Perturbations of embedded eigenvalues of asymptotically periodic magnetic Schr\"odinger operators on a cylinder

TL;DR

This paper studies how embedded eigenvalues of magnetic Schrödinger operators on a cylinder persist under perturbations, using advanced mathematical tools to characterize the set of potentials maintaining these eigenvalues.

Contribution

It demonstrates that under certain conditions, embedded eigenvalues persist in a smooth manifold of potentials, extending understanding of spectral stability in magnetic Schrödinger operators.

Findings

Embedded eigenvalues persist under specific perturbations.

The set of potentials preserving eigenvalues forms a smooth finite-codimension manifold.

An explicit example satisfying the theorem's assumptions is provided.

Abstract

We investigate the persistence of embedded eigenvalues for a class of magnetic Laplacians on an infinite cylindrical domain. The magnetic potential is assumed to be $C^2$ and asymptotically periodic along the unbounded direction, with an algebraic decay rate towards a periodic background potential. Under the condition that the embedded eigenvalue of the unperturbed operator lies away from the thresholds of the continuous spectrum, we show that the set of nearby potentials for which the embedded eigenvalue persists forms a smooth manifold of finite and even codimension. The proof employs tools from Floquet theory, exponential dichotomies, and Lyapunov--Schmidt reduction. Additionally, we give an example of a potential which satisfies the assumptions of our main theorem.