Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Heat Kernel Estimates for Schrödinger Operators in the Domain Above a Bounded Lipschitz Function

Anthony Graves-McCleary

TL;DR

This work derives sharp two-sided heat-kernel bounds for the Dirichlet Schrödinger operator $\Delta+W$ in a domain $\Omega$ situated above a bounded Lipschitz graph, with $W$ decaying like $\langle x\rangle^{-(2+\epsilon)}$ away from the boundary. The approach combines uniform-domain theory, boundary Harnack principles, harmonic profiles, and Doob's $h$-transform to reduce the problem to a weighted heat kernel $p_\mu$ and a corresponding Green’s function, yielding estimates of the form $p^W(t,x,y) \asymp \frac{h(x)h(y)}{h(x+\sqrt{t})h(y+\sqrt{t})t^{N/2}} e^{-c d(x,y)^2/t}$. A key contribution is the demonstration of existence and comparability of the harmonic profile for $\Delta+W$ with the original harmonic profile, under a conditional gaugeability framework, together with robust integral bounds for Green’s functions weighted by $W$. The paper also outlines obstacles in extending these results to unbounded Lipschitz graphs, underscoring the role of domain geometry in heat-kernel behaviour.

Abstract

We give matching upper and lower bounds for the Dirichlet heat kernel of a Schrödinger operator $Δ+W$ in the domain above the graph of a bounded Lipschitz function, in the case when $W$ decays away from the boundary faster than quadratically.

Heat Kernel Estimates for Schrödinger Operators in the Domain Above a Bounded Lipschitz Function

TL;DR

This work derives sharp two-sided heat-kernel bounds for the Dirichlet Schrödinger operator in a domain situated above a bounded Lipschitz graph, with decaying like away from the boundary. The approach combines uniform-domain theory, boundary Harnack principles, harmonic profiles, and Doob's -transform to reduce the problem to a weighted heat kernel and a corresponding Green’s function, yielding estimates of the form . A key contribution is the demonstration of existence and comparability of the harmonic profile for with the original harmonic profile, under a conditional gaugeability framework, together with robust integral bounds for Green’s functions weighted by . The paper also outlines obstacles in extending these results to unbounded Lipschitz graphs, underscoring the role of domain geometry in heat-kernel behaviour.

Abstract

We give matching upper and lower bounds for the Dirichlet heat kernel of a Schrödinger operator in the domain above the graph of a bounded Lipschitz function, in the case when decays away from the boundary faster than quadratically.
Paper Structure (10 sections, 24 theorems, 88 equations)

This paper contains 10 sections, 24 theorems, 88 equations.

Table of Contents

  1. Introduction and Overview
  2. Notation
  3. Uniform Domains and the Boundary Harnack Principle
  4. Heat Kernel Estimates of the Laplacian in a Lipschitz Domain
  5. Estimates in a Lipschitz Domain
  6. Conditional Gaugeability
  7. Dirichlet Forms and Doob's $h$-Transform
  8. Heat Kernel Estimates: Main Results
  9. Obstacles to the Unbounded Case: Example in $\mathbf R^2$
  10. Acknowledgements

Key Result

Theorem 2.3

Let $\Omega$ be an unbounded uniform domain in $\mathbf R^N$. There exist constants $A_0, A_1>1$ such that for any $\xi\in \partial \Omega$, any $r>0$, and any positive harmonic functions $u$ and $v$ in $B(\xi, A_0r)\cap \Omega$ with Dirichlet boundary conditions along $\partial \Omega$, we have tha

Theorems & Definitions (28)

  • Definition 2.1
  • Definition 2.2
  • Theorem 2.3
  • Definition 2.4
  • Lemma 2.5
  • Proposition 2.6
  • Proposition 3.1
  • Lemma 3.2
  • Theorem 3.3
  • Corollary 3.4
  • ...and 18 more