On the eigenvalue counting function for weighted Laplace-Beltrami operators

Abstract

A weighted Laplace-Beltrami operator on a Riemannian manifold M is an operator of the form $$H = - \sigma ^{ - 2} \nabla \cdot (\sigma \nabla )$$ where σ is a positive, locally bounded function defined on M. We obtain upper and lower bounds on the eigenvalue counting function of H for a class of incomplete manifolds with locally bounded geometry and for certain weights σ. Our method relies upon Dirichlet-Neumann bracketing and so no smoothness assumption on the metric or on σ is needed. Our results apply, in particular, to the Dirichlet Laplacian on unbounded domains in Rn satisfying Hardy’s inequality and to certain elliptic operators with degenerate coefficients.