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Frontiers of Mathematics in China

Front. Math. China    2017, Vol. 12 Issue (4) : 907-919     DOI: 10.1007/s11464-017-0612-9
Parity-decomposition and moment analysis for stationary Wigner equation with inflow boundary conditions
Ruo LI1,2, Tiao LU1,2,3(), Zhangpeng SUN1
1. School of Mathematical Sciences, Peking University, Beijing 100871, China
2. HEDPS & CAPT, LMAM, Peking University, Beijing 100871, China
3. Collaborative Innovation Center of IFSA (CICIFSA), Shanghai Jiao Tong University, Shanghai 200240, China
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We study the stationary Wigner equation on a bounded, onedimensional spatial domain with inflow boundary conditions by using the parity decomposition of L. Barletti and P. F. Zweifel [Transport Theory Statist. Phys., 2001, 30(4-6): 507–520]. The decomposition reduces the half-range, two-point boundary value problem into two decoupled initial value problems of the even part and the odd part. Without using a cutoff approximation around zero velocity, we prove that the initial value problem for the even part is well-posed. For the odd part, we prove the uniqueness of the solution in the odd L2-space by analyzing the moment system. An example is provided to show that how to use the analysis to obtain the solution of the stationary Wigner equation with inflow boundary conditions.

Keywords Stationary Wigner equation      inflow boundary conditions      wellposedness      parity-decomposition      moment analysis     
Corresponding Authors: Tiao LU   
Issue Date: 06 July 2017
 Cite this article:   
Ruo LI,Tiao LU,Zhangpeng SUN. Parity-decomposition and moment analysis for stationary Wigner equation with inflow boundary conditions[J]. Front. Math. China, 2017, 12(4): 907-919.
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Ruo LI
Tiao LU
Zhangpeng SUN
1 ArnoldA, LangeH, ZweifelP F. A discrete-velocity, stationary Wigner equation. J Math Phys, 2000, 41(11): 7167–7180
doi: 10.1063/1.1318732
2 BarlettiL. A mathematical introduction to the Wigner formulation of quantum mechanics. Boll Unione Mat Ital, 2003, 6-B(3): 693–716
3 BarlettiL, ZweifelP F. Parity-decomposition method for the stationary Wigner equation with inflow boundary conditions. Transport Theory Statist Phys, 2001, 30(4-6): 507–520
doi: 10.1081/TT-100105935
4 CaiZ, FanY, LiR, LuT, WangY. Quantum hydrodynamics models by moment closure of Wigner equation. J Math Phys, 2012, 53: 103503
doi: 10.1063/1.4748971
5 CostolanskiA S, KelleyC T. Efficient solution of the Wigner-Poisson equations for modeling resonant tunneling diodes. IEEE Trans Nanotechnology, 2010, 9(6): 708–715
doi: 10.1109/TNANO.2010.2053214
6 EvansL C. Partial Differential Equations. 2nd ed. Providence: Amer Math Soc, 2010
doi: 10.1090/gsm/019
7 FerryD K, GoodnickS M. Transport in Nanostructures. Cambridge: Cambridge Univ Press, 1997
doi: 10.1017/CBO9780511626128
8 FrensleyW R. Wigner function model of a resonant-tunneling semiconductor device. Phys Rev B, 1987, 36: 1570–1580
doi: 10.1103/PhysRevB.36.1570
9 GehringA, KosinaH. Wigner function-based simulation of quantum transport in scaled DG-MOSFETs using a Monte Carlo method. J Comput Electr, 2005, 4: 67–70
doi: 10.1007/s10825-005-7109-6
10 HilleryM, ÓConnellR F, ScullyM O, WignerE P. Distribution functions in physics: Fundamentals. Phys Rep, 1984, 106(3): 121–167
doi: 10.1016/0370-1573(84)90160-1
11 JensenK L, BuotF A. Numerical aspects on the simulation of I-V characteristics and switching times of resonant tunneling diodes. J Appl Phys, 1990, 67: 2153–2155
doi: 10.1063/1.345551
12 JiangH, CaiW, TsuR. Accuracy of the Frensley inflow boundary condition for Wigner equations in simulating resonant tunneling diodes. J Comput Phys, 2011, 230: 2031–2044
doi: 10.1016/
13 JiangH, LuT, CaiW. A device adaptive inflow boundary condition for Wigner equations of quantum transport. J Comput Phys, 2014, 248: 773–786
doi: 10.1016/
14 LiR, LuT, SunZ-P. Convergence of semi-discrete stationary Wigner equation with inflow boundary conditions. Preprint, 2014
15 LiR, LuT, SunZ-P. Stationary Wigner equation with inflow boundary conditions: Will a symmetric potential yield a symmetric solution? SIAM J Appl Math, 2014, 70(3): 885–897
doi: 10.1137/130941754
16 QuerliozQ, Saint-MartinJ, DoV-N, BournelA, DollfusP. A study of quantum transport in end-of-roadmap DG-MOSFETs using a fully self-consistent Wigner Monte Carlo approach. IEEE Trans Nanotechnology, 2006, 5(6): 737–744
doi: 10.1109/TNANO.2006.883477
17 ShaoS, LuT, CaiW. Adaptive conservative cell average spectral element methods for transient Wigner equation in quantum transport. Commun Comput Phys, 2011, 9: 711–739
doi: 10.4208/cicp.080509.310310s
18 ShihJ J, HuangH C, WuG Y. Effect of mass discontinuity in the Wigner theory of resonant-tunneling diodes. Phys Rev B, 1994, 50(4): 2399–2405
doi: 10.1103/PhysRevB.50.2399
19 TsuchiyaH, OgawaM. Simulation of quantum transport in quantum device with spatially varying effective mass. IEEE Trans Electron Devices, 1991, 38(6): 1246–1252
doi: 10.1109/16.81613
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