A sharp lower bound for some neumann eigenvalues of the hermite operator

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(odd)(1)(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.