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\centerline{\tit Schr\"odinger operators generated by substitutions
${}^*$\footnote{}{${}^*$\ftn Work partially supported by EU contract CHRX-CT93-0411}}

\vskip.5truecm

\centerline{\aut Anton Bovier${}^{1}$\footnote{}{${}^{1}$\ftn
Weierstrass-Institut f\"ur Angewandte Analysis und Stochastik, Mohrenstrasse 39, D-10117
Berlin, Germany}
\footnote{}{\ftn e-mail bovier@iaas-berlin.d400.de} and
\aut \underbar {Jean-Michel Ghez}${}^{2,3}$\footnote{}{${}^{2}$\ftn Centre de Physique
Th\'eorique - CNRS, Luminy Case 907, F-13288 Marseille Cedex 9}
\footnote{}{${}^{3}$\ftn Phymat, D\'epartement de Math\'ematiques,
Universit\'e de Toulon et du Var,}
\footnote{}{\ftn B.P. 132 - F-83957 La Garde Cedex, France}}
\footnote{}{\ftn e-mail: ghez@cpt.univ-mrs.fr}

\vskip1truecm\rm

\noindent{\bf Abstract: }{Schr\"odinger operators with potentials generated 
by  primitive substitutions are simple models for one dimensional quasi-crystals. 
We review recent results on their spectral properties. These include in particular 
an algorithmically verifiable sufficient condition for their spectrum to be
singular continuous and supported on a Cantor set of zero Lebesgue measure.
Applications to specific examples are discussed.}

\vskip.5truecm\rm

We consider one dimensional Schr\"odinger operators $H$ defined by
$$
H\psi_n=\psi_{n+1}+\psi_{n-1}+V_n\psi_n,\quad\psi\in l^2(\Z),
\tag {1}
$$
where $(V_n)_{n\in\Z}$ is an aperiodic sequence generated by a substitution.

A {\it substitution} is a map $\xi$ from a finite alphabet $\A$ to the set
$\A^*$ of words on $\A$, which can be naturally extended to a map from $\A^*$
to $\A^*$ and then to a map from $\A^{\N}$ to $\A^{\N}$. We also define the
free group $\widehat\A^*$, extension of $\A^*$ obtained by addition of
the formal inverses of the letters in $\A$ as generators. $\xi$ is said
{\it primitive} if $\exists k\in\N$ s.t. $\forall (\a,\b)\in \A^2$, $\xi^k(\a)$
contains $\b$. A {\it substitution sequence} or {\it automatic sequence} is a
$\xi$-right fixpoint $u_r=\a_r...$ given by indefinite iteration of $\xi$ [1].
By choosing a $\xi$-left fixpoint $u_l=...\a_l$ such that the word $\a_l\a_r$ is
contained in $u_r$, we define by concatenation a doubly infinite word $w=u_lu_r$.
We say that a Schr\"odinger operator of type (1) is {\it generated}
by $\xi$ if the sequence $(V_n)_{n\in\Z}$ is defined by $V_n=v(w_n)$, where
$v$ is a map $\A\rightarrow\R$.

Such operators have become popular as simple models for electron transport in
one-dimensional quasi-crystals [2]. The study of some particular examples, namely 
the Fibonacci [3-6], Thue-Morse [7,8] and period-doubling [8] sequences, has led 
to the following common expectations for this type of operators:
\hfill\break (i) Their spectrum is purely singular continuous and supported on a
Cantor set of zero Lebesgue measure [3-5,7,8].
\hfill\break (ii) The spectral gaps are labelled by a countable set of algebraic
numbers, depending on the substitution [6-8].

By means of the K-theory of $C^*$-algebras, (ii) has been proven to be correct
for all operators generated by primitive substitutions [9], with the limitation 
that in general one cannot exclude that some gaps are closed. A general 
perturbative approach to compute the opening of gaps has been proposed in [10].

The basic tool to establish (i) is the transfer-matrix formalism. That is, the 
study of the properties of solutions of Eq. (1) lead to the analysis of products 
of two-by two matrices of the form
$$
P_n(E)=\prod_{k=1}^{n} T_E(w_{n-k})\tag {2}
$$
where $w_k$ is the $k$-th letter in the substitution sequence $w$ and 
$T_E:\A\rightarrow SL(2)$ is a map that assigns, for fixed energy $E$, to each 
letter in the alphabet a unimodular two-by-two matrix.  In the case of Eq. (1),
$
T_E(w_n)=\left(\matrix{E-v(w_n)&-1\cr1&0}\right)
$
but the precise form of this map is not important, and therefore the same
results hold for all types of second order equations that can be reduced to a 
problem of products of unimodular $2\times 2$-matrices. This includes in 
particular the continuous Laplacian with piecewise constant potential of a finite 
number of different shapes (Kronig-Penney model, see [12]). The main idea of the 
proof of the singularity of the spectrum $\s(H)$ of $H$ is its identification with 
the set $\OO$ of zero Lyapunov exponents 
$\g(E)=\lim_{n\uparrow\infty}\frac{1}{n}\log||T_E^{(n)}||$, known to be of zero 
Lebesgue measure by a general result in [8] based on a theorem of Kotani [13].
To do this requires a rather careful analysis of the asymptotic properties of 
$P_n(E)$, which is made possible by the self-similar structure of the potential. 
Let for any word $\o\in\A^*$
$$
T_E(\o)\equiv \prod_{\a\in\o}T_E(\a)\tag{3}
$$
and
$$
T^{(k)}_E(\o)\equiv T_E^{(k-1)}(\xi(\o)),\text {with} T_E^{(0)}\equiv T_E\tag {4}
$$
>From this recursion one can obtain an even more useful system of recursive equations 
for the traces of these transfer matrices. In general there exists a finite subset 
of words $\B\subset \A^*$ containing $\A$ for which (4) yields a closed set of recursive 
polynomial equations for the quantities $x_E^{(k)}(\b)\equiv \tr T^{(k)}_E(\b)$,
$\b\in \B$, which is called the {\it trace map} [14,15,11]. It turns out that to 
each trace map one can associate a {\it reduced trace map} that is {\it monomial} 
[11], and to this a substitution $\f$ on $\B$ whose properties are ultimately crucial 
for the spectral analysis. We call such a substitution {\it semi-primitive} if:
\hfill\break (i) There exists $\C\subset\B$ such that $\f|_{\C}$ is a primitive
substitution from $\C$ to $\C^*$;
\hfill\break (ii) There exists $k$ such that for all $\b\in\B$, $\f^k(\b)$ contains
at least one letter from $\C$.

With this notation, the main result proven in [11] is the following.

\vfill\eject

\theo [11]: {\it Let $H$ a one-dimensional Schr\"odinger operator of type (1)
generated by a primitive substitution $\xi$ on a finite alphabet $\A$. Assume
that its reduced trace map is associated to a semi-primitive substitution
$\f$. Assume also that there is a $k$ such that $\xi^k(0)$ contains $\b\b$
for some $\b\in\B$. Then the spectrum of $H$ is singular and supported on a set
of zero Lebesgue measure. If moreover there exist $n_0,m<\infty$ such that
$\xi^{n_0}(0)=\xi^m(\g_0)\Gamma\o$, where $\g_0\in\C$, $\G,\o\in\widehat\A^*$,
and $\G=\xi^m(\g_0)\d$ for some $\d\in\widehat\A^*$, then $H$ has no eigenvalues. 
Therefore, the spectrum of $H$ is purely singular continuous and supported 
on a Cantor set of zero Lebesgue measure.}

This is in fact an algorithmic procedure to prove the singularity or the singular 
continuity of the spectrum of $H$. Note that the supplementary hypothesis for the 
second result is probably not necessary, since there is at least one example 
(Thue-Morse) for which it is not satisfied although $\s(H)$ is singular continuous.

The hypothesis of the theorem have been shown to be satisfied in a large number of 
particular cases in [11]; more recently, in [16] a new class of examples was provided 
by sequences derived from circle maps with rotation numbers obtained from precious means.
The only known example in which even the hypothesis necessary to guarantee the 
singularity of the spectrum are not satisfied is the Rudin-Shapiro sequence [9].

Let us emphasize that out theorem provides explicit examples of operators with 
singular continuous spectrum. Complementary results have recently been obtained by 
Hof et al.[17] in which for certain types of so-called `palindromic' substitutions it 
was shown that there exists an infinite number of unspecified translates of the 
original sequence for which the corresponding operator has singular continuous spectrum.

\vskip.5truecm
We end this note with some remarks on the nature of the solutions of the Schr\"odinger 
equation (1). The proof of the theorem shows in fact that for energies in the spectrum, 
no solution tends to zero at infinity, but these leaves room for a variety of behaviours.
However, in most cases, there are countable subsets of energies in the spectrum, 
characterized by the fact that the transfer matrices $T_E^{(n)}(\a)$, for given $n$, 
commute for all $\a\in \A$, at which the solutions are extended in a very regular way 
in that they consist of pieces of repeating patterns arranged according to the 
substitution [12]. Such solutions have actually been discovered already in [7] and [8],
but they have been recently re-discovered in numerical investigations several times and 
have given rise to erroneous claims of coexisting absolutely continuous spectrum. While 
this is nonsense, it is not unlikely that these states will be quite important for the 
transport properties of systems with singular continuous spectrum, a problem that still 
has not been satisfactorily been investigated.
%in the last years in the interpretation of numerical results concerning the type
%of models we are dealing with.

\vfill\eject

{\bf References}
\item {[1]} M. Queff\'elec, {\it Substitution dynamical systems. Spectral
analysis}, Springer Lecture Notes in Mathematics {\bf 1294} (1987)
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{\bf 53}, 1951-1953 (1984)
\item {[3]} M. Casdagli, Comm. Math. Phys. {\bf 107}, 295-318 (1986)
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\item {[6]} L. Raymond: {\it A dynamical approach to the gap-labelling of a
quasi-periodic Schr\"odinger operator} in {\it Dynamique des syst\`emes
complexes, aspects classique et quantique}, Marseille, 13-17/6/1994
\item {[7]} J. Bellissard, in {\it Number theory and physics}, J.-M. Luck,
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\item {[8]} J. Bellissard, A. Bovier, J.-M. Ghez, Comm. Math. Phys. {\bf 135},
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 (1992)
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\item {[12]} A. Bovier, J.-M. Ghez: {\it Remarks on the spectral properties
of tight binding and Kronig-Penney models with substitution sequences},
Preprint CPT-94/P.3003
\item {[13]} S. Kotani, Rev. Math. Phys. {\bf 1}, 129-133 (1990)
\item {[14]} J.-P. Allouche, J. Peyri\`ere, C.R. Acad. Sci. Paris {\bf 302},
No 18, s\'erie II, 1135-1136 (1986)
\item {[15]} M. Kol\'ar, F. Nori, Phys. Rev  {\bf B42}, 1062-1065 (1990)
\item {[16]} M. H\"ornquist, M. Johansson: {\it Singular continuous electron
spectrum for a class of circle sequences}, Preprint Link\"oping University
(1994), to appear in J. Phys. A
\item{[17]} A. Hof, O. Knill, B. Simon: {\it Singular continuous spectrum for
palindromic Schr\"odinger operators}, Preprint Caltech (1994).
\end