A Brunn-Minkowski inequality for Schr\"odinger operators with Kato class potentials

TL;DR

This paper establishes a Brunn-Minkowski inequality for the first Dirichlet eigenvalue of Schrödinger operators with convex, Kato class potentials, leading to log-concavity results for the ground state.

Contribution

It introduces a novel Brunn-Minkowski inequality for Schrödinger operators with Kato potentials, extending geometric inequalities to spectral properties.

Findings

Proved a Brunn-Minkowski inequality for the first eigenvalue of Schrödinger operators.

Established log-concavity of the ground state wave function.

Derived strong log-concavity under additional geometric and potential conditions.

Abstract

In this paper we prove a Brunn-Minkowski inequality for the first Dirichlet eigenvalue of a Schr\"odinger type operator $\mathcal{H}_V:=-\operatorname{div}(A\nabla)+V$, where $V$ is convex and Kato decomposable, using the trace class property of the generated semigroup. As a consequence, we obtain the log-concavity of the ground state using the ultracontractivity of the semigroup, and also the strong log-concavity under additional assumptions on $\Omega$ and $V$.