Spectrally Unstable Domains

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 SA 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 D0 ∈ SA, then ζ is not in the spectrum of A with domain D ∈ SA near D0. But SA 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) 6= ∅. We characterize these “spectrally unstable” domains as being those satisfying a nontrivial relation with the domain of the Friedrichs extension of Ac.