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 $-\Delta -{(d-2)}^2/\left(4\right|x{|}^2)-W(x)$ on $L^2\left(R^d\right)$. 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.