00-323 Oleg Safronov

The discrete spectrum in the spectral gaps of semibounded operators with non-signdefinite perturbations (188K, Postscript) Aug 25, 00

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Abstract. Given two selfadjoint operators $A$ and $V=V_+-V_-$, we study the motion of the eigenvalues of the operator $A(t)=A-tV$ as $t$ increases. Let $\alpha>0$ and let $\lambda$ be a regular point for $A$. We consider the quantities $N_+(V;\lambda,\alpha),\ N_-(V;\lambda,\alpha),\ N_0(V;\lambda,\alpha)$ defined as the number of the eigenvalues of the operator $A(t)$ that pass point $\lambda$ from the right to the left, from the left to the right or change the direction of their motion exactly at point $\lambda$, respectively, as $t$ increases from $0$ to $\alpha>0.$ We study asymptotic characteristics of these quantities as $\alpha\to \infty.$ In the present paper we extend the results obtained in \cite{S2} and give new applications to differential operators.

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