A bloch decomposition-based split-step pseudospectral method for quantum dynamics with periodic potentials

SIAM Journal on Scientific Computing

Volumn 29, Issue 2, 2007, Pages 515-538

  Huang, Zhongyi ^a   Jin, Shi ^b   Markowich, Peter A ^c   Sparber, Christof ^c  

^a TSINGHUA UNIVERSITY   (China)

^b UNIVERSITY OF WISCONSIN MADISON   (United States)

^c UNIVERSITY OF VIENNA   (Austria)

Author keywords

Bloch decomposition; Lattice potential; Schr dinger equation; Semiclassical asymptotics; Time splitting spectral method

Indexed keywords

ALGORITHMS; ASYMPTOTIC ANALYSIS; COMPUTATION THEORY; NUMERICAL METHODS; PERTURBATION TECHNIQUES; SOLID STATE PHYSICS;

BLOCH DECOMPOSITION; LATTICE POTENTIAL; SEMICLASSICAL ASYMPTOTICS; TIME-SPLITTING SPECTRAL METHOD;

SCHRODINGER EQUATION;

EID: 41449101890     PISSN: 10648275     EISSN: None     Source Type: Journal    
DOI: 10.1137/060652026     Document Type: Article

References (34)

1. J. ASCH AND A. KNAUF, Motion in periodic potentials, Nonlinearity, 11 (1998), pp. 175-200.
2. N. W. ASHCROFT AND N. D. MERMIN, Solid State Physics, Saunders, New York, 1976.
3. W. Z. BAO, S. JIN, AND P. MARKOWICH, On time-splitting spectral approximations for the Schrödinger equation in the semiclassical regime, J. Comput. Phys., 175 (2002), pp. 487-524.
4. W. Z. BAO, S. JIN, AND P. A. MARKOWICH, Numerical study of time-splitting spectral discretizations of nonlinear Schrödinger equations in the semiclassical regimes, SIAM J. Sci. Comput., 25 (2003), pp. 27-64.
5. P. BECHOUCHE AND F. POUPAUD, Semi-classical limit of a Schrödinger equation for a stratified material, Monatsh. Math., 129 (2000), pp. 281-301.
6. A. BENSOUSSAN, J. L. LIONS, AND G. P APANICOLAOU, Asymptotic Analysis for Periodic Structures, North-Holland, Amsterdam, 1978.
7. F. BLOCH, Über die Quantenmechanik der Elektronen in Kristallgittern, Z. Phys., 52 (1928), pp. 555-600.
8. E. I. BLOUNT, Formalisms of band theory, in Solid State Physics, Vol. 13, Academic Press, New York, 1962, pp. 305-373.
9. K. BUSCH, Photonic band structure theory: Assessment and perspectives, Compte Rendus Phys., 3 (2002), pp. 53-66.
10. R. CARLES, WKB analysis for nonlinear Schrödinger equations with a potential, Comm. Math. Phys., 69 (2007), pp. 195-221.
11. R. CARLES. P. A. MARKOWICH, AND C SPARBER, Semiclassical asymptotics for weakly nonlinear Bloch waves, J. Statist. Phys., 117 (2004), pp. 369-401.
12. C. CONCA, R. ORIVE, AND M. VANNINATHAN, Bloch approximation in homogenization on bounded domains, Asymptot. Anal., 41 (2005), pp. 71-91.
13. C. CONCA, N. SRINIVASAN, AND M. VANNINATHAN, Numerical solution of elliptic partial differential equations by Bloch waves method, in Congress on Differential Equations and Applications/VII CMA, Universidad de Salamanca, 2001, pp. 63-83.
14. C. CONCA AND M. VANNINATHAN, Homogenization of periodic structures via Bloch decomposition, SIAM J. Appl. Math., 57 (1997), pp. 1639-1659.
15. M. V. FISCHETTI AND S. E. LAUX, Monte Carlo analysis of electron transport in small semiconductor devices including band-structure and space-charge effects, Phys. Rev. B, 38 (1998), pp. 9721-9745.
16. P. GÉRARD, P. MARKOWICH, N. MAUSER, AND F. POUPAUD, Homogenization limits and Wigner transforms, Comm. Pure Appl. Math, 50 (1997), pp. 323-378.
17. L. GOSSE, Multiphase semiclassical approximation of an electron in a one-dimensional crystalline lattice. II. Impurities, confinement and Bloch oscillations, J. Comput. Phys., 201 (2004), pp. 344-375.
18. L. GOSSE AND P. A. MARKOWICH, Multiphase semiclassical approximation of an electron in a one-dimensional crystalline lattice - I. Homogeneous problems, J. Comput. Phys., 197 (2004), pp. 387-417.
19. L. GOSSE AND N. MAUSER, Multiphase semiclassical approximation of an electron in a one-dimensional crystalline lattice. III. From ab initio models to WKB for Schrödinger-Poisson, J. Comput. Phys., 211 (2006), pp. 326-346.
20. J. C. GUILLOT, J. RALSTON, AND E. TRUBOWITZ, Semiclassical asymptotics in solid-state physics, Comm. Math. Phys., 116 (1998), pp. 401-415.
21. D. HERMANN. M. FRANK, K. BUSCH, AND P. WÖLFLE, Photonic band structure computations, Optics Express, 8 (2001), pp. 167-173.
22. R. HORN AND C. JOHNSON, Matrix Analysis, Cambridge University Press, Cambridge, UK, 1985.
23. Z. HUANG, S. JIN, P. MARKOWICH, C. SPARSER, AND C. ZHENG, A time-splitting spectral scheme for the Maxwell-Dirac system, J. Comput. Phys., 208 (2005), pp. 761-789.
24. S. JIN AND Z. XIN, Numerical passage from systems of conservation laws to Hamilton-Jacobi equations, and relaxation schemes, SIAM J. Numer. Anal., 35 (1998), pp. 2385-2404.
25. J. D. JOANNOPOULOS AND M. L. COHEN, Theory of short range order and disorder in tetrahedrally bonded semiconductors. Solid State Phys., 31 (1974), pp. 71-148.
26. H. J. KORSCH AND M. GLÜCK, Computing quantum eigenvalues made easy, European J. Phys., 23 (2002), pp. 413-425.
27. S. E. LAUX, M. V. FISCHETTI, AND D. J. FRANK, Monte Carlo analysis of semiconductor devices: The DAMOCLES program, IBM J. Res. Develop., 34 (1990), pp. 466-494.
28. J. M. LUTTINGER, The effect of a magnetic field on electrons in a periodic potential, Phys. Rev., 84 (1951), pp. 814-817.
29. G. PANATI, H. SPOHN, AND S. TEUFEL, Effective dynamics for Bloch electrons: Peierls substitution and beyond, Comm. Math. Phys., 242 (2003), pp. 547-578.
30. M. REED AND B. SIMON, Methods of Modern Mathematical Physics IV. Analysis of Operators, Academic Press, New York, 1978.
31. S. TEUFEL, Adiabatic Perturbation Theory in Quantum Dynamics, Lecture Notes in Math. 1821, Springer-Verlag, Berlin, 2003.
32. C. H. WILCOX, Theory of Bloch waves, J. Anal. Math., 33 (1978), pp. 146-167.
33. J. ZAK, Dynamics of electrons in solids in external fields, Phys. Rev., 168 (1968), pp. 686-695.
34. A. ZETTEL, Sturm-Liouville problems, in Spectral Theory and Computational Methods for Sturm-Liouville Problems, Lecture Notes in Pure and Appl. Math. 191, D. Hinton and P. W. Schäfer, eds., Dekker, New York, 1997, pp. 1-104.