Notes on Rank One Perturbed Resolvent. Perturbation of Isolated Eigenvalue

Details Notes on Rank One Perturbed Resolvent. Perturbation of Isolated Eigenvalue Notes on Rank One Perturbed Resolvent. Perturbation of Isolated Eigenvalue

2003-12-27

arxiv.org/abs/math-ph/0312070v2

Physics

Mathematical Physics

Latex 2.09

Scientific paper

This paper is a didactic commentary (a transcription with variations) to the paper of S.R. Foguel {\it Finite Dimensional Perturbations in Banach Spaces}. Addressed, mainly: postgraduates and related readers. Subject: Suppose we have two linear operators, A, B, so that B - A is rank one. Let \lambda_o be an {\it isolated} point of the spectrum of A. In addition, let \lambda_o be an {\it eigenvalue} of A: \lambda_o \in \sigma_{pp}(A) . The question is: Is \lambda_o an eigenvalue of B ? And, if so, is the multiplicity of \lambda_o in \sigma_{pp}(B) equal to the multiplicity of \lambda_o in \sigma_{pp}(A) ? -- or less? -- or greater? Keywords: M.G.Krein's Formula, Finite Rank Perturbation

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