Wellposedness of a Nonlinear, Logarithmic Schr?dinger Equation of Doebner-Goldin Type Modeling Quantum Dissipation

摘要

This paper is concerned with the modeling and analysis of quantum dissipation phenomena in the Schr?dinger picture. More precisely, we do investigate in detail a dissipative, nonlinear Schr?dinger equation somehow accounting for quantum Fokker-Planck effects, and see 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 easily 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.