Removal of the resolvent-like dependence on the spectral parameter from perturbations

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dc.creatorMotovilov, A. K.
dc.date1998-10-07
dc.date.accessioned2026-07-07T05:26:20Z
dc.date.available2026-07-07T05:26:20Z
dc.descriptionThe spectral problem (A + V(z))ψ=zψis considered with A, a self-adjoint operator. The perturbation V(z) is assumed to depend on the spectral parameter z as resolvent of another self-adjoint operator A': V(z)=-B(A'-z)^{-1}B^{*}. It is supposed that the operator B has a finite Hilbert-Schmidt norm and spectra of the operators A and A' are separated. Conditions are formulated when the perturbation V(z) may be replaced with a ``potential'' W independent of z and such that the operator H=A+W has the same spectrum and the same eigenfunctions (more precisely, a part of spectrum and a respective part of eigenfunctions system) as the initial spectral problem. The operator H is constructed as a solution of the non-linear operator equation H=A+V(H) with a specially chosen operator-valued function V(H). In the case if the initial spectral problem corresponds to a two-channel variant of the Friedrichs model, a basis property of the eigenfunction system of the operator H is proved. A scattering theory is developed for H in the case where the operator A has continuous spectrum.
dc.descriptionLaTeX, 4 pages, no figures
dc.identifierhttps://arxiv.org/abs/math/9810044
dc.identifierhttp://arxiv.org/abs/math/9810044
dc.identifierZeitschrift fuer Angewandte Mathematik und Mechanik (ZAMM), Special Issue 2 (ISBN 3-05-501745-5), 229-232 (1996)
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/77516
dc.subjectSpectral Theory
dc.subjectPrimary 47A56, 47Nxx; Secondary 47N50, 47A40
dc.titleRemoval of the resolvent-like dependence on the spectral parameter from perturbations
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