A quantum-transport model for semiconductors: The Wigner-Poisson problem on a bounded Brillouin zone. (English) Zbl 0742.35046

Summary: We analyse a quantum-mechanical model for the transport of electrons in semiconductors. The model consists of the quantum Liouville (Wigner) equation posed on the bounded Brillouin zone corresponding to the semiconductor crystal lattice, with a self-consistent potential determined by a Poisson equation. A global existence and uniqueness proof for this model is the main result of the paper.


35Q40 PDEs in connection with quantum mechanics
35A05 General existence and uniqueness theorems (PDE) (MSC2000)
Full Text: DOI EuDML


[1] M. CESSENAT, Théorèmes de Trace pour des Espaces des Fonctions de la Neutronique. C.R. Acad. Sc. Paris, tome 300, série l, n^\circ 3, 1985. Zbl0648.46028 MR777741 · Zbl 0648.46028
[2] R. DAUTRAY and J. L. LIONS, Analyse Mathématique et Calcul Numérique pour les Sciences et les Techniques. Tome 3, Masson, Paris, 1985. Zbl0642.35001 MR902802 · Zbl 0642.35001
[3] F. GOLSE, P. L. LIONS, B. PERTHAME and R. SENTIS, Regularity of the Moments of the Solution of a Transport Equation. J. Funct. Anal. 88, pp. 110-125, 1988. Zbl0652.47031 MR923047 · Zbl 0652.47031
[4] J. C. GUILLOT, J. RALSTON and E. TRUBOWITZ, Semi-Classical Asymptotics in Solid State Physics. Communications in Math. Phys., vol. 116, n^\circ 3, pp. 401-415, 1988. Zbl0672.35014 MR937768 · Zbl 0672.35014
[5] C. KITTEL, Introduction to Solid States Physics, J. Wiley and Sons, New York, 1968. Zbl0052.45506 · Zbl 0052.45506
[6] P. A. MARKOWICH and C. RINGHOFER, An Analysis of the Quantum Liouville Equation. To appear in ZAMM, 1988. Zbl0682.46047 MR990011 · Zbl 0682.46047
[7] P. A. MARKOWICH, On the Equivalence of the Schrödinger and the Quantum Liouville Equations. To appear in Math. Meth. In the Appl. Sci., 1988. Zbl0696.47042 MR1001097 · Zbl 0696.47042
[8] A. PAZY, Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer Verlag, New York-Berlin-Heidelberg-Tokyo, 1983. Zbl0516.47023 MR710486 · Zbl 0516.47023
[9] V. I. TATARSKII, The Wigner Representation of Quantum Mechanics. Sov. Phys. Usp., vol. 26, n^\circ 4, pp. 311-327, 1983. MR730012
[10] A. ARNOLD, P. DEGOND, P. A. MARKOWICH and H. STEINRÜCK, The Wigner-Poisson Equation in a Crystal, to appear in : Applied Mathematics Letters, 1989. Zbl0822.58070 MR1003856 · Zbl 0822.58070
[11] P. DEGOND, P. A. MARKOWICH and H. STEINRÜCK, A Mathematical Derivation of the Wigner-Poisson Problem on a bounded Brillouin Zone from the Schrödinger Equation, manuscript.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.