Journal of Nonlinear Science

, Volume 22, Issue 5, pp 631–663 | Cite as

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

  • P. Guerrero
  • J. L. López
  • J. Montejo-Gámez
  • J. Nieto


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.


Wigner–Fokker–Planck equation Doebner–Goldin equations Dissipative quantum mechanics Nonlinear Schrödinger equation Logarithmic nonlinearities Local solvability Madelung transformation Reconstruction of the wavefunction 

Mathematics Subject Classification

35A07 35G30 35Q55 76Y05 81Q05 



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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Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  • P. Guerrero
    • 1
  • J. L. López
    • 2
  • J. Montejo-Gámez
    • 2
  • J. Nieto
    • 2
  1. 1.Centre de Recerca MatemáticaUniversitat Autónoma de BarcelonaBarcelonaSpain
  2. 2.Departamento de Matemática Aplicada, Facultad de CienciasUniversidad de GranadaGranadaSpain

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