Convexity inequalities for eigenvalues and log-concavity of eigenfunctions

TL;DR

This paper presents simplified proofs of key inequalities related to Schrödinger operators, including the Brunn--Minkowski inequality for eigenvalues and the log-concavity of the first eigenfunction.

Contribution

It introduces new, straightforward proofs for these classical results, applicable to a broad class of domains and potentials.

Findings

Proof of Brunn--Minkowski inequality for Dirichlet eigenvalues

Log-concavity of the first Dirichlet eigenfunction in convex domains

Applicability to $C^{1,1}$ connected domains and convex potentials

Abstract

We give simple new proofs of two well-known results for the Schr\"odinger operator: first, the Brunn--Minkowski inequality for Dirichlet eigenvalues and, second, the log-concavity of the first Dirichlet eigenfunction. Our proof of the first applies to a class of domains including $C^{1,1}$ connected domains and convex potentials. In the special case of convex domains, the second result is a simple corollary.