WIAS Preprint No. 1179, (2006)

Scattering theory for open quantum systems


  • Behrndt, Jussi
  • Malamud, Mark
  • Neidhardt, Hagen

2010 Mathematics Subject Classification

  • 47A40 47B25 47A55 47B44 47E05


  • Scattering theory, open quantum system, maximal dissipative operator, pseudo-Hamiltonian, quasi-Hamiltonian, Lax-Phillips scattering, scattering matrix, characteristic function, boundary triplet, Weyl function, Sturm-Liouville operator




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 Schrödinger-Poisson systems.

Appeared in

  • Math. Phys. Anal. Geom., 10 (2007) pp. 313--358.

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