×

A variational approach to the Schrödinger-Poisson system: asymptotic behaviour, breathers, and stability. (English) Zbl 0999.82062

Summary: A variational formulation of the three-dimensional Schrödinger-Poisson system is proposed with the aim of solving the open problem of the asymptotic behaviour in time of the solutions in the case of attractive Coulomb forces. A dispersive equation relating density and linear moment dispersions is found. Optimal bounds for the kinetic energy are obtained which leads to study the asymptotic behaviour in time for the solutions in the attractive case with positive energy. The description of the asymptotic behaviour properties of the solutions such as a the existence of breathing mode solution, i.e. a changing size oscillatory wave function, in the case of attractive potential with negative kinetic energy are also given. A study of the stability of stationary solutions is proposed using a Lyapunov functional and also starting from a perturbation of an associated time-independent solution of the Schrödinger Poisson equation (linear stability).

MSC:

82D37 Statistical mechanics of semiconductors
82C70 Transport processes in time-dependent statistical mechanics
81S30 Phase-space methods including Wigner distributions, etc. applied to problems in quantum mechanics
81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
35Q40 PDEs in connection with quantum mechanics
PDFBibTeX XMLCite
Full Text: DOI