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

V. A. Sloushch

Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 28, Tome 270 (2000), pp. 317-324

Résumé

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.