Spectrally unstable domains

TL;DR

This paper investigates the spectral stability of selfadjoint extensions of unbounded operators in Hilbert spaces, identifying conditions under which certain domains cause spectral instability, especially relating to the Friedrichs extension.

Contribution

It characterizes spectrally unstable domains of selfadjoint extensions in terms of their relation to the Friedrichs extension's domain.

Findings

Spectrally unstable domains are characterized by their relation to the Friedrichs extension.

The set of selfadjoint extension domains forms a finite-dimensional manifold.

Spectral stability is preserved near certain domains, but instability can occur arbitrarily close to others.

Abstract

Let $H$ be a separable Hilbert space, $A_c:\mathcal D_c\subset H\to H$ a densely defined unbounded operator, bounded from below, let $\mathcal D_{\min}$ be the domain of the closure of $A_c$ and $\mathcal D_{\max}$ that of the adjoint. Assume that $\mathcal D_{\max}$ with the graph norm is compactly contained in $H$ and that $\mathcal D_{\min}$ has finite positive codimension in $\mathcal D_{\max}$. Then the set of domains of selfadjoint extensions of $A_c$ has the structure of a finite-dimensional manifold $\mathfrak {SA}$ and the spectrum of each of its selfadjoint extensions is bounded from below. If $\zeta$ is strictly below the spectrum of $A$ with a given domain $\mathcal D_0\in \mathfrak {SA}$, then $\zeta$ is not in the spectrum of $A$ with domain $\mathcal D\in \mathfrak {SA}$ near $\mathcal D_0$. But $\mathfrak {SA}$ contains elements $\mathcal D_0$ with the property that for every neighborhood $U$ of $\mathcal D_0$ and every $\zeta\in \mathbb R$ there is $\mathcal D\in U$ such that $\mathrm{spec}(A_\mathcal D)\cap (-\infty,\zeta)\ne \emptyset$. We characterize these "spectrally unstable" domains as being those satisfying a nontrivial relation with the domain of the Friedrichs extension of $A_c$.