Sharp Semiclassical Bounds for the Moments of Eigenvalues for Some Schrödinger Type Operators with Unbounded Potentials

University of Cape Town, Department of Mathematics, Private Bag, Rondebosch 7701, South Africa

^∗ Corresponding author. E-mail: Vitali.Vougalter@uct.ac.za

Abstract

We establish sharp semiclassical upper bounds for the moments of some negative powers for the eigenvalues of the Dirichlet Laplacian. When a constant magnetic field is incorporated in the problem, we obtain sharp lower bounds for the moments of positive powers not exceeding one for such eigenvalues. When considering a Schrödinger operator with the relativistic kinetic energy and a smooth, nonnegative, unbounded potential, we prove the sharp Lieb-Thirring estimate for the moments of some negative powers of its eigenvalues.