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Quantum Dynamical Semigroups for Diffusion Models with Hartree Interaction

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Abstract

We consider a class of evolution equations in Lindblad form, which model the dynamics of dissipative quantum mechanical systems with mean-field interaction. Particularly, this class includes the so-called Quantum Fokker-Planck-Poisson model. The existence and uniqueness of global-in-time, mass preserving solutions is proved, thus establishing the existence of a nonlinear conservative quantum dynamical semigroup. The mathematical difficulties stem from combining an unbounded Lindblad generator with the Hartree nonlinearity.

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Authors and Affiliations

  1. Institut für Numerische Mathematik, Universität Münster, Einsteinstr. 62, 48149, Münster, Germany

    A. Arnold

  2. Institut für Mathematik, Universität Wien, Nordbergstr. 15, 1090, Vienna, Austria

    C. Sparber

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  1. A. Arnold
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  2. C. Sparber
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Communicated by H. Spohn

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Arnold, A., Sparber, C. Quantum Dynamical Semigroups for Diffusion Models with Hartree Interaction. Commun. Math. Phys. 251, 179–207 (2004). https://doi.org/10.1007/s00220-004-1172-x

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