A Brunn-Minkowski inequality for Schrödinger operators with Kato class potentials

Abstract

In this paper we prove a Brunn-Minkowski inequality for the first Dirichlet eigenvalue of a Schrödinger 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 $Ω$ and $V$.