  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.
