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 $\operatorname{spec}(A_{\mathcal{D} })\cap(-\infty,\zeta)\neq\emptyset$. We characterize these “spectrally unstable” domains as being those satisfying a nontrivial relation with the domain of the Friedrichs extension of $A_{c}$.