Ground state representation for the fractional Laplacian with Hardy potential in angular momentum channels

TL;DR

This paper develops a ground state representation for the fractional Laplacian with Hardy potential in angular momentum channels, aiding the understanding of relativistic atomic models through spectral analysis.

Contribution

It introduces a novel ground state representation for the Hardy operator with fractional Laplacian in angular momentum channels, using subordinated Bessel heat kernels.

Findings

Provides a ground state representation on the half-line

Utilizes subordinated Bessel heat kernels for proofs

Enhances spectral analysis of relativistic atoms

Abstract

Motivated by the study of relativistic atoms, we consider 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 $L^2(\mathbb{R}^d)$, when $\kappa\geq0$ and $\alpha\in(0,2]\cap(0,d+2\ell)$. We give a ground state representation of the corresponding form on the half-line (Theorem 1.5). For the proof we use subordinated Bessel heat kernels.