Bulk behaviour of ground states for relativistic Schrödinger operators with compactly supported potentials

Abstract

We show a probabilistic representation of the ground state of a massive or massless Schrödinger operator with a potential well which leads to precise two-sided approximations inside the well in terms of the moment generating function of the first exit time from the well, and outside the well in terms of the Laplace transform of the first entrance time into the well. Combining scaling properties and heat kernel estimates, we derive explicit local rates both inside and outside the well. We also show how this approach extends to fully supported decaying potentials. Since these estimates rely on a monotone decreasing behaviour of the ground state, we develop a moving planes-type argument, which also covers the massive operator having an exponentially light Lévy measure.