A remark on the discrete spectrum of non-self-adjoint Jacobi operators
aa r X i v : . [ m a t h . SP ] J a n A REMARK ON THE DISCRETE SPECTRUM OFNON-SELF-ADJOINT JACOBI OPERATORS
L. GOLINSKII
Abstract.
We study the trace class perturbations of the whole-line,discrete Laplacian and obtain a new bound for the perturbation deter-minant of the corresponding non-self-adjoint Jacobi operator. Based onthis bound, we refine the Lieb–Thirring inequality due to Hansmann–Katriel. The spectral enclosure for such operators is also discussed.
Introduction
In the last two decades there was a splash of activity around the spectraltheory of non-self-adjoint perturbations of some classical operators of mathe-matical physics, such as the Laplace and Dirac operators on the whole space,their fractional powers, and others. Recently, there has been some interestin studying certain discrete models of the above problem. In particular,the structure of the spectrum for compact, non-self-adjoint perturbations ofthe free Jacobi and the discrete Dirac operators has attracted much atten-tion lately. Actually the problem concerns the discrete component of thespectrum and the rate of its accumulation to the essential spectrum. Suchtype of results under various assumptions on the perturbations are unitedunder a common name
Lieb–Thirring inequalities . In the case of the freewhole-line Jacobi operator, such inequalities include the distance from aneigenvalue to the whole essential spectrum [ − , J = J ( { a j } , { b j } , { c j } ) j ∈ Z = . . . . . . . . . a − b c a b c a b c . . . . . . . . . , with uniformly bounded complex entries and a n c n = 0. The spectral theoryof the underlying non-self-adjoint Jacobi operator includes, among others,the structure of their spectra. We denote by J the discrete Laplacian, i.e., Date : January 7, 2021.2010
Mathematics Subject Classification.
Key words and phrases.
Non-self-adjoint Jacobi matrices; discrete spectrum; perturba-tion determinant; Jost solutions. J = J ( { } , { } , { } ). If J − J is a compact operator, that is,lim n →±∞ a n − n →±∞ c n − n →±∞ b n = 0 , the geometric image of the spectrum is utterly clear σ ( J ) = σ ess ( J ) ∪ σ d ( J ) = [ − , ∪ σ d ( J ) , the discrete component σ d ( J ) is an at most countable set of points in C \ [ − ,
2] with the only possible limit points on [ − , traceclass perturbations of the discrete Laplacian(0.2) J − J ∈ S ⇔ ∞ X n = −∞ ( | − a n | + | b n | + | − c n | ) < ∞ . Now the discrete spectrum is the set of isolated eigenvalues of finite algebraicmultiplicity.The currently best result which governs the behavior of the discrete spec-trum is due to Hansmann–Katriel [9, Theorem 1]. It states that for each ε ∈ (0 ,
1) there is a constant C ( ε ) > X λ ∈ σ d ( J ) dist( λ, [ − , ε | λ − | + ε ≤ C ( ε ) k J − J k . The result is known to be sharp [1] in the sense that (0.3) is false for ε = 0.Yet the question arises naturally whether it is possible to drop at leastone of the two small parameters on the left side. We answer this questionaffirmatively in this note. The price we pay is a constant on the right side. Theorem 0.1.
Let J − J ∈ S . Then for each ε ∈ (0 , there is a constant C ( ε ) > so that (0.4) X λ ∈ σ d ( J ) dist( λ, [ − , | λ − | − ε ≤ C ( ε )∆ , ∆ := ∞ X n = −∞ ( | b n | + | − a n c n | ) . If J is a discrete Schr¨odinger operator, that is, a n = c n ≡
1, then(0.5) X λ ∈ σ d ( J ) dist( λ, [ − , | λ − | − ε ≤ C ( ε ) k J − J k . Remark 0.2.
The appearance of the value ∆ in place of k J − J k mightseem reasonable. Indeed, given a Jacobi matrix J , consider a class S ( J ) ofJacobi matrices S ( J ) = { b J := T − J T, T = diag( t j ) j ∈ Z is a diagonal isomorphism of ℓ ( Z ) } , b J = J (cid:0) { a j r j } , { b j } , { c j r − j } (cid:1) , r n = t n t n +1 , n ∈ Z . As b J is similar to J , the equality σ d ( b J ) = σ d ( J ) holds. So the left side of(0.4) does not alter within the class S ( J ). The same is true for the value∆, in contrast to k J − J k . For the class S ( J ) both sides of (0.4) vanish,whereas k J − J k , J ∈ S ( J ), can be arbitrarily large. ACOBI MATRICES 3
Next, | − a n c n | ≤ | − a n | + | − c n | + | − a n || − c n | , and so∆ ≤ k J − J k + k J − J k . We see that for small perturbations the value ∆ has at least the same orderas k J − J k .The so-called perturbation determinant L ( λ, J ) := det( I + ( J − J )( J − λ ) − ) , introduced by M.G. Krein [8] in the late 50-th, comes in as a principalanalytic tool. The main feature of this analytic function on the resolvent set ρ ( J ) = C \ [ − ,
2] is that the zero divisor agrees with the discrete spectrumof the perturbed operator J , and moreover, the multiplicity of each zeroequals the algebraic multiplicity of the corresponding eigenvalue. So theoriginal problem of spectral theory can be restated as the classical problemof the zero distributions of analytic functions, which goes back to Jensenand Blaschke.The arguments in [9] pursue in two steps. The first one results in a certainbound for the perturbation determinant, typical for the functions of non-radial growth. The classes of such analytic (and subharmonic) functions inthe unit disk were introduced and studied in [2, 6] (for some advances see[3]). The Blaschke-type conditions for the zero sets (Riesz measures) wereproved therein, with an important amplification in [9, Theorem 4], betteradapted for applications. The second step is just the latter result applied tothe bound mentioned above.In our approach to the problem the argument in the first step is totallydifferent. Instead of certain operator-theoretic means and the Fourier trans-form, we deal with the associated three-term recurrence relation(0.6) a k − u k − + b k u k + c k u k +1 = λ ( z ) u k , k ∈ Z , λ ( z ) = z + 1 z , and its modifications. Here λ ( · ) is the Zhukovsky function which maps theunit disk onto the resolvent set ρ ( J ) = C \ [ − , u = ( u k ) k ∈ Z from ℓ ( Z ) is exactly the eigenvector of J with the eigenvalue λ . Next, the solutions u ± = ( u ± k ) k ∈ Z are called the Jost solutions at ±∞ if(0.7) lim n →±∞ z ∓ n u ± n ( z ) = 1 , z ∈ D := D \{ } . We study the Jost solutions by reducing the difference equation (0.6) toa Volterra-type discrete integral equation, see, e.g. [12, Section 7.5] and[5]. The bounds for the Jost solutions stem from successive approximationsmethod. The perturbation determinant arises as the Wronskian of the Jostsolutions, so its bound is then straightforward.Note also, that the relation(0.8) | L ( z, J ) − | ≤ (4 x + 5 x ) e x , x := 2 | z || − z | (∆ / + ∆)for z in the open unit disk D := {| z | < } provides some information aboutthe spectral enclosure, see (2.5) and Remark 2.1. L. GOLINSKII Jost solutions and discrete Volterra equations
The following two companions of the main difference equation (0.6) areof particular concern(1.1) v k − ( z ) + b k v k ( z ) + a k c k v k +1 ( z ) = (cid:16) z + 1 z (cid:17) v k ( z ) , k ∈ Z , and(1.2) a k − c k − w k − ( z ) + b k w k ( z ) + w k +1 ( z ) = (cid:16) z + 1 z (cid:17) w k ( z ) , k ∈ Z ,z ∈ D . Put(1.3) α n := n − Y j = −∞ a j , γ n := n − Y j = −∞ c − j , n ∈ Z . It is easy to see that u = ( u k ) is a solution of (0.6) if and only if u k = α k v k , (cid:16) u k = γ k w k (cid:17) , k ∈ Z , where v = ( v k ) ( w = ( w k )) is a solution of (1.1) ((1.2)), respectively. Inparticular, if u ± = ( u ± k ) are the Jost solutions of (0.6), then u + n = ∞ Y j = n a − j v + n = ∞ Y j = n c j w + n ,u − n = n − Y j = −∞ c − j w − n = n − Y j = −∞ a j v − n , (1.4)where v ± = ( u ± k ) ( w ± = ( w ± k )) are the Jost solutions of (1.1) ((1.2)), re-spectively.We are aimed at obtaining the bounds for the Jost solutions v + and w − by reducing the difference equations to the Volterra-type discrete integralequations. The unity of the corresponding coefficients (the first one in (1.1)and the third one in (1.2)) appears to be crucial.Define the (non-symmetric) Green kernels by G r ( n, m ; z ) := (cid:26) z m − n − z n − m z − z − , m ≥ n, , m ≤ n,G l ( n, m ; z ) := (cid:26) , m ≥ n, z n − m − z m − n z − z − , m ≤ n, n, m ∈ Z , z ∈ D . (1.5)The basic property of the kernels can be verified directly G r,l ( n, m − z ) + G r,l ( n, m + 1; z ) − (cid:16) z + 1 z (cid:17) G r,l ( n, m ; z ) = δ n,m ,G r,l ( n − , m ; z ) + G r,l ( n + 1 , m ; z ) − (cid:16) z + 1 z (cid:17) G r,l ( n, m ; z ) = δ n,m . (1.6)The kernels T r ( n, m ; z ) := − b m G r ( n, m ; z ) + (1 − a m − c m − ) G r ( n, m − z ) ,T l ( n, m ; z ) := − b m G l ( n, m ; z ) + (1 − a m c m ) G l ( n, m + 1; z ) , z ∈ D , (1.7) n, m ∈ Z , are the key players of the game. ACOBI MATRICES 5
Theorem 1.1.
The Jost solution v + = ( v + k ) of the difference equation (1.1) at + ∞ satisfies the discrete Volterra equation (1.8) v + n ( z ) = z n + ∞ X m = n +1 T r ( n, m ; z ) v + m ( z ) , n ∈ Z , z ∈ D . Conversely, each solution v = ( v n ) of (1.8) solves (1.1) .Similarly, the Jost solution w − = ( w − k ) of (1.2) at −∞ satisfies the dis-crete Volterra equation (1.9) w − n ( z ) = z − n + n − X m = −∞ T l ( n, m ; z ) w − m ( z ) , n ∈ Z , z ∈ D . Conversely, each solution w = ( w n ) of (1.9) solves (1.2) .Proof. We multiply the first relation (1.6) for G r by v + m , (1.1) by G r ( n, m ),and subtract the later from the former h G r ( n, m + 1) v + m − G r ( n, m ) v + m − i + h − b m G r ( n, m ) + G r ( n, m − i v + m − a m c m G r ( n, m ) v + m +1 = δ n,m v + m . Next, taking into account that G r ( n, n + 1) = 1, G r ( n, n ) = 0, we sum upover m from n + 1 to NG r ( n, N + 1) v + N + N X m = n +1 h − b m G r ( n, m ) + G r ( n, m − i v + m − N X m = n a m c m G r ( n, m ) v + m +1 = v + n , or v + n = G r ( n, N + 1) v + N − a N c N G r ( n, N ) v + N +1 + N X m = n +1 T r ( n, m ) v + m . The latter equality holds for an arbitrary solution of (1.1). If v + is the Jostsolution at + ∞ , then, by (1.5),lim N →∞ G r ( n, N + 1) v + N − a N c N G r ( n, N ) v + N +1 = z n , and (1.8) follows.The direct reasoning for (1.2) is the same. We multiply the first relation(1.6) for G l by w − m , (1.2) by G l ( n, m ), and subtract the later from the former h G l ( n, m − w − m − G l ( n, m ) w − m − i + h − b m G l ( n, m ) + G r ( n, m + 1) i w − m − a m − c m − G l ( n, m ) w − m − = δ n,m w − m . The summation over m from − N to n − w − n = G l ( n, − N − w −− N − a − N − c − N − G l ( n, − N ) w −− N − + n − X m = − N T l ( n, m ) w − m . If w − is the Jost solution of (1.2) at −∞ , thenlim N →∞ G l ( n, − N − w −− N − a − N − c − N − G l ( n, − N ) w −− N − = z − n , L. GOLINSKII and (1.9) follows.To prove the converse statements, let v = ( v n ) be a solution of (1.8).Then v n − + v n +1 = (cid:16) z + 1 z (cid:17) z n + T r ( n − , n ) v n + T r ( n − , n + 1) v n +1 + ∞ X m = n +2 h T r ( n − , m ) + T r ( n + 1 , m ) i v m . But T r ( n − , n ) v n = − b n v n ,T r ( n − , n + 1) v n +1 = h − b n +1 G r ( n − , n + 1) + (1 − a n c n ) G r ( n − , n ) i v n +1 = − (cid:16) z + 1 z (cid:17) b n +1 v n +1 + (1 − a n c n ) v n +1 = (cid:16) z + 1 z (cid:17) T r ( n, n + 1) v n +1 + (1 − a n c n ) v n +1 ,T r ( n − , m ) + T r ( n + 1 , m ) = (cid:16) z + 1 z (cid:17) T r ( n, n + 1) . Finally, v n − + v n +1 = − b n v n + (1 − a n c n ) v n +1 + (cid:16) z + 1 z (cid:17) (cid:16) z n + ∞ X m = n +1 T r ( n, m ) v m (cid:17) , which is (1.1). The proof for the second converse statement is identical. (cid:3) It is convenient and instructive to introduce new variables in both (1.8)and (1.9) f rm := v + m z − m − , e T r ( n, m ; z ) := T r ( n, m ; z ) z m − n ,f lm := w − m z m − , e T l ( n, m ; z ) := T l ( n, m ; z ) z n − m , (1.10)so the Volterra equations turn into f rn ( z ) = g rn ( z ) + ∞ X m = n +1 e T r ( n, m ; z ) f rm ( z ) ,g rn ( z ) := ∞ X m = n +1 e T r ( n, m ; z ) , (1.11)and f ln ( z ) = g ln ( z ) + n − X m = −∞ e T l ( n, m ; z ) f lm ( z ) ,g ln ( z ) := n − X m = −∞ e T l ( n, m ; z ) . (1.12)These are better than the original ones owing to the simple analytic prop-erties of the kernels e T r,l . Indeed, it is not hard to verify that e T r,l ( n, m ; · ) are ACOBI MATRICES 7 polynomials of z , and (cid:12)(cid:12) e T r ( n, m ; z ) (cid:12)(cid:12) ≤ δ rm min n ( m − n ) + , | z || z − | o ,δ rm := | b m | + | − a m − c m − | , n, m ∈ Z , z ∈ D , (1.13) (cid:12)(cid:12) e T l ( n, m ; z ) (cid:12)(cid:12) ≤ δ lm min n ( n − m ) + , | z || z − | o ,δ lm := | b m | + | − a m c m | , n, m ∈ Z , z ∈ D . (1.14)In particular,(1.15) (cid:12)(cid:12) e T r,l ( n, m ; z ) (cid:12)(cid:12) ≤ δ r,lm | ω ( z ) | , ω ( z ) := 2 z − z , z ∈ D := D \{± } . So, the series for g r,ln converge absolutely and uniformly on each compactsubset of D , which omits ±
1, and | g r,ln ( z ) | ≤ | ω ( z ) | ∆ r,ln , ∆ rn := ∞ X m = n +1 δ rm , ∆ ln := n − X m = −∞ δ lm . According to the general result [12, Lemma 7.8] concerning the discreteVolterra equations, we have for n ∈ Z and z ∈ D | f rn ( z ) | = | z − n v + n − | ≤ | ω ( z ) | ∆ rn exp (cid:8) | ω ( z ) | ∆ rn (cid:9) , | f ln ( z ) | = | z n w − n − | ≤ | ω ( z ) | ∆ ln exp (cid:8) | ω ( z ) | ∆ ln (cid:9) , (1.16)or | v + n − z n | ≤ | z | n | ω ( z ) | ∆ rn exp (cid:8) | ω ( z ) | ∆ rn (cid:9) , | w − n − z − n | ≤ | z | − n | ω ( z ) | ∆ ln exp (cid:8) | ω ( z ) | ∆ ln (cid:9) . (1.17) 2. The Wronskian and the Lieb–Thirring inequality
Let us go back to the main equation (0.6). Given its two solutions u ′ =( u ′ n ) and u ′′ = ( u ′′ n ), the equality below is obvious a n − ( u ′ n − u ′′ n − u ′ n u ′′ n − ) = c n ( u ′ n u ′′ n +1 − u ′ n +1 u ′′ n ) . The Wronskian W ( u ′ , u ′′ ) is naturally defined as W ( u ′ , u ′′ ) := β n ( u ′ n u ′′ n +1 − u ′ n +1 u ′′ n ) , β n := a n n Y j = −∞ c j a j . Such choice of β n makes the Wronskian independent of n .From now on we put u ′ = u + , u ′′ = u − . By the transition formulas (1.4),we can express W ( u + , u − ) in terms of v + and w − : W ( u + , u − ) = β n ( u + n u − n +1 − u + n +1 u − n )= β n (cid:16) ∞ Y j = n a − j n Y j = −∞ c − j v + n w − n +1 − ∞ Y j = n +1 a − j n − Y j = −∞ c − j v + n +1 w − n (cid:17) = ∞ Y j = −∞ a − j ( v + n w − n +1 − a n c n v + n +1 w − n ) . L. GOLINSKII
So the bound for the Wronskian will follows from the inequalities (1.17).Note also that∆ r,ln ≤ ∆ := ∞ X j = −∞ ( | b j | + | − a j c j | ) , n ∈ Z . We are now ready for
Proof of Theorem 0.1 . Put U ( z ) := ω ( z )2 ∞ Y j = −∞ a j W ( u + , u − )= ω ( z )2 (cid:0) v +0 ( z ) w − ( z ) − v +1 ( z ) w − ( z ) + (1 − a c ) v +1 ( z ) w − ( z ) (cid:1) . (2.1)If p j ( z ) := v + j ( z ) − z j , q j ( z ) := w − j ( z ) − z − j , j = 0 , , then U ( z ) := ω ( z )2 h (1 + p ( z ))( z − + q ( z )) − ( z + p ( z ))(1 + q ( z )) + (1 − a c ) v +1 ( z ) w − ( z ) i = 1 + ω ( z )2 d ( z ) , d ( z ) = d ( z ) − d ( z ) + d ( z ) , where d ( z ) := q ( z ) + z − p ( z ) + p ( z ) q ( z ) ,d ( z ) := p ( z ) + zq ( z ) + p ( z ) q ( z ) ,d ( z ) := (1 − a c ) v +1 ( z ) w − ( z ) . We proceed with the upper bound for the function U term by term.1. For d we have | ω ( z ) | | d ( z ) | ≤ | ω ( z ) | (cid:0) | q ( z ) | + | z − p ( z ) | + | p ( z ) q ( z ) | (cid:1) . In view of (1.17) | ω ( z ) | (cid:0) | q ( z ) | + | z − p ( z ) | (cid:1) ≤ | ω ( z ) | | z | ∆ e | ω ( z ) | ∆ , | ω ( z ) | | p ( z ) q ( z ) | ≤ | ω ( z ) | | z | ∆ e | ω ( z ) | ∆ ≤ | ω ( z ) | | z | ∆ e | ω ( z ) | ∆ , and so | ω ( z ) | | d ( z ) | ≤ | ω ( z ) | | z | ∆ e | ω ( z ) | ∆ . Next, it is clear that | − z | + | z | ≥ , | ω ( z ) | ≥ | ω ( z ) | | z | or (cid:12)(cid:12)(cid:12)(cid:12) ω ( z ) z (cid:12)(cid:12)(cid:12)(cid:12) ≤ | ω ( z ) | ) . Hence, 3 | ω ( z ) | | z | ≤ | ω ( z ) | (1 + | ω ( z ) | ) ACOBI MATRICES 9 and finally | ω ( z ) | | d ( z ) | ≤ | ω ( z ) | + | ω ( z ) | )∆ e | ω ( z ) | ∆ ≤ (cid:8) | ω ( z ) | (∆ / + ∆) + | ω ( z ) | (∆ / + ∆) (cid:9) e | ω ( z ) | (∆ / +∆) . (2.2)2. For d we have | ω ( z ) | | d ( z ) | ≤ | ω ( z ) | (cid:0) | p ( z ) | + | zq ( z ) | + | p ( z ) q ( z ) | (cid:1) . It is immediate from (1.17) that | p ( z ) | ≤ | ω ( z ) | ∆ e | ω ( z )∆ , | q ( z ) | ≤ | ω ( z ) | ∆ e | ω ( z )∆ , | p ( z ) q ( z ) | ≤ | ω ( z ) | ∆ e | ω ( z )∆ , and so | ω ( z ) | | d ( z ) | ≤ | ω ( z ) | ∆ e | ω ( z ) | ∆ + | ω ( z ) | e | ω ( z ) | ∆ ≤ | ω ( z ) | ∆ e | ω ( z ) | ∆ ≤ | ω ( z ) | (∆ / + ∆) e | ω ( z ) | (∆ / +∆) . (2.3)3. For d we have by (1.17), | ω ( z ) | | d ( z ) | ≤ | ω ( z ) | (cid:16) | ω ( z ) | ∆ e | ω ( z ) | ∆ (cid:17) , and since 1 + xe x ≤ e x for x ≥
0, then(2.4) | ω ( z ) | | d ( z ) | ≤ | ω ( z ) | e | ω ( z ) | ∆ ≤ | ω ( z ) | / + ∆) e | ω ( z ) | (∆ / +∆) . A combination of (2.2) – (2.4) produces the following bound for U | U ( z ) − | ≤ (4 x + 5 x ) e x , x := | ω ( z ) | (∆ / + ∆) , | U ( z ) | ≤ (1 + 4 x + 5 x ) e x ≤ e x . (2.5)By the non-self-adjoint version of [11, Proposition 10.6] (the calculationthere is algebraic and so immediately extends to the non-self-adjoint case), U ( · ) = L ( · , J ), so we come to the bound for the perturbation determinant(2.6) log | L ( z, J ) | ≤ | z || − z | (∆ / + ∆) , L (0 , J ) = 1 . The rest is standard nowadays. According to [9, Theorem 4], for each ε ∈ (0 ,
1) there is a constant C ( ε ) > Z ( L ) X ζ ∈ Z ( L ) (1 − | ζ | ) | ζ − | ε | ζ | ε ≤ C ( ε )(∆ / + ∆) , (each zero is taken with its multiplicity). The latter inequality turns into(0.4) when we go over to the Zhukovsky images and take into account thedistortion for the Zhukovsky function [9, Lemma 7]. The proof is complete.For the discrete Schr¨odinger operators J ( a j = c j ≡
1) (0.5) follows from∆ = ∞ X j = −∞ | b j | = k J − J k . Remark 2.1.
As a byproduct, the first bound in (2.5) for the perturba-tion determinant provides some information on the location of the discretespectrum (spectral enclosure). Indeed, let κ be a unique positive root of theequation (4 x + 5 x ) e x = 1 , κ ≈ . . Then L = 0 in D as long as | ω ( z ) | (∆ / + ∆) < κ, | z || − z | < κ / + ∆) , or in terms of the Zhukovsky images(2.7) σ d ( J ) ⊂ λ ∈ C \ [ − ,
2] : | λ − | ≤ / + ∆) κ ! . So, the discrete spectrum lies in a certain Cassini oval.The spectral enclosure is normally derived from the Birman–Schwingerprinciple. Precisely, λ ( z ) ∈ σ d ( J ) ⇒ k K ( z ) k ≤ ,K is the Birman–Schwinger operator. In our case one has(2.8) σ d ( J ) ⊂ (cid:8) λ ∈ C \ [ − ,
2] : | λ − | ≤ k J − J k (cid:9) . It might be curious comparing the ovals in (2.7) and (2.8).For the discrete Schr¨odinger operators the sharp oval which contains thediscrete spectrum is known [10](2.9) σ d ( J ) ⊂ (cid:8) λ ∈ C \ [ − ,
2] : | λ − | ≤ k J − J k (cid:9) . References [1] S. B¨ogli, F. Stampach, On Lieb–Thirring inequalities for one-dimensional non-self-adjoin Jacobi and Schr¨odinger operators. Preprint (2020), arXiv:2004.09794.[2] A. Borichev, L. Golinskii, S. Kupin, A Blaschke-type condition and its applicationto complex Jacobi matrices. Bull. Lond. Math. Soc. (2009), no. 1, 117–123.[3] A. Borichev, L. Golinskii, S. Kupin, On zeros of analytic functions satisfying non-radial growth conditions. Rev. Mat. Iberoam. (2018), no. 3, 1153–1176.[4] M. Demuth, M. Hansmann, G. Katriel, Eigenvalues of non-selfadjoint operators:a comparison of two approaches. In: Mathematical physics, spectral theory andstochastic analysis, Oper. Theory Adv. Appl., vol. 232, Birkh¨auser/Springer BaselAG, Basel, 2013, pp. 107–163.[5] I. Egorova, L. Golinskii, On the location of the discrete spectrum for complexJacobi matrices. Proc. Amer. Math. Soc. (2005), no. 12, 3635–3641.[6] S. Favorov, L. Golinskii, A Blaschke-type condition for analytic and subharmonicfunctions and application to contraction operators. Advances in Mathematical Sci-ences, Linear and Complex Analysis (dedicated to V. P. Havin), (2009), 37–47.[7] R. Frank, The Lieb–Thirring inequalities: recent results and open problems.Preprint (2020), arXiv:2007.09326.[8] I.C. Gohberg, M.G. Krein, Introduction to the Theory of Linear NonselfadjointOperators , Translations of Mathematical Monographs, vol. 18, AMS, ProvidenceRI, 1969.[9] M. Hansmann, G. Katriel, Inequalities for the eigenvalues of non-selfadjoint Jacobioperators. Complex Anal. Oper. Theory (2011), no. 1, 197–218.[10] O. Ibrogimov, F. Stampach, Spectral enclosures for non-self-adjoint discreteSchr¨odinger operators. Preprint (2019), arXiv:1903.08620. ACOBI MATRICES 11 [11] R. Killip, B. Simon, Sum rules for Jacobi matrices and their applications to spectraltheory. Ann. of Math. (2) (2003), no. 1, 253–321.[12] G. Teschl,
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