Removal of the Resolvent-like Dependence on the Spectral Parameter from Perturbations A
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چکیده
The spectral problem (A + V (z))ψ = zψ 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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تاریخ انتشار 1998