Heat kernel estimates of fractional Schrödinger operators with Hardy potential on half-line

TL;DR

The paper analyzes sharp two-sided heat-kernel estimates for the Dirichlet fractional Laplacian on the half-line perturbed by a Hardy potential $q(x)=\kappa x^{-\alpha}$, with $\kappa=\kappa_{\delta}$ and $\delta\in(0,1/2]$. It constructs the perturbed kernel $\tilde{p}$ via a ground-state–type perturbation series using the weighted density $p_D^{(w)}$ and proves invariant relations for $x^{-\delta}$, leading to precise two-sided bounds of the form $\tilde{p}(t,x,y)\approx \left( 1 \wedge \frac{x}{t^{1/\alpha}} \right)^{\alpha/2-\delta}\left( 1 \wedge \frac{y}{t^{1/\alpha}} \right)^{\alpha/2-\delta}\left( t^{-1/\alpha}\wedge \frac{t}{|x-y|^{1+\alpha}} \right)$. The authors establish joint continuity of $\tilde{p}^{(w)}(t,x,y)$ and derive a matching lower bound, while also identifying a blow-up threshold: if $\kappa>\kappa^{*}$, the perturbed kernel blows up, with a saturated kernel at the critical value $\kappa^{*}$. This work extends local Hardy-inequality-based results to a fractional nonlocal setting on the half-line, providing a fractional counterpart to classical and nonlocal Hardy-perturbed heat-kernel estimates with potential-analytic implications.

Abstract

We provide sharp two-sided estimates of the heat kernel of the Dirichlet fractional Laplacian on the half-line perturbed by the Hardy potential.

Theorems & Definitions (39)

• Theorem 1.1
• Lemma 2.1
• proof
• Theorem 3.1
• Lemma 3.2
• proof
• Lemma 3.3
• proof
• Lemma 3.4
• proof