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

## TL;DR

This paper investigates the solvability complexity index (SCI) for unbounded selfadjoint operators, showing that convexity of the extended essential spectrum ensures SCI=1, with applications to Schrödinger operators.

## Contribution

It establishes conditions under which the SCI for spectrum computation of unbounded selfadjoint operators is equal to 1, extending results to perturbations and Schrödinger operators.

## Key findings

- SCI=1 for operators with convex extended essential spectrum
- SCI=1 for certain perturbed operators
- Application to Schrödinger operators with compactly supported potentials

## Abstract

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.