Ukrainian Mathematical Journal

, Volume 55, Issue 9, pp 1532–1541 | Cite as

On the Point Spectrum of Self-Adjoint Operators That Appears under Singular Perturbations of Finite Rank

  • M. E. Dudkin
  • V. D. Koshmanenko

Abstract

We discuss purely singular finite-rank perturbations of a self-adjoint operator A in a Hilbert space ℋ. The perturbed operators \(\tilde A\) are defined by the Krein resolvent formula \((\tilde A - z)^{ - 1} = (A - z)^{ - 1} + B_z \), Im z ≠ 0, where Bz are finite-rank operators such that dom Bz ∩ dom A = |0}. For an arbitrary system of orthonormal vectors \(\{ \psi _i \} _{i = 1}^{n < \infty } \) satisfying the condition span |ψi} ∩ dom A = |0} and an arbitrary collection of real numbers \({\lambda}_i \in {\mathbb{R}}^1\), we construct an operator \(\tilde A\) that solves the eigenvalue problem \(\tilde A\psi _i = {\lambda}_i {\psi}_i , i = 1, \ldots ,n\). We prove the uniqueness of \(\tilde A\) under the condition that rank Bz = n.

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Copyright information

© Plenum Publishing Corporation 2003

Authors and Affiliations

  • M. E. Dudkin
    • 1
  • V. D. Koshmanenko
    • 2
  1. 1.Kiev Polytechnic InstituteKiev
  2. 2.Institute of MathematicsUkrainian Academy of SciencesKiev

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