Integral Equations and Operator Theory

, Volume 82, Issue 4, pp 533–554 | Cite as

Spectral Approximation for Quasiperiodic Jacobi Operators

Article

Abstract

Quasiperiodic Jacobi operators arise as mathematical models of quasicrystals and in more general studies of structures exhibiting aperiodic order. The spectra of these self-adjoint operators can be quite exotic, such as Cantor sets, and their fine properties yield insight into the associated quantum dynamics, that is, the one-parameter unitary group that solves the time-dependent Schrödinger equation. Quasiperiodic operators can be approximated by periodic ones, the spectra of which can be computed via two finite dimensional eigenvalue problems. Since long periods are necessary for detailed approximations, both computational efficiency and numerical accuracy become a concern. We describe a simple method for numerically computing the spectrum of a period-K Jacobi operator in O(K2) operations, then use the algorithm to investigate the spectra of Schrödinger operators with Fibonacci, period doubling, and Thue–Morse potentials.

Mathematics Subject Classification

Primary 47B36 65F15 81Q10 Secondary 15A18 47A75 

Keywords

Jacobi operator Schrödinger operator quasicrystal Fibonacci period doubling Thue–Morse 

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References

  1. 1.
    IEEE Standard for Floating-Point Arithmetic (IEEE Standard 754–2008). Institute of Electrical and Electronics Engineers, Inc., (2008)Google Scholar
  2. 2.
    Anderson E., Bai Z., Bischof C., Blackford S., Demmel J., Dongarra J., Du Croz J., Greenbaum A., Hammarling S., McKenney A., Sorensen D.: LAPACK User’s Guide, third edn. SIAM, Philadelphia (1999)CrossRefMATHGoogle Scholar
  3. 3.
    Avila A., Jitomirskaya S.: The ten martini problem. Ann. Math. 170, 303–342 (2009)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Avila A., Krikorian R.: Reducibility or nonuniform hyperbolicity for quasiperiodic Schrödinger cocycles. Ann. Math. 164, 911–940 (2006)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Avishai Y., Berend D.: Trace maps for arbitrary substitution sequences. J. Phys. A 26, 2437–2443 (1993)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Avishai Y., Berend D., Glaubman D.: Minimum-dimension trace maps for substitution sequences. Phys. Rev. Lett. 72, 1842–1845 (1994)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Bellissard, J.: Spectral properties of Schrödinger’s operator with a Thue–Morse potential. In: Springer Proceedings in Physics, Number Theory and Physics, vol. 47, pp. 140–150. Springer, Berlin (1990)Google Scholar
  8. 8.
    Bellissard J., Iochum B., Scoppola E., Testard D.: Spectral properties of one-dimensional quasicrystals. Commun. Math. Phys. 125, 527–543 (1989)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Bellissard J., Bovier A., Ghez J.-M.: Spectral properties of a tight binding Hamiltonian with period doubling potential. Commun. Math. Phys. 135, 379–399 (1991)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Bischof C.H., Lang B., Sun X.: A framework for symmetric band reduction. ACM Trans. Math. Softw. 26, 581–601 (2000)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Damanik D.: Singular continuous spectrum for a class of substitution Hamiltonians II. Lett. Math. Phys. 54, 25–31 (2000)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Damanik D.: Uniform singular continuous spectrum for the period doubling Hamiltonian. Ann. Henri Poincaré 2, 101–118 (2001)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Damanik, D.: Strictly ergodic subshifts and associated operators. In: Proceedings of Symposium Pure Mathematical Spectral Theory and Mathematical Physics: a Festschrift in Honor of Barry Simon’s 60th Birthday, vol. 76, pp. 539–563. American Mathematical Society, Providence, RI (2007)Google Scholar
  14. 14.
    Damanik, D., Embree, M., Gorodetski, A.: Spectral properties of Schrödinger operators arising in the study of quasicrystals, (2012). arXiv:1210.5753 [math.SP]
  15. 15.
    Damanik D., Embree M., Gorodetski A., Tcheremchantsev S.: The fractal dimension of the spectrum of the Fibonacci Hamiltonian. Commun. Math. Phys. 280, 499–516 (2008)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Damanik, D., Gorodetski, A.: Almost sure frequency independence of the dimension of the spectrum of Sturmian Hamiltonians, (2014). arXiv:1406.4810 math.SP]
  17. 17.
    Damanik D., Lenz D.: Uniform spectral properties of one-dimensional quasicrystals, I. Absence of eigenvalues. Commun. Math. Phys. 207, 687–696 (1999)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Damanik, D., Fillman, J.: Spectral Theory of Discrete One-Dimensional Ergodic Schrödinger Operators. In preparationGoogle Scholar
  19. 19.
    Damanik, D., Gorodetski, A., Yessen, W.: The Fibonacci Hamiltonian, (2014). arxiv:1403.7823 [math.SP]
  20. 20.
    Delyon F., Peyrière J.: Recurrence of the eigenstates of a Schrödinger operator with automatic potential. J. Stat. Phys. 64, 363–368 (1991)CrossRefMATHGoogle Scholar
  21. 21.
    Even-Dar Mandel S., Lifshitz R.: Electronic energy spectra of square and cubic Fibonacci quasicrystals. Philos. Mag 88, 2261–2273 (2008)CrossRefGoogle Scholar
  22. 22.
    Falconer K.: Fractal Geometry: Mathematical Foundations and Applications, 2nd edn. Wiley, Chichester (2003)CrossRefGoogle Scholar
  23. 23.
    Gear C.W.: A simple set of test matrices for eigenvalue programs. Math. Comp. 23, 119–125 (1969)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Halsey T.C., Jensen M.H., Kadanoff L.P., Procaccia I., Shraiman B.I.: Fractal measures and their singularities: the characterization of strange sets. Phys. Rev. A 33, 1141–1151 (1986)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Harper P.G.: Single band motion of conduction electrons in a uniform magnetic field. Proc. Phys. Soc. A 68, 874–878 (1955)CrossRefMATHGoogle Scholar
  26. 26.
    Hof A., Knill O., Simon B.: Singular continuous spectrum for palindromic Schrödinger operators. Commun. Math. Phys 174, 149–159 (1995)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Kohmoto M., Kadanoff L.P., Tang C.: Localization problem in one dimension: mapping and escape. Phys. Rev. Lett. 50, 1870–1872 (1983)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Kotani S.: Jacobi matrices with random potentials taking finitely many values. Rev. Math. Phys. 1, 129–133 (1989)MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    Lamoureux M.P.: Reflections on the almost Mathieu operator. Integral Equ. Oper. Theory 28, 45–59 (1997)MathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    Last Y.: Zero measure spectrum for the almost Mathieu operator. Commun. Math. Phys. 164, 421–432 (1994)MathSciNetCrossRefMATHGoogle Scholar
  31. 31.
    Last Y.: Quantum dynamics and decompositions of singular continuous spectra. J. Funct. Anal. 142, 406–445 (1996)MathSciNetCrossRefMATHGoogle Scholar
  32. 32.
    Liu, Q., Qu, Y.: Iteration of polynomial pair under Thue–Morse dynamic (2014). arXiv:1403.2257 [math.DS]
  33. 33.
    Marin L.: On- and off-diagonal Sturmian operators: Dynamic and spectral dimension. Rev. Math. Phys. 24(05), 1250011 (2012)MathSciNetCrossRefMATHGoogle Scholar
  34. 34.
    Mendes P., Oliveira F.: On the topological structure of the arithmetic sum of two Cantor sets. Nonlinearity 7, 329–343 (1994)MathSciNetCrossRefMATHGoogle Scholar
  35. 35.
    Moreira C.G., Morales E.M., Rivera-Letelier J.: On the topology of arithmetic sums of regular Cantor sets. Nonlinearity 13, 2077–2087 (2000)MathSciNetCrossRefMATHGoogle Scholar
  36. 36.
    Ostlund S., Pandit R., Rand D., Schellnhuber H.J., Siggia E.D.: One-dimensional Schrödinger equation with an almost periodic potential. Phys. Rev. Lett. 50, 1873–1876 (1983)MathSciNetCrossRefGoogle Scholar
  37. 37.
    Palis J., Takens F.: Hyperbolicity and Sensitive Chaotic Dynamics at Homoclinic Bifurcations. Cambridge University Press, Cambridge (1993)MATHGoogle Scholar
  38. 38.
    Parlett, B.N.: The Symmetric Eigenvalue Problem, SIAM Classics edition. SIAM, Philadelphia (1998)Google Scholar
  39. 39.
    Queffélec M.: Substitution Dynamical Systems—Spectral Analysis. Springer, Berlin (1987)MATHGoogle Scholar
  40. 40.
    Reed M., Simon B.: Methods of Modern Mathematical Physics I: Functional Analysis, revised and enlarged edition. Academic Press, San Diego (1980)Google Scholar
  41. 41.
    Rudin W.: Some theorems on Fourier coefficients. Proc. Am. Math. Soc. 10, 855–859 (1959)MathSciNetCrossRefMATHGoogle Scholar
  42. 42.
    Rutishauser, H.: On Jacobi rotation patterns. In: Proceedings of Symposia in Applied Mathematics Experimental Arithmetic, High Speed Computing and Mathematics, vol. 15, pp. 219–239. American Mathematical Society, Providence, RI (1963)Google Scholar
  43. 43.
    Shapiro, H.S.: Extremal problems for polynomials and power series. Master’s thesis, Massachusetts Institute of Technology (1951)Google Scholar
  44. 44.
    Simon B.: Szegő’s Theorem and Its Descendants. Princeton University Press, Princeton (2011)Google Scholar
  45. 45.
    Tél T., Fülöp Á., Vicsek T.: Determination of fractal dimensions for geometrical multifractals. Phys. A 159, 155–166 (1989)CrossRefGoogle Scholar
  46. 46.
    Teschl G.: Jacobi Operators and Completely Integrable Nonlinear Lattices. American Mathematical Society, Providence (2000)MATHGoogle Scholar
  47. 47.
    Thouless D.J.: Bandwidths for a quasiperiodic tight-binding model. Phys. Rev. B 28, 4272–4276 (1983)MathSciNetCrossRefGoogle Scholar
  48. 48.
    Toda M.: Theory of Nonlinear Lattices, 2nd edn. Springer, Berlin (1989)CrossRefGoogle Scholar
  49. 49.
    van Moerbeke, P.: The spectrum of Jacobi matrices. Invent. Math. 37, 45–81 (1976)MathSciNetCrossRefMATHGoogle Scholar
  50. 50.
    Yessen W.N.: Spectral analysis of tridiagonal Fibonacci Hamiltonians. J. Spectr. Theory 3, 101–128 (2013)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer Basel 2015

Authors and Affiliations

  1. 1.Department of Computational and Applied MathematicsRice UniversityHoustonUSA
  2. 2.Department of MathematicsVirginia TechBlacksburgUSA
  3. 3.Department of MathematicsRice UniversityHoustonUSA

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