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Invariance of intrinsic hypercontractivity under perturbation of Schrödinger operators

Leonard Gross

TL;DR

The paper establishes that intrinsic hypercontractivity for Schrödinger operators is preserved under a broad class of perturbations by potentials, in a dimension‑free Dirichlet‑form setting. It develops a ground‑state transform framework, derives L^p hyperboundedness and moment‑product bounds using Aida’s WKB identity, and constructs defective logarithmic Sobolev inequalities for the ground‑state measure that can be tightened to obtain a spectral gap. The main contribution is a quantitative, parameter‑dependent perturbation theorem that yields explicit bounds on the ground‑state entropy and the spectral gap, valid in arbitrary dimensions. The results unify Bakry–Émery convexity with perturbation theory for Schrödinger operators and provide a versatile toolkit for generating logarithmic Sobolev inequalities across a broad class of models.

Abstract

A Schrödinger 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 Schrödinger operator ``intrinsically hypercontractive". I will show that if one adds a suitable potential onto an intrinsically hypercontractive Schrödinger 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.

Invariance of intrinsic hypercontractivity under perturbation of Schrödinger operators

TL;DR

The paper establishes that intrinsic hypercontractivity for Schrödinger operators is preserved under a broad class of perturbations by potentials, in a dimension‑free Dirichlet‑form setting. It develops a ground‑state transform framework, derives L^p hyperboundedness and moment‑product bounds using Aida’s WKB identity, and constructs defective logarithmic Sobolev inequalities for the ground‑state measure that can be tightened to obtain a spectral gap. The main contribution is a quantitative, parameter‑dependent perturbation theorem that yields explicit bounds on the ground‑state entropy and the spectral gap, valid in arbitrary dimensions. The results unify Bakry–Émery convexity with perturbation theory for Schrödinger operators and provide a versatile toolkit for generating logarithmic Sobolev inequalities across a broad class of models.

Abstract

A Schrödinger 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 Schrödinger operator ``intrinsically hypercontractive". I will show that if one adds a suitable potential onto an intrinsically hypercontractive Schrödinger 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.
Paper Structure (9 sections, 7 theorems, 34 equations)

This paper contains 9 sections, 7 theorems, 34 equations.

Table of Contents

  1. Introduction
  2. Statements
  3. The main theorem
  4. Non-standard hyperboundedness in $L^p(m)$
  5. A product of moments
  6. A defective LSI for $m_\psi$.
  7. Spectral gap
  8. Hyperboundedness of $\nabla^*\nabla +V$ in $L^p(m)$
  9. Interval of validity

Key Result

Theorem 2.2

$($Main theorem$)$. Assume that $m(X) =1$ and that Then a. $\nabla^*\nabla + V$ is essentially self-adjoint on ${\mathcal{D}}(\nabla^*\nabla) \cap L^\infty$. Let $H =$ closure of $\nabla^*\nabla + V$. b. $\lambda_0 \equiv \inf$ spectrum $H$ is an isolated eigenvalue of multiplicity one. It has an eigenfunction $\psi > 0\ a.e.$ with $\int_X \psi^2 dm There is a constant $c_1$ depending only on $c,

Theorems & Definitions (11)

  • Theorem 2.2
  • Remark 2.3
  • Remark 2.4
  • Theorem 2.6
  • Corollary 2.7
  • Remark 2.8
  • Corollary 2.9
  • Theorem 2.10
  • Remark 2.11
  • Theorem 2.12
  • ...and 1 more