On stability in the Borg–Hochstadt theorem for periodic Jacobi matrices (original) (raw)

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.

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.

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 ...

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.

A Note on the Eigenvalues for Periodic Three-Dimensional Jacobi-Perron Algorithms

Sitzungsberichte und Anzeiger der mathematisch-naturwissenschaftlichen Klasse, 2010

In their profound study on the connections between Lyapunov theory and approximation properties of Jacobi-Perron algorithm BROISE-ALAMICHEL and GUIVARC'H 2001 proved a generalization of an inequality due to PALEY and URSELL [2]. In this note this inequality is slightly refined for dimension n ¼ 3. This shows that for the eigenvalues 0 > j 1 j ! j 2 j ! j 3 j of a periodic expansion the inequality j 1 2 j < 1 is true. Furthermore it allows a more direct proof for the inequality 1 þ 2 < 0 where 0 > 1 > 2 > 3 are the Lyapunov exponents of the algorithm.

Spectral properties of Jacobi matrices by asymptotic analysis

Journal of Approximation Theory, 2003

We consider two classes of Jacobi matrix operators in l 2 with zero diagonals and with weights of the form n a þ c n for 0oap1 or of the form n a þ c n n aÀ1 for a41; where fc n g is periodic. We study spectral properties of these operators (especially for even periods), and we find asymptotics of some of their generalized eigensolutions. This analysis is based on some discrete versions of the Levinson theorem, which are also proved in the paper and may be of independent interest.

Perturbation of eigenvalues for periodic matrix pairs via the Bauer–Fike theorems

Linear Algebra and its Applications, 2004

We shall link the well-established concept of joint spectrum with the interesting and practical periodic eigenvalue problems (PEVPs). A Bauer-Fike perturbation theory, incorporating a Clifford algebra technique, for joint spectrum is applied to PEVPs, producing new perturbation results.

Relative oscillation theory for Jacobi matrices

Arxiv preprint arXiv:0810.5648, 2008

We develop relative oscillation theory for Jacobi matrices which, rather than counting the number of eigenvalues of one single matrix, counts the difference between the number of eigenvalues of two different matrices. This is done by replacing nodes of solutions associated with one matrix by weighted nodes of Wronskians of solutions of two different matrices.

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.