Dissipative Schroedinger-Poisson systems

We deal with a stationary, dissipative Schroedinger-Poisson system which allows for a current flow through an open, spatially one-dimensional quantum system determined by a dissipative Schroedinger operator. This dissipative Schroedinger operator can be regarded as a pseudo-Hamiltonian of the corresponding open quantum system. The (self-adjoint) dilation of the dissipative operator serves as a quasi-Hamiltonian of the system which is used to define physical quantities such as density and current for the open quantum system. The thus defined charge density in its dependence on the electrostatic potential is the nonlinear term in Poisson's equation. We prove that the dissipative Schroedinger-Poisson system always admits a solution and all solutions are included in a ball the radius of which depends only on the data of the problem