Eigenvalues of Schrödinger operators with potential asymptotically homogeneous of degree $-2$

We strengthen and generalise a result of Kirsch and Simon on the behaviour of the function $N_L(E)$, the number of bound states of the operator $L = \Delta+V$ in $\mathbb{R}^d$ below $-E$. Here $V$ is a bounded potential behaving asymptotically like $P(\omega)r^{-2}$ where $P$ is a function on the sphere. It is well known that the eigenvalues of such an operator are all nonpositive, and accumulate only at 0. If the operator $\Delta_{S^{d-1}}+P$ on the sphere $S^{d-1}$ has negative eigenvalues $-\mu_1,\ldots,-\mu_n$ less than $-(d-2)^2/4$, we prove that $N_L(E)$ may be estimated as

$$N_L(E) = \frac{\log(E^{-1})}{2\pi}\sum_{i=1}^n \sqrt{\mu_i-(d-2)^2/4} +O(1).$$

Thus, in particular, if there are no such negative eigenvalues, then $L$ has a finite discrete spectrum. Moreover, under some additional assumptions including the fact that $d=3$ and that there is exactly one eigenvalue $-\mu_1$ less than $-1/4$, with all others $> -1/4$, we show that the negative spectrum is asymptotic to a geometric progression with ratio $\exp(-2\pi/\sqrt{\mu_1 - \frac{1}{4}})$.