On the discrete spectrum of complex banded matrices (original) (raw)

Limit sets for the discrete spectrum of complex Jacobi matrices

Sbornik: Mathematics, 2005

The discrete spectrum of complex banded matrices that are compact perturbations of the standard banded matrix of order p is under consideration. The rate of stabilization for the matrix entries sharp in the sense of order which provides finiteness of the discrete spectrum is found. The pbanded matrix with the discrete spectrum having exactly p limit points on the interval (−2, 2) is constructed. The results are applied to the study of the discrete spectrum of asymptotically periodic Jacobi matrices.

The Spectrum of Periodic Jacobi Matrices with Slowly Oscillating Diagonal Terms

We study the spectrum of periodic Jacobi matrices. We concentrate on the case of slowly oscillating diagonal terms and study the behaviour of the zeros of the associated orthogonal polynomials in the spectral gap. We find precise estimates for the distance from single eigenvalues of truncated matrices in the spectral gap to the diagonal entries of the matrix. We include a brief numerical example to show the exactness of our estimates.

Discrete spectra for some complex infinite band matrices

Opuscula Mathematica, 2021

Under suitable assumptions the eigenvalues for an unbounded discrete operator \(A\) in \(l_2\), given by an infinite complex band-type matrix, are approximated by the eigenvalues of its orthogonal truncations. Let \[\Lambda (A)=\{\lambda \in {\rm Lim}_{n\to \infty} \lambda _n : \lambda _n \text{ is an eigenvalue of } A_n \text{ for } n \geq 1 \},\] where \({\rm Lim}_{n\to \infty} \lambda_n\) is the set of all limit points of the sequence \((\lambda_n)\) and \(A_n\) is a finite dimensional orthogonal truncation of \(A\). The aim of this article is to provide the conditions that are sufficient for the relations \(\sigma(A) \subset \Lambda(A)\) or \(\Lambda (A) \subset \sigma (A)\) to be satisfied for the band operator \(A\).

Spectral results for perturbed periodic Jacobi matrices using the discrete Levinson technique

Studia Mathematica, 2018

For an arbitrary Hermitian period-T Jacobi operator, we assume a perturbation by a Wigner-von Neumann type potential to devise subordinate solutions to the formal spectral equation for a (possibly infinite) real set, S, of the spectral parameter. We employ discrete Levinson type techniques to achieve this, which allow the analysis of the asymptotic behaviour of the solution. This enables us to construct infinitely many spectral singularities on the absolutely continuous spectrum of the periodic Jacobi operator, which are stable with respect to an l 1-perturbation. An analogue of the quantisation conditions from the continuous case appears, relating the frequency of the oscillation of the potential to the quasi-momentum associated with the purely periodic operator.

Singular Continuous Spectrum for a Class of Almost Periodic Jacobi MATRICES1

AMERICAN MATHEMATICAL SOCIETY, 1982

We consider the operator// on l2(Z) depending upon three parameters, X, a, 0, ... (1) [#(X, a, 0)u] (n) = u(n + 1) + u(n - 1) + X cos(2iran + 6)u(n). ... In this note we will sketch the proof of the following result whose detailed proof will appear elsewhere [3]. ... THEOREM 1. Fix a, an ...

Asymptotics of the discrete spectrum for complex Jacobi matrices

Opuscula Mathematica, 2014

The spectral properties and the asymptotic behaviour of the discrete spectrum for a special class of infinite tridiagonal matrices are given. We derive the asymptotic formulae for eigenvalues of unbounded complex Jacobi matrices acting in l 2 (N).

Eigenvalues for Perturbed Periodic Jacobi Matrices by the Wigner-von Neumann Approach

Integral Equations and Operator Theory, 2016

The Wigner-von Neumann method, which has previously been used for perturbing continuous Schrödinger operators, is here applied to their discrete counterparts. In particular, we consider perturbations of arbitrary T-periodic Jacobi matrices. The asymptotic behaviour of the subordinate solutions is investigated, as too are their initial components, together giving a general technique for embedding eigenvalues, λ, into the operator's absolutely continuous spectrum. Introducing a new rational function, C(λ; T), related to the periodic Jacobi matrices, we describe the elements of the a.c. spectrum for which this construction does not work (zeros of C(λ; T)); in particular showing that there are only finitely many of them.

On real banded Toeplitz matrices whose limiting eigenvalue distribution has real support

arXiv (Cornell University), 2017

We introduce and investigate a class of complex semi-infinite banded Toeplitz matrices satisfying the condition that the spectra of their principal submatrices accumulate onto a real interval when the size of the submatrix grows to ∞. We prove that a banded Toeplitz matrix belongs to this class if and only if its symbol has real values on a Jordan curve located in C \ {0}. Surprisingly, it turns out that, if such a Jordan curve is present, the spectra of all the submatrices have to be real. The latter claim is also proven for matrices given by a more general symbol. Further, the limiting eigenvalue distribution of a real banded Toeplitz matrix is related to the solution of a determinate Hamburger moment problem. We use this to derive a formula for the limiting measure using a parametrization of the Jordan curve. We also describe a Jacobi operator, whose spectral measure coincides with the limiting measure. We show that this Jacobi operator is a compact perturbation of a tridiagonal Toeplitz matrix. Our main results are illustrated by several concrete examples; some of them allow an explicit analytic treatment, while some are only treated numerically. Update: The proof of Theorem 8 contains an error. An erratum is attached in the end.