Motovilov A.K.
Removal of the resolvent-like energy dependence from
interactions and invariant subspaces of a total Hamiltonian.
20 pages, RevTeX style, published version of the paper.
Journal-ref: J. Math. Phys., 1995, vol. 36 (12), pp. 6647-6664
(87K, LaTeX)
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.