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Removal of the Energy Dependence from the Resolvent Like Energy Dependent Interactions

Technical Report:

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.
Authors:
Publication Date:
Dec 31, 1994
Product Type:
Technical Report
Report Number:
JINR-E-5-94-259
Reference Number:
SCA: 661100; PA: AIX-26:012170; EDB-95:031939; SN: 95001324647
Resource Relation:
Other Information: DN: Submitted to Teoreticheskaya i Matematicheskaya Fizika.; PBD: 1994
Subject:
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; MANY-BODY PROBLEM; EIGENFUNCTIONS; ENERGY DEPENDENCE; HAMILTONIANS; SCATTERING; SERIES EXPANSION; 661100; CLASSICAL AND QUANTUM MECHANICS
OSTI ID:
10112423
Research Organizations:
Joint Inst. for Nuclear Research, Dubna (Russian Federation). Lab. of Theoretical Physics
Country of Origin:
JINR
Language:
English
Other Identifying Numbers:
Other: ON: DE95613385; TRN: XJ9406555012170
Availability:
OSTI; NTIS (US Sales Only); INIS
Submitting Site:
INIS
Size:
20 p.
Announcement Date:
Jun 30, 2005

Technical Report:

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}
}