Eigenvalue Problem and Estimates for a Class of Nilpotent Lie Group

Abstract:

      In this paper we discuss the spectral properties of the Laplacian on the nilpotentLie group H_n×R^k.We show that the nonexistence of eigenvalue on the whole space by usingthe unitary representation theory and the existence of Dirichlet’s eigenvalue on the boundeddomain of H_n×R^k with the extension of Friedrichs'method to classical Laplacian.We alsogive the estimates of the gap by establishing some useful relation of invariant vector fields.