A sharp lower bound for some Neumann eigenvalues of the Hermite operator

Abstract

This paper deals with the Neumann eigenvalue problem for the Hermite operator defined in a convex, possibly unbounded, planar domain $\Omega$, having one axis of symmetry passing through the origin. We prove a sharp lower bound for the first eigenvalue ${\mu }_1^{odd}(\Omega )$ with an associated eigenfunction odd with respect to the axis of symmetry. Such an estimate involves the first eigenvalue of the corresponding one-dimensional problem. As an immediate consequence, in the class of domains for which ${\mu }_1(\Omega )={\mu }_1^{odd}(\Omega )$, we get an explicit lower bound for the difference between $\mu (\Omega )$ and the first Neumann eigenvalue of any strip.