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 = Δ+V in ℝ^d below -E. Here V is a bounded potential behaving asymptotically like P(ω)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 Δ_S ^d-1 +P on the sphere S^d-1 has negative eigenvalues -μ1, ⋯ ,-μ_n less than -(d-2)²/4, we prove that N_L(E) may be estimated as N_L(E) =log(E^-1)/ 2πn∑i=1 √μi - (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 -μ1 less than -1/4, with all others > -1/4, we show that the negative spectrum is asymptotic to a geometric progression with ratio exp(-2π/ √μ1 - 1/4 ).