ProceedingsofSymposiainPureMathematics 3 1 Finite Gap Jacobi Matrices: A Review 0 2 Jacob S. Christiansen, Barry Simon, and Maxim Zinchenko n a J 3 1. Introduction 2 Perhaps the most common theme in Fritz Gesztesy’s broad opus is the study ] P of problems with periodic or almost periodic finite gap differential and difference S equations, especially those connected to integrable systems. The present paper . reviewsrecentprogressintheunderstandingoffinitegapJacobimatricesandtheir h t perturbations. We’d like to acknowledge our debt to Fritz as a collaborator and a friend. We hope Fritz enjoys this birthday bouquet! m We consider Jacobi matrices, J, on ℓ2( 1,2,..., ) indexed by a ,b ∞ , [ an >0, bn R, where (u0 0) { } { n n}n=1 ∈ ≡ 2 (Ju) =a u +b u +a u (1.1) v n n n+1 n n n−1 n−1 3 oritstwo-sidedanalogonℓ2(Z)wherea ,b ,u areindexedbyn ZandJ isstill n n n ∈ 7 givenby(1.1)(wereferto“Jacobimatrix”fortheone-sidedobjectsand“two-sided 0 Jacobi matrix” for the Z analog). Here the a’s and b’s parametrize the operator J 5 and u ℓ2. 1. W{enr}e∈call that associated to each bounded Jacobi matrix, J, there is a unique 0 probability measure, µ, of compact support in R characterized by either of the 3 equivalent 1 : (a) J is unitarily equivalentto multiplicationby x onL2(R,dµ) bya unitary with v (Uδ )(x) 1. i 1 ≡ X (b) a ,b ∞ aretherecursionparametersfortheorthogonalpolynomialsforµ. { n n}n=1 r We’ll call µ the spectral measure for J. a ByafinitegapJacobimatrix,wemeanonewhoseessentialspectrumisafinite union σ (J)=e [α ,β ] [α ,β ] (1.2) ess 1 1 ℓ+1 ℓ+1 ≡ ∪···∪ where α <β < <α <β (1.3) 1 1 ℓ+1 ℓ+1 ··· 2010 Mathematics Subject Classification. 47B36, 42C05,58J53,34L15. Key words and phrases. Isospectral torus, Orthogonal polynomials, Szego˝’s theorem, Szego˝ asymptotics,Lieb–Thirringbounds. The first author was supported in part by a Steno Research Grant (09-064947) from the DanishResearchCouncilforNatureandUniverse. ThesecondauthorwassupportedinpartbyNSFgrantDMS-0968856. ThethirdauthorwassupportedinpartbyNSFgrantDMS-0965411. (cid:13)c0000(copyrightholder) 1 2 JACOBS.CHRISTIANSEN,BARRYSIMON,ANDMAXIMZINCHENKO 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 [56]. We’ll present the theory of perturbations of such J that decay but not too slowly. Ourinterestwillbe inspectraltypes,Lieb–Thirringbounds onthe discrete eigenvaluesandonorthogonalpolynomialasymptotics. We begininSection2with a discussion of the case ℓ = 0 where we may as well take e = [ 2,2], in which the − (0-dimensional) torus is the single point with a 1, b 0. We’ll discuss the n n ≡ ≡ theory in that case as background. Section 3 describes the isospectral torus. Section 4 discusses the results for general finite gap sets with a mention of the special results that occur if each [α ,β ]hasrationalharmonicmeasure,inwhichcasetheisospectraltoruscontains j j only periodic J’s. Section 5 discusses a method for the general finite gap case whichreliesontherealizationofC easthequotientoftheunitdiskinCby ∪{∞}\ a Fuchsian group—a method pioneered by Peherstorfer–Sodin–Yuditskii [42, 55], who were motivated by earlier work of Widom [64] and Aptekarev [4]. Whilewefocusonthefinitegapcase,wenotetherearesomeresultsongeneral compact e’s in R with various restrictive conditions on e (e.g., Parreau–Widom). Peherstorfer–Yuditskii[42]discusshomogeneoussetsandChristiansen[8,9]proves versions 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 suitable for generalization. Let ρe be the equilibrium measure for e and Ge(z) its Green’s function (−E(ρe)−Φρe(z) in terms of (3.1)/(3.2)). Then (4.5) should read N Ge(xn)< (1.4) ∞ nX=1 (which for finite gap e is equivalent to (4.5)). Similarly, (4.6) should read log[f(x)]dρe(x)> (1.5) Z −∞ (again, for finite gap e equivalent to (4.6)). J.S.C.andM.Z.wouldliketothankCaltechforitshospitalitywherethis man- uscript was written. 2. The Zero Gap Case The Jacobi matrix, J , with a 1, b 0 is called the free Jacobi matrix. It 0 n n ≡ ≡ is easy to see that the solutions of J u=λu are given by solving 0 α+α−1 =λ (2.1) for λ C and setting ∈ 1 u = (αn α−n) (2.2) n 2i − This is polynomially bounded in n if and only if α =1. If α=eik, then | | λ=2cosk, u =sin(kn) (2.3) n Thus, σ(J )=[ 2,2], λ ( 2,2) all eigenfunctions bounded (2.4) 0 − ∈ − ⇒ FINITE GAP JACOBI MATRICES: A REVIEW 3 (by all eigenfunctions here, we mean without the boundary condition u =0). 0 In identifying the spectral type, the following is useful: Theorem 2.1. Let J be a Jacobi matrix with a +a−1+ b bounded. Suppose allsolutionsof(Ju) =λu (whereu ,u arearbitrnary)anrebo|unn|dedforλ S R. n n 0 1 ∈ ⊂ Then the spectrum of J on S is purely a.c. in the sense that if µ is the spectral measure of J and is Lebesgue measure, then |·| µ (S)=0, T S and T >0 µ (T)>0 (2.5) s ac ⊂ | | ⇒ Remark. The modern approachto this theorem would use the inequalities of Jitomirskaya–Last[28,29]orGilbert–Pearsonsubordinacytheory[23,24,30,40] tohandleµ andtheresultsofLast–Simon[36]forthea.c.spectrum. Thesimplest s proof for this special case (where the above ideas are overkill) is perhaps Simon [49]. A simple variation of parameters in the difference equation implies that under ℓ1 perturbations, eigenfunctions remain bounded when λ ( 2,2), that is, ∈ − Theorem 2.2. Let J be a Jacobi matrix with ∞ a 1 + b < (2.6) n n | − | | | ∞ nX=1 Then σ (J)=[ 2,2] and the spectrum on ( 2,2) is purely a.c. ess − − Remark. ThecontinuumanalogofTheorem2.2goesbacktoTitchmarsh[60]. Thus, the spectrum outside [ 2,2] is a set of eigenvalues x N where N N . (2.6) has implications f−or these eigenvalues. { n}n=1 ∈ ∪{∞} Theorem 2.3. Let x N be the eigenvalues of a Jacobi matrix. Then { n}n=1 N ∞ ∞ (x2 4)1/2 b +4 a 1 (2.7) n− ≤ | n| | n− | nX=1 nX=1 nX=1 Remarks. 1. This implies N ∞ ∞ 1 dist(x ,[ 2,2])1/2 b +4 a 1 (2.8) n n n − ≤ 2(cid:18) | | | − |(cid:19) nX=1 nX=1 nX=1 2. The analog of (2.8) in the continuum case is due to Lieb–Thirring [37] who provedit when the power 1/2 is replaced by p>1/2 and the right side is replaced by b p+1/2, a 1p+1/2 and1/2 by a suitable constant. They provedthe analog n n | | | − | isfalseifp<1/2andconjecturedtheresultifp=1/2. Thisconjecturewasproven by Weidl [63] with an alternate proof andoptimal constant by Hundermark–Lieb– Thomas [25]. (2.8) and its p > 1/2 analogs are called Lieb–Thirring inequalities after [37]. 3. This theorem is a result of Hundertmark–Simon [26] who used a method inspired by [25]. 4. (2.7) is optimal in the sense that its p<1/2 analog is false and one cannot put a constant γ < 1 in front of neither the b sum nor the a 1 sum. The same − also applies to (2.8). 5. (2.7) implies p>1/2 analogs by an argument of Aizenman–Lieb [3]. 4 JACOBS.CHRISTIANSEN,BARRYSIMON,ANDMAXIMZINCHENKO 6. The one-half power in (2.7)/(2.8) is especially significant for the following reason: x(z)=z+z−1 (2.9) maps D to C [ 2,2]. Its inverse ∪{∞}\ − z(x)= 1 x x2 4 (2.10) 2 − − (cid:0) p (cid:1) 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 (1 z(x ))< (2.11) n −| | ∞ nX=1 Theorem 2.4. Let J be a Jacobi matrix with σ (J) = [ 2,2] and Jacobi ess − parameters a ,b ∞ . Suppose its spectral measure has the form { n n}n=1 dµ=f(x)dx+dµ (2.12) s where dµ is singular with respect to dx. Suppose that x N are its pure points s { n}n=1 outside [ 2,2]. Consider the three conditions: − N (a) dist(x ,[ 2,2])1/2 < (2.13) n − ∞ nX=1 2 (b) (4 x2)−1/2log[f(x)]dx> (2.14) Z − −∞ −2 (c) lim a ...a exists in (0, ) (2.15) 1 n n→∞ ∞ Then any two conditions imply the third. Moreover, in that case, ∞ (d) (a 1)2+b2 < (2.16) n− n ∞ nX=1 K K (e) lim (a 1) and lim b exist (2.17) n n K→∞nX=1 − K→∞nX=1 Remarks. 1. (2.13) is called a critical Lieb–Thirring inequality. (2.14) is the Szego˝ condition. 2. Since f L1, the integral in (2.14) can only diverge to . That is, the ∈ −∞ integraloverlog isalwaysfiniteand(2.14)isequivalenttotheintegralconverging + absolutely. 3. By a result of Ullman [62], σ (J) = [ 2,2] and f(x) > 0 for a.e. x in ess − [ 2,2] implies lim (a ...a )1/n = 1, so (2.15) can be thought of as a second n→∞ 1 n − term in the asymptotics of 1 log(a ...a ). n 1 n 4. Condition(c)canbethoughtofasthree statements: limsup< , liminf > ∞ 0,andlimsup=liminf. Thefullstrengthof(c)isnotalwaysneeded. Forexample, (a) plus limsup>0 implies (b) and the rest of (c). 5. This result can be thought of as an analog of a theorem of Szego˝ for OPUC [57] (see also [50, Ch. 2]). That (b) (c), if there are no eigenvalues, is due to ⇒ Shohat[47] and that (b) (c), if there are finitely many x’s, is due to Nevai [38]. ⇔ The general (a) + (b) (c) is due to Peherstorfer–Yuditskii [41] and the essence ⇒ FINITE GAP JACOBI MATRICES: A REVIEW 5 of this theorem is from Killip–Simon [32], although the precise theorem is from Simon–Zlatoˇs [54]. Corollary 2.5. If (2.6) holds, then so does (2.14). Proof. (2.6) implies ∞ a converges absolutely and, by Theorem 2.3, it n=1 n implies (2.13). Thus, (2.14)Qholds by Theorem 2.4. (cid:3) Remarks. 1. This result was a conjecture of Nevai [39]. 2. It was proven by Killip–Simon [32]. It was the need to complete the proof of this that motivated Hundertmark–Simon [26]. There is a close connection between these conditions and asymptotics of the OPRL: Theorem 2.6. Let p (x) ∞ be the orthonormal polynomials for a Jacobi { n }n=0 matrix, J, obeying the conditions (a)–(c) of Theorem 2.4. Then uniformly for x in compact subsets of C [ 2,2], ∪{∞}\ − p (x) n lim (2.18) n→∞ 1(x+√x2 4) n 2 − (cid:2) (cid:3) exists and is analytic with zeros only at the x ’s. n Remarks. 1. When there are no x ’s, this is essentially a result of Szego˝ n [57, 58]. For the general case, see Peherstorfer–Yuditskii[41]. 2. This is called Szego˝ asymptotics. 3. The reason for the different sign in (2.10) and (2.18) is that, as n , → ∞ p (x) , z(x) < 1 so z(x)np (x) is bounded. The other solution of (2.9) is n n → ∞ | | z(x)−1 and it is that solution that appears in the denominator of (2.18). While conditions (a)–(c) of Theorem 2.4 are sufficient for Szego˝ asymptotics, they are not necessary: Theorem 2.7. Let J be a Jacobi matrix whose parameters obey (2.16) and (2.17). Then (2.18) holds on compact subsets of C [ 2,2]. Conversely, if ∪{∞}\ − (2.18) holds uniformly on the circle x =R for some R>2, then (2.16) and (2.17) | | hold. Remarks. 1. This is a result of Damanik–Simon [14]. 2. Thereexistexampleswhere(2.16)and(2.17)holdbutboth(2.13)and(2.14) fail. Theorem 2.8. For a Jacobi matrix, J, with parameters a ,b ∞ , spectral { n n}n=1 measure obeying (2.12), and discrete eigenvalues x N , one has { n}n=1 ∞ (a 1)2+b2 < (2.19) n− n ∞ nX=1 if and only if (a) σ (J)=[ 2,2] (2.20) ess − N (b) dist(x ,[ 2,2])3/2 < (2.21) n − ∞ nX=1 6 JACOBS.CHRISTIANSEN,BARRYSIMON,ANDMAXIMZINCHENKO 2 (c) (4 x2)+1/2log[f(x)]dx> (2.22) Z − −∞ −2 Remarks. 1. This theorem is due to Killip–Simon [32]. 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 [37]. ⇒ Theorem 2.9. Let J be a Jacobi matrix with σ (J) = [ 2,2] and spectral ess − measure, dµ, given by (2.12). Suppose f(x)>0 for a.e. x in [ 2,2]. Then − lim a 1 + b =0 (2.23) n n n→∞| − | | | Remark. This is oftencalledthe Denisov–Rakhmanovtheoremafter [44, 45, 15]. The result is due to Denisov. Rakhmanov had the analog for OPUC which implies the weak version of Theorem 2.9, where σ (J) = [ 2,2] is replaced by ess − σ(J) = [ 2,2]. That the result as stated was true was a long-standing conjecture − settled by Denisov. Conditions on the spectrum combined with weak conditions on the Jacobi pa- rametershavestrongconsequences. Forexample,the existenceoflim a ...a n→∞ 1 n clearly has no implication for the b’s, but if combined with σ(J) =[ 2,2] implies, by Theorems 2.4 and 2.8, that ∞ b2 < . Similarly, one has − n=1 n ∞ P Theorem 2.10. Suppose σ (J)=[ 2,2] and ess − lim (a ...a )1/n =1 (2.24) 1 n n→∞ Then N 1 lim (a 1)2+b2 =0 (2.25) N→∞N nX=1 n− n Remarks. 1. (2.24) says that the underlying measure is regular in the sense of Ullman–Stahl–Totik; see the discussion in Section 3. 2. This theorem is a result of Simon [52]. 3. The Isospectral Torus Let e be a finite gap set with ℓ gaps and ℓ + 1 components, e = [α ,β ], j j j j = 1,...,ℓ +1. There is associated to e a natural ℓ-dimensional torus, e, of T almost periodic Jacobi matrices. If a ,b ∞ are almost periodic sequences, { n n}n=−∞ they are determined by their values for n 1 so we can view the elements of e as ≥ T either one- or two-sidedJacobi matrices. There are at least three different ways to think of e: (a) As rTeflectionless two-sided Jacobi matrices, J, with σ(J) = e. This is the approachof [5, 7, 21, 22, 42, 53, 55, 59]. (b) As one-sided Jacobi matrices whose m-functions are minimal Herglotz func- tionsontheRiemannsurfaceof ℓ+1(z α )(z β ) 1/2. Thisistheapproach j=1 − j − j of [10]. (cid:2)Q (cid:3) (c) As two-sided almost periodic J which are regular in the sense of Stahl–Totik [56] with σ(J)=e. This is the approach of [35]. FINITE GAP JACOBI MATRICES: A REVIEW 7 In understanding these notions, some elementary aspects of potential theory are relevant, so we begin by discussing them. For discussion of potential theory ideas in spectral theory, see Stahl–Totik [56] or Simon [51]. On our finite gap set, e, there is a unique probability measure, ρe, called the equilibrium measure which minimizes (ρ)= logx y −1dρ(x)dρ(y) (3.1) E Z | − | among all probability measures supported on e. The corresponding equilibrium potential is Φρe(x)=Z log|x−y|−1dρe(x) (3.2) The capacity, C(e), is defined by C(e)=exp( (ρe)) (3.3) −E A Jacobi matrix with σ (J)=e has ess limsup(a ...a )1/n C(e) (3.4) 1 n ≤ J is called regular if one has equality in (3.4). We call a two-sided Jacobi matrix regular if each of the (one-sided) Jacobi matrices J (resp. J ) with parameters a ,b ∞ (resp. a ,b ∞ ) (3.5) + − { n n}n=1 { −n −n+1}n=1 is regular. ρe is the density of zeros for any regular J with σess(J)=e. The ℓ+1 numbers ρe([αj,βj]), j =1,...,ℓ+1, which sum to 1 are called the harmonic measures of the bands. We also recall that a bounded function, ψ, on Z is called almost periodic if Skψ k∈Z, where (Skψ)n =ψn−k, has compact closure { } in ℓ∞ (see the appendix to Section 5.13 in [53] for more on this class). Such ψ’s areassociatedto acontinuousfunction, Ψ,onatorusoffinite orcountablyinfinite dimension so that ψ =Ψ(e2πinω1,e2πinω2,...) (3.6) n The set of n + K n ω : n ,n Z, K n < is called the frequency module of {ψ 0whePn kt=h1erekiskno0prokp∈er subPmko=d1u|lek(|ove∞r Z}) that includes all the nonvanishingBohr–Fouriercoefficients. Thissetforarbitrary ω K iscalledthe { k}k=1 frequency module generated by ω K . With J given by (3.5), we{defik}nke=m1 (z) for z C R by ± ± ∈ \ m (z)= δ ,(J z)−1δ (3.7) ± 1 ± 1 h − i One has for a two-sided Jacobi matrix that δ ,(J z)−1δ = (a2m (z) m (z)−1)−1 (3.8) h 0 − 0i − 0 + − − AnimportantfactisthatJ aredeterminedbym ,essentiallybecausem deter- ± ± ± mine the spectral measures µ via their Herglotz representations, ± dµ (x) ± m (z)= (3.9) ± Z x z − andµ determinethea’sandb’sviarecursioncoefficientsforOPRL.Alternatively, ± the Jacobi parameters can be read off a continued fraction expansion of m (z) at ± z = . ∞ 8 JACOBS.CHRISTIANSEN,BARRYSIMON,ANDMAXIMZINCHENKO It is sometimes useful to let J have parameters a ,b ∞ , in which − { −n−1 −n}n=1 case e δ ,(J z)−1δ = (z b +a2m (z)+a2 m˜ (z))−1 (3.10) h 0 − 0i − − 0 0 + −1 − We can now turn to the descriptions of the isospectral torus. A two-sided Jacobi matrix, J, is called reflectionless on e if for a.e. λ e and all n, ∈ Re δ ,(J (λ+i0))−1δ =0 (3.11) n n h − i (g(λ+i0) means lim g(λ+iε)). It is known that this is equivalent to ε↓0 a2m (λ+i0)m (λ+i0)=1 for a.e. λ e (3.12) 0 + − ∈ First Definition of the Isospectral Torus. A two-sided Jacobi matrix, J, is said to lie in the isospectral torus, e, for e if σ(J)=e and J is reflectionless on e. T G (z) = δ ,(J z)−1δ is determined by Imlog(G (x+i0)) via an expo- 00 0 0 00 nential Herglohtz repre−sentationi. This argument is π/2 on e, 0 on ( ,α ), and π 1 −∞ on (β , ). G is real in each gap and monotone, so G has at most one zero ℓ+1 00 00 ∞ andthatzerodetermines Imlog(G (x+i0))onthat gap. If G >0 on(β ,α ) 00 00 j j+1 we’ll say the zero is at β and if G < 0 on (β ,α ) the zero is at α . Thus, j 00 j j+1 j+1 the zeros of G determine G and so ImG (λ+i0) on e. 00 00 00 By (3.10), G has a zero at λ if and only if m or m˜ has a pole at λ , 00 0 + − 0 and one can show that m and m˜ have no common poles. The residue of the + − pole is determined by the derivative of G at λ=λ . The reflectionless condition 00 0 determinesImm andImm˜ one,soa ,a ,b ,m ,m˜ ,andthusJ,areuniquely + − 0 −1 0 + − determined by knowing the position of the zero and if they are in the gaps (as opposedtothe edges)whetherthe polesareinm orm˜ . Hence,foreachgap,we + − have the two copies of (β ,α ) glued at the ends, that is, a circle. Thus, given j j+1 that one can show each possibility occurs, e is a product of ℓ circles, that is, a T torus. It is not hard to show that the Jacobi parameters depend continuously on the positions of the zeros of G and m /m˜ data. 00 + − We turnto the secondapproach. Any G as aboveis purely imaginaryonthe 00 bands which implies, by the reflection principle, that it can be meromorphically continued to a matching copy of C e. This suggests meromorphic + functions on , two copies of Sglue≡d tog∪e{th∞er}a\long e, will be important. is + S S S precisely the compactified Riemann surface of R(z), where p ℓ+1 R(z)= (z α )(z β ) (3.13) j j − − jY=1 is a Riemann surface of genus ℓ. Meromorphic functions on the surface that are Snot functions symmetric under interchange of the sheets (i.e., meromorphic on C) have degree at least ℓ+1. ByaminimalmeromorphicHerglotzfunction,wemeanameromorphicfunction of degree ℓ+1 on that obeys (i) Imf >0 on S C (C = z: Imz >0 ) + + + S ∩ { } (ii) f has a zero at on and a pole at on . + − ∞ S ∞ S Such functions must have their ℓ other poles on R in the gaps on one sheet or the other and are uniquely, up to a constant, determined by these ℓ poles, one per gap. Each “gap,” when you include the two sheets and branch points at the gap edges,is a circle. So if we normalize by m(z)= z−1+O(z−2) near on , the + − ∞ S FINITE GAP JACOBI MATRICES: A REVIEW 9 set of such minimal normalized Herglotz functions is an ℓ-dimensional torus. Each such Herglotz function can be written on C as + + S ∩ dµ(x) m(z)= (3.14) Z x z − whereµissupportedoneplusthepolesofminthegapson . µthendetermines + S a Jacobi matrix. Second Definition of the Isospectral Torus. The isospectral torus, e, is T the set of one-sided J’s whose m-functions are minimal Herglotz functions on the compact Riemann surface of √R given by (3.13). S Therelationbetweenthetwodefinitionsisthattherestrictionsofthetwo-sided J’s to the one-sided are these J given by minimal Herglotz functions. In the other direction, each J is almost periodic and so has a unique almost periodic two-sided extension. Third Definition of the Isospectral Torus. The isospectral torus is the almost periodic two-sided J’s with σ(J)=e and which are regular. This is equivalent to the reflectionless definition since regularity implies the Lyapunovexponent is zeroandthen Kotanitheory [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 generatedby the harmonic measuresof 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 are bounded for any λ eint. ∈ Szego˝ asymptotics is more complicated than in the ℓ = 0 case. One has for the OPRL associated to a point in the isospectral torus (thought of as a one-sided Jacobi matrix) that for all z C σ(J), ∈ \ p (z)exp( nΦ (z)) (3.15) n − ρe is asymptoticallyalmostperiodic as a function ofn withmagnitude bounded away from 0 for all n. The frequency module is z-dependent (as written, this is even true if ℓ = 0 as can bee seen from the free case): the frequencies come from the harmonic measures of the bands plus one that comes from the conjugate harmonic function of Φ (z) in C (which gives the z-dependence of the frequency module). ρe + The limit of (3.15) on e, where Φ (x) = 0, yields the boundedness of solutions ρe of (J λ)u = 0. There is also a limit at z = : a ...a /C(e)n which is almost 1 n − ∞ periodic. 4. Results in the Finite Gap Case Aswe’veseen,ifJ˜isintheisospectraltorusforeandλ eint,thenallsolutions of J˜u=λu are bounded. This remains true under ℓ1 pertu∈rbations by a variation of parameters, so Theorem 2.1 is applicable and we have Theorem 4.1. Let e be a finite gap set and J˜, with parameters a˜ ,˜b ∞ , an element of e, the isospectral torus for e. Let J be a Jacobi matrix{winth n}n=1 T ∞ a a˜ + b ˜b < (4.1) n n n n | − | | − | ∞ nX=1 10 JACOBS.CHRISTIANSEN,BARRYSIMON,ANDMAXIMZINCHENKO Then σ (J)=e and the spectrum on eint is purely a.c. ess 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: \ Theorem4.2. ThereisaconstantC dependingonlyonesothatforanyJacobi matrix, J, obeying (4.1) for some J˜∈ Te, we have, with {xn}Nn=1 the eigenvalues of J, N ∞ dist(x ,e)1/2 C +C a a˜ + b ˜b (4.2) n 0 n n n n ≤ (cid:18) | − | | − |(cid:19) nX=1 nX=1 where ℓ α β 1/2 j+1 j C = − (4.3) 0 (cid:12) 2 (cid:12) Xj=1 (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) Remarks. 1. This result is essentially in Frank–Simon [18]. They are only explicit about perturbations of two-sidedJacobimatrices where J˜has no eigenval- ues. Theymentionthatonecanuseinterlacingtothengetresultsfortheone-sided case—this makes that idea explicit. 2. Prior to [18], Frank–Simon–Weidl [19] proved such a bound on the x in n R [α ,β ] and Hundertmark–Simon [27] if 1/2 in the power of dist(...)1/2 is 1 ℓ+1 \ replaced by p>1/2 and 1 in the power of a a˜ and b ˜b by p+1/2, that n n n n | − | | − | is, noncritical Lieb–Thirring bounds. Theorem4.3. LetJ beaJacobimatrixwithσ (J)=eandJacobiparameters ess a ,b ∞ . Suppose its spectral measure has the form { n n}n=1 dµ=f(x)dx+dµ (4.4) s where dµ is singular with respect to dx. Suppose x N are the pure points of dµ outsidse e. Consider the three conditions: { n}n=1 N (a) dist(x ,e)1/2 < (4.5) n ∞ nX=1 (b) dist(x,R e)−1/2log[f(x)]dx> (4.6) Ze \ −∞ a ...a (c) For some constant R>1, R−1 1 n R (4.7) ≤ C(e)n ≤ Then any two imply the third, and if they hold, there exists J˜ e so that ∈T lim a a˜ + b ˜b =0 (4.8) n n n n n→∞| − | | − | Moreover, a ...a (d) lim 1 n exists in (0, ) (4.9) n→∞a˜1...a˜n ∞ K (e) lim (b ˜b ) exists in R (4.10) n n K→∞nX=1 −