## Abstract

The spectral problem (A + V(z)){psi} = z{psi} is considered where main Hamiltonian A is a self-adjoint operator of sufficiently arbitrary nature. The perturbation V(z) = -B(A`-z){sup -1} B{sup *} 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 for the Hamiltonian H = A{sub B}{sup *} B{sub A}`. 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 when operator A has continuous spectrum. (author). 26 refs.

## Citation Formats

Motovilov, A K.
Removal of the Energy Dependence from the Resolvent Like Energy Dependent Interactions.
JINR: N. p.,
1994.
Web.

Motovilov, A K.
Removal of the Energy Dependence from the Resolvent Like Energy Dependent Interactions.
JINR.

Motovilov, A K.
1994.
"Removal of the Energy Dependence from the Resolvent Like Energy Dependent Interactions."
JINR.

@misc{etde_10112423,

title = {Removal of the Energy Dependence from the Resolvent Like Energy Dependent Interactions}

author = {Motovilov, A K}

abstractNote = {The spectral problem (A + V(z)){psi} = z{psi} is considered where main Hamiltonian A is a self-adjoint operator of sufficiently arbitrary nature. The perturbation V(z) = -B(A`-z){sup -1} B{sup *} 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 for the Hamiltonian H = A{sub B}{sup *} B{sub A}`. 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 when operator A has continuous spectrum. (author). 26 refs.}

place = {JINR}

year = {1994}

month = {Dec}

}

title = {Removal of the Energy Dependence from the Resolvent Like Energy Dependent Interactions}

author = {Motovilov, A K}

abstractNote = {The spectral problem (A + V(z)){psi} = z{psi} is considered where main Hamiltonian A is a self-adjoint operator of sufficiently arbitrary nature. The perturbation V(z) = -B(A`-z){sup -1} B{sup *} 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 for the Hamiltonian H = A{sub B}{sup *} B{sub A}`. 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 when operator A has continuous spectrum. (author). 26 refs.}

place = {JINR}

year = {1994}

month = {Dec}

}