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

Abstract

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