Extended weak coupling limit for Friedrichs Hamiltonians
Abstract
We study a class of selfadjoint 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 offdiagonal 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:
 Department of Mathematical Methods in Physics, Warsaw University, Hoza 74, 00682 Warsaw (Poland)
 (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 selfadjoint 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 offdiagonal 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}
}

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