Limit sets for the discrete spectrum of complex Jacobi matrices (original) (raw)
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On the discrete spectrum of complex banded matrices
Journal of Mathematical Physics, Analysis, Geometry
The discrete spectrum of complex banded matrices which are compact perturbations of the standard banded matrix of order ppp 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 ppp-banded matrix with the discrete spectrum having exactly ppp preassigned points on the interval (−2,2)(-2,2)(−2,2) is constructed. The results are applied to the study of the discrete spectrum of asymptotically periodic complex 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.
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.
2005
For all hyperbolic polynomials we proved in [11] a Lipschitz estimate of Jacobi matrices built by orthogonalizing polynomials with respect to measures in the orbit of classical Perron-Frobenius-Ruelle operators associated to hyperbolic polynomial dynamics (with real Julia set). Here we prove that for all sufficiently hyperbolic polynomials this estimate becomes exponentially better when the dimension of the Jacobi matrix grows. In fact, our main result asserts that a certain natural non-linear operator on Jacobi matrices built by a hyperbolic polynomial with real Julia set is a contraction in operator norm if the polynomial is sufficiently hyperbolic. This allows us to get for such polynomials the solution of a problem of Bellissard, in other words, to prove the limit periodicity of the limit Jacobi matrix. This fact does not require the iteration of the same fixed polynomial, and therefore it gives a wide class of limit periodic Jacobi matrices with singular continuous spectrum.
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).
Limit periodic Jacobi matrices with a prescribed ppp-adic hull and a singular continuous spectrum
Mathematical Research Letters, 2006
For all hyperbolic polynomials we proved in [9] a Lipschitz estimate of Jacobi matrices built by orthogonalizing polynomials with respect to measures in the orbit of classical Perron-Frobenius-Ruelle operators associated to hyperbolic polynomial dynamics (with real Julia set). Here we prove that for all sufficiently hyperbolic polynomials this estimate becomes exponentially better when the dimension of the Jacobi matrix grows. In fact, our main result asserts that a certain natural non-linear operator on Jacobi matrices built by a hyperbolic polynomial with real Julia set is a contraction in operator norm if the polynomial is sufficiently hyperbolic. This allows us to get for such polynomials the solution of a problem of Bellissard, in other words, to prove the limit periodicity of the limit Jacobi matrix. This fact does not require the iteration of the same fixed polynomial, and therefore it gives a wide class of limit periodic Jacobi matrices with singular continuous spectrum. such a polynomial is well-defined by the position of its critical values CV (T ) := {t i = T (c i ) : T (c i ) = 0, c i > c j for i > j}.
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 \(A\).