Heat kernel bounds for the fractional Laplacian with Hardy potential in angular momentum channels

TL;DR

This paper establishes precise heat kernel bounds for a fractional Laplacian operator with Hardy potential, relevant to relativistic atomic models, by analyzing angular momentum channels and deriving sharp estimates.

Contribution

It provides the first sharp heat kernel bounds for the fractional Hardy operator in angular momentum channels, extending previous results to a broader class of operators.

Findings

Derived sharp heat kernel bounds for the fractional Hardy operator.

Extended heat kernel estimates to functions with angular momentum components.

Results applicable to models of relativistic atoms.

Abstract

Motivated by the study of relativistic atoms, we prove sharp heat kernel bounds for the Hardy operator $(-\Delta)^{\alpha/2}-\kappa|x|^{-\alpha}$ acting on functions of the form $u(|x|) |x|^{\ell} Y_{\ell,m}(x/|x|)$ in $L^2(\R^d)$, when $\alpha\in(0,2]\cap(0,d+2\ell)$.