The discrete spectrum of differential operators in the spectral gaps in the case of nonnegative perturbations of higher order

Let $A$ be a selfadjoint, elliptic second order differential operator, let $(\alpha,\beta)$ be the inner gap in the spectrum of $A$; let $B(t)=A+tW^*W$, where $W$ is a differential operator of higher order. Conditions are obtained that guarantee that the spectrum of the operator $B(t)$ in the gap $(\alpha,\beta)$ be discrete, or do not accumulate to the right edge of the spectral gap, or be finite. The quantity $N(\lambda,A,W,\tau)$, $\lambda\in(\alpha,\beta)$, $\tau>0$ (the number of eigenvalues of the operator $B(t)$ having passed the point $\lambda\in(\alpha,\beta)$ as $t$ increases from 0 to $\tau$) is considered. Estimates for $N(\lambda,A,W,\tau)$ are obtained. For the perturbation $W^*W$ of special from, the asymptotics of $N(\lambda,A,W,\tau)$, $\tau\to+\infty$, is given.