Intrinsic ultracontractivity and eigenfunction estimates for Schrödinger operators (Q1178622)

Let \(H_ 0\) be an uniformly elliptic operator in \(D\subset\mathbb{R}^ n\), \(n\geq 2\) with Dirichlet boundary conditions. Intrinsic ultraconductivity is proved for the semigroup of the Schrödinger operator \(H=H_ 0+V\), where \(V\) belongs to the Kato class of potentials and \(D\) is a Hölder domain of order 0 or a uniform Hölder domain of order \(\alpha\), \(0<\alpha<2\). For every \(\alpha\geq 2\), there exists a uniform Hölder domain of order \(\alpha\) for which the Dirichlet Laplacian is not intrinsically ultraconductive. For a certain class of domains it is shown that the heat kernels of \(H_ 0\) and \(H\) decay at the same rate at the boundary.