Invariance of intrinsic hypercontractivity under perturbation of Schr\"odinger operators

Abstract

A Schr\"odinger operator that is bounded below and has a unique positive ground state can be transformed into a Dirichlet form operator by the ground state transformation. If the resulting Dirichlet form operator is hypercontractive, Davies and Simon call the Schr\"odinger operator ``intrinsically hypercontractive". I will show that if one adds a suitable potential onto an intrinsically hypercontractive Schr\"odinger operator it remains intrinsically hypercontractive. The proof uses a fortuitous relation between the WKB equation and logarithmic Sobolev inequalities. All bounds are dimension independent. The main theorem will be applied to several examples.