Gaussian beam methods for the dirac equation in the semi-classical regime

Communications in Mathematical Sciences

Volumn 10, Issue 4, 2012, Pages 1301-1315

  Wu, Hao ^a   Huang, Zhongyi ^a   Jin, Shi ^b,c   Yin, Dongsheng ^a  

^a TSINGHUA UNIVERSITY   (China)

^b SHANGHAI JIAO TONG UNIVERSITY   (China)

^c UNIVERSITY OF WISCONSIN MADISON   (United States)

Author keywords

Dirac equation; Eulerian formulations; Gaussian beam method; Lagrangian; Semi classical regime

Indexed keywords

EID: 84878989671     PISSN: 15396746     EISSN: 19450796     Source Type: Journal    
DOI: 10.4310/CMS.2012.v10.n4.a14     Document Type: Article

References (40)

1. J. Cserti and G. Dávid, Unified description of Zitterbewegung for spintronic, graphene, and superconducting systems, Phys. Rev. B, 74, 172305, 2006.
2. P. Dirac, The Quantum Theory of the Electron, Proc. R. Soc. Lond. A, 117, 610-624, 1928.
3. B. Engquist and O. Runborg, Computational high frequency wave propagation, Acta Numer., 12, 181-266, 2003.
4. E.J. Heller, Time-dependent approach to semiclassical dynamics, J. Chem. Phys., 62, 1544- 1555, 1975.
5. D. Hsieh, D. Qian, L. Wray, Y. Xia, Y.S. Hor, R.J. Cava, and M.Z. Hasan, A topological Dirac insulator in a quantum spin Hall phase, Nature, 452, 970-974, 2008.
6. Z.Y. Huang, S. Jin, P.A. Markowich, C. Sparber, and C.X. Zheng, A time-splitting spectral scheme for the Maxwell-Dirac system, J. Comput. Phys., 208, 761-789, 2005.
7. S. Jin, H. Liu, S. Osher, and R. Tsai, Computing multi-valued physical observables the semiclassical limit of the Schrödinger equations, J. Comput. Phys., 205, 222-241, 2005.
8. S. Jin, P. Markowich, and C. Sparber, Mathematical and computational methods for semiclassical Schrödinger equations, Acta Numer., 20, 121-209, 2011.
9. S. Jin and S. Osher, A level set method for the computation of multivalued solutions to quasilinear hyperbolic PDEs and Hamilton-Jacobi equations, Commun. Math. Sci., 1, 575-591, 2003.
10. S. Jin, H. Wu, and X. Yang, Gaussian beam methods for the Schrödinger equation in the semi-classical regime: Lagrangian and Eulerian formulations, Commun. Math. Sci., 6, 995-1020, 2008.
11. S. Jin, H. Wu, and X. Yang, A numerical study of the Gaussian beam methods for onedimensional Schrödinger-Poisson equations, J. Comput. Math., 28, 261-272, 2010.
12. S. Jin, H. Wu, and X. Yang, Semi-Eulerian and high order Gaussian Beam methods for the Schrödinger equation in the semiclassical regime, Commun. Comput. Phys., 9, 668-687, 2011.
13. S. Jin, H. Wu, X. Yang, and Z. Huang, Bloch decomposition-based Gaussian beam method for the Schrödinger equation with periodic potentials, J. Comput. Phys., 229, 4869-4883, 2010.
14. W. Johnson, S. Blundell, and J. Sapirstein, Finite basis sets for the Dirac equation constructed from B splines, Phys. Rev. A, 37, 307-315, 1988.
15. O. Klein, Die Reflexion von Elektronen an einem Potentialsprung nach der relativistischen Dynamik von Dirac, Z. Phys. A, 53, 157-165, 1929.
16. S. Leung and J. Qian, Eulerian Gaussian beams for Schrödinger equations in the semi-classical regime, J. Comput. Phys., 228, 2951-2977, 2009.
17. S. Leung, J. Qian, and R. Burridge, Eulerian Gaussian beams for high-frequency wave propagation, Geophys., 72, 61-76, 2007.
18. H. Liu, O. Runborg, and N.M. Tanushev, Error estimates for Gaussian Beam superpositions, Math. Comput., to appear.
19. J. Lu and X. Yang, Frozen Gaussian approximation for high frequency wave propagation, Commun. Math. Sci., 9, 663-683, 2011.
20. J. Lu and X. Yang, Convergence of frozen Gaussian approximation for high frequency wave propagation, Commun. Pure Appl. Math., 65, 759-789, 2012.
21. C. Min, Simplicial isosurfacing in arbitrary dimension and codimension, J. Comput. Phys., 190, 295-310, 2003.
22. O. Morandi and F. Schürrer, Wigner model for quantum transport in graphene, J. Phys. A, 44, 265301, 2011.
23. M. Motamed and O. Runborg, Taylor expansion errors in Gaussian beam summation, Wave Motion, 47, 421-439, 2010.
24. A.H. Castro Neto, F. Guinea, N.M.R. Peres, K.S. Novoselov, and A.K. Geim, The electronic properties of graphene, Rev. Modern Phys., 81, 109-162, 2009.
25. D. Peng, B. Merriman, S. Osher, H. Zhao, and M. Kang, A PDE based fast local level set method, J. Comput. Phys., 155, 410-438, 1999.
26. M.M. Popov, A new method of computation of wave fields using Gaussian beams, Wave Motion, 4, 85-97, 1982.
27. J. Qian and L. Ying, Fast Gaussian wavepacket transforms and Gaussian beams for the Schröinger equation, J. Comput. Phys., 229, 7848-7873, 2010.
28. J. Ralston, Gaussian beams and the propagation of singularities, Studies in PDEs, MAA Stud. Math., 23, 206-248, 1982.
29. E. Schrödinger, über die kräftefreie Bewegung in der relativistischen Quantenmechanik, Sitzungsber. Preuss. Akad. Wiss., Phys. Math. Kl., 24, 418-428, 1930.
30. C. Sparber and P.A. Markowich, Semiclassical asymptotics for the Maxwell-Dirac system, J. Math. Phys., 44, 4555-4572, 2003.
31. C. Sparber and P.A. Markowich, Erratum: Semiclassical asymptotics for the Maxwell-Dirac system, J. Math. Phys., 45, 5101, 2003.
32. H. Spohn, Semiclassical limit of the Dirac equation and spin precession, Ann. Physics, 282, 420-431, 2000.
33. N.M. Tanushev, B. Engquist, and R. Tsai, Gaussian beam decomposition of high frequency wave fields, J. Comput. Phys., 228, 8856-8871, 2009.
34. N.M. Tanushev, Superpositions and higher order Gaussian beams, Commun. Math. Sci., 6, 449-475, 2008.
35. Thaller B. The Dirac Equation 1992 Springer.
36. X. Wen, High order numerical methods to two dimensional delta function integrals in level set methods, J. Comput. Phys., 228, 4273-4290, 2009.
37. X. Wen, High order numerical methods to three dimensional delta function integrals in level set methods, SIAM J. Sci. Comput., 32, 1288-1309, 2010.
38. Y. Xia, D. Qian, D. Hsieh, L.Wray, A. Pal, H. Lin, A. Bansil, D. Grauer, Y.S. Hor, R.J. Cava, and M.Z. Hasan, Observation of a large-gap topological-insulator class with a single Dirac cone on the surface, Nature Physics, 5, 398-402, 2009.
39. D. Yin and C. Zheng, Gaussian beam formulations and interface conditions for the onedimensional linear Schrödinger equation, Wave Motion, 48, 310-324, 2011.
40. D. Yin and C. Zheng, Composite coherent states approximation for one-dimensional multiphased wave functions, Commun. Comput. Phys., 11, 951-984, 2012.