Abstract
A study is made of the spectral problem (A+V(z)ψ=zψ, in which the main HamiltonianA is a self-adjoint operator of fairly general nature, while the perturbationV(z)=−B(A′−z)−1B* depends on the energyz as the resolvent of some other self-adjoint operatorA′. It is assumed that the operatorB has a finite Hilbert—Schmidt norm and, in addition, that the spectra of the operatorsA andA′ are separated. Conditions are formulated under which it is possible to replace the perturbationV(z) by an energy-independent “potential”W such that the HamiltonianH=A+W has the same spectrum (more precisely, part of the spectrum) and the same eigenfunctions as the original spectral problem. Completeness and orthogonality theorems are proved for the systems of eigenfunctions of the operatorH=A+W. A scattering theory is constructed for the HamiltonianH in the case when the operatorA has a continuous spectrum. Applications of the results to the few-body problem are discussed.
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