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Sharp two-sided heat kernel estimates for Schrödinger operators with decaying potentials

Xin Chen, Jian Wang

Abstract

We establish global two-sided heat kernel estimates (for full time and space) of the Schrödinger operator $-\frac{1}{2}Δ+V$ on $\R^d$, where the potential $V(x)$ is locally bounded and behaves like $c|x|^{-α}$ near infinity with $α\in (0,2)$ and $c> 0$, or with $α>0$ and $c<0$.Our results improve all known results in the literature, and it seems that the current paper is the first one where consistent two-sided heat kernel bounds for the long range potentials are established.

Sharp two-sided heat kernel estimates for Schrödinger operators with decaying potentials

Abstract

We establish global two-sided heat kernel estimates (for full time and space) of the Schrödinger operator on , where the potential is locally bounded and behaves like near infinity with and , or with and .Our results improve all known results in the literature, and it seems that the current paper is the first one where consistent two-sided heat kernel bounds for the long range potentials are established.
Paper Structure (10 sections, 25 theorems, 196 equations)

This paper contains 10 sections, 25 theorems, 196 equations.

Table of Contents

  1. Introduction
  2. The case for positive potentials
  3. Preliminaries
  4. Proof of Theorem \ref{['t1-1']}: the case that $0<t\le C_0t_0(\max\{|x|,|y|\})$
  5. Proof of Theorem \ref{['t1-1']}: the case that $t>C_0t_0(\max\{|x|,|y|\})$
  6. Proof of Theorem \ref{['t1-3']}: estimates of Green function
  7. The case for negative potentials
  8. Upper bound
  9. Lower bound
  10. Appendix

Key Result

Theorem 1.1

(Z1) Suppose that $d\ge 3$ and that there exist constants $\alpha>0$ and $K_1,K_2>0$ such that Then the following statements hold.

Theorems & Definitions (45)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Proposition 1.5
  • Remark 1.6
  • Lemma 2.1
  • proof
  • Remark 2.2
  • Lemma 2.3
  • ...and 35 more