Perturbations of embedded eigenvalues for self-adjoint ODE systems

TL;DR

This paper investigates how small perturbations affect embedded eigenvalues in self-adjoint differential operators, showing that the set of perturbations preserving the embedded eigenvalue forms a smooth manifold with a specified co-dimension.

Contribution

The paper introduces a detailed analysis of the structure of perturbations that keep eigenvalues embedded, utilizing exponential dichotomies and Lyapunov-Schmidt reduction techniques.

Findings

The set of small perturbations preserving the embedded eigenvalue is a smooth manifold.

The co-dimension of this manifold is explicitly characterized.

The methods involve exponential dichotomies and their roughness properties.

Abstract

We consider a perturbation problem for embedded eigenvalues of a self-adjoint differential operator in $L^2(\mathbb R;\mathbb R^n)$. In particular, we study the set of all small perturbations in an appropriate Banach space for which the embedded eigenvalue remains embedded in the continuous spectrum. We show that this set of small perturbations forms a smooth manifold and we specify its co-dimension. Our methods involve the use of exponential dichotomies, their roughness property and Lyapunov-Schmidt reduction.