00-403 Oleg Safronov

The discrete spectrum of selfadjoint operators under perturbations of variable sign (256K, Postscript) Oct 13, 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 quantity $N(\lambda,A,W_+,W_-,\alpha)$ defined as the difference between the number of the eigenvalues of $A(t)$ that pass the point $\lambda$ from right to left and the number of the eigenvalues passing $\lambda$ from left to right as $t$ increases from $0$ to $\alpha.$ We study the asymptotic behavior of $N(\lambda,A,W_+,W_-,\alpha)$ as $\alpha\to \infty.$ Applications to Schr\"odinger and Dirac operators are given.

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