The discrete spectrum asymptotics with large coupling constant in the case of strong nonnegative perturbations

Let $A$ be a selfadjoint operator, $(\alpha,\beta)$ the inner gap in the spectrum of the operator $A$; let $B(t)=A+tW^*W$, where the operator $W(A-iI)^{-1}$ is not necessarily bounded. Conditions are obtained that guarantee that the spectrum of $B(t)$ in $(\alpha,\beta)$ be discrete. Let $N(\lambda,A,W,\tau)$, $\lambda\in(\alpha,\beta)$, $\tau>0$ be the number of eigenvalues of the operator $B(t)$ having passed the point $\lambda\in(\alpha,\beta)$ as $t$ increases from 0 to $\tau$. The asymptotics $N(\lambda,A,W,\tau)$, $\tau\to+\infty$, is obtained in terms of the spectral asymptotics of a certain selfadjoint compact operator.