Persistence of embedded eigenvalues

TL;DR

This paper investigates conditions ensuring that embedded eigenvalues of self-adjoint operators persist under small perturbations, revealing that in certain cases, the set of perturbations preserving the eigenvalue forms a smooth submanifold.

Contribution

It provides a geometric characterization of the set of perturbations that preserve embedded eigenvalues in self-adjoint operators.

Findings

Embedded eigenvalues can remain under small perturbations in specific conditions.

The set of perturbations preserving an embedded eigenvalue forms a smooth submanifold of co-dimension m.

The results apply to simple eigenvalues with finite multiplicity embedded in the continuous spectrum.

Abstract

We consider conditions under which an embedded eigenvalue of a self-adjoint operator remains embedded under small perturbations. In the case of a simple eigenvalue embedded in continuous spectrum of multiplicity m < \infty we show that in favorable situations the set of small perturbations of a suitable Banach space which do not remove the eigenvalue form a smooth submanifold of co-dimension m.