On a rigorous interpretation of the quantum Schrödinger–Langevin operator in bounded domains with applications

AbstractIn this paper we make it mathematically rigorous the formulation of the following quantum Schrödinger–Langevin nonlinear operator for the wavefunctionAQSL=iℏ∂t+ℏ22mΔx−λ(Sψ−〈Sψ〉)−Θℏ[nψ,Jψ] in bounded domains via its mild interpretation. The a priori ambiguity caused by the presence of the multi-valued potential λSψ, proportional to the argument of the complex-valued wavefunctionψ=|ψ|exp{iℏSψ}, is circumvented by subtracting its positional expectation value,〈Sψ〉(t):=∫ΩSψ(t,x)nψ(t,x)dx, as motivated in the original derivation (Kostin, 1972 [45]). The problem to be solved in order to find Sψ is mostly deduced from the modulus-argument decomposition of ψ and dealt with much like in Guerrero et al. (2010) [37]. Here ℏ is the (reduced) Planck constant, m is the particle mass, λ is a friction coefficient, nψ=|ψ|2 is the local probability density, Jψ=ℏmIm(ψ¯∇xψ) denotes the electric current density, and Θℏ is a general operator (eventually nonlinear) that only depends upon the macroscopic observables nψ and Jψ. In this framework, we show local well-posedness of the initial-boundary value problem associated with the Schrödinger–Langevin operator AQSL in bounded domains. In particular, all of our results apply to the analysis of the well-known Kostin equation derived in Kostin (1972) [45] and of the Schrödinger–Langevin equation with Poisson coupling and enthalpy dependence (Jüngel et al., 2002 [41])

Wellposedness of a nonlinear, logarithmic Schrödinger equation of Doebner-Goldin type modeling quantum dissipation

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 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 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

A wavefunction description of stochastic-mechanics tion: Fokker-Planck derivation, dissipastationary dynamics and numerical approximations

A nonlinear Schr"odinger equation describing how a quantum particle interacts with its surrounding reservoir is derived from the Wigner--\linebreak Fokker--Planck equation (WFPE) via stochastic (Nelsonian) mechanical techniques. This model can be reduced just to a logarithmic Schr"odinger equation (LSE) through a suitable gauge transformation that allows to explore its steady state dynamics and makes its mathematical and numerical analysis more tractable. The transient behaviour of the standard deviation from the mean position associated with its solutions is also studied numerically and compared with that stemming from the WFPE

Wellposedness of a nonlinear, logarithmic Schrödinger equation of Doebner-Goldin type modeling quantum dissipation

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 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 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