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Corrections to the classical behavior of the number of bound states of Schrödinger operators. (English) Zbl 0646.35019
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 $$d=3$$, $$\lim_{E\downarrow 0}(N_ E/| \ln E|)=\sum^{[\sqrt{c}-]}_{l=0}(2l+1)\sqrt{c-(l+)^ 2}$$.
Reviewer: W.Kirsch

##### MSC:
 35J10 Schrödinger operator, Schrödinger equation 35P05 General topics in linear spectral theory for PDEs
##### Keywords:
bound states; Schrödinger operator
Full Text:
##### References:
 [1] \scW. Kirsch and B. Simon, J. Funct. Anal., to appear. [2] Müller, C., () [3] Reed, M.; Simon, B., () [4] Reed, M.; Simon, B., ()
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