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All issues Volume 9 / No 5 (2014) Math. Model. Nat. Phenom., 9 5 (2014) 204-238 References
Free Access
Issue
Math. Model. Nat. Phenom.
Volume 9, Number 5, 2014
Spectral problems
Page(s) 204 - 238
DOI https://doi.org/10.1051/mmnp/20149514
Published online 17 July 2014
  1. S. Astels. Cantor sets and numbers with restricted partial quotients. Trans. Amer. Math. Soc., 352 (2000), 133–170. [CrossRef] [MathSciNet] [Google Scholar]
  2. A. Avila, S. Jitomirskaya. The Ten Martini Problem, Annal. Math., 170 (2009), 303–342. [CrossRef] [MathSciNet] [Google Scholar]
  3. J. Avron, V. Mouche, P. H. M., B. Simon. On the measure of the spectrum for the almost Mathieu operator. Commun. Math. Phys., 132 (1990), 103–118. [CrossRef] [Google Scholar]
  4. S. Beckus, F. Pogorzelski. Spectrum of lebesgue measure zero for jacobi matrices of quasicrystals. Mathematical Physics, Analysis and Geometry, 16 (2013), 289–308. [CrossRef] [MathSciNet] [Google Scholar]
  5. J. Bellissard. Spectral properties of Schrödinger’s operator with a Thue-Morse potential. Number Theory and Physics (Les Houches, 1989), 140–150, Springer Proc. Phys., 47, Springer, Berlin 1990. [Google Scholar]
  6. J. Bellissard, A. Bovier, J.-M. Ghez. Spectral properties of a tight binding Hamiltonian with period doubling potential. Commun. Math. Phys. 135 (1991), 379–399. [CrossRef] [Google Scholar]
  7. J. Bellissard, A. Bovier, J.-M. Ghez. Gap labelling theorems for one-dimensional discrete Schrödinger operators. Rev. Math. Phys., 4 (1992), 1–37. [CrossRef] [MathSciNet] [Google Scholar]
  8. J. Bellissard, B. Iochum, E. Scoppola, D. Testard. Spectral properties of one dimensional quasi-crystals. Commun. Math. Phys., 125 (1989), 527–543. [CrossRef] [Google Scholar]
  9. T. C. Brown. A characterization of the quadratic irrationals. Canad. Math. Bull., 34 (1991), no. 1, 36–41. [CrossRef] [MathSciNet] [Google Scholar]
  10. S. Cantat. Bers and Hénon, Painlevé and Schrödinger. Duke Math. J., 149 (2009), 411–460. [CrossRef] [MathSciNet] [Google Scholar]
  11. R. Carmona, J. Lacroix. Spectral theory of random Schrödinger operators. Boston: Birkhäuser, 1990. [Google Scholar]
  12. M. Casdagli. Symbolic dynamics for the renormalization map of a quasiperiodic Schrödinger equation. Commun. Math. Phys., 107 (1986), 295–318. [CrossRef] [Google Scholar]
  13. M.-D. Choi, G. A. Elliottt, N. Yui. Gauss polynomials and the rotation algebra. Invent. Math., 99 (1990), 225–246. [CrossRef] [MathSciNet] [Google Scholar]
  14. D. Crisp, W. Moran, A. Pollington, P. Shiue. Substitution invariant cutting sequences. J. Théor. Nombres Bordeaux, 5 (1993), no. 1, 123–137. [Google Scholar]
  15. H. L. Cycon, R. G. Froese, W. Kirsch, B. Simon. Schrödinger operators. Books and monographs in physics. Berlin, Heidelberg, New York: Springer, 1987. [Google Scholar]
  16. J. M. Dahl. The spectrum of the off-diagonal Fibonacci operator. Ph.D. thesis, Rice University, 2010-2011. [Google Scholar]
  17. D. Damanik. Substitution Hamiltonians with Bounded Trace Map Orbits. J. Math. Anal. App., 249 (2000), 393–411. [CrossRef] [Google Scholar]
  18. D. Damanik. Uniform singular continuous spectrum for the period doubling Hamiltonian. Annal. Henri Poincaré, 20 (2001), 101–108. [CrossRef] [Google Scholar]
  19. D. Damanik. Strictly ergodic subshifts and associated operators. Spectral theory and mathematical physics: a Festschrift in honor of Barry Simon’s 60th birthday, Sympos. Pure Math., 76, Part 2, Amer. Math. Soc., Providence, RI, 2007. [Google Scholar]
  20. D. Damanik, M. Embree, A. Gorodetski. Spectral properties of the Schrödinger operators arising in the study of quasicrystals. (preprint) arXiv:1210.5753. [Google Scholar]
  21. D. Damanik, M. Embree, A. Gorodetski, S. Tcheremchantsev. The fractal dimension of the spectrum of the Fibonacci Hamiltonian. Commun. Math. Phys., 280 (2008), 499–516. [CrossRef] [Google Scholar]
  22. D. Damanik, A. Gorodetski. Hyperbolicity of the trace map for the weakly coupled Fibonacci Hamiltonian. Nonlinearity, 22 (2009), 123–143. [CrossRef] [Google Scholar]
  23. D. Damanik, A. Gorodetski. Spectral and quantum dynamical properties of the weakly coupled Fibonacci Hamiltonian. Commun. Math. Phys., 305 (2011), 221–277. [CrossRef] [Google Scholar]
  24. D. Damanik, A. Gorodetski. The density of states measure of the weakly coupled Fibonacci Hamiltonian. Geom. Funct. Anal., 22 (2012), 976–989. [CrossRef] [MathSciNet] [Google Scholar]
  25. D. Damanik, A. Gorodetski, B. Solomyak. Absolutely continuous convolutions of singular measures and an application to the square fibonacci hamiltonian. preprint (arXiv:1306.4284). [Google Scholar]
  26. D. Damanik, P. Munger, W. N. Yessen. Orthogonal polynomials on the unit circle with Fibonacci Verblunsky coefficients, I. The essential support of the measure. J. Approx. Theory, 173 (2013), 56–88. [CrossRef] [MathSciNet] [Google Scholar]
  27. D. Damanik, P. Munger, W. N. Yessen. Orthogonal polynomials on the unit circle with Fibonacci Verblunsky coefficients, II. Applications. J. Stat. Phys., 153 (2013), 339–362. [CrossRef] [Google Scholar]
  28. D. Damanik, D. Lenz. Uniform spectral properties of one-dimensional quasicrystals. IV. Quasi-Sturmian potentials. J. Anal. Math., 90 (2003), 115–139. [CrossRef] [MathSciNet] [Google Scholar]
  29. C. R. de Oliveira. Intermediate spectral theory and quantum dynamics. Progress in Mathematical Physics, vol. 54, Birkhäuser Verlag, Basel, 2009. [Google Scholar]
  30. B. Farb, D. Margalit. A primer on mapping class groups. Princeton University Press, Princeton, NJ., 2012. [Google Scholar]
  31. A. Fathi, F. Laudenbach, V. Poénaru. Travaux de Thurston sur les surfaces. Asterisque, 66, 67 (1979), (Translation by Kim, D. and Margalit, D., Thurston’s work on surfaces, Princeton University Press, 2012). [Google Scholar]
  32. N. P. Fogg. Substitutions in dynamics, arithmetics and combinatorics. Lecture Notes in Mathematics, vol. 1794, Springer-Verlag, Berlin, 2002, Edited by V. Berthé, S. Ferenczi, C. Mauduit and A. Siegel. [Google Scholar]
  33. W. H. Gottschalk. Substitution minimal sets. Trans. Amer. Math. Soc., 109 (1963), 467–491. [CrossRef] [MathSciNet] [Google Scholar]
  34. B. C. Hall. Quantum theory for mathematicians. Graduate Texts in Mathematics, vol. 267, Springer, New York, 2013. [Google Scholar]
  35. E. Hamza, R. Sims, G. Stolz. Dynamical localization in disordered quantum spin systems. Comm. Math. Phys., 315 (2012), 215–239. [CrossRef] [MathSciNet] [Google Scholar]
  36. P. G. Harper. Single bond motion of conducting electros in a uniform magnetic field. Proc. Phys. Soc. A, 68 (1955), 874–878. [Google Scholar]
  37. B. Hasselblatt. Handbook of Dynamical Systems: Hyperbolic Dynamical Systems. vol. 1A, Elsevier B. V., Amsterdam, The Netherlands, 2002. [Google Scholar]
  38. B. Hasselblatt, A. Katok. Handbook of Dynamical Systems: Principal Structures. vol. 1A, Elsevier B. V., Amsterdam, The Netherlands, 2002. [Google Scholar]
  39. B. Hasselblatt, Ya. Pesin. Partially hyperbolic dynamical systems. Handbook of dynamical systems, 1B (2006), 1–55, Elsevier B. V., Amsterdam (Reviewer: C. A. Morales). [Google Scholar]
  40. M. W. Hirsch, C. C. Pugh. Stable Manifolds and Hyperbolic Sets, Proc. Symp. Pure Math., 14 (1968), 133–163. [CrossRef] [Google Scholar]
  41. A. Hof. Some remarks on discrete aperiodic Schrödinger operators. J. Stat. Phys., 72 (1993), 1353–1374. [CrossRef] [Google Scholar]
  42. A. Katok, B. Hasselblatt. Introduction to the Modern Theory of Dynamical Systems. Cambridge University Press, New York, NY, 1995. [Google Scholar]
  43. M. Kohmoto, L. P. Kadanoff, C. Tang. Localization problem in one dimension: Mapping and escape. Phys. Rev. Lett., 50 (1983), 1870–1872. [CrossRef] [MathSciNet] [Google Scholar]
  44. S. Kotani. Jacobi matrices with random potentials taking finitely many values. Rev. Math. Phys., 1 (1989), 129–133. [CrossRef] [MathSciNet] [Google Scholar]
  45. D. Lenz. Singular spectrum of Lebesgue measure zero for one-dimensional quasicrystals. Comm. Math. Phys., 227 (2002), 119–130. [CrossRef] [MathSciNet] [Google Scholar]
  46. D. Lenz, P. Stollmann. An ergodic theorem for Delone dynamical systems and exitense of the integrated density of states. J. Anal. Math., 97 (2005), 1–24. [CrossRef] [MathSciNet] [Google Scholar]
  47. E. Lieb, T. Schultz, D. Mattis. Two soluble models of an antiferromagnetic chain. Ann. Phys., 16 (1961), 407–466. [CrossRef] [Google Scholar]
  48. Q.-H. Liu, J. Peyrière, Z.-Y. Wen. Dimension of the spectrum of one-dimensional discrete Schrödinger operators with Sturmian potentials. C. R. Math. Acad. Sci. Paris, 345 (2007), 667–672. [CrossRef] [MathSciNet] [Google Scholar]
  49. Q.-H. Liu, B. Tan, Z.-X. Wen, J. Wu. Measure zero spectrum of a class of Schrödinger operators. J. Statist. Phys., 106 (2002), 681–691. [CrossRef] [MathSciNet] [Google Scholar]
  50. J. M. Luttinger. The effect of a magnetic field on electros in a periodic potential. Phys. Rev., 94 (1951), 814–817. [CrossRef] [Google Scholar]
  51. R. Mañé. The Hausdorff dimension of horseshoes of diffeomorphisms of surfaces, Boletim da Sociedade Brasileira de Matemática, 20 (1990), 1–24. [CrossRef] [MathSciNet] [Google Scholar]
  52. M. Mei. Spectral properties of discrete Schrödinger operators with primitive invertible substitution potentials. preprint (arXiv:1311.0954) (2013). [Google Scholar]
  53. M. Morse, G. A. Hedlund. Symbolic dynamics II. Sturmian trajectories. Amer. J. Math., 62 (1940), 1–42. [Google Scholar]
  54. S. Newhouse. Nondensity of Axiom A on S2. Global Analysis (Proc. Sympos. Pure Math., Vol. XIV, Berkeley, Calif., 1968), 191–202, Amer. Math. Soc., Providence, RI, 1970. [Google Scholar]
  55. S. Ostlund, R. Pandit, D. Rand, H. J. Schellnhuber, E. D. Siggia. One-dimensional Schrödinger equation with an almost periodic potential. Phys. Rev. Lett., 50 (1983), 1873–1876. [CrossRef] [Google Scholar]
  56. J. C. Oxtoby. Ergodic sets. Bull. Amer. Math. Soc., 58 (1952), 116–136. [CrossRef] [MathSciNet] [Google Scholar]
  57. J. Palis, F. Takens. Hyperbolicity and Sensetive Chaotic Dynamics at Homoclinic Bifurcations. Cambridge University Press, Cambridge, 1993. [Google Scholar]
  58. L. Pastur, A. Figotin. Spectra of random and almost-periodic operators. Grundlehren der mathematischen Wissenschaften, Vol. 297, Springer, 1992. [Google Scholar]
  59. R. C. Penner. A construction of pseudo-Anosov homeomorphisms. Trans. Amer. Math. Soc., 310 (1988), 179–197. [CrossRef] [MathSciNet] [Google Scholar]
  60. Ya. Pesin. Dimension Theory in Dynamical Systems. Chicago Lect. Math. Series, 1997. [Google Scholar]
  61. Ya. Pesin. Lectures on Partial Hyperbolicity and Stable Ergodicity. Zurich Lect. Adv. Math., European Mathematical Society, 2004. [Google Scholar]
  62. M. Pollicott. Analyticity of dimensions for hyperbolic surface diffeomorphisms. preprint. [Google Scholar]
  63. C. Pugh, M. Shub, A. Wilkinson. Hölder foliations. Duke Math. J., 86 (1997), 517–546. [CrossRef] [MathSciNet] [Google Scholar]
  64. L. Raymond. A constructive gap labelling for the discrete Schrödinger operator on a quasiperiodic chain. preprint (1997). [Google Scholar]
  65. C. Remling. The absolutely continuous spectrum of Jacobi matrices. Ann. Math., 174 (2011), 125–171. [CrossRef] [Google Scholar]
  66. J. A. G. Roberts. Escaping orbits in trace maps. Physica A: Stat. Mech. App., 228 (1996), 295–325. [CrossRef] [Google Scholar]
  67. D. Schechtman, I. Blech, J. W. Gratias, D. Cahn. Meallic phase with long-range orientational order and no translational symmetry. Phys. Rev. Lett., 53 (1984), 1951–1953. [CrossRef] [Google Scholar]
  68. B. Simon. Equilibrium measures and capacities in spectral theory. Inverse problems and imaging, 1 (2007), 713–772. [CrossRef] [MathSciNet] [Google Scholar]
  69. S. Smale. Differentiable Dynamical Systems. Bull. Amer. Math. Soc., 73 (1967), 747–817. [Google Scholar]
  70. A. Sütő. The spectrum of a quasiperiodic Schrödinger operator. Commun. Math. Phys., 111 (1987), 409–415. [CrossRef] [Google Scholar]
  71. L. A. Takhtajan. Quantum mechanics for mathematicians, Graduate Studies in Mathematics, vol. 95, American Mathematical Society, Providence, RI, 2008. [Google Scholar]
  72. B. Tan, Z.-X. Wen, Y. Zhang. Invertible substitutions on a three-letter alphabet. C. R. Math. Acad. Sci. Paris, 336 (2003), 111–116. [CrossRef] [MathSciNet] [Google Scholar]
  73. G. Teschl. Jacobi operators and completely integrable nonlinear lattices. AMS mathematical surveys and monographs, vol. 72, American Mathematical Society, Providence, RI. [Google Scholar]
  74. G. Teschl. Mathematical methods in quantum mechanics. Graduate Studies in Mathematics, vol. 99, American Mathematical Society, Providence, RI, 2009, With applications to Schrödinger operators. [Google Scholar]
  75. W. P. Thurston. On the geometry and dynamics of diffeomorphisms of surfaces. Bull. Amer. Math. Soc., 19 (1988), 417–431. [CrossRef] [MathSciNet] [Google Scholar]
  76. M. Toda. Theory of Nonlinear Lattices. Solid-State Sciences 20, Berlin-Heidelberg-New York, Springer-Verlag, 1981. [Google Scholar]
  77. Z. Wen, Y. Zhang. Some remarks on invertible substitutions on three letter alphabet. Chinese Sci. Bull., 44 (1999). [Google Scholar]
  78. W. N. Yessen. Spectral analysis of tridiagonal Fibonacci Hamiltonians. J. Spectr. Theory, 3 (2013), 101–128. [CrossRef] [MathSciNet] [Google Scholar]
  79. W. N. Yessen. On the energy spectrum of 1d quantum ising quasicrystal. Annal. H. Poincaré, 15 (2014), 419–467. [CrossRef] [Google Scholar]
  80. W. N. Yessen. Properties of 1D Classical and Quantum Ising Models: Rigorous Results. Ann. Henri Poincaré, 15 (2014), 793–828. [CrossRef] [MathSciNet] [Google Scholar]

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