Cwikel-Lieb-Rozenblum type inequalities for Hardy-Schrödinger operator

We prove a Cwikel-Lieb-Rozenblum type inequality for the number of negative eigenvalues of the Hardy-Schrödinger operator $-\mathrm{\Delta }-(d-2{)}^2\mathrm{/}(4\mathrm{\mid }x{\mathrm{\mid }}^2)-W(x)$ on $L^2({\mathbb{R}}^d)$. The bound is given in terms of a weighted $L^{d\mathrm{/}2}-$norm of $W$ which is sharp in both large and small coupling regimes. We also obtain a similar bound for the fractional Laplacian.