Scattering Theory for Open Quantum Systems

Quantum systems which interact with their environment are often modeled by maximal dissipative operators or so-called Pseudo-Hamiltonians. In this paper the scattering theory for such open systems is considered. First it is assumed that a single maximal dissipative operator \(A_D\) in a Hilbert spac...

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Bibliographic Details
Published in:arXiv.org 2006-10
Main Authors: Behrndt, J, Malamud, M M, Neidhardt, H
Format: Article
Language:English
Subjects:
Embedded systems
Energy dissipation
Hilbert space
Open systems
Operators (mathematics)
Quantum theory
Scattering
Online Access:Get full text
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Summary:Quantum systems which interact with their environment are often modeled by maximal dissipative operators or so-called Pseudo-Hamiltonians. In this paper the scattering theory for such open systems is considered. First it is assumed that a single maximal dissipative operator \(A_D\) in a Hilbert space \(\sH\) is used to describe an open quantum system. In this case the minimal self-adjoint dilation \(\widetilde K\) of \(A_D\) can be regarded as the Hamiltonian of a closed system which contains the open system \(\{A_D,\sH\}\), but since \(\widetilde K\) is necessarily not semibounded from below, this model is difficult to interpret from a physical point of view. In the second part of the paper an open quantum system is modeled with a family \(\{A(\mu)\}\) of maximal dissipative operators depending on energy \(\mu\), and it is shown that the open system can be embedded into a closed system where the Hamiltonian is semibounded. Surprisingly it turns out that the corresponding scattering matrix can be completely recovered from scattering matrices of single Pseudo-Hamiltonians as in the first part of the paper. The general results are applied to a class of Sturm-Liouville operators arising in dissipative and quantum transmitting Schr\"{o}dinger-Poisson systems.
ISSN:2331-8422