An inverse problem for the Hermite operator

Abstract

We prove that, for any convex planar set $\Omega$, the first non-trivial Neumann eigenvalue ${\mu }_1\left(\Omega \right)$ of the Hermite operator is greater than or equal to 1. Furthermore, under some additional assumptions on $\Omega$, we show that ${\mu }_1\left(\Omega \right)=1$ if and only if $\Omega$ is any strip. The study of the equality case requires, among other things, an asymptotic analysis of the Neumann eigenvalues of the Hermite operator in thin domains.