Eigenvalue bounds for non-self-adjoint Schr\"odinger operators and pseudodifferential generalizations

TL;DR

This survey paper reviews spectral bounds for complex potential Schrödinger operators on various spaces and introduces a new theorem extending these bounds to fractional Laplacians on compact manifolds.

Contribution

It provides a comprehensive collection of spectral bounds results and introduces a novel theorem for fractional Laplacians on compact manifolds.

Findings

Spectral bounds are established for Schrödinger operators with complex potentials.

Extension of spectral bounds to fractional Laplacians on compact manifolds.

Abstract

This is mostly a survey paper, where we collect results concerning the spectral bounds of deterministic and random Schr\"odinger operators with complex potentials, both on \(\mathbb{R}^d\) and on compact manifolds. The survey part is complemented by a new theorem, where we extend the result on spectral bounds on compact manifolds to the case of fractional Laplacians, applying methods by Cuenin and Sogge. These bounds are formulated in terms of the \(L^p\)-norms of the corresponding potentials.