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

摘要

This paper deals with the Neumann eigenvalue problem for the Hermite operatordefined in a convex, possibly unbounded, planar domain $\Omega$, having oneaxis of symmetry passing through the origin. We prove a sharp lower bound forthe first eigenvalue $\mu_1^{odd}(\Omega)$ with an associated eigenfunction oddwith respect to the axis of symmetry. Such an estimate involves the firsteigenvalue of the corresponding one-dimensional problem. As an immediateconsequence, in the class of domains for which$\mu_1(\Omega)=\mu_1^{odd}(\Omega)$, we get an explicit lower bound for thedifference between $\mu(\Omega)$ and the first Neumann eigenvalue of any strip.