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 AD in a Hilbert space $\sH$ is used to describe an open quantum system. In this case the minimal self-adjoint dilation K of AD can be regarded as the Hamiltonian of a closed system which contains the open system $\{A_D,\sH\}$, but since 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 Schrödinger-Poisson systems.
In this paper, we consider a class of Unitary Quantum Walks on arbitrary graphs, parameterized by a family of scattering matrices. These Scattering Quantum Walks model the discrete dynamics of a system on the graph’s edges, where a scattering process at the vertices is governed by the scattering matrices assigned to each vertex. We show the Scattering Quantum Walks encompass several known Quantum Walks. We further introduce two classes of Scattering Open Quantum Walks on arbitrary graphs, also parameterized by scattering matrices: one defined on the edges, the other on the vertices of the graph. We show these walks give rise to proper quantum channels and describe their main spectral and dynamical properties, relating them to naturally associated classical Markov chains.
Quantum dissipation is studied for a discrete system that linearly interacts with a reservoir of harmonic oscillators at thermal equilibrium. Initial correlations between system and reservoir are assumed to be absent. The dissipative dynamics as determined by the unitary evolution of system and reservoir is described by a Kraus map consisting of an infinite number of matrices. For all Laplace-transformed Kraus matrices exact solutions are constructed in terms of continued fractions that depend on the pair correlation functions of the reservoir.