On The Solvability Complexity Index for Unbounded Selfadjoint Operators and Schr\"{o}dinger Operators

We study the solvability complexity index (SCI) for unbounded selfadjoint operators on separable Hilbert spaces and perturbations thereof. In particular, we show that if the extended essential spectrum of a selfadjoint operator is convex, then the SCI for computing its spectrum is equal to 1. This result is then extended to relatively compact perturbations of such operators and applied to Schr\"{o}dinger operators with compactly supported (complex valued) potentials to obtain SCI=1 in this case, as well.