Lieb–Thirring inequalities for complex finite gap Jacobi matrices

Christiansen, Jacob S.; Zinchenko, Maxim

doi:10.1007/s11005-017-0961-z

Cited by 4 publications

(4 citation statements)

References 41 publications

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“…Quantitative approaches based on Lieb-Thirring inequalities, and the identification of the distribution law of the discrete spectrum around the limit points, have also been developed, see e.g. [8,10,18,4,12,17,19,22,26,36] and references therein. These results have been established mostly for specific models like the Schrödinger operators and Jacobi matrices.…”

Section: Introductionmentioning

confidence: 99%

On the spectral properties of non-selfadjoint discrete Schrödinger operators

Bourget¹,

Sambou²,

Taarabt³

2018

Preprint

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Let H0 be a purely absolutely continuous selfadjoint operator acting on some separable infinite-dimensional Hilbert space and V be a compact non-selfadjoint perturbation. We relate the regularity properties of V to various spectral properties of the perturbed operator H0 + V . The structure of the discrete spectrum and the embedded eigenvalues are analysed jointly with the existence of limiting absorption principles in a unified framework. Our results are based on a suitable combination of complex scaling techniques, resonance theory and positive commutators methods. Various results scattered throughout the literature are recovered and extended. For illustrative purposes, the case of the one-dimensional discrete Laplacian is emphasized.

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Section: Introductionmentioning

confidence: 99%

On the spectral properties of non-selfadjoint discrete Schrödinger operators

Bourget¹,

Sambou²,

Taarabt³

2018

Preprint

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“…We can mention the following directions of investigations: perturbations and spectral analysis (see [10] and references therein); quadrature rules ( [8]); eigenvalue problems ( [7]); determinacy questions ( [3]). Two-sided Jacobi matrices are also studied intensively: for the real case we refer to [4], and for recent developments see [9], [5] and references therein.…”

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confidence: 99%

On bounded complex Jacobi matrices and related moment problems

Zagorodnyuk¹

2023

Preprint

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In this paper we study the linear functional S on complex polynomials which is associated to a bounded complex Jacobi matrix J. The associated moment problem is considered: find a positive Borel measure µ on C subject to conditions z n dµ = s n , where s n are prescribed complex numbers (moments). This moment problem may be viewed as an extension of the Stieltjes and Hamburger moment problems to the complex plane. Sufficient conditions for the solvability of the moment problem are provided. As a corollary, we obtain conditions for the existence of an integral representation S(p) = C p(z)dµ, with a positive Borel measure µ. An interrelation of the associated to the complex Jacobi matrix operator A 0 , acting in l 2 on finite vectors, and the multiplication by z operator in L 2 µ is discussed as well.

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“…Let us consider the case when α n > 0 and β n ∈ R. In [8] were formulated conditions for N = 2 assuring that the discrete spectrum of A is empty. For general N the asymptotics of discrete spectrum of A was studied in [6], see also [9] for the case N = 1.…”

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confidence: 99%

“…For example, in [3] there was considered in detail the case when sequences defining A are periodic. In [6,8] there was considered the behaviour of the point spectrum of compact perturbations of the periodic case. In [4] Mourre commutator method has been applied to the study of continuous spectrum of discrete Schrödinger operators.…”

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Spectral Properties of Some Complex Jacobi Matrices

Świderski

2020

Integr. Equ. Oper. Theory

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A. We study spectral properties of bounded and unbounded complex Jacobi matrices. In particular, we formulate conditions assuring that the spectrum of the studied operators is continuous on some subsets of the complex plane and we provide uniform asymptotics of their generalised eigenvectors. We illustrate our results by considering complex perturbations of real Jacobi matrices belonging to several classes: asymptotically periodic, periodically modulated and the blend of these two. Moreover, we provide conditions implying existence of a unique closed extension. The method of the proof is based on the analysis of a generalisation of shifted Turán determinants to the complex setting.2010 Mathematics Subject Classification. Primary: 47B36.

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Lieb–Thirring inequalities for complex finite gap Jacobi matrices

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.

Cited by 4 publications

References 41 publications

On the spectral properties of non-selfadjoint discrete Schrödinger operators

On the spectral properties of non-selfadjoint discrete Schrödinger operators

On bounded complex Jacobi matrices and related moment problems

Spectral Properties of Some Complex Jacobi Matrices

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