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

V. A. Sloushch

Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 28, Tome 270 (2000), pp. 325-335

Résumé

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.