Optimal design problems for Schr\"odinger operators with noncompact resolvents

TL;DR

This paper studies the optimization of potentials in Schrödinger operators to minimize cost functions related to their negative spectra, proving the existence of optimal solutions under certain conditions.

Contribution

It establishes the existence of optimal potentials for Schrödinger operators with prescribed support, extending to cases involving finitely many negative eigenvalues.

Findings

Existence of optimal potentials under certain assumptions

Applicable to cost functions involving finitely many negative eigenvalues

Provides a framework for spectral optimization in quantum mechanics

Abstract

We consider optimization problems for cost functionals which depend on the negative spectrum of Schr\"odinger operators of the form $-\Delta+V(x)$, where $V$ is a potential, with prescribed compact support, which has to be determined. Under suitable assumptions the existence of an optimal potential is shown. This can be applied to interesting cases such as costs functions involving finitely many negative eigenvalues.