Spectrally unstable domains

Abstract

© 2017 University of Illinois. Let H be a separable Hilbert space, Ac : Dc ⊂ H →H a densely defined unbounded operator, bounded from below, let Dmin be the domain of the closure of Ac and Dmax that of the adjoint. Assume that Dmax with the graph norm is compactly contained in H and that Dmin has finite positive codimension in Dmax. Then the set of domains of selfadjoint extensions of Ac has the structure of a finite-dimensional manifold (Formula Presented) and the spectrum of each of its selfadjoint extensions is bounded from below. If ζ is strictly below the spectrum of A with a given domain (Formula Presented), then ζ is not in the spectrum of A with domain (Formula Presented) near D0. But (Formula Presented) contains elements D0 with the property that for every neighborhood U of D0 and every ζ ∈ R there is D∈U such that spec(AD) ∩ (−∞, ζ) ≠ Ø. We characterize these “spectrally unstable” domains as being those satisfying a nontrivial relation with the domain of the Friedrichs extension of Ac.