WIAS Preprint No. 728, (2002)

Density and current of a dissipative Schrödinger operator


  • Kaiser, Hans-Christoph
  • Neidhardt, Hagen
  • Rehberg, Joachim

2010 Mathematics Subject Classification

  • 47A20 34B24 47A55 47B44


  • open Schrödinger-Poisson systems, carrier and current density, dissipative Schrödinger operator, self-adjoint dilation, generalized eigenfunctions, characteristic function, density matrix


A net current flow through an open 1-dimensional Schrödinger-Poisson system is modeled by replacing self-adjoint boundary conditions by dissipative ones. To give a rigorous definition of carrier and current densities the well-known dilation theory of dissipative operators is used where the self-adjoint dilation is regarded as the Hamiltonian of a larger closed system which contains the open one. The carrier density turns out to be performed by the generalized eigenstates of the dilation while the current density is related to the characteristic function of the dissipative operator. A rigorous setup of a dissipative Schrödinger-Poisson system is outlined.

Appeared in

  • Journal of Mathematical Physics, 2002, 43, 11, p 5325-5350

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