Perturbations of embedded eigenvalues for the planar bilaplacian

TL;DR

This paper investigates how embedded eigenvalues of the planar bilaplacian with potential behave under small perturbations, revealing that the set of potentials preserving such eigenvalues forms a complex infinite-dimensional structure.

Contribution

It demonstrates that the set of potentials maintaining embedded eigenvalues in the planar bilaplacian is a highly intricate infinite-dimensional manifold with infinite codimension.

Findings

Embedded eigenvalues are generally unstable under perturbations.

The set of potentials preserving embedded eigenvalues forms an infinite-dimensional manifold.

Persistence of embedded eigenvalues is linked to the multiplicity of the essential spectrum.

Abstract

Operators on unbounded domains may acquire eigenvalues that are embedded in the essential spectrum. Determining the fate of these embedded eigenvalues under small perturbations of the underlying operator is a challenging task, and the persistence properties of such eigenvalues is linked intimately to the multiplicity of the essential spectrum. In this paper, we consider the planar bilaplacian with potential and show that the set of potentials for which an embedded eigenvalue persists is locally an infinite-dimensional manifold with infinite codimension in an appropriate space of potentials.