Removal of the dependence on energy from interactions depending on it as a resolvent

A. K. Motovilov

Teoretičeskaâ i matematičeskaâ fizika, Tome 104 (1995) no. 2, pp. 281-303

Résumé

The spectral problem $(A + V(z))\psi =z\psi$ 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.