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arXiv · 2412.20282

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

Leonard Gross
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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.

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Leonard Gross. 2024-12-28. Invariance of intrinsic hypercontractivity under perturbation of Schr\"odinger operators. https://arxiv.org/abs/2412.20282

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