Lieb-Thirring Inequalities for Finite and Infinite Gap Jacobi Matrices

We establish Lieb-Thirring power bounds on discrete eigenvalues of Jacobi operators for Schatten class perturbations under very general assumptions. Our results apply, in particular, to perturbations of reflectionless Jacobi operators with finite gap and Cantor-type essential spectrum.


Introduction
Let A be a self-adjoint operator on some Hilbert space H and define where σ d is the discrete and σ ess the essential spectrum. Each term in the sum is repeated according to the multiplicity of the eigenvalue λ. Upper bounds on S p (A) for various choices of A and values of p have shown to be useful in studies of quantum mechanics, differential equations, and dynamical systems. The reader is referred to, e.g., [9] for history and reviews. The original Lieb-Thirring inequalities deal with perturbations of the Laplacian on L 2 (R d ) and assert that where V − = max{0, −V } and L p,d is a constant independent of V . This was proven by Lieb and Thirring in 1976 for p > 1/2 if d = 1 and for p > 0 if d ≥ 2. Their motivation was a rigorous proof of the stability of matter, see [14,15]. When d = 1, the bound in (1.2) fails to hold for p < 1/2 and the endpoint result for p = 1/2 was proven by Weidl [24] some 20 years later.
In this paper we consider selfadjoint Jacobi operators on ℓ 2 (Z) represented by the tridiagonal Jacobi matrices with bounded parameters a n > 0 and b n ∈ R. Our main goal is to obtain Lieb-Thirring inequalities for perturbations of almost periodic Jacobi matrices. In the general setting of almost periodic parameters, the spectrum is typically a Cantor set. We are motivated by the recent developments in spectral theory of Jacobi matrices, see [2,3,6,7], and in particular by the finite gap results of Frank and Simon [8] and also Hundertmark and Simon [11]. Before explaining our new results, let us briefly go through what is already known. The spectral theory for perturbations of the free Jacobi matrix, J 0 , (i.e., the case of a n ≡ 1 and b n ≡ 0) is well-understood and developed in much detail, see [21]. When J = {a n , b n } ∞ n=1 is a compact perturbation of J 0 , Hundertmark and Simon [10] proved that S p (J) ≤ L p, J 0 ∞ n=1 4|a n − 1| p+1/2 + |b n | p+1/2 , p ≥ 1/2, (1.4) with some explicit constants L p, J 0 that are independent of J. As in the continuous case, the inequality is false for p < 1/2. More recently, the p = 1/2 case of (1.4) was extended to finite gap Jacobi matrices in [5,8,11]. In the setting of periodic and almost periodic parameters, the role of J 0 as a natural limiting point is taken over by the so-called isospectral torus, denoted T E . See, e.g., [3,4,22] for a deeper discussion of this object. The finite gap version of (1.4) with p = 1/2 says that if E is a finite gap set (i.e., a finite union of disjoint, compact intervals) and J is a trace class perturbation of an element As before, the constant L 1/2, E is independent of J, J ′ and only depends on the underlying set E. In comparison with previous attempts, the novelty of [8] lies in a clever reduction of the Lieb-Thirring bound for eigenvalues in a single gap to the previously known case of no gaps. However, the method yields little information about the constants that come with each gap. As a result, this approach is hard to generalize to sets with infinitely many gaps.
In the present paper we improve and extend the eigenvalue bounds of [11] to infinite gap Jacobi matrices and obtain Lieb-Thirring bounds for Schatten class perturbations (i.e., non trace class perturbations) of finite and infinite gap matrices. Our new abstract results can be described in the following way. Let J ′ be a two-sided Jacobi matrix with σ(J ′ ) = σ ess (J ′ ) and suppose J = J ′ + δJ is a compact perturbation of J ′ . While compact perturbations do not change the essential spectrum, they usually produce a number of discrete eigenvalues. By a general result of Kato [12] specialized to the present setting, we have the following bound where · 1 denotes the trace norm. In contrast to the Lieb-Thirring bounds, the power on the eigenvalues in (1.6) is the same as on the perturbation. Kato's inequality is optimal for perturbations with large sup norm. On the other hand, the Lieb-Thirring bound with p = 1/2 is optimal for perturbations with small sup norm (cf. [10]). Our first main result (Theorem 3.1) in Section 3 can be thought of as an interpolation between Kato's bound (1.6) and the Lieb-Thirring bound (1.5). More precisely, we show that under certain assumptions on the unperturbed matrix J ′ , a Lieb-Thirring bound of the form holds for any trace class perturbation J. The constant L p, J ′ is independent of δJ and can be specified explicitly. Our second main result (Theorem 3.2) is more general, but has slightly stronger assumptions on J ′ . We show that whenever δJ = J − J ′ belongs to the Schatten class S p+1/2 . As before, the explicit constant L p, J ′ does not depend on δJ. We mention in passing that for trace class perturbations and 1/2 < p < 1, one has both (1.7) and (1.8) since S 1 ⊂ S p+1/2 . The latter bound is slightly better for small perturbations. As for the classical Lieb-Thirring bounds, our proofs of (1.7) and (1.8) rely on a version of the Birman-Schwinger principle and a new estimate for with D ≥ 0 being a diagonal matrix. We establish the latter in Section 2. Using the functional calculus, one can express the positive and negative parts of (J ′ − x) −1 as Cauchy-type integrals. This fact enables us (see Theorem 2.1) to give an upper bound on (1.9) in terms of D 1 and a slight variation of the m-functions for the spectral measures dρ n of (J ′ , δ n ). To estimate further, we impose absolute continuity of dρ n and the reflectionless condition (to be defined in Section 2). If E is a homogeneous set in the sense of Carleson [1] (i.e., there is an ε > 0 so that |(x − δ, x + δ) ∩ E| ≥ δε for all x ∈ E and all δ < diam(E)), then both conditions are fulfilled for every J ′ in the isospectral torus T E . Theorem 2.2 then gives an upper bound that only involves the ordinary m-function, but for all reflectionless measures on E. This result is the key to our Lieb-Thirring bounds.
The second part of the paper focuses on explicit examples of infinite gap sets for which our results apply. This has so far been unexplored territory although the issue is quite natural from an almost periodic point of view. In Section 4, a detailed study of infinite band sets with one accumulation point is followed by a thorough investigation of fat Cantor sets. For both types of structure, which are defined from a sequence {ε k } ∞ k=1 with 0 < ε k < 1, we obtain Lieb-Thirring bounds as in (1.7) The summability condition in question is nearly optimal as it is, in fact, a necessary condition for the Lieb-Thirring bound in the case p = 1/2.
We also provide alternative versions of our bounds where the distance to the essential spectrum is measured by the potential theoretic Green function. Since the infinite gap sets discussed in Section 4 are homogeneous and hence regular for potential theory, the Green function g is the unique continuous function which is positive and harmonic in C E, vanishes on E, and for which g(z) − log |z| is harmonic at ∞. Our alternative Lieb-Thirring bounds hold for J = J ′ + δJ with J ′ from the isospectral torus T E and take the form (1.10) where the constant L p, E is independent of J, J ′ and only depends on p and the underlying set E. In the case of an infinite band set, a sufficient condition for (1.10) is ∞ k=1 ε k < ∞. This, in turn, is shown to be a necessary condition for the alternative bound (1.10) in the case p = 1. For the middle ε-Cantor sets of Section 4.2, a stronger condition seems to be needed and we show that (1.10) is satisfied provided ε k ≤ C/2 k for all large k.

Trace Norm Estimates
In this section we obtain trace norm estimates which will play a crucial role in the proofs of our main results. Theorem 2.1. Suppose D ≥ 0 is a diagonal matrix of trace class and J ′ is a self-adjoint Jacobi matrix. Let E = σ(J ′ ), then where dρ n is the spectral measure of (J ′ , δ n ), that is, the measure from the Herglotz representation of the nth diagonal entry of (J ′ − z) −1 , Proof. Fix x ∈ R E and let E ± = E ∩ (x, ±∞). In addition, let R ± be the positive and negative parts of (J ′ − x) −1 defined by where P E ± (J ′ ) are the spectral projections of J ′ onto the sets E ± . Then and hence D 1/2 R ± D 1/2 ≥ 0. This yields the trace norm estimate, Let Γ ± be non-intersecting rectangular contours around E ± . Using the functional calculus we can express the RHS of (2.3) as a Cauchy-type integral, Multiplying by D 1/2 from the left and from the right and taking the trace then gives Finally, deforming the contours Γ ± into E ± traversed twice in the opposite directions and noting that 1 2πi we obtain Combining (2.9) with (2.5) yields (2.1).
A natural question is how to estimate the integrals in (2.1), but first some notation. Throughout the paper, E ⊂ R will denote a compact set. We let β 0 = inf E and α 0 = sup E. Since [β 0 , α 0 ] E is an open set, it can be written a disjoint union of open intervals, hence (2.10) For convenience we define (α, β) with β < α by With this convention, we shall refer to (α j , β j ), j ≥ 0, as the gaps of E. We also call (α j , β j ), j ≥ 1, the inner gaps and (α 0 , β 0 ) the outer gap of E. For a probability measure dρ supported on E, define the associated Herglotz function by The measure dρ is called reflectionless (on E) if Re[m(x + i0)] = 0 for a.e. x ∈ E. (2.13) When E is essentially closed, we will denote the set of all reflectionless probability measures supported on E by R E . Reflectionless measures appear prominently in spectral theory of finite and infinite gap Jacobi matrices (see, e.g., [3,19,21,22]). In particular, the isospectral torus T E associated to E is the set of all Jacobi matrices J ′ that are reflectionless on E (i.e., the spectral measure of (J ′ , δ n ) belongs to R E for every n ∈ Z) and for which σ(J ′ ) = E. It is well known (see for example [22]) that dρ is a reflectionless probability measure on E if and only if m(z) is of the form For absolutely continuous reflectionless measures we have the following upper bound (2.15) for the integrals that appear on the RHS of our trace norm estimate (2.1). This result is the key to our Lieb-Thirring bounds for perturbations of reflectionless Jacobi matrices in Section 4. Theorem 2.2. Let E ⊂ R be an essentially closed compact set and suppose dρ is a reflectionless absolutely continuous probability measure on E. Denote the gaps of E as in (2.10). Then, for every k ≥ 1, Since dρ is absolutely continuous, we have with m(z) as in (2.14). By the reflectionless assumption, Im[m(t + i0)] = |m(t + i0)| a.e. on E, and hence and define Then Definew(t) andp ± (t) as above, but with γ k replaced byγ k . Then The fact that is a reflectionless probability measures on E + then gives Similarly, noting that Thus, combining (2.27) and (2.29) with (2.25) giveŝ We estimate the integral by considering two cases.
In the next to last inequality we utilized the Cauchy-Schwarz inequality in the form √ a + √ b ≤ 2(a + b). Combining (2.30) with (2.31)-(2.32), and noting that the estimate in (2.32) is larger than the one in (2.31) and that 2 + log 2 < 3, then giveŝ In a similar way, one obtains an upper bound for the integral over E − , The final step is to note that the integral on the LHS of (2.15) and (2.18) can be estimated in two ways, namelŷ Combining these estimates with (2.34) and (2.35), respectively, and choosing the better bound then yield the result.

Abstract Lieb-Thirring Bounds
In this section we obtain Lieb-Thirring bounds for trace class and, more generally, Schatten class perturbations of a wide range of Jacobi matrices. In particular, our results apply to perturbations of periodic and finite gap Jacobi matrices as well as to several infinite gap Jacobi matrices.
Theorem 3.1. Let J and J ′ be two-sided Jacobi matrices such that δJ = J −J ′ is in the trace class, that is, and denote the gaps of E as in (2.10). In addition, suppose there exist non-negative constants {C k } k≥0 such that for some 1/2 < p < 1, and such that the spectral measures dρ n of (J ′ , δ n ) satisfy Then σ ess (J) = E and the discrete eigenvalues of J satisfy the Lieb-Thirring bound, where the constant L p, J ′ is independent of δJ and explicitly given by Proof. Assumption (3.3) implies that the spectral measures dρ n of J ′ cannot have point masses at the endpoints of E (i.e., {α k , β k } k≥0 ). Thus, J ′ has no isolated eigenvalues and hence σ ess (J ′ ) = σ(J ′ ) = E. Weyl's theorem then yields σ ess (J) = E since J is a compact perturbation of J ′ . Let (c) ± = max(±c, 0) and define tridiagonal matrices δJ ± and diagonal matrices D ± by Let N(J ∈ I) denote the number of eigenvalues of J contained in an interval I ⊂ R E. Then by a version of the Birman-Schwinger principle [8, Theorem where the last inequality follows from the fact that D ± ≥ δJ ± ≥ 0. By assumption (3.3) and Theorem 2.1, we get that Then writing the LHS of (3.4) as we can estimate using (3.10) and (3.11) to get As the first integral is bounded bŷ we have Combining this with (3.7) then yields (3.4).
In the next theorem we extend our Lieb-Thirring bounds to non trace class perturbations.
Theorem 3.2. Let J and J ′ be two-sided Jacobi matrices such that δJ = J −J ′ is in the Schatten class S p for some p > 1, that is, (cf. [13, Lemma 2.3]) n∈Z |δa n | p + |δb n | p < ∞. (3.16) Let E = σ(J ′ ) and denote the gaps of E as in (2.10). In addition, suppose there exist non-negative constants {C k } k≥0 such that and such that the spectral measures dρ n of (J ′ , δ n ) satisfy Then σ ess (J) = E and the discrete eigenvalues of J satisfy the Lieb-Thirring bound

19)
where the constant L p, J ′ is independent of δJ and explicitly given by Proof. As in the previous theorem, assumption (3.18) implies that J ′ has no isolated eigenvalues. Since J is a compact perturbation of J ′ , it follows that σ ess (J) = E. Define compact operators δJ ± and D ± as in (3.6)-(3.7). Then δJ = δJ + − δJ − and 0 ≤ δJ ± ≤ D ± . Let N(J ∈ I) denote the number of eigenvalues of J contained in an interval I ⊂ R E. For λ ∈ R E, we denote by N ± λ (J ′ , δJ ± ) the number of eigenvalues of J ′ ± xδJ ± that pass through λ as x runs through the interval (0, 1). By a version of the Birman-Schwinger principle [8, Theorem To handle non trace class perturbations, we estimate further in terms of finite rank truncated versions of D ± . For this, let 0 < r < dist(λ, E) and define the finite rank diagonal matrices D ±,r by (D ±,r ) n,n = ((D ± ) n,n − r) + .

(3.24)
Then D ± − D ±,r ≤ r so the eigenvalues of J ′ + D ±,r + x(D ± − D ±,r ) can move a distance of no more than r as x ranges from 0 to 1. Thus, Estimating the RHS by the trace norm, applying Theorem 2.1, and using the assumption (3.18) then yields Let ℓ 0 = ∞, ℓ k = (β k − α k )/2 for k ≥ 1, and d ± n = (D ± ) n,n for n ∈ Z. Applying (3.23) to an interval [α k + x, β k − x] and using (3.26) with r = x/2 then gives for a.e. x ∈ (0, ℓ k ), k ≥ 0, Write the LHS of (3.19) as an integral and estimate by use of (3.27) to get Rearranging the integral and the sum over n by the monotone convergence theorem and estimating the integrals bŷ Recalling (3.7) and using Jensen's convexity inequality lead to (3.19).
Several remarks pertaining to the previous two theorems are in order. (c) If E is a homogeneous set and J ′ ∈ T E , then the spectral measures dρ n of J ′ are absolutely continuous (cf., e.g., [17,18]) and hence by Theorem 2.2 it is possible to replace while simultaneously changing Thus, a necessary condition for the following Lieb-Thirring bound is bounded in each gap, the conditions for some constants C k > 0 are necessary for (3.34) to hold. Thus, the assumptions (3.3) and (3.18) in our theorems are close to being necessary.

Examples
In this section we obtain Lieb-Thirring bounds for perturbations of Jacobi matrices from the isospectral tori, T E , for two explicit classes of homogeneous infinite gap sets. The isospectral torus associated with a homogeneous set E is known to consist of almost periodic Jacobi matrices, see [3,22]. We also recall that reflectionless measures on homogeneous sets are necessarily absolutely continuous [17,18]. 4.1. Infinite Band Example. In this subsection we consider an explicit example of a compact set E which consists of infinitely many disjoint intervals that accumulate at inf E. Suppose {ε k } ∞ k=1 ⊂ (0, 1) and let where E 0 = [β 0 , α 0 ] and E k is the compact set obtained from E k−1 by removing the middle ε k portion from the first of the k bands in E k−1 . We will denote the gap at level k by (α k , β k ), that is, It is easy to see that E is a homogeneous set if and only if sup k≥1 ε k < 1.

4)
then ∞ k=1 ε k < ∞. Proof. First assume ∞ k=1 ε k < ∞ and let dρ be a reflectionless probability measure on E. Fix k ≥ 1 and define where γ j ∈ [α j , β j ], j ≥ 1, are chosen in such a way that In addition, let b 0 = α 0 − β 0 and be the band and gap lengths at level j. Then it follows from the construction of E j that Letting c = min j≥1 (1 − ε j )(1 − ε j+1 ), we can estimate p ± (x) as follows and similarly, Now suppose x ∈ (α k , β k ). Then since γ k ∈ [α k , β k ] and the estimates (4.10)-(4.11) combined with (4.7) yield where C is a constant that depends only on E. This proves the second and more involved part of (4.3).
To handle the case of x ∈ (α 0 , β 0 ), let p + (x) and p − (x) be defined as in (4.5) but with k = ∞ and k = 0, respectively. Then and just as for the above estimates, we get p For the converse direction assume that (4.4) holds. Let dρ be the reflectionless measure on E that corresponds to γ j = β j for every j ≥ 1 and let p + (t) be defined as in (4.5) with k = ∞. Then p + (x) → p + (β 0 ) as x ր β 0 and since 1 + x ≥ exp(x/2) for x ∈ [0, 2], we have for every p > 1/2. In either case, the constant L p, E is independent of J and J ′ and only depends on p and E.
Proof. Recall that every reflectionless measures on E is absolutely continuous since E is a homogeneous set. By construction of E, Thus, (4.3) combined with (2.15) yields (3.3) and (3.18) for the gap at level k ≥ 1 with a constant C k = C √ ε k log(1/ε k ), (4.19) where C > 0 is sufficiently large and independent of k. Since In addition to Theorem 4.2, we have the following result in which the distance to the essential spectrum is measured by the potential theoretic Green function g instead of the usual distance function. The proof relies on the well known relation between the Green function and the equilibrium measure for E, denoted dµ E , where γ(E) = − log cap(E) is the so-called Robin constant for E.
Theorem 4.3. Let E be the infinite band set constructed in (4.1) and suppose J, J ′ are two-sided Jacobi matrices such that J ′ ∈ T E and J = J ′ + δJ is a compact perturbation of J ′ . If ∞ k=1 ε k < ∞, then for every p > 1, where the constant L p, E is independent of J, J ′ and only depends on p and E.
Proof. Let ∂ = 1 2 ( ∂ ∂x − i ∂ ∂y ), then for any analytic function f (z) we have 2∂ Ref (z) = f ′ (z) by the Cauchy-Riemann equations. Combining this observation with (4.21) yields, For convenience, we define ε 0 = 1/e. Then since the equilibrium measure dµ E is reflectionless on E, it follows from (4.3) that Recalling that the Green function vanishes on E, integration over the gaps then gives As in the proofs of Theorems 3.2 and 4.2, we hence get where C k = C √ ε k log(1/ε k ). Thus, for each k ≥ 0, and since ε (p−1)/2 k log(1/ε k ) is a bounded sequence, summing over k yields (4.22). 3 to also cover the endpoint results p = 1/2, respectively, p = 1. In this regard we point out that ∞ k=1 ε k < ∞ is a necessary condition. Indeed, let J ′ ∈ T E be such that the spectral measure dρ of (J ′ , δ 0 ) has the form (4.6) with γ j = β j for all j ≥ 1, equivalently, and consider the rank one perturbation J = J ′ + δb 0 δ 0 , · δ 0 . Then, as in Assume that δb 0 < 0 and denote by λ 0 the eigenvalue of J below β 0 = inf E. It is known (cf. [23]) that the Green function satisfies for some c > 0 and all x < β 0 sufficiently close to β 0 . Hence, it follows from (4.17) with p = 1/2, respectively, (4.22) with p = 1 that |λ 0 − β 0 | 1/2 ≤ C|δb 0 | for some constant C < ∞ and all δb 0 < 0 sufficiently close to zero. Thus, and hence, ∞ k=1 ε k < ∞ follows from the converse direction of Theorem 4.1. The above considerations also lead to the new insight that there are several homogeneous sets for which the endpoint Lieb-Thirring bounds (i.e., (4.17) with p = 1/2, respectively, (4.22) with p = 1) cannot hold. For example, every infinite band set of the form (4.1) with sup k≥1 ε k < 1 and ∞ k=1 ε k = ∞. (4.32) Moreover, we see that the endpoint results do not even need to hold for homogeneous sets with optimally smooth Green function (i.e., Hölder continuous of order 1/2). Indeed, in our setting a result of Totik [23, Corollary 3.3] implies that g ∈ Lip(1/2) precisely when ∞ k=1 ε 2 k < ∞. So the infinite band set E with ε k = 1/(k + 1) is homogeneous and the Green function for C E is optimally smooth. Yet, the endpoint Lieb-Thirring bounds do not hold for perturbations of some element in T E .
4.2. ε-Cantor Set Example. In this subsection we consider fat Cantor sets (i.e., those of positive Lebesgue measure). Suppose {ε k } ∞ k=1 ⊂ (0, 1) and let be the middle ε-Cantor set, that is, E 0 = [β 0 , α 0 ] and E k is obtained from E k−1 by removing the middle ε k portion from each of the 2 k−1 bands in E k−1 . It is known (cf. [16, p. 125]) that E is a homogeneous set (in particular, E is of positive measure) if and only if ∞ k=1 ε k < ∞. Our first main result is then ∞ k=1 kε k < ∞. Remark 4.6. By symmetry, the condition in (4.35) is equivalent to lim sup Proof. Assume that ∞ k=1 kε k < ∞. Since the first inequality in (4.34) follows directly from Lemma 4.7 below (with i = 0 and m = 0), we merely focus on establishing the estimate for the inner gaps. As for notation, denote by (α j , β j ), j ≥ 0, the gaps of E and let γ j be an arbitrary point in [α j , β j ] for j ≥ 1. Moreover, let and be the band and gap lengths at level k. Fix a gap, say (α j k , β j k ), at level k ≥ 1 (i.e., an interval in E k−1 E k ). We claim that it suffices to show that when x ∈ (α j k , β j k ). For it readily follows that and Suppose that x ∈ (α j k , β j k ) and set B 0 = E 0 . If k > 1, then x belongs to precisely one of the two bands in E 1 . Denote this band by B 1 . Similarly, if k > 2, denote by B 2 the unique band in E 2 ∩ B 1 which contains x. We may continue in this way to obtain a finite sequence of bands each of which contains x. As for further notation, let ( We observe that B i and B i+1 always have precisely one endpoint in common. Our estimation now splits into three parts. We start by estimating the product corresponding to all the gaps of E which are contained in (E i ∩B i−1 ) B i for i = 1, . . . , k − 1. As follows from Lemma 4.8, this infinite product is bounded as long as ∞ k=1 kε k < ∞. Then we estimate the finite product corresponding to the endpoints α 0 , β 0 and the gaps (α j i , β j i ) for i = 1, . . . , k−1. This product is bounded by some constant divided by √ b k , see Lemma 4.9 below. The final step is to estimate the product corresponding to the gaps in B k−1 (α j k , β j k ). But this can be done as in Lemma 4.7 (with i = k and m = 0).
For the converse direction we mimic the proof of Theorem 4.1 and take dρ to be the reflectionless measure on E which corresponds to γ j = β j for all j ≥ 1. It then suffices to show that Convergence of the above product implies that the factors are bounded. Hence, for some constant d > 0 and all j ≥ 1. Our aim is thus to show that for some constant c > 0. This will immediately imply (4.43). For the sake of clarity, we shall refer to the following figure.
The idea is to estimate the terms from all the gaps in D 1 , all the gaps in D 2 , etc., as well as the term from the gap between D 1 and D 2 , the gap between D 2 and D 3 , etc. Start by noting that j: (α j ,β j )⊂Dn for every n ≥ 1. If (α jn , β jn ) denotes the gap between D n and D n+1 , it follows from (4.41) that Hence, we obtain (4.45) with 2c = ∞ i=1 (1 − ε i ) > 0. This completes the proof.
We now formulate and prove the three technical lemmas that are needed in the proof of Theorem 4.5.
Lemma 4.7. Suppose ∞ j=1 jε j < ∞ and consider the infinite products where c = ∞ j=1 (1 − ε j ). Proof. Let us assume that the point x lies to the left of the band A i . Then With the figure below in mind, the idea is for every n ≥ 1 to estimate the term from the gap G n and the terms from all the gaps in F n .
If dist(x, A i ) ≥ mb i and G n = (α n , β n ), we have and j: (α j ,β j )⊂Fn It hence follows that and (4.49) is obtained by interchanging the order of summation.
Lemma 4.8. Suppose ∞ j=1 jε j < ∞ and let A denote the set given by The set A is the union of 2 k−1 − 1 bands in E k−1 (namely, all bands except for B k−1 ) and 2 i−1 − 1 gaps at level i for i = 2, . . . , k − 1. Let F 1 , F 2 , . . . , F 2 k−1 −1 be an ordering of the bands in E k−1 B k−1 so that (4.56) and since we have Here, the term j + 1 − k comes from m = 0 and the inner sum is bounded by 1/2 i for the 2 i terms corresponding to m = 2 i , . . . , 2 i+1 − 1. When i runs from 0 to k − 3, we get the entire sum for m ≥ 1. By Lemma 4.7, it follows that To finish the proof, fix a level i ∈ {2, . . . , k − 1} and order the 2 i−1 − 1 gaps at this level according to their distance to x. The mth gap in this ordering, say G m = (α m , β m ), then satisfies that it follows that The proof of (4.55) is now an immediate consequence of (4.60) and (4.64) for i = 2, . . . , k − 1 .
In order to estimate the product over i in (4.68), we rearrange the factors in the denominator. Letγ j σ(i) be the endpoint of B i which is farthest from x (this happens to be the endpoint of B i which is not an endpoint of B i+1 ). Then x is closer to the other endpoint of B i and we have |x −γ j i | |x −γ j σ(i) | ≤ g i + b i /2 b i /2 ≤ 1 + 4ε i c (4.70) for i = 1, . . . , k − 2. Sinceγ j σ(k−1) is an endpoint of B k−1 , we also have (4.71) Hence, and the result follows from (4.68).
As a direct consequence of Theorems 4.5, 2.2, 3.1, and 3.2, we get the following Lieb-Thirring bounds.
Theorem 4.10. Let E be the middle ε-Cantor set constructed in (4.33) and suppose J, J ′ are two-sided Jacobi matrices such that J ′ ∈ T E and J = J ′ + δJ is a compact perturbation of J ′ . If ε k ≤ C/4 k for some C > 0 and all large k, then λ∈σ(J)\E dist(λ, E) p ≤ L p, E n∈Z |δa n | + |δb n | (4.73) for 1/2 < p < 1. If ε k ≤ C/a k for some a > 4, C > 0, and all large k, then λ∈σ(J)\E dist(λ, E) p ≤ L p, E n∈Z |δa n | p+1/2 + |δb n | p+1/2 (4.74) for all p > 1/2. In either case, the constant L p, E is independent of J and J ′ and only depends on p and E.
Proof. By construction of E, we have (1 − ε j ). Here, C > 0 is sufficiently large and independent of k. Since there are 2 k−1 gaps at level k, each of length g k ≤ 2 1−k ε k (α 0 − β 0 ), the exponential decay assumptions on ε k yield (3.2) and (3.17). The result now follows from Theorems 3.1 and 3.2.
As before, let g denote the potential theoretic Green function for the domain C E with logarithmic pole at infinity. The counterpart of Theorem 4.3 for middle ε-Cantor sets reads Theorem 4.11. Let E be the middle ε-Cantor set constructed in (4.33) and suppose J, J ′ are two-sided Jacobi matrices such that J ′ ∈ T E and J = J ′ + δJ is a compact perturbation of J ′ . If ε k ≤ C/2 k for some C > 0 and all large k, then for every p > 1, λ∈σ(J)\E g(λ) p ≤ L p, E n∈Z |δa n | (p+1)/2 + |δb n | (p+1)/2 , (4.77) where the constant L p, E is independent of J, J ′ and only depends on p and E.