Abstract:
Let $A$ be a selfadjoint operator, $(\alpha,\beta)$ a gap in the spectrum of $A$, $B=A+V$, where, in general, the perturbation operator $V$ is unbounded. We establish some abstract conditions under which the
spectrum of $B$ on $(\alpha,\beta)$ is discrete; does not accumulate to $\beta$; is finite. An estimate of the number of the eigenvalues of $B$ on $(\alpha,\beta)$ is obtained.