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Title: Extended weak coupling limit for Friedrichs Hamiltonians

Abstract

We study a class of self-adjoint operators defined on the direct sum of two Hilbert spaces: a finite dimensional one called sometimes a 'small subsystem' and an infinite dimensional one called a 'reservoir'. The operator, which we call a 'Friedrichs Hamiltonian', has a small coupling constant in front of its off-diagonal term. It is well known that under some conditions in the weak coupling limit the appropriately rescaled evolution in the interaction picture converges to a contractive semigroup when restricted to the subsystem. We show that in this model, the properly renormalized and rescaled evolution converges on the whole space to a new unitary evolution, which is a dilation of the above mentioned semigroup. Similar results have been studied before ( 1990) in more complicated models under the name of 'stochastic limit'.

Authors:
;  [1];  [2]
  1. Department of Mathematical Methods in Physics, Warsaw University, Hoza 74, 00-682 Warsaw (Poland)
  2. (Belgium)
Publication Date:
OSTI Identifier:
20929606
Resource Type:
Journal Article
Resource Relation:
Journal Name: Journal of Mathematical Physics; Journal Volume: 48; Journal Issue: 1; Other Information: DOI: 10.1063/1.2405402; (c) 2007 American Institute of Physics; Country of input: International Atomic Energy Agency (IAEA)
Country of Publication:
United States
Language:
English
Subject:
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; COUPLING CONSTANTS; HAMILTONIANS; HILBERT SPACE; INTERACTIONS; MATHEMATICAL EVOLUTION; RENORMALIZATION

Citation Formats

Derezinski, Jan, Roeck, Wojciech de, and Instituut voor Theoretische Fysica, K.U. Leuven. Extended weak coupling limit for Friedrichs Hamiltonians. United States: N. p., 2007. Web. doi:10.1063/1.2405402.
Derezinski, Jan, Roeck, Wojciech de, & Instituut voor Theoretische Fysica, K.U. Leuven. Extended weak coupling limit for Friedrichs Hamiltonians. United States. doi:10.1063/1.2405402.
Derezinski, Jan, Roeck, Wojciech de, and Instituut voor Theoretische Fysica, K.U. Leuven. Mon . "Extended weak coupling limit for Friedrichs Hamiltonians". United States. doi:10.1063/1.2405402.
@article{osti_20929606,
title = {Extended weak coupling limit for Friedrichs Hamiltonians},
author = {Derezinski, Jan and Roeck, Wojciech de and Instituut voor Theoretische Fysica, K.U. Leuven},
abstractNote = {We study a class of self-adjoint operators defined on the direct sum of two Hilbert spaces: a finite dimensional one called sometimes a 'small subsystem' and an infinite dimensional one called a 'reservoir'. The operator, which we call a 'Friedrichs Hamiltonian', has a small coupling constant in front of its off-diagonal term. It is well known that under some conditions in the weak coupling limit the appropriately rescaled evolution in the interaction picture converges to a contractive semigroup when restricted to the subsystem. We show that in this model, the properly renormalized and rescaled evolution converges on the whole space to a new unitary evolution, which is a dilation of the above mentioned semigroup. Similar results have been studied before ( 1990) in more complicated models under the name of 'stochastic limit'.},
doi = {10.1063/1.2405402},
journal = {Journal of Mathematical Physics},
number = 1,
volume = 48,
place = {United States},
year = {Mon Jan 15 00:00:00 EST 2007},
month = {Mon Jan 15 00:00:00 EST 2007}
}