The spectrum of a quasiperiodic Schrödinger operator. (English) Zbl 0624.34017

The spectrum \(\sigma\) (H) of the tight binding Fibonacci Hamiltonian \((H_{mn}=\delta_{m,n+1}+\delta_{m+1,n}+\delta_{m,n}\mu v(n)\), \(v(n)=\chi_{[-\omega^ 3,\omega^ 2[}((n-1)\omega)\), 1/\(\omega\) is the golden number) is shown to coincide with the dynamical spectrum, the set on which an infinite subsequence of traces of transfer matrices is bounded. The point spectrum is absent for any \(\mu\), and \(\sigma\) (H) is a Cantor set for \(| \mu | \geq 4\). Combining this with Casdagli’s earlier result, one finds that the spectrum is singular continuous for \(| \mu | \geq 16\).


34L99 Ordinary differential operators
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[1] Kohmoto, M., Kadanoff, L. P., Tang, Ch.: Localization problem in one dimension: Mapping and escape. Phys. Rev. Lett.50, 1870–1876 (1983)
[2] 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)
[3] de Bruijn, N. G.: Sequences of zeros and ones generated by special production rules. Indagationes Math.84, 27–37 (1981) · Zbl 0471.10007
[4] Rand, D., Ostlund, S., Sethna, J., Siggia, E. D.: Universal transition from quasiperiodicity to chaos in dissipative systems. Phys. Rev. Lett.49, 132–135 (1982) · Zbl 0538.58025
[5] Feigenbaum, M. J., Hasslacher, B.: Irrational decimations and path integrals for external noise. Phys. Rev. Lett.49, 605–609 (1982)
[6] Kohmoto, M., Oono, Y.: Cantor spectrum for an almost periodic Schrödinger equation and a dynamical map. Phys. Lett.102A, 145–148 (1984)
[7] Casdagli, M.: Symbolic dynamics for the renormalization map of a quasiperiodic Schrödinger equation. Commun. Math. Phys.107, 295–318 (1986) · Zbl 0606.39004
[8] Lang, S.: Introduction to diophantine approximations. Reading, MA: Addison-Wesley 1966 Chap. 1 · Zbl 0144.04005
[9] Simon, B.: Almost periodic Schrödinger operators: A review. Adv. Appl. Math.3, 463–490 (1982) · Zbl 0545.34023
[10] Delyon, F., Petritis, D.: Absence of localization in a class of Schrödinger operators with quasiperiodic potential. Commun. Math. Phys.103, 441–444 (1986) · Zbl 0604.35072
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