Removal of the resolvent-like energy dependence from interactions and invariant subspaces of a total Hamiltonian

TL;DR

This paper develops a method to replace energy-dependent interactions in a Hamiltonian with an energy-independent potential, preserving spectral properties and eigenfunctions, and explores invariant subspaces and scattering theory for such systems.

Contribution

It introduces conditions and a construction for replacing resolvent-like energy dependence with a static potential, linking the problem to invariant subspaces and scattering theory.

Findings

Equivalent spectrum and eigenfunctions achieved with the potential W

Established orthogonality and expansion theorems for eigenfunctions

Developed scattering theory for Hamiltonians with continuous spectrum

Abstract

The spectral problem $(A + V(z))\psi=z\psi$ is considered where the main Hamiltonian $A$ is a self-adjoint operator of sufficiently arbitrary nature. The perturbation $V(z)=-B(A'-z)^{-1}B^{*}$ depends on the energy $z$ as resolvent of another self-adjoint operator $A'$. The latter is usually interpreted as Hamiltonian describing an internal structure of physical system. The operator $B$ is assumed to have a finite Hilbert-Schmidt norm. The conditions are formulated when one can replace the perturbation $V(z)$ 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 Hamiltonian $H$ is constructed as a solution of the non-linear operator equation $H=A+V(H)$. It is established that this equation is closely connected with the problem of searching for invariant subspaces of the Hamiltonian $ {\bf H}=\left[ \begin{array}{lr} A & B B^{*} & A' \end{array}\right]. $ The orthogonality and expansion theorems are proved for eigenfunction systems of the Hamiltonian $ H=A + W $. Scattering theory is developed for this Hamiltonian in the case where the operator $A$ has continuous spectrum.