Intrinsic ultracontractivity and eigenfunction estimates for Schrödinger operators

Abstract

Let D be a domain in R N , n ⩾ 2, and let H = H 0 + V, where H 0 is a divergence form uniformly elliptic operator with Dirichlet boundary conditions and V is in the Kato class. Intrinsic ultracontractivity is proved for the semigroup of H when D is Hölder domain of order 0 or a uniformly Hölder domain of order α for 0 < α < 2. For every α ⩾ 2, there exists a uniformly Hölder domain of order α for which the Dirichlet Laplacian is not intrinsically ultracontractive. For a large class of domains it is shown that the heat kernels for H 0 and H decay at the same rate at the boundary. Applications are given to the lifetime of conditioned Brownian motion. Some of our results seem to be new even for smooth domains.