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
Article

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.

Keywords

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 

Notes

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.

References

  1. Amrouche, C., Girault, V.: Decomposition of vector spaces and application to the Stokes problem in arbitrary dimension. Checoslov. Math. J. 44, 109–140 (1994) MathSciNetMATHGoogle Scholar
  2. Amrouche, C., Ciarlet, P.G., Ciarlet, P., Jr.: Vector and scalar potentials, Poincaré’s theorem and Korn’s inequality. C. R. Acad. Sci. Paris, Ser. I 345, 603–608 (2007) MathSciNetMATHCrossRefGoogle Scholar
  3. Antonelli, P., Marcati, P.: On the finite energy weak solutions to a system in quantum fluid dynamics. Commun. Math. Phys. 287, 657–686 (2009) MathSciNetMATHCrossRefGoogle Scholar
  4. Arnold, A., López, J.L., Markowich, P.A., Soler, J.: An analysis of quantum Fokker–Planck models: a Wigner function approach. Rev. Mat. Iberoam. 20, 771–814 (2004) MATHCrossRefGoogle Scholar
  5. Arnold, A., Dhamo, E., Mancini, C.: The Wigner–Poisson–Fokker–Planck system: global-in-time solutions and dispersive effects. Ann. Inst. Henri Poincaré C, Anal. Non Linéaire 24, 645–676 (2007) MATHCrossRefGoogle Scholar
  6. Auberson, G., Sabatier, P.C.: On a class of homogeneous nonlinear Schrödinger equations. J. Math. Phys. 35, 4028–4040 (1994) MathSciNetMATHCrossRefGoogle Scholar
  7. Babin, A., Figotin, A.: Some mathematical problems in a neoclassical theory of electric charges. Discrete Contin. Dyn. Syst., Ser. A 27, 1283–1326 (2010) MathSciNetMATHCrossRefGoogle Scholar
  8. Bacciagaluppi, G.: Nelsonian mechanics revisited. Found. Phys. Lett. 12, 1–16 (1999) MathSciNetCrossRefGoogle Scholar
  9. Bialynicki-Birula, I., Mycielski, J.: Nonlinear wave mechanics. Ann. Phys. 100, 62–93 (1976) MathSciNetCrossRefGoogle Scholar
  10. Brezis, H.: Analyse Fonctionnelle. Masson, Paris (1983) MATHGoogle Scholar
  11. Cañizo, J.A., López, J.L., Nieto, J.: Global L 1 theory and regularity of the 3D nonlinear Wigner–Poisson–Fokker–Planck system. J. Differ. Equ. 198, 356–373 (2004) MATHCrossRefGoogle Scholar
  12. Cazenave, T.: Stable solutions of the logarithmic Schrödinger equation. Nonlinear Anal. TMA 7, 1127–1140 (1983) MathSciNetMATHCrossRefGoogle Scholar
  13. Cazenave, T.: An Introduction to Nonlinear Schrödinger Equations. Textos de Métodos Matemáticos, vol. 22 (1989). Rio de Janeiro Google Scholar
  14. Cazenave, T., Haraux, A.: Equations d’évolution avec non linéarité logarithmique. Ann. Fac. Sci. Univ. Toulouse 2, 21–55 (1980) MathSciNetMATHCrossRefGoogle Scholar
  15. Cid, C., Dolbeault, J.: Defocusing nonlinear Schrödinger equation: confinement, stability and asymptotic stability. Technical report (2001) Google Scholar
  16. Colliander, J., Keel, M., Staffilani, G., Takaoka, H., Tao, T.: Global well-posedness for Schrödinger equations with derivative. SIAM J. Math. Anal. 33, 649–669 (2001) MathSciNetMATHCrossRefGoogle Scholar
  17. Cufaro Petroni, N., De Martino, S., De Siena, S., Illuminati, F.: Stochastic–hydrodynamic model of halo formation in charged particle beams. Phys. Rev. Spec. Top., Accel. Beams 6, 034206 (2003) CrossRefGoogle Scholar
  18. Davidson, M.P.: A model for the stochastic origins of Schrödinger’s equation. J. Math. Phys. 20, 1865–1869 (1979) MathSciNetCrossRefGoogle Scholar
  19. Davidson, M.P.: Comments on the nonlinear Schrödinger equation. Il Nuovo Cimento B V116B, 1291–1296 (2001) Google Scholar
  20. De Martino, S., Lauro, G.: Soliton-like solutions for a capillary fluid. In: Proceedings of the 12th Conference on WASCOM, pp. 148–152 (2003) Google Scholar
  21. De Martino, S., Falanga, M., Godano, C., Lauro, G.: Logarithmic Schrödinger-like equation as a model for magma transport. Europhys. Lett. 63, 472–475 (2003) CrossRefGoogle Scholar
  22. Doebner, H.D., Goldin, G.A.: On a general nonlinear Schrödinger equation admitting diffusion currents. Phys. Lett. A 162, 397–401 (1992) MathSciNetCrossRefGoogle Scholar
  23. Doebner, H.D., Goldin, G.A., Nattermann, P.: A family of nonlinear Schrödinger equations: linearizing transformations and resulting structure. In: Antoine, J.-P., et al. (eds.) Quantization, Coherent States and Complex Structures, pp. 27–31. Plenum, New York (1996) Google Scholar
  24. Fényes, I.: Eine wahrscheinlichkeitstheoretische begrundung und interpretation der Quantenmechanik. Z. Phys. 132, 81–103 (1952) MATHGoogle Scholar
  25. Garbaczewski, P.: Modular Schrödinger equation and dynamical duality. Phys. Rev. E 78, 031101 (2008) MathSciNetCrossRefGoogle Scholar
  26. Guerra, F.: Structural aspects of stochastic mechanics and stochastic field theory. Phys. Rep. 77, 263–312 (1981) MathSciNetCrossRefGoogle Scholar
  27. Guerra, F., Pusterla, M.: A nonlinear Schrödinger equation and its relativistic generalization from basic principles. Lett. Nuovo Cimento 34, 351–356 (1982) MathSciNetCrossRefGoogle Scholar
  28. Guerrero, P., López, J.L., Nieto, J.: Global H 1 solvability of the 3D logarithmic Schrödinger equation. Nonlinear Anal., Real World Appl. 11, 79–87 (2010) MathSciNetMATHCrossRefGoogle Scholar
  29. Guerrero, P., López, J.L., Montejo-Gámez, J., Nieto, J.: A wavefunction description of stochastic–mechanical Fokker–Planck dissipation: derivation, stationary dynamics, and numerical approximation. Preprint (2011) Google Scholar
  30. Jüngel, A., Mariani, M.C., Rial, D.: Local existence of solutions to the transient quantum hydrodynamic equations. Math. Models Methods Appl. Sci. 12, 485–495 (2002) MathSciNetMATHCrossRefGoogle Scholar
  31. Kostin, M.D.: On the Schrödinger-Langevin equation. J. Chem. Phys. 57, 3589–3591 (1972) CrossRefGoogle Scholar
  32. Kostin, M.D.: Friction and dissipative phenomena in quantum mechanics. J. Stat. Phys. 12, 145–151 (1975) CrossRefGoogle Scholar
  33. Lauro, G.: A note on a Korteweg fluid and the hydrodynamic form of the logarithmic Schrödinger equation. Geophys. Astrophys. Fluid Dyn. 102, 373–380 (2008) MathSciNetCrossRefGoogle Scholar
  34. Lindblad, G.: On the generators of quantum dynamical semigroups. Commun. Math. Phys. 48, 119–130 (1976) MathSciNetMATHCrossRefGoogle Scholar
  35. López, J.L.: Nonlinear Ginzburg–Landau-type approach to quantum dissipation. Phys. Rev. E 69, 026110 (2004) CrossRefGoogle Scholar
  36. López, J.L., Montejo-Gámez, J.: A hydrodynamic approach to multidimensional dissipation-based Schrödinger models from quantum Fokker–Planck dynamics. Physica D 238, 622–644 (2009) MathSciNetMATHCrossRefGoogle Scholar
  37. Nattermann, P., Scherer, W.: Nonlinear gauge transformations and exact solutions of the Doebner–Goldin equation. In: Doebner, H.D., et al. (eds.) Nonlinear, Deformed and Irreversible Quantum Systems, pp. 188–199. World Scientific, Singapore (1995) Google Scholar
  38. Nelson, E.: Derivation of the Schrödinger equation from Newtonian mechanics. Phys. Rev. 150, 1079–1085 (1966) CrossRefGoogle Scholar
  39. Ozawa, T.: On the nonlinear Schrödinger equations of derivative type. Indiana Univ. Math. J. 45, 137–163 (1996) MathSciNetMATHCrossRefGoogle Scholar
  40. Pazy, A.: Semigroups of Linear Operators and Applications to Partial Differential Equations. Applied Mathematical Sciences, vol. 44. Springer, New York (1983) MATHCrossRefGoogle Scholar
  41. Rudin, W.: Real and Complex Analysis. McGraw-Hill, New York (1966) MATHGoogle Scholar
  42. Sanin, A.L., Smirnovsky, A.A.: Oscillatory motion in confined potential systems with dissipation in the context of the Schrödinger–Langevin–Kostin equation. Phys. Lett. A 372, 21–27 (2007) MATHCrossRefGoogle Scholar
  43. Teismann, H.: Square-integrable solutions to a family of nonlinear Schrödinger equations from nonlinear quantum theory. Rep. Math. Phys. 56, 291–310 (2005) MathSciNetMATHCrossRefGoogle Scholar
  44. Wallstrom, T.C.: Inequivalence between the Schrödinger equation and the Madelung hydrodynamic equations. Phys. Rev. A 49, 1613–1617 (1994) MathSciNetCrossRefGoogle Scholar

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

Personalised recommendations