Modular Schr'{o}dinger equation and dynamical duality

📝 Abstract

We discuss quite surprising properties of the one-parameter family of modular (Auberson and Sabatier (1994)) nonlinear Schr"{o}dinger equations. We develop a unified theoretical framework for this family. Special attention is paid to the emergent \it dual \rm time evolution scenarios which, albeit running in the \it real time \rm parameter of the pertinent nonlinear equation, in each considered case, may be mapped among each other by means of an “imaginary time” transformation (more seriously, an analytic continuation in time procedure).

💡 Analysis

We discuss quite surprising properties of the one-parameter family of modular (Auberson and Sabatier (1994)) nonlinear Schr"{o}dinger equations. We develop a unified theoretical framework for this family. Special attention is paid to the emergent \it dual \rm time evolution scenarios which, albeit running in the \it real time \rm parameter of the pertinent nonlinear equation, in each considered case, may be mapped among each other by means of an “imaginary time” transformation (more seriously, an analytic continuation in time procedure).

📄 Content

arXiv:0805.1536v2 [quant-ph] 8 Aug 2008 Modular Schr¨odinger equation and dynamical duality Piotr Garbaczewski∗ Institute of Physics, University of Opole, 45-052 Opole, Poland August 22, 2021 Abstract We discuss quite surprising properties of the one-parameter family of modular (Auberson and Sabatier (1994)) nonlinear Schr¨odinger equations. We develop a unified theoretical framework for this family. Special attention is paid to the emergent dual time evolution scenarios which, albeit running in the real time parameter of the pertinent nonlinear equation, in each considered case, may be mapped among each other by means of a suitable analytic continuation in time procedure. This dynamical duality is characteristic for non-dissipative quantum motions and their dissipative (diffusion-type processes) partners, and naturally extends to classical motions in confining and scattering potentials. PACS numbers: 02.50.Ey, 05.20.-y, 05.40.Jc 1 Motivation An inspiration for the present paper comes from the recent publication [1] discussing effects of various scale transformations upon the free Schr¨odinger picture dynamics. In particular, it has been noticed that an appropriate definition of the scale covariance induces Hamiltonians which mix, with a pertinent scale exponent as a hyperbolic ro- tation angle, an original free quantum dynamics with its free dissipative counterpart (effectively, a suitable version of the free Brownian motion), [1] c.f. also [2]. The two disparately different time evolution patterns do run with respect to the same real time label. However, the ultimate ”mixing” effect of the above mentioned scale transformations takes the form of the the Lorentz-like transformation Eq. (35) where one Hamiltonian takes the role of a regular ”time” while another of the ”space” dimension labels. This obvious affinity with the Euclidean ”space-time” notion and the involved complex analysis methods (e. g. Wick rotation, imaginary time transforma- tion, analytic continuation in time) sets both conceptual and possibly phenomenological obstacles/prospects pertaining to an existence of such dynamical patterns in nature. ∗Electronic address: pgar@uni.opole.pl 1 Most puzzling for us is the apparently dual notion of the time label which while a priori referring to the real time evolution, may as well be interpreted as a Euclidean (imaginary time) evolution. It is our aim to address an issue in its full generality, by resorting to a one-parameter family of modular Schr¨odinger equations, where external conservative potentials admitted. The adopted perspective relies on standard approaches to nonlinear dynamical sys- tems where a sensitive dependence on a control (here, coupling strength) parameter may arise, possibly inducing global changes of properties of solutions to the equations of motion. That is exactly the case in the present analysis. We wish to demonstrate that a duality property (realized by an imaginary time transformation) does relate solutions of the modular nonlinear Schr¨odinger equation for various coupling constant regimes. In Section 2 we set a general Lagrangian and Hamiltonian framework for the sub- sequent discussion and indicate that effectively the dynamics can be reduced to three specific coupling value choices, [3, 4], each of them being separately discussed in the lit- erature. We aim at a stationary action principle formulation of the modular Schr¨odinger dynamics and the related hydrodynamics (the special cases are: the familiar Bohm- type quantum hydrodynamics [5] and the hydrodynamical picture of diffusion-type processes [2]). Basic principles of the action principle workings are patterned after the classical hydrodynamics treatises, [7, 6]. In Section 3, in Hamiltonian dynamics terms, we give a new derivation and further generalize the original arguments of [1]. We employ scaling properties of the Shannon entropy of a continuous probability density ρ .= ψ∗ψ. The scale covariant patterns of evolution, non-dissipative with an admixture of a dissipative component, are estab- lished in external potential fields. Section 4 is devoted to a detailed analysis of the emergent time duality notion for the three special values 0, 1, 2 of the coupling parameter. There, we pay more atten- tion to a specific intertwine between confining and scattering dynamical systems. A properly addressed sign issue for the potential function appears to be vital for a math- ematical consistency of the formalism (links with the theory of dynamical semigroups that underlies the dissipative dynamics scenarios). As a byproduct of a discussion we establish the Lyapunov functionals for the considered dynamical patterns of behavior and the (Shannnon) entropy production time rate is singled out. Section 5 addresses more specific problems that allow to grasp the duality concept from a perspective of the theory of classical conservative systems and diffusion-type (specifically - Smoluchowski) stochastic processes. A number of illustrative examp

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