Comptes Rendus Mathématique

Antoine Detaille

Petru Mironescu

Quantitative suboptimal Sobolev embeddings

Alain Haraux

The universal bound property

Giorgos Kapetanakis

Lucas Reis

Normal points on Artin–Schreier curves over finite fields

Qi Li

Jianliang Zhai

Tusheng Zhang

Wong–Zakai approximations and support theorems for SDEs under Lyapunov conditions

Lê Vǎn Thành

Nguyen Ngoc Tu

Non-uniform Berry–Esseen bounds for exchangeable pairs with applications to the mean-field classical $N$-vector models and Jack measures

Stefaan Vaes

Factoriality of twisted locally compact group von Neumann algebras

Dmitrii Zakharov

Spherical sets avoiding orthonormal bases

Roberta Bianchini

Vincent Laheurte

Frank Sueur

A new look at the controllability cost of linear evolution systems with a long gaze at localized data

Walid Refai

Comptes Rendus
Mathématique
Walid Refai
On the point spectrum of quasi-periodic Schrödinger operators
Volume 363 (2025), p. 425-428
Online since: 20 May 2025
https://doi.org/10.5802/crmath.730
This article is licensed under the
Creative Commons Attribution 4.0 International License.
http://creativecommons.org/licenses/by/4.0/
C EN T R E
MER S ENN E
The Comptes Rendus. Mathématique are a member of the
Mersenne Center for open scientific publishing
www.centre-mersenne.org — e-ISSN : 1778-3569
Comptes Rendus. Mathématique
2025, Vol. 363, p. 425-428
https://doi.org/10.5802/crmath.730
Research article / Article de recherche
Dynamical systems / Systèmes dynamiques
On the point spectrum of quasi-periodic
Schrödinger operators
Sur le spectre ponctuel des opérateurs de Schrödinger
quasi-périodiques
Walid Refai a
a Department of mathematics, College of Addair, Jazan university, Jizan, Saudi Arabia
E-mail: wrefai@jazanu.edu.sa
Abstract. In this paper we study the spectral properties of a family of discrete one-dimensional quasi-
periodic Schrödinger operators with discontinuous potential. In large disorder with piecewise analytic
potential, under some suitable transversality condition and a Diophantine rotation number, we prove using
basically KAM theory that the spectrum of this operator have a large punctual component. Moreover, the
most of its eigenvectors are ℓ2-localized in one site. We give an explicit estimation of the measure of the
punctual spectrum and we prove that the set of phases for which this spectrum is empty have zero Hausdorff
dimension.
Résumé. Dans cet article, nous étudions les propriétés spectrales d’une famille d’opérateurs de Schrödinger
quasi-périodiques discrets unidimensionnels avec potentiel discontinu. Pour un désordre important avec
un potentiel analytique par morceaux, sous des conditions de transversalité appropriées et un nombre de
rotation diophantien, nous démontrons essentiellement à l’aide de la théorie KAM que le spectre de cet
opérateur possède une grande composante ponctuelle. De plus, la majorité de ses fonctions propres sont
ℓ2-localisées en un seul site. Nous fournissons une estimation explicite de la mesure du spectre ponctuel et
prouvons que l’ensemble des phases pour lesquelles ce spectre est vide a une dimension de Hausdorff nulle.
Keywords. Quasi-periodic Schrödinger operators, pure point spectrum, eigenfunctions.
Mots-clés. Opérateurs de Schrödinger quasi-périodiques, spectre purement ponctuel, fonctions propres.
2020 Mathematics Subject Classification. 35P30, 37K55, 37K60.
Manuscript received 21 November 2024, accepted 5 February 2025.
1. Introduction
We consider the Schrödinger operator Hθ defined in ℓ
2(Z) by
(Hθu)n :=−ε(un+1 +un−1)+ v(θ+nω)un , n ∈Z, (1)
where ε and ω are real numbers and θ in the torus T = R/2πZ. The function v : T→ R is called
potential and un+1 +un−1 = (∆u)n is the discrete Laplacian.
ISSN (electronic): 1778-3569 https://comptes-rendus.academie-sciences.fr/mathematique/
426 Walid Refai
In order to properly set up hypotheses, the following objects are defined:
Definition 1.
(1) A finite partition P is defined as a finite subdivision of intervals of T.
(2) ∂P := {∂∆; ∆ ∈P }.
(3) P is said to be an ω-partition if and only if ∂P ∈ωZ.
(4) [P ]ω is the smallest integer M such that:
∂P ⊆ω(Z∩ [−M , M ]).
Assume that:
• ω is a Diophantine number satisfying
∥nω∥ := inf
m∈Z
∣∣nω−2πm∣∣Ê κ|n|τ ∀ n ∈Z\ {0}, κ> 0, τ> 1; (2)
• the potential v is smooth on the pieces of P , where P is an ω-partition, and satisfies for
θ ∈T
max
0ÉℓÉk
sup
θ∈T
∣∣∣∣ 1ℓ!∂ℓθv(θ)
∣∣∣∣Éβγk , ∀ k Ê 0. (3)
Furthermore, assume that v satisfies the two following transversality conditions, there exists a
non zero integer p such that:
max
0ÉkÉp
∣∣∣∂kx (v(θ+x)− v(θ))∣∣∣Êσ> 0, ∀ θ, ∀ x, (4)
max
0ÉkÉp
∣∣∣∂kθ(v(θ+x)− v(θ))∣∣∣Êσ∥x∥, ∀ θ, ∀ x. (5)
The main result of this paper is the following:
Theorem 2. There exists a constant ε0 > 0, which only depends on β, γ, p, σ, M, κ and τ such that
if |ε| É ε0 then the spectrum of Hθ has a large punctual component.
More precisely, there exists a set Zε ⊆ T, such that Leb(T \ Zε) goes to 0 as ε tends to 0, and a
vector V : Zε→ ℓ2(Z) such that:
(1) HθV (θ) = E(θ)V (θ), ∀ θ ∈ Zε;
(2) ∥V −e0∥ℓ2(Z) ÉO (ε);
(3) Leb
{
E(θ); θ ∈ Zε
}
goes to 4 as ε tends to 0;
(4) The Hausdorff dimension of T\ Ẑ ε, where Ẑ ε =⋃n∈Z(Zε−nω), is zero.
To our knowledge, the only discontinuous model studied in details is with a piecewise constant
potential (a paradigmatic example of this fact is the Fibonacci model), which admits a singular
spectrum. According to the result of Damanik and Killip in 2005 with a discontinuous quasi-
periodic potential, where they showed the absence of the continuous spectrum, this work allows
us to have another type of spectrum, notably the localization of most of the eigenvectors at one
site of ℓ2(Z).
From the previous theorem, the following corollary proceeds:
Corollary 3.
(1) E(Ẑ ε) ⊂σp (Hθ), ∀ θ ∈ Ẑ ε;
(2)
{
θ ∈T; σp (Hθ) =;
}
has zero Hausdorff dimension;
where σp (Hθ) is the punctual spectrum of Hθ , it is defined as being the closure of the set of
eigenvalues of Hθ.
Proof. This is an immediate consequence of Theorem 2. Indeed, we define for all m ∈ Z the
application, τm : ℓ2(Z) → ℓ2(Z ), by (τmu)n = un−m . So, according to the first two assertions
of Theorem 2, there exists a family of eigenvectors V n(θ) = τnV (θ + nω) which are linearly
Walid Refai 427
independent, hence (1). The assertion (2) follows from Theorem 2(4), since
{
θ ∈T; σp (Hθ) =;
}
is included in T\ Ẑ ε. □
The presented results are on the spectral properties of the quasi-periodic Schrödinger oper-
ators with a discontinuous potential. The proof is inspired from [2,3]. It is a more global anal-
ysis than in [2], and follows the scheme presented in [3], all in considering an analytical norm
instead of Gevrey. This work also generalizes the previous one [6]. For around forty years, the
study of quasi-periodic Schrödinger operators has been the subject of extensive research, in the
strong coupling regime, the breakthrough was in the mid-1980’s, through the work of J. Fröhlich,
T. Spencer, P. Wittver [4] and Ya. G. Sinaı̆ [7]. The model with discontinuities was mentioned in [3],
but without proof. We give the proof in full detail, with an explicit estimation of the measurement
of the point spectrum. We also show that most eigenvectors are ℓ2-localized at one site. So a nat-
ural question arises, can we have exponential localization for this model? In [1], J. Bourgain and
M. Goldstein showed exponential localization, but for a model without discontinuities. Finally,
S. Jitomirskaya and B. Simon showed in [5] that
{
θ; σp (Hθ) =;
}
is a dense Gδ. We improved this
result by showing that this set has zero Hausdorff dimension.
2. On the proof
For this problem we identify the family of operators Hθ as a symmetric infinite-dimensional
matrix that depends on the parameter θ:
(D +F )(θ) =

. . . 0
v(θ−ω) −ε
−ε v(θ) −ε
−ε v(θ+ω)
0
. . .
 .
The goal is to conjugate this matrix to a diagonal matrix D∞(θ) by an orthogonal matrix made
up of a complete set of orthonormal eigenvectors. The way to construct such a matrix is by an
iterative procedure
(U j )
∗ · · · (U 1)∗(D +F )U 1 · · ·U j = D j+1 +F j+1 (6)
where ∥F j (θ)∥ = ε j goes to 0 as j tends to ∞ and D j (θ) is a block matrix. More precisely, D j (θ)
has an invariant eigensubspace S j (θ) ⊂ RΛ j (θ), where Λ j (θ) is an increasing sequence of subsets
of Z, such that diam
(
Λ j (θ)
)Éλ j and #(Λ j (θ))Éµ j . Note that this is valid for all θ. In this process
we follow the approach of [3], with important differences, notably the construction of another
decomposition into blocks, which permits to control the size of partition.
From this analysis we deduce the following estimates.
Proposition 4. Consider the set Z j =
{
θ; Λ j (θ) = {0}
}
, then for each j , Z j \ Z j+1 is a union of at
most K j+1 intervals Ii , where K1 is given, K j+1 É
( 1
ε j
)28(τ+1)d j a j , Leb(Ii ) É ν j É εa j /4µ jj , with
a j = 1
16τ(τ+1)µ9j+1
, d j = 1
200τµ2j
and ε j+1 É ε
µ99j+2
j .
Now let’s take Zε =∩ jÊ1Z j , it follows that T\ Ẑ ε has zero Haussdorff dimension. That is
∀ s > 0, lim
ν→0Hν,s (T\ Ẑ ε) = 0,
where Hν,s (T\ Ẑ ε) = inf
(∑∞
i=1|Ui |s
)
and the infimum is taken over all ν-coverings of T\ Ẑ ε.
Indeed, putting W j = Z j \ Z j+1 for all j Ê 1, we have W j = ∑K j+1i=1 Ii . It follows that T \ Ẑ ε ⊆{
θ; θ+nω ∈ ⋃∞j=1 W j , ∀ n ∈ Z}, this implies that T \ Ẑ ε ⊆ ⋃∞j=1 W j which is a ν1-cover of T \ Ẑ ε.
428 Walid Refai
For ε sufficiently small and ω Diophantine, we show that T \ Ẑ ε ⊆ ⋃∞j=2 ⋃|n|ÉK 22 (W j −nω) which
is a ν2-cover of T\ Ẑ ε. Then we show inductively that T\ Ẑ ε ⊆⋃∞j=N ⋃|n|ÉK 2N (W j −nω) which is a
νN -cover of T\ Ẑ ε. This gives for all s ∈ ]0,1[ that
Hν,s (T\ Ẑ ε) É
∞∑
j=N
3K 2j ν
s
j .
Thus, the rapid decrease of ε j to 0 allows us to conclude that the series of terms 3K 2j ν
s
j is
convergent because 3K 2j ν
s
j É ε
saN /8µN
N for all j Ê js . This proves that Hν,s (T \ Ẑ ε) É ε
saN /16µN
N
for all N Ê js , which implies the result.
Acknowledgments
The author wishes to thank H. Eliasson for many ideas which have been decisive in this paper.
Declaration of interests
The author does not work for, advise, own shares in, or receive funds from any organization
that could benefit from this article, and has declared no affiliations other than their research
organizations.
References
[1] J. Bourgain and M. Goldstein, “On nonperturbative localization with quasi-periodic poten-
tial”, Ann. Math. (2) 152 (2000), no. 3, pp. 835–879.
[2] L. H. Eliasson, “Discrete one-dimensional quasi-periodic Schrödinger operators with pure
point spectrum”, Acta Math. 179 (1997), no. 2, pp. 153–196.
[3] L. H. Eliasson, S. B. Kuksin, S. Marmi and J.-C. Yoccoz, Dynamical systems and small divisors,
Lecture Notes in Mathematics, Springer, 2002, pp. viii+191.
[4] J. Fröhlich, T. C. Spencer and P. Wittwer, “Localization for a class of one-dimensional quasi-
periodic Schrödinger operators”, Commun. Math. Phys. 132 (1990), no. 1, pp. 5–25.
[5] S. Y. Jitomirskaya and B. Simon, “Operators with singular continuous spectrum. III. Almost
periodic Schrödinger operators”, Commun. Math. Phys. 165 (1994), no. 1, pp. 201–205.
[6] W. Refai, “Localization of the discrete one-dimensional quasi-periodic Schrödinger opera-
tors”, Math. Methods Appl. Sci. 47 (2024), no. 12, pp. 10435–10443.
[7] Y. G. Sinaı̆, “Anderson localization for one-dimensional difference Schrödinger operator
with quasiperiodic potential”, J. Stat. Phys. 46 (1987), no. 5-6, pp. 861–909.


Not a member yet

Quantitative suboptimal Sobolev embeddings

The universal bound property

Normal points on Artin–Schreier curves over finite fields

Wong–Zakai approximations and support theorems for SDEs under Lyapunov conditions

Non-uniform Berry–Esseen bounds for exchangeable pairs with applications to the mean-field classical $N$ -vector models and Jack measures

Factoriality of twisted locally compact group von Neumann algebras

Spherical sets avoiding orthonormal bases

A new look at the controllability cost of linear evolution systems with a long gaze at localized data

On the point spectrum of quasi-periodic Schrödinger operators

Non-uniform Berry–Esseen bounds via Malliavin–Stein method

5,369

6,067