Faber–Krahn inequalities for Schrödinger operators with point and with Coulomb interactions

Abstract

We obtain new Faber–Krahn-type inequalities for certain perturbations of the Dirichlet Laplacian on a bounded domain. First, we establish a two- and three-dimensional Faber–Krahn inequality for the Schrödinger operator with point interaction: the optimizer is the ball with the point interaction supported at its center. Next, we establish three-dimensional Faber–Krahn inequalities for a one- and two-body Schrödinger operator with attractive Coulomb interactions, the optimizer being given in terms of Coulomb attraction at the center of the ball. The proofs of such results are based on symmetric decreasing rearrangement and Steiner rearrangement techniques; in the first model, a careful analysis of certain monotonicity properties of the lowest eigenvalue is also needed.