Singular continuous spectrum on a cantor set of zero Lebesgue ...

Journal of Statistical Physics, Vol. 56, Nos. 3/4, 1989

Singular Continuous Spectrum on a Cantor Set of Zero Lebesgue Measure for the Fibonacci Hamiltonian Andrfis Slit61'2 Received March 2, 1989 It is rigorously proven that the spectrum of the tight-binding Fibonacci Hamiltonian, Hm. = 6 . . . . 1 + 6 . . . . t + 6m.#([(n + 1)c~] - [n~]) where c~= ( x / 5 - 1 ) / 2 and [ . ] means integer part, is a Cantor set of zero Lebesgue measure for all real nonzero #, and the spectral measures are purely singular continuous. This follows from a recent result by Kotani, coupled with the vanishing of the Lyapunov exponent in the spectrum. KEY W O R D S : Schr6dinger equation; Cantor spectrum; singular continuity; Lyapunov exponent.

In the present paper we settle a long-standing problem about the spectrum of a one-dimensional discrete Schr6dinger operator, called the Fibonacci Hamiltonian. The operator is defined on 12(2~) by (H~p)(n) = t~(n + 1) + ~(n - 1) + #v(n) ~b(n)

(1)

where # # 0 is a real number, v(n) = [(n + 1)c~] - [n~]

(2)

with c~= (x/5 - 1)/2 and [- ] meaning integer part. The spectral problem of H was addressed for the first time independently by Kohmoto et alJ 1) and by Ostlund et al. ~2) Nonrigorous but physically appealing arguments, supported by numerical work, led these authors to suggest that the spectrum of H is a Cantor set of zero Lebesgue measure. In a subsequent work,

1 Universit6 de Lausanne, Institut de Physique Thdorique, CH-1015 Lausanne, Switzerland. z O n leave from the Central Research Institute for Physics, Budapest, Hungary. 525 0022-4715/89/0800-0525506.00/0 9 1989 Plenum PuNishing Corporation

526

Silt6

Kohmoto and Oono (3) found further arguments in favor of a Cantor spectrum, and they also conjectured that the spectral measures should be purely singular continuous for any nonvanishing potential strength Ix. Since then, these conjectures about the Cantor spectrum and singular continuity have been widely accepted and propagated in the physical literature as facts, although no rigorous proof has been given. Apparently the first rigorous result was obtained by Casdagli. (4) The main tool of refs. 1-3 was to pass from the Schr6dinger equation H 0 = E0 to the study of a dynamical map (r,_ 1, Zn, T, + 1) ~ (rn, % + 1, rn + 2) in R3. Here r, is the trace of the transfer matrix M , defined by qs(F, + 1)'] (qs(1)~ ~s(r,) ] = M " \ 0 ( 0 ) )

(3)

where Fn is the n th Fibonacci number. Casdagli proved that B = {Ee ~l {r.(E)} is a bounded sequence}

(4)

is a Cantor set of zero Lebesgue measure if IIx]/> 16. The threshold was improved to IIx]~>4 by Silt6 (5) (without obtaining that the Lebesgue measure ]B] = 0), who also showed that B coincides with the spectrum of H and that the spectrum is purely continuous for all #. As a combination of the findings in refs. 4 and 5, one then gets: The spectrum of H is Cantor set of zero Lebesgue measure and it is purely singular continuous if I/x[ >~ 16. A new development was reached in a recent work by Kotani, (6) who studies operators of the type (1) (Jacobi matrices) with potentials taking values from a finite set. Let 0 ~0, then lAB>0, and by Kotani-Simon, H(O) has a nonvanishing absolutely continuous spectrum for a.e. 0. This contradicts the theorem of Kotani, so that Ispec(H)] = 0. i

Fibonacci Hamiltonian

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ACKNOWLEDGMENT

This research was partially supported by the Fonds National Suisse, grant 2.483-0.87. NOTE A D D E D IN PROOF

The results of ref. 5 and of the present paper have recently been extended to potentials (2) with any irrational e.(15) REFERENCES 1. M. Kohmoto, L. P. Kadanoff, and C. Tang, Localization problem in one dimension: Mapping and escape, Phys. Rev. Lett. 50:1870-1872 (1983). 2. S. Ostlund, R. Pandit, D. Rand, H. J. Schnellnhuber, and E. D. Siggia, One-dimensional Schr6dinger equation with an almost periodic potential, Phys. Rev. Lett. 50:1873-1876 (1983). 3. M. Kohmoto and Y. Oono, Cantor spectrum for an almost periodic Schr6dinger equation and a dynamical map, Phys. Lett. 102A:145-148 (1984). 4. M. Casdagli, Symbolic dynamics for the renormalization map of a quasiperiodic Schr6dinger equation, Commun. Math. Phys. 107:295-318 (1986). 5. A. Sfit6, The spectrum of a quasiperiodic Schr6dinger operator, Commun. Math. Phys. 111:409-415 (1987). 6. S. Kotani, Jacobi matrices with random potentials taking finitely many values, Preprint, University of Tokyo (1988). 7. J. Avron and B. Simon, Almost periodic Schr6dinger operators II. The integrated density of states, Duke Math. J. 50:369-391 (1983). 8. A. S. Besicovitch, Almost Periodic Functions (Cambridge University Press, Cambridge, 1932). 9. H. Furstenberg and H. Kesten, Products of random matrices, Ann. Math. Stat. 31:457-469 (1960). 10. D. Herbert and R. Jones, Localized states in disordered systems, J. Phys. C 4:1145-1161 (1971). 11. D. J. Thouless, A relation between the density of states and the range of localization in one dimensional random systems, J. Phys. C 5:77-81 (1972). 12. W. Craig and B. Simon, Subharmonicity of the Lyapunov index, Duke Math. J. 50:551-560 (1983). 13. S. Kotani, Lyapunov indices determine absolutely continuous spectra of stationary random one-dimensional Schr6dinger operators, in Stochastic Analysis, K. Ito, ed. (North-Holland, Amsterdam, 1984), pp. 225-247. 14. El. Simon, Kotani theory of one-dimensional stochastic Jacobi matrices, Commun. Math. Phys. 89:227-234 (1983). 15. J. Bellissard, B. Jochum, E. Scoppola, and D. Testard, Spectral properties of one dimensional quasi-crystals, preprint, March 1989. Communicated by Ch.-E. Pfister