Universal inequalities and bounds for weighted eigenvalues of the Schr¨odinger operator on the Heisenberg group

For a bounded domain Ω in the Heisenberg group Hn, we investigate the Dirichlet weighted eigenvalue problem of the Schrodinger operator −ΔHn + V, where ΔHn 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 ΔHn 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 Δ and the Schr¨odinger operator −Δ+ V on a domain in Rn and other manifolds.