Scattering Theory for Open Quantum Systems

DOI10.1007/S11040-008-9035-XzbMath1144.81480arXivmath-ph/0610088MaRDI QIDQ6469813

Mark M. Malamud, Jussi Behrndt, Hagen Neidhardt

Publication date: 31 October 2006

Abstract: 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 $widetildeK$ of $A_D$ can be regarded as the Hamiltonian of a closed system which contains the open system $A_D,sH$, but since $widetildeK$ 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.

Mathematics Subject Classification ID

Perturbation theory of linear operators (47A55) Linear symmetric and selfadjoint operators (unbounded) (47B25) Quantum stochastic calculus (81S25) Quantum scattering theory (81U99) Scattering theory of linear operators (47A40)

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