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

TL;DR

This paper demonstrates that the property of intrinsic hypercontractivity of Schrödinger operators remains stable under certain potential perturbations, with proofs leveraging relations between WKB equations and logarithmic Sobolev inequalities.

Contribution

It establishes the invariance of intrinsic hypercontractivity under potential perturbations for Schrödinger operators, extending previous understanding of their stability.

Findings

Intrinsic hypercontractivity is preserved under suitable potential additions.

Bounds are independent of the dimension.

The main theorem applies to various examples.

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.