Schrödinger Equations with Logarithmic Self-Interactions: From Antilinear PT-Symmetry to the Nonlinear Coupling of Channels

Abstract

Schrödinger equations with non-Hermitian, but P T -symmetric quantum potentials V(x) 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, V(x) → W(ψ(x), x). Here, the state-dependence in W(ψ(x), x) 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 W(ψ(x), x) is regularized via a deformation of the real line of x into a self-consistently constructed complex contour C. The new role played by P T -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.

Solution of the logarithmic Schrödinger equation with a Coulomb potential

“…The logarithmic Schrödinger equation (log SE) is one of the nonlinear modifications of Schrödinger's equation, with applications in quantum optics [1,2], nuclear physics [3,4], diffusion phenomena [5], stochastic quantum mechanics [6], effective quantum gravity [7] and Bose-Einstein condensation [8,9]. A relativistic version in the form of a Klein-Gordon type equation displaying dilatation/conformal covariance was first proposed by G. Rosen [10].…”

Multi-well log-anharmonic oscillators

“…Due to the left-right symmetry of our illustrative double-well soft-core model (2) (cf. also its square-well predecessor in [6]), multiple qualitative features of the bound-state spectra were found predictable a priori, without any extensive ad hoc numerical or perturbative calculations which only confirmed the expectations. In our paper we intend to follow and extend this direction of research, therefore.…”

“…The analytic, smooth logarithmic barrier of Refs. [4,6] emerges as one of the other eligible candidates for a partially penetrable barrier, therefore. In loc.…”

Kinks in the relativistic model with logarithmic nonlinearity

“…Currently they find numerous applications in different areas of physics, including the particle physics and classical and quantum gravity [25,26,[30][31][32][33][34][35][36][37][38][39]. Modern statistical mechanical justifications of logarithmic models and their connections to fundamental quantum-mechanical notions can be found in [40], and their applicability to other branches of physics is discussed in [41,42].…”