Scattering Theory for Open Quantum Systems with Finite Rank Coupling

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 ${\mathfrak H}$ 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 $\{A_{\!D},{\mathfrak H}\}$ , 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(μ)} of maximal dissipative operators depending on energy μ, 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 Schrodinger–Poisson systems.