We show that discrete one-dimensional Schr\"odinger operators on the
half-line with ergodic potentials generated by the doubling map on the circle,
$V_\theta(n) = f(2^n \theta)$, may be realized as the half-line restrictions of
a non-deterministic family of whole-line operators. As a consequence, the
Lyapunov exponent is almost everywhere positive and the absolutely continuous
spectrum is almost surely empty.Comment: 4 page

Damanik, David

Killip, Rowan

English

arXiv

We show that discrete one-dimensional Schrödinger operators on the half-line with ergodic potentials generated by the doubling map on the circle, V_θ(n)=f(2^nθ), may be realized as the half-line restrictions of a non-deterministic family of whole-line operators. As a consequence, the Lyapunov exponent is almost everywhere positive and the absolutely continuous spectrum is almost surely empty

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ALMOST EVERYWHERE POSITIVITY OF THE LYAPUNOV
EXPONENT FOR THE DOUBLING MAP
DAVID DAMANIK AND ROWAN KILLIP
Abstract. We show that discrete one-dimensional Schro¨dinger operators on
the half-line with ergodic potentials generated by the doubling map on the
circle, Vθ(n) = f(2
nθ), may be realized as the half-line restrictions of a non-
deterministic family of whole-line operators. As a consequence, the Lyapunov
exponent is almost everywhere positive and the absolutely continuous spec-
trum is almost surely empty.
Consider discrete Schro¨dinger operators
(1) [Hθφ](n) = φ(n+ 1) + φ(n− 1) + λf(2
nθ)φ(n)
on ℓ2(Z+), Z+ = {1, 2, . . .}, with a Dirichlet boundary condition, φ(0) = 0. Here,
λ > 0 denotes the coupling constant, θ ∈ T = R/Z, and f : T → R is measurable,
bounded, and non-constant.
Since the doubling map, T : T → T, θ 7→ 2θ mod 1, is ergodic with respect to
Lebesgue measure on T, the Lyapunov exponent exists for every energy E, that is,
there is a function γ : R→ [0,∞) such that
γ(E) = lim
n→∞
1
n
log ‖M(n,E, θ)‖ for almost every θ,
where M(n,E, θ) is the transfer matrix from 1 to n for the operator Hθ at energy
E ∈ R, that is,
M(n,E, θ) =
(
E − λf(2nθ) −1
1 0
)
× · · · ×
(
E − λf(2θ) −1
1 0
)
.
It is expected that γ(E) > 0 for every E and that Hθ has pure point spectrum
with exponentially decaying eigenfunctions for almost every θ. The reason for this is
that the potentials are strongly mixing and similar to random potentials (generated
by i.i.d. random variables) for which these results were established in the 1980’s.
There is a huge literature on this subject; we refer the reader to [4, 11] for detailed
references.
However, very few results are known for the operators Hθ and these are limited
to the case of small coupling: Chulaevsky and Spencer, [5], proved an asymptotic
formula for γ(E), in the limit λ → 0. This implied positivity of γ(E) for all
0 < |E| < 2 and λ small, which in turn was used by Bourgain and Schlag, [3],
to prove that for λ < λ0 and almost every θ, Hθ has pure point spectrum in
0 < δ ≤ |E| ≤ 2 − δ with exponentially decaying eigenfunctions. This is usually
referred to as Anderson localization.
Date: December 31, 2014.
D. D. was supported in part by NSF grant DMS–0227289.
1
2 D. DAMANIK, R. KILLIP
The results just described require a regularity assumption on f (e.g., Ho¨lder
continuity). Beyond small coupling, no spectral results have been established. In
particular, it was not known, for any λ > 0, whether there can be any absolutely
continuous spectrum (for a.e. θ).1 In this note, we show that there is none.
Theorem 1. Suppose that λ > 0 and f is measurable, bounded, and non-constant.
Then, the Lyapunov exponent γ(E) is positive for almost every E ∈ R and the
absolutely continuous spectrum of Hθ is empty for almost every θ ∈ T.
Remark. Note that if the binary expansion of θ is periodic, the absolutely continuous
spectrum of Hθ is not empty so that the second statement cannot be improved.
As we mentioned above, it is expected that γ(E) is strictly positive for every
energy E. Once this can be shown, the next natural step will then be to prove
Anderson localization, using, for example, methods from [3]. Since such a result is
currently out of reach, we note that, by general principles, the almost everywhere
positivity of γ(E) yields the following consequence in terms of a localization result.
Given α ∈ [0, π), let Hθ,α denote the operator which acts on ℓ2(Z+) as in (1),
but with φ(0) given by cos(α)φ(0) + sin(α)φ(1) = 0. Thus, the original operator
(with a Dirichlet boundary condition) corresponds to α = 0. By [11, Theorem 13.4],
Theorem 1 implies the following:
Corollary 1. Suppose that λ > 0 and f is measurable, bounded, and non-constant.
Then, for almost every α ∈ [0, π) and almost every θ ∈ T, the operator Hθ,α has
pure point spectrum, and all eigenfunctions decay exponentially at infinity.
It is interesting to note that there are two possible viewpoints and both have
been used successfully. The work of Chulaevsky and Spencer showed that methods
from the random case (specifically an approach developed by Figotin and Pastur
[11]) could be extended to establish the asymptotic formula and positivity for the
Lyapunov exponent. On the other hand, Anderson localization was then proven
by Bourgain and Schlag by adapting methods originally developed for models with
very little randomness, namely, with underlying dynamics given by the shift [1, 6, 7]
and the skew-shift on the torus [2].
The main ingredient in the proof of Theorem 1 will be a result of Kotani, [9]
(see [12] for an adaptation to the discrete case), which shows that for whole-line
operators with non-deterministic potentials, the Lyapunov exponent is almost ev-
erywhere positive and, consequently, the absolutely continuous spectrum is almost
surely empty. A potential is non-deterministic if it is not determined uniquely by
its restriction to a half-line. Thus, the natural strategy will be to define a non-
deterministic family of whole-line operators whose restrictions to the right half-line
yield the family {Hθ}. Note that the Lyapunov exponents for the two families
are the same and that the half-line operators cannot have absolutely continuous
spectrum if the whole-line operators do not have any.
Proof of Theorem 1. The first step is to conjugate the doubling map T to a symbolic
shift via the binary expansion. Let Ω+ = {0, 1}Z+ and define D : Ω+ → T by
1It should be possible to exclude absolutely continuous spectrum using Kotani’s support the-
orem [10]; see [5, 13] for remarks indicating this possibility and [8] for related applications of the
support theorem. However, this would not give absence of absolutely continuous spectrum in full
generality, that is, this approach would only work for some f ’s and some values of λ.
POSITIVE LYAPUNOV EXPONENTS FOR THE DOUBLING MAP 3
D(ω) =
∑∞
n=1 ωn2
−n. The shift transformation, S : Ω+ → Ω+, is given by (Sω)n =
ωn+1. Clearly, D ◦ S = T ◦D.
Next we introduce a family of whole-line operators as follows. Let Ω = {0, 1}Z
and define, for ω ∈ Ω, the operator
[Hωφ](n) = φ(n+ 1) + φ(n − 1) + Vω(n)φ(n)
in ℓ2(Z), where
Vω(n) = f [D({ωn, ωn+1, ωn+2, . . .})].
The family {Hω}ω∈Ω is non-deterministic since Vω restricted to Z+ only depends
on {ωn}n≥1 and hence, by non-constancy of f , we cannot determine the values of
Vω(n) for n ≤ 0 uniquely from the knowledge of Vω(n) for n ≥ 1. It follows from
[9, 12] that the Lyapunov exponent for {Hω}ω∈Ω is almost everywhere positive
and σac(Hω) is empty for almost every ω ∈ Ω with respect to the (
1
2
, 1
2
)-Bernoulli
measure on Ω.
Finally, let us consider the restrictions of Hω to ℓ
2(Z+), that is, let H
+
ω =
E∗HωE, where E : ℓ
2(Z+)→ ℓ2(Z) is the natural embedding. Observe that H+ω =
Hθ, where θ = D({ω1, ω2, ω2, . . .}). This immediately implies the statement on
the positivity of the Lyapunov exponent for the family {Hθ}θ∈T. As finite-rank
perturbations preserve absolutely continuous spectrum, σac(H
+
ω ) ⊆ σac(Hω) for
every ω ∈ Ω. This proves that σac(H+ω ) = ∅ for almost every ω ∈ Ω. 
Remark. The proof relies on only two properties of the doubling map: it is not
one-to-one and it can be extended to a dynamical system over Z. Consequently,
Theorem 1 extends to other models with these properties. For example, θ 7→ mθ
mod 1 for any integer m ≥ 2.
Acknowledgments. We thank Svetlana Jitomirskaya and Barry Simon for useful
conversations.
References
[1] J. Bourgain and M. Goldstein, On nonperturbative localization with quasi-periodic potential,
Ann. of Math. 152 (2000), 835–879
[2] J. Bourgain, M. Goldstein, and W. Schlag, Anderson localization for Schro¨dinger operators
on Z with potentials given by the skew-shift, Commun. Math. Phys. 220 (2001), 583–621
[3] J. Bourgain and W. Schlag, Anderson localization for Schro¨dinger operators on Z with
strongly mixing potentials, Commun. Math. Phys. 215 (2000), 143–175
[4] R. Carmona and J. Lacroix, Spectral Theory of Random Schro¨dinger Operators, Birkha¨user,
Boston (1990)
[5] V. Chulaevsky and T. Spencer, Positive Lyapunov exponents for a class of deterministic
potentials, Commun. Math. Phys. 168 (1995), 455–466
[6] M. Goldstein and W. Schlag, Ho¨lder continuity of the integrated density of states for quasi-
periodic Schro¨dinger equations and averages of shifts of subharmonic functions, Ann. of Math.
154 (2001), 155–203
[7] S. Jitomirskaya, Metal-insulator transition for the almost Mathieu operator, Ann. of Math.
150 (1999), 1159–1175
[8] W. Kirsch, S. Kotani, and B. Simon, Absence of absolutely continuous spectrum for some
one-dimensional random but deterministic Schro¨dinger operators, Ann. Inst. H. Poincare´
Phys. The´or. 42 (1985), 383–406
[9] S. Kotani, Ljapunov indices determine absolutely continuous spectra of stationary random
one-dimensional Schro¨dinger operators, Stochastic analysis (Katata/Kyoto, 1982), pp. 225–
247, North-Holland Math. Library, 32, North-Holland, Amsterdam (1984)
[10] S. Kotani, Support theorems for random Schro¨dinger operators, Comm. Math. Phys. 97
(1985), 443–452
4 D. DAMANIK, R. KILLIP
[11] L. Pastur and A. Figotin, Spectra of Random and Almost-Periodic Operators, Grundlehren
der Mathematischen Wissenschaften 297, Springer-Verlag, Berlin (1992)
[12] B. Simon, Kotani theory for one-dimensional stochastic Jacobi matrices, Commun. Math.
Phys. 89 (1983), 227–234
[13] T. Spencer, Ergodic Schro¨dinger operators, in Analysis, et cetera, pp. 623–637, Academic
Press, Boston (1990)
Mathematics 253–37, California Institute of Technology, Pasadena, CA 91125, USA
E-mail address: damanik@its.caltech.edu
Department of Mathematics, University of California, Los Angeles, CA 90055, USA
E-mail address: killip@math.ucla.edu


Almost Everywhere Positivity of the Lyapunov Exponent for the Doubling Map

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