Removal of the Energy Dependence from the Resolvent-like Energy-dependent Interactions

The spectral problem (A+V (z))ψ = zψ is considered with A, a self-adjoint Hamiltonian of sufficiently arbitrary nature. The perturbation V (z) is assumed to depend on the energy z as resolvent of another self-adjoint operator A ′ : V (z) = −B(A ′ −z) −1 B *. It is supposed that operator B has a finite Hilbert-Schmidt norm and spectra of operators A and A ′ are separated. The conditions are formulated when the perturbation V (z) may be replaced with an energy-independent " potential " W such that the Hamiltonian H = A + W has the same spectrum (more exactly a part of spectrum) and the same eigenfunctions as the initial spectral problem. The orthogonality and expansion theorems are proved for eigenfunction systems of the Hamiltonian H = A + W. Scattering theory is developed for H in the case when operator A has continuous spectrum. Applications of the results obtained to few-body problems are discussed.