Advertisement

Acta Applicandae Mathematica

, Volume 1, Issue 3, pp 241–261 | Cite as

A review of recent work on almost periodic differential and difference operators

  • Russell A. Johnson
Article

Abstract

We discuss recent work on almost periodic differential and difference operators, especially the one-dimensional Schrödinger equation and its finite-difference analogue.

AMS (MOS) subject classifications (1980)

34A30 47A10 82A42 

Key words

Almost periodic Schrödinger operator random operator 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Ablowitz M.: Stud. Appl. Math. 58 (1978), 17–94.Google Scholar
  2. 2.
    Ablowitz M., Kaup D., Newell A. and Segur H.: Stud. Appl. Math. 53 (1974), 249–315.Google Scholar
  3. 3.
    Anderson P.: Phys. Rev. 109, (1958), 1492–1501.Google Scholar
  4. 4.
    Andre G. and Aubrey S.: Ann. Israel Phys. Soc. 3 (1980), 133.Google Scholar
  5. 5.
    Aubrey S.: The new concept of transition by breaking of analyticity, in Solid State Sciences: Solitons and Condensed Matter Physics, Vol. 8, Springer-Verlag, Berlin, New York, 1978.Google Scholar
  6. 6.
    Avron J. and Simon B.: Comm. Math. Phys. 82 (1981), 101–120.Google Scholar
  7. 7.
    Avron J. and Simon B.: Duke Math. J. 50 (1983), 369–391.Google Scholar
  8. 8.
    Avron J. and Simon B.: Bull. Amer. Math. Soc. 6 (1982), 81–86.Google Scholar
  9. 9.
    Avron J. and Simon B.: J. Func. Anal. 43 (1981), 1–31.Google Scholar
  10. 10.
    Avron J. and Simon B.: Phys. Rev. Lett. 46 (1981), 1166.Google Scholar
  11. 11.
    A'bell M.: Phys. Rev. Lett. 43 (1979), 1954–1957.Google Scholar
  12. 12.
    Bellissard J. and Scoppola E.: Comm. Math. Phys. 85 (1982), 301–308.Google Scholar
  13. 13.
    Bellissard J. and Simon B.: J. Func. Anal. 48 (1982), 408–419.Google Scholar
  14. 14.
    Bellissard J., Bessis D. and Moussa P.: Phys. Rev. Lett. 49 (1982), 701–704.Google Scholar
  15. 15.
    Bellissard J., Lima R. and Scoppola E.: Comm. Math. Phys. 88 (1983), 465–478.Google Scholar
  16. 16.
    Bellissard, J., Lima, R. and Testard, D.: ‘Almost random operators, K-theory, and spectral properties’, Marseilles preprint, 1981.Google Scholar
  17. 17.
    Bellissard J., Lima R. and Testard D.: Comm. Math. Phys. 88 (1983), 207–234.Google Scholar
  18. 18.
    Carmona R.: Duke Math. J. 49 (1982), 191–213.Google Scholar
  19. 19.
    Carmona, R.: Lecture Notes (in preparation).Google Scholar
  20. 20.
    Chulaevsky V.: Uspekhi Math. Nauk. 36 (1981), 203–204.Google Scholar
  21. 21.
    Coddington E. and Levinson N.: Theory of Ordinary Differential Equations, McGraw-Hill, New York, 1955.Google Scholar
  22. 22.
    Connes A.: ‘Sur la theorie noncommutative de l'integration’, Lecture Notes in Math. Vol. 725, Springer-Verlag, Berlin, New York, 1980, pp. 19–143.Google Scholar
  23. 23.
    Connes A.: Adv. Math. 39 (1981), 31–55.Google Scholar
  24. 24.
    Coppel W.: ‘Dichotomies in Stability Theory’, Lecture Notes in Math. Vol. 629, Springer-Verlag, Berlin, New York, 1978.Google Scholar
  25. 25.
    Craig W.: Comm. Math. Phys. 88 (1983), 113–130.Google Scholar
  26. 26.
    Craig, W. and Simon, B.: ‘Log Hölder continuity of the integrated density of states for stochastic Jacobi matrices’, to appear in Comm. Math. Phys. Google Scholar
  27. 27.
    Craig, W. and Simon, B.: ‘Subharmonicity of the Lyapounov index’, to appear in Duke Math. J. Google Scholar
  28. 28.
    Deift, P. and Simon, B.: ‘Almost periodic Schrödinger operators III. The absolutely continuous spectrum in one dimension’, to appear in Comm. Math. Phys. Google Scholar
  29. 29.
    Delyon F. and Souillard B.: Comm. Math. Phys. 89 (1983), 415–426.Google Scholar
  30. 30.
    Delyon F., Kunz H. and Souillard B.: J. Phys. A 16 (1983), 25–42.Google Scholar
  31. 31.
    Dinaburg E. and Sinai Ya.: Func. Anal. Appl. 9 (1975), 8–21.Google Scholar
  32. 32.
    Dubrovin B., Matveev V. and Novikov S.: Russian Math. Surv. 31 (1976), 59–146.Google Scholar
  33. 33.
    Ellis R. and Johnson R.: J. Diff. Eqns. 44 (1982), 21–39.Google Scholar
  34. 34.
    Fishman S., Grempel D. and Prange R.: Phys. Rev. Lett. 49 (1982), 833.Google Scholar
  35. 35.
    Flaschka H. and Newell A.: ‘Integrable systems of non-linear evolutions’, Lecture Notes in Phys. Vol. 38, Springer-Verlag, New York, Berlin, 1975, pp. 355–440.Google Scholar
  36. 36.
    Forest M. and McLaughlin D.: J. Math. Phys. 23 (1982), 1248–1277.Google Scholar
  37. 37.
    Fröhlich J. and Spencer T.: Comm. Math. Phys. 88 (1983), 151–184.Google Scholar
  38. 38.
    Furstenberg H.: Trans. Math. Soc. 108 (1963), 377–428.Google Scholar
  39. 39.
    Gardner C., Greene J., Kruskal M. and Miura R.: Phys. Rev. Lett. 19 (1967), 1095–1097.Google Scholar
  40. 40.
    Giachetti, R. and Johnson, R.: ‘Spectral theory of second order almost periodic differential operators, and its relation to certain non-linear evolution equations’, USC preprint.Google Scholar
  41. 41.
    Goldscheid I.: Sov. Math. Dokl. 22 (1980), 670–675.Google Scholar
  42. 42.
    Goldscheid I., Molchanov S. and Pastur L.: Funct. Anal. Appl. 11 (1977), 1–10.Google Scholar
  43. 43.
    Gordon A.: Usp. Math. Nauk. 31 (1976), 257–260.Google Scholar
  44. 44.
    Herman, M.: ‘Une methode pur minoriser les exposants de Lyapounov et quelques exemples montrant le charactere local d'un theoreme d'Arnold et de Moser sur le tore de dimension 2’, Palaiseau preprint.Google Scholar
  45. 45.
    Herman, M.: ‘Rotation number for skew product homeomorphisms’, handwritten notes.Google Scholar
  46. 46.
    Hofstadter D.: Phys. Rev. B 14 (1976), 2239–2249.Google Scholar
  47. 47.
    Ishii K.: Supp. Theor. Phys. 53 (1973), 77–138.Google Scholar
  48. 48.
    Johnson, R.: ‘An almost periodic Schrödinger equation with an eigenvalue’, unpublished.Google Scholar
  49. 49.
    Johnson, R.: ‘Lyapounov exponents for the almost periodic Schrödinger equation’, to appear in Illinois J. of Math. Google Scholar
  50. 50.
    Johnson R.: J. Diff. Eqns. 46 (1982), 165–194.Google Scholar
  51. 51.
    Johnson, R.: in preparation.Google Scholar
  52. 52.
    Johnson R. and Moser J.: Comm. Math. Phys. 84 (1982), 403–438.Google Scholar
  53. 53.
    Kotani, S.: ‘Lyapounov indices determine absolutely continuous spectra of stationary random one-dimensional Schrödinger operators’, Jyoto preprint.Google Scholar
  54. 54.
    Krichever I. and Novikov S.: Russ. Math. Surv. 35 (1980), 53–79.Google Scholar
  55. 55.
    Kunz, H. and Souillard, B.: in preparation.Google Scholar
  56. 56.
    LaCroix, J.: in ‘Marches Aleatoires, Proc. Nancy 1981’, to appear in Lecture Notes in Math., Springer-Verlag, Berlin, New York.Google Scholar
  57. 57.
    Ledrappier F. and Royer G.: Acad. Sci. Paris, Serie A, 290 (1980), 513–514.Google Scholar
  58. 58.
    Little W.: Phys. Rev. A 134 (1964), 1416.Google Scholar
  59. 59.
    Lloyd P.: J. Phys. C 2 (1969), 1717.Google Scholar
  60. 60.
    Magnus W. and Winkler S.: Hill's Equation, Dover, New York, 1979.Google Scholar
  61. 61.
    McKean H.: Comm. Pure Appl. Math. 34 (1981), 599–691.Google Scholar
  62. 62.
    McKean H. and Trubowitz E.: Comm. Pure Appl. Math. 29 (1976), 143–226.Google Scholar
  63. 63.
    McKean H. and van Moerbeke P.: Comm. Pure Appl. Math. 33 (1980), 23–42.Google Scholar
  64. 64.
    McKean H. and van Moerbeke P.: Invent. Math. 30 (1975), 217–274.Google Scholar
  65. 65.
    Molchanov S.: Math. USSR Izv. 12 (1978), 69–101.Google Scholar
  66. 66.
    Moser J.: Comm. Math. Helv. 56 (1981), 198–224.Google Scholar
  67. 67.
    Novikov S.: Func. Anal. Appl. 8 (1974), 236–246.Google Scholar
  68. 68.
    Oseledec V.: Trans. Moscow Math. Soc. 19 (1968), 197–231.Google Scholar
  69. 69.
    Pastur L.: Comm. Math. Phys. 75 (1980), 179–196.Google Scholar
  70. 70.
    Peierls R.: Z. Phys. 80 (1933), 763.Google Scholar
  71. 71.
    Pimsner M. and Voiculescu D.: J. Operator Theory, 4 (1980), 201–218.Google Scholar
  72. 72.
    Pöschl J.: Comm. Math. Phys. 88 (1983), 447–464.Google Scholar
  73. 73.
    Ruelle D.: Publ. IHES, 50 (1979), 275–320.Google Scholar
  74. 74.
    Sacker R. and Sell G.: J. Diff. Eqns. 15 (1974), 429–458.Google Scholar
  75. 75.
    Sacker R. and Sell G.: J. Diff. Eqns. 27 (1978), 320–358.Google Scholar
  76. 76.
    Sarnak P.: Comm. Math. Phys. 84 (1982), 377–401.Google Scholar
  77. 77.
    Scott A., Chu F. and McLaughlin D.: Proc. IEEE, 61 (1973), 1443–1483.Google Scholar
  78. 78.
    Selgrade J.: Isolated invariant sets for flows on vector bundles, Trans. Amer. Math. Soc. 203 (1975). 359–390.Google Scholar
  79. 79.
    Shubin M.: Trudy Sem. Petrovskii, 3 (1978), 243–270.Google Scholar
  80. 80.
    Simon B.: Adv. Appl. Math. 3 (1982), 463–490.Google Scholar
  81. 81.
    Simon, B.: ‘Kotani theory for one-dimensional stochastic Jacobi matrices’, to appear in Comm. Math. Phys. Google Scholar
  82. 82.
    Simon, B.: ‘On the equality of the density of states in the Lloyd and Maryland models’, Phys. Rev. B, 1983.Google Scholar
  83. 83.
    Thouless D.: J. Phys. C 5 (1972), 77–81.Google Scholar
  84. 84.
    Zakharov V. and Shabat A.: Soc. Phys. JETP, 34 (1972), 62–69.Google Scholar
  85. 85.
    Kac, M.: ‘Nonlinear dynamics and inverse problems’, in Proceedings of the 1977 Summer School in Statistical Mechanics, Warsaw, 1978.Google Scholar
  86. 86.
    McKean H.: ‘Integrable systems and algebraic curves’, Lecture Notes in Math. Vol. 755, Springer-Verlag, New York, Berlin, 1979, pp. 83–220.Google Scholar
  87. 87.
    McKean H.: Comm. Pure Appl. Math. 34 (1981), 197–257.Google Scholar
  88. 88.
    Elliot G.: C.r. Math. Acad. Sci. Canada, 4 (1982), 255–258.Google Scholar
  89. 89.
    Barnsely M., Geronimo J., Harrington A.: Comm. Math. Phys. 88 (1983), 479–502.Google Scholar
  90. 90.
    Scharf G.: Helv. Phys. Acta 24 (1965), 573–605.Google Scholar
  91. 91.
    Herbert D. and Jones R.: J. Phys. C 4 (1971), 1145.Google Scholar

Copyright information

© D. Reidel Publishing Company 1983

Authors and Affiliations

  • Russell A. Johnson
    • 1
    • 2
  1. 1.Universität HeidelbergHeidelbergFederal Republic of Germany
  2. 2.Dept. of MathematicsUniversity of Southern CaliforniaLos AngelesU.S.A.

Personalised recommendations