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

Abstract:

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 $\exp (-2\pi /\sqrt {\mu _1 - \frac {1}{4}})$.