Mathematics – Spectral Theory
Scientific paper
1998-10-07
Zeitschrift fuer Angewandte Mathematik und Mechanik (ZAMM), Special Issue 2 (ISBN 3-05-501745-5), 229-232 (1996)
Mathematics
Spectral Theory
LaTeX, 4 pages, no figures
Scientific paper
The spectral problem (A + V(z))\psi=z\psi is 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.
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