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Wellposedness of a Nonlinear, Logarithmic Schrödinger Equation of Doebner–Goldin Type Modeling Quantum Dissipation

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Abstract

This paper is concerned with the modeling and analysis of quantum dissipation phenomena in the Schrödinger picture. More precisely, we do investigate in detail a dissipative, nonlinear Schrödinger equation somehow accounting for quantum Fokker–Planck effects, and see how it is drastically reduced to a simpler logarithmic equation via a nonlinear gauge transformation in such a way that the physics underlying both problems keeps unaltered. From a mathematical viewpoint, this allows for a more easily achievable analysis regarding the local wellposedness of the initial-boundary value problem. This simplification requires the performance of the polar (modulus argument) decomposition of the wavefunction, which is rigorously attained (for the first time to the best of our knowledge) under quite reasonable assumptions.

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Acknowledgements

This work has been partially supported by Ministerio de Ciencia e Innovación (Spain), Project MTM2008-05271 and Project MTM2011-23384, and Junta de Andalucía, Project FQM-316.

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Authors and Affiliations

  1. Centre de Recerca Matemática, Universitat Autónoma de Barcelona, Campus de Bellaterra, edifici C, Bellaterra, 08193, Barcelona, Spain

    P. Guerrero

  2. Departamento de Matemática Aplicada, Facultad de Ciencias, Universidad de Granada, 18071, Granada, Spain

    J. L. López, J. Montejo-Gámez & J. Nieto

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  1. P. Guerrero
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  2. J. L. López
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  3. J. Montejo-Gámez
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  4. J. Nieto
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Correspondence to J. L. López.

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Communicated by D. Zenkov.

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Guerrero, P., López, J.L., Montejo-Gámez, J. et al. Wellposedness of a Nonlinear, Logarithmic Schrödinger Equation of Doebner–Goldin Type Modeling Quantum Dissipation. J Nonlinear Sci 22, 631–663 (2012). https://doi.org/10.1007/s00332-012-9123-8

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  • DOI: https://doi.org/10.1007/s00332-012-9123-8

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