A sharp lower bound for some Neumann eigenvalues of the Hermite operator (Q2444521)

The authors study the Neumann eigenvalue problem for the Hermite operator on planar, convex domains that are symmetric with respect to an axis crossing the origin. They prove a lower bound for the lowest eigenvalue corresponding to eigenfunctions that are odd with respect to the symmetry axis. This estimate is sharp when the domain is a rectangle, or a right strip, unbounded in one direction, along the symmetry axis. The result is proved firstly in the case of bounded domains using a dimension-reduction argument which involves a comparison with the first nontrivial eigenvalue of a corresponding one-dimensional problem. The case of unbounded domains is also treated by a suitable extension procedure.