The distribution of negative eigenvalues of Schrödinger operators on asymptotically hyperbolic manifolds

We study the asymptotic behavior of the counting function of negative eigenvalues of Schrödinger operators with real valued potentials which decay at infinity on asymptotically hyperbolic manifolds. We establish conditions on the rate of decay of the potential that determine if there are finitely or infinitely many negative eigenvalues. In the latter case, they may only accumulate at zero and we obtain the asymptotic behavior of the counting function of eigenvalues in an interval (-\infty, -E) as E\rightarrow 0 .