 00323 Oleg Safronov
 The discrete spectrum in the spectral gaps of semibounded operators with
nonsigndefinite perturbations
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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
$A(t)=AtV$ 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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