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

Peter A. Markowich; Gerhard Rein; Gershon Wolansky

doi:10.1023/A:1014050206769

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 journal › Article › peer-review

22 Scopus citations

Abstract

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 language	English (US)
Pages (from-to)	1221-1239
Number of pages	19
Journal	Journal of Statistical Physics
Volume	106
Issue number	5-6
DOIs	https://doi.org/10.1023/A:1014050206769
State	Published - 2002
Externally published	Yes

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.

Keywords

Hartree problem
Nonlinear stability
Schrödinger-Poisson system
Stationary solutions

ASJC Scopus subject areas

Statistical and Nonlinear Physics
Mathematical Physics

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10.1023/A:1014050206769

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title = "Existence and nonlinear stability of stationary states of the Schr{\"o}dinger-Poisson system",

abstract = "We consider the Schr{\"o}dinger 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 Schr{\"o}dinger-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.",

keywords = "Hartree problem, Nonlinear stability, Schr{\"o}dinger-Poisson system, Stationary solutions",

author = "Markowich, {Peter A.} and Gerhard Rein and Gershon Wolansky",

note = "Funding Information: This research was supported by the OEAD, the International Erwin Schr{\"o}dinger 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.",

year = "2002",

doi = "10.1023/A:1014050206769",

language = "English (US)",

volume = "106",

pages = "1221--1239",

journal = "Journal of Statistical Physics",

issn = "0022-4715",

publisher = "Springer New York",

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T1 - Existence and nonlinear stability of stationary states of the Schrödinger-Poisson system

AU - Markowich, Peter A.

AU - Rein, Gerhard

AU - Wolansky, Gershon

N1 - 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.

PY - 2002

Y1 - 2002

N2 - 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.

AB - 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.

KW - Hartree problem

KW - Nonlinear stability

KW - Schrödinger-Poisson system

KW - Stationary solutions

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U2 - 10.1023/A:1014050206769

DO - 10.1023/A:1014050206769

M3 - Article

AN - SCOPUS:0141672043

SN - 0022-4715

VL - 106

SP - 1221

EP - 1239

JO - Journal of Statistical Physics

JF - Journal of Statistical Physics

IS - 5-6

ER -

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

Abstract

Bibliographical note

Keywords

ASJC Scopus subject areas

Access to Document

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