Spectrally unstable domains

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 $A_c$.