Scattering Theory for Open Quantum Systems

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 $\widetilde K$ 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.