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Heat Kernel Estimates for Schrödinger Operators with Decay at Infinity on Parabolic Manifolds

Anthony Graves-McCleary, Laurent Saloff-Coste

TL;DR

The paper addresses heat-kernel estimates for Schrödinger operators $\Delta_\mu+W$ on parabolic weighted manifolds where $W$ decays faster than quadratically. It develops an integrated framework combining Doob's $h$-transform, gaugeability, and subcritical/critical theory to relate the Schrödinger heat kernel to transformed kernels, yielding sharp upper and lower bounds. The main contributions include explicit subcritical and critical heat-kernel bounds expressed via growth functions $H$ and volume $V_μ$, extending Murata–Grigor'yan type results from $\mathbb{R}^2$ to broad parabolic manifolds, and providing concrete examples (model manifolds, infinite half-cylinder). The results have broad implications for Green's functions, potential theory, and parabolic heat-kernel analysis on manifolds with density, illustrating how decay at infinity governs long-time behavior of heat flow under Schrödinger perturbations.

Abstract

We give estimates for positive solutions for the Schrödinger equation $(Δ_μ+W)u=0$ on a wide class of parabolic weighted manifolds $(M, dμ)$ when $W$ decays to zero at infinity faster than quadratically. These can be combined with results of Grigor'yan to give matching upper and lower bounds for the heat kernel of the corresponding Schrödinger operator $Δ_μ+W$. In particular, this appears to complement known results for Schrödinger operators on $\mathbf{R}^2$.

Heat Kernel Estimates for Schrödinger Operators with Decay at Infinity on Parabolic Manifolds

TL;DR

The paper addresses heat-kernel estimates for Schrödinger operators on parabolic weighted manifolds where decays faster than quadratically. It develops an integrated framework combining Doob's -transform, gaugeability, and subcritical/critical theory to relate the Schrödinger heat kernel to transformed kernels, yielding sharp upper and lower bounds. The main contributions include explicit subcritical and critical heat-kernel bounds expressed via growth functions and volume , extending Murata–Grigor'yan type results from to broad parabolic manifolds, and providing concrete examples (model manifolds, infinite half-cylinder). The results have broad implications for Green's functions, potential theory, and parabolic heat-kernel analysis on manifolds with density, illustrating how decay at infinity governs long-time behavior of heat flow under Schrödinger perturbations.

Abstract

We give estimates for positive solutions for the Schrödinger equation on a wide class of parabolic weighted manifolds when decays to zero at infinity faster than quadratically. These can be combined with results of Grigor'yan to give matching upper and lower bounds for the heat kernel of the corresponding Schrödinger operator . In particular, this appears to complement known results for Schrödinger operators on .
Paper Structure (15 sections, 39 theorems, 132 equations)

This paper contains 15 sections, 39 theorems, 132 equations.

Table of Contents

  1. Introduction
  2. Notation
  3. Background and Overview
  4. Setting: Weighted Manifolds
  5. Parabolic Manifolds and Harmonic Functions Outside a Ball
  6. Doob's $h$-transform
  7. Iterated $h$-Transform
  8. Compactly Supported Potentials
  9. Gaugeability in Non-Parabolic Manifolds
  10. Subcriticality and Criticality Theory
  11. Potentials with Faster Than Quadratic Decay: Main Results
  12. Applications and Examples
  13. Existence of Critical Operators
  14. More Examples
  15. Acknowledgements

Key Result

Theorem 1.1

(Murata murata2) Let $W$ be such that $\left< x \right>^{\alpha-\frac{2}{p}}\left\vert W(x)\right\vert\in L^p(\mathbf R^2, dx)$ for $p>1$, $\alpha>2$. Then if $\Delta+W$ is subcritical, the profile $h>0$ for $\Delta+W$ satisfies If $\Delta+W$ is critical, the profile $h>0$ for $\Delta+W$ satisfies

Theorems & Definitions (58)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 1.3
  • Definition 1.4
  • Definition 1.5
  • Definition 1.6
  • Definition 1.7
  • Definition 1.8
  • Theorem 1.9
  • Theorem 1.10
  • ...and 48 more