Existence and nonlinear stability of stationary states of the Schrödinger-Poisson system

Peter A. Markowich*, Gerhard Rein, Gershon Wolansky

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

21 Scopus citations


We consider the Schrödinger Poisson system in the repulsive (plasma physics) Coulomb case. Given a stationary state from a certain class we prove its non-linear stability, using an appropriately defined energy-Casimir functional as Lyapunov function. To obtain such states we start with a given Casimir functional and construct a new functional which is in some sense dual to the corresponding energy-Casimir functional. This dual functional has a unique maximizer which is a stationary state of the Schrödinger-Poisson system and lies in the stability class. The stationary states are parameterized by the equation of state, giving the occupation probabilities of the quantum states as a strictly decreasing function of their energy levels.

Original languageEnglish (US)
Pages (from-to)1221-1239
Number of pages19
JournalJournal of Statistical Physics
Issue number5-6
StatePublished - 2002


  • Hartree problem
  • Nonlinear stability
  • Schrödinger-Poisson system
  • Stationary solutions

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Mathematical Physics


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