Corrections to the classical behavior of the number of bound states of Schrödinger operators (Q1103777)

Let us denote by \(N_ E\) the number of bound states of the Schrödinger operator \(H=-\Delta -c/(1+| x|^ 2)+V_ 0\) below -E. \(V_ 0\) is a potential decaying at infinity sufficiently fast. We prove that, for dimension \(d=1\), \(\lim_{E\downarrow 0}(N_ E/| \ln E|)=(1/\pi)\sqrt{c-1/4}\) and for \(d=3\), \(\lim_{E\downarrow 0}(N_ E/| \ln E|)=\sum^{[\sqrt{c}-]}_{l=0}(2l+1)\sqrt{c-(l+)^ 2}\).