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Author: Neidhardt, Hagen × End date: 2019 × Start date: 2018 × End date: 2009 × Start date: 2000 × Has files: Yes × Type: Text × Subject: Sturm-Liouville operator ×

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    Scattering theory for open quantum systems
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2006) Behrndt, Jussi; Malamud, Mark M.; Neidhardt, Hagen; Exner, Pavel
    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.
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    Scattering matrices and Weyl functions
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2006) Behrndt, Jussi; Malamud, Mark M.; Neidhardt, Hagen
    For a scattering system consisting of two selfadjoint extensions of a symmetric operator A with finite deficiency indices, the scattering matrix and the spectral shift function are calculated in terms of the Weyl function associated with the boundary triplet for A* and a simple proof of the Krein-Birman formula is given. The results are applied to singular Sturm-Liouville operators with scalar- and matrix-valued potentials, to Dirac operators and to Schroedinger operators with point interactions.