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
Externally publishedYes

Bibliographical note

Funding Information:
This research was supported by the OEAD, the International Erwin Schrödinger Institute in Vienna, the Wittgenstein 2000 Award of P. A. M. funded by the Austrian FWF, the FWF Project Fokker–Planck und Mittlere-Feld-Gleichungen (nr. P14876), and the EU-funded TMR-network.


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

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Mathematical Physics


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