LIEB–THIRRING INEQUALITIES FOR COMPLEX FINITE GAP JACOBI MATRICES JACOB S. CHRISTIANSEN AND MAXIM ZINCHENKO Abstract. We establish Lieb–Thirring power bounds on discrete eigenvalues of Jacobi operators for Schatten class complex perturbations of periodic and more generally finite gap almost periodic Jacobi matrices.

1. Introduction In this paper we consider bounded non-selfadjoint Jacobi operators on `2 (Z) represented by tridiagonal matrices   .. .. .. . . .   a0 b 1 c 1     a1 b 2 c 2 (1.1) J =    a2 b 3 c 3   ... ... ... with bounded complex parameters {an , bn , cn }n∈Z . Our goal is to obtain Lieb– Thirring inequalities for complex perturbations of periodic and, more generally, almost periodic Jacobi operators with absolutely continuous finite gap spectrum. Lieb–Thirring inequalities for selfadjoint and complex perturbations of the discrete Laplacian have been studied extensively in the last decade [1, 7, 15, 17, 18, 22]. The original work of Lieb and Thirring [25, 26] was done in the context of continuous Schr¨odinger operators, motivated by their study of the stability of matter. We refer to [6, 9, 10, 12, 13, 21, 31] for more recent developments on Lieb–Thirring inequalities for Schr¨odinger operators and to [20] for a review and history of the subject. Date: March 16, 2017. 2010 Mathematics Subject Classification. 34L15, 47B36. Key words and phrases. Finite gap Jacobi matrices, Complex perturbations, Eigenvalues estimates. JSC is supported in part by the Research Project Grant DFF–4181-00502 from the Danish Council for Independent Research. MZ is supported in part by Simons Foundation Grant CGM-281971. 1

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J. S. CHRISTIANSEN AND M. ZINCHENKO

Much less is known for perturbations (especially complex ones) of operators with gapped spectrum. Lieb–Thirring inequalities for selfadjoint perturbations of periodic and almost periodic Jacobi operators with absolutely continuous finite gap spectrum have been established only recently [4, 5, 11, 23]. Analogs of these finite gap Lieb–Thirring inequalities for complex perturbations are not known. The aim of the present work is to fill this gap. What is currently known in the case of complex perturbations is the closely related class of Kato inequalities [16, 19]. Such inequalities have larger exponents on the eigenvalue side when compared to Lieb–Thirring inequalities (cf. (1.3) vs. (1.2) and (1.6) vs. (1.7)) and hence are not optimal for small perturbations of Jacobi operators. To put our new results in perspective, we first discuss the best currently known results on eigenvalue power bounds for Jacobi operators in more detail. The spectral theory for perturbations of the free Jacobi matrix, J0 , (i.e., the case of an = cn ≡ 1 and bn ≡ 0) is well-understood, see [29]. Let E = σ(J0 ) = [−2, 2] and suppose J is a selfadjoint Jacobi matrix (i.e., an = cn > 0) such that δJ = J − J0 is a compact operator, that is, J is a compact selfadjoint perturbation of J0 . Hundertmark and Simon [22] proved the following Lieb– Thirring inequalities, X

∞ X
p− 21 dist λ, E |δan |p + |δbn |p , ≤ Lp, E

p ≥ 1,

(1.2)

n=−∞

λ∈σd (J)

with some explicit constants Lp, E independent of J. Here, σd (J) is the discrete spectrum of J. It was also shown in [22] that the inequality is false for p < 1. More recently, (1.2) was extended to selfadjoint perturbations of periodic and almost periodic Jacobi matrices with absolutely continuous finite gap spectrum [23, 5, 11, 4]. When E is a finite gap set (i.e., a finite union of disjoint, compact intervals), the role of J0 as a natural background operator is taken over by the so-called isospectral torus, denoted TE . See, e.g., [3, 2, 30, 27, 29] for a deeper discussion of this object. For J 0 ∈ TE and a compact selfadjoint perturbation J = J 0 + δJ, Frank and Simon [11] proved (1.2) for p = 1 while the case of p > 1 is established in [4]. The constant Lp, E is now independent of J and J 0 and only depends on p and the underlying set E. As alluded to above, there is a general result of Kato [24] which applies to compact selfadjoint perturbations of arbitrary bounded selfadjoint operators. Specialized to the case of perturbations of Jacobi matrices from TE , it states that X λ∈σd (J)

dist λ, E


p

≤

kδJkpp

≤

∞ X n=−∞

(4|δan | + |δbn |)p ,

p ≥ 1,

(1.3)

LIEB–THIRRING INEQUALITIES FOR COMPLEX JACOBI MATRICES

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where k · kp denotes the Schatten norm. In contrast to the Lieb–Thirring bounds, the power on the eigenvalues in (1.3) is the same as on the perturbation and so is larger than the power on the eigenvalues in (1.2) by 1/2. Kato’s inequality is optimal for perturbations with large sup norm. On the other hand, the Lieb–Thirring bound with p = 1 is optimal for perturbations with small sup norm (cf. [22]). A fact that seemingly went unnoticed is that one can combine (1.2) and (1.3) into one ultimate inequality which is optimal for both large and small perturbations (at least when p = 1). This inequality takes the form ∞ X X
p− 12 1 2 dist λ, E (1 + |λ|) ≤ Cp, E |δan |p + |δbn |p , p ≥ 1, (1.4) n=−∞

λ∈σd (J)

where the constant Cp, E is independent of J and J 0 , J = J 0 +δJ, δJ is compact, J 0 ∈ TE , and E is a finite gap set. In recent years, several results have also been established for non-selfadjoint perturbations of selfadjoint Jacobi matrices [1, 17, 18, 16, 19]. For compact non-selfadjoint perturbations J = J0 + δJ of the free Jacobi matrix J0 , a near generalization (with an extra ε) of the Lieb–Thirring bound (1.2) was obtained by Hansmann and Katriel [18] using the complex analytic approach developed in [1]. Their non-selfadjoint version of the Lieb–Thirring inequalities takes the following form: For every 0 < ε < 1,
∞ X dist z, [−2, 2] p+ε X |δan |p + |δbn |p + |δcn |p , p ≥ 1, (1.5) ≤ Lp, ε 1 2 − 4| 2 |z n=−∞ z∈σd (J) where the eigenvalues are repeated according to their algebraic multiplicity and the constant Lp, ε is independent of J. Whether or not this inequality continues to hold for ε = 0 is an open problem. For non-selfadjoint perturbations of Jacobi matrices from finite gap isospectral tori TE , an eigenvalue power bound was first obtained by Golinskii and Kupin in [16]. Shortly thereafter, this bound was superseded by a generalization of Kato’s inequality to non-selfadjoint perturbations of arbitrary bounded selfadjoint operators (see Hansmann [19]). In the special case of a compact non-selfadjoint perturbation J = J 0 + δJ of J 0 ∈ TE , Hansmann’s result reads X z∈σd (J)

dist z, E


p

≤ Kp

∞ X

|δan |p + |δbn |p + |δcn |p ,

p > 1,

(1.6)

n=−∞

where the eigenvalues are repeated according to their algebraic multiplicity and Kp is a universal constant that depends only on p. The purpose of the present article is to generalize the Lieb–Thirring bound (1.4) to the case of compact non-selfadjoint perturbations J = J 0 + δJ of

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J. S. CHRISTIANSEN AND M. ZINCHENKO

Jacobi matrices J 0 from finite gap isospectral tori TE . Let ∂E denote the set of endpoints of the intervals that form E. Then our main result can be formulated as follows: For every p ≥ 1 and any ε > 0,
∞ X dist z, E p+ε (1 + |z|) 1−3ε X 2 ≤ Lε, p, E |δan |p + |δbn |p + |δcn |p , (1.7) 1 2 dist(z, ∂E) n=−∞ z∈σd (J) where the eigenvalues are repeated according to their algebraic multiplicity and the constant Lε, p, E is independent of J 0 and J. We note that for the eigenvalues that accumulate to ∂E, the inequality (1.7) gives a qualitatively better estimate than (1.6). We also point out that (1.7) is new even for perturbations of the free Jacobi matrix J0 since, unlike (1.5), it is nearly optimal not only for small but also for large perturbations. As with (1.5), it is an open problem whether or not (1.7) remains true for ε = 0. 2. Schatten Norm Estimates In this section we establish the fundamental estimates that are needed to prove our main result, Theorem 3.3. Throughout, Sp will denote the Schatten class and k · kp the corresponding Schatten norm for p ≥ 1. To clarify our application of complex interpolation, we occasionally use k · k∞ to denote the operator norm. Theorem 2.1. Suppose J 0 is a selfadjoint Jacobi matrix and D ≥ 0 is a diagonal matrix of Schatten class Sp for some p ≥ 1. Denote by dρn the spectral measure of (J 0 , δn ), that is, the measure in the Herglotz representation of the nth diagonal entry of (J 0 − z)−1 , ˆ

 dρn (t) 0 −1 δn , (J − z) δn = , z ∈ Crσ(J 0 ). (2.1) σ(J 0 ) t − z Then

√

kD

1/2

0

−1

(J − z)

D1/2 kpp

2 kDkpp ≤ sup dist(z, σ(J 0 ))p−1 n∈Z

ˆ σ(J 0 )

dρn (t) |t − z|

(2.2)

for z ∈ Crσ(J 0 ). Proof. We consider first the case p = 1. Let {P (t)}t∈R denote the projectionvalued spectral family of the selfadjoint operator J 0 . Recall that for any measurable and bounded function f on σ(J 0 ), the bounded operator f (J 0 ) is given by the functional calculus, ˆ 0 f (J ) = f (t)dP (t). (2.3) σ(J 0 )

LIEB–THIRRING INEQUALITIES FOR COMPLEX JACOBI MATRICES

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Taking f (t) = 1/(t − z) in (2.3), substituting into (2.1), and recalling that the measure in the Herglotz representation is unique yield hδn , dP (t)δn i = dρn (t). Applying (2.3) to f (t) = 1/|t − z| and using (2.4) then imply ˆ dρn (t) 0 −1 hδn , |J − z| δn i = , z ∈ Crσ(J 0 ). |t − z| σ(J 0 )

(2.4)

(2.5)

We also note that if, in addition, the function f (t) in (2.3) is nonnegative, then f (J 0 ) is a bounded, selfadjoint, and nonnegative operator. Fix z ∈ Crσ(J 0 ). In the following we assume without loss of generality that Im(z) ≥ 0. Define the nonnegative functions  1   1  f (t) = Im , f+ (t) = Re χ(Re(z),∞) (t), t−z t−z  1  f− (t) = −Re χ(−∞,Re(z)] (t), (2.6) t−z and note that 1 , t−z √  1   1  2 f+ (t) + f− (t) + f (t) = Re , + Im ≤ t−z t−z |t − z|

f+ (t) − f− (t) + if (t) =

(2.7) t ∈ R. (2.8)

Then we have f (J 0 ) ≥ 0, f± (J 0 ) ≥ 0, and (J 0 − z)−1 = f+ (J 0 ) − f− (J 0 ) + if (J 0 ), √ f+ (J 0 ) + f− (J 0 ) + f (J 0 ) ≤ 2 |J 0 − z|−1 .

(2.9) (2.10)

Using (2.9), the triangle inequality, and the fact that for nonnegative operators the trace norm coincides with the trace, we obtain the estimate kD1/2 (J 0 − z)−1 D1/2 k1 ≤ kD1/2 f+ (J 0 )D1/2 k1 + kD1/2 f− (J 0 )D1/2 k1 + kD1/2 f (J 0 )D1/2 k1


= tr D1/2 f+ (J 0 )D1/2 + tr D1/2 f− (J 0 )D1/2 + tr D1/2 f (J 0 )D1/2 . (2.11) Let Dn,n denote the diagonal entries of D. Since D is nonnegative and diagonal, we have X Dn,n = tr(D) = kDk1 . (2.12) n∈Z

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J. S. CHRISTIANSEN AND M. ZINCHENKO

Hence, by linearity of the trace it follows from (2.11), (2.10), and (2.5) that
kD1/2 (J 0 − z)−1 D1/2 k1 ≤ tr D1/2 [f+ (J 0 ) + f− (J 0 ) + f (J 0 )]D1/2 √
≤ 2 tr D1/2 |J 0 − z|−1 D1/2 ˆ √ X dρn (t) = 2 Dn,n σ(J 0 ) |t − z| n∈Z ˆ √ dρn (t) ≤ 2 kDk1 sup . (2.13) n∈Z σ(J 0 ) |t − z| This is exactly the case p = 1 in (2.2). To obtain (2.2) for p > 1, we use complex interpolation. Define the map ζ 7→ T (ζ) = Dζp/2 |J 0 − z|−1 Dζp/2

(2.14)

from the strip 0 ≤ Re(ζ) ≤ 1 into the space of bounded operators. Then for any u, v ∈ `2 (Z), the scalar function ζ 7→ hu, T (ζ)vi

(2.15)

is continuous on the strip 0 ≤ Re(ζ) ≤ 1, analytic in its interior, and bounded. In addition, since kDiy k ≤ 1 and Dx+iy = Diy Dx = Dx Diy for all x ≥ 0, y ∈ R,

(2.16)

it follows that kT (iy)k∞ ≤ k|J 0 − z|−1 k ≤

1 , dist(z, σ(J 0 ))

y ∈ R,

(2.17)

y ∈ R.

(2.18)

and by [28, Theorem 2.7] and (2.13), kT (1 + iy)k1 ≤ kDp/2 |J 0 − z|−1 Dp/2 k1 ˆ √ dρn (t) p ≤ 2 kD k1 sup , n∈Z σ(J 0 ) |t − z|

Thus, by the complex interpolation theorem (see [28, Theorem 2.9], [14, Theorem III.13.1]), we have 1−x kT (x)k1/x ≤ sup kT (iy)k∞ sup kT (1 + iy)kx1 , y∈R

0 < x < 1.

(2.19)

y∈R

Taking x = 1/p, raising both sides to the power p, and noting that T (1/p) = D1/2 (J 0 − z)−1 D1/2 and kDp k1 = kDkpp finally yields (2.2).  In what follows, E ⊂ R will denote a finite gap set, that is, E=

N [

[αn , βn ],

n=1

α1 < β1 < α2 < · · · < αN < βN ,

N ≥ 1,

(2.20)

LIEB–THIRRING INEQUALITIES FOR COMPLEX JACOBI MATRICES

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and ∂E will be the set of endpoints of E, that is, ∂E = {αn , βn }N n=1 .

(2.21)

For a probability measure dρ supported on E, we define the associated mfunction by ˆ dρ(t) m(z) = , z ∈ CrE. (2.22) E t−z The measure dρ is called reflectionless (on E) if Re[m(x + i0)] = 0 for a.e. x ∈ E.

(2.23)

Reflectionless measures appear prominently in spectral theory of finite and infinite gap Jacobi matrices (see, e.g., [2, 27, 29, 30]). In particular, the isospectral torus associated to E is the set of all Jacobi matrices J 0 that are reflectionless on E (i.e., the spectral measure of (J 0 , δn ) is reflectionless for every n ∈ Z) and for which σ(J 0 ) = E. It is well known (see for example [30]) that dρ is a reflectionless probability measure on E if and only if m(z) is of the form −1

m(z) = p (z − βN )(z − α1 )

N −1 Y j=1

z − γj p , (z − βj )(z − αj+1 )

(2.24)

for some γj ∈ [βj , αj+1 ], j = 1, . . . , N − 1. We now provide an estimate for the variant of the m-function for dρn that appear in Theorem 2.1. Theorem 2.2. Let E ⊂ R be a finite gap set and suppose dρ is a reflectionless probability measure on E. Then for every p > 1, ˆ dρ(t) Kp, E ≤ z ∈ CrE, (2.25) 1 1 , p dist(z, E)p−1 dist(z, ∂E) 2 (1 + |z|) 2 E |t − z| where the constant Kp, E is independent of dρ. In addition, for every ε > 0, ˆ dρ(t) Kε, E ≤ , z ∈ CrE, (2.26) 1 1 dist(z, E)ε dist(z, ∂E) 2 (1 + |z|) 2 −ε E |t − z| where the constant Kε, E is independent of dρ. Proof. Denote the bands of E as in (2.20). Since dρ is reflectionless on a finite gap set, it follows from the Stieltjes inversion formula and (2.24) that dρ is absolutely continuous with density given by w(t) =

N −1 Y |t − γj | 1 1/π p Im[m(t + i0)] = p (2.27) π |t − βN | |t − α1 | j=1 |t − βj | |t − αj+1 |

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J. S. CHRISTIANSEN AND M. ZINCHENKO

for some γj ∈ [βj , αj+1 ], j = 1, . . . , N − 1. Fix 1 ≤ k ≤ N and rearrange the terms in (2.27) as follows N Y |t − γj−1 | |t − γj | p p w(t) = p . |t − αk | |t − βk | j=1 |t − αj | |t − βj | j=k+1 |t − αj | |t − βj | (2.28)

1/π

k−1 Y

Since αj < βj ≤ γj ≤ αj+1 < βj+1 for j = 1, . . . , N − 1, the two products in (2.28) are at most 1 for every t ∈ [αk , βk ] and thus 1/π w(t) ≤ p , |t − αk | |t − βk |

t ∈ [αk , βk ].

Applying this estimate for the individual bands of E implies that ˆ N ˆ dρ(t) 1 1 X βk dt p ≤ , z ∈ CrE. p p π k=1 αk |t − z| |t − αk | |t − βk | E |t − z|

(2.29)

(2.30)

By [18, Lemma 11], each integral in the sum can be estimated by ˆ βk Kp dt 1 p p ≤ . p p−1 |t − αk | |t − βk | dist(z, [αk , βk ]) |z − αk | |z − βk | αk |t − z| (2.31) Since the function z 7→ dist(z, ∂E)(1 + |z|)/|z − αk ||z − βk | is continuous on Cr{αk , βk } and bounded near αk , βk , and ∞, it is bounded on CrE, and therefore ˆ βk 1 dt Kp, E p ≤ (2.32) 1 1 . p |t − αk | |t − βk | dist(z, E)p−1 dist(z, ∂E) 2 (1 + |z|) 2 αk |t − z| Combining (2.32) with (2.30) yields (2.25). In order to obtain (2.26), note that since E is a bounded set we have the trivial bound |t − z| |t| + |z| ≤ ≤ KE , t ∈ E, z ∈ CrE. (2.33) 1 + |z| 1 + |z| This inequality yields the estimate ˆ ˆ dρ(t) |t − z|ε dρ(t) = 1+ε E |t − z| E |t − z| ˆ ε ε ≤ KE (1 + |z|) E

and (2.26) hence follows from (2.25).

dρ(t) , |t − z|1+ε

z ∈ CrE

(2.34) 

LIEB–THIRRING INEQUALITIES FOR COMPLEX JACOBI MATRICES

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3. Lieb–Thirring Bounds We start this section by recalling some results on the distribution of zeros of analytic functions with restricted growth towards the boundary of the domain of analyticity. Let a+ denote the maximum of a and 0. The following theorem for analytic functions on the unit disk is an alternative form of the extension [18, Theorem 4] of the earlier result [1, Theorem 0.2]. Theorem 3.1. Let S ⊂ ∂D be a finite collection of points and suppose h(z) is an analytic function on D such that |h(0)| = 1 and for some K, α, β, γ ≥ 0, K|z|γ log |h(z)| ≤ , z ∈ D. (3.1) (1 − |z|)α dist(z, S)β Then for every ε > 0, there exists a constant Cα,β,γ,ε independent of h(z) such that the zeros of h(z) satisfy X (1 − |z|)α+1+ε dist(z, S)(β−1+ε)+ ≤ Cα,β,γ,ε K, (3.2) |z|(γ−ε)+ z∈D, h(z)=0

where each zero is repeated according to its multiplicity. In [16], an analogous result on the distribution of zeros of analytic functions on Ω = CrE was obtained via a reduction to the unit disk case. For our purposes we will need the following extension of [16, Theorem 0.1] where an additional decay assumption at infinity is imposed in exchange for a stronger conclusion. The extension follows from the reduction to the unit disk case developed in [16] combined with the above version (Theorem 3.1) of the unit disk result. We omit the proof as it is a straightforward modification of the one presented in [16]. Theorem 3.2. Let E ⊂ R be a finite gap set and suppose f (z) is an analytic function on Ω = CrE such that |f (∞)| = 1 and for some K, p, q, r ≥ 0, K , z ∈ Ω. (3.3) log |f (z)| ≤ p dist(z, E) dist(z, ∂E)q (1 + |z|)r Then for every ε > 0, there exists a constant Cp,q,r,ε independent of f (z) such that the zeros of f (z) satisfy X 0 0 0 dist(z, E)p dist(z, ∂E)q (1 + |z|)r ≤ Cp,q,r,ε K, (3.4) z∈Ω, f (z)=0 0

where p = p + 1 + ε, q 0 = 21 [(p + 2q − 1 + ε)+ − p0 ], r0 = (p + q + r − ε)+ − p0 − q 0 , and each zero is repeated according to its multiplicity. We are now ready to present our finite gap version of the Lieb–Thirring inequalities for non-selfadjoint perturbations of Jacobi matrices from the isospectral torus TE .

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Theorem 3.3. Let E ⊂ R be a finite gap set and suppose J, J 0 are two-sided Jacobi matrices such that J 0 ∈ TE and J = J 0 + δJ is a compact perturbation of J 0 . Then for every p ≥ 1 and any ε > 0, ∞ X dist(z, E)p+ε (1 + |z|) 1−3ε X 2 ≤ Lε, p, E |δan |p + |δbn |p + |δcn |p , (3.5) 1 2 dist(z, ∂E) n=−∞ z∈σd (J) where the eigenvalues are repeated according to their algebraic multiplicity and the constant Lε, p, E is independent of J and J 0 . Proof. Suppose that δJ ∈ Sp for some p ≥ 1 and define D ≥ 0 to be the diagonal matrix with the entries  Dn,n = max |δan−1 |, |δan |, |δbn |, |δcn−1 |, |δcn | , n ∈ Z. (3.6) A straightforward verification shows that δJ = D1/2 BD1/2 , where B is a bounded tridiagonal matrix whose entries lie in the unit disk. This in particular means that kBk ≤ 3. Define
f (z) = detdpe I + D1/2 (J 0 − z)−1 D1/2 B , (3.7) where dpe is the smallest integer ≥ p. This regularized perturbation determinant is analytic on Ω = CrE (see, e.g., [14, Chapter IV. §3]). More importantly, the zeros of f coincide with the discrete eigenvalues of J and the multiplicity of the zeros matches the algebraic multiplicity of the corresponding eigenvalues (see [18] and [13, Appendix C] for a proof). By [8, Lemma XI.9.22(d)], we have log |f (z)| ≤ Kp kD1/2 (J 0 − z)−1 D1/2 Bkpp ≤ 3Kp kD1/2 (J 0 − z)−1 D1/2 kpp (3.8) for some constant Kp . It thus follows from Theorems 2.1 and 2.2 (with ε/2 instead of ε) that log |f (z)| ≤

Kε,p,E kDkpp ε

1

dist(z, E)p+ 2 −1 dist(z, ∂E) 2 (1 + |z|)

1−ε 2

,

z ∈ Ω.

(3.9)

Applying Theorem 3.2 (with ε/2 instead of ε) and noting that (1 − 3ε)/2 ≤ r0 p and Dn,n ≤ |δan−1 |p + |δan |p + |δbn |p + |δcn−1 |p + |δcn |p then yield (3.5).  References [1] A. Borichev, L. Golinskii, and S. Kupin, A Blaschke-type condition and its application to complex Jacobi matrices, Bull. Lond. Math. Soc. 41 (2009), no. 1, 117–123. MR 2481997 [2] J. S. Christiansen, Dynamics in the Szeg˝ o class and polynomial asymptotics, J. Anal. Math. (to appear). [3] J. S. Christiansen, B. Simon, and M. Zinchenko, Finite gap Jacobi matrices, I. The isospectral torus, Constr. Approx. 32 (2010), 1–65. MR 2659747 [4] J. S. Christiansen and M. Zinchenko, Lieb–Thirring inequalities for finite and infinite gap Jacobi matrices, Ann. Henri Poincar´e (2017), doi:10.1007/s00023-016-0546-x.

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