The Discrete Spectrum in the Spectral Gaps of Semibounded Operators with Non-sign-definite Perturbations |
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Authors: | O.L. Safronov |
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Affiliation: | a Department of Mathematics, Royal Institute of Technology, 10044, Stockholm, Swedenf1 |
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Abstract: | Given two self-adjoint operators A and V = V+ − V− , we study the motion of the eigenvalues of the operator A(t) = A − tV as t increases. Let α > 0 and let λ be a regular point for A. We consider the quantities N+ (V; λ, α), N− (V; λ, α), and N0(V; λ, α) defined as the number of eigenvalues of the operator A(t) that pass point λ from the right to the left, from the left to the right, or change the direction of their motion exactly at point λ, respectively, as t increases from 0 to α > 0. We study asymptotic characteristics of these quantities as α → ∞. In the present paper, the results obtained previously [O. L. Safronov, Comm. Math. Phys.193 (1998), 233–243] are extended and given new applications to differential operators. |
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