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THE MODIFIED QUANTUM WIGNER SYSTEM IN WEIGHTED $L^{2}$ -SPACE

Part of: Elliptic equations and systems Special classes of linear operators Parabolic equations and systems Equations of mathematical physics and other areas of application

Published online by Cambridge University Press:  13 October 2016

BIN LI  and
HAN YANG
BIN LI*
Affiliation:
Department of Mathematics, Jincheng College of Sichuan University, Chengdu 611731, PR China email [email protected]
HAN YANG
Affiliation:
College of Mathematics, Southwest Jiaotong University, Chengdu 611756, PR China email [email protected]
*
✉ [email protected]
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Abstract

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This paper is concerned with the modified Wigner (respectively, Wigner–Fokker–Planck) Poisson equation. The quantum mechanical model describes the transport of charged particles under the influence of the modified Poisson potential field without (respectively, with) the collision operator. Existence and uniqueness of a global mild solution to the initial boundary value problem in one dimension are established on a weighted $L^{2}$ -space. The main difficulties are to derive a priori estimates on the modified Poisson equation and prove the Lipschitz properties of the appropriate potential term.

Keywords

modified Wigner–Poisson equationmodified Wigner–Poisson–Fokker–Planck equationinitial boundary value problemglobal mild solution

MSC classification

Primary: 35Q40: PDEs in connection with quantum mechanics
Secondary: 35K55: Nonlinear parabolic equations 35J60: Nonlinear elliptic equations 47B44: Accretive operators, dissipative operators, etc.
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 95 , Issue 1 , February 2017 , pp. 73 - 83
Copyright
© 2016 Australian Mathematical Publishing Association Inc. 

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