



αk v(x − k)

(3)

k∈Zd

a single site potential of generalised step function form. Note that the sum contains only finitely many non-vanishing terms. Due to the fact that the coefficients αk , k ∈ Zd , may change sign, the random potential Vω can have negative correlations between values at different sites. Now we are in the position to formulate our main result: Theorem 1 Let Hω = − + Vper + Vω be an Anderson model on 2 (Zd ) or an alloy-type model on L2 (Rd ) with a single site potential u of generalised step function form. Assume that the density f of the coupling constants has compact support and bounded variation. Then there is a continuous function CW : R → R such that for all E1 , E2  E and for all l ∈ N the Wegner bound   E Nωl (E2 ) − Nωl (E1 )  CW (E) (2l + 1)d |E2 − E1 | holds.

(4)

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Remark 2 (i) A possible choice of the function CW is CW (E) = κ1 C(E, V) f BV B 1 . Here f BV denotes the total variation norm of f , C(E, V) equals ⎛ ⎞ d 2   π e E+V∞ exp ⎝ n2j ⎠ , 2 d j=1 n∈Z ,n j 0

V∞ := supx∈X,ω∈ |Vper (x) + Vω (x)|, and B 1 is the column sum norm of the inverse of the multi-dimensional Laurent matrix {α j−k } j,k∈Zd . In the case of the discrete Anderson model one can choose CW (E) = 1

f BV B 1 . κ (ii) Taking into account the discussion following inequality (2) we see that estimate (4) implies that the integrated density of states N : R → R is actually locally uniformly Lipschitz-continuous. This in turn implies that the derivative n(E) := dN(E) exists almost everywhere on R and is dE locally uniformly bounded by CW (E). The function n is called density of states. (iii) As mentioned above the Wegner estimate (4) can be used in a proof of localisation, along the so called multiscale analysis. This is an induction argument over increasing lenght scales lk (k ∈ N), where on each scale one establishes exponential decay properties of the Green’s function of the random operator restricted to a box, with probability sufficiently close to one. A crucial ingredient needed to prove the induction step is the control of spectral resonances of Hamiltonians restricted to disjoint boxes. This in turn can be reduced to the control of clustering properties of eigenvalues of a single finite box restriction of the random Hamiltonian. The Wegner estimate (4) is a sufficient bound for this purpose. Remark 3 (Similar results in earlier literature) For Anderson/alloy-type models where the single site potential has fixed sign, Wegner estimates are well understood by now, see e.g. [8, 18, 27] and the references therein. Let us discuss earlier theorems in the literature which establish Wegner estimates for single site potentials that change sign. Theorem 1 recovers the main result of [26] where the same statement was proven under two additional  conditions: It was assumed that there is an index j ∈ Zd such that |α j| > k∈Zd ,k= j |αk | and that the density f belongs to the Sobolev space Wc1,1 (R). Exactly the same statement as in Theorem 1 above, but only for dimensions d = 1 and d = 2 was proven in [22] in a joint paper with V. Kostrykin. There is another method to prove Wegner estimates for single site potentials that are allowed to change sign which is based on certain vector fields in the parameter space underlying the alloy-type model. It was introduced in [17] by F. Klopp and improved by Hislop and Klopp in [14]. Its advantage

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is that it applies to arbitrary continuous, compactly supported single site potentials (which are not identically equal to zero). The regularity requirement on the density f is slightly more restrictive than in the Theorem 1. However, this method applies only to certain energy intervals [E1 , E2 ]: sufficiently low energies are allowed, but arbitrary high energies are not allowed. The papers [14, 17, 22] contain various other results, which we do not state here, because they cannot be directly compared with our theorem above. Additional aspects of Wegner estimates for sign-nondefinite single site potentials are discussed e.g. in [9], Section 5.5. of [27], and [8]. There is an alternative approach to control resonances of finite box Hamiltonians which applies to models with analytic, but not necessarily monotone parameter dependence (cf. [2–5, 13, 19]. It uses variants of Cartan’s Theorem to derive bounds which resemble Wegner estimates. Remark 4 (i) Let us briefly discuss the relevance of the condition that the Fourier transform αˆ does not vanish on [0, 2π [d . It ensures that the multidimensional Laurent matrix A with coefficients α j−k , j, k ∈ Zd , when considered as an operator from  p (Zd ) to  p (Zd ) has a bounded inverse B. However, in the proof of the Wegner estimate above we encounter not the infinite matrix A, but rather finite size matrices A which need to have bounded inverses B with norms uniformly bounded in  = l , l ∈ N. The relevant norm is the column sum norm, corresponding to the operator norm on 1 (). If A is chosen to be a finite section multi-dimensional Toeplitz operator this leads to nontrivial open questions concerning the invertibility of truncated Toeplitz matrices, see for instance [6]. This is the reason why the results of [22] are restricted to dimension one and two, cf. also [20]. However, it turns out that one has a certain freedom in the choice of the finite volume matrices A . In particular, one can choose them to be finite multi-dimensional circulant matrices (rather than finite Toeplitz matrices), which have much better invertibility properties and can be used to complete the proof of Theorem 1. (ii) Let us explain how the Toeplitz, respectively circulant, matrices A enter the proof of the Wegner estimate. To deal with the (possible) non-monotonicity of the dependence of the potential Vω on the random variable ωk one uses a suitable transformation of coordinates. This transformation acts on a restricted probability space associated to a finite sub-collection of variables ωk , k ∈ Q. To be suitable the transformation has to satisfy two conditions. On the one hand, the transformed random variables should couple to non-negative transformed single site potentials. On the other hand, the inverses of the transformations should be uniformly bounded with respect to the finite subset Q ⊂ Zd . That a linear transformation with the desired properties exists is the content of Proposition 9.

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(iii) At this point it should be emphasised that the use of the method indicated in the above two items is not restricted to Wegner estimates. There are other situations where it can be used to cope with a nonmonotoneous dependence ωk → Vω . An instance would be the proof that the fractional moment of the Green’s function of the Anderson model on 2 (Zd ) is uniformly bounded, see Theorem 2.3 in [24] for a precise statement. (iv) Let us compare the results of the present paper to the ones of the recent work [28]. We first discuss those aspects which are common to both works. They treat alloy-type models with sign-changing single site potentials. Both use a transformation of coordinates in the (finite dimensional) probability space associated to compact cubes in configuration space to extract monotone families from the random potential. This leads to situations where methods to prove Wegner estimates for fixed-sign single site potentials can be applied. So much for the aspects common to both papers. Now for the differences: The Wegner estimates obtained in the present paper apply to operators in the continuous as well as in the discrete setting, and are actually in both cases optimal in the sense that the right hand side of (4) depends linearly on the volume of the box and on the length of the energy interval. The results of [28] concern only discrete alloy-type models (albeit the methods should be amenable to derive similar, but weaker Wegner estimates for analogous models in the continuum) and yield, depending on the specific assumptions, Wegner estimates linear or polynomial in the volume of the box. This difference in the final results reflects the fact that the proofs use different arguments once monotone random parameters have been identified: [28] relies on the scheme of proof established in [16] (see also [29] for an earlier version of this argument), while the present paper invokes [7] (see also [21] for an earlier version of this argument). However, the most important difference of the two approaches lie in the choice of the transformation of variables on the probability space. In the present paper the transformation produces a family (ηm ) ot random variables which are stationary (like the the original coupling constants (ωk )). The non-vanishing condition on αˆ ensures that one has a good control of the transformation (ωk ) → (ηm ) and its inverse. This allows one to go in the proof back  and forth between  the two representations of the potential Vω (x) = k∈Zd ωk u(x − k) = m ηm v(x − m), cf. (7). On the other hand, the transformation of variables in [28] can be interpreted as a one-dimensional substitution rule. In particular the resulting family of random variables is no longer stationary. Moreover the proof does not use (and does not allow) to go back to the original random variables (ωk ). (v) Let us mention that a third variant of the choice of transformation on the probability space can be used to prove Wegner estimates and the Lipschitz-continuity of the integrated density of states for random potentials which are given by a Gaussian random field with sign-changing

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covariance function. More precisely the transformation allows one to establish a simple criterion when condition (4) in [11] is satisfied. This is established in [25]. Example 5 To apply the theorem one has to know that αˆ vanishes nowhere on the torus [0, 2π [d . Let us give a variety of instances where this condition holds. (i)  The first case is when there is an index j ∈ Zd such that |α j| > k∈Zd ,k= j |αk |, cf. [26]. (ii) If d = 1 and the diameter N of the support of α is kept fixed, the property holds for a dense, open subset of α ∈ R N+1 . (iii) An explicit example for d = 2 is provided in [10]. Let α2,−2 = 16, α1,−1 = 36, and α−1,1 = 27. Then the corresponding Fourier transform α(θ ˆ 1 , θ2 ) = 16 e−2iθ1 e2iθ2 − 36 e−iθ1 eiθ2 + 27 eiθ1 e−iθ2

(iv)

(v)

(vi)

(vii)

does not vanish anywhere on [0, 2π [2 . In fact this is just a special case of an infinite family of trigonometric polynomials αˆ satisfying the non-vanishing condition (in any dimension d ∈ N). We introduce this class  of examples next. Let N ∈ N be arbitrary and ϕ : R → C, ϕ(x) := Nj=−N ϕ j e−i jx a trigonometric polynomial which does not vanish anywhere. By Example (ii) above we know that lots of such function ϕ exist. Obviously ϕ is continuous and 2π -periodic. In particular, infx∈R |ϕ(x)| is strictly positive. Let c ∈ Zd be arbitrary and define the function αˆ : Rd → C by α(θ) ˆ := ϕ( c, θ ). Then αˆ is 2π periodic in each coordinate and has the form k∈Zd αk e−ik·θ with only finitely many coefficients αk different from zero. As ϕ is bounded away from zero the same holds for the function α. ˆ The procedure described in the last item can be generalised to higher dimensions. Let d < D be two integers. Suppose that one has the a priori information that the trigonometric polynomial  : Rd → C, (x) :=  −i j·x does not vanish. Let C : R D → Rd be a linear map and j∈Zd  j e set α(θ) ˆ := ϕ(Cθ). As above one concludes that αˆ vanishes nowhere on the D-dimensional torus. Let αˆ and βˆ be two trigonometric polynomials which vanish nowhere on the d-dimensional torus. Then the same property holds for their product. The Fourier coeffcients of the product are simply given by a convolution. This way we can combine examples of the type (i) and (iv) to obtain new ones. Let αˆ be a trigonometric polynomial which vanishes nowhere on the d-dimensional torus. Then the same property holds for trigonometric polynomials which are obtained from αˆ by a sufficiently small perturbation of the Fourier coefficients. This allows us again to broaden the class of examples obtained in the previous items of the list.

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2 Proof of Theorem 1 Remark 6 (Cubes) Theorem 1 concerns Hamiltonians restricted to a cube l ⊂ X d of side length 2l + 1. However, in the proof we will have to deal with several modifications of this cube. First, we need to consider cubes which are subsets of Zd and not of X d (which may be either Rd or Zd ). Second, modified cubes will be of larger side length than the original cube l . At this point we list the various cube sizes which will appear at various stages of the proof. To avoid confusion, cubes in X d will be denoted by the symbol  while the letter Q will be reserved for cubes in Zd . In the following we assume that l ∈ N is fixed and that  = l . Since the single site potential u has compact support, there is some R ∈ N such that supp u is contained in  R . This implies that the set of the lattice sites k such that the coupling constants ωk influence the potential Vω inside the cube  is contained in the set Ql+R = l+R ∩ Zd . Similarly, there is some r ∈ N such that the support of v is contained in r . Consequently, the set {k ∈ Zd | supp vk ∩ l } is contained in Ql+r . Here we used the abbreviation vk (x) = v(x − k). Finally, since we assumed that α : Zd → R is compactly supported, there is some D ∈ N such that supp α ⊂ Q D . The relation between v, u, and α implies R  r + D. Let us point point out that all cubes listed depend on the reference cube  of side length 2l + 1. The proof of Theorem 1 uses results on spectral averaging for non-negative single site potentials established in [7, Section 4] (see also [21]). These facts are formulated in Proposition 7. The subsequent Propositions 8 and 9 contain the estimates which are needed to deal with single site potentials of changing sign. Proposition 7 ([7]) Let I = [E1 , E2 ] be an interval. Then (a)    



E χ j Pl (I)χ j

E Tr Plω (I)  C(E2 , V) ω j∈Ql

 2 d  2 Here C(E2 , V) := e E2 +V∞ n∈Zd ,n j 0 exp π2 and V∞ := j=1 n j supx∈X,ω∈ |Vper (x) + Vω (x)|. (b) Let H = − + W be a Schrödinger operator with a bounded potential W, w a function satisfying w  χ j for some j ∈ Zd , and t → Ht = H + tw a one parameter family of operators. Denote by Ht a selfadjoint restriction of Ht to a cube  and by Pt (I) the associated spectral projection on to 2 the interval I. Then, for any g ∈ L∞ c (R) and any φ ∈ L () with φ = 1 we have

dt g(t) φ, χ j Pt (I)χ jφ  |I| g ∞ The first statement allows one to decompose the expectation value of the trace of the spectral projector into local contributions of unit cubes. The local

Wegner Estimates for Sign-Changing Single Site Potentials

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contributions do not depend on the trace, but rather on the norm of certain restricted operators. Spectral averaging is easier to perform of norms than on traces. Estimate (a) is proven using Dirichlet-Neumann-bracketing and Jensen’s inequality in the case of operators on L2 (Rd ). For operators on 2 (Z2 ) the estimate is trivial and the constant C(E2 , V) can be chosen equal to one. Statement (b) is a spectral averaging estimate which is based on a contour integral in the complex plane and the residue theorem. The next two propositions contain the main technical results of the paper. For a cube  = l let us denote by A and B matrices with coefficients in Ql+R  . If A is invertible, B will denote its inverse. The column sum norm supk j |B ( j, k)| of B will be denoted by B 1 . Proposition 8 Let I = [E1 , E2 ] be an interval. If there exists an invertible matrix A : 1 (Ql+R ) → 1 (Ql+R ) such that A ( j, k) = α j−k

for all j ∈ Ql+r and k ∈ Ql+R

(5)

then for any j ∈ Ql and φ ∈ L2 (l ) with φ = 1   E φ, χ j Plω (I)χ jφ  |I| f BV B 1 Here f BV denotes the total variation norm of the density f of the random variables ωk . Let us point out that condition (5) fixes the coefficients of A only in a multi-dimensional rectangle. The coefficients outside the rectangle are arbitrary, up to the invertibility condition. Denote by A : 1 (Zd ) → 1 (Zd ) the linear operator whose coefficients in the canonical orthonormal basis are A( j, k) = α j−k for j, k ∈ Zd . Since the function α has compact support, the operator A is bounded. Moreover, αˆ is an element of the Wiener algebra of functions on the d-dimensional torus with absolutely convergent Fourier series. The so-called ‘1/ f Theorem’ of Wiener states that if αˆ vanishes nowhere on the torus, then the inverse 1/αˆ has an absolutely convergent Fourier series as well. Wiener’s original result concerns the case d = 1, but Gelfand’s proof of the theorem (cf. e.g. [1, 15]) extends directly ˆ then the to arbitrary d. If we denote by βn the Fourier coefficients of 1/α, Laurent matrix B = (β j−k ) j,k∈Zd has finite column sum norm, i.e. is bounded as an operator 1 (Zd ) → 1 (Zd ). Furthermore, since 1/αˆ is the inverse of α, ˆ the operator B is the inverse of A. For a given cube  = l and a site m ∈ Zd we consider the associated lattice l+R (m) = m + l+R , where l+R = (2l + 2R + 1)Zd , and the projection πl+R : Zd → l+R , πl+R (m) = l+R ∩ l+R (m). Note that the map πl+R is pe˜ ∈ Zd is such that m − m ˜ ∈ l+R , riodic with respect to the lattice l+R , i. e. if m ˜ = πl+R (m). If there is no danger of confusion we will drop the then πl+R (m) subscript l + R denoting the period.

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Proposition 9 Let l > R and  = l . Def ine the matrix A by A ( j, k) = απ( j−k) for j, k ∈ Ql+r . Then A satisf ies condition (5), is invertible and

B 1  B 1 < ∞ where B is the inverse of A and B the inverse of A. Note that A (and thus its inverse B , as well) is a multi-dimensional circulant matrix. Now we prove the two propositions before completing the proof of Theorem 1 at the end of this section. The next proof is an adaptation of results in [22, 26]. Proof of Proposition 8 Let us first reduce the model to the case κ = 1. (Recall that κ > 0 was a uniform lower bound for v on the unit cube.) Obviously we can write the random potential as     1 ωk u(x − k) = (κ ωk ) u(x − k) . κ d d k∈Z

k∈Z



Now κ1 u(x) = j∈Zd α j( κ1 v(x − j)) where by assumption κ1 v  χ. The distribution of the random variable κ ωk has the density x → h(x) := κ1 f (x/κ). The total variation norm of h equals κ1 f BV . Thus we can replace κ by one, if we keep in mind that the variation norm of the density gets multiplied by 1/κ. As pointed out earlier E { φ, χ j Plω (I)χ jφ } depends only on a finite number of random variables ωk . More precisely if R is such that the compact support of u is contained in  R then only the coupling constants ωk with index k in Ql+R influence the scalar product φ, χ j Plω (I)χ jφ . Thus we can express the expectation value E { φ, χ j Plω (I)χ jφ } as a finite dimensional integral

  dω F(ω ) φ, χ j Plω (I)χ jφ . (6) 

RL

Here F(ω ) = k∈Ql+R f (ωk ) is the common density of the random variables ωk with index in Ql+R and L is the cardinality of this index set. In the sequel it will be convenient to have two alternative representations for the random potential Vω . This will be presented next. For any x ∈ R we have      Vω (x) = ωk u(x−k) = ωk v(x− j− k) = v(x−m) αm−k ωk . k∈Zd

k∈Zd

j∈Zd

m∈Zd

k∈Zd

(7)   If x ∈ l , the last sum equals m∈Ql+r v(x − m) k∈Ql+R αm−k ωk . Here r ∈ N is such that the compact support of v is contained in r . Thus, we can

Wegner Estimates for Sign-Changing Single Site Potentials

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conveniently express Vω (x) for x ∈ l using new random variables ηm :=  k∈Ql+R αm−k ωk , m ∈ Ql+r as  ηm v(x − m). Vω (x) = m∈Ql+r

By assumption there exists an invertible matrix A : 1 (Ql+R ) → 1 (Ql+R ) such that for all m ∈ Ql+r  ηm = A (m, k) ωk (8) k∈Ql+R

We define the random variables ηm for m ∈ Ql+R \ Ql+r by requiring that the relation (8) is true for these indices as well. Thus η = A ω and ω = B η , where η = (ηk )k∈Ql+R . If we express the random potential in the η-variables, the ‘effective’ single site potentials v j satisfy v j  χ j. Now we fix a lattice site j ∈ Ql and a configuration ηk , k ∈ Ql+r \ { j} and set H := − + Vper + m∈Ql+r \{ j} ηm v(· − m). To the one-parameter family of operators η j → H + η jv(· − j) the spectral averaging result of Proposition 7(b) applies: For any g ∈ L∞ c (R) and φ ∈ L2 (l ) with φ = 1 we have

  (9) dη j g(η j) φ, χ j PlB η (I)χ jφ  |I| g ∞ . However, the η-random variables are no longer independent. To understand their dependence we have to analyse the common density. It can be compactly written in the form k(η ) = | det B | F(B η ) where F(ω ) is the original common density of the ωk , k ∈ Ql+R . Thus (6) equals

  | det B | dη k(η ) φ, χ j Plω (I)χ jφ RL

= | det B |

⊥j

R L−1

dη

R

  dη jk(η ) φ, χ j PlB η (I)χ jφ .

⊥j

(10)

Here we denote by η the sub-collection of random variables indexed by Ql+R \ { j}. If we apply (9) to the one dimensional integral appearing in (10), we obtain the upper bound |I| supη j ∈R |k(η )|. Assume for the moment that f    )  is continuously differentiable. Then sup |k(η )|  R  ∂k(η dη j. Since ∂η j  η j ∈R

 ∂k(η ) = ∂η j



f ((B η ) p )

k∈Ql+R p∈Ql+R

=



k∈Ql+R

B (k, j) f  ((ω )k )

∂ f ((B η )k ) ∂η j  p∈Ql+R

f ((ω ) p )

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we can pass back to the ω-variables and establish the bound

 ⊥j dη sup |k(η )|  | det A | |B (k, j)| f  L1 . η j ∈R

R L−1

k∈Ql+R

Thus (10) is bounded by |I| B 1 f BV . To extend this estimate to general densities f of bounded variation, let { fk }k be an approximation sequence of smooth, nonnegative, compactly supported functions such that fk 1 = 1 for all k ∈ N, supk∈N fk BV  f BV and limk→∞ fk − f 1 = 0, cf. e.g. Theorems 1.6.1 and 5.3.1. in [30]. Then we have

  dω F(ω ) φ, χ j Plω (I)χ jφ RL

=

dω

 k∈

RL



+

  fk (ωk ) φ, χ j Plω (I)χ jφ

RL

dω

 k∈

f (ωk ) −



 fk (ωk )

  φ, χ j Plω (I)χ jφ .

k∈

As we have shown, the first integral on the right is bounded by |I| B 1 fk BV . This expression is bounded by |I| B 1 f BV uniformly in k ∈ N. A telescoping argument shows that the norm of the second integral is bounded by L f − fk 1 , which tends to zero as k → ∞. Thus the proposition is proven.   The following lemma will be used in the proof of Proposition 9. Lemma 10 Let N ∈ N and π = π N . Then, for all j ∈ Q N and k ∈ Q N+D the equality απ( j−k) = α j−k holds. Proof We first show that απ( j−k) = 0 implies α j−k = 0: Set m = j − k. If αm = 0 then m ∈ Q D , since the support of α is contained in Q D . This implies π(m) = m and thus απ(m) = αm . Now we consider the case απ(m) = 0. In this case m is an element of Q D +  N+D . Since j ∈ Q N and k ∈ Q N+D , i. e. j ∞  N and k ∞  N + D the triangle inequality implies j − k ∞  2N + D. Once we have shown Q2N+D ∩ (Q D +  N+D ) = Q D

(11)

it follows immediately that π( j − k) = j − k. To see (11) notice that any x ∈ Q2N+D ∩ (Q D +  N+D ) satisfies the following conditions: x ∈ Zd , x ∞  2N + D and ∃ y ∈ (2N + 2D + 1)Zd with y − x ∞  D. Assume that x ∈ Q D . Then y = 0 and consequently y ∞  2N + 2D + 1. The second triangle inequality forces y − x ∞  y ∞ − x ∞  D + 1, a contradiction.  

Wegner Estimates for Sign-Changing Single Site Potentials

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Proof of Proposition 9 An application of Lemma 10 with the choice N = l + r (recall that R  r + D) shows that the matrix A with coefficients A ( j, k) = απ( j−k) for j, k ∈ Ql+R satisfies condition (5). Now we have to identify the in verse of A . Define B : 1 (Ql+R ) → 1 (Ql+R ) by B ( j, k) := B( p, k) p∈l+R ( j)

where B( p, k) are the coefficients of the inverse B of A : 1 (Zd ) → 1 (Zd ). Recall that the projection π : Zd → l+R is defined by π(m) = l+R ∩ l+R (m). Let us calculate the product B A . For any i, j ∈ Ql+R we have by the very definition of A and B :     (B A )(i, j) = B (m, j)απ(i−m) = B(m + n, j) αi−m+ p m∈Ql+R

=





m∈Ql+R n∈l+R



B(m + n, j)

m∈Ql+R n∈l+R

p∈l+R

αi−m+ p−n .

p∈l+R

In the lastline we  used that n + l+R = l+R for n ∈ l+R . By the decomposition Zd = m∈Ql+R n∈l+R {m + n} and the definition of the Laurent matrix A, the last expression equals      B(k, j)αi+ p−k = B(k, j)A(i + p, k) = δi+ p, j p∈l+R k∈Zd

p∈l+R k∈Zd

p∈l+R

since B is the inverse of A. Note that for i, j ∈ l+R we have δi+ p, j = 0 for all p ∈ l+R \ {0}. It follows that  δi+ p, j = δi, j. p∈l+R

Thus we have checked that B is the inverse of A . The last step in the proof is to establish that B 1  B 1 . Indeed, for all  we have           

B 1  |B ( j, k)|  B( j + p, k) |B(m, k)|  B 1 .    m∈Zd j∈l+R j∈l+R  p∈l+R   Completion of the Proof of Theorem 1 We collect the estimates established so far:    



E χ j Pl (I)χ j

by Proposition 7(a) E Tr Plω (I)  C(E2 , V) ω  C(E2 , V)

 j∈Ql

j∈Ql

sup E

φ =1

  φ, χ j Plω (I)χ jφ

 C(E2 , V) |Ql | f BV B 1 |I|

by Proposition 8

 C(E2 , V) (2l + 1)d f BV B 1 |I|

by Proposition 9  

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Remark 11 Daniel Lenz pointed out to us that the calculation in the proof of Proposition 9 can be understood from a more abstract point of view. If G is a discrete abelian group and H a subgroup (with appropriate invariant measures) then the projection map induces a homomorphism between the convolution algebras 1 (G) and 1 (H).

References 1. Arveson, W.: A short course on spectral theory. Graduate Texts in Mathematics, vol. 209. Springer, New York (2002) 2. Bourgain, J., Goldstein, M., Schlag W.: Anderson localization for Schrödinger operators on Z2 with quasi-periodic potential. Acta Math. 188(1), 41–86 (2002) 3. Bourgain, J.: Green’s function estimates for lattice Schrödinger operators and applications. Annals of Mathematics Studies, vol. 158. Princeton University Press, Princeton (2005) 4. Bourgain, J.: Anderson localization for quasi-periodic lattice Schrödinger operators on Zd , d arbitrary. Geom. Funct. Anal. 17(3), 682–706 (2007) 5. Bourgain, J.: An approach to Wegner’s estimate using subharmonicity. J. Stat. Phys. 134(5–6), 969–978 (2009) 6. Böttcher, A., Silbermann, B.: Introduction to large truncated Toeplitz matrices. Springer (1999) 7. Combes, J.-M., Hislop, P.D.: Localization for some continuous, random Hamiltionians in ddimensions. J. Funct. Anal. 124, 149–180 (1994) 8. Combes, J.-M., Hislop, P.D., Klopp, F.: An optimal Wegner estimate and its application to the global continuity of the integrated density of states for random Schrödinger operators. Duke Math. J. 140(3), 469–498 (2007) 9. Combes, J.-M., Hislop, P.D., Klopp, F., Nakamura, S.: The Wegner estimate and the integrated density of states for some random operators. Proc. Indian Acad. Sci. Math. Sci. 112(1), 31–53 (2002) 10. Douglas, R.G., Howe, R.: On the C∗ -algebra of Toeplitz operators on the quarter-plane. Trans. Am. Math. Soc. 158, 203–217 (1971) 11. Fischer, W., Hupfer, T., Leschke, H., Müller, P.: Existence of the density of states for multidimensional continuum Schrödinger operators with Gaussian random potentials. Commun. Math. Phys. 190(1), 133–141 (1997) 12. Germinet, F., Klein, A.: A characterization of the Anderson metal-insulator transport transition. Duke Math. J. 124(2), 309–350 (2004) 13. Goldstein, M., Schlag, W.: Hölder continuity of the integrated density of states for quasiperiodic Schrödinger equations and averages of shifts of subharmonic functions. Ann. Math. (2), 154(1), 155–203 (2001) 14. Hislop, P.D., Klopp, F.: The integrated density of states for some random operators with nonsign definite potentials. J. Funct. Anal. 195(1), 12–47 (2002) 15. Katznelson, Y.: An Introduction to Harmonic Analysis. Cambridge University Press (2004) 16. Kirsch, W.: Wegner estimates and Anderson localization for alloy-type potentials. Math. Z. 221, 507–512 (1996) 17. Klopp, F.: Localization for some continuous random Schrödinger operators. Commun. Math. Phys. 167, 553–569 (1995) 18. Kirsch, W., Metzger, B.: The integrated density of states for random Schrödinger operators. In: Spectral Theory and Mathematical Physics, pp. 649–698. AMS (2007) 19. Krüger, H.: Localization for random operators with non-monotone potentials with exponentially decaying correlations. http://arxiv.org/1006.5233 20. Kozak, A.V., Simonenko, I.B.: Projection methods for solving multidimensional discrete convolution equations. Sib. Mat. Z. 21(2), 119–127, 237 (1980) 21. Kotani, S., Simon, B.: Localization in general one-dimensional random systems. II. Continuum Schrödinger operators. Commun. Math. Phys. 112(1), 103–119 (1987)

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22. Kostrykin, V., Veseli´c, I.: On the Lipschitz continuity of the integrated density of states for sign-indefinite potentials. Math. Z. 252(2), 367–392 (2006) 23. Stollmann, P.: Caught by disorder: bound states in random media. Progress in Mathematical Physics, vol. 20. Birkhäuser (2001) 24. Tautenhahn, M., Veseli´c, I.: Spectral properties of discrete alloy-type models. In: XVth International Conference on Mathematical Physics, Prague, 2009. World Scientific (2010) 25. Veseli´c, I.: Lipschitz-continuity of the integrated density of states for Gaussian random potentials. (Preprint) 26. Veseli´c, I.: Wegner estimate and the density of states of some indefinite alloy type Schrödinger operators. Lett. Math. Phys. 59(3), 199–214 (2002) 27. Veseli´c, I.: Existence and regularity properties of the integrated density of states of random Schrödinger Operators. Lecture Notes in Mathematics, vol. 1917. Springer, Verlag (2007) 28. Veseli´c, I.: Wegner estimate for discrete alloy-type models. Ann. Henri Poincaré. doi:10.1007/s00023-010-0052-5. Earlier version available at http://www.ma.utexas.edu/mp_arc/ a/09-100. 29. Wegner, F.: Bounds on the DOS in disordered systems. Z. Phys. B. 44, 9–15 (1981) 30. Ziemer, W.P.: Weakly Differentiable Functions. Springer, New York (1989)

Math Phys Anal Geom (2010) 13:315–330 DOI 10.1007/s11040-010-9082-y

Numerical Range for Orbits Under a Central Force Mao-Ting Chien · Hiroshi Nakazato

Received: 30 November 2008 / Accepted: 2 August 2010 / Published online: 13 August 2010 © Springer Science+Business Media B.V. 2010

Abstract We present an explicit form for the central force that describes the orbit of some roulette curve, and interpret the orbit of the roulette curve as an algebraic curve F(1, x, y) = 0 associated to the homogeneous polynomial F(t, x, y) of a matrix A. The hodograph of the orbit is obtained as the boundary generating curve of the numerical range of A. Keywords Numerical range · Algebraic curve · Central force · Roulette curve Mathematics Subject Classifications (2010) 15A60 · 70F05 · 14Q05 1 Introduction Let A be an n × n complex matrix. The numerical range of A is defined and denoted as   W(A) = ξ ∗ Aξ : ξ ∈ Cn , ξ ∗ ξ = 1 . It is well known that W(A) is convex. Consider the homogeneous polynomial F(t, x, y) associated with A defined as       (1) F(t, x, y) = det tIn + x A + A∗ /2 + y A − A∗ /(2i) ,

M.-T. Chien (B) Department of Mathematics, Soochow University, Taipei 11102, Taiwan e-mail: [email protected] H. Nakazato Department of Mathematical Sciences, Faculty of Science and Technology, Hirosaki University, Hirosaki 036-8561, Japan e-mail: [email protected]

316

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The numerical range of A is the convex hull of the real part of dual curve G(1, X, Y) = 0 of F(t, x, y) = 0 (cf. [10]). The dual curve is also called the boundary generating curve or Kippenhahn curve of W(A). Thus, there is a duality between the boundary generating curve G(1, X, Y) = 0 of the numerical range of a matrix A and the algebraic curve F(t, x, y) = 0 associated with A. Another duality lies between the orbit of a point mass moving under a central force and its hodograph. In a celebrated book [9], Kepler gave a primitive form of the conservation law of the angular momentum. Some historians of science view that he clearly established this formula (so called Kepler’s second law) in his “Epitome of Copernican Astronomy” (1617, 1620, 1621) (cf. [1]). In a classical model of Hipparchus and Ptolemaeus, the Sun describes an eccentric circle (x − e)2 + y2 = 1(0 < e < 1) around the Earth (0, 0) (cf. [12]). It is assumed a circular motion around the center of the circle is uniform. Such a motion is not realized under a central force with the center at the Earth, since it violates the conservation law of angular momentum. We remove the assumption of the uniformity on the motion. Can we realize an eccentric circle orbit under some central force at the Earth? Newton already gave an affirmative answer by using Euclidean geometry. Ptolemaeus provided epicycle or roulette curve models of planetary motions in which the orbit is not assumed to be closed. We consider a closed orbit as its approximation. In a normal form, Ptolemaeus model is expressed as x = x(t) = cos(nt) + a cos(mt), y = y(t) = sin(nt) + a sin(mt), where m, n are distinct non-zero integers and 0 < a and t is a normalized time parameter. Under the condition |n/m| < a < 1 or 1 < a < |n/m|, the angular momentum M(t) = x(t)y (t) − y(t)x (t) of the planet changes its sign. Using this model, Ptolemaeus explained the retrogradation of the apparent movement of planets. Kepler’s second law implies that the retrogradation does not occur for the movement under a central force. Motivated by Newton’s result for eccentric circle orbit, we treat another classical model, the composition of two circular motions for n = −(m − 1), a > 1 in Theorem 2, and present the orbit in terms of the associated homogeneous polynomial F(t, x, y) of some matrix in Theorem 4. For λ ∈ R, the straight line {(x, λ x) : x ∈ R} and the curve (cos((m − 1)t) + a cos(mt), − sin((m − 1)t) + a sin(mt)). a > 1, have 2m common points in the real plane R2 because of the hyperbolicity of F(t, x, y). By choosing suitable λ, we may assume that these 2m common points are non-singular points of the curve, and these 2m points lie on one real analytic curve. Thus the form F(t, x, y) is an irreducible algebraic

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317

curve of order 2m with (2m − 1)(2m − 2)/2 (real) double points (cf. [14]). This number is the maximum of the singular points of an irreducible algebraic curve of order 2m. Even in the case n = −(m − 1), the above position curve over time t violates Kepler’s second law. We treat, in Section 2, its orbit {(x(t), y(t)) : 0 ≤ t ≤ 2π } as a geometry set in the plane, not as a position over ordered-time t. Our study is related with the dynamical system under a central force. The dynamical system under a central force provides simplified models of two body problems. Newton proved Kepler’s second law for a point mass moving under a central force. He also treated some special interesting central forces. We consider the movement of a point mass with unit mass under a central force f (x, y : x , y , t). The equation of motion is given by x (t) =  y (t) = 

x(t) x(t)2

+

y(t)2

y(t) x(t)2 + y(t)2

  f x, y : x , y , t

(2)

  f x, y : x , y , t

(3)

where f (x, y : x , y , t) is a real valued smooth function on the domain   (x, y, a, b , t) : α < x2 + y2 < β, (a, b ) ∈ R2 , −∞ < t < ∞ for 0 ≤ α < β ≤ ∞ (cf. [2, p. 33], [7]). We usually assume the analyticity of f (x, y : x , y , t). If f (x, y : x , y , t) < 0, the force is attractive toward the origin (0, 0). If f (x, y : x , y , t) > 0, the force is repulsive against the origin. We mainly treat time independent and velocity independent central  force f (x, y : x , y , t) = f (x, y) satisfying circular symmetry f (x, y) = f ( x2 + y2 , 0). We abbreviate f (r) for f (r, 0). In modern notations, Newton provided circular orbits under the central force 8r fe (r) = −  3 , 2 r + 1 − e2 where 0 < e ≤ 1 (cf. [5, p. 80]). Under the initial condition   (x(0), y(0)) = (1 + e, 0), x (0), y (0) = (0, 1/(1 + e)), the point mass under the central force fe describes the circle (x − e)2 + y2 = 1. In Section 2, We obtain central force fm,a (r) with initial conditions    (x(0), y(0)) = (a + 1, 0), x (0), y (0) = (1/(a + 1), 0), (4) in which a point mass describes the roulette curve z(s) = exp(−i(m − 1)s) + a exp(ims).

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In this model, the rotations of exp(−i(m − 1)s) and exp(ims) are opposite. We write the roulette curve in xy-coordinates: x(φ(s)) = cos((m − 1)s) + a cos(ms),

(5)

y(φ(s)) = − sin((m − 1)s) + a sin(ms),

(6)



for some analytic function φ satisfying φ(−s) = −φ(s), φ (0) = 0. The parameter m satisfying 1 < m < ∞. In Theorem 2, we present an explicit formula for the central force of the orbit of the roulette curves (5) and (6). In Theorem 4, we interpret the orbit for an integer m > 1 as an algebraic curve F(1, x, y) = 0 associated to a matrix A, and characterize the hodograph as the boundary generating curve of W(A) rotated with angle π/2 around the origin. 2 Angular Momentum and Central Force We assume that the orbit of a point mass is given by (x(s), y(s)) = (ξ(t), η(t)) in analytic parameter s and the time parameter t. Suppose that x(s)y (s) − y(s)x (s) > 0 for every s. The (signed) curvature K(s) of the orbit at the point (x(s), y(s)) is defined as K(s) =

x (s)y (s) − y (s)x (s)  3/2 x (s)2 + y (s)2

(cf. [8]). The orbit (ξ(t), η(t)) of the movement of a point mass satisfying (2) and (3) assures M = x(t)y (t) − y(t)x (t) = const since dM = x (t)y (t) − y (t)x (t) + x(t)y (t) − y(t)x (t) dt =

 x(t)y(t) − x(t)y(t)   f x, y : x , y , t x(t)2 + y(t)2

= 0. We have no interests in case M = 0. We assume that M > 0. Under this assumption, the condition K(s) = 0, that is, x (s)y (s) − y (s)x (s) = 0 implies that (x (s), y (s)) = 0 and hence f (x(s), y(s)) = 0. We assume that K(s) = 0. The osculating circle of the orbit at the point (x(s), y(s)) is given by

2

2 y (s) x (s)   x − (x(s) − ) + y − (y(s) + ) K(s) x (s)2 + y (s)2 K(s) x (s)2 + y (s)2 =

1 , K(s)2

provided x(s)y (s) − y(s)x (s) > 0. Denote

 y (s) x (s)   , y(s) + (x0 (s), y0 (s)) := x(s) − . K(s) x (s)2 + y (s)2 K(s) x (s)2 + y (s)2

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Let 0 ≤ θ(s) ≤ π , θ(s) = π/2 be the angle between the two vectors (x(s), y(s)) and    y (s), −x (s)  (x(s) − x0 (s), y(s) − y0 (s)) = . K(s) x (s)2 + y (s)2 We have the equation cos θ(s) = sgn(K(s)) 

x(s)y (s) − y(s)x (s)  , x(s)2 + y(s)2 x (s)2 + y (s)2

where sgn(K(s)) = K(s)/|K(s)|. The condition K(s) > 0 implies 0 ≤ θ(s) < π/2, and K(s) < 0 implies π/2 < θ(s) ≤ π . Under the assumption x(s)y (s) − y(s)x (s) > 0, the central force f (x(s), y(s)) is given by f (x(s), y(s)) = −|K(s)| × 

M2  , x(s)2 + y(s)2 cos3 θ(s)

(7)

(cf. [5, p. 79]), where M = ξ(0)η (0) − η(0)ξ  (0). The factor M2 /((x(s)2 + y(s)2 ) cos2 θ(s)) in (7) gives the square of the velocity v(s) of the point mass at (x(s), y(s)) by Kepler’s second law. Equation (7) is a general formula for central forces. The orbit may have multiple points. Suppose that (x0 , y0 ) is an ordinary k-ple point and the orbit has k analytic branches. Then we have k quantities K1 , . . . , Kk and cos θ1 , . . . , cos θk at the point (x0 , y0 ), and the quantities Kj , j = 1, 2, . . . k cos3 θ j have the common value independent of j if the central force f (x, y : x , y , t) is independent of the time t and the velocity. In the following, we give an explicit expression of the central force for the roulette curves (5) and (6). If we assume the initial condition (4) then the angular momentum M = 1. Lemma 1 Let m, a be real numbers greater than 1. Then the quantities r(s), K(s), cos θ(s), v(s) for the orbits (5) and (6) with the condition M = 0 are given as the following:  1/2 r(s) = a2 + 1 + 2a cos((2m − 1)s) ,  3  2 a − 1 m + 3m2 − 3m + 1 − am(m − 1) cos((2m − 1)s) K(s) =   3/2 , a2 + 1 m2 − 2m + 1 − 2am(m − 1) cos((2m − 1)s)   2 a − 1 m + 1 + a cos((2m − 1)s) , cos θ(s) = α M2 + y(s)2 ) cos2 θ(s)  2  1/2 a + 1 m2 − 2m + 1 − 2am(m − 1) cos ((2m − 1) s) = M2 ,  2  a2 − 1 m + 1 + a cos ((2m − 1) s)

v(s)2 =

(x(s)2

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where

1/2  α = a2 + 1 + 2a cos ((2m − 1) s)   1/2 × a2 + 1 m2 − 2m + 1 − 2am(m − 1) cos ((2m − 1) s) .

Proof By the addition formula of cosines, r(s)2 = x(s)2 + y(s)2 = a2 + 1 + 2a (cos((m − 1)s) cos(ms) − sin((m − 1)s) sin(ms)) = a2 + 1 + 2a cos((2m − 1)s), and

  x (s)2 + y (s)2 = a2 + 1 m2 − 2m + 1 − 2am(m − 1) cos((2m − 1)s),   x (s)y (s) − y (s)x (s) = a2 − 1 m3 + 3m2 − 3m +1 − am(m − 1) cos((2m − 1)s).

It follows that K(s) > 0 for every s. Since θ(s) is the angle between the vectors (−y(s), x(s)) and (x (s), y (s)), we have  r(s) x (s)2 + y (s)2 cos θ(s) = −y(s)x (s) + x(s)y (s), and hence  a2 + 1 + 2a cos((2m − 1)s)   × a2 + 1 m2 − 2m + 1 − 2am(m − 1) cos((2m − 1)s) cos θ(s)   = a2 − 1 m + 1 + a cos((2m − 1)s). The conclusion formulae are then immediate.



Apply the formulae in Lemma 1, we obtain an explicit form for the central force f (r) = fm,a (r) in (7). Theorem 2 Let m, a be real number greater than 1. Then the roulette curves (5) and (6) can be realized as the orbit of a point mass under a central force   4rM2 (2m − 1)(m(m + 1)a2 − m(m − 3) − 2) − m(m − 1)r2 fm,a (r) = −   3 r2 + (2m − 1) a2 − 1 under the initial condition x(0) = 1, y(0) = 0, x (0) = 0, y (0) = M = 0. Proof Note that K(s) > 0 for every s. Equation (7) is rewritten as − fm,a (r) = M2

K(s) . r2 cos3 θ(s)

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321

To assert the conclusion equality, it suffices to prove 4r3

K(s) cos3 θ(s) =

(2m − 1)(m(m + 1)a2 − m(m − 3) − 2) − m(m − 1)r2 . (r2 + (2m − 1)(a2 − 1))3

This can be achieved by substituting the formulate of Lemma 1 into the above equation, and showing that both sides of the equation are exactly the same as   3m2 − 3m + 1 + a2 − 1 m3 − am(m − 1) cos((2m − 1)s) .  3 4 (a2 − 1)m + 1 + a cos((2m − 1)s)  Remark The roulette curves (5) and (6) is symmetric with respect to the line joining every double point and the origin (0, 0). Hence every double point has two analytic branches for which K1 / cos3 θ1 = K2 / cos3 θ2 . If m is an integer, the curves (5) and (6) is viewed as the real affine part of an irreducible complex projective curve F(t, x, y) = 0 of order 2m with genus g = 0, that is, the curve is a rational curve. Every singular point of the curve in the projective plane CP2 lies on the real affine plane R2 . Historical background of the curves (5) and (6) for an integer m is given in [3]. If m is a non-integral rational number, the curves (5) and (6) lies on an algebraic curve. However, singular points in the plane CP2 do not necessarily lie on the real affine plane. Interesting orbits related with central forces can be found in [4]. Classical examples of algebraic curves under some central forces are given in [13, 15].

3 Orbits and Numerical Ranges In matrix analysis, the numerical range W(A) of an n × n matrix A is the convex hull of the real part of the dual curve of the algebraic curve F(1, x, y) = 0 defined by (1). This result is a foundation of the numerical approximation of W(A) given in [11]. On the analogy, we provide an interpretation in Theorem 4 that the rational algebraic roulette curve can be regarded as the curve F(1, x, y) = 0 associated to a matrix A. The graph of the velocity vector (X(t), Y(t)) = (x (t), y (t)) is known as the hodograph. We show that the hodograph can be obtained as the boundary generating curve of W(A) rotated with angle π/2 around the origin. We begin with the construction of a family of matrices T(2m, H). For m = 2, 3, . . . and a parameter H > 1, we define a 2m × 2m matrix T = T(2m, H) = (tij) in the following way: √ (i) tm 2m = H 4m−2 − 1/H m−1 , (ii) ti 2m = 0 for i = m,

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t2m j = 0 for every 1 ≤ j ≤ 2m, tij = 0 for 1 ≤ i ≤ 2m − 2, 1 ≤ j ≤ 2m − 1, j = i + 1, t2m−1 j = 0 for 2 ≤ j ≤ 2m, tm−1+k m+k = tm−k m+1−k for k = 1, 2, . . . , m − 1, tk k+1 = t12 if 1 ≤ k ≤ m − 1 and k is odd, tk k+1 = t23 if 1 ≤ k ≤ m − 1 and k is even, (viii) t2m−1 1 = t23 . (iii) (iv) (v) (vi) (vii)

The entries t12 = 1/H m−1 , t23 = H m if m is even, and t12 = H m , t23 = 1/H m−1 if m is odd. The matrix T(2m, c) for odd m = 3 and even m = 4 look like ⎛ ⎞ 0 0 0 0 H3 0 ⎜ 0 ⎟ 0 1/H 2 0 0 √ 0 ⎜ ⎟ 2 2⎟ ⎜ 0 10 0 0 1/H 0 H − 1/H ⎟ T(6, H) = ⎜ ⎜ 0 ⎟, 0 0 0 H3 0 ⎜ ⎟ ⎝1/H 2 0 ⎠ 0 0 0 0 0 0 0 0 0 0 and

⎞ 0 0 0 0 0 0 1/H 3 0 ⎜ 0 ⎟ 0 H4 0 0 0 0 0 ⎜ ⎟ 3 ⎜ 0 ⎟ 0 0 1/H 0 0 0 √ 0 ⎜ ⎟ 3 3⎟ ⎜ 0 14 0 0 0 1/H 0 0 H − 1/H ⎟ T(8, H) = ⎜ ⎜ 0 ⎟. 0 0 0 0 H4 0 0 ⎟ ⎜ ⎟ ⎜ 0 0 0 0 0 0 1/H 3 0 ⎟ ⎜ ⎝ H4 0 ⎠ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎛

The construction of T is partly treated in [6]. Next, for 0 ≤ θ ≤ 2π and H > 1, we introduce a 2m × 2m Hermitian matrix:   A(θ, H) := H 4m−2 − 1 I2m   +2H m−1 (−1)m−1 cos((m − 1)θ)(T) − sin((m − 1)θ ) (T)   +2H 3m−2 (−1)m−1 cos(mθ)(T) + sin(mθ) (T)      = H 4m−2 − 1 I2m + H m−1 (−1)m−1 exp −(−1)m i(m − 1)θ T +(−1)m−1 exp((−1)m i(m − 1)θ)T ∗ }    +H 3m−2 (−1)m−1 exp (−1)m imθ T    +(−1)m−1 exp −(−1)m imθ T ∗ . We shall show that the matrix A(θ, H) is singular by finding a row vector ξ(θ, H) satisfying ξ(θ, H)A(θ, H) = 0,

Numerical Ranges for Central Force Orbits

323

or equivalently T

A(θ, H)ξ(θ, H) = 0. Theorem 3 Let Ak = Ak (θ, H) be the k-th row of the matrix A(θ, H). If m ≥ 2 is even, the equation 2m−1 

ak (θ)Ak (θ, H)

k=1

−

exp (−i(2m − 1)/2 θ ) + H 2m−1 exp (i(2m − 1)/2 θ ) A2m (θ, H) = 0 √ H 4m−2 − 1

holds for the coef f icient functions a1 (θ), . . . , a2m−1 (θ ) given by (i) am+2k (θ) = −(−1)k exp(i(2k − 1)/2 θ) for k = 0, ±1, ±2, . . . with 2 ≤ m + 2k ≤ 2m − 2. (ii) am−1−2k = (−1)k exp(i(2m − 2k − 3)/2 θ) for k = 0, 1, 2, . . . with 1 ≤ m − 1 − 2k ≤ m − 1. (iii) am+1+2k = (−1)k exp(−i(2m − 2k − 1)/2 θ) for k = 0, 1, 2, . . . with m + 1 ≤ m + 1 + 2k ≤ 2m − 1. If m is odd, the equation 2m−1 

ak (θ)Ak (θ, H)

k=1

−

exp (i(2m − 1)/2 θ ) + H 2m−1 exp (−i(2m − 1)/2 θ ) A2m (θ, H) = 0 √ H 4m−2 − 1

holds for the coef f icient functions a1 (θ), . . . , a2m−1 (θ ) given by (i) am+2k (θ) = (−1)k exp(−i(2k − 1)/2 θ) for k = 0, ±1, ±2, . . . with 1 ≤ m + 2k ≤ 2m − 1. (ii) am−1−2k = (−1)k exp(−i(2m − 2k − 3)/2 θ) for k = 0, 1, 2, . . . with 2 ≤ m − 1 − 2k ≤ m − 1. (iii) am+1+2k = (−1)k exp(i(2m − 2k − 1)/2 θ) for k = 0, 1, 2, . . . with m + 1 ≤ m + 1 + 2k ≤ 2m − 2. Proof Consider a row vector S=

2m 

ak Ak (θ, H)

k=1

for some undetermined coefficients a1 , . . . , a2m . We denote by S( j) the j-th coordinate of the row vector S.

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Firstly, we deal with the 2m-th coordinate of the row vector S using the coefficients am and a2m as in the hypotheses. In the case m is even, √ S(2m) = am

H 4m−2 − 1 (−H m−1 exp(−i(m − 1)θ ) − H 3m−2 exp(imθ)) H m−1

+a2m (H 4m−2 − 1)  = exp(−iθ/2) H 4m−2 − 1(exp(−i(m − 1)θ ) + H 2m−1 exp(imθ)) − =



exp(−i(2m − 1)/2 θ) + H 2m−1 exp(i(2m − 1)/2 θ) 4m−2 (H − 1) √ H 4m−2 − 1 H 4m−2 − 1(exp(−i(2m − 1)/2 θ) + H 2m−1 exp(i(2m − 1)/2 θ)

− exp(−i(2m − 1)/2 θ) − H 2m−1 exp(i(2m − 1)/2 θ)) = 0. In the case m is odd, √ S(2m) = am

H 4m−2 − 1 m−1 (H exp(i(m − 1)θ ) + H 3m−2 exp(−imθ)) H m−1

+a2m (H 4m−2 − 1)  = exp(iθ/2) H 4m−2 − 1(exp(i(m − 1)θ ) + H 2m−1 exp(−imθ)) − =



exp(i(2m − 1)/2 θ) + H 2m−1 exp(−i(2m − 1)/2 θ) 4m−2 (H − 1) √ H 4m−2 − 1 H 4m−2 − 1(exp(i(2m − 1)/2θ) + H 2m−1 exp(−i(2m − 1)/2θ)

− exp(i(2m − 1)/2θ) − H 2m−1 exp(−i(2m − 1)/2θ)) = 0. Secondary, we continue on the m-th coordinate of S using the coefficients of am , a2m and am−1 , am+1 in the hypotheses. In the case m is even,   S(m) = am H 4m−2 − 1 + am−1

1



H m−1

 −H 3m−2 exp(imθ) + am+1

1

− H m−1 exp(−i(m − 1)θ ) 

− H m−1 exp(i(m − 1)θ ) H m−1 √  H 4m−2 − 1 3m−2 exp(−imθ) + a2m −H H m−1   × − H m−1 exp (i(m − 1)θ) − H 3m−2 exp(−imθ)    = − exp(−iθ/2) H 4m−2 − 1 + am−1 − exp(−i(m − 1)θ )

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325

  −H 2m−1 exp(imθ) + am+1 − exp(i(m − 1)θ )  −H 2m−1 exp(−imθ) + exp(−iθ/2)(exp(−i(m − 1)θ )   +H 2m−1 exp(imθ)) exp(i(m − 1)θ) + H 2m−1 exp(−imθ) = 2 exp(−iθ/2) − am−1 exp(−i(m − 1)θ) − am+1 exp(i(m − 1)θ )  +H 2m−1 exp(i(4m − 3)/2θ) + exp(−i(4m − 1)/2θ)  −am−1 exp(imθ) − am+1 exp(−imθ) .

(8)

Substituting am−1 = exp(i(2m − 3)/2 θ) and am+1 = exp(−i(2m − 1)/2 θ) into (8), we have S(m) = 0. In the case m is odd,   S(m) = am H 4m−2 − 1 + am−1



1 H m−1

 +H 3m−2 exp(−imθ) + am+1

H m−1 exp(i(m − 1)θ ) 1

H m−1



H m−1 {exp(−i(m − 1)θ )

√  H 4m−2 − 1  m−1 +H H exp(imθ) + a2m exp(−i(m − 1)θ ) H m−1  +H 3m−2 exp(imθ)    = exp (iθ/2) H 4m−2 − 1 − exp(i(2m − 1)/2 θ)  +H 2m−1 exp(−i(2m − 1)/2 θ) exp(−i(m − 1)θ )  +H 2m−1 exp(imθ) + am−1 (exp(i(m − 1)θ ) 3m−2

+H 2m−1 exp(−imθ)) + am+1 (exp(−i(m − 1)θ ) +H 2m−1 exp(imθ)) = − exp(iθ/2) + am−1 exp(i(m − 1)θ) + am+1 exp(−i(m − 1)θ )  +H 2m−1 − exp(−i(2m − 1)/2 θ) exp(−i(m − 1)θ ) − exp(i(2m − 1)/2 θ) exp(imθ) + am−1 exp(−imθ)  +am+1 exp(imθ) = −2 exp(iθ/2) + am−1 exp(i(m − 1)θ) + am+1 exp(−i(m − 1)θ )  +H 2m−1 − exp(−i(4m − 3)/2 θ) − exp(i(4m − 1)/2 θ)  +am−1 exp(−imθ) + am+1 exp(imθ) .

(9)

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M.-T. Chien, H. Nakazato

Substituting the coefficients am−1 = exp(−i(2m−3)/2 θ) and am+1 = exp(i(2m− 1)/2 θ) into (9), we obtain S(m) = 0. Finally, we treat the coordinates S(m − 1), S(m + 1), S(m − 2), S(m + 2), . . . of the row vector S successively. For instance, in the case m is even, we have   S(m − 1) = am−1 (H 4m−2 − 1) + am − exp(i(m − 1)θ ) − H 2m−1 exp(−imθ)   +am−2 − H 2m−2 exp(−i(m − 1)θ) − H 4m−2 exp(imθ)    = exp(i(2m − 3)/2 θ) −1 + H 4m−2 + exp(−iθ/2) exp(i(m − 1)θ )   +H 2m−1 exp(−imθ) − am−2 H 2m−2 exp(−i(m − 1)θ )  +H 4m−2 exp(imθ) = H 4m−2 (exp(i(2m − 3)θ/2) − am−2 exp(imθ)) − exp(i(2m − 3)θ/2 + exp(i(2m − 3)θ/2)   +H 2m−1 exp(−i(2m + 1)θ/2) − am−2 exp(−i(m − 1)θ )

(10)

Substituting the coefficients am−2 = exp(−3iθ/2) into (10), we have S(m − 1) = 0. The equation ξ(θ, H)A(θ, H) = 0 is then obtained by the recursive process.  We interpret the orbit of the roulette curves (5) and (6) as an algebraic curve F(1, x, y) = 0 associated to the homogeneous polynomial F(t, x, y) of a matrix A, and the hodograph of the orbit as the boundary generating curve of the numerical range of A. Theorem 4 The rational algebraic roulette curve x = cos((m − 1)s) + a cos(ms),

y = − sin((m − 1)s) + a sin(ms)

0 ≤ s ≤ 2π for m = 2, 3, . . ., a > 1, is the curve def ined by the form       F(t, x, y) = det tIn + x A + A∗ /2 + y A − A∗ /(2i) associated to the (2m) × (2m) matrix A = (−1)m−1

H 4m−2 − 1 T(2m, H), 2H m−1

where H = a1/(2m−1) . Moreover, the curve of the hodograph {(x (t), y (t)) : t ∈ R} is obtained as the boundary generating curve of W(A) rotated with an angle −π/2 around the origin. Proof For a non-zero real number λ, the form associated to λA satisfies       FλA (t, x, y) = det tI2m +λ x A+ A∗ /2+λy A− A∗ /(2i) = F A (t, λx, λy).

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327

Hence we have     (x, y) ∈ R2 : FλA (1, x, y) = 0 = (x/λ, y/λ) ∈ R2 : F A (1, x, y) = 0 . By Theorem 3, A(θ, H) is singular, and thus    det H 4m−2 − 1 I2m + 2(−1)m−1 H m−1 cos((m − 1)θ )   + H 3m−2 cos(mθ) (T) + 2 − H m−1 sin((m − 1)θ )   + H 3m−2 sin(mθ) (T) = 0. Therefore, the complex curve x = cos((m − 1)s) + a cos(ms),

y = − sin((m − 1)s) + a sin(ms)

for s ∈ C satisfies the equation F(1, x(s), y(s)) = 0. Conversely, if (x, y) ∈ C2 satisfies F(1, x, y) = 0, we shall show that there exists s ∈ C satisfying x = cos((m − 1)s) + a cos(ms),

y = − sin((m − 1)s) + a sin(ms).

This follows from the irreducibility of the form F[t, x, y] in the polynomial ring C[t, x, y]. For the irreducibility, we see that the curve F(1, x, y) = 0 of degree 2m and the line y = 0 meet at m − 1 ordinary double points and 2 non-singular points. Since these intersection points are connected by a real analytic arc {(cos((m − 1)s) + a cos(ms), − sin((m − 1)s) + a sin(ms)) : 0 ≤ s ≤ 2π } , it follows that these intersection points belong to the same component of the curve F(t, x, y) = 0. Hence the form F(t, x, y) consists of a single irreducible factor. The hodograph is the velocity vector (X(t), Y(t)) := (x (t), y (t)) of the orbit. The equation x(t)(−Y(t)/M) + y(t)(X(t)/M) + 1 = 0 shows that if (x(t), y(t)) lies on an algebraic curve, then the curve (−Y(t)/M, X(t)/M) lies on its dual curve. Hence, for M = 1, the hodograph (X(t), Y(t)) is obtained as the dual curve of the orbit by rotating an angle −π/2 around the origin, and the dual curve is precisely the boundary generating curve of W(A).  Remark We show, in Theorem 4, that the orbit of some roulette curve is algebraic. It is interesting to ask the existence of transcendental closed orbits under some central force. In the following, we construct an analytic transcendental closed curve which is realized as an orbit under a central force.

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Consider the curve x = cos θ log(3 + cos θ), y = sin θ log(3 + cos θ),  0 ≤ θ ≤ 2π . Then the function r = x2 + y2 satisfies the equation

(11)

er = 3 + cos θ.

(12)

Suppose the curve (11) lies on a real algebraic curve g(x, y) = 0. Note that {(x, y) ∈ R2 : g(x, y) = 0} = R2 . By the symmetry of the curve with respect to the x-axis, we may assume that  g(x, y) = cn,m xn y2m , n,m

for some real coefficients cn,m . It follows that   m cn,m rn+2m cosn θ 1 − cos2 θ = 0.

(13)

n,m

Since (13) holds for every θ ∈ C, it is rewritten as  b p,q r p cosq θ = 0.

(14)

p,q

for some real coefficients b p,q . Equation (12) is equivalent to cos θ = er − 3.

(15)

By (14), cos θ is an implicit function of r, and | cos θ| ≤ M|r| p0 for some 0 < M < ∞ and some positive integer p0 . But this contradicts (15). Thus the the curve (11) is a transcendental closed curve. This curve is realized as the orbit under a central force depending only on r.

Fig. 1 Roulette curve 4

2

-4

-2

2

-2

-4

4

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329

Fig. 2 Dual curve (X(s), Y(s))

0.3 0.2 0.1

-0.3 -0.2 -0.1

0.1

0.2

0.3

-0.1 -0.2 -0.3

4 An Example We illustrate Theorem 4 by considering the roulette curve for m = 3 and a = 4: x = cos(2s) + 4 cos(3s), y = − sin(2s) + 4 sin(3s).

(16)

The associated matrix A=

15  1/5  T 6, 4 , 29/5

and the curve (16) coincides with the curve F(1, x, y) = det(I6 + x(A + A∗ )/2 + y(A − A∗ )/(2i)) = 0. The curve (16), displayed in Fig. 1, is a sextic algebraic curve. Its dual curve (X(s), Y(s)) is obtained by a general formula X(s) =

Y(s) =

2 cos(2s) − 12 cos(3s) −y (s) = , x(s)y (s) − y(s)x (s) 46 + 4 cos(5s)

−2 sin(2s) − 12 sin(3s)) x (s) = . x(s)y (s) − y(s)x (s) 46 + 4 cos(5s)

This dual curve, displayed in Fig. 2, is an algebraic curve of degree 10, and the convex hull of the dual curve is the numerical range of A. Acknowledgments The authors would like to express their thanks to an anonymous referee for his (or her) helpful and valuable suggestions on central forces which have led to the improvement of the present version of the paper. The research of the first author was supported in part by Taiwan National Science Council.

References 1. Aiton, E.J.: How Kepler discovered the elliptical orbit. Math. Gaz. 59, 250–260 (1975) 2. Arnold, V.I.: Mathematical methods of classical mechanics. In: Graduate Text in Mathematics, vol. 60. Springer (1978)

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3. Brieskorn, E., Knörrer, H.: Algebraic Curves. Birkhäuser, Basel (1986) 4. Calin, O., Chang, D.-C., Greiner, P.: Geometric Analysis on the Heisenberg Group and Its Generalizations. American Mathematical Society, International Press, Boston (2007) 5. Chandrasekar, S.: Newton’s Principia for the Common Reader. Clarendon Press, Oxford (1995) 6. Chien, M.T., Nakazato, H.: Boundary generating curve of the c-numerical range. Linear Algebra Appl. 294, 67–84 (1999) 7. Goldstein, H., Poole, C., Safko, J.: Classical Mechanics, 3rd edn. Addison-Wesley, New York (2002) 8. Gray, A.: Modern Differential Geometry of Curves and Surfaces. CRC Press, Boca Raton (1993) 9. Kepler, J.: Astronomia Nova: New Astronomy. English Translation by W. Donahue. Cambridge University Press, New York (1992) 10. Kippenhahn, R.: Über den Wertevorrat einer Matrix. Math. Nachr. 6, 193–228 (1951) 11. Li, C.K., Sung, C.H., Tsing, N.K.: c-Convex matrices: characterizations inclusions relations and normality. Linear Multilinear Algebra 25, 275–287 (1989) 12. Linton, C.M.: From Eudoxus to Einstein. A History of Mathematical Astronomy. Cambridge University Press, Cambridge (2004) 13. Moulton, F.R.: An Introduction to Celestial Mechanics, 2nd edn. Dover, New York (1984) 14. Walker, R.J.: Algebraic Curves. Dover, New York (1950) 15. Whittaker, E.T.: A Treatise on the Analytical Dynamics of Particles and Rigid Bodies. Cambridge University Press, Cambridge (1944)

Math Phys Anal Geom (2010) 13:331–355 DOI 10.1007/s11040-010-9083-x

Mechanism of Energy Transfers to Smaller Scales Within the Rotational Internal Wave Field Ranis N. Ibragimov

Received: 27 January 2009 / Accepted: 28 September 2010 / Published online: 8 October 2010 © Springer Science+Business Media B.V. 2010

Abstract We discuss the effect of the earth rotation on the two-triad interaction and the oceanic energy distribution processes that occur between five coupled internal gravity waves. The system we study is a two-triad test wave system consisting of an initial wave of the tidal M2 frequency interacting with four recipient waves forming two resonant triads. It is shown that the general mechanism of an arbitrarily large number of internal wave interactions can be described by a three classes of interactions which we call the sum, middle and difference interaction classes. The four latitude singularities are distinguished for the particular case of five interacting waves and all three classes of resonant interactions are studied separately at those critical values. It is shown that the sum and difference interaction classes represent the latitudeinferior and latitude-predominant classes respectively. The phenomenon of coalescence of the middle and difference interaction classes is observed along latitude 48.25◦ N. It shown that at this value of latitude, the coalescence phenomenon provides the analogy between rotating and reflecting internal waves from slopes. Keywords Critical latitude · Resonant interactions · Internal waves · Energy spectrum Mathematics Subject Classifications (2010) 76B55 · 67B70

R. N. Ibragimov (B) Department of Mathematics, University of Texas at Brownsville & Texas Southmost College, 80 Fort Brown, Brownsville, TX 78520, USA e-mail: [email protected]

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1 Introduction Internal waves are found everywhere in the ocean. They represent a random superposition of many waves with different amplitudes, wave numbers, and frequencies. Typical velocities of internal waves are about 5 cm s−1 and typical vertical displacements range from a few meters to a few tens of kilometers, and the vertical wavelengths from about one meter to about one kilometer. Because of the complexity of the internal wave field, comparatively little is known about the dynamic processes which govern the internal wave field in the ocean. From the practical standpoint, understanding of such processes is important by many reasons. One of them is that internal waves play an important climatic role, permitting information about changes in one region of the ocean to be transmitted to another. They also advect and disperse pollutants, chemical and biological tracers and affect the transmission of sound. Internal waves are ubiquitous in the ocean and they play a dominant role in mixing due to turbulence and energy transfers in the sea water [12]. Because of nonlinear interactions, it is difficult to model the ocean’s energy spectrum theoretically, despite of many attempts (see e.g., [6, 16]). Nevertheless, from a practical standpoint, understanding the mixing processes in the ocean is important for a variety of applied problems. For example, mixing is responsible for advection and dispersion of chemical and biological traces [19] and it plays an important role in climate variability and the global ocean circulation. Understanding dynamic processes of internal wave interactions and their energy exchange constitute the major basis for the construction of the complete dynamics [18]. Today, resonant wave interaction theories is an active research topic (for review, see [15, 23]). Resonant triad interactions have been studied in many physical systems including electronics [10], nonlinear optics [1], and among a variety of waves in plasmas [8]. In the context of waves in fluids, weaker quadratic interactions among sets of four surface gravity waves were discovered first in Phillips [20], after which resonant triads were found to exist among sets of three gravity-capillary waves [17], two surface waves and one internal gravity wave [24], three internal gravity waves [21], and three Rossby waves [13]. Because of its stochastic nature, internal wave field is best described statistically. Comparatively little is known, however, about the dynamic processes which govern the internal wave field in the ocean and determine the energy level and the spectral shape. Consequently, the current estimates of breaking are not precise enough to provide estimates of mixing in the deep ocean. So far the energy distribution is well understood for the case of a single resonant triad only. At present there are still no any detailed enough theoretical studies of this process for two (and, respectively, few) resonant triads. The main difficulty in modeling several resonant triads is due to the presence of waves of different modes and frequencies in a wave field. To the best of our knowledge, the only exception is a study of the stochasticities of two-triad interactions that occur in two-degree-of-freedom autonomous Hamiltonian systems that ignore the effects of the earth’s rotation reported in Kim and West [11].

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In this article the occurrence of latitude-dependent two resonant triads is reported for the first time. The crucial difference between the present analysis and previous studies is that the model presented here is based on the dynamics of internal waves, i.e., the results do not depend on “black box modeling”. The main goal is to provide tractable analysis of interactions between several resonant triads with the further goal of understanding the redistribution of energy within the oceanic energy spectrum at different latitudes. The present analysis is based on the latitude-dependent classification of resonant interactions between discrete mode internal waves. Some extension to the case of large number of resonantly interaction waves is presented in Section 6 of this article.

2 Experimental Model The simulations were done using non-hydrostatic analytical model which describes two-dimensional flow vertically confined to lie between two horizontal, rigid boundaries at z = 0 (rigid lid approximation for the free surface) and z = −H at the bottom, where H is the range independent depth of the water column. Explicit viscosity and diffusion terms are ignored and the Boussinesq approximation is employed. Without loss of generality, it is assumed hereafter that the horizontal x coordinate is increasing eastward and the transverse horizontal coordinate y northward. Additionally, the stratification considered here is highly idealized, having constant buoyancy frequency N defined by 1  −g dρ 2 N= , (1) ρ0 dz where g is the gravitational acceleration, ρ0 the mean density of the medium and ρ is the background density stratification. While N is commonly used in laboratory and theoretical experiments, it is not common in the ocean. In two dimensions, the incompressibility condition is taken into account by introducing a stream function ψ which is related to the two velocity components u and w by u = ψz , w = −ψx . The model is given by the system of three nonlinear equations for the stream function ψ, density ρ = ρ0 + ρ + ρ  and the velocity component v in the y direction as follows: ∇ 2 ψt − gρx − f vz = ψx ∇ 2 ψz − ψz ∇ 2 ψx , vt + f ψz = ψx v − ψz vx , ρt +

N2 ψ x = ψ x ρz − ψz ρ x , g

(2) (3) (4)

where ρ  is the change from the state of rest, f = 2 sin θ is the inertial frequency, which depends on the rotation rate of the earth (angular velocity ) and the latitude θ. The boundary conditions for system (2)–(4), ψ (0) = ψ (H) = 0 are appropriate for a flat bottom at z = 0 and rigid lid at z = −H. To model weakly nonlinear interactions, the latter governing equations are

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nondimensionalized by introducing the characteristic length scale l∗ , typical velocity scale u∗ , and the time scale ω∗−1 . Then the nonlinear terms will be small if the ratio u∗ δ= (5) ω∗ /l∗ of typical particle speed to the typical phase speed of the waves is small. Thus, since internal gravity waves have frequencies limited to the range between the minimum of inertial frequency f and the maximum buoyancy frequency N, the Fourier components of the superposition of solution for 0 (δ) linear problem can were found in the separable form ψ (x, z, t) =

N  f

An (ω, n) ω,n (z) ei(kω,n x−ωt+ϕ ) dω,

(6)

n

where An is the amplitude of the n-th mode, kω,n is the corresponding horizontal wave number and each frequency has its own set of discrete modes ω,n providing the eigenfunctions of the system, where n indicates the order of the mode. The modes are oscillating only in the region of the water column where the internal wave frequency satisfies f  ω  N. Outside of this region the solutions can’t satisfy both boundary conditions and they are evanescent. The nonlinear terms give rise to the sinusoidal waves with wave numbers and frequencies equal to the sum (or difference) of the wave numbers and frequencies of the primary waves and thus producing forcing terms of the form cos(φi + φ j) and sin(φi + φ j) in the 0(δ 2 ) terms, where φl = kl x − ωl t + ϕl is the phase of the lth wave in which ϕl is the phase constant.

3 Classification of Wave Interactions We consider the superposition of five phase-locked interacting waves φ1 + φ2 + φ0 = φ3 + φ4 + φ0 = 0

(7)

− → and which form two resonant triads whose wavenumber vectors k j = (kj, m j) and the frequencies ω j satisfy the resonant conditions − → − → − → − → − → − → k1+ k2+ k0= k3+ k4+ k0=0 (8) and ω1 + ω2 + ω0 = ω3 + ω4 + ω0 = 0

(9) − → where the wave frequencies ω1 ± ω2 , ω3 ± ω4 and wavenumber vectors k 1 ± − → − → − → k 2 , k 3 ± k 4 satisfy the dispersion relation  2  2 2  N 2 ki ± kj + f 2 mi ± m j , (10) ωi ± ω j =  2  2 ki ± kj + mi ± m j

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− → in which ωμ is the frequency of the μth wave with wavenumber vector k μ . Frequencies ωi and mode numbers ni of the waves composing the resonant triads satisfy the requirement (ωi , ni ) ∈ (ω,n) , where  (ω,n) = (ωi , n, ); f (θ)  ωi  N, |ni | ∈ [nmin , nmax ]

(11)

and i = 0, 1, 2, 3, 4. If ω0 < 0 and (or mi < 0) for the primary wave (k0 , m0 , ω0 ) then we take an absolute value of ω0 and (or m0 ). It follows from (6) that we can take m0 > 0, ω0 > 0 when computing triads. Since the inertial frequency is latitude-dependent, it is assumed hereafter that f = f (θ ). The simulations were done using the uniform stratification with constant buoyancy frequency N = 10−3 [s−1 ], variable latitude θ ∈ [1◦ , 74.2◦ ] and the deep-water depth of H = 5 × 103 [m]. We classify the resonant interactions as follows: For the sum interaction class, the resonance condition (8) and (9) is written in the form ω1,3 + ω2,4 = ω0 ,

− → − → − → k 1,3 + k 2,4 = k 0 ,

(12)

 − →  where the high frequency of the primary wave θ0 , k 0 and the vertical wave number m1 of the recipient wave 1 are treated as known such that |ω0 | = max |ωi |. Without loss of generality, it can be assumed that the vertical

i=0,1,2,3,4

mode of the primary wave is positive, i.e., n0 > 0 implying that either n1 or n2 is positive. In present simulations n1 is chosen to be positive. Additionally, ω0 and ω1 are set to be be a mode n0 and n1 waves respectively, i.e., m0 = n0 π/H and ω1 be a mode n1 wave, i.e., m1 = n1 π/H, where ni ∈ N. The horizontal wave number of the primary wave k0 is determined from the dispersion relation. Then, the horizontal wave numbers k1,3 (and corresponding frequencies ω1 and ω3 ) are found by solving (12) simultaneously for given k0 , m0 , ω0 and m1 . − → This procedure allows to determine as many as two wave vectors k 2,4 from the second resonant condition (12). Finally, ω2 (k2 , m2 ) and ω4 (k4 , m4 ) are found from the the dispersion relation. For the dif ference and middle triad interaction classes, one of the low fre − → quency waves is assumed to be known (we choose θ1 , k 1 to be known) and the vertical mode number m0 of the recipient wave 0 is treated as known. This allows to rearrange e.g., the first resonance condition (12) for one of the − → − → → − triads, i.e., k 0 , k 1 , k 2 to the form ω0 − ω1 = ω2 , which, after switching the indices on waves 0 and 1 becomes ω0 + ω2 = ω1 . And the same procedure can be repeated for the second triad. After the indices are switched, for the both triads, two situations differ in details; either ωi > ω0 > ω j or ω0 < ωi < ω j, where ωi and ω j are the frequencies of the recipient waves forming the parts of the corresponding resonant triads which ω0 belongs to. In the former case, we classify the resonant interaction as the middle interaction class and in the latter

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case, we call it the dif ference interaction class. Thus, after the index switch, for the both middle and difference interaction classes the vertical mode number and the wave vector of the wave 0 is considered to be known and the mode number of wave 1 or 3 has been chosen as in the sum reactions. However, we note that while we could assume n1 > 0 for the sum interactions, this is not true for both difference and middle interaction classes. Therefore positive and negative values of n1 are considered. Additionally, in contrast with the sum interaction class, there may be as many as four solutions for the horizontal component of the wave vector. However, the present analysis is limited to the case of two solutions only denoted here by k1 and k3 (some generalization to the case when all four solutions are included is presented in Conclusions of this article). The rest of the procedure of determining the middle and difference resonant triads is the same as for the sum triads. Thus all coupled discrete mode resonant triads for each of the above three classes are generated starting from the primary single wave of the frequency ω0 . In present simulations we choose n0 = 1 and the frequency of the primary wave to be the M2 tidal frequency, i.e., ω0 = ω M2 = 1.4075 × 10−4 [s−1 ] which allows to associate the current model with the internal wave field generated by a barotropic tidal current oscillating with the M2 tidal frequency. This choice for the primary wave is motivated by the following reason: abstract internal tides are internal waves generated in stratified waters by the interaction of barotropic tidal currents with variable bottom topography. They play an important role in dissipating tidal energy and lead to mixing in the deep ocean. In particular, it was estimated by Polzin et al. [22] and St. Laurent and Nash [23] that roughly half of internal wave energy in the ocean is produced by the movement of the barotropic tide around topography, which generates internal waves with the same frequency. The simulations presented here were done with nmin = 1 and nmax = 2. An example of the energy distribution for 25,000 resonant triads with nmax = 50 is presented in Section 6.

4 Latitude Singularities Two coupled discrete mode resonant triads − → − →  − → − → − → → − k 0, k 1, k 2 , k 0, k 3, k 4

(13)

are generated in (ω,n) space starting from the primary single wave of the frequency ω0 = ω M2 . The structure of the dispersion relation (10) defines whether or not the resonance conditions can be met, i.e., both resonant triad in (13) are obtained by solving the resonance conditions (12) and, according to the the wave interaction classification in Section 3, the obtained triads are classified as the sum, middle or difference interaction class. The existence of each of the above classes depends on the earth’s latitude.

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4.1 Existence of the Sum Interaction Class Figure 1 represents the distribution of five resonant waves in (ω,n) with frequencies (ω M2 , ω1 , ..., ω5 ) forming two triads (13) which belong to the sum interaction class, i.e., |ω0 | = max |ωi |. Numerical results of computations i=0,1,2,3,4

show that for this class of wave interactions, (12) do not have solutions for θ  θ1∗ ≈ 23.5◦ . In particular, as it is shown in Fig. 1, the frequencies ∗ ωi (i = 1, 2, 3, 4) approach some critical value ωsum as θ increases monotoni∗ ∗− ∗+ as θ → θ1∗ . cally to the critical value θ1 such that ω1,4 → ωsum and ω2,3 → ωsum The non-existence of this class of resonant interactions for θ  θ1∗ can be explained as follows: the equation ω1(3) + ω2(4) = ω M2 yields the following restrictive condition: 2 f < ω M2 < 2N.

(14)

From the dispersion relation (10) and the resonant conditions (12) for the sum interaction class, the inequality (14) can be written in the equivalent form

N2α2 + f 2 2f < < 2N, (15) α2 + 1

Sum interactions 2 o

θ = 10 N o

θ = 20 N

1.9 wave 3

wave 1 1.8

Mode numbers

1.7

1.6

1.5

1.4

1.3

1.2 wave 2

wave 4 1.1

wave 0 1

0

0.2

0.4

0.6

0.8

Frequency [s- 1] Equator

1

1.2

1.4 -4

x 10

North Pole

Fig. 1 Resonant wave distribution in frequency–mode number space (ω,n) for sum interaction class. All five waves are generated from a single primary wave of frequency ω M2 and vertical mode number n0 = 1

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where α = k M2 /m M2 is the aspect ratio. It follows from (15) that there are no solutions for the sum interaction class if the following inequality holds: F ( f, ω M2 , k M2 , m M2 ) :=

f2 α2 N2

−

1 >0 3 + 4α 2

(16)

or equivalently, the restrictive condition (14) can also be expressed in terms of four recipient waves of frequencies ω1 , ω2 or ω3 , ω4 as ω1(3) , +ω2(4) − 2 f < 0.

(17)

Numerical computations of solutions for resonance conditions (12) satisfying the dispersion relation (10) shown in Fig. 2 indicate that both inequalities (16) and (17) are valid if θ < θ1∗ ≈ 23.5◦ . In particular, condition (16) is illustrated in panel (a) and condition (17) in panel (b). For the reason that the resonant triads belonging to the sum interaction class exist for very limited latitude range, we call the sum interaction class a latitude-inferior class. Preliminary analysis shows that both resonant triads of the sum interaction class exist for higher ∗ ≈ ω M2 /2 which corresponds mode number waves when θ > θ1∗ and that ωsum to the parametric subharmonic resonant triad interaction mechanism (PSI) originally reported by Craik [2] in the context of the boundary layers. The PSI mechanism is known as the class of resonant wave interactions wherein energy is transferred from a primary wave to two recipient waves of half the primary frequency and smaller vertical scale, and the range of interactions where the secondary waves are near this half-frequency. Although not clarified yet, we suggest that the limiting case for the higher mode resonant triads of

Existence of sum triads −0.1 −0.15

F(f, ωM2, k0, m0) < 0

−0.2

(a)

− 0.25 −0.3 −0.35

4

6

8

10

12

14

16

18

20

21

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24

−4

10

(b) ω1,2 + ω3,4 - 2 f > 0 −5

10

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16

18

20

21

22

23

24

Latitude [deg]

Fig. 2 Conditions for existence of two resonant riads belonging to the sum interaction class

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sum interaction class at increasing values of latitude can be utilized to model the PSI mechanism. Some discussion and related questions are also presented in Section 6. 4.2 Coalescence of the Middle and Difference Interaction Classes The resonant wave distribution of two resonant triads (13) in (ω,n) which belong to the both middle and difference interaction classes is shown in Fig. 3. Initially, at lower latitude values, e.g., at θ = 10◦ , two resonant triads are − − → − → → separated into two classes: middle interaction class k M2 , k 1 , k 2 for which ω2 < ω M2 < ω1 and the difference interaction class

(18)

  − − → − → → k M2 , k 3 , k 4 for which ω M2  = min

i=0.3,4

|ωi | (see Fig. 3). Numerical calculations of frequencies versus a latitude presented in Fig. 4 show that when the latitude approaches the value θ = θ2∗ ≈ 48.25◦ , the middle triad class is lost since the condition (18) is not valid for θ  θ2∗ (see Figs. 3 and 4). In particular, as Fig. 4 shows, ω2 → ω M2 as θ → θ2∗ and ω2 > ω M2 for

Difference interactions 2 o

θ = 10 N o

θ = 20 N

1.9

o

θ = 30 N o

θ = 45 N θ = 60oN

1.8

o

wave 2

θ = 70 N

wave 4

o

θ = 74 N

Mode numbers

1.7

1.6

1.5

1.4

1.3 wave 1

wave 3

1.2 wave 0

1.1

1

0

0.1 Equator

0.2 North Pole

0.3

0.4

0.5

Frequency [s−1]

0.6

0.7

0.8

0.9

1 −3

x 10

Buoyancy

Fig. 3 Resonant wave distribution in frequency–mode number space (ω,n) for the middle and difference interaction classes. All five waves are generated from a single primary wave of frequency ω M2 and vertical mode number n0 = 1

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1

x 10

ω

ω

0.9

0

3

ω

1

ω

2

0.8

ω

ω4

3

ω

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0.7

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0.6

0.5

0.4

0.3

ω1

0.2

ω0

0.1

ω2 0

01

5

10

15

20

30

35

40

4548.25

55 5860

65

70 74.3

Latitude [deg] Fig. 4 This plot shows the frequencies of five resonantly interaction waves for the difference interaction class as functions of the latitude. Note the jump for waves 3 and 4 at the value of latitude θ = 58.4◦ N

θ > θ2∗ . The latter fact suggests that two resonant triads belong to two different classes: middle and difference interaction classes if θ < θ2∗ and the both triads belong to the difference interaction class only if θ > θ2∗ . 4.2.1 Analogy with the Critical Slope The question of particular interest is how the middle and difference interaction classes coalesce into one class in the vicinity of the critical latitude value θ = θ2∗ . Additionally we are interested in finding a convenient parameterization which would manifest the distinction between these two classes classes for two resonant triads sufficiently close to the limiting case θ → θ2∗ . We find that the middle and difference interaction classes are easily distinguishable in terms of two parameters ε1 and ε2 defined via ε1 =

α(k2 − k M2 ) −α(m2 + m M2 ) , ε2 = . k M2 + k2 m M2 − m2

(19)

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The parameters ε1 and ε2 are defined from the following reason: the reso− → − → − →  nance condition (12) for the middle class interaction k M2 , k 1 , k 2 can be written as ⎛ ⎞ 12 ⎛ ⎞ 12 2 2 1  2 2 2 k1 2 2 k2 2 2 2 N N + f + f 2 2 N α + f m1 m2 ⎠ −⎝ ⎠ . =⎝ (20) k21 k22 α2 + 1 + 1 + 1 2 2 m1

m2

θ2∗ ,

Since ω2 → ω M2 as θ → it is possible to associate the primary wave of the phase φ M2 with the wave propagating eastward and impinging on a slopping boundary with the slope ε p to the horizontal. Whence, the corresponding reflected wave can be associated with the wave of the phase φ2 . If the group velocity of the primary wave is directed downward then (since the frequencies of the incident and reflected waves and the same, i.e., ω M2 = ω2 )), in the limiting case when θ = θ2∗ , relation (20) yields  1  1 1 + ε2p γ 2 α −2 2 1 + γ 2 α −2 2 =2 , (21) 1 + ε2p α −2 1 + α −2 where γ (θ) = f/N is a given value at each latitude. The condition (21) is valid only if the vertical and the horizontal wave numbers of the incident and reflected waves are related to each other via (see also [24]) k2 = λε p k M2 , m2 = −λε p m M2 ,

(22)

where we denote λε p =

1 + ε p α −1 . 1 − ε p α −1

(23)

The resonance condition (21) determines uniquely the value of the latitudedependent critical slope ε p (i.e., the slope at which the incident and reflected waves of the corresponding phases φ M2 and φ2 form the resonant triad of the middle class interaction) via the equation      ε p γ 2 − 4 α −4 − 3γ 2 α −6 + 1 − 4γ 2 α −2 − 3 = 0 (24) which follows directly from (21). Remark 1 There are two interpretations for (24). Namely, it can be regarded as the polynomial equation for α for the given value of the slope ε p or it can be regarded as the equation for ε p for the given aspect ratio k M2 /m M2 . In the present analysis, since the latter approach has been employed, we use the index p for the critical slope ε p indicating that the value of the slope is determined uniquely from the polynomial equation (24). Numerical results of computations of εi (i = 1, 2) and solution ε p of (24) at different values of the latitude are shown in Fig. 5. Numerical values of ε1 , ε2 and ε p are equal to the same value 3.00593 × 10−3 [rad] (which is approximately is 1.72 [deg]) at θ = θ2∗ = 48.25◦ , i.e., at this critical value of the

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0.08

ε

0.06

Critical slope ε p [rad] 2

0.04

0.02

0 o

θ = 48.25 −0.02 −0.04 −0.06 −0.08 −0.1 −0.12

ε1 0

5

10

15

20

25

30

35

40

45

50

55

60

Latitude [deg]

Fig. 5 Convergence of two parameters to a critical slope value

latitude, formulae (19) and (22) give the same numerical values for k2 and m2 associated with the vertical and horizontal wave numbers of the wave reflected off the inclined plane with the slope ε p such that the both incident and reflected − − → − →  → waves are the part of the same resonant triad k M2 , k 1 , k 2 belonging to the middle interaction class. Thus the distinction and coalescence mechanism for the middle and difference interaction classes in the vicinity of the critical latitude θ2∗ can be parameterized in terms of εi (i = 1, 2) and ε p as shown in the table below: εi < ε p εi > ε p εi = ε p

Wave interaction classification Both middle and difference interaction classes exist. Only difference interaction class exists. Analogy with the critical slope for the middle interaction class.

4.3 Latitude-predominant Interaction Class The results of Sections 4.1 and 4.2 suggest that for the values of the latitude θ > θ2∗ = 48.25◦ , there is only one class of resonant interactions remaining. Namely, the both resonant triads − − → − →  − − → − → → → k M2 , k 1 , k 2 , k M2 , k 3 , k 4 (25)   belong to the difference interaction class for which ω M2  = min |ωi | (see i=0.1,2,3,4

Figs. 3, 4 and 5) which is manifested as the latitude-predominant class of

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resonant interactions. There are three remarkable features in the behavior of these two resonant triads. In order to illustrate them, in addition to Fig. 4, we plot more detailed behavior of the frequencies versus θ for values of θ > θ2∗ in Fig. 6. The first feature, as Figs. 4 and 6 show, is that when θ < θ3∗− ≈ 58.415◦ , the frequencies ω3 and ω4 are slowly decreasing functions of the latitude. For example, the table below shows the numerical values of ω3 and ω4 at some points of this interval: 1◦ 9.4279 × 10−4 8.0204 × 10−4

ω3 ω4

30◦ 9.4145 × 10−4 8.0068 × 10−4

58◦ 9.3765 × 10−4 7.9691 × 10−4

The second feature is that both ω3 and ω4 have a jump discontinuity when θ3∗− < θ <θ3∗+ ≈ 58.42◦ , such that [ω3 ] ≈ −6 × 10−4 and [ω4 ] ≈ −4.5 × 10−4 , where ω j = ω−j − ω+j in which ω−j = lim∗− ω j and ω+j = lim∗+ ω j (see Fig. 6). θ→θ3

θ →θ3

Finally, Fig. 6 (see also Fig. 4) displays the third feature manifested by the convergence ω1 → ω3+ and ω2 → ω4+ as θ → θ4∗ ≈ 74.5◦ which is known as the critical latitude [3]. The third feature illustrated in Fig. 6 is that near the critical latitude θ ≈ θ4∗ , lim∗ ω1 = ω3+ and lim∗ ω2 = ω4+ . Another interpretation of this feature can also θ→θ4

θ→θ4

−4

10

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3

8

ω

4

6

4

2 58.4

58.405

58.41

58.415

58.42

58.425

−4

x 10 4

ω

−1

Frequency [s ]

3

ω1 3

ω4 ω

2

1 60

2

ω

0

65

70

75

Latitude [deg]

− − → − → − →  − → − →  → Fig. 6 The behaviour of two resonant triads k M2 , k 1 , k 2 and k M2 , k 3 , k 4 belonging to ∗ the difference interaction class in the range θ2 < θ < critical latitude

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− → be provided in terms of the resultant wavenumber vectors k i (i = 0, ..., 4) which are plotted numerically in Fig. 7. Because of the coalescence of the middle and difference interaction classes, the both triads (25) are called hereafter the first and the second dif ference triads. Panel (a) shows the wavenumber vectors of the first difference triad while the wavenumbers of the second difference triad are shown in panel (b). As Fig. 7 shows, at the critical latitude, the limiting case k0 << m0 occurs for the both resonant triads corresponding to the horizontal particle motion. Since the particle velocities associated with the primary wave of the phase φ M2 are all in lines perpendicular − → to k M2 , this situation is analogous to the formation of the Taylor columns in rotating stratified fluids [26]. The latter phenomenon has also been reported

−3

1

x 10

−3

First didderence triad 1

Second didderence triad

x 10

(a) 74o

o

74o

1

(b) o

o

o

1

1

74o

K

1

74o

1

0.5

0.5 K

3

K0

Vertical wave number

K0 0

0

−0.5

−0.5

K4

K2 −1

−1

1o −1.5

−2

45o 58o 74o

−1 0 1 Horizontal wave number

74o −1.5 −2

2 -4

x 10

−1 0 1 Horizontal wave number

1o 2 -3

x 10

− − → − → − →  − → − →  → Fig. 7 Wave vectors of two triads k M2 , k 1 , k 2 and k M2 , k 3 , k 4 belonging to the middle and difference interaction classes. Because of the coalescence of the two classes, the triad of waves 0, 1 and 2 is called the first difference traid and the triad of waves 0, 3 and 4 is called the second difference triad. Panel a shows the wavenumber vectors of the first difference traid while − → the wavenumbers of the second difference traid are shown in panel b. Wavenumber vectors k 3 and − → k 4 almost don’t change until θ < θ3∗− ≈ 58.415◦ and there is a very sharp changes between them for θ3∗− < θ < θ3∗+ ≈ 58.42◦ . The arrows at the top and the bottom of each plot are used indicate schematically the latitude evolution of wavenumber vectors

Energy Transfers Within the Rotational Internal Wave Field

345

in the context of the latitude-dependent time evolution of the Great Red Spot on Jupiter (GRS) under very restrictive conditions that the GRS is strictly confined in the domain where the zonal wind varies with latitude only (see [25]).

5 Preliminary Analysis of the Latitude-Dependent Energy Distribution To describe the physics of the background internal wave spectrum in the ocean, the dynamics of a random superposition of many waves with different amplitudes, wave numbers and frequencies spanning the range of possible frequencies should be modeled. However, such physical models had hitherto been developed because of the complexity of realistic internal wave field in the ocean. The goal of this section is to use the above narrow modeling scenario of two resonant triads (25) to model the dynamics of the latitude-dependent oceanic energy spectrum. To this end, on the slow interaction time scale τ = δt, the dynamics of interaction for five weakly internal waves forming two triads (25) is simulated by the following set of the coupled fluctuations of the wave amplitudes that governs the time evolution of resonant triads: dA0 dτ dA1 dτ dA2 dτ dA3 dτ dA4 dτ

= γ120 A1 A2 + γ340 A34 ,

(26a)

= γ021 A0 A2 ,

(26b)

= γ012 A0 A1

(26c)

= γ043 A0 A4

(26d)

= γ034 A0 A3 ,

(26e)

where the nonlinear coupling coefficients γijl are given by   bij + bji kl γijl = |kl |2 ωl2

(27)

in which the constants bij are defined via     2 N 2 ωi    ωi m j  1 ωi ω j  bij = ωi + ω j k j + ki + kj + f 2 mi + m j 2 ki kj g kj ki kj   × mi kj − m jki . (28) The wave amplitudes Ai (τ ) (i = 0, ..., 4) are initialized based on a choice of the initial energy spectrum EGM in (ω,n) domain, i.e., Ai (0) = Ai |τ =0 , where Ai (0) are determined in terms of the initial energy of ith wave.

346

R.N. Ibragimov

5.1 Initialization of the Model In further simulations, the initial wave amplitudes are chosen to give an energy distribution in (ω,n) domain which matches the Garret and Munk (GM) energy spectrum, i.e.,  Ai (0) =

2ki2 EGM N 2 ki2 + f 2 (θ) mi2

 12

.

(29)

The choice of the GM model spectrum is motivated for the following reason: away from direct forcing, intermediate-scale internal waves are empirically described by the universal GM spectrum [4, 5], which continues to be a useful description of the oceanic internal wave field, particularly, of the deep, open ocean. The GM spectrum is often used as a representative statistical description of the internal wave field in studies of nonlinear wave interactions (e.g., [7, 18, 24]) and mixing parameterization (e.g., [6, 22]). Observations of the internal wave spectrum in the deep ocean indicate the remarkable fact that it has much the same shape wherever it is observed, unless the observations are made close to a strong source of internal waves. A fundamental role in spanning the universal deep ocean spectrum could be attributed to weakly nonlinear-wave interactions within the internal wave field which smooth out any spectral irregularity by redistributing energy within the spectrum

80

80

A0|τ=0

74 70

2 τ=0

74 70

60

A |

A0|τ=0

A |

3 τ=0

A4|τ=0

60 A |

1 τ=0

50 Latitude

Latitude

50

40

40

30

30

20

20

10

10

0

0

2

4 6 Initial amplitudes

8

0

0

1

2 3 Initial amplitudes

4

Fig. 8 Initial wave amplitudes matching the Garret and Munk (GM) oceanic enery spectum versus the latitude

Energy Transfers Within the Rotational Internal Wave Field

347

−4

10

θ = 20o 0

θ = 48.25 0

θ = 50

−5

θ = 74.20

Energy

10

−6

10

−7

10

−8

10

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

−1

1 −3

Frequency [s ]

x 10

Fig. 9 GM energy distribution at different latitudes

θ = 30o

Two resonant triads 4

(a)

Amplitude A0

2

0 −2 −4

0

50

100

150

200

250

300

350

400 o

θ = 74.2 0.81

(b) Amplitude A0

0.8 0.79 0.78 0.77 0.76 0.75

0

500

1000

1500 Inertial periods

2000

2500

3000

Fig. 10 Time evolution of A0 for two difference triads at 30◦ and 74.2◦ . The amplitude A0 is initialized to match the GM energy spectrum

348

R.N. Ibragimov

[15, 18]. The GM spectrum provides an important link in the overall energy cascade from the large generation scale to the small scale dissipation scale. The energy available for the mixing processes is originally supplied at large scales (a few tens of kilometers) and then transferred across the internal wave spectrum down to small dissipation scales (a few meters) by nonlinear resonant wave interactions among internal waves. Numerical results of computations of the initial wave amplitudes Ai (0) matching the GM spectrum are shown in Fig. 8. In particular, Fig. 8, panel (b) reveals a sharp change of A3 (0) and A4 (0)  when θ ∈ θ3∗− , θ3∗+ . Additionally, the initial value of the amplitude of the primary wave A0 (0) is increasing function of θ for θ < θ3∗− and A0 (0) starts to decrease for θ > θ3∗+ which is also shown in the table below A0 (0) θ [deg]

1.97 10

2.61 20

2.93 30

3.02 45

2.83 60

0.90 74

The initial energy of each wave is given by ei (0) =

(a)

  m2 1 2 Ai (0) N 2 + f 2 (θ) 2i , 4 ki

(b)

Amplitudes A and A time evolution 1

3

5

3

(30)

Amplitudes A2 and A4 time evolution

2 θ = 58.4 o

1 0

0

A3

A

3

−1

A

1

−5

A1 0

100

200

300

400

500

−2 −3

0

(c)

(d)

5

3

100

200

300

400

500

A3

2

θ = 60 o

1 A3 0

0 −1

A1

−2 −5

A1 0

100

200 300 Inertial periods

400

500

−3

0

100

200 300 Inertial periods

400

500

Fig. 11 Time evolution of aplitudes A1 , A2 , A3 , A4 for two fifference traids in the vicinity of the latritude singularity θ ∈ (θ3∗− , θ3∗+ ), where ω3 and ω4 have a jump discontinuity and close to the critical latitude θ ≈ θ4∗ ≈ 74.5◦ , where ω1 → ω3 and ω2 ω4 . The amplitudes are initialized to match the GM energy spectrum

Energy Transfers Within the Rotational Internal Wave Field

349

where, at each corresponding frequency band, ei (0) matches the numerical values of EGM shown in Fig. 9 for different values of the latitude, i.e., Ai (0) in (30) are determined numerically from (29). 5.2 Dynamics of the Model The numerical solutions of initial value problem (26a–26e) and (29) for wave amplitudes are shown in Figs. 10 and 11 at different latitudes. In particular, Fig. 10 is used to compare the dynamics of the primary wave amplitude A0 (τ ) at one of the lower values of latitude (θ = 30◦ ) for 400 inertial periods and at the value close to the critical latitude, θ = θ4∗ ≈ 74.5◦ for 3,000 inertial periods. The effects of the jump discontinuity for ω3 and ω4 when θ3∗− < θ < θ3∗+ and the convergence ω1 → ω3+ and ω2 → ω4+ as θ → θ4∗ are illustrated in Fig. 11. The numerical simulation giving the quantitative prediction of the resulting time evolution of the energy spectrum is shown in Fig. 12. In particular, the panel (b) in Fig. 12 is used to illustrate the effect of coalescence of middle and difference class interactions at θ = θ2∗ . Additionally, panel (d) is used to illustrate that there no resonant waves in the higher frequency bands when approaching the critical latitude. This also can be seen from Fig. 3 showing

θ = 20 o

−4

10

θ = 48.25o

−5

10

τ=0 τ = 100

−5

10

τ=0 τ = 100 −6

Energy

10

(b)

(a)

−6

10

−7

10

−7

10

−8

10

−8

0

0.2

0.4

0.6

0.8

10

1

0

0.2

0.4

0.6

0.8

−3

x 10

θ = 50o

−4

10

o

θ = 74.2

−4

10 τ=0 τ = 100

−6

10

1 −3

x 10

τ=0 τ = 100 −5

Energy

10

(c)

−8

10

(d) −6

10

−10

10

−12

10

−7

0

0.2

0.4

0.6 −1

Frequency [s ]

0.8

1 −3

x 10

10

0

0.2

0.4

0.6

0.8

−1

Frequency [s ]

Fig. 12 Dynamics of the energy distribution two difference triads at different latitudes

1 −3

x 10

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R.N. Ibragimov

the latitude-dependent resonant wave distribution in (ω,n) domain and Fig. 9 in which the initial energy distribution for the same two difference triads is shown. Moreover, comparing the latitude-dependent energy distribution shown in panels (a)–(d) of Fig. 12, we can conclude that the spectrum is “more stable” at low latitudes in the sense that a smaller time variation in the spectrum dynamics occurs at lower latitudes.

6 Conclusions, Questions and Some Generalization This article is devoted to discussion and numerical investigation of the latitude singularities for resonant wave interactions described schematically in Fig. 13 below. For the purpose of illustration and qualitative description, five discrete mode resonant waves decomposed into two latitude dependent triads were investigated. The resonant triads were classified as a sum, middle and difference interaction classes. Latitude Singularity at θ = θ1∗ ≈ 23◦ For the given choice of vertical mode numbers, we have shown that the resonant triads belonging to the sum interaction class do not exist for θ  θ1∗ ≈ 23.5◦ (see (16), (17) and Fig. 2). The latter fact suggests that this class of resonant interaction is a latitudeinferior class. It is one of our goals in further studies is to use the results of this study to model the PSI mechanism as a limiting case for the higher mode (n0 > 1 and nmax > 2) resonant triads belonging to the sum interaction class at the values of latitude θ > θ1∗ . The question of particular interest to understand how the PSI mechanism rules out the energy transfers within two higher modes triads. It has been suggested in recent studies by MacKinnon and Winters [14] that the PSI could be a significant mechanism of energy transfer to smaller scales. However, it has not been clarified yet and is still remains and open question how exactly the energy transfer is af fected by latitude.

Fig. 13 Discussion of “critical points”

θ = 90 o (North Pole)

θ = 74.5 o (Critical latitude) o

θ = 58.4 (Jump) o θ = 48.25 (Middle triad class lost)

o

θ = 23 (Sum triad class lost)

o

θ = 0 (Equator)

Energy Transfers Within the Rotational Internal Wave Field

351

Latitude Singularity at θ = θ2∗ ≈ 48.25◦ When the latitude approaches the value θ > θ2∗ ≈ 48.25◦ , the middle interaction class is “lost”, i.e., two remaining resonant triads belonging to the middle and difference reaction classes coalesce into one difference interaction class to the difference interaction class. The later class interactions exists for all possible range of latitudes,   of resonant i.e., for θ ∈ 0◦ , θ4∗ and, by this reasoning, it is called a latitude-predominant interaction class. Furthermore, at the critical value θ = θ2∗ , the triad belonging to the middle interaction class can be associated with the triad composed of the incident and reflected waves off the sloping boundary with the latitudedependent slope ε p which is determined from (26).   Latitude Singularity at θ = θ3∗± ≈ 58.4◦ When θ ∈ θ3∗− , θ3∗+ , where θ3∗− ≈ 58.415◦ and θ3∗+ ≈ 58.42◦ , the frequencies ω3 and ω4 of the second resonant triad to the latitude-persistent class, have the jump discontinuity,   belonging ω j = ω−j − ω+j , where ω−j = lim∗− ω j and ω+j = lim∗+ ω j, ( j = 3, 4). θ→θ3

θ →θ3

Latitude Singularity at the Critical Latitude θ = θ4∗ ≈ 74.5◦ In the latitude range θ3∗+ < θ < θ4∗ , computations of the frequencies of two remaining latitude-persistent difference resonant triads show the convergence ω1 → ω3+ and ω2 → ω4+ . The time behaviors of the amplitudes of the recipient waves corresponding to these frequencies also approach each other, i.e., A1 → A3 and A2 → A4 for θ3∗+ < θ < θ4∗ (see also Fig. 11). It is one of our goals in further studies is to investigate the energy exchange between resonant triads in the vicinity of the critical latitude. In particular, one of the questions recently been raised in MacKinnon and Winters [14] is: why the ef f iciency of energy exchange between internal waves increase so much near the critical latitude? Generalization to Larger Number of Resonant Triads The attempt to apply the above analysis were made to investigate a larger number of resonant triads generated from the primary wave of the phase φ M2 and with nmax = 50. To this end, each of the resonant wave obtained from the primary wave of frequency ω M2 and mode number n0 = 1 (called Level 0 waves) is modelled as a primary wave to get the Level 1 waves, i.e., the waves obtained from the Level 0 waves. Next, the waves of Level 2 from the base waves of Level 1 are generated and so on (such model of multi level resonant triads spanning the range of possible frequencies has recently been reported in Ibragimov [9]). In generating Level n waves (n = 1, 2, 3, ...n, n = 0), we obtain the family of resonant curves determined by the resonant conditions − → − → − → k 2η−1 + k 2η + k 0 = 0

(31)

ω2η−1 + ω2η + ω0 = 0

(32)

and

352

R.N. Ibragimov

where η = 1, ..., P in which P is the number of the resonant waves in the model. The resonant curves (31) and (32) are obtained provided that the dispersion relation (10) is satisfied for each of the triad for different values aspect ratios α corresponding to each of the   resonant triads and which monotonically increases from min mk00 to max mk00 for ωi ∈ f, N and i = 0, ..., P. The dynamics of the model (26a–26e) and (29) generalized to the arbitrarily large number P of internal waves is given by system of coupled equations dAη  (33) = γijη Ai A j, dτ j,k

where the coefficients γijη are found by formula (27) and the initial conditions are determined from the GM spectrum from by (29). The simulations below were done for P = 50,001 resonant waves forming 25,000 resonant triads. To understand how each of the three classes of resonant interactions alters the initial distribution and time evolution of the energy spectrum, we solve the initial value problem (33) and (29) for 25,000 resonant triads belonging to all three classes and belonging to the middle and difference interaction classes separately. The numerical results for the energy spectrum corresponding to former case (all three classes) are shown in Fig. 14. Figure 15 shows the θ = 20o

−3

o

(a)

τ=0 τ=100

−4

−4

−5

10

−6

−5

10

−6

10

10

−7

10

0

0.2

0.4

0.6

0.8

−7

10

1

0

−3

0.2

0.4

0.6

0.8

x 10 o

θ = 50

−3

10

(c)

τ=0 τ=100

−4

−4

Energy

−5

−6

(d)

τ=0 τ=100

10

10

−5

10

−6

10

10

−7

10

θ = 74.2o

−3

10

1 −3

x 10

10

Energy

(b)

τ=0 τ=100

10 Energy

10 Energy

θ = 48.5

−3

10

10

−7

0

0.2

0.4

0.6 −1

Frequency [s ]

0.8

1 −3

x 10

10

0

0.2

0.4

0.6 −1

Frequency [s ]

Fig. 14 Sum and difference energy spectrum for 25,000 triads with nmax = 50

0.8

1 −3

x 10

Energy Transfers Within the Rotational Internal Wave Field

353

o

θ = 20

−3

−4

10

−5

10

−5

10

−6

−6

10

−7

10

10

−7

0

0.2

0.4

0.6

0.8

0

1

0.2

0.4

0.6

0.8

x 10

o

θ = 74.2

o

θ = 50

−3

1 −3

x 10

−3

-3

10

−4

10

(c)

τ=0 τ=100

10

−5

10

(d)

tau=0 τ=100

-4

Energy

10 Energy

(b)

τ=0 τ=100

−4

Energy

Energy

10

(a)

τ=0 τ=100

10

10

θ = 48.5o

−3

10

-5

10

-6

10

−6

10

-7

10

−7

10

0

0.2

0.4

0.6

Frequency [s−1]

0.8

1 −3

0

0.2

0.4

0.6

Frequency [s−1]

0.8

1 −3

x 10

x 10

Fig. 15 Difference energy spectrum for 25,000 triads with nmax = 50

dynamics of the energy spectrum for 25,000 resonant triads belonging to the middle and difference interaction class only (i.e., the resonant triads belonging to the sum interaction class are excluded). Comparison of the time evolution of the resulting energy spectrums shown in Figs. 14 and 15 suggests that the sum interaction class is responsible for the spectrum distortion as observed in Fig. 14. In particular, we note that stability of the energy spectrum also depends on the latitude. For example, the comparison of the numerical results in panels (a) and (d) in Fig. 15 shows that stability of the energy spectrum is mostly upset at the high frequency band near the critical latitude. One factor could be the fact that, near the critical latitude, the waves with frequencies close to the subhormonic frequencies have group velocities near zero (which is manifested in Fig. 7), and, by breaking close to where they are generated, where mixing is several times stronger than at lower latitudes. At this point, the question why the sum interaction class is signif icant in the spectrum distortion remains unclarified. We hope that better understanding how the time evolution of wave amplitudes at different latitudes is affected by the presence of many resonant triads will help to clarify the latter question. In particular, Fig. 10 shows that the time evolution of the amplitude A0 affected

354

R.N. Ibragimov θ = 30o

Fifty three resonant triads 4

(a) Amplitude A0

2

0

–2

–4

0

50

100

150

200

250

300

350

400 o

θ = 74.2 0.57

(b)

Amplitude A

0

0.56 0.55 0.54 0.53 0.52

0

500

1000

1500 Inertial periods

2000

2500

3000

Fig. 16 Time evolution of A0 for 53 difference triads at 30◦ and 74.2◦

by two resonant triads at the latitudes θ = 30◦ for 400 and θ = 74.2◦ for 3,000 inertial periods looks periodic. However, Fig. 16 illustrating the time evolution of the amplitude A0 affected by 53 triads at the same latitudes and for the same corresponding inertial periods shows that adding additional resonant triads to the model breaks down the periodicity, although the time behavior of A0 in Fig. 16 is still close to the periodic state. We also note that the number of resonant triads is also latitude-dependent. In particular, the following table shows the number of triads of Levels 0 and 1 waves at 30◦ and 74◦ Sum and difference Number of triads S0θ=30◦ S0θ=74.2◦ S1θ=30◦ S1θ=74.2◦

Difference only

6

6

2

2

1,132

75

222

26

where we denote the number of triads by Slθ=T ◦ , in which l is the number of levels and T is the value of latitude.

Energy Transfers Within the Rotational Internal Wave Field

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References 1. Armstrong, J.A., Bloembergen, N., Ducuing, J. Pershan, P.S.: Interactions between light waves in a nonlinear dielectric. Phys. Rev. 127, 1918–1939 (1962) 2. Craik, A.D.D.: Nonlinear resonant instability in boundary layers. J. Fluid Mech. 50, 393–413 (1971) 3. Furevik, T., Foldvik, A.: Stability at M2 critical latitude in the Barents Sea. Geophys. Res. 101(C4), 8823–8837 (1996) 4. Garrett, C.J.R, Munk, W.H.: Space time scale of internal waves. A progress report. J. Geophys. Res. 80, 291–297 (1975) 5. Garrett, C.J.R, Munk, W.H.: Internal waves in the ocean. Annu. Rev. Fluid Mech. 11, 339–369 (1979) 6. Heney, F.S., Wright, J., Flatte, S.M.: Energy and action flow through the internal wave field— an eikonal approach. J. Geophys. Res. 91(C7), 8487–8495 (1986) 7. Hibiya, T., Niwa, Y., Fujiwara, K.: Direct numerical simulation of the roll-off range of internal wave shear spectra in the ocean. J. Geophys. Res. 101(C6), 123–129 (1996) 8. Hur, M.S., Hwang, I., Jang, H.J., Suk, H.: Effects of the frequency detuning in Raman backscattering of infinitely long laser puleses in plasmas. Phys. Plasmas 13, 73–103 (2006) 9. Ibragimov, R.N., Ibragimov, N.H.: Effects of rotation on self-resonant internal gravity waves in the ocean. Ocean Model. 31, 80–87 (2010) 10. Jurkus, A., Robinson, P.N.: Saturation effects in a traveling-wave parametric amplifier. Proc. IEE Part B 107, 119–122 (1960) 11. Kim, W., West, B.J.: Chaotic properties of internal wave triad interactions. Phys. Fluids 9(3), 632–641 (1997) 12. Laurent, L., Garrett, C.: The role of internal tides in mixing the deep ocean. J. Phys. Oceanogr. 32, 2882–2899 (2000) 13. Longuet-Higgins, M.S., Gill, A.E.: Resonant interactions between planetary waves. Proc. R. Soc. Lond. A 299, 120–128 (2004) 14. MacKinnon, J., Winters, K.: Tidal mixing hotspots governed by rapid parametric subharmonic instability. J. Phys. Oceanogr. (2008, to appear) 15. McComas, C.H.: Equilibrium mechanisms within the internal wave field. J. Phys. Oceanogr. 7, 836–845 (1977) 16. McComas, C.H., Bretherton, F.: Resonant interaction of oceanic internal waves. J. Geophys. Res. 82(9), 1397–1412 (1977) 17. McGoldrick, L.F.: Resonant interactions among capillary-gravity waves J. Fluid Mech. 21, 305–332 (1965) 18. Muller, P., Naratov, A.: The internal wave action model (IWAM). In: Proceedings, Aha Huliko’a Hawaiian Winter Workshop, School of Ocean and Earth Science and Technology, Special Publication (2003) 19. Needler, G.T.: Dispersion in the ocean by physical, geochemical and biological processes. Philos. Trans. R. Soc. Lond., A 319, 177–187 (1986) 20. Phillips, O.M.: On the dynamics of unsteady gravity waves of finite amplitude. J. Fluid Mech. 9, 193–217 (1960) 21. Phillips, O.M.: The Dynamics of the Upper Ocean. Cambridge University Press, New York (1966) 22. Polzin, K.L., Toole, J.M., Ledwell, J.R.: Spatial variability of turbulent mixing in the abyssal ocean. Science 276, 93–96 (1997) 23. St. Laurent, L., Nash, J.: An examination of the radiative and dissipative properties of the internal tide. Deep-Sea Res. 51, 3029–3042 (2004) 24. Thorpe, S.A.: On wave interaction in a stratified fluid. J. Fluid Mech. 24(4), 737–751 (1966) 25. Titman, C.W., Davis, P.A., Hilton, P.A.: Taylor columns in a shear flow and Jupiter’s Great Red Spot. Nature 255, 538–549 (1975) 26. Veronis, G.: The analogy between rotating and stratified fluids. Annu. Rev. Fluid Mech. 2, 37–66 (1970)

Math Phys Anal Geom (2010) 13:357–390 DOI 10.1007/s11040-010-9084-9

D-bar Operators on Quantum Domains Slawomir Klimek · Matt McBride

Received: 3 February 2010 / Accepted: 8 October 2010 / Published online: 12 November 2010 © Springer Science+Business Media B.V. 2010

Abstract We study the index problem for the d-bar operators subject to AtiyahPatodi-Singer boundary conditions on noncommutative disk and annulus. Keywords Quantum spaces · Noncommutative geometry · Dirac operators · APS boundary conditions Mathematics Subject Classification (2010) 46L87

1 Introduction It this paper we consider noncommutative analogs of the d-bar operator on simple complex plane domains with boundary: disk and annulus. In both cases the corresponding quantum domain, its boundary, a d-bar operator, and an analog of the L2 Hilbert space of functions on the domain is constructed using a weighted shift, subject to suitable assumptions. The weighted shift plays the role of the complex coordinate z. For such d-bar operators we consider boundary conditions of Atiyah, Patodi, Singer (APS) type [1]. This can be done so that both the commutative and the noncommutative setup appear in close analogy. The main result of

S. Klimek (B) · M. McBride Department of Mathematical Sciences, Indiana University-Purdue University Indianapolis, 402 N. Blackford St., Indianapolis, IN 46202, USA e-mail: [email protected] M. McBride e-mail: [email protected]

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the paper is that the quantum d-bar operators subject to APS conditions are unbounded Fredholm operators. Additionally we compute their index. Let us recall that an unbounded operator D is called a Fredholm operator if D is closed, has closed range, and finite dimensional kernel and cokernel. Equivalently, see [2], a closed operator D is Fredholm if it has a bounded parametrix Q such that both QD − I and DQ − I are compact. The technical part of the paper consist of finding such a parametrix. The celebrated APS boundary condition was introduced in [1] to handle the index theory for geometrical operators on manifolds with boundary when usual local boundary conditions were not available. Because it is non-local, the APS condition seems to be naturally suited to consider in noncommutative geometry. A more general class of APS-type boundary conditions was described in [3]. Here we consider only simple APS-type boundary conditions given by spectral projections. This paper is a continuation and an extension of [4], which considered APS theory on the noncommutative unit disk. Here we present somewhat different and more detailed treatment of the disk case as well as a similar theory on the cylinder. In particular the modifications we consider here yield a compact parametrix for the d-bar operators, which was not the case in [4]. The present paper will be followed by a separate note containing a construction of a parametrix for the quantum d-bar operator on the semi-infinite cylinder i.e., a punctured disk. Noncommutative domains considered in this paper were previously discussed in [5, 6]. Other papers that studied d-bar operator in similar situations (but not the APS boundary conditions) are: [7–14]. A related study of an example of APS boundary conditions in the context of noncommutative geometry is contained in [15], another one is in [16]. The ideas in this paper can be further extended in several directions. The present setup fits into deformation-quantization scheme and so it will be desirable to consider classical limit of the quantum d-bar operators. Other, different, possibly higher dimensional examples should also be constructed. Because of the compact parametrix, the d-bar operators of this paper can be used to define Fredholm modules over quantum domains (with boundary), which will be interesting to explore. While the computation of the index in the present work is fairly straightforward, it is a challenging question to find a noncommutative framework for such calculations in general. The paper is organized as follows. In the preliminary (Section 2) we describe the classical d-bar operators on domains in complex plane subject to APS-type boundary conditions and compute their index. Section 3 contains the main constructions of the paper: quantum disk, quantum annulus, Hilbert spaces, dbar operators, APS-type boundary conditions. The main results are also stated in this section. Section 4 is the longest of the paper. It contains detailed analysis of some finite difference operators in weighted 2 spaces. The operators are essentially unbounded Jacobi operators, see [17]. That analysis constitutes

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the technical backbone of the paper. Section 5 introduces noncommutative Fourier transform on our quantum domains. The Fourier transform essentially diagonalizes the d-bar operators and thus reduces their analysis to the analysis of the difference operators of the previous section. Finally, Section 6 describes proofs of the main results.

2 The D-bar Operator on Domains in the Complex Plane It this section we review the basic aspects of the APS theory for the d-bar operator on simple domains in the complex plane C. We start by introducing some notation. The first domain is the disk: D = {z ∈ C : |z|  ρ+ }

∂ D = {z ∈ C : |z| = ρ+ }  S1 . The second domain is an annulus in the complex plane C: Aρ− ,ρ+ = {z ∈ C : 0 < ρ−  |z|  ρ+ }

∂ Aρ− ,ρ+ = {z ∈ C : |z| = ρ± }  S1 ∪ S1 , which can also be viewed as a finite cylinder. For each of those domains we will consider the d-bar operator: ∂ ∂z defined on the space of smooth functions. First we will concentrate on the unit disk. In this case we have the short exact sequence: D=

r

0 −→ C0∞ (D) −→ C∞ (D) −→ C∞ (∂ D) −→ 0 ∞

(1)

∞

where r : C (D) → C (∂ D) is the restriction map to the boundary, r f (ϕ) = f (1 · eiϕ ). Here C0∞ (D) is the space of smooth functions on D vanishing at the boundary and z ∈ D has polar representation z = ρeiϕ . Now we consider the APS-like boundary conditions on D. Notice that the APS theory cannot be applied directly in this case since the operator D does not quite decompose into tangential (boundary) and transverse parts near boundary. However this is only a minor technical annoyance, and it is clear that −i∂/∂ϕ is the correct boundary operator. The APS-type boundary conditions considered in this paper are given in terms of the spectral projections of the boundary operator −i∂/∂ϕ as follows. Let π A (I) be the spectral projection of a self-adjoint operator A onto interval I. For an integer N we introduce P N : P N = π 1 ∂ (−∞, N]. i ∂ϕ

(2)

In other words P N is the orthogonal projection in L2 (S1 ) onto span{einϕ }n N .

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The main object of the APS theory is the operator D N defined to be the operator D with the domain:   dom(D N ) = f ∈ C∞ (D) ⊂ L2 (D) : r f ∈ Ran P N . We have the following theorem, see [4] for details. Theorem 2.1 The closure of the operator D N is an unbounded Fredholm operator in L2 (D) and it has the following index: Index(D N ) = N + 1. Now we will discuss the annulus. While we skip some functional analytic details, we show the index calculation in a similar fashion to what was done in [4] in the disk case. If one lets r± be the restriction to the boundary map i.e., r± f (ϕ) = f (ρ± eiϕ ), then one has the short exact sequence:     r=r+ ⊕r−     0 −→ C0∞ Aρ− ,ρ+ −→ C∞ Aρ− ,ρ+ −→ C∞ S1 ⊕ C∞ S1 −→ 0

(3)

where C0∞ (Aρ− ,ρ+ ) is the space of smooth functions on Aρ− ,ρ+ which are zero on the boundary. The key to index calculation of the d-bar operator is the following proposition. In what follows we use the usual inner product on L2 (Aρ− ,ρ+ ):  dz ∧ dz f (z)g(z) f, g = . −2iπ Aρ− ,ρ+ Proposition 2.2 Let D be the operator D=

∂ ∂z

on C∞ (Aρ− ,ρ+ ). Then the kernel of D is the set of bounded holomorphic functions on Aρ− ,ρ+ . Moreover 

2π

Df, g = f, Dg +  −

r+ f (ϕ)r+ g(ϕ)ρ+ e−iϕ

0 2π

r− f (ϕ)r− g(ϕ)ρ− e−iϕ

0

dϕ 2π

dϕ 2π

where f, g ∈ C∞ (Aρ− ,ρ+ ) and D=−

∂ . ∂z

Proof The first conclusion is clear. The integration by parts formula follows immediately from Stokes’ Theorem.



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In order to define APS-type boundary conditions here we take extra caution since the boundary has two components. Let P± N be the spectral projections in ∂ onto interval (−∞, N] i.e.: L2 (S1 ) of the boundary operators ± 1i ∂ϕ P± N = π± 1

∂ i ∂ϕ

(−∞, N]

(4)

where ± is introduced due to the boundary orientations of the inner circle and outer circle. Then, for integers M, N, we define the operator D M,N to be equal to D with domain     − dom(D M,N ) = f ∈ C∞ Aρ− ,ρ+ : r+ f ∈ Ran P+ M , r− f ∈ Ran P N . An immediate corollary of this definition is the description of the kernel of D M,N . Corollary 2.3 Let D M,N be as def ined above, then 

M f : f (z) = n=−N cn zn Ker(D M,N ) = 0

if N + M  0 otherwise.

It follows from Proposition 2.2 that the adjoint of D M,N , is (the closure of) the operator D M,N which is equal to D but with the following domain     −iϕ dom(D M,N ) = f ∈ C∞ Aρ− ,ρ+ : e−iϕ r+ f ∈ Ker P+ r− f ∈ Ker P− M, e N . Moreover, one has the following description of the kernel of D M,N 

f : f (z) = −(M+2) cn zn if N + M < 0 n=N Ker(D M,N ) = 0 otherwise. The following theorem is the corresponding index theorem for the commutative cylinder. Theorem 2.4 The closure of the operator D M,N is an unbounded Fredholm operator. Its index is given by: Index(D M,N ) = M + N + 1. Proof To show the Fredholm property one follows [1]. If f ∈ C∞ (Aρ− ,ρ+ ) then f (z) has the following Fourier representation:  f (z) = fn (ρ)einϕ . n∈Z

This Fourier representation is exactly the spectral decomposition of [1] using the eigenvectors of the boundary operators ±i∂/∂ϕ. In the Fourier transform the operator D decomposes into sum of ordinary differential operators which allows for explicit calculation of a parametrix just like in [1].

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The index computation is as follows. We have: dim Ker(D M,N ) = #{n | − N  n  M} 0 if M + N < 0 = M + N + 1 if M + N  0. In a similar fashion dim Ker(D∗M,N ) = #{n | N  n  −(M + 2)} −(M + N + 1) if M + N < 0 = 0 if M + N  0. Consequently Index(D M,N ) = dim Ker(D M,N ) − dim Ker(D∗M,N ) = M + N + 1,



and the proof is finished.

We now turn our attention to the d-bar operator in the quantum domains.

3 The D-bar Operator on the Non-Commutative Domains In this section we define the main objects of this paper: quantum disk, quantum annulus, Hilbert spaces of L2 “functions”, and d-bar operators. The main results are also stated at the end of this section. In the following definitions we let S be either N or Z. The main input of the theory is a weighted shift U W in 2 (S). Conceptually, U W is a noncommutative complex coordinate on the corresponding noncommutative domain. Definition 3.1 Let {ek }, k ∈ S be the canonical basis for 2 (S). Given a bounded sequence of numbers {wk }, called weights, the weighted shift U W is an operator in 2 (S) defined by: U W ek = wk ek+1 . We will also need the usual shift operator U which is defined by Uek = ek+1 and the diagonal operator W defined by Wek = wk ek .

(5)

∗ Note that U W decomposes to U W = U W and W = (U W U W )1/2 as in the polar decomposition. If S = N then the shift U W is called unilateral and it will be used to define a quantum disk. If S = Z then the shift U W is called bilateral and it will be used to define a quantum annulus (also called a quantum cylinder).

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We require the following conditions on U W : Condition 1 The weights are uniformly positive wk   > 0, for every k ∈ S. Condition 2 The shift U W is hyponormal, i.e.,

∗  S = UW , U W  0. Condition 3 The operator S defined in Condition 2 is injective. Let us remark on some implications of these conditions. First note how S acts on the basis {ek }  ∗  ∗ Sek = U W UW − UW UW ek   2 2 ek = sk ek , = wk − wk−1 (6) 2 where sk := wk2 − wk−1 . It follows that the Conditions 2 and 3 mean that the weights wk form a strictly increasing sequence. Hence the following limits exist and are positive numbers:

w± := lim wk . k→±∞

Secondly, observe that S is a trace class operator with easily computable trace: tr(S) = (w+ )2 in the unilateral case and tr(S) = (w + )2 − (w− )2 in the bilateral case. Moreover S is invertible with unbounded inverse. Let C∗ (W) be the C∗ − algebra generated by U W . Then it is known that there are short exact sequences analogous to (1) and (3). Let K be the ideal of compact operators. Then in the unilateral case the C∗ − algebra generated by U W is the Noncommutative Disk of [5] with the following short exact sequence:   r 0 −→ K −→ C∗ (W) −→ C S1 −→ 0. Similarly, in the bilateral case the C∗ − algebra generated by U W is the Noncommutative Cylinder, see [6], with the following short exact sequence:     r=r+ ⊕r− 0 −→ K −→ C∗ (W) −→ C S1 ⊕ C S1 −→ 0. In the above we let again, abusing notation, r be the restriction map in the disk case and r± in the cylinder case. These two sequences are described in [18]. Now we proceed to the definitions of the quantum d-bar operators. With slight abuse, we will use the same notation for both classical and quantum operators. We define the Hilbert space H as the completion of C∗ (W) with respect to the inner product , S defined as follows:   a, b S = tr S1/2 b S1/2 a∗ where a, b ∈ C∗ (W). It is easy to verify that a, a S is well-defined and positive. Note that the inner product , S is slightly different than the one defined in [4]. This is done (among other reasons) to make definitions more symmetric.

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The basic idea of the definition of a quantum d-bar operator, explained in [4], is to replace derivatives with commutators and so to consider operators of the form a → P[Q, a]R, where P, Q, R are possibly unbounded operators affiliated with C∗ (W). We make our choices so that it is possible to impose APS like boundary conditions, prove the Fredholm property and compute the index. Additionally we would like the operator D to have algebraic relations with U W and U W similar to the relations of the complex partial derivative with z and z¯ . With that in mind, we make the following definition of a quantum d-bar operator D in H : Da = S−1/2 [a, U W ] S−1/2 where the domain of D is the set of those a ∈ H for which S1/2 DaS1/2 (Da)∗ is trace class. It will be verified later that Dom(D) is dense and that for a ∈ Dom(D), r(a) is a square integrable function on the boundary of the domain. This definition is again somewhat different than the one considered in [4]: it is symmetric with respect to left/right multiplication, and the operator D has better functional-analytic properties. A straightforward computation shows the following identities:  n =0 D UW  ∗ D UW = 1  ∗ n 

 ∗ n  , U W S−1/2 D UW = S−1/2 U W  ∗ n−1 1/2  ∗ n−2 ∗ −1/2 = S−1/2 U W S − S−1/2 U W SU W S − ···   n−1 ∗ − S1/2 U W S−1/2 . ∂ if U W was z and the The first two computations show that D looks like ∂z third computation illustrates the non-commutativity of the situation. We proceed to the definitions of the APS-type boundary conditions on D. Let again P N be the orthogonal projection in L2 (S1 ) defined in (2), and let P± N be the orthogonal projections defined in (4). Now we can define D N , D M,N in full analogy with the previous section. The operator D N equals the unilateral operator D with domain

dom(D N ) = {a ∈ Dom(D) : r(a) ∈ Ran P N } . Similarly, the operator D M,N equals the bilateral operator D with domain   − dom(D M,N ) = a ∈ Dom(D) : r+ (a) ∈ Ran P+ N , r− (a) ∈ Ran P M . We are now in a position to state the main results of this paper. Theorem 3.1 For the non-commutative disk case, the operator D N is an unbounded Fredholm operator. Moreover ind(D N ) = N + 1. This is a slight modification from [4], where a somewhat different version of D N was considered. We additionally have:

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Theorem 3.2 For the non-commutative cylinder case, the operator D M,N is an unbounded Fredholm operator. Moreover ind(D M,N ) = M + N + 1. The proofs are contained in the last section.

4 Analysis of Finite Difference Operators In this section we present a detailed analysis of certain finite difference operators related to Jacobi matrices. While the literature on finite difference operators is quite vast, there seems to be no specific reference that deals with Fredholm properties of such operators with unbounded coefficients and boundary conditions at infinity. As indicated in the introduction, these type of operators come up us components of D and its adjoint in Fourier transforms. This will be fully explained in the following section. As before S is either Z or N. Given a sequences of positive numbers a = {an }n∈S called weights, the Hilbert Space a2 (S) is defined by    1 2 2 a (S) = f = { fn }n∈S : | fn | < ∞ a n∈S n  1 fn gn . If a sequence { fn } ∈ a2 (S) has a n∈S n limits, lim fn , they will be denoted f±∞ . with inner product given by f, g = n→±∞

Given two weight sequences a and a we will be studying throughout this section the following unbounded Jacobi type difference operators between a2 (S) and a2 (S): A fn = an ( fn − cn−1 fn−1 ) where   dom(A) = f ∈ a2 (S) : A f a2 (S) < ∞ and A fn = an ( fn − cn fn+1 ) where 

dom(A) = f ∈ a2 (S) : A f 2 (S) < ∞ a

for n ∈ S. If S = N we assume in the above that f−1 = 0. The coefficients an , an , and cn ∈ C are assumed to satisfy:  1  1  1  0 < |cn |  1 , = C < ∞ , = C < ∞ , < ∞, a a c n∈S n n∈S n n∈S n

(7)

where the infinite product above is, as usual, the limit of the partial products. We also define:  1 . K= |c | n∈S n

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The goal of this section is to establish the Fredholm properties of the operators A, A and related operators obtained by imposing conditions at infinities. This is done by constructing a parametrix for each operator. Our discussion will be split into two separate but similar cases: unilateral and bilateral. 4.1 Unilateral Case We first study the kernels of A and A, in order to see if these operators have inverses or not. Proposition 4.1 Given A and A above we have Ker A = {0} dim Ker A = 1. Proof First consider the equation A fn = 0 which is an ( fn − cn−1 fn−1 ) = 0 for n = 0, 1, 2 . . . Then solving recursively one can see that the only solution to the equation is f0 = f1 = · · · = fn = 0 for all n. This shows that Ker A is trivial and thus A is an invertible operator. Secondly consider the equation A fn = 0 which is an ( fn − cn fn+1 ) = 0 for n = 0, 1, 2 . . . Then solving recursively one has n = 0 ⇒ f1 = c10 f0 n = 1 ⇒ f2 = c01c1 f0 .. .. . . which in general gives fn =

1 f0 , c0 c1 · · · cn−1

thus showing that A has a one dimensional kernel provided that fn ∈ a2 (N). Notice the following ∞

| fn | =

 1 1 | f0 |  | f0 | = K| f0 | |c0 · · · cn−1 | |ci | i=0

since |ci |  1 for all i = 0, 1, . . . From this it follows that  f 2 

∞  1 2 K | f0 |2 = CK2 | f0 |2 < ∞ a n n=0

with the constants defined at the beginning of the section. Thus this completes the proof.

Next we show how to find the inverse T of A and we study its properties.

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Proposition 4.2 There exists an operator T ∈ B(a2 (N), a2 (N)) such that T A = Ia2 (N) and AT = I2 (N) . Indeed it is given by the formula (8) below. In particular a A is an unbounded Fredholm operator with zero index. Proof Let {gn } ∈ a2 (N) and consider the equation A fn = gn which is an ( fn − cn−1 fn−1 ) = gn for { fn } ∈ dom(A), n = 0, 1, 2 . . .. As above, solving for each n recursively one arrives at the following formula ⎛ ⎞ n n−1  1 ⎝ ⎠ c j gi , (8) (Tg)n = fn = a i=0 i j=i where in the above we set, for convenience: n−1 

c j = 1.

j=n

Next we show that T ∈ B(a2 (N), a2 (N)). We divide and multiply each term as follows 1 cn−1 cn−1 · · · c0 gn + gn−1 + · · · + g0 an an−1 a0 √ √ √ cn−1 an−1 gn−1 cn−1 · · · c0 a0 g0 a n gn = + ··· + √ + √ . √ an an an−1 an−1 a0 a0

(Tg)n =

∞  1 |(Tg)n |2 and since |cn |  1 for every n, using the Cauchy–  a n=0 n Schwarz inequality one has  √    √ 2   2 a a0 1 1 n 2 2 2 |(Tg)n |  + ··· + |gn | + · · · + |g0 | an a0 an a0 ∞   1  g2 = Cg2 . a n n=0

Since Tg2 =

Consequently: Tg2 

∞  1 Cg2  a n n=0

= C Cg2 ,

√ which implies that T  C C, thus one has T ∈ B(a2 (N), a2 (N)). A straightforward calculation shows that T A = Ia2 (N) and AT = I2 (N) .

a

An important corollary from this proposition is the existence of limits at infinity for sequences which are in the domain of A.

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Corollary 4.3 Let f = { fn } ∈ dom(A), then lim fn = f∞ exists and is given by n→∞

the following formula

f∞

⎛ ⎞ ∞ ∞  1 ⎝ ⎠ = c j A fi . a i=0 i j=i

(9)

Proof If f ∈ dom(A), then write f as, f = T(A f ), then one has the following ⎛ ⎞ n n−1   1 ⎝ fn = c j⎠ A fi . a i=0 i j=i Using assumptions (7) and estimating as above, we see that the formula (9) is well defined. Now the fact that lim fn = f∞ follows from a simple /2 n→∞ argument

We now wish to consider the operator A and determine if it has bounded right inverse since Proposition 4.1 tells us that A has a one dimensional kernel. The next proposition will show this. We will be using the following notation: if V be a closed subspace of a Hilbert space H, then we denote ProjV , to be the orthogonal projection onto V. Proposition 4.4 Given A from above then there exists a T ∈ B(a2 (N), a2 (N)) such that AT = I2 (N) and T A = Ia2 (N) − ProjKer A . In particular A is an una bounded Fredholm operator with index equal to one. Proof From Proposition 4.1 we know that A has a one dimensional kernel spanned by the following vector  ∈ Ker(A): n−1   ∞  1  n = ci = ci . c i=n i=0 i i=0 ∞ 

Next consider the equation Agn = an (gn − cn gn+1 ) = fn for n = 0, 1, 2, . . . As before solve the equation recursively and one will arrive at the formula

gn+1

⎛ ⎞ n n n    1 1 ⎝ 1⎠ = g0 − fi .  c a c i=0 i i=0 i j=i j

where g0 is arbitrary. To finish the construction of T we need to choose g0 so that T A = Ia2 (N) − ProjKer A as it’s clear that AT = I2 (N) . a

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The disadvantage of the above formula for T is that it does not translate easily to the bilateral case. Anticipating it, we rewrite the above solution in an equivalent but different looking form: ⎛ ⎞ ∞  ∞ i−1    1 ⎝ (T f )n = gn = c j⎠ fi − ci L( f )  a j=n i=n i i=n = T0 fn − n L( f ),

(10)

 where we set n−1 j=n c j = 1 and L( f ) is an arbitrary constant. This form of solution is also explained conceptually when considering bilateral case. For T f to be orthogonal to Ker A, one needs , T f = 0 for the above  ∈ Ker A. From this one can deduce that L( f ) is the following following linear functional of f :  ∞ ∞ 1 1 i−1  ∞ n=0 i=n an ai j=n c j k=n ck fi , T0 f

 . L( f ) := = ∞ 1 ∞ 2 ||||2 n=0 a i=n |ci | n

It is straightforward to verify now that T A = Ia2 (N) − ProjKer A and that AT = I2 (N) . All that remains is to show the boundedness of T. The operator a √ T0 is bounded by CC in exactly the same way as the operator T is Proposition 4.2. To estimate L( f ) we notice that ∞  ∞  ∞ ∞   1  2 1  2 C  2 C  |||| = |c |  |c | = , i i a i=n a i=0 K2 n=0 n n=0 n √ √ √ which implies that |L( f )| ≤ K C|| f || and ||T|| ≤ CC + K CC . This completes the proof.

We again get a corollary on the existence of limits at infinity for sequences which are in the domain of A. Corollary 4.5 Let f ∈ dom(A), then f∞ exists and is given by the following formula f∞ = −L(A f ). Proof The proof for the T0 term is identical to the proof of the Corollary 4.3. To compute the limit of the other term we note: n =

∞  i=n

as n → ∞.

∞

i=0 ci = n−1 i=0

ci ci

→1



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The above corollaries allow us to consider “boundary” conditions on A and A. We define the operators A0 and A0 as follows: A0 is the operator A but with domain dom(A0 ) = { f ∈ dom(A) : f∞ = 0}, and A0 is the operator A with domain dom(A0 ) = { f ∈ dom(A) : f∞ = 0}. The four operators are closely related as shown by the following computation of the adjoint of A. Proposition 4.6 The adjoint of A has the following formula A∗ = A0 . Moreover the adjoint of A has the following formula ∗

A = A0 . Proof Computing the inner product one has: ∞ ∞   1 an ( fn − cn−1 fn−1 )gn = ( fn − cn−1 fn−1 )gn a n=0 n n=0  N  N N    = lim fn gn − cn−1 fn−1 gn . ( fn − cn−1 fn−1 )gn = lim

A f, g =

N→∞

N→∞

n=0

n=0

n=0

Then, setting n − 1 → n one arrives at  N   A f, g = lim fn (gn − cn gn+1 ) − c N f N g N+1 N→∞

=

∞  n=0



n=0

1 fn an (gn − cn gn+1 ) − f∞ g∞ an

= f, Ag − f∞ g∞ .

Here note that c−1 n < ∞ and |cn |  1 implies that the cn converge to 1. The functional f → f∞ is not continuous thus implying that if f ∈ ∗ dom(A∗ ), then f∞ = 0 and if g ∈ dom(A ), then g∞ = 0. This completes the proof.

It follows that all four operators are Fredholm operators where parametrix in each case is T, T, or their adjoints. For completeness we compute the adjoint of T and of T: this is not necessary for the main argument but may possibly be useful in future applications.

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Proposition 4.7 The adjoint of T is equal to T0 of (10), i.e., it has the following formula: ⎛ ⎞ ∞ k−1   1 ⎝ c j⎠ fk . (T ∗ f )n = (T0 f )n = ak j=n k=n

Similarly: ∗

T f = Tf −

, f

T. ||||

Proof Looking at the inner product one has   ∞ ∞   1 1 cn−1 cn−1 · · · c0 1 Tg, f = (Tg)n fn = gn + gn−1 + · · · + g0 fn a a an an−1 a0 n=0 n n=0 n      ∞ ∞  1 1 1 cn−1 = gn fn + gn−1 fn + · · · a an a an−1 n=0 n n=0 n +

  ∞  1 cn−1 · · · c0 g0 fn . a a0 n=0 n

Then using n → j + 1 in the second sum, n → j + 2 in the third sum and so on and relabeling the indices, one has      ∞ ∞  1 1 cn 1 Tg, f = gn fn + gn fn+1 + · · · a an a an+1 n=0 n n=0 n      ∞ ∞  1 1 cn 1 gn fn + gn fn+1 + · · · = a an a an+1 n=0 n n=0 n   ∞  1 cn · · · cn+k + gn fn+(k+1) + · · · a an+(k+1) n=0 n   ∞  1 1 cn cn · · · cn+k = gn fn +  fn+1 + · · · +  fn+(k+1) + · · · a an an+1 an+(k+1) n=0 n = g, T ∗ f . This then shows the first result. For the second formula we notice that we just ∗ showed that T0 = T and the second term comes from an easy computation of the adjoint of the projection f → L( f ).

Combining Propositions 4.2, 4.4, and 4.6 we get the following results about A0 and A0 .

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Corollary 4.8 A0 is an unbounded Fredholm operator with index equal to minus one. We have A0 T0 = I2 (N) − ProjCoker(A0 ) a

T0 A0 = Ia2 (N) where T0 := T

∗

We also have: Corollary 4.9 A0 is an unbounded Fredholm operator with index zero, and A0 T0 = Ia2 (N) T0 A0 = I2 (N) . a

It turns out that we can say more about the parametrices introduced above. Proposition 4.10 Each of the parametrix operators: T, T0 , T, T0 is a Hilbert– Schmidt operator. Proof We present the details for the operator T, other cases are similar. In fact the proposition already follows from the way we estimated the norm of T since T is an integral operator. We give an alternative proof here. First ∞  Tei 2 where {ei } is the canonical basis for note that T2HS = tr(T ∗ T) = i=0

a2 (N). So (Tei )n =

1 cn−1 cn−1 · · · c0 (ei )n + (ei )n−1 + · · · + (ei )0 . an an−1 a0

It follows that (Tei )n = 0 ∀n < i, and

√ ai (Tei )i = ai ci √ (Tei )i+1 = ai ai ci+1 ci √ (Tei )i+2 = ai ai .. .

Then we estimate Tei 2 =

∞ 1  1 1 |ci ci+1 · · · ci+k |2  C ,  ai ai+k ai k=0

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373

and consequently T2HS =

∞  i=0

Tei 2  C

∞  √ 1  CC ⇒ T HS  CC . a i=0 i

We now shift our attention to the bilateral case and study the same type of properties as considered in the unilateral case. It turns out that both A and A have one dimensional kernels in that case, one has to use infinite products for some expressions, and there are more options of imposing conditions at infinities. However the analytic aspects of the theory are no different then the unilateral case and so we provide less detail in some estimates to avoid repetitiveness. 4.2 Bilateral Case As in the unilateral case we start with the study of the kernels of A and A. It turns out that both A and A have one dimensional kernels. First recall the constants defined at the beginning of this section C=

 1  1  1 < ∞. < ∞ , C = < ∞ and K =  a a |c | n∈Z n n∈Z n n∈Z n

Proposition 4.11 Given A and A above we have: dim KerA = 1 dim KerA = 1. Proof First we investigate the kernel of A. To this end we need to solve the equation A fn = an ( fn − cn−1 fn−1 ) = 0 for n ∈ Z. This is done recursively and, for n  0, one arrives at the following  n−1   ci f−1 , n  0. fn = i=−1

Next, in a similar fashion, solve the equation for n < 0 to get the following  −n   1 f−n = f−1 , n  1. c i=−2 i The two formulas above can be written compactly in the following semi-infinite product  n−1   fn = ci α i=−∞

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for any constant α. To see that the kernel of A is indeed one dimensional, we need to verify that { fn } ∈ a2 (Z). Using the fact that |ci |  1 for all i one has that  2  1  n−1    1 2 2 c |α|  |α| = |α|2 C < ∞,  f 22 (Z) =   i  a an  an n∈Z

n∈Z

i=−∞

thus { fn } ∈ a2 (Z). Next we study the equation A fn = an ( fn − cn fn+1 ) = 0 for n ∈ Z. We get n−1  1 fn = f0 for n ≥ 0 c i=0 i and the similar formula for n < 0  −n   f−n = ci f0

for n  1.

i=1

We also have the same type of semi-infinite product for A: ∞   ci β fn = i=n

for any constant β. As with A, to guarantee that the kernel of A is one dimensional, we need to verify that { fn } ∈ a2 (Z). Using the fact that |ci |  1 for all i one has that  ∞ 2   1   1  2 ci  |β|2  |β|2 = |β|2 C < ∞.  f 2 (Z) =  a  an  an n∈Z

i=n

This completes the proof.

n∈Z



Next we construct a parametrix for A. Proposition 4.12 There exists an operator T ∈ B(a2 (Z), a2 (Z)) such that AT = Ia2 (Z) and T A = I2 (Z) − ProjKer A . In particular A is an unbounded Fredholm a operator with index equal to one. Proof We start by looking at the equation A fn = an ( fn − cn−1 fn−1 ) = gn which can be written as: gn fn − cn−1 fn−1 = . (11) an We use variation of constants method to solve (11). First observe that the homogeneous equation fn − cn−1 fn−1 = 0 has the following solution by the

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375

  n−1 α for some constant kernel calculation in Proposition 4.11: fn = c i i=−∞ α. Consequently we set ⎛ ⎞ n−1  fn = ⎝ c j⎠ αn j=−∞

and substitute this into (11). This leads to the following equation for αn : ⎛ ⎞ n−1  1 gn ⎠ αn − αn−1 = ⎝ c an j=−∞ j which has a solution given by:

⎛ ⎞ n i−1  1 ⎝  1⎠ αn = gi . a j=−∞ c j i=−∞ i

Therefore one has a particular solution of (11): ⎛ ⎞ n n−1   1 ⎝ fn = c j⎠ gi , a i i=−∞ j=i and the general solution is

⎛ ⎞  n−1  n n−1   1 ⎝ ⎠ fn = c j gi − ci α. a i=−∞ i j=i i=−∞

The above expression gives the formula for T: (Tg)n = (T1 g)n − α(g)− n, where

(12)

⎛ ⎞ n n−1  1 ⎝ ⎠ (T1 g)n := c j gi a i=−∞ i j=i

(13)

n−1 and − n := i=−∞ ci , and α(g) arbitrary. It’s is clear from our construction that AT = Ia2 (Z) . To make sure that we get T A = I2 (Z) − ProjKer A , we must make a choice on α(g) just as in the unilateral a case:     n n−1 1 1 n−1 n∈Z i=−∞ an ai k=−∞ ck j=i c j gi − , T1 g

  = α(g) := . 1 n−1 ||− ||2 |c |2 n∈Z an

i=−∞

i

Convergence of the sums and products and the boundedness of T √ is established just as in the unilateral case. The operator T1 is bounded by CC in

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essentially the same way as the operator T is Proposition 4.2. To see that we write √ √ cn−1 an−1 1 an 1 gn−1 + · · · − (T1 g)n = √ gn + √ an an an−1 an−1 and estimate using the Cauchy–Schwarz inequality and the fact that the |ci |  1 for all i:  √   √ 2  2 a a 1 1 n−1 n 2 2 2 |(T1 g)n |  + + ··· |gn | + |gn−1 | + · · · an an−1 an an−1  n   1  g2 . a i i=−∞ Consequently T1 g2 =

 1  1 |(T1 g)n |2  Cg2 = (C C)g2 .   a a n∈Z n n∈Z n

To estimate α(g) we notice that ∞     1   1  C 2 2 = C  ||||2 = |c |  |c | , i i a i=n a i∈Z K2 n∈Z n n∈Z n

√ √ √ which implies that |α(g)| ≤ K C||g|| and ||T|| ≤ CC + K CC . This completes the proof.

An important corollary from this proposition is the existence of limits at infinities for the sequences which are in the domain of A. Corollary 4.13 Let f ∈ dom(A), then f±∞ exist and are given by the following formulas: ⎛ ⎞  ∞  ∞ ∞   1 ⎝ ⎠ c j A fi − ci α(A f ) f∞ = a i=−∞ i j=i i=−∞ f−∞ = α(A f ). Proof Using the previous proposition and the methods invoked in Corollaries 4.3 and 4.5 yields the desired result.

Next we state analogous results about the A. Proposition 4.14 There exists a T ∈ B(a2 (Z), a2 (Z)) such that AT = I2 (Z) and a

T A = I2 (Z) − ProjKer A . In particular A is an unbounded Fredholm operator a with index equal to one.

D-bar Operators on Quantum Domains

377

Proof The solution of the equation an ( fn − cn fn+1 ) = gn

for n ∈ Z

is given the following formula ⎛ ⎞ ∞  ∞ i−1    1 ⎝ c j⎠ gi − ci β(g) = T0 gn − β(g)+ (Tg)n = n,  a i=n i j=n i=n where we set

n−1 j=n

c j = 1 and β(g) is an arbitrary constant. Here ⎛ ⎞ ∞ i−1  1 ⎝ ⎠ (T0 g)n := c j gi , a j=n i=n i

(14)

(15)

∞ and + n := i=n ci . One has the relation AT = I2 (Z) , however to make sure one has T A = a Ia2 (Z) − ProjKer A , we need to make the following choice of β(g):  ∞ 1 1 i−1  ∞ n∈Z i=n an ai j=n c j k=n ck gi + , T0 g

  β(g) = = . 1 ∞ 2 ||+ ||2 n∈Z a i=n |ci | n

The previous methods yield T 

√ √ CC + K CC < ∞,

and the statement of the proposition follows.



An immediate corollary is the following: Corollary 4.15 Let f ∈ dom(T), then f±∞ exist and f∞ = β(A f ) f−∞

⎛ ⎞  ∞  ∞ i−1   1 ⎝  ⎠ = c j A fi − ci β(A f ).  a j=−∞ i=−∞ i i=−∞

Imposing vanishing conditions at infinities we can construct the following six operators. A0 is the operator A but with domain dom(A0 ) = { f ∈ dom(A) : f∞ = 0} and A0 is the operator A with domain dom(A0 ) = { f ∈ dom(A) : f∞ = 0}. A1 is the operator A with domain dom(A1 ) = { f ∈ dom(A) : f−∞ = 0}

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and A1 is the operator A with domain dom(A1 ) = { f ∈ dom(A) : f−∞ = 0}. Finally A2 is the operator A with domain dom(A2 ) = { f ∈ dom(A) : f±∞ = 0} and A2 is the operator A with domain dom(A2 ) = { f ∈ dom(A) : f±∞ = 0}. The above operators are related by the calculation of adjoints of A and A. Proposition 4.16 With the above def initions we have: A∗ = A2 , A∗0 = A1 , A∗1 = A0 , A∗2 = A, ∗

∗

∗

∗

A = A2 , A0 = A1 , A1 = A0 , A2 = A. Proof This easily follows from the integration by parts formula: A f, g = f, Ag − f∞ g∞ + f−∞ g−∞ .

It follows from the definitions and the kernel calculations for A and A that the just introduced six operators A0 , A1 , A2 , A0 , A1 , A2 have no kernel, while the adjoint calculation shows that only A2 , A2 have cokernel (of dimension one). Next we find a parametrix for each of the above operators. So far we have constructed T, formula (12), and T, formula (14). In view of the above ∗ proposition we set T2 := T and T2 := T ∗ . We have also introduced T1 , formula (13), and T0 , formula (15) and one can verify like in Proposition 4.7 that T1∗ = T0 . We introduce similar looking operators: ⎛ ⎞ n n−1  1 ⎝ ⎠ c j gi (T1 g)n := a i=−∞ i j=i and

⎛ ⎞ ∞ i−1   1 ⎝ c j⎠ gi , (T0 g)n :=  a i=n i j=n

for which we have T0∗ = T1 . Then we get the following summary of the Fredholm properties of our operators.

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Proposition 4.17 With the above def initions we have A0 T0 = Ia2 (Z) and T0 A0 = I2 (Z) a

A1 T1 = Ia2 (Z) and T1 A1 = I2 (Z) a

A2 T2 = Ia2 (Z) − ProjCoker(A2 ) and T2 A2 = I2 (Z) a

T0 A0 = I2 (Z) and A0 T0 = Ia2 (Z) a

T1 A1 = I2 (Z) and A1 T1 = Ia2 (Z) a

T2 A2 = I2 (Z) and A2 T2 = Ia2 (Z) − ProjCoker(A2 ) . a

In particular all six operators are unbounded Fredholm operators with index zero for A0 , A1 , A0 , A1 and index minus one for A2 , A2 . We conclude this section with a simple observation on functional-analytic properties of the parametrices. Proposition 4.18 Each of the eight parametrix operators: T, T0 , T1 , T2 , T, T0 , T1 , T2 is a Hilbert–Schmidt operator.

5 Fourier Transform in Quantum Domains In this section we consider the Fourier Transform in the quantum domains, and get decomposition theorems for the Hilbert Space H and the operator D, defined in Section 3. The following discussion covers both cases S = N and S = Z in a fairly uniform manner: there are only a few places where the difference between the unilateral and the bilateral cases needs to be covered separately. We will make an extensive use of the label operator defined as: Kek = kek , where {ek }, k ∈ S is the canonical basis for a2 (S). The label operator lets us write different diagonal operators as its functions. For example two previously introduced operators can be expressed, with some notational abuse, as W = W(K), and S = S(K), see (5) and (6), with W(k) = wk , and S(k) = sk = wk2 − 2 wk−1 . Additionally, the elements of a2 (S) will also be written using the function notation i.e., { fk } = { f (k)}. If { f (k)} has a limit at ±∞ it is denoted by f (±∞). For the purpose of the following discussion we define a(n) (k) = S−1/2 (k)S−1/2 (k + n).

(16)

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Then one has the following lemma which is essentially a Fourier decomposition of the Hilbert space H . Lemma 5.1 Let a(n) = {a(n) (k)} be the of positive numbers def ined ∞sequence 2 2 above. The map I : ∞ m=0 a(m) (S) ⊕ n=1 a(n) (S) → H given by ∞

∞

{ fm (k)}k∈S ⊕ m=0

I

{gn (k)}k∈S → n=1

∞ 

U m fm (K) +

m=0

∞ 

gn (K)(U ∗ )n

n=1

is well-def ined and is an isomorphism of Hilbert spaces. Proof First we need to show that I is an isometry. We will only do this for the gn (K) terms as the calculation for the fn (K) terms is essentially identical. We have !∞ !2   ∞ ∞ ! !   ! ∗ n! 1/2 ∗ n 1/2 n gn (K)(U ) ! = tr S (K) gn (K)(U ) S (K) U gl (K) ! ! ! n=1 n=1 l=1 H   ∞  = tr S1/2 (K)S1/2 (K + n) |gn (K)|2 n=1

=

∞  ∞ 

1

n=1 k=0

a(n) k

|gn (k)|2 =

∞  n=1

{gn (k)}22

a(n)

= {gn (k)}2∞

2 n=1 a(n)

and thus the norms are the same and I is an isometry on its range. To show that Ran I = H we need to demonstrate that Ran I is dense in H . First note that C∗ (W) is dense in H by construction. Define δl (k) to be the following function: 1 k=l δl (k) = 0 k = l. Then the (not normalized) canonical basis in a2(m) (S) corresponds through the map I to U m δl (K) and similarly the canonical basis in a2(n) (S) corresponds to δl (K)(U ∗ )n . Note that U m δl (K) and δl (K)(U ∗ )n sit inside C∗ (W), so all that is required is to show they generate a dense set in C∗ (W) in the topology induced by H (they do not in the usual topology of C∗ (W)). However this is clear since  l L

δl (K) → I L→∞

in H

because the operator S is trace class. It follows that U, U ∗ are in Ran I, and thus Ran I is a dense subspace of H .



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381

In what follows it will be convenient sometimes to write the Fourier series for a ∈ H in one of two ways: a=

∞ 

U m fm (K) +

m=0

∞ 

gn (K)(U ∗ )n =

n=1

∞ 

U m fm (K) +

m=1

∞ 

gn (K)(U ∗ )n (17)

n=0

where we always set f0 (k) = g0 (k). We will now use the Fourier transform described in the above lemma to find a decomposition of D in terms of the operators A and A defined in the previous section. Recall that those operators depend on sequences of weights a, a and coefficients c subject to conditions (7). Since in the following the parameters vary, we will need appropriate decorations on A and A. To do that, in addition to sequences (16), we introduce: c(n) (k) := W(k)W −1 (k + n + 1).

(18)

Now we define the operators A(n) as follows: A(n) : dom(A(n) ) ⊂ a2(n+1) (S) → a2(n) (S) where dom(A(n) ) = { f ∈ a2(n+1) (S) : A f 2(n) (S) < ∞} a   (n) (n) (n) A f (k) = a (k) f (k) − c (k − 1) f (k − 1) . The corresponding formal adjoints A previous section i.e., A

(n)

(n)

are defined in the same way as in the

(n)

: dom(A ) ⊂ a2(n) (S) → a2(n+1) (S) (n)

where dom(A ) = { f ∈ a2(n) (S) : A f 2(n+1) (S) < ∞} a

A

(n)

f (k) = a(n+1) (k)( f (k) − c(n) (k) f (k + 1)).

Additionally we will need the following diagonal operator W (m) (K) := W(K + m) i.e., W (m) f (k) := W(k + m) f (k) for f ∈ a2(n) (S). Clearly W (m) is a bounded, invertible, self-adjoint operator with a bounded inverse. Now we can state the main decomposition theorem. A minor difficulty here is that D is not diagonal with respect to the Fourier decomposition of the Hilbert space but rather shifts the components by one. Theorem 5.2 With the above notation the operator D has the following decomposition: Da =

∞  m=1

U m fm (K) +

∞  n=0

gn (K)(U ∗ )n

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S. Klimek, M. McBride

 where a is given by formula (17) and fm+1 = −A (n−1) (n−1) W A gn . We write symbolically:

(m)

 W (m) fm and gn−1 =

  (m) (n−1) (n−1) ∞ D∼ A )n=1 . = (−A W (m) )∞ m=0 , (W Proof We compute the expression Da = S−1/2 (K) [a, U W(K)] S−1/2 (K) using ∞ ∞   U m fm (K) + gn (K)(U ∗ )n . We use the the Fourier decomposition: a = m=0

n=1

following commutation relation f (K)U = U f (K + 1). Then one obtains, setting in the unilateral case W(−1) = fn (−1) = gn (−1) = 0, Da = S−1/2 (K) [a, U W(K)] S−1/2 (K) ∞ 

=

  S−1/2 (K) U m fm (K)U W(K) − U W(K)U m fm (K) S−1/2 (K)

m=0

+

∞ 

  S−1/2 (K) gn (K)(U ∗ )n−1 W(K) − U W(K)gn (K)(U ∗ )n S−1/2 (K).

n=1

The above expression is equal to −

∞ 

U m+1 S−1/2 (K)S−1/2 (K + m + 1) (W(K + m) fm (K) − W(K) fm (K + 1))

m=0

+

∞ 

S−1/2 (K)S−1/2 (K + n − 1) (W(K + n − 1)gn (K) − W(K − 1)gn (K − 1))

n=1

× (U ∗ )n−1 , which can be written as: −

∞  m=0

U m+1 a(m+1) (K)

 × W(K + m) fm (K) −

 W(K) W(K + m + 1) fm (K + 1) W(K + m + 1)   ∞  W(K − 1) (n−1) + W(K + n − 1)a (K) gn (K) − ( gn (K − 1) (U ∗ )n−1 . W K + n − 1) n=1

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383

This is equal to: −

∞ 

  U m+1 a(m+1) (K) W (m) (K) fm (K) − c(m) (K)W (m) (K + 1) fm (K + 1)

m=0

+

∞ 

  W (n−1) (K)a(n−1) (K) gn (K) − c(n−1) (K − 1)gn (K − 1) (U ∗ )n−1 .

n=1

Consequently Da = −

∞ 

U m+1 A

(m)

W (m) fm (K) +

m=0

∞ 

W (n−1) A(n−1) gn (K)(U ∗ )n−1 .

n=1

Next we need to verify that the a(n) , see (16), and the c(n) , see (18), satisfy the conditions (7). Note that  wk is an increasing sequence converging to  since  wk  + (n) w > 0 one has |c (k)| =  wk+n+1   1. In the unilateral case, S = N, we compute K(n) :=

∞ 

1

k=0

c(n) (k)

Next note that C(n) :=

∞  k=0

1 a(n) (k)

=

(w + )n+1 < ∞. w0 · · · wn

" " #∞ #∞ ∞   # # √ = sk sk+n ≤ $ sk $ sk+n k=0

k=0

" #∞ # +$ =w sk < ∞,

k=0

k=n

with the constant C(n) going to zero as n → ∞. In the bilateral case (k ∈ Z) we have ∞ 

1

k=−∞

c(n) (k)

K(n) := Next we estimate ∞  C(n) := k=−∞

≤

%

1 a(n) (k)

=

(w+ )n+1 < ∞. (w − )n+1

 √  √ sk sk+n + sk sk+n

=

k≤−n/2

(w + )2 − (w− )2

& 

k>−n/2

& % sk + (w + )2 − (w − )2 sk < ∞,

k≤−n/2

and again the constant C

(n)

k>n/2

goes to zero as n → ∞.



As we will see later on, the significance of lim C(n) = 0 is that it implies n→∞

compactness of a parametrix of D, subject to APS boundary conditions.

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We state here without a proof the analogous result for the formal adjoint D of D. We define Db := S−1/2 (K)[b , W(K)U ∗ ]S−1/2 (K). on the maximal domain, like the operator D. We have the following decomposition. Theorem 5.3 With the above notation the operator D can be written as   (n−1) D∼ W (n−1) )∞ = (−W (m) A(m) )∞ m=0 , (A n=1 .

6 Results We are now in a position to consider the proofs of the main results of this paper. We rephrase here the statements of the theorems from Section 3 adding more detail. The operator D N equals the unilateral operator D with domain dom(D N ) = {a ∈ Dom(D) : r(a) ∈ Ran P N } . We will now prove the first of the main results of this paper. Theorem 6.1 The operator D N def ined above is an unbounded Fredholm operator with index given by ind(D N ) = N + 1. In fact, there is a bounded operator Q N (a parametrix) such that Ker(Q N ) = Coker(D N ), D N Q N = I − ProjCoker(DN ) , and Q N D N = I − ProjKer(DN ) . Moreover the parametrix Q N is a compact operator. Proof All the hard work has been done. It’s now just a matter of piecing together appropriate results from the sections. the previous ∞First we analyze n ∗ n APS boundary conditions. Let a = ∞ n=0 U fn (K) + n=1 gn (K)(U ) be in dom(D N ). Then the restriction r(a) from Section 3 is well defined. We note that r acts on U, U ∗ , and f (K) in the following way r(U) = eiϕ r(U ∗ ) = e−iϕ r( f (K)) = f (∞) · I := lim f (k) · I. k→∞

The third equation holds because the difference f (K) − f (∞) · I is a compact operator, and r vanishes on compact operators. Consequently we see that r acts on a ∈ Dom(D) in the following way: r(a) =

∞  m=0

eimϕ fn (∞) +

∞  n=1

gn (∞)e−inϕ .

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385

This means that for r(a) to be in the range of P N , where Ran P N = span{einϕ }, n N

one has the following: if N  0, then fn (∞) = 0 for n > N, and if N < 0, then fn (∞) = 0 for all n and gn (∞) = 0 for n < −N. Thus from Theorem 5.2 and from Proposition 4.6 one can represent D N subject to the APS boundary conditions as follows ⎧   (m) (m) N (n−1) (n−1) ∞ ⎪ (−A W (m) )m=0 , (−A0 W (m) )∞ A )n=1 ⎪ m=N+1 , (W ⎪ ⎪ ⎨ for N  0  DN =  (m) (m) ∞ (n−1) (n−1) −N−1 (n−1) (n−1) ∞ ⎪ (−A W ) , (W A ) , (W A ) ⎪ 0 n=1 0 m=0 n=−N ⎪ ⎪ ⎩ for N < 0 Also note from Theorem 5.2, Proposition 4.6 and the above analysis of the APS conditions, one can represent D∗N as follows  ⎧  (n−1) N ⎪ )m=0 , (−W (m) A(m) )∞ , (A0 W (n−1) )∞ (−W (m) A(m) ⎪ n=1 0 m=N+1 ⎪ ⎪ ⎨ for N  0 ∗  DN =  (n−1) (n−1) ⎪ (m) (m) ∞ (n−1) −N−1 (n−1) ∞ ⎪ A ) , (A W ) , (A W ) (−W 0 ⎪ m=0 n=1 n=−N ⎪ ⎩ for N < 0 From these representations and from Proposition 4.1, one gets the following  N + 1 for N ≥ 0 dim KerD N = 0 for N < 0 and

 ∗

dim KerD N =

0 for N ≥ 0 −(N + 1) for N < 0

and thus the index calculation follows. To conclude that D N is a Fredholm operator we need to construct a parametrix. We build Q N in the following fashion:  ⎧  (m) N (m) ⎪ (−V (m) T )m=0 , (−V (m) T0 )∞ , (T (n−1) V (n−1) )∞ ⎪ n=1 m=N+1 ⎪ ⎪ ⎨ for N  0  QN =  (m) ⎪ ⎪ (−V (m) T0 )∞ , (T0 (n−1) V (n−1) )−N−1 , (T (n−1) V (n−1) )∞ ⎪ n=1 m=0 n=−N ⎪ ⎩ for N < 0 where T (n) , T (n)

(n)

, T0(n) , and T0

A , A(n) 0 and A0

(n)

(n)

are, correspondingly, the parametrices for A(n) ,

, as defined in Section 3, and  −1 V (m) := W (m) .

From Corollary 4.8 and Propositions 4.4 and 4.2, it follows that I − ProjKer DN for N ≥ 0 QN DN = I for N < 0

386

S. Klimek, M. McBride

and

DN QN =

I I − ProjKer

DN

∗

for N ≥ 0 for N < 0.

From the construction, the kernel of each T operator is the cokernel of the corresponding A operator, which implies that Ker(Q N ) = Coker(D N ). Finally all that remains is to show that Q N is a bounded, and in fact, (m) are compact a compact operator. Notice that T (n−1) V (n−1) and −V (m) T0 operators (in fact Hilbert-Schmidt operators) with norms that can be estimated as follows: 1 √ (n−1) (n) ||T (n−1) V (n−1) || ≤ C C w0 and similarly ||V (m) T0

(m)

|| ≤

1 √ (m) (m+1) C C . w0

Since C(n) → 0 as n → ∞, it follows from the decomposition that Q N is compact as a uniform limit of compact operators. Thus this completes the proof.

Now we consider the non-commutative cylinder case. The operator D M,N equals the bilateral operator D with domain   − dom(D M,N ) = a ∈ Dom(D) : r+ (a) ∈ Ran P+ N , r− (a) ∈ Ran P M . Theorem 6.2 The operator D M,N above is an unbounded Fredholm operator with index given by ind(D M,N ) = M + N + 1. In fact, there is a bounded operator Q M,N such that the following holds: D M,N Q M,N = I − ProjCoker(DM,N ) , Q M,N D M,N = I − ProjKer(DM,N ) , and additionally Ker(Q M,N ) = Coker(D M,N ). Moreover the parametrix Q M,N is a compact operator. Proof The proof is analogous to the previous proof, however there are more cases to consider. This is due to the way we treated both the disk and the cylinder in complete parallel so far. A different Fourier transform of the Hilbert space could also have been considered leading to an easier index calculation. However that would have made the corresponding decompositions of D different complicated to analyze. and more ∞ n ∗ n Let a = ∞ n=0 U fn (K) + n=1 gn (K)(U ) be in dom(D M,N ). Then we have ∞ ∞   eimϕ fn (±∞) + gn (±∞)e−inϕ . r± (a) = m=0

We need r+ (a) to be in Ran

P+ N

n=1

= span{e

inϕ

n N

}, and for r− (a) to be in Ran P− M =

span {einϕ }, so one is led to consider the following six cases. In each case we list

−Mn

D-bar Operators on Quantum Domains

387

the decomposition of the operator D M,N (in the first line), its adjoint D M,N ∗ (in the second line), and the parametrix Q M,N (in the third line). Case 1 M + N ≥ 0 Case 1(a)

N ≥ 0, M > 0  (m) (m) N (−A W (m) )m=0 , (−A0 W (m) )∞ m=N+1 , M , (W (n−1) A(n−1) )∞ (W (n−1) A(n−1) )n=1 n=M+1 1





N , (−W (m) A1 (m) )∞ (−W (m) A2 (m) )m=0 m=N+1 ,

(A2

(n−1)

M W (n−1) )n=1 , (A0

(n−1)

W (n−1) )∞ n=M+1



 (m) N (m) , (−V (m) T0 )∞ (−V (m) T )m=0 m=N+1 , M (T (n−1) V (n−1) )n=1 , (T1(n−1) V (n−1) )∞ n=M+1

Case 1(b)



N < 0, M > 0  (m) (n−1) (n−1) −N−1 (−A0 W (m) )∞ A0 )n=1 , m=0 , (W M (W (n−1) A(n−1) )n=−N , (W (n−1) A(n−1) )∞ n=M+1 1



 (n−1) W (n−1) )−N−1 (−W (m) A1 (m) )∞ m=0 , (A1 n=1 , (A2

(n−1)

M W (n−1) )n=−N , (A0

(n−1)

W (n−1) )∞ n=M+1

 (m) (n−1) (n−1) −N−1 (−V (m) T0 )∞ V )n=1 , m=0 , (T0 M (T (n−1) V (n−1) )n=−N , (T1(n−1) V (n−1) )∞ n=M+1



In the formulas above there is no second term when N = −1. Case 1(c)

M ≤ 0, N ≥ 0 

(−A1

(m)

(−A0

W (m) )−M−1 m=0 , (−A

(m)

(m)

N W (m) )m=−M ,

(n−1) (n−1) ∞ W (m) )∞ A1 )n=1 m=N+1 , (W





388

S. Klimek, M. McBride



(m) N (−W (m) A0 (m) )−M−1 A2 (m) )m=−M , m=0 , (−W

(−W (m) A1 (m) )∞ m=N+1 , (A0

(n−1)

W (n−1) )∞ n=1

 (m) (m) (m) N (−V (m) T1 )−M−1 T )m=−M , m=0 , (−V (−V (m) T0

(m) ∞ )m=N+1 , (T1(n−1) V (n−1) )∞ n=1





When M = 0 in the above formulas we simply omit the first term. Case 2 M + N < 0 Case 2(a)

N < 0, M ≤ 0  (m) (m) (−A2 W (m) )−M−1 W (m) )∞ m=−M , m=0 , (−A0 (n−1) (n−1) ∞ )−N−1 A1 )n=−N (W (n−1) A(n−1) n=1 , (W 2



 (m) (−W (m) A(m) )−M−1 A1 (m) )∞ m=−M , m=0 , (−W (A

(n−1)

W (n−1) )−N−1 n=1 , (A0

(n−1)

W (n−1) )∞ n=−N

 (m) (m) (m) (−V (m) T2 )−M−1 T0 )∞ m=−M , m=0 , (−V (n−1) (n−1) ∞ (T2(n−1) V (n−1) )−N−1 V )n=−N n=1 , (T1





In the formulas above there is no first term when M = 0. Case 2(b)

N < 0, M > 0  (m) (n−1) (n−1) M A0 )n=1 , (−A0 W (m) )∞ m=0 , (W (n−1) (n−1) ∞ (W (n−1) A(n−1) )−N−1 A1 )n=−N 2 n=M+1 , (W



 (n−1) M W (n−1) )n=1 , (−W (m) A1 (m) )∞ m=0 , (A1 (A

(n−1)

W (n−1) )−N−1 n=M+1 , (A0

(n−1)

W (n−1) )∞ n=−N



D-bar Operators on Quantum Domains

389

 (m) (n−1) (n−1) M V )n=1 , (−V (m) T0 )∞ m=0 , (T0 (n−1) (n−1) ∞ (T2(n−1) V (n−1) )−N−1 V )n=−N n=M+1 , (T1

Case 2(c)



N ≥ 0, M < 0 

(−A1

(m)

(−A0 

N−1 W (m) )m=0 , (−A2

(m)

(m)

W (m) )−M−1 m=N ,

(n−1) (n−1) ∞ W (m) )∞ A1 )n=1 m=−M , (W



N−1 , (−W (m) A(m) )−M−1 (−W (m) A0 (m) )m=0 m=N ,

(−W (m) A1 (m) )∞ m=−M , (A0

(n−1)

W (n−1) )∞ n=1

 (m) N−1 (m) , (−V (m) T2 )−M−1 (−V (m) T1 )m=0 m=N , (−V (m) T0

(m) ∞ )m=−M , (T1(n−1) V (n−1) )∞ n=1





In the formulas above there is again no first term when N = 0. From these representations and from Proposition 4.11, one gets the following M + N + 1 for M + N ≥ 0 dim Ker(D M,N ) = 0 for M + N < 0, and



∗

dim Ker(D M,N ) =

0 for M + N ≥ 0 −(M + N + 1) for M + N < 0.

Thus index calculation follows. Using the analysis done in Section 4, we get the following two relations I − ProjKerDM,N for M + N  0 Q M,N D M,N = I for M + N < 0, and

D M,N Q M,N =

I for M + N  0 I − ProjKerDM,N ∗ for M + N < 0.

The relation Ker(Q M,N ) = Coker(D M,N ) follows from the property of the parametrix of each component of Q M,N .

The proof that Q M,N is compact is the same as in the unilateral case.

390

S. Klimek, M. McBride

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