Capacitary Estimates for Dirichlet Eigenvalues

LetHbe the non-negative definite selfadjoint operator associated to a regular irreducible Dirichlet form onL2(X, m). Assume thatHhas discrete spectrum. We study perturbations of this operator which arise through the imposition of Dirichlet boundary conditions on a compact subset ofX. The eigenvalues of the perturbed operator are of course shifted to the right. Under an ultracontractivity condition, we show that the magnitude of this shift can be estimated by the capacity. We also obtain a capacitary lower bound for the ground-state shift under suitable conditions. An application to the “crushed ice” problem is described.