Heat kernels and Green functions for fractional Schrödinger operators with confining potentials

TL;DR

The paper develops global, two-sided estimates for the heat kernel $p(t,x,y)$ and Green function $G^V(x,y)$ of the fractional Schrödinger operator $\mathcal{L}^V=\mathcal{L}-V$ with confining potential $V$ that grows at infinity. By imposing a doubling radial profile $g$ for $V$ and introducing the scale $t_0(s)$ from $e^{-t g(s)}=\frac{t}{(1+s)^{\alpha}}$, the authors derive sharp spatial bounds and qualitatively sharp time-behavior bounds, valid for all times, and distinguish short-time and large-time regimes. The large-time behavior is governed by the ground state with weight $H(x)=\frac{1}{g(|x|)(1+|x|)^{d+\alpha}}$, leading to $p(t,x,y)\asymp e^{-\lambda_1 t} H(x)H(y)$ under suitable monotonicity assumptions on $t_0$. An explicit example $V(x)=\log^{\beta}(1+|x|)$ is analyzed to illustrate the regime-switching and the corresponding Green-function bounds, highlighting the method's blend of probabilistic and analytic techniques via the Feynman–Kac formula and Lévy systems.

Abstract

We give two-sided, global (in all variables) estimates of the heat kernel and the Green function of the fractional Schrödinger operator with a non-negative and locally bounded potential $V$ such that $V(x) \to \infty$ as $|x| \to \infty$. We assume that $V$ is comparable to a radial profile with the doubling property. Our bounds are sharp with respect to spatial variables and qualitatively sharp with respect to time. The methods we use combine probabilistic and analytic arguments. They are based on the strong Markov property and the Feynman--Kac formula.

Theorems & Definitions (42)

• Theorem 1.1
• Example 1.2
• Theorem 1.3
• Lemma 2.1
• Lemma 2.2
• proof
• Lemma 2.3
• proof
• Lemma 2.4
• proof