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

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 $u\left(\right|x\left|\right)\left|x{|}^{\ell }{Y}_{\ell ,m}\right(x/\left|x\right|)$ in $L^2\left(R^d\right)$, when $\kappa \ge 0$ 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.