Embedded eigenvalues for perturbed periodic Jacobi operators using a geometric approach

TL;DR

This paper introduces a geometric method to embed infinitely many eigenvalues into the essential spectrum of periodic Jacobi operators using oscillating potentials, relaxing previous rational dependence constraints.

Contribution

It extends a geometric approach to periodic Jacobi operators, allowing for the embedding of infinitely many eigenvalues with fewer restrictions on quasi-momenta.

Findings

Successfully embeds infinitely many eigenvalues into the spectrum.

Relaxes rational dependence conditions on quasi-momenta.

Provides conditions for subordinate solutions and eigenvalue embedding.

Abstract

We consider the problem of embedding eigenvalues into the essential spectrum of periodic Jacobi operators, using an oscillating, decreasing potential. To do this we employ a geometric method, previously used to embed eigenvalues into the essential spectrum of the discrete Schr\"{o}dinger operator. For periodic Jacobi operators we relax the rational dependence conditions on the values of the quasi-momenta from this previous work. We then explore conditions that permit not just the existence of infinitely many subordinate solutions to the formal spectral equation but also the embedding of infinitely many eigenvalues.