Finite Gap Jacobi Matrices: A Review
aa r X i v : . [ m a t h . SP ] J a n Proceedings of Symposia in Pure Mathematics
Finite Gap Jacobi Matrices: A Review
Jacob S. Christiansen, Barry Simon, and Maxim Zinchenko
1. Introduction
Perhaps the most common theme in Fritz Gesztesy’s broad opus is the studyof problems with periodic or almost periodic finite gap differential and differenceequations, especially those connected to integrable systems. The present paperreviews recent progress in the understanding of finite gap Jacobi matrices and theirperturbations. We’d like to acknowledge our debt to Fritz as a collaborator andfriend. We hope Fritz enjoys this birthday bouquet!We consider Jacobi matrices, J , on ℓ ( { , , . . . , } ) indexed by { a n , b n } ∞ n =1 , a n > b n ∈ R , where ( u ≡ Ju ) n = a n u n +1 + b n u n + a n − u n − (1.1)or its two-sided analog on ℓ ( Z ) where a n , b n , u n are indexed by n ∈ Z and J is stillgiven by (1.1) (we refer to “Jacobi matrix” for the one-sided objects and “two-sidedJacobi matrix” for the Z analog). Here the a ’s and b ’s parametrize the operator J and { u n } ∈ ℓ .We recall that associated to each bounded Jacobi matrix, J , there is a uniqueprobability measure, µ , of compact support in R characterized by either of theequivalent(a) J is unitarily equivalent to multiplication by x on L ( R , dµ ) by a unitary with( U δ )( x ) ≡ { a n , b n } ∞ n =1 are the recursion parameters for the orthogonal polynomials for µ .We’ll call µ the spectral measure for J .By a finite gap Jacobi matrix, we mean one whose essential spectrum is a finiteunion σ ess ( J ) = e ≡ [ α , β ] ∪ · · · ∪ [ α ℓ +1 , β ℓ +1 ] (1.2)where α < β < · · · < α ℓ +1 < β ℓ +1 (1.3) Mathematics Subject Classification.
Key words and phrases.
Isospectral torus, Orthogonal polynomials, Szeg˝o’s theorem, Szeg˝oasymptotics, Lieb–Thirring bounds.The first author was supported in part by a Steno Research Grant (09-064947) from theDanish Research Council for Nature and Universe.The second author was supported in part by NSF grant DMS-0968856.The third author was supported in part by NSF grant DMS-0965411. c (cid:13) ℓ counts the number of gaps.We will see that for each such e , there is an ℓ -dimensional torus of two-sided J ’s with σ ( J ) = e and J almost periodic and regular in the sense of Stahl–Totik[ ]. We’ll present the theory of perturbations of such J that decay but not tooslowly. Our interest will be in spectral types, Lieb–Thirring bounds on the discreteeigenvalues and on orthogonal polynomial asymptotics. We begin in Section 2 witha discussion of the case ℓ = 0 where we may as well take e = [ − , a n ≡ b n ≡
0. We’ll discuss thetheory in that case as background.Section 3 describes the isospectral torus. Section 4 discusses the results forgeneral finite gap sets with a mention of the special results that occur if each[ α j , β j ] has rational harmonic measure, in which case the isospectral torus containsonly periodic J ’s. Section 5 discusses a method for the general finite gap casewhich relies on the realization of C ∪ {∞} \ e as the quotient of the unit disk in C bya Fuchsian group—a method pioneered by Peherstorfer–Sodin–Yuditskii [
42, 55 ],who were motivated by earlier work of Widom [ ] and Aptekarev [ ].While we focus on the finite gap case, we note there are some results on generalcompact e ’s in R with various restrictive conditions on e (e.g., Parreau–Widom).Peherstorfer–Yuditskii [ ] discuss homogeneous sets and Christiansen [
8, 9 ] provesversions of Theorems 4.3 and 4.5 below for suitable infinite gap e ’s. See [
16, 65 ]for discussion of properties of some e ’s and examples relevant to this area.These works suggest forms of two conditions in the finite gap case suitablefor generalization. Let ρ e be the equilibrium measure for e and G e ( z ) its Green’sfunction ( −E ( ρ e ) − Φ ρ e ( z ) in terms of (3.1)/(3.2)). Then (4.5) should read N X n =1 G e ( x n ) < ∞ (1.4)(which for finite gap e is equivalent to (4.5)). Similarly, (4.6) should read Z log[ f ( x )] dρ e ( x ) > −∞ (1.5)(again, for finite gap e equivalent to (4.6)).J.S.C. and M.Z. would like to thank Caltech for its hospitality where this man-uscript was written.
2. The Zero Gap Case
The Jacobi matrix, J , with a n ≡ b n ≡ J u = λu are given by solving α + α − = λ (2.1)for λ ∈ C and setting u n = 12 i ( α n − α − n ) (2.2)This is polynomially bounded in n if and only if | α | = 1. If α = e ik , then λ = 2 cos k, u n = sin( kn ) (2.3)Thus, σ ( J ) = [ − , , λ ∈ ( − , ⇒ all eigenfunctions bounded (2.4) INITE GAP JACOBI MATRICES: A REVIEW 3 (by all eigenfunctions here, we mean without the boundary condition u = 0).In identifying the spectral type, the following is useful: Theorem . Let J be a Jacobi matrix with a n + a − n + | b n | bounded. Supposeall solutions of ( Ju ) n = λu n ( where u , u are arbitrary ) are bounded for λ ∈ S ⊂ R .Then the spectrum of J on S is purely a.c. in the sense that if µ is the spectralmeasure of J and | · | is Lebesgue measure, then µ s ( S ) = 0 , T ⊂ S and | T | > ⇒ µ ac ( T ) > Remark.
The modern approach to this theorem would use the inequalities ofJitomirskaya–Last [
28, 29 ] or Gilbert–Pearson subordinacy theory [
23, 24, 30, 40 ]to handle µ s and the results of Last–Simon [ ] for the a.c. spectrum. The simplestproof for this special case (where the above ideas are overkill) is perhaps Simon[ ].A simple variation of parameters in the difference equation implies that under ℓ perturbations, eigenfunctions remain bounded when λ ∈ ( − , Theorem . Let J be a Jacobi matrix with ∞ X n =1 | a n − | + | b n | < ∞ (2.6) Then σ ess ( J ) = [ − , and the spectrum on ( − , is purely a.c. Remark.
The continuum analog of Theorem 2.2 goes back to Titchmarsh [ ].Thus, the spectrum outside [ − ,
2] is a set of eigenvalues { x n } Nn =1 where N ∈ N ∪ {∞} . (2.6) has implications for these eigenvalues. Theorem . Let { x n } Nn =1 be the eigenvalues of a Jacobi matrix. Then N X n =1 ( x n − / ≤ ∞ X n =1 | b n | + 4 ∞ X n =1 | a n − | (2.7) Remarks.
1. This implies N X n =1 dist( x n , [ − , / ≤ (cid:18) ∞ X n =1 | b n | + 4 ∞ X n =1 | a n − | (cid:19) (2.8)2. The analog of (2.8) in the continuum case is due to Lieb–Thirring [ ] whoproved it when the power 1 / p > / | b n | p +1 / , | a n − | p +1 / and 1 / p < / p = 1 /
2. This conjecture was provenby Weidl [ ] with an alternate proof and optimal constant by Hundermark–Lieb–Thomas [ ]. (2.8) and its p > / ].3. This theorem is a result of Hundertmark–Simon [ ] who used a methodinspired by [ ].4. (2.7) is optimal in the sense that its p < / γ < b sum nor the a − p > / ]. JACOB S. CHRISTIANSEN, BARRY SIMON, AND MAXIM ZINCHENKO
6. The one-half power in (2.7)/(2.8) is especially significant for the followingreason: x ( z ) = z + z − (2.9)maps D to C ∪ {∞} \ [ − , z ( x ) = (cid:0) x − p x − (cid:1) (2.10)has a square root singularity at x = ±
2. Thus, the finiteness of the left side of(2.7)/(2.8) is equivalent to a Blaschke condition N X n =1 (1 − | z ( x n ) | ) < ∞ (2.11) Theorem . Let J be a Jacobi matrix with σ ess ( J ) = [ − , and Jacobiparameters { a n , b n } ∞ n =1 . Suppose its spectral measure has the form dµ = f ( x ) dx + dµ s (2.12) where dµ s is singular with respect to dx . Suppose that { x n } Nn =1 are its pure pointsoutside [ − , . Consider the three conditions: (a) N X n =1 dist( x n , [ − , / < ∞ (2.13)(b) Z − (4 − x ) − / log[ f ( x )] dx > −∞ (2.14)(c) lim n →∞ a . . . a n exists in (0 , ∞ ) (2.15) Then any two conditions imply the third. Moreover, in that case, (d) ∞ X n =1 ( a n − + b n < ∞ (2.16)(e) lim K →∞ K X n =1 ( a n − and lim K →∞ K X n =1 b n exist (2.17) Remarks.
1. (2.13) is called a critical Lieb–Thirring inequality. (2.14) is theSzeg˝o condition.2. Since f ∈ L , the integral in (2.14) can only diverge to −∞ . That is, theintegral over log + is always finite and (2.14) is equivalent to the integral convergingabsolutely.3. By a result of Ullman [ ], σ ess ( J ) = [ − ,
2] and f ( x ) > x in[ − ,
2] implies lim n →∞ ( a . . . a n ) /n = 1, so (2.15) can be thought of as a secondterm in the asymptotics of n log( a . . . a n ).4. Condition (c) can be thought of as three statements: lim sup < ∞ , lim inf >
0, and lim sup = lim inf. The full strength of (c) is not always needed. For example,(a) plus lim sup > ] (see also [ , Ch. 2]). That (b) ⇒ (c), if there are no eigenvalues, is due toShohat [ ] and that (b) ⇔ (c), if there are finitely many x ’s, is due to Nevai [ ].The general (a) + (b) ⇒ (c) is due to Peherstorfer–Yuditskii [ ] and the essence INITE GAP JACOBI MATRICES: A REVIEW 5 of this theorem is from Killip–Simon [ ], although the precise theorem is fromSimon–Zlatoˇs [ ]. Corollary . If (2.6) holds, then so does (2.14) . Proof. (2.6) implies Q ∞ n =1 a n converges absolutely and, by Theorem 2.3, itimplies (2.13). Thus, (2.14) holds by Theorem 2.4. (cid:3) Remarks.
1. This result was a conjecture of Nevai [ ].2. It was proven by Killip–Simon [ ]. It was the need to complete the proofof this that motivated Hundertmark–Simon [ ].There is a close connection between these conditions and asymptotics of theOPRL: Theorem . Let { p n ( x ) } ∞ n =0 be the orthonormal polynomials for a Jacobimatrix, J , obeying the conditions (a)–(c) of Theorem 2.4. Then uniformly for x incompact subsets of C ∪ {∞} \ [ − , , lim n →∞ p n ( x ) (cid:2) ( x + √ x − (cid:3) n (2.18) exists and is analytic with zeros only at the x n ’s. Remarks.
1. When there are no x n ’s, this is essentially a result of Szeg˝o[
57, 58 ]. For the general case, see Peherstorfer–Yuditskii [ ].2. This is called Szeg˝o asymptotics.3. The reason for the different sign in (2.10) and (2.18) is that, as n → ∞ , p n ( x ) → ∞ , | z ( x ) | < z ( x ) n p n ( x ) is bounded. The other solution of (2.9) is z ( x ) − and it is that solution that appears in the denominator of (2.18).While conditions (a)–(c) of Theorem 2.4 are sufficient for Szeg˝o asymptotics,they are not necessary: Theorem . Let J be a Jacobi matrix whose parameters obey (2.16) and (2.17) . Then (2.18) holds on compact subsets of C ∪ {∞} \ [ − , . Conversely, if (2.18) holds uniformly on the circle | x | = R for some R > , then (2.16) and (2.17) hold. Remarks.
1. This is a result of Damanik–Simon [ ].2. There exist examples where (2.16) and (2.17) hold but both (2.13) and (2.14)fail. Theorem . For a Jacobi matrix, J , with parameters { a n , b n } ∞ n =1 , spectralmeasure obeying (2.12) , and discrete eigenvalues { x n } Nn =1 , one has ∞ X n =1 ( a n − + b n < ∞ (2.19) if and only if (a) σ ess ( J ) = [ − ,
2] (2.20)(b) N X n =1 dist( x n , [ − , / < ∞ (2.21) JACOB S. CHRISTIANSEN, BARRY SIMON, AND MAXIM ZINCHENKO (c) Z − (4 − x ) +1 / log[ f ( x )] dx > −∞ (2.22) Remarks.
1. This theorem is due to Killip–Simon [ ]. They call (a)Blumenthal–Weyl, (b) Lieb–Thirring, and (c) quasi-Szeg˝o.2. The continuous analog of (2.19) ⇒ (2.21) is due to Lieb–Thirring [ ]. Theorem . Let J be a Jacobi matrix with σ ess ( J ) = [ − , and spectralmeasure, dµ , given by (2.12) . Suppose f ( x ) > for a.e. x in [ − , . Then lim n →∞ | a n − | + | b n | = 0 (2.23) Remark.
This is often called the Denisov–Rakhmanov theorem after [
44, 45,15 ]. The result is due to Denisov. Rakhmanov had the analog for OPUC whichimplies the weak version of Theorem 2.9, where σ ess ( J ) = [ − ,
2] is replaced by σ ( J ) = [ − , n →∞ a . . . a n clearly has no implication for the b ’s, but if combined with σ ( J ) = [ − ,
2] implies,by Theorems 2.4 and 2.8, that P ∞ n =1 b n < ∞ . Similarly, one has Theorem . Suppose σ ess ( J ) = [ − , and lim n →∞ ( a . . . a n ) /n = 1 (2.24) Then lim N →∞ N N X n =1 ( a n − + b n = 0 (2.25) Remarks.
1. (2.24) says that the underlying measure is regular in the senseof Ullman–Stahl–Totik; see the discussion in Section 3.2. This theorem is a result of Simon [ ].
3. The Isospectral Torus
Let e be a finite gap set with ℓ gaps and ℓ + 1 components, e j = [ α j , β j ], j = 1 , . . . , ℓ + 1. There is associated to e a natural ℓ -dimensional torus, T e , ofalmost periodic Jacobi matrices. If { a n , b n } ∞ n = −∞ are almost periodic sequences,they are determined by their values for n ≥ T e aseither one- or two-sided Jacobi matrices. There are at least three different ways tothink of T e :(a) As reflectionless two-sided Jacobi matrices, J , with σ ( J ) = e . This is theapproach of [
5, 7, 21, 22, 42, 53, 55, 59 ].(b) As one-sided Jacobi matrices whose m -functions are minimal Herglotz func-tions on the Riemann surface of (cid:2) Q ℓ +1 j =1 ( z − α j )( z − β j ) (cid:3) / . This is the approachof [ ].(c) As two-sided almost periodic J which are regular in the sense of Stahl–Totik[ ] with σ ( J ) = e . This is the approach of [ ]. INITE GAP JACOBI MATRICES: A REVIEW 7
In understanding these notions, some elementary aspects of potential theoryare relevant, so we begin by discussing them. For discussion of potential theoryideas in spectral theory, see Stahl–Totik [ ] or Simon [ ].On our finite gap set, e , there is a unique probability measure, ρ e , called theequilibrium measure which minimizes E ( ρ ) = Z log | x − y | − dρ ( x ) dρ ( y ) (3.1)among all probability measures supported on e . The corresponding equilibriumpotential is Φ ρ e ( x ) = Z log | x − y | − dρ e ( x ) (3.2)The capacity, C ( e ), is defined by C ( e ) = exp( −E ( ρ e )) (3.3)A Jacobi matrix with σ ess ( J ) = e haslim sup( a . . . a n ) /n ≤ C ( e ) (3.4) J is called regular if one has equality in (3.4). We call a two-sided Jacobi matrixregular if each of the (one-sided) Jacobi matrices J + (resp. J − ) with parameters { a n , b n } ∞ n =1 (resp. { a − n , b − n +1 } ∞ n =1 ) (3.5)is regular. ρ e is the density of zeros for any regular J with σ ess ( J ) = e .The ℓ + 1 numbers ρ e ([ α j , β j ]), j = 1 , . . . , ℓ + 1, which sum to 1 are called theharmonic measures of the bands. We also recall that a bounded function, ψ , on Z is called almost periodic if { S k ψ } k ∈ Z , where ( S k ψ ) n = ψ n − k , has compact closurein ℓ ∞ (see the appendix to Section 5.13 in [ ] for more on this class). Such ψ ’sare associated to a continuous function, Ψ, on a torus of finite or countably infinitedimension so that ψ n = Ψ( e πinω , e πinω , . . . ) (3.6)The set of { n + P Kk =1 n k ω k : n , n k ∈ Z , P Kk =1 | n k | < ∞} is called the frequencymodule of ψ when there is no proper submodule (over Z ) that includes all thenonvanishing Bohr–Fourier coefficients. This set for arbitrary { ω k } Kk =1 is called thefrequency module generated by { ω k } Kk =1 .With J ± given by (3.5), we define m ± ( z ) for z ∈ C \ R by m ± ( z ) = h δ , ( J ± − z ) − δ i (3.7)One has for a two-sided Jacobi matrix that h δ , ( J − z ) − δ i = − ( a m + ( z ) − m − ( z ) − ) − (3.8)An important fact is that J ± are determined by m ± , essentially because m ± deter-mine the spectral measures µ ± via their Herglotz representations, m ± ( z ) = Z dµ ± ( x ) x − z (3.9)and µ ± determine the a ’s and b ’s via recursion coefficients for OPRL. Alternatively,the Jacobi parameters can be read off a continued fraction expansion of m ± ( z ) at z = ∞ . JACOB S. CHRISTIANSEN, BARRY SIMON, AND MAXIM ZINCHENKO
It is sometimes useful to let e J − have parameters { a − n − , b − n } ∞ n =1 , in whichcase h δ , ( J − z ) − δ i = − ( z − b + a m + ( z ) + a − ˜ m − ( z )) − (3.10)We can now turn to the descriptions of the isospectral torus. A two-sidedJacobi matrix, J , is called reflectionless on e if for a.e. λ ∈ e and all n ,Re h δ n , ( J − ( λ + i − δ n i = 0 (3.11)( g ( λ + i
0) means lim ε ↓ g ( λ + iε )). It is known that this is equivalent to a m + ( λ + i m − ( λ + i
0) = 1 for a.e. λ ∈ e (3.12) First Definition of the Isospectral Torus.
A two-sided Jacobi matrix, J ,is said to lie in the isospectral torus, T e , for e if σ ( J ) = e and J is reflectionless on e . G ( z ) = h δ , ( J − z ) − δ i is determined by Im log( G ( x + i π/ e , 0 on ( −∞ , α ), and π on ( β ℓ +1 , ∞ ). G is real in each gap and monotone, so G has at most one zeroand that zero determines Im log( G ( x + i G > β j , α j +1 )we’ll say the zero is at β j and if G < β j , α j +1 ) the zero is at α j +1 . Thus,the zeros of G determine G and so Im G ( λ + i
0) on e .By (3.10), G has a zero at λ if and only if m + or ˜ m − has a pole at λ ,and one can show that m + and ˜ m − have no common poles. The residue of thepole is determined by the derivative of G at λ = λ . The reflectionless conditiondetermines Im m + and Im ˜ m − on e , so a , a − , b , m + , ˜ m − , and thus J , are uniquelydetermined by knowing the position of the zero and if they are in the gaps (asopposed to the edges) whether the poles are in m + or ˜ m − . Hence, for each gap, wehave the two copies of ( β j , α j +1 ) glued at the ends, that is, a circle. Thus, giventhat one can show each possibility occurs, T e is a product of ℓ circles, that is, atorus. It is not hard to show that the Jacobi parameters depend continuously onthe positions of the zeros of G and m + / ˜ m − data.We turn to the second approach. Any G as above is purely imaginary on thebands which implies, by the reflection principle, that it can be meromorphicallycontinued to a matching copy of S + ≡ C ∪ {∞} \ e . This suggests meromorphicfunctions on S , two copies of S + glued together along e , will be important. S isprecisely the compactified Riemann surface of p R ( z ), where R ( z ) = ℓ +1 Y j =1 ( z − α j )( z − β j ) (3.13) S is a Riemann surface of genus ℓ . Meromorphic functions on the surface that arenot functions symmetric under interchange of the sheets (i.e., meromorphic on C )have degree at least ℓ + 1.By a minimal meromorphic Herglotz function, we mean a meromorphic functionof degree ℓ + 1 on S that obeys(i) Im f > S + ∩ C + ( C + = { z : Im z > } )(ii) f has a zero at ∞ on S + and a pole at ∞ on S − .Such functions must have their ℓ other poles on R in the gaps on one sheet orthe other and are uniquely, up to a constant, determined by these ℓ poles, one pergap. Each “gap,” when you include the two sheets and branch points at the gapedges, is a circle. So if we normalize by m ( z ) = − z − + O ( z − ) near ∞ on S + , the INITE GAP JACOBI MATRICES: A REVIEW 9 set of such minimal normalized Herglotz functions is an ℓ -dimensional torus. Eachsuch Herglotz function can be written on S + ∩ C + as m ( z ) = Z dµ ( x ) x − z (3.14)where µ is supported on e plus the poles of m in the gaps on S + . µ then determinesa Jacobi matrix. Second Definition of the Isospectral Torus.
The isospectral torus, T e , isthe set of one-sided J ’s whose m -functions are minimal Herglotz functions on thecompact Riemann surface S of √ R given by (3.13).The relation between the two definitions is that the restrictions of the two-sided J ’s to the one-sided are these J given by minimal Herglotz functions. In the otherdirection, each J is almost periodic and so has a unique almost periodic two-sidedextension. Third Definition of the Isospectral Torus.
The isospectral torus is thealmost periodic two-sided J ’s with σ ( J ) = e and which are regular.This is equivalent to the reflectionless definition since regularity implies theLyapunov exponent is zero and then Kotani theory [
33, 48 ] implies J is reflection-less.As noted, the J ’s in the isospectral torus are all almost periodic. Their fre-quency module is generated by the harmonic measures of the bands. In particular,the elements of the isospectral torus are periodic if and only if all harmonic mea-sures are rational. Their spectra are purely a.c. and all solutions of Ju = λu arebounded for any λ ∈ e int .Szeg˝o asymptotics is more complicated than in the ℓ = 0 case. One has forthe OPRL associated to a point in the isospectral torus (thought of as a one-sidedJacobi matrix) that for all z ∈ C \ σ ( J ), p n ( z ) exp( − n Φ ρ e ( z )) (3.15)is asymptotically almost periodic as a function of n with magnitude bounded awayfrom 0 for all n . The frequency module is z -dependent (as written, this is eventrue if ℓ = 0 as can bee seen from the free case): the frequencies come from theharmonic measures of the bands plus one that comes from the conjugate harmonicfunction of Φ ρ e ( z ) in C + (which gives the z -dependence of the frequency module).The limit of (3.15) on e , where Φ ρ e ( x ) = 0, yields the boundedness of solutionsof ( J − λ ) u = 0. There is also a limit at z = ∞ : a . . . a n /C ( e ) n which is almostperiodic.
4. Results in the Finite Gap Case
As we’ve seen, if ˜ J is in the isospectral torus for e and λ ∈ e int , then all solutionsof ˜ Ju = λu are bounded. This remains true under ℓ perturbations by a variationof parameters, so Theorem 2.1 is applicable and we have Theorem . Let e be a finite gap set and ˜ J , with parameters { ˜ a n , ˜ b n } ∞ n =1 ,an element of T e , the isospectral torus for e . Let J be a Jacobi matrix with ∞ X n =1 | a n − ˜ a n | + | b n − ˜ b n | < ∞ (4.1) Then σ ess ( J ) = e and the spectrum on e int is purely a.c. Remark.
We are not aware of this appearing explicitly in the literature, al-though it follows easily from results in [
42, 10 ].As for eigenvalues in R \ e : Theorem . There is a constant C depending only on e so that for any Jacobimatrix, J , obeying (4.1) for some ˜ J ∈ T e , we have, with { x n } Nn =1 the eigenvaluesof J , N X n =1 dist( x n , e ) / ≤ C + C (cid:18) ∞ X n =1 | a n − ˜ a n | + | b n − ˜ b n | (cid:19) (4.2) where C = ℓ X j =1 (cid:12)(cid:12)(cid:12)(cid:12) α j +1 − β j (cid:12)(cid:12)(cid:12)(cid:12) / (4.3) Remarks.
1. This result is essentially in Frank–Simon [ ]. They are onlyexplicit about perturbations of two-sided Jacobi matrices where ˜ J has no eigenval-ues. They mention that one can use interlacing to then get results for the one-sidedcase—this makes that idea explicit.2. Prior to [ ], Frank–Simon–Weidl [ ] proved such a bound on the x n in R \ [ α , β ℓ +1 ] and Hundertmark–Simon [ ] if 1 / . . . ) / isreplaced by p > / | a n − ˜ a n | and | b n − ˜ b n | by p + 1 /
2, thatis, noncritical Lieb–Thirring bounds.
Theorem . Let J be a Jacobi matrix with σ ess ( J ) = e and Jacobi parameters { a n , b n } ∞ n =1 . Suppose its spectral measure has the form dµ = f ( x ) dx + dµ s (4.4) where dµ s is singular with respect to dx . Suppose { x n } Nn =1 are the pure points of dµ outside e . Consider the three conditions: (a) N X n =1 dist( x n , e ) / < ∞ (4.5)(b) Z e dist( x, R \ e ) − / log[ f ( x )] dx > −∞ (4.6)(c) For some constant
R > , R − ≤ a . . . a n C ( e ) n ≤ R (4.7) Then any two imply the third, and if they hold, there exists ˜ J ∈ T e so that lim n →∞ | a n − ˜ a n | + | b n − ˜ b n | = 0 (4.8) Moreover, (d) lim n →∞ a . . . a n ˜ a . . . ˜ a n exists in (0 , ∞ ) (4.9)(e) lim K →∞ K X n =1 ( b n − ˜ b n ) exists in R (4.10) INITE GAP JACOBI MATRICES: A REVIEW 11
Remarks.
1. Depending on which implications one looks at, only part of (c)is needed. For example, if (a) holds,(b) ⇔ lim sup n →∞ a . . . a n C ( e ) n > ], but parts of it were known. While [ ] focus on Szeg˝oasymptotics (see below), the work of Widom [ ] and Aptekarev [ ] implied ifthere are no or finitely many x n ’s, then (b) ⇒ (c), and Peherstorfer–Yuditskii [ ]proved (a) + (b) ⇒ (c) (and as noted to us privately by Peherstorfer, combiningtheir results and an idea of Garnett [ ] yields (4.11)).3. That (e) holds does not seem to have been noted before, although it followseasily from the results in [ ]. For g n ( z ) ≡ p n ( z ) / ˜ p n ( z ) has a limit as n → ∞ on C \ [ α , β ℓ +1 ] and that limit also exists and is analytic and nonzero at infinity (seeTheorem 4.5 below). Since z − n p n ( z ) = ( a . . . a n ) − (cid:18) − (cid:18) n X j =1 b j (cid:19) z − + O ( z − ) (cid:19) (4.12)near z = ∞ ,log( g n ( z )) = − log (cid:18) a . . . a n ˜ a . . . ˜ a n (cid:19) − (cid:20) n X j =1 ( b j − ˜ b j ) (cid:21) z − + O ( z − ) (4.13)so convergence of the analytic functions uniformly near ∞ implies convergence ofthe O ( z − ) term.Theorems 4.2 and 4.3 immediately imply: Corollary . If (4.1) holds, so does (4.6) . Proof.
Since ˜ a . . . ˜ a n /C ( e ) n is almost periodic bounded away from 0 and ∞ ,and P ∞ n =1 | a n − ˜ a n | < ∞ and ˜ a n , ˜ a − n bounded imply P ∞ n =1 | − a n / ˜ a n | < ∞ , wehave (4.9), which implies (4.7). By Theorem 4.2, (4.1) ⇒ (4.5), so Theorem 4.3implies (4.6). (cid:3) Remark.
This is a result of [ ], although [ ] conjectured Theorem 4.2 andnoted it would imply this corollary. Theorem . If the conditions (a)–(c) of Theorem 4.3 hold, then for all z ∈ C ∪ {∞} \ [ α , β ℓ +1 ] , lim n →∞ p n ( z ) / ˜ p n ( z ) exists and the limit is analytic with zerosonly at the x n in R \ [ α , β ℓ +1 ] . Remarks.
1. In this form, this result is from [ ], although earlier it appearedimplicitly in Peherstorfer–Yuditskii [
42, 43 ], and special cases (with stronger as-sumptions on the x n ’s) are in [
64, 4 ]. See also [ ].2. There is also an asymptotic result on e not pointwise but in L ( dµ ) sense;see [ ].3. Asymptotics results for orthogonal polynomials on finite gap sets have beenpioneered by Akhiezer and Tomˇcuk [
1, 2 ].We do not know an analog of the “if and only if” statement of Theorem 2.7,but there is one direction:
Theorem . Let { ˜ a n , ˜ b n } ∞ n =1 be an element of the isospectral torus, T e , of afinite gap set, e . Let { a n , b n } ∞ n =1 be another set of Jacobi parameters and δa n , δb n given by δa n = a n − ˜ a n , δb n = b n − ˜ b n Suppose that (a) ∞ X n =1 | δa n | + | δb n | < ∞ (4.14)(b) For any k ∈ Z ℓ , N X n =1 e πi ( k · ωωω ) n δa n and N X n =1 e πi ( k · ωωω ) n δb n (4.15) have ( finite ) limits as N → ∞ . (c) For every ε > , sup N (cid:26)(cid:12)(cid:12)(cid:12)(cid:12) N X n =1 e πi ( k · ωωω ) n δa n (cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12) N X n =1 e πi ( k · ωωω ) n δb n (cid:12)(cid:12)(cid:12)(cid:12)(cid:27) ≤ C ε exp( ε | k | ) (4.16) Let p n ( z ) ( resp. ˜ p n ( z )) be the orthonormal polynomials for { a n , b n } ∞ n =1 ( resp. { ˜ a n , ˜ b n } ∞ n =1 ) . Then for any z ∈ C \ R , lim n →∞ p n ( z )˜ p n ( z ) (4.17) exists and is finite and nonzero. Remarks.
1. Here ωωω = ( ω , . . . , ω ℓ ) is the ℓ -tuple of harmonic measures (i.e., ω j = ρ e ([ α j , β j ])) and k · ωωω = P ℓj =1 k j ω j . We thus require infinitely many condi-tions.2. This result is from [ ].3. If the torus consists of period p elements (i.e., each ρ e ([ α j , β j ]) is k j /p , wherethere is no common factor for p, k , . . . , k ℓ ), then the infinity of conditions (4.15)reduces to the finitely many conditions that for j = 1 , , . . . , p , P Nn =0 δa np + j and P Nn =0 δb np + j have finite limits and (4.16) becomes automatic.4. [ ] uses this theorem to construct examples where Szeg˝o asymptotics holds,but both (4.5) and (4.6) fail to hold.An analog of Theorem 2.8 is not known for general e but is known in one specialcase. We say e is p -periodic with all gaps open if ℓ = p −
1, and for j = 1 , . . . , p , ρ e ([ α j , β j ]) = 1 /p .We also need a notion of approach to the isospectral torus rather than a singleelement. Given two Jacobi matrices, we define d m ( J, J ′ ) = ∞ X k =0 e −| k | ( | a m + k − a ′ m + k | + | b m + k − b ′ m + k | ) (4.18)and d m ( J, T e ) = inf J ′ ∈T e d m ( J, J ′ ) (4.19) INITE GAP JACOBI MATRICES: A REVIEW 13
Theorem . Let e be p -periodic with all gaps open. Let J be a Jacobi matrixwith spectral measure obeying (4.4) and eigenvalues { x n } Nn =1 outside e . Then ∞ X m =1 d m ( J, T e ) < ∞ (4.20) if and only if (a) σ ess ( J ) = e (4.21)(b) N X n =1 dist( x n , e ) / < ∞ (4.22)(c) Z e dist( x, R \ e ) +1 / log[ f ( x )] dx > −∞ (4.23) Remark.
This theorem is due to Damanik–Killip–Simon [ ]. Their methodis specialized to the periodic case, and in that case, proves some of the earlier resultsof this section, such as Theorem 4.2. Theorem . Suppose J is a Jacobi matrix with σ ess ( J ) = e and so that the f of (4.4) is a.e. strictly positive on e . Then lim m →∞ d m ( J, T e ) = 0 (4.24) Remarks.
1. This is a result of Remling [ ]. For the periodic case, it wasproven earlier by [ ], who conjecture the result for general e .2. Remling replaces (4.24) by the assertion that every right limit of J (i.e.,limit point of { a n + r , b n + r } ∞ n =1 as r → ∞ ) is in T e . By a compactness argument, itis easy to see that this is equivalent to (4.24). Theorem . Let e be a finite gap set and J a Jacobi matrix so that (a) σ ess ( J ) = e (4.25)(b) J is regular, i.e., lim n →∞ ( a . . . a n ) /n = C ( e ) (4.26) Then lim M →∞ M M X m =1 d m ( J, T e ) = 0 (4.27) Remarks.
1. This result was proven in case all harmonic measures are rationalby Simon [ ], who conjectured the result in general. It was proven by Kr¨uger [ ].2. By the Schwarz inequality, (4.27) is equivalent tolim M →∞ M M X m =1 d m ( J, T e ) = 0 (4.28)We close this section on results with a list of some open questions:(1) Do (a)–(c) of Theorem 4.3 imply that ∞ X n =1 ( a n − ˜ a n ) + ( b n − ˜ b n ) < ∞ (4.29)as is true in the case e = [ − , e case? (3) Is there a converse to Theorem 4.6? This would be interesting even in theperiodic case.
5. Methods
The theory of regular Jacobi matrices says one expects the leading growth of P n ( z ) as n → ∞ to be exp( n Φ ρ e ( z )). Φ ρ e is harmonic on C ∪ {∞} \ e so we canlocally define a harmonic conjugate and so e Φ ρ e ( z ) analytic with Re e Φ ρ e = Φ ρ e . Ifyou circle around x , log( z − x ) changes by 2 πi , so circling around the band [ α j , β j ],we expect R log( z − x ) dρ e ( x ) to change by 2 πiρ e ([ α j , β j ]) and exp( − e Φ ρ e ( z )) to havea change of phase by exp( − πiρ e ([ α j , β j ])). Thus, we are led to consider analyticfunctions on C + which we can continue along any curve in C ∪ {∞} \ e .To get a single-valued function, we need to lift to the universal covering spaceof C ∪ {∞} \ e and exp( − e Φ ρ e ( z )) will transform under the homotopy group via acharacter of this group.So long as ℓ = 0, this cover, as a Riemann surface, is the disk, D , and the decktransformations act as a family of fractional linear transformations on the disk,that is, a Fuchsian group. The use of these Fuchsian groups is thus critical to thetheory and used to prove several of the theorems of Section 4 (Theorems 4.7, 4.8,and 4.9 are exceptions).For more on Fuchsian groups, see Beardon [ ], Ford [ ], Katok [ ], Simon[ ], and Tsuji [ ]. The pioneers in this approach were Sodin–Yuditskii [ ]. See[
42, 10, 11, 12, 53 ] for applications of these techniques.
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