The Saxon-Hutner theorem, its converse theorem and their applications to localization and delocalization problems in binary quasiperiodic lattices

International Journal of Modern Physics B

Volumn 11, Issue 18, 1997, Pages 2157-2182

  Iguchi, Kazumoto ^a  

^a NONE   (Japan)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0005880524     PISSN: 02179792     EISSN: None     Source Type: Journal    
DOI: 10.1142/S0217979297001118     Document Type: Article

References (87)

1. C. Kittel, Introduction to Solid State Physics, 5th Ed., (Academic Press, New York, 1995); N. W. Ashcroft and N. D. Mermin, Solid State Physics, (Saunders College New York, 1976).
2. C. Kittel, Introduction to Solid State Physics, 5th Ed., (Academic Press, New York, 1995); N. W. Ashcroft and N. D. Mermin, Solid State Physics, (Saunders College New York, 1976).
3. P. W. Anderson, Phys. Rev. 109, 1492 (1958).
4. J. Hori, Spectral Properties of Disordered Chains and Lattices, (Pergamon Oxford, 1968); J. M. Ziman, Models of Disorder, (Cambridge University Press, Cambridge, 1979).
5. J. Hori, Spectral Properties of Disordered Chains and Lattices, (Pergamon Oxford, 1968); J. M. Ziman, Models of Disorder, (Cambridge University Press, Cambridge, 1979).
6. A. Ja. Gordon, Uspekhi Mat. Nauk 31, 257 (1976); B. Simon, Adv. Appl. Math. 3, 463 (1982).
7. A. Ja. Gordon, Uspekhi Mat. Nauk 31, 257 (1976); B. Simon, Adv. Appl. Math. 3, 463 (1982).
8. M. Kohmoto, L. P. Kadanoff and C. Tang, Phys. Rev. Lett. 50, 1870 (1983); S. Ostlund, D. Pandit, D. Rand, H. J. Schelmhuber and E. D. Siggia, Phys. Rev. Lett. 50, 1873 (1983); Also see J. B. Sokoloff, Phys. Rep. 126, 189 (1985).
9. M. Kohmoto, L. P. Kadanoff and C. Tang, Phys. Rev. Lett. 50, 1870 (1983); S. Ostlund, D. Pandit, D. Rand, H. J. Schelmhuber and E. D. Siggia, Phys. Rev. Lett. 50, 1873 (1983); Also see J. B. Sokoloff, Phys. Rep. 126, 189 (1985).
10. M. Kohmoto, L. P. Kadanoff and C. Tang, Phys. Rev. Lett. 50, 1870 (1983); S. Ostlund, D. Pandit, D. Rand, H. J. Schelmhuber and E. D. Siggia, Phys. Rev. Lett. 50, 1873 (1983); Also see J. B. Sokoloff, Phys. Rep. 126, 189 (1985).
11. M. Kohmoto and Y. Oono, Phys. Lett. A102, 145 (1984); Also see L. P. Kadanoff and C. Tang, Proc. Natl. Acad. Sci. (USA) 81, 1276 (1984).
12. M. Kohmoto and Y. Oono, Phys. Lett. A102, 145 (1984); Also see L. P. Kadanoff and C. Tang, Proc. Natl. Acad. Sci. (USA) 81, 1276 (1984).
13. M. Kohmoto and J. R. Banavar, Phys. Rev. B34, 563 (1986).
14. M. Kohmoto, B. Sutherland and C. Tang, Phys. Rev. B35, 1020 (1987).
15. F. Delyon and D. Petritis, Commun. Math. Phys. 103, 441 (1986).
16. M. Casdagli, Commun. Math. Phys. 107, 295 (1986).
17. A. Sütö, Commun. Math. Phys. 111, 409 (1987).
18. A. Sütö, J. Stat. Phys. 56, 525 (1989).
19. J. Bellissard, B. lochum, E. Scoppola and D. Testard, Commun. Math. Phys. 125, 527 (1989).
20. P. A. Kalugin, A. Yu. Kitaev and L. S. Levitov, Zh. Eksp. Teor. Fiz. 91, 692 (1986) [Sov. Phys. JETP 64 (2), 410 (1986)].
21. P. A. Kalugin, A. Yu. Kitaev and L. S. Levitov, Zh. Eksp. Teor. Fiz. 91, 692 (1986) [Sov. Phys. JETP 64 (2), 410 (1986)].
22. B. Sutherland, in Disorder and Nonlinearity, ed. A. R. Bishop, D. K. Campbell and S. Pnevmatikos (Springer-Verlag, Berlin, 1989), p. 62.
23. K. Iguchi, Phys. Rev. B43, 5915 (1991); ibid. B43, 5919 (1991); J. Math. Phys. 33, 3938 (1992); Mater. Sci. Eng. B15, L13 (1992); Phys. Lett. A174, 38 (1993); Int. J. Mod. Phys. B8, 1931 (1994).
24. K. Iguchi, Phys. Rev. B43, 5915 (1991); ibid. B43, 5919 (1991); J. Math. Phys. 33, 3938 (1992); Mater. Sci. Eng. B15, L13 (1992); Phys. Lett. A174, 38 (1993); Int. J. Mod. Phys. B8, 1931 (1994).
25. K. Iguchi, Phys. Rev. B43, 5915 (1991); ibid. B43, 5919 (1991); J. Math. Phys. 33, 3938 (1992); Mater. Sci. Eng. B15, L13 (1992); Phys. Lett. A174, 38 (1993); Int. J. Mod. Phys. B8, 1931 (1994).
26. K. Iguchi, Phys. Rev. B43, 5915 (1991); ibid. B43, 5919 (1991); J. Math. Phys. 33, 3938 (1992); Mater. Sci. Eng. B15, L13 (1992); Phys. Lett. A174, 38 (1993); Int. J. Mod. Phys. B8, 1931 (1994).
27. K. Iguchi, Phys. Rev. B43, 5915 (1991); ibid. B43, 5919 (1991); J. Math. Phys. 33, 3938 (1992); Mater. Sci. Eng. B15, L13 (1992); Phys. Lett. A174, 38 (1993); Int. J. Mod. Phys. B8, 1931 (1994).
28. K. Iguchi, Phys. Rev. B43, 5915 (1991); ibid. B43, 5919 (1991); J. Math. Phys. 33, 3938 (1992); Mater. Sci. Eng. B15, L13 (1992); Phys. Lett. A174, 38 (1993); Int. J. Mod. Phys. B8, 1931 (1994).
29. M. Baake, U. Grimm and D. Joseph, Int. J. Mod. Phys. B7, 1527 (1993); J. A. G. Roberts and M. Baake, J. Stat. Phys. 74, 829 (1994).
30. M. Baake, U. Grimm and D. Joseph, Int. J. Mod. Phys. B7, 1527 (1993); J. A. G. Roberts and M. Baake, J. Stat. Phys. 74, 829 (1994).
31. D. S. Saxon and R. A. Hutner, Philips Res. Rep. 4, 81 (1949); J. M. Luttinger, ibid. 6, 303 (1951).
32. D. S. Saxon and R. A. Hutner, Philips Res. Rep. 4, 81 (1949); J. M. Luttinger, ibid. 6, 303 (1951).
33. K. Iguchi, J. Phys. Soc. Jpn. 64, 177 (1995); (Errata) ibid. 64, 3124 (1995): Phys. Rev. B51, 12268 (1995); K. Iguchi and T. Yoshikawa, Int. J. Mod. Phys. B9, 3033 (1995).
34. K. Iguchi, J. Phys. Soc. Jpn. 64, 177 (1995); (Errata) ibid. 64, 3124 (1995): Phys. Rev. B51, 12268 (1995); K. Iguchi and T. Yoshikawa, Int. J. Mod. Phys. B9, 3033 (1995).
35. K. Iguchi, J. Phys. Soc. Jpn. 64, 177 (1995); (Errata) ibid. 64, 3124 (1995): Phys. Rev. B51, 12268 (1995); K. Iguchi and T. Yoshikawa, Int. J. Mod. Phys. B9, 3033 (1995).
36. B. Y. Tong, Phys. Rev. 175, 710 (1968).
37. D. Schechtman, I. Blech, D. Gratias and J. W. Cahn, Phys. Rev. Lett. 53, 1951 (1984); R. Merlin, K. Bajema, R. Clarke, F. -T. Juang and P. K. Bhattacharya, Phys. Rev. Lett. 55, 1768 (1985).
38. D. Schechtman, I. Blech, D. Gratias and J. W. Cahn, Phys. Rev. Lett. 53, 1951 (1984); R. Merlin, K. Bajema, R. Clarke, F. -T. Juang and P. K. Bhattacharya, Phys. Rev. Lett. 55, 1768 (1985).
39. J.-M. Luck and D. Petritis, J. Stat. Phys. 42, 289 (1986); B. Sutherland, Phys. Rev. Lett. 57, 770 (1986); M. Kohmoto, B. Sutherland and K. Iguchi, ibid. 58, 2436 (1987); W. Gellermann, M. Kohmoto, B. Sutherland and P. C. Taylor, ibid. 72, 633 (1994); B. Sutherland and M. Kohmoto, Phys. Rev. B36, 5877 (1987); P. K. Ghosh, Phys. Lett. A161, 153 (1991); M. Baake, D. Joseph and P. Kramer, ibid. A168, 199 (1992).
40. J.-M. Luck and D. Petritis, J. Stat. Phys. 42, 289 (1986); B. Sutherland, Phys. Rev. Lett. 57, 770 (1986); M. Kohmoto, B. Sutherland and K. Iguchi, ibid. 58, 2436 (1987); W. Gellermann, M. Kohmoto, B. Sutherland and P. C. Taylor, ibid. 72, 633 (1994); B. Sutherland and M. Kohmoto, Phys. Rev. B36, 5877 (1987); P. K. Ghosh, Phys. Lett. A161, 153 (1991); M. Baake, D. Joseph and P. Kramer, ibid. A168, 199 (1992).
41. J.-M. Luck and D. Petritis, J. Stat. Phys. 42, 289 (1986); B. Sutherland, Phys. Rev. Lett. 57, 770 (1986); M. Kohmoto, B. Sutherland and K. Iguchi, ibid. 58, 2436 (1987); W. Gellermann, M. Kohmoto, B. Sutherland and P. C. Taylor, ibid. 72, 633 (1994); B. Sutherland and M. Kohmoto, Phys. Rev. B36, 5877 (1987); P. K. Ghosh, Phys. Lett. A161, 153 (1991); M. Baake, D. Joseph and P. Kramer, ibid. A168, 199 (1992).
42. J.-M. Luck and D. Petritis, J. Stat. Phys. 42, 289 (1986); B. Sutherland, Phys. Rev. Lett. 57, 770 (1986); M. Kohmoto, B. Sutherland and K. Iguchi, ibid. 58, 2436 (1987); W. Gellermann, M. Kohmoto, B. Sutherland and P. C. Taylor, ibid. 72, 633 (1994); B. Sutherland and M. Kohmoto, Phys. Rev. B36, 5877 (1987); P. K. Ghosh, Phys. Lett. A161, 153 (1991); M. Baake, D. Joseph and P. Kramer, ibid. A168, 199 (1992).
43. J.-M. Luck and D. Petritis, J. Stat. Phys. 42, 289 (1986); B. Sutherland, Phys. Rev. Lett. 57, 770 (1986); M. Kohmoto, B. Sutherland and K. Iguchi, ibid. 58, 2436 (1987); W. Gellermann, M. Kohmoto, B. Sutherland and P. C. Taylor, ibid. 72, 633 (1994); B. Sutherland and M. Kohmoto, Phys. Rev. B36, 5877 (1987); P. K. Ghosh, Phys. Lett. A161, 153 (1991); M. Baake, D. Joseph and P. Kramer, ibid. A168, 199 (1992).
44. J.-M. Luck and D. Petritis, J. Stat. Phys. 42, 289 (1986); B. Sutherland, Phys. Rev. Lett. 57, 770 (1986); M. Kohmoto, B. Sutherland and K. Iguchi, ibid. 58, 2436 (1987); W. Gellermann, M. Kohmoto, B. Sutherland and P. C. Taylor, ibid. 72, 633 (1994); B. Sutherland and M. Kohmoto, Phys. Rev. B36, 5877 (1987); P. K. Ghosh, Phys. Lett. A161, 153 (1991); M. Baake, D. Joseph and P. Kramer, ibid. A168, 199 (1992).
45. J.-M. Luck and D. Petritis, J. Stat. Phys. 42, 289 (1986); B. Sutherland, Phys. Rev. Lett. 57, 770 (1986); M. Kohmoto, B. Sutherland and K. Iguchi, ibid. 58, 2436 (1987); W. Gellermann, M. Kohmoto, B. Sutherland and P. C. Taylor, ibid. 72, 633 (1994); B. Sutherland and M. Kohmoto, Phys. Rev. B36, 5877 (1987); P. K. Ghosh, Phys. Lett. A161, 153 (1991); M. Baake, D. Joseph and P. Kramer, ibid. A168, 199 (1992).
46. M. Kolár and M. K. Ali, Phys. Rev. B39, 426 (1989); ibid. B39, 6538 (1989).
47. M. Kolár and M. K. Ali, Phys. Rev. B39, 426 (1989); ibid. B39, 6538 (1989).
48. M. Inoue, T. Takemori and H. Miyazaki, Solid State Commun. 77, 751 (1991); J. Phys. Soc. Jpn. 60, 3460 (1990); ibid. 61, 969 (1992).
49. M. Inoue, T. Takemori and H. Miyazaki, Solid State Commun. 77, 751 (1991); J. Phys. Soc. Jpn. 60, 3460 (1990); ibid. 61, 969 (1992).
50. M. Inoue, T. Takemori and H. Miyazaki, Solid State Commun. 77, 751 (1991); J. Phys. Soc. Jpn. 60, 3460 (1990); ibid. 61, 969 (1992).
51. F. Axel, J. P. Allouche, M. Kleman and M. Mendes-France, J. Phys. (Paris) Colloq. 47, C3-181 (1986): Y. Avishai and D. Berend, Phys. Rev. B45, 2717 (1992).
52. F. Axel, J. P. Allouche, M. Kleman and M. Mendes-France, J. Phys. (Paris) Colloq. 47, C3-181 (1986): Y. Avishai and D. Berend, Phys. Rev. B45, 2717 (1992).
53. J.-M. Luck, Phys. Rev. B39, 5834 (1989); M. Dulea, M. Johansson and R. Riklund, ibid. B45, 105 (1992); ibid. B46, 3296 (1992).
54. J.-M. Luck, Phys. Rev. B39, 5834 (1989); M. Dulea, M. Johansson and R. Riklund, ibid. B45, 105 (1992); ibid. B46, 3296 (1992).
55. J.-M. Luck, Phys. Rev. B39, 5834 (1989); M. Dulea, M. Johansson and R. Riklund, ibid. B45, 105 (1992); ibid. B46, 3296 (1992).
56. S. Aubry, C. Godrèche and J.-M. Luck, Europhys. Lett. 4, 639 (1987); J. Stat. Phys. 51, 1033 (1988).
57. S. Aubry, C. Godrèche and J.-M. Luck, Europhys. Lett. 4, 639 (1987); J. Stat. Phys. 51, 1033 (1988).
58. A. Bovier and J.-M. Ghez, Commun. Math. Phys. 158, 45 (1993).
59. M. K. Ali and G. Gumbs, Phys. Rev. B38, 7091 (1988); K. Iguchi and H. Su, J Phys. Soc. Jpn. 63, 184 (1994).
60. M. K. Ali and G. Gumbs, Phys. Rev. B38, 7091 (1988); K. Iguchi and H. Su, J Phys. Soc. Jpn. 63, 184 (1994).
61. K. Iguchi, J. Phys. Soc. Jpn. 62, 1802 (1993); Mod. Phys. Lett. B7, 39 (1993); HyperSpace 3, 36 (1994).
62. K. Iguchi, J. Phys. Soc. Jpn. 62, 1802 (1993); Mod. Phys. Lett. B7, 39 (1993); HyperSpace 3, 36 (1994).
63. K. Iguchi, J. Phys. Soc. Jpn. 62, 1802 (1993); Mod. Phys. Lett. B7, 39 (1993); HyperSpace 3, 36 (1994).
64. K. Iguchi, Phys. Lett. A190, 243 (1994); J. Math. Phys. 35, 1008 (1994); T. Yoshikawa and K. Iguchi, Int. J. Mod. Phys. B10, 2081 (1996).
65. K. Iguchi, Phys. Lett. A190, 243 (1994); J. Math. Phys. 35, 1008 (1994); T. Yoshikawa and K. Iguchi, Int. J. Mod. Phys. B10, 2081 (1996).
66. K. Iguchi, Phys. Lett. A190, 243 (1994); J. Math. Phys. 35, 1008 (1994); T. Yoshikawa and K. Iguchi, Int. J. Mod. Phys. B10, 2081 (1996).
67. M. Kolář and F. Nori, Phys. Rev. B42, 1062 (1990); Y. Avishai and D. Berend, J. Phys. A26, 2437 (1993); Y. Avishai, D. Berend and D. Glaubman, Phys. Rev. Lett. 72, 1842 (1994); K. Iguchi, J. Math. Phys. 34, 3481 (1993); Mod. Phys. Lett. B8, 29 (1994); Phys. Lett. A205, 25 (1995); Phys. Lett. A216, 37 (1996); J. Bellissard, in Number Theory and Physics, eds. J.-M. Luck, P. Moussa and M. Waldschmidt, (Springer-Verlag, Berlin, 1990); in From Number Theory to Physics, eds. M. Waldschmidt, P. Moussa, J.-M. Luck and C. Itzykson, (Springer-Verlag, Berlin, 1992).
68. M. Kolář and F. Nori, Phys. Rev. B42, 1062 (1990); Y. Avishai and D. Berend, J. Phys. A26, 2437 (1993); Y. Avishai, D. Berend and D. Glaubman, Phys. Rev. Lett. 72, 1842 (1994); K. Iguchi, J. Math. Phys. 34, 3481 (1993); Mod. Phys. Lett. B8, 29 (1994); Phys. Lett. A205, 25 (1995); Phys. Lett. A216, 37 (1996); J. Bellissard, in Number Theory and Physics, eds. J.-M. Luck, P. Moussa and M. Waldschmidt, (Springer-Verlag, Berlin, 1990); in From Number Theory to Physics, eds. M. Waldschmidt, P. Moussa, J.-M. Luck and C. Itzykson, (Springer-Verlag, Berlin, 1992).
69. M. Kolář and F. Nori, Phys. Rev. B42, 1062 (1990); Y. Avishai and D. Berend, J. Phys. A26, 2437 (1993); Y. Avishai, D. Berend and D. Glaubman, Phys. Rev. Lett. 72, 1842 (1994); K. Iguchi, J. Math. Phys. 34, 3481 (1993); Mod. Phys. Lett. B8, 29 (1994); Phys. Lett. A205, 25 (1995); Phys. Lett. A216, 37 (1996); J. Bellissard, in Number Theory and Physics, eds. J.-M. Luck, P. Moussa and M. Waldschmidt, (Springer-Verlag, Berlin, 1990); in From Number Theory to Physics, eds. M. Waldschmidt, P. Moussa, J.-M. Luck and C. Itzykson, (Springer-Verlag, Berlin, 1992).
70. M. Kolář and F. Nori, Phys. Rev. B42, 1062 (1990); Y. Avishai and D. Berend, J. Phys. A26, 2437 (1993); Y. Avishai, D. Berend and D. Glaubman, Phys. Rev. Lett. 72, 1842 (1994); K. Iguchi, J. Math. Phys. 34, 3481 (1993); Mod. Phys. Lett. B8, 29 (1994); Phys. Lett. A205, 25 (1995); Phys. Lett. A216, 37 (1996); J. Bellissard, in Number Theory and Physics, eds. J.-M. Luck, P. Moussa and M. Waldschmidt, (Springer-Verlag, Berlin, 1990); in From Number Theory to Physics, eds. M. Waldschmidt, P. Moussa, J.-M. Luck and C. Itzykson, (Springer-Verlag, Berlin, 1992).
71. M. Kolář and F. Nori, Phys. Rev. B42, 1062 (1990); Y. Avishai and D. Berend, J. Phys. A26, 2437 (1993); Y. Avishai, D. Berend and D. Glaubman, Phys. Rev. Lett. 72, 1842 (1994); K. Iguchi, J. Math. Phys. 34, 3481 (1993); Mod. Phys. Lett. B8, 29 (1994); Phys. Lett. A205, 25 (1995); Phys. Lett. A216, 37 (1996); J. Bellissard, in Number Theory and Physics, eds. J.-M. Luck, P. Moussa and M. Waldschmidt, (Springer-Verlag, Berlin, 1990); in From Number Theory to Physics, eds. M. Waldschmidt, P. Moussa, J.-M. Luck and C. Itzykson, (Springer-Verlag, Berlin, 1992).
72. M. Kolář and F. Nori, Phys. Rev. B42, 1062 (1990); Y. Avishai and D. Berend, J. Phys. A26, 2437 (1993); Y. Avishai, D. Berend and D. Glaubman, Phys. Rev. Lett. 72, 1842 (1994); K. Iguchi, J. Math. Phys. 34, 3481 (1993); Mod. Phys. Lett. B8, 29 (1994); Phys. Lett. A205, 25 (1995); Phys. Lett. A216, 37 (1996); J. Bellissard, in Number Theory and Physics, eds. J.-M. Luck, P. Moussa and M. Waldschmidt, (Springer-Verlag, Berlin, 1990); in From Number Theory to Physics, eds. M. Waldschmidt, P. Moussa, J.-M. Luck and C. Itzykson, (Springer-Verlag, Berlin, 1992).
73. M. Kolář and F. Nori, Phys. Rev. B42, 1062 (1990); Y. Avishai and D. Berend, J. Phys. A26, 2437 (1993); Y. Avishai, D. Berend and D. Glaubman, Phys. Rev. Lett. 72, 1842 (1994); K. Iguchi, J. Math. Phys. 34, 3481 (1993); Mod. Phys. Lett. B8, 29 (1994); Phys. Lett. A205, 25 (1995); Phys. Lett. A216, 37 (1996); J. Bellissard, in Number Theory and Physics, eds. J.-M. Luck, P. Moussa and M. Waldschmidt, (Springer-Verlag, Berlin, 1990); in From Number Theory to Physics, eds. M. Waldschmidt, P. Moussa, J.-M. Luck and C. Itzykson, (Springer-Verlag, Berlin, 1992).
74. M. Kolář and F. Nori, Phys. Rev. B42, 1062 (1990); Y. Avishai and D. Berend, J. Phys. A26, 2437 (1993); Y. Avishai, D. Berend and D. Glaubman, Phys. Rev. Lett. 72, 1842 (1994); K. Iguchi, J. Math. Phys. 34, 3481 (1993); Mod. Phys. Lett. B8, 29 (1994); Phys. Lett. A205, 25 (1995); Phys. Lett. A216, 37 (1996); J. Bellissard, in Number Theory and Physics, eds. J.-M. Luck, P. Moussa and M. Waldschmidt, (Springer-Verlag, Berlin, 1990); in From Number Theory to Physics, eds. M. Waldschmidt, P. Moussa, J.-M. Luck and C. Itzykson, (Springer-Verlag, Berlin, 1992).
75. M. Kolář and F. Nori, Phys. Rev. B42, 1062 (1990); Y. Avishai and D. Berend, J. Phys. A26, 2437 (1993); Y. Avishai, D. Berend and D. Glaubman, Phys. Rev. Lett. 72, 1842 (1994); K. Iguchi, J. Math. Phys. 34, 3481 (1993); Mod. Phys. Lett. B8, 29 (1994); Phys. Lett. A205, 25 (1995); Phys. Lett. A216, 37 (1996); J. Bellissard, in Number Theory and Physics, eds. J.-M. Luck, P. Moussa and M. Waldschmidt, (Springer-Verlag, Berlin, 1990); in From Number Theory to Physics, eds. M. Waldschmidt, P. Moussa, J.-M. Luck and C. Itzykson, (Springer-Verlag, Berlin, 1992).
76. G. B. Arfken and H. J. Weber, Mathematical Methods for Physicists, 4th Ed., (Academic Press, New York, 1995).
77. M. Toda, Theory of Nonlinear Lattices, 2nd Enlarged Ed., (Springer-Verlag, New York, 1989), p. 104.
78. R. Fricke and F. Klein, Vorlesungen über Theorie der Automorphen Functionen, (Teubner, Leipzig, 1897): W. Magnus, Noneuclidean Tessellations and Their Groups, (Academic Press, New York, 1974); W. Magnus. A. Karrass and D. Solitar, Combinatorial Group Theory, (Dover, New York, 1976), p. 120.
79. R. Fricke and F. Klein, Vorlesungen über Theorie der Automorphen Functionen, (Teubner, Leipzig, 1897): W. Magnus, Noneuclidean Tessellations and Their Groups, (Academic Press, New York, 1974); W. Magnus. A. Karrass and D. Solitar, Combinatorial Group Theory, (Dover, New York, 1976), p. 120.
80. R. Fricke and F. Klein, Vorlesungen über Theorie der Automorphen Functionen, (Teubner, Leipzig, 1897): W. Magnus, Noneuclidean Tessellations and Their Groups, (Academic Press, New York, 1974); W. Magnus. A. Karrass and D. Solitar, Combinatorial Group Theory, (Dover, New York, 1976), p. 120.
81. E. Yagi and K. Niizeki, Talk at the Fourth Workshop on Quasicrystals, Tsukuba, Dec., 1995. Unpublished.
82. S. Iyanaga and Y. Kawada, Encyclopedic Dictionary of Mathematics, (The MIT Press, Cambridge, 1997).
83. W. Kohn, Phys. Rev. 133, A171 (1964); "Metals and Insulators", in Many-Body Physics, eds. C. Dewitt and R. Balian, (Gordon and Breach, New York, 1968), p. 353. References therein; Also see, C. N. Yang, Rev. Mod. Phys. 34, 694 (1962).
84. W. Kohn, Phys. Rev. 133, A171 (1964); "Metals and Insulators", in Many-Body Physics, eds. C. Dewitt and R. Balian, (Gordon and Breach, New York, 1968), p. 353. References therein; Also see, C. N. Yang, Rev. Mod. Phys. 34, 694 (1962).
85. W. Kohn, Phys. Rev. 133, A171 (1964); "Metals and Insulators", in Many-Body Physics, eds. C. Dewitt and R. Balian, (Gordon and Breach, New York, 1968), p. 353. References therein; Also see, C. N. Yang, Rev. Mod. Phys. 34, 694 (1962).
86. D. J. Thouless, Phys. Rep. 13, 93 (1974); "Percolation and Localization", in III Condensed Matter, Les Houches, Session XXXI, (North-Holland, Amsterdam, 1978), p. 5. References therein.
87. D. J. Thouless, Phys. Rep. 13, 93 (1974); "Percolation and Localization", in III Condensed Matter, Les Houches, Session XXXI, (North-Holland, Amsterdam, 1978), p. 5. References therein.