Universal inequalities and bounds for weighted eigenvalues of the Schrödinger operator on the Heisenberg group

Abstract

For a bounded domain \Omega in the Heisenberg group H^n, we investigate the Dirichlet weighted eigenvalue problem of the Schrödinger operator - \Delta_{H^n} +V, where \Delta_{H^n} is the Kohn Laplacian and V is a nonnegative potential. We establish a Yang-type inequality for eigenvalues of this problem. It contains the sharpest result for \Delta_{H^n} in [17] of Soufi, Harrel II and Ilias. Some estimates for upper bounds of higher order eigenvalues and the gaps of any two consecutive eigenvalues are also derived. Our results are related to some previous results for the Laplacian \Delta and the Schrödinger operator -\Delta+V on a domain in R^n and other manifolds.