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 $V_0$.