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APPEARED IN BULLETIN OF THE

arXiv:math/9307232v1 [math.SP] 1 Jul 1993APPEARED IN BULLETIN OF THEAMERICAN MATHEMATICAL SOCIETYVolume 29, Number 1, July 1993, Pages 85-87ABSENCE OF CANTOR SPECTRUM FOR A CLASSOF SCHR¨ODINGER OPERATORSNorbert RiedelAbstract. It is shown that the complete localization of eigenvectors for the almostMathieu operator entails the absence of Cantor spectrum for this operator.1Among almost periodic Schr¨odinger operators the almost Mathieu operator hasenjoyed the majority of attention in the last ten to fifteen years for the followingthree reasons: First, it is the simplest nontrivial operator of its kind, and there-fore it is deemed to be more accessible than others.

Second, subjecting the almostMathieu operator to the Fourier transformation leads to an operator of the samekind (duality property). Third, and probably most important, depending on theparameters involved, the almost Mathieu operator displays nearly all the conceiv-able features operators of this kind could have, such as point spectrum, absence ofpoint spectrum, absolute continuous spectrum, singular spectrum, etc.

Accordingto most researchers who have contributed to this field and who have stated theiropinion, however, the only exception from this pattern seems to concern the natureof the spectrum of the almost Mathieu operator, considered as a bounded selfad-joint operator. As manifested by repeatedly stated conjectures (see [9] for one ofthe earliest sources and [5] for one of the latest), the almost Mathieu operator isexpected to have a nowhere dense spectrum (Cantor spectrum) whenever the op-erator is not periodic.

We shall argue that the absence of Cantor spectrum for thealmost Mathieu operator not only does occur, but we shall also identify the reasonas to why it happens.2The operator in question is defined on the Hilbert space ℓ2(Z) as(H(α, β, θ)ξ)n = ξn+1 + ξn−1 + 2β cos(2παn + θ)ξn ,where α, β, θ are real constants. We will be concerned exclusively with the case thatα is an irrational number.

In this case the spectrum Sp(α, β) of H(α, β, θ) does notdepend on the parameter θ. For a full set of α’s (in the sense of Lebesgue measure)which are sufficiently badly approximable by rational numbers and for sufficientlylarge β, Fr¨ohlich et al.

in [3] and Sinai in [10] have shown independently, with1991 Mathematics Subject Classification. Primary 47B39, 47C15; Secondary 31A15.Received by the editors July 15, 1992 and, in revised form, December 9, 1992c⃝1993 American Mathematical Society0273-0979/93 $1.00 + $.25 per page1

2NORBERT RIEDELquite involved techniques, that H(α, β, θ) has, for almost all θ, a complete set ofeigenvectors that decay exponentially. What is actually shown in these papers isthat any generalized eigenvector of H(α, β, θ) which grows sufficiently slowly as|n| approaches infinity decays exponentially.

This fact, combined with some otherelementary observations, leads us to assume the validity of the following conditionfor the parameters α and β in question:(L)For every χ ∈Sp(α, β) there is a θ such that H(α, β, θ) has aneigenvector that decays exponentially as |n| →∞.This condition entails that the average Lyapunov index for the operator H(α, β, θ)takes the constant value log |β| on Sp(α, β). By virtue of the “showless formula”,which establishes a connection between the average Lyapunov index and the log-arithmic potential associated with the integrated density of states, it follows thatSp(α, β) is a regular compactum (i.e., the Dirichlet problem is solvable in (C ∪{∞})\Sp(α, β) for any continuous function on Sp(α, β)), and the integrated den-sity of states τ for H(α, β, θ) is the equilibrium distribution of Sp(α, β) (cf.

[11] forthe potential theoretic terminology being used). The level curves of the conductorpotential associated with Sp(α, β) can be identified with the spectra of perturbedalmost Mathieu operators.Theorem 1 [7].

Assume that (L) holds. A complex number z is contained in thespectrum of the operator(Hδ(α, β)ξ)n = ξn+1 + ξn−1 + β(δe2παni + δ−1e−2παni)ξn,ξ ∈ℓ2(Z) ,if and only ifRlog |z −s| dτ(s) = log |β| + | log |δ||.The gist in the setup of the proof of this theorem is to consider the operatorsHδ(α, β) as elements of the irrational rotation C∗-algebra Aα associated with thenumber α.The C∗-algebra Aα is generated by two unitary operators u and vsatisfying the commutation relation uv = e2παivu.The operator Hδ(α, β) canbe identified with the element u + u−1 + β(δv + δ−1v−1).

The resolvent of theoperator Hδ(α, β) on each connected component of the resolvent set can now beexpanded into absolutely convergent series in the monomials upvq. This in turnleads to the definition of subharmonic functions measuring the order of decay forthese expansions in every point of the resolvent set.

The proof of Theorem 1 is thenestablished by exhibiting interrelationships between these subharmonic functions.It should be noted that the conclusion of Theorem 1 remains true if the condition (L)is replaced by the assumption that α is sufficiently well approximable by rationalsand that |β| ≥1 [8].This seems to suggest that the statement in Theorem 1may be true for all irrational numbers α and for all |β| ≥1. By a basic dualityargument it is readily seen that a similar characterization of the level curves of theconductor potential associated with Sp(α, β) in terms of the spectra of perturbedalmost Mathieu operators holds for any β with 0 < |β| < 1, whenever the statementin Theorem 1 holds for β−1.

The condition (L), in conjunction with Theorem 1, isindispensible for the following assertion.Theorem 2 [7]. Assume that (L) holds.

Then Sp(α, β) is not a Cantor set.The issue of the nature of Sp(α, β) has been taken up in a number of papers.Choi et al. [2], inspired by earlier work by Bellissard and Simon [1], show that for

ABSENCE OF CANTOR SPECTRUM3numbers α which are sufficiently well approximable by rationals, Sp(α, β) is indeeda Cantor set and, moreover, they show that all possible gaps in Sp(α, β) do actuallyoccur. The most penetrating analysis to date of the nature of Sp(α, β) has beenconducted by Helffer and Sj¨ostrand in three memoirs [4].

Most of their investigationis limited to the case β = 1 for α’s which are sufficiently badly approximable byrational numbers. Even though this work produces a wealth of detailed information,it leaves the main question unanswered.

In [10] Sinai claims that Sp(α, β) is aCantor set in those cases where he establishes complete localization of eigenvectors.As we have seen, this claim does not conform with our Theorem 2.The proof of Theorem 2 heavily relies on some C∗-algebraic machinery that wasdeveloped in [6] in order to address spectral problems for the almost Mathieu oper-ator. In conjunction with Theorem 1, Theorem 2 provides for an argument whichtranslates the condition (L) into a smoothness condition for the logarithmic poten-tial involved.

In this context condition (L) appears as a device to control the growthof the (noncommutative) Fourier expansions of certain functionals associated withpoints in Sp(α, β). That forementioned smoothness of the logarithmic potential isshown to conflict with the assumption that Sp(α, β) is a Cantor set.References1.

J. Bellissard and B. Simon, Cantor spectrum for the almost Mathieu equation, J. Funct.Anal. 48 (1982), 408–519.2.

M.-D. Choi, G. A. Elliott, and N. Yui, Gauss polynomials and the rotation algebra, Invent.Math. 99 (1990), 225–246.3.

J. Fr¨ohlich, T. Spencer, and P. Wittwer, Localization for a class of one-dimensional quasi-periodic Schr¨odinger operators, Comm. Math.

Phys. 132 (1990), 5–25.4.

B. Helffer and J. Sj¨ostrand, Analyse semi-classique pour l’´equation de Harper. I-III, M´em.Soc.

Math. France (N.S.) (4) 116, (4) 117, (1) 118, (1988–1990).5.

L. Pastur and A. Figotin, Spectra of random and almost-periodic operators, Springer-Verlag, New York, 1992.6. N. Riedel, Almost Mathieu operators and rotation C∗-algebras, Proc.

London Math. Soc.

(3) 56 (1988), 281–302.7., The spectrum of a class of almost periodic operators, preprint.8., Regularity of the spectrum for the almost Mathieu operator, preprint.9. B. Simon, Almost periodic Schr¨odinger operators : a review, Adv.

in Appl. Math.

3 (1982),463–490.10. Ya.

G. Sinai, Anderson localization for one-dimensional difference Schr¨odinger operatorwith quasiperiodic potential, J. Stat. Phys.

46 (1987), 861–909.11. M. Tsuji, Potential theory in modern function theory, Chelsea, New York, 1959.Department of Mathematics, Tulane University, New Orleans, Louisiana 70118

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