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 NL(E), the number of bound states of the operator L = ∆ + V in Rd 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 ∆Sd−1 +P on the sphere S d−1 has negative eigenvalues −μ1, . . . ,−μn less than −(d−2)2/4, we prove that NL(E) may be estimated as NL(E) = log(E−1) 2π n ∑ 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 − 14 ).