Eigenvalues for perturbed periodic Jacobi matrices by the Wigner-von Neumann approach

TL;DR

This paper extends the Wigner-von Neumann perturbation method from continuous Schrödinger operators to discrete periodic Jacobi matrices, analyzing eigenvalue embedding and spectrum characteristics.

Contribution

It introduces a new technique for embedding eigenvalues into the spectrum of periodic Jacobi matrices using the Wigner-von Neumann approach.

Findings

Identifies conditions for eigenvalue embedding in the spectrum.

Introduces a rational function C(λ;T) related to the spectrum.

Shows only finitely many spectral elements where the method fails.

Abstract

The Wigner-von Neumann method, which was previously used for perturbing continuous Schr\"{o}dinger 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, $\lambda$, into the operator's absolutely continuous spectrum. Introducing a new rational function, $C(\lambda;T)$, related to the periodic Jacobi matrices, we describe the elements of the a.c. spectrum for which this construction does not work (zeros of $C(\lambda;T)$); in particular showing that there are only finitely many of them.