Abstract
We study the nonlinear logarithmic Schrödinger equation in three dimensions. We establish the existence of the solutions of general quasi-linear Schrödinger equations. Finally, we show the convergence of the logarithmic quantum mechanics to the linear regime.
Similar content being viewed by others
References
Adams R. (1975). Sobolev Spaces. Academic, New York
Białynicki-Birula I. and Mycielski J. (1975). Wave equations with logarithmic nonlinearities. Bull. Acad. Pol. Sc. 23: 461–466
Białynicki-Birula I. and Mycielski J. (1976). Nonlinear wave mechanics. Ann. Phys. 100: 62–93
Białynicki-Birula, I., Sowiński, T.: Solutions of the logarithmic Schrodinger equation in rotating harmonic trap. In: Abdullaev F.Kh., Konotop V.V. (eds.) Nonlinear Waves: Classical and Quantum Aspects, vol. 153, Kluwer, Amsterdam (2004)
Bu C., Tsutaya K. and Zhang C. (2005). Nonlinear Schrödinger Equation with Inhomogeneous Dirichlet Boundary Data. J. Math. Phys. 46: 6
Buljan H., Šiber A., Soljačic M., Schwartz T., Segev M. and Christodoulides D.N. (2003). Incoherent white light solitons in logarithmically saturable noninstantaneous nonlinear media. Phys. Rev. E 68: 036607
Carlen E. (1991). Superadditivity of Fisher’s information and logarithmic Sobolev inequality. J. Func. Anal. 101: 194–211
Cazenave T. (2003). Semilinear Schrodinger Equations. AMS, New York
Cazenave T. (1983). Stable solutions of the logarithmic Schrödinger equation. Nonlinear. Anal. 7: 1127–1140
Cazenave T. and Haraux A. (1979). Équation de Schrödinger avec non-linéarité logarithmique. C.R. Acad. Sci. Paris Sér. A–B. 288: A253–A256
Cazenave T. and Haraux A. (1980). Équations d’v́olution avec non linéarité logarithmique. Ann. Fac. Sci. Toulouse Math. 2: 21–51
De Martino S., Falanga M., Godano C. and Lauro G. (2003). Logarithmic Schrödinger-like equation as a model for magma transport. Europhys. Lett. 63: 472
Gähler R., Klein A.G. and Zeilinger A. (1981). Neutron optical tests of nonlinear wave mechanics. Phys. Rev. A 23: 1611
Górka P. (2006). Logarithmic quantum mechanics: existence of the ground state. Found. Phys. Lett. 19: 591–601
Gross L. (1975). Logarithmic Sobolev inequalities. Am. J. Math. 97: 1061–1083
Hefter E.F. (1985). Application of the nonlinear Schrödinger equation with logarithmic inhomogeneous term to nuclear physics. Phys. Rev. A 32: 1201
Królikowski W., Edmundson D. and Bang O. (2000). Unified Model for partially coherent solitons in logarithmically nonlinear media. Phys. Rev. E 61: 3122
Shimony A. (1979). Proposed neutron interferometer test of some nonlinear variants of wave mechanics. Phys. Rev. A 20: 394
Shull C.G., Atwood D.K., Arthur J. and Horne M.A. (1980). Search for a nonlinear variant of the Schrödinger equation by neutron interferometry. Phys. Rev. Lett. 44: 765
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Górka, P. Convergence of Logarithmic Quantum Mechanics to the Linear One. Lett Math Phys 81, 253–264 (2007). https://doi.org/10.1007/s11005-007-0183-x
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11005-007-0183-x