Multiphase semiclassical approximation of an electron in a one-dimensional crystalline lattice I. Homogeneous problems

Journal of Computational Physics

Volumn 197, Issue 2, 2004, Pages 387-417

  Gosse, Laurent ^a   Markowich, Peter A ^b  

^a CNR   (Italy)

^b UNIVERSITY OF VIENNA   (Austria)

Author keywords

Homogenization; Moment method; Non strictly hyperbolic systems; Periodic potential; Semiclassical limit; Vlasov equation

Indexed keywords

HOMOGENIZATION METHOD; HYPERBOLIC FUNCTIONS; VLASOV EQUATION; WAVE FUNCTIONS;

HOMOGENIZATION; HYPERBOLIC SYSTEM; MOMENTS METHOD; MULTIPHASES; NON-STRICTLY HYPERBOLIC; NON-STRICTLY HYPERBOLIC SYSTEM; PERIODIC POTENTIALS; SEMI-CLASSICAL APPROXIMATION; SEMICLASSICAL APPROXIMATIONS; SEMICLASSICAL LIMIT;

METHOD OF MOMENTS;

EID: 3242737651     PISSN: 00219991     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.jcp.2003.12.004     Document Type: Article

References (53)

1. Amadori D. Gosse L. Guerra G. Global BV entropy solutions and uniqueness for hyperbolic systems of balance laws Arch. Rational Mech. Anal. 162 2002 327-366
2. Ashcroft N.W. Mermin N.D. Solid-state Physics 1976 Rinehart and Winston Holt
3. Ancona F. Marson A. A wave front tracking algorithm for N×N non genuinely nonlinear conservation laws J. Differential Equations 177 2001 454-493
4. Bal G. Fannjiang A. Papanicolaou G. Ryzhik L. Radiative transfer in periodic structure J. Stat. Phys. 95 1999 479-494
5. Bao W.Z. Jin S. Markowich P.A. On time-splitting spectral approximations for the Schrödinger equation in the semiclassical regime J. Comput. Phys. 175 2002 487-524
6. Benamou J.D. Direct computation of multivalued phase space solutions for Hamilton-Jacobi equations Commun. Pure Appl. Math. 52 1999 1443-1475
7. Bensoussan A. Lions J.L. Papanicolaou G. Asymptotic Analysis for Periodic Structures 1978 North-Holland Amsterdam
8. Bloch F. Über die quantenmechanik der electronen in kristallgittern Z. Phys. 52 1928 555-600
9. Bouchut F. James F. One-dimensional transport equations with discontinuous coefficients Nonlinear Anal. TMA 32 1998 891-933
10. Bouchut F. James F. Duality solutions for pressureless gases, monotone conservation laws and uniqueness Commun. PDE 24 1999 2173-2189
11. Bouchut F. Jin S. Li X. Numerical approximations of pressureless and isothermal gas dynamics SIAM J. Numer. Anal. 41 2003 135-158
12. Brassart M. Limite semi-classique de transformée de Wigner dans des milieux périodiques ou aléatoires, Ph.D. Thesis, Univ. de Nice, France, 2002
13. Brenier Y. Averaged multivalued solutions for scalar conservation laws SIAM J. Numer. Anal. 21 1984 1013-1037
14. Brenier Y. Corrias L. A kinetic formulation for multibranch entropy solutions of scalar conservation laws Ann. IHP Nonlinear Anal. 15 1998 169-190
15. Brenier Y. Grenier E. Sticky particles and scalar conservation laws SIAM J. Numer. Anal. 35 1998 2317-2328
16. Cheng L.-T. Liu H. Osher S.J. High-frequency wave propagation in Schrödinger equations using the level set method Commun. Math. Sci. 1 3 2003 593-621
17. Christ M. Kiselev A. WKB and spectral analysis of one-dimensional Schrödinger operators for slowly varying potentials Commun. Math. Phys. 218 2001 245-262
18. Colin de Verdière Y. Parisse B. Singular Bohr-Sommerfeld rules Commun. Math. Phys. 205 1999 459-500
19. Concordel M.C. Periodic homogenization of Hamilton-Jacobi equations: additive eigenvalues and variational formula Indiana Univ. Math. J. 45 1996 1095-1118
20. Concordel M.C. Periodic homogenization of Hamilton-Jacobi equations. II. Eikonal equations Proc. Roy. Soc. Edinburgh A 127 1997 665-689
21. Connor J.N.L. Uzer T. Marcus R.A. Eigenvalues of the Schrödinger equation for a periodic potential with nonperiodic boundary conditions; a uniform semiclassical analysis J. Chem. Phys. 80 10 1984 5095-5106
22. Crandall M. Lions P.L. Viscosity solutions of Hamilton-Jacobi equations Trans. Am. Math. Soc. 282 1984 487-502
23. Engquist B. Runborg O. Tornberg A.K. High frequency wave propagation by the segment projection method J. Comput. Phys. 178 2002 373-390
24. Evans L.C. Effective Hamiltonians and quantum states, Séminaire X-EDP 2000-2001. Available from http://math.polytechnique.fr/ seminaires/seminaires-edp/2000-2001/evans.ps
25. Evans L.C. Gomes D. Effective Hamiltonians and averaging for Hamiltonian dynamics, I Arch. Rational Mech. Anal. 157 2001 1-33
26. Fatemi E. Engquist B. Osher S. Numerical solution of the high frequency asymptotic expansion for the scalar wave equation J. Comput. Phys. 120 1995 145-155
27. Gérard P. Markowich P.A. Mauser N.J. Poupaud F. Homogenization limits and Wigner transforms Commun. Pure Appl. Math. 50 4 1997 323-379
28. Goatin P. Oleinik estimates and uniqueness for hyperbolic systems of balance laws Math. Models Meth. Appl. Sci. 13 2003 527-543
29. Goatin P. Gosse L. Decay of positive waves for n×n hyperbolic systems of balance laws, Proc. Amer. Math. Soc. (to appear)
30. Gosse L. Using K-branch entropy solutions for multivalued geometric optics computations J. Comput. Phys. 180 2002 155-182
31. Gosse L. James F. Convergence results for an inhomogeneous system arising in various high frequency approximations Math. 90 2002 721-753
32. Gosse L. Jin S. Li X. Two moment systems for computing multiphase semiclassical limits of the Schrödinger equation, Math. http://dx.doi.org/10.1142/S0218202503003082
33. Guillot J.C. Ralston J. Trubowitz E. Semiclassical asymptotics in solid-state physics Commun. Math. Phys. 116 1988 401-415
34. Hövermann F. Spohn H. Teufel S. Semiclassical limit for the Schrödinger equation for a short scale periodic potential Commun. Math. Phys. 215 2001 609-629
35. Izumiya S. Kossioris G.T. Geometric singularities for solutions of single conservation laws Arch. Rational Mech. Anal. 139 1997 255-290
36. Jin S. Li X. Multi-phase computations of the semiclassical limit of the Schrödinger equation and related problems: Whitham vs. Wigner Physica D 182 2003 46-85
37. Jin S. Osher S. A level set method for the computation of multivalued solutions to quasi-linear hyperbolic PDEs and Hamilton-Jacobi equations Commun. Math. Sci. 1 3 2003 575-591
38. Keller J.B. Semiclassical mechanics SIAM Rev. 27 1985 485-504
39. Korsch H.J. Glück M. Computing quantum eigenvalues made easy Eur. J. Phys. 23 2002 413
40. Kronig R. Penney W.G. Quantum mechanics of electrons in crystal lattices Proc. R. Soc. A 144 1931 101
41. Lions P.L. Papanicolaou G. Varadhan S.R.S. Homogenization of Hamilton-Jacobi equations, unpublished manuscript, circa 1988
42. Lions P.L. Paul T. Sur les measures de Wigner Revista Mat. Iberoamericana 9 1993 553-618
43. Lions P.L. Perthame B. Tadmor E. A kinetic formulation of multidimensional scalar conservation laws and related equations J. Am. Math. Soc. 7 1994 169-191
44. Lions P.L. Perthame B. Tadmor E. Kinetic formulation of the isentropic gas dynamics and p-systems Commun. Math. Phys. 163 1994 415-431
45. Ludwig D. Uniform expansions at a caustic Commun. Pure Appl. Math. 19 1966 215-250
46. Markowich P. Mauser N.J. Poupaud F. A Wigner-function approach to semiclassical limits: electrons in a periodic potential J. Math. Phys. 35 1994 1066-1094
47. Poupaud F. Ringhofer C. Semi-classical limits in a crystal with external potentials and effective mass theorems Commun. Partial Differential Equations 21 1996 1897-1918
48. Runborg O. Some new results in multiphase geometrical optics Math. Mod. Numer. Anal. 34 2000 1203-1231
49. Sparber C. Markowich P. Mauser N. Multivalued geometrical optics: Wigner functions vs. WKB methods Asymptotic Anal. 33 2003 153-187
50. Symes W.W. Qian J. A slowness matching Eulerian method for multivalued solutions of Eikonal equations J. Sci. Comp. 19 2003 501-526
51. Tadmor E. Tassa T. On piecewise regularity of entropy solutions to scalar conservation laws Commun. PDE 18 1993 1631-1652
52. Vasseur A. Convergence of a semi-discrete kinetic scheme for the system of isentropic gas dynamics with γ = 3 Indiana Univ. Math. J. 48 1999 347-364
53. Zettl A. Spectral theory and computational methods for Sturm-Liouville problems In: Hinton D. Schaefer P.W. (Eds.) Lecture Notes in Pure and Applied Mathematics vol. 191 1997 Marcel Dekker New York