Eigenvalues of Schroedinger 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$\R^d$below$-E$\mathrm{.}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 ${\mathrm{\Delta }}_{S^{d-1}}+P$ on the sphere has negative eigenvalues $-{\mu }_1,\mathrm{.}\mathrm{.}\mathrm{.},-{\mu }_n$ less than $-(d-2{)}^2\mathrm{/}4$, we prove that $N_L(E)$ may be estimated as $$N_L(E))=\frac{\log (E^{-1})}{2\pi }\sum _{i=1}^n\sqrt{{\mu }_i-(d-2{)}^2\mathrm{/}4}+O(1);$$ thus, in particular, if there are no such negative eigenvalues then $L$ has a finite discrete spectrum. Moreover, under some additional assumptions including that $d=3$ and that there is exactly one eigenvalue $-{\mu }_1$ less than -1/4, with all others $>-1\mathrm{/}4$, we show that the negative spectrum is asymptotic to a geometric progression with ratio $\exp (-2\pi \mathrm{/}\sqrt{{\mu }_1-\text{qtr}})$.