Oleg Safronov
The discrete spectrum of the Schr\"odinger operator 
in the large coupling constant limit
(238K, Postscript)

ABSTRACT.  In a simple quantum mechanical model for impurities in a crystal, 
one studies the spectrum of a periodic Schr\"odinger operator 
$A=-\Delta+p(x)$, 
perturbed by a decaying potential $-\alpha V_0$, 
which models the impurity ($\alpha>0$ is a coupling constant). 
The basic assumptions are that $H$ has a spectral gap $\Lambda$ 
in its essential spectrum; eigenvalues of $A(\alpha)=A-\alpha V_0$ 
 inside $\Lambda$ 
correspond to the eigenstates of the electrons observed in solid state physics. 
We study the number of eigenvalues of $A(\alpha)$ 
inside an interval $(\lambda_1,\lambda_2)\subset \Lambda$. 
The main purpose of our study is to obtain an asymptotics 
of this number, as $\alpha\to \infty$. 
We point out that the problem 
has been solved only for one-dimensional 
Schr\"odinger operators in \cite{Sob}. 
The results of the present paper deal with 
 multidimensional Schr\"odinger operators. 
However we consider 
only the case of spherically symmetric 
potentials $V_0$.