Schrödinger Equations with Logarithmic Self-Interactions: From Antilinear PT-Symmetry to the Nonlinear Coupling of Channels
Nuclear Physics Institute of the CAS, Hlavní 130, 250 68 Řež, Czech Republic
Institute of Systems Science, Durban University of Technology, Durban 4000, South Africa
Author to whom correspondence should be addressed.
Received: 20 June 2017 / Revised: 31 July 2017 / Accepted: 8 August 2017 / Published: 18 August 2017
Schrödinger equations with non-Hermitian, but
-symmetric quantum potentials
found, recently, a new field of applicability in classical optics. The potential acquired there a new physical role of an “anomalous” refraction index. This turned attention to the nonlinear Schrödinger equations in which the interaction term becomes state-dependent,
. Here, the state-dependence in
is assumed logarithmic, and some of the necessary mathematical assumptions, as well as some of the potential phenomenological consequences of this choice are described. Firstly, an elementary single-channel version of the nonlinear logarithmic model is outlined in which the complex self-interaction
is regularized via a deformation of the real line of x
into a self-consistently constructed complex contour C
. The new role played by
-symmetry is revealed. Secondly, the regularization is sought for a multiplet of equations, coupled via the same nonlinear self-interaction coupling of channels. The resulting mathematical structures are shown to extend the existing range of physics covered by the logarithmic Schrödinger equations.
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Znojil, M.; Růžička, F.; Zloshchastiev, K.G. Schrödinger Equations with Logarithmic Self-Interactions: From Antilinear PT-Symmetry to the Nonlinear Coupling of Channels. Symmetry 2017, 9, 165.
Znojil M, Růžička F, Zloshchastiev KG. Schrödinger Equations with Logarithmic Self-Interactions: From Antilinear PT-Symmetry to the Nonlinear Coupling of Channels. Symmetry. 2017; 9(8):165.
Znojil, Miloslav; Růžička, František; Zloshchastiev, Konstantin G. 2017. "Schrödinger Equations with Logarithmic Self-Interactions: From Antilinear PT-Symmetry to the Nonlinear Coupling of Channels." Symmetry 9, no. 8: 165.
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