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

For a bounded domain W in the Heisenberg group Hn, we investigate the Dirichlet weighted eigenvalue problem of the Schrödinger operator - DHn +V, where DHn 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 DHn 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 D and the Schrödinger operator -D+V on a domain in Rn and other manifolds.