The Schrödinger with variable mass model: mathematical analysis and semi-classical limit
Author:
Jihène Kefi
Journal:
Quart. Appl. Math. 62 (2004), 201-220
MSC:
Primary 82C70; Secondary 34B15
DOI:
https://doi.org/10.1090/qam/2054596
MathSciNet review:
MR2054596
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Abstract: In this paper, we propose and analyze a one-dimensional stationary quantum-transport model: the Schrödinger with variable mass. In the first part, we prove the existence of a solution for this model, with a self-consistent potential determined by the Poisson problem, whereas, in the second part, we rigorously study its semi-classical limit which gives us the kinetic model limit. The rigorous limit was based on the analysis of the support of the Wigner transform.
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D. J. Ben Daniel and C. B. Duke, Space-Charge Effects on Electron Tunneling, Phys. Rev., 152, 683–693 (1966)
H. Brezzi and P. A. Markowich, The Three Dimensional Wigner-Poisson Model: Existence Uniqueness and Approximation, Math. Meth. Appl. Sci., 14, 35–61 (1991)
H. Brezzi and P. A. Markowich, A mathematical analysis of quantum transport in three dimensional crystals, Anna. di Matematica Pura Applicata, 160, 171–191 (1991)
P. Degond and P. A. Markowich, A Quantum Transport Model for Semiconductors: The Wigner-Poisson Problem on a bounded Brillouin zone, M2AN, 24, no. 6, 697–710 (1990)
F. Nier, A stationary Schrödinger-Poisson system arising from the modeling of electronic devices, Forum Mathematicum 2, 5, 489–551 (1990)
F. Nier, A variational formulation of Schrödinger-Poisson systems in dimension $d \le 3$, Comm. Part. Diff. Equations, 18, 1125–1147 (1993)
N. Ben Abdallah, On multi-dimensional Schrödinger-Poisson Scattering model for semiconductors, J. Math. Phys. 41, no. 7, 4241–4261 (2000)
P. Gérard, Mesures semiclassiques et ondes de bloch, Sém. Ecole Polytechnique XVI, 1–19 (1990–1991)
P. Gérard, P. A. Markowich, N. Mauser, and F. Poupaud, Homogenization Limits and Wigner Transforms, Comm. Pure Appl. Math., 50, no. 4, 323–379 (1997)
P. L. Lions and T. Paul, Sur les mesures de Wigner, Revista Mathematica Iberoamericana, 9, 553–618 (1993)
P. A. Markowich and N. J. Mauser, The classical limit of a self-consistent quantum-Vlasov equation in $3 - D$, Math. Meth. Mod., 16, no. 6, 409–442 (1993)
P. A. Markowich, N. J. Mauser, and F. Poupaud, A Wigner function approach to semi-classical limits: electrons in a periodic potential, J. Math. Phys., 35, no. 3, 1066–1094 (1994)
F. Poupaud and C. Ringhofer, Semi-classical limits in a crystal with exterior potentials and effective mass theorems, Comm. Partial Differential Equations, 21, no. 11–12, 1897–1918 (1996)
D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer, New York (1977)
N. Ben Abdallah, P. Degond, and P. A. Markowich, On a one-dimensional Schrödinger-Poisson Scattering model, ZAMP, 48, 135–155 (1997)
H. Brézis, Analyse Fonctionnelle, Théorie et Applications, Masson, Paris (1983)
N. Ben Abdallah and J. Kefi, Limite semi-classique du problème de Schrödinger avec masse variable, C. R. Acad. Sci. Paris, 331, Série I, 165–170 (2000)
E. P. Wigner, On the quantum correction for the thermodynamic equilibrium, Phys. Rev., 40, 749–759 (1932)
N. Ben Abdallah, A Hybrid kinetic-Quantum model for stationary electron transport in a Resonant Tunneling Diode, J. Statis. Phys. 90, no. 3–4, 627–662 (1998)
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