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\TopicFR{SystÃ¨mes dynamiques}
\TopicEN{Dynamical systems}

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\title{On the point spectrum of quasi-periodic Schr\"odinger operators}
\alttitle{Sur le spectre ponctuel des opÃ©rateurs de SchrÃ¶dinger quasi-pÃ©riodiques}

\author{\firstname{Walid} \lastname{Refai}}

\address{Department of mathematics, College of Addair, Jazan university, Jizan, Saudi Arabia}

\email{wrefai@jazanu.edu.sa}

\begin{abstract}
In this paper we study the spectral properties of a family of discrete one-dimensional quasi-periodic Schr\"odinger 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 $\ell^2$-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.
\end{abstract}

\begin{altabstract}
Dans cet article, nous Ã©tudions les propriÃ©tÃ©s spectrales d'une famille d'opÃ©rateurs de SchrÃ¶dinger quasi-pÃ©riodiques discrets unidimensionnels avec potentiel discontinu. Pour un dÃ©sordre important avec un potentiel analytique par morceaux, sous des conditions de transversalitÃ© appropriÃ©es et un nombre de rotation diophantien, nous dÃ©montrons essentiellement Ã&nbsp; l'aide de la thÃ©orie KAM que le spectre de cet opÃ©rateur possÃ¨de une grande composante ponctuelle. De plus, la majoritÃ© de ses fonctions propres sont $\ell^2$-localisÃ©es 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.
\end{altabstract}

\keywords{\kwd{Quasi-periodic Schr\"odinger operators} \kwd{pure point spectrum} \kwd{eigenfunctions}}
\altkeywords{\kwd{OpÃ©rateurs de SchrÃ¶dinger quasi-pÃ©riodiques} \kwd{spectre purement ponctuel} \kwd{fonctions propres}}

\subjclass{35P30, 37K55, 37K60}

\begin{document}

\maketitle

\section{Introduction}

We consider the Schr{\"o}dinger operator $H_\theta$ defined in $\ell^2(\Z)$ by
\begin{equation}
(H_\theta u)_{n}\coloneqq -\epsilon(u_{n+1}+u_{n-1})+v(\theta + n\omega)u_n, \quad n\in{\mathbb Z},
\end{equation}
where $\epsilon$ and $\omega$ are real numbers and $\theta$ in the torus $\T=\R/2\pi\Z$.
The function $v \colon \T \to \R$ is called potential and $u_{n+1}+u_{n-1}=(\Delta u)_n$ is the discrete Laplacian.

\goodbreak

In order to properly set up hypotheses, the following objects are defined:
\begin{defi}\mbox{}
\begin{enumerate}
\item A finite partition $\P$ is defined as a finite subdivision of intervals of\/ $\T$.
\item $\partial\P\coloneqq\braces[\big]{\partial\Delta; \ \Delta\in\P}.$
\item $\P$ is said to be an $\omega$-partition if and only if\/ $\partial\P\in\omega\Z$.
\item $[\P]_\omega$ is the smallest integer $M$ such that:
\[
\partial\P\subseteq\omega\parens[\big]{\Z\cap[-M,M]}.
\]
\end{enumerate}
\end{defi}

Assume that:
\begin{itemize}
\item $\omega$ is a Diophantine number satisfying
\begin{equation}\label{diofdesc}
\norm{n\omega}\coloneqq \inf_{m\in\mathbb{Z}}\abs[\big]{n\omega - 2\pi m}\geq\frac{\kappa}{\abs{n}^{\tau}} \quad \forall n \in \mathbb{Z}\setminus\braces{0}, \ \kappa&gt;0, \ \tau&gt;1;
\end{equation}
\item the potential $v$ is smooth on the pieces of $\P$, where $\P$ is an $\omega$-partition, and satisfies for $\theta\in\T$
\begin{equation}
\max_{0\leq\ell\leq k}\sup_{\theta\in \mathbb{T}}\abs[\Bigg]{\frac{1}{\ell !}\partial_{\theta}^{\ell}v(\theta)}\leq \beta\gamma^{k}, \quad \forall k\geq 0.
\end{equation}
\end{itemize}

Furthermore, assume that $v$ satisfies the two following transversality conditions, there exists a non zero integer $p$ such that:
\begin{gather}
\max_{0\leq k\leq p}\abs[\bigg]{\partial_{x}^{k}\parens[\big]{v(\theta + x)-v(\theta)}}\geq \sigma&gt;0, \quad \forall \theta, \ \forall x, \label{trans1desc}\\
\max_{0\leq k\leq p}\abs[\bigg]{\partial_{\theta}^{k}\parens[\big]{v(\theta + x)-v(\theta)}}\geq \sigma\norm{x}, \quad \forall \theta, \ \forall x. \label{trans2desc}
\end{gather}

The main result of this paper is the following:
\begin{theo}\label{t1}
There exists a constant $\epsilon_0&gt;0$, which only depends on $\beta$, $\gamma$, $p$, $\sigma$, $M$, $\kappa$ and $\tau$ such that if\/ $\abs{\epsilon}\leq\epsilon_0$ then the spectrum of $H_\theta$ has a large punctual component.

More precisely, there exists a set $Z_\epsilon\subseteq\T$, such that $\Leb(\T\setminus Z_\epsilon)$ goes to $0$ as $\epsilon$ tends to $0$, and a vector $V \colon Z_\epsilon \to \ell^2(\Z)$ such that:
\begin{enumerate}
\item $H_\theta V(\theta)=E(\theta)V(\theta)$, $\forall \theta\in Z_\epsilon$;
\item $\norm{V-e^0}_{\ell^2(\Z)}\leq\mathcal O(\epsilon)$;
\item $\Leb\braces[\big]{E(\theta); \ \theta\in Z_\epsilon}$ goes to $4$ as $\epsilon$ tends to $0$;
\item \label{t1_4} The Hausdorff dimension of\/ $\T\setminus\hat Z_\epsilon$, where $\hat Z_\epsilon=\bigcup_{n\in\Z}(Z_\epsilon-n\omega)$, is zero.
\end{enumerate}
\end{theo}

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 $\ell^2(\Z)$.

From the previous theorem, the following corollary proceeds:

\begin{coro}\label{cor}\mbox{}
\begin{enumerate}
\item \label{cor_1} $E(\hat Z_\epsilon) \subset \overline{\sigma_p(H_\theta)}$, $\forall\theta\in\hat Z_\epsilon$;
\item \label{cor_2} $\braces[\big]{\theta\in\T; \ \sigma_p(H_\theta)=\emptyset}$ has zero Hausdorff dimension;
\end{enumerate}
where $\sigma_p(H_\theta)$ is the punctual spectrum of $H_\theta$, it is defined as being the closure of the set of eigenvalues of $H_\theta$.
\end{coro}
\begin{proof}
This is an immediate consequence of Theorem~\ref{t1}. Indeed, we define for all $m\in\Z$ the application, $\tau_m \colon \ell^2(\Z) \to \ell^2(Z)$, by $(\tau_m u) _n=u_{n-m}$. So, according to the first two assertions of Theorem~\ref{t1}, there exists a family of eigenvectors $V^n(\theta)=\tau_n V(\theta+n\omega)$ which are linearly independent, hence~\eqref{cor_1}. The assertion~\eqref{cor_2} follows from Theorem~\ref{t1}$\MK$\eqref{t1_4}, since $\braces[\big]{\theta\in\T; \ \sigma_p(H_\theta)=\emptyset}$ is included in $\T\setminus\hat Z_\epsilon$.
\end{proof}

The presented results are on the spectral properties of the quasi-periodic Schr\"odinger operators with a discontinuous potential. The proof is inspired from~\cite{1,2}. It is a more global analysis than in~\cite{1}, and follows the scheme presented in~\cite{2}, all in considering an analytical norm instead of Gevrey. This work also generalizes the previous one~\cite{WR}. For around forty years, the study of quasi-periodic Schr\"odinger 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.\ Fr\"ohlich, T.\ Spencer, P.\ Wittver~\cite{ 7} and Ya.\ G.\ Sina\u\i~\cite{14}. The model with discontinuities was mentioned in~\cite{2}, 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 $\ell^2$-localized at one site. So a natural question arises, can we have exponential localization for this model? In~\cite{00}, J.\ Bourgain and M.\ Goldstein showed exponential localization, but for a model without discontinuities. Finally, S.\ Jitomirskaya and B.\ Simon showed in~\cite{Gdelta} that $\braces[\big]{\theta;\ \sigma_p(H_\theta)=\emptyset}$ is a dense $G_\delta$. We improved this result by showing that this set has zero Hausdorff dimension.

\section{On the proof}

For this problem we identify the family of operators $H_\theta$ as a symmetric infinite-dimensional matrix that depends on the parameter $\theta$:
\[
(D + F)(\theta)=
\begin{pmatrix}
\ddots 	&amp; 					&amp; 			&amp; 					&amp; 0 		\\
		&amp; v(\theta -\omega) 	&amp; -\epsilon 	&amp; 					&amp; 		\\
		&amp; -\epsilon 			&amp; v(\theta)	&amp; -\epsilon 			&amp; 		\\
		&amp; 					&amp; -\epsilon 	&amp; v(\theta +\omega) 	&amp; 		\\
0 		&amp; 					&amp; 			&amp; 					&amp; \ddots	\\
\end{pmatrix}.
\]
The goal is to conjugate this matrix to a diagonal matrix $D_\infty (\theta)$ 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
\begin{equation}\label{123}
{(U^j)}^* \cdots {(U^1)}^* (D + F) U^1 \cdots U^j = D^{j+1} + F^{j+1}
\end{equation}
where $\norm{F^j(\theta)}=\epsilon_j$ goes to $0$ as $j$ tends to $\infty$ and $D^j(\theta)$ is a block matrix. More precisely, $D^j(\theta)$ has an invariant eigensubspace $S_j(\theta)\subset\R^{\La_j(\theta)}$, where $\La_j(\theta)$ is an increasing sequence of subsets of $\Z $, such that $\diam\parens[\big]{\La_j(\theta)}\leq\lambda_j$ and $\#\parens[\big]{\La_j(\theta)}\leq\mu_j$. Note that this is valid for all $\theta$. In this process we follow the approach of~\cite{2}, 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.
\begin{prop}
Consider the set $Z_j=\braces[\big]{\theta; \ \La_j(\theta)=\braces{0}}$, then for each $j$, $Z_j\setminus Z_{j+1}$ is a union of at most $K_{j+1}$ intervals $I_i$, where $K_1$ is given, $K_{j+1}\leq\parens[\big]{\frac{1}{\epsilon_j}}^{28(\tau+1)d_ja_j}$, $\Leb(I_i)\leq\nu_j\leq\epsilon_j^{a_j/4\mu_j}$, with
\[
a_j=\frac{1}{16\tau(\tau+1)\mu_{j+1}^9}, \quad d_j=\frac{1}{200\tau\mu_j^2} \quad and \quad \epsilon_{j+1}\leq\epsilon_j^{\mu_{j+2}^{99}}.
\]
\end{prop}

Now let's take $Z_\epsilon=\cap_{j\geq1}Z_j$, it follows that $\T\setminus\hat Z_\epsilon$ has zero Haussdorff dimension. That is
\[
\forall s&gt;0, \ \lim_{\nu\to 0}\mathcal H_{\nu,s}(\T\setminus\hat Z_\epsilon)=0,
\]
where $\mathcal H_{\nu,s}(\T\setminus\hat Z_\epsilon)=\inf\parens[\big]{\sum_{i=1}^\infty\abs{U_i}^s}$ and the infimum is taken over all $\nu$-coverings of $\T\setminus\hat Z_\epsilon.$

Indeed, putting $W_j=Z_j\setminus Z_{j+1}$ for all $j\geq1$, we have $W_j=\sum_{i=1}^{K_{j+1}}I_i$. It follows that $\T\setminus\hat Z_\epsilon\subseteq\braces[\big]{\theta; \ \theta+n\omega\in\bigcup_{j=1}^\infty W_j, \ \forall n\in\Z}$, this implies that $\T\setminus\hat Z_\epsilon\subseteq\bigcup_{j=1}^\infty W_j$ which is a $\nu_1$-cover of $\T\setminus\hat Z_\epsilon$. For $\epsilon$ sufficiently small and $\omega$ Diophantine, we show that $\T\setminus\hat Z_\epsilon\subseteq\bigcup_{j=2}^\infty\bigcup_{\abs{n}\leq K_2^2}(W_j-n\omega)$ which is a $\nu_2$-cover of $\T\setminus\hat Z_\epsilon$. Then we show inductively that $\T\setminus\hat Z_\epsilon\subseteq\bigcup_{j=N}^\infty\bigcup_{\abs{n}\leq K_N^2}(W_j-n\omega)$ which is a $\nu_N$-cover of $\T\setminus\hat Z_\epsilon$. This gives for all $s\in\mathopen{]}0,1\mathclose{[}$ that
\[
\mathcal H_{\nu,s}(\T\setminus\hat Z_\epsilon)\leq\sum_{j=N}^\infty3K_j^ 2\nu_j^s.
\]
Thus, the rapid decrease of $\epsilon_j$ to $0$ allows us to conclude that the series of terms $3K_j^2\nu_j^s$ is convergent because $3K_j^2 \nu_j^s\leq\epsilon_N^{sa_N/8\mu_N}$ for all $j\geq j_s$. This proves that $\mathcal H_{\nu,s}(\T\setminus\hat Z_\epsilon)\leq\epsilon_N^{sa_N/16\mu_N}$ for all $N\geq j_s$, which implies the result.

\section*{Acknowledgments}

The author wishes to thank H.\ Eliasson for many ideas which have been decisive in this paper.

\section*{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.

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