Uniform spectral properties of one-dimensional quasicrystals, IV. Quasi-sturmian potentials

Journal d'Analyse Mathematique

Volumn 90, Issue , 2003, Pages 115-139

  Damanik, David ^a   Lenz, Daniel ^b,c  

^a UNIVERSITY OF CALIFORNIA   (United States)

^b HEBREW UNIVERSITY OF JERUSALEM   (Israel)

^c GOETHE UNIVERSITY   (Germany)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0242267001     PISSN: 00217670     EISSN: 15658538     Source Type: Journal    
DOI: 10.1007/BF02786553     Document Type: Article

References (43)

1. J.-P. Allouche, M. Baake, J. Cassaigne, and D. Damanik, Palindrome complexity, Theoret. Comput. Sci., to appear.
2. J.-P. Allouche and D. Damanik, Applications of combinatorics on words to physics, in preparation.
3. J. Bellissard, A. Bovier and J.-M. Ghez, Spectral properties of a tight binding Hamiltonian with period doubling potential. Commun. Math. Phys. 135 (1991), 379-399.
4. J. Bellissard, B. Iochum, E. Scoppola and D. Testard, Spectral properties of one-dimensional quasi-crystals. Commun. Math. Phys. 125 (1989), 527-543.
5. J. Berstel, Recent results in Sturmian words, in Developments in Language Theory (J. Dassow and A. Salomaa, eds.), World Scientific, Singapore, 1996, pp. 13-24.
6. A. Bovier and J.-M. Ghez, Spectral properties of one-dimensional Schrödinger operators with potentials generated by substitutions, Commun. Math. Phys. 158 (1993), 45-66; Erratum Commun. Math. Phys. 166 (1994), 431-432.
7. R. Carmona, A. Klein and F. Martinelli, Anderson localization for Bernoulli and other singular potentials. Commun. Math. Phys. 108 (1987), 41-66.
8. R. Carmona and J. Lacroix, Spectral Theory of Random Schrödinger Operators, Birkhäuser, Boston, 1990.
9. J. Cassaigne, Sequences with grouped factors, in Developments in Language Theory III, Aristotle University of Thessaloniki, 1998, pp. 211-222.
10. E. M. Coven, Sequences with minimal block growth, II, Math. Systems Theory 8 (1975), 376-382.
11. E. M. Coven and G. A. Hedlund, Sequences with minimal block growth, Math. Systems Theory 7 (1973), 138-153.
12. H. L. Cycon, R. Froese, W. Kirsch and B. Simon, Schrödinger Operators, with Application to Quantum Mechanics and Global Geometry, Springer, Berlin-Heidelberg-New York, 1987.
13. D. Damanik, α-continuity properties of one-dimensional quasicrystals, Commun. Math. Phys. 192 (1998), 169-182.
14. D. Damanik, Singular continuous spectrum for the period doubling Hamiltonian on a set of full measure, Commun. Math. Phys. 196 (1998), 477-483.
15. D. Damanik, Singular continuous spectrum for a class of substitution Hamiltonians, Lett. Math. Phys. 46 (1998), 303-311.
16. D. Damanik, Gordon-type arguments in the spectral theory of one-dimensional quasicrystals, in Directions in Mathematical Quasicrystals (M. Baake and R. V. Moody, eds.), CRM Monograph Series 13, AMS, Providence, RI, 2000, pp. 277-305.
17. D. Damanik, Substitution Hamiltonians with bounded trace map orbits, J. Math. Anal. Appl. 249 (2000), 393-411.
18. D. Damanik, R. Killip and D. Lenz, Uniform spectral properties of one-dimensional quasicrystals, III. α-continuity, Commun. Math. Phys. 212 (2000), 191-204.
19. D. Damanik and D. Lenz, Uniform spectral properties of one-dimensional quasicrystals, I. Absence of eigenvalues, Commun. Math. Phys. 207 (1999), 687-696.
20. D. Damanik and D. Lenz, Uniform spectral properties of one-dimensional quasicrystals, II. The Lyapunov exponent, Lett. Math. Phys. 50 (1999), 245-257
21. D. Damanik and D. Lenz, Half-line eigenfunction estimates and purely singular continuous spectrum of Lebesgue measure zero, Forum Math., to appear.
22. D. Damanik and B. Solomyak, Some high-complexity Hamiltonians with purely singular continuous spectrum, Ann. Inst. H. Poincaré 3 (2002), 99-105.
23. F. Delyon and D. Petritis, Absence of localization in a class of Schrödinger operators with quasiperiodic potential, Commun. Math. Phys. 103 (1986), 441-444.
24. A. Gordon, On the point spectrum of the one-dimensional Schrödinger operator, Uspekhi Mat. Nauk 31 (1976), 257-258.
25. G. A. Hedlund and M. Morse, Symbolic dynamics, Amer. J. Math. 60 (1938), 815-866.
26. A. Hof, O. Knill and B. Simon, Singular continuous spectrum for palindromic Schrödinger operators, Commun. Math. Phys. 174 (1995), 149-159.
27. B. Iochum, L. Raymond and D. Testard, Resistance of one-dimensional quasicrystals, Physica A 187 (1992), 353-368.
28. S. Jitomirskaya and Y. Last, Power-law subordinacy and singular spectra. I. Half-line operators, Acta Math. 183 (1999), 171-189.
29. M. Kaminaga, Absence of point spectrum for a class of discrete Schrödinger operators with quasiperiodic potential, Forum Math. 8 (1996), 63-69.
30. A. Ya. Khinchin, Continued Fractions, Dover Publications, Mineola, 1997.
31. S. Kotani, Jacobi matrices with random potentials taking finitely many values, Rev. Math. Phys. 1 (1989), 129-133.
32. Y. Last, Quantum dynamics and decompositions of singular continuous spectra, J. Funct. Anal. 142 (1996), 406-445.
33. Y. Last and B. Simon, Eigenfunctions, transfer matrices, and absolutely continuous spectrum of one-dimensional Schrödinger operators, Invent. Math. 135 (1999), 329-367.
34. D. Lenz, Aperiodische Ordnung und gleichmässige spektrale Eigenschaften von Quasikristallen, Dissertation, Frankfurt am Main, Logos, 2000.
35. D. Lenz, Hierarchical structures in Sturmian dynamical systems, Theoret. Comput. Sci., to appear.
36. W. Magnus, A. Karrass and D. Solitar, Combinatorial Group Theory, Dover, New York, 1976.
37. M. E. Paul, Minimal symbolic flows having minimal block growth. Math. Systems Theory 8 (1975), 309-315.
38. M. Queffélec, Substitution Dynamical Systems - Spectral Analysis, Lecture Notes in Math. 1284, Springer, Berlin, Heidelberg, New York, 1987.
39. M. Reed and B. Simon, Methods of Modem Mathematical Physics, Vol. I: Functional Analysis, Academic Press, New York, 1972.
40. J. A. G. Roberts, Escaping orbits in trace maps, Physica A 228 (1996), 295-325.
41. D. Shechtman, I. Blech, D. Gratias and J. V. Cahn, Metallic phase with long-range orientational order and no translational symmetry, Phys. Rev. Lett. 53 (1984), 1951-1953.
42. A. Sütö, The spectrum of a quasiperiodic Schrödinger operator, Commun. Math. Phys. 111 (1987), 409-415.
43. A. Sütö, Singular continuous spectrum on a Cantor set of zero Lebesgue measure for the Fibonacci Hamiltonian, J. Stat. Phys. 56 (1989), 525-531.