Sfb 288 Differential Geometry and Quantum Physics

Abstract for Sfb Preprint No. 387

The discrete spectrum of perturbed selfadjoint operators under non-signdefinite perturbations with a large coupling constant

Oleg Safronov

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 $alpha o infty.$ Applications to Schr"odinger and Dirac operators are given.

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