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

Motivated by the study of relativistic atoms, we consider the Hardy operator $(-\mathrm{\Delta }{)}^{\alpha \mathrm{/}2}-\kappa \mathrm{\mid }x{\mathrm{\mid }}^{-\alpha }$ acting on functions of the form $u(\mathrm{\mid }x\mathrm{\mid })\mathrm{\mid }x{\mathrm{\mid }}^{\mathrm{\ell }}{Y}_{\mathrm{\ell },m}(x\mathrm{/}\mathrm{\mid }x\mathrm{\mid })$ in $L^2({\mathbb{R}}^d)$, when $\kappa \ge 0$ and $\alpha \in (0,2]\cap (0,d+2\mathrm{\ell })$. 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.