%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Almost Everything about the Almost Mathieu Operator I %
% Author: Y. Last %
% This is a (plain) TeX document %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\baselineskip 15pt
\magnification 1200
\mathsurround=1pt
\font\bigrm=cmr10 scaled\magstep3
\font\smallrm=cmr8
\def\ref#1#2#3#4#5#6{#1:\ #2.\ #3\ {\bf #4},\ #5 \ (#6) }
{ $ { } $ }\vskip 0.3in
\centerline{\bigrm Almost Everything}
\medskip
\centerline{\bigrm about the Almost Mathieu Operator
I\footnote{*}{\smallrm To appear in the Proceedings of the XIth
International Congress of Mathematical Physics, Paris 1994}}
\bigskip\bigskip\bigskip
\centerline{Y. Last\footnote{**}{\smallrm Work partially
supported by the GIF}}
\bigskip\bigskip
\centerline{Department of Physics}
\centerline{Technion - Israel Institute of Technology}
\centerline{32000 Haifa, Israel\footnote{***}{\smallrm Current address:
Division of Physics, Mathematics and Astronomy, California Institute
of Technology, Pasadena, California 91125}}
\bigskip\bigskip\bigskip\bigskip\bigskip\noindent
{\bf Abstract: }
We review some aspects of the spectral theory of the Almost Mathieu
operator (acting on $\ell^2(Z)$):
$$(H_{\alpha, \lambda, \theta} u)(n) =
u(n+1)+u(n-1)+\lambda\cos (2\pi\alpha n+\theta)u(n)\;,$$
where $\alpha,\lambda,\theta \in R$. We concentrate on the spectrum
as a set, and describe the partial results obtained so far on the
conjectures that, for irrational $\alpha$, it is a Cantor set and has
Lebesgue measure $|4-2|\lambda||$.
\vfill\eject
\noindent
{\bf 1. Introduction}
\medskip\noindent
The Almost Mathieu operator is the discrete one dimensional Schr\"odinger
operator (acting on $\ell^2(Z)$) given by:
$$(H_{\alpha, \lambda, \theta} u)(n) =
u(n+1)+u(n-1)+\lambda\cos (2\pi\alpha n+\theta)u(n)\;,
\eqno (1.1)$$
where $\alpha,\lambda,\theta \in R$. Its name comes from the similarity to
the Mathieu equation: $$ -y''(x)+\lambda\cos (x)\, y(x) =
Ey(x)\;. \eqno (1.2)$$
It is often also called Harper's operator, although
many think that Harper's name should be reserved for the special case of
$H_{\alpha, \lambda, \theta}$ with $\lambda=2$.
$H_{\alpha, \lambda, \theta}$ is a
tight binding model for the Hamiltonian of an electron in a one
dimensional lattice, subject to a commensurate
(if $\alpha$ is rational) or
incommensurate (if $\alpha$ is irrational) potential.
It is also related to the
Hamiltonian of an electron in a two dimensional lattice, subject to a
perpendicular (uniform) magnetic field. There are two different ways (or
limits) in which this relation can be obtained. The first, going back
to Harper [11], is to start with a tight binding model of a two
dimensional rectangular lattice
(which only takes into account nearest neighbors
interaction), and then to consider a Landau gauge for the magnetic field.
Namely, the vector potential is taken to be in one direction,
parallel to one of
the directions of the lattice and perpendicular to the other.
This makes the
Hamiltonian separable, such that the eigenfunctions are plain waves in the
direction which is perpendicular to the vector potential, and one obtains
$H_{\alpha, \lambda, \theta}$ for the
direction of the vector potential. $\alpha$ is the magnetic
flux per unit cell (in quantum flux units), $\theta$ is the wave-number of
the plane waves in the transversal direction, and $\lambda/2$ is the ratio
between the length of a unit cell in the direction of the vector potential
and its length in the transversal direction. In particular, $\lambda =2$
corresponds to a square lattice. This approach is closely related to the
standard Landau gauge solution for free electrons (in the plane) in a
uniform magnetic field,
where one gets plane waves in one direction and the
harmonic oscillator (which gives rise to Landau levels) in the other
direction. In this sense,
$H_{\alpha, \lambda, \theta}$ appears as a tight binding analog of the
harmonic oscillator. The second way, going back at least to Rauh [17], is
to start with free electrons
(in the plane) in a uniform magnetic field, and
to consider the perturbation of a single Landau level arising from a weak
periodic (sinusoidal) potential. The magnetic flux per unit cell is
$1/\alpha$ in this case.
In both of these ways, the relevant energy spectrum
is the union over $\theta$ of the individual energy
spectra of $H_{\alpha, \lambda, \theta}$.
Namely, it is the set $S(\alpha, \lambda)$ defined by:
$$S(\alpha, \lambda)\;\equiv\;
\bigcup_\theta\sigma (\alpha, \lambda, \theta)\;\;, \eqno (1.3)$$
where
$$\sigma (\alpha, \lambda, \theta)\;\equiv\;
{\rm Spec\,}(H_{\alpha, \lambda, \theta})\;\;. \eqno (1.4)$$
Other
than its relations with some fundamental physical problems,
$H_{\alpha, \lambda, \theta}$ is
fascinating also because of the incredible spectral richness obtained by
varying the parameters $\alpha,\lambda,\theta$,
along with the fact that, to
a large extent, this richness can be rigorously analyzed. It serves as a
primary example for many spectral phenomena.
This is the first of two papers, in these proceedings,
which are intended to
survey the spectral theory of the Almost Mathieu
operator, and it will concentrate on the spectrum as set. The other paper,
by S. Jitomirskaya, will concentrate on aspects of the
spectral decomposition, and the remarkable metal-insulator transition
occurring (for irrational $\alpha$) at $|\lambda|=2$. In Section 2, we
review some elementary spectral properties of
$H_{\alpha, \lambda, \theta}$. In Section 3, we
discuss the ``Ten Martini Problem,'' namely,
we describe the partial results
obtained so far on the conjecture that the spectrum (for irrational
$\alpha$) is a Cantor set. In Section 4,
we discuss the Lebesgue measure of
the spectrum.
\bigskip\bigskip\noindent
{\bf 2. Some Elementary Spectral Properties}
\medskip\noindent
If $\alpha$ is a rational number, $\alpha = p/q$, where $p$ and $q$ are
relatively prime,
$H_{\alpha, \lambda, \theta}$ is a periodic Jacobi matrix,
and by Bloch-Floquet
theory, the structure of the spectrum
$\sigma (\alpha, \lambda, \theta)$ is well understood. It
consists of $q$ bands (closed intervals), which are usually separated by
gaps. As $\theta$ is varied, these bands move,
and their length may change,
but this happens in such a way that different
bands never overlap (other than in band edges).
Namely, an energy which is inside a given band (suppose that we
always label the bands by their order of occurrence on the real line) for
some $\theta$, will never be in a
different band for any other $\theta$ (see,
e.g., [3]). Thus, the set $S(\alpha, \lambda)$
is similar to $\sigma (\alpha, \lambda, \theta)$ for any
individual $\theta$, and also consists of $q$ bands. If $\alpha$ is
irrational, it follows from general principles [7] that the spectrum is
independent of $\theta$. Thus, we have in such case
$S(\alpha, \lambda)=\sigma (\alpha, \lambda, \theta)$ for
any $\theta$.
An important property of
$H_{\alpha, \lambda, \theta}$ is the Aubry duality [1], which allows
to relate eigenfunctions and spectra of the Almost Mathieu operator with
some given $\lambda$ to those of the Almost Mathieu operator with
$\lambda$ replaced by $4/\lambda$. This duality can be understood from a
physical viewpoint: If we consider the magnetic field problem described
above, and change the gauge of the vector potential into a Landau gauge in
the transversal direction,
we obtain again the Almost Mathieu operator, but
with $\lambda$ replaced by $4/\lambda$,
and the whole operator rescaled by a
factor of $\lambda/2$. Since the energy spectrum of the magnetic field
Hamiltonian must be gauge independent, we should expect:
$$S(\alpha, \lambda)\;=\;
{\lambda\over 2}S(\alpha, {4/\lambda})\; . \eqno (2.1)$$
Indeed, (2.1) was
rigorously proven (to hold for any real $\alpha$) by Avron and Simon [2].
>From (2.1), we see, in particular,
that it is sufficient to study $S(\alpha, \lambda)$
for $0\leq\lambda\leq 2$, since
for $\lambda > 2$, $S(\alpha, \lambda)$ is obtained immediately from the
$\lambda <2$ case.
In the rational case $\alpha = p/q$, $S(\alpha, \lambda)$
(which consists of $q$
bands) can have at most $q-1$ gaps. It turns out that these gaps are
always open, other than the middle gap for even $q$.
This fact was recently
proven by van Mouche [25], and also, independently, by Choi, Elliot, and
Yui [6]. In fact, Choi, Elliot, and Yui have even obtained an explicit
lower bound on the size of each open gap, such that the theorem is:
\bigskip\noindent
{\bf Theorem 2.1. } {\it
If $Q\ni\alpha = {p/ q}$, and $|\lambda|\leq 2$, then all gaps
(except the middle gap, for $q$ even)
of $S(\alpha, \lambda)$ are open, and have width larger than
$(|\lambda|/16)^q$ .}
\bigskip\medskip
Since $H_{\alpha, \lambda, \theta}$ is strongly continuous in
$\alpha$, $\sigma (\alpha, \lambda, \theta)$
is also continuous, in the sense that if
$E\in\sigma (\alpha, \lambda, \theta)$ and $\alpha_n\to
\alpha$, then there are points $E_n\in\sigma(\alpha_n,\lambda,\theta)$
such that $E_n\to E$. The set $S(\alpha, \lambda)$
has even better continuity
properties, and various results on this continuity have been obtained.
The strongest (at least as far as the implications we discuss below are
concerned) is due to Avron, van Mouche, and Simon,
who have shown uniform
(in both $\alpha$ and the energy ) H\"older continuity of order $1/2$.
Namely,
\bigskip\noindent
{\bf Theorem 2.2. } {\it
For fixed $\lambda$ and $|\alpha-\alpha'| < C(\lambda)$,
each $E\in S(\alpha, \lambda)$ has $E'\in S(\alpha',\lambda)$
with: $|E-E'|< 6|\lambda(\alpha-\alpha')|^{1/2}$.}
\bigskip\medskip\noindent
In particular, Theorem 2.2 implies that for every gap
in $S(\alpha,\lambda)$ with
measure $|g|$ larger than
$12|\lambda(\alpha-\alpha')|^{1/2}$, there is a corresponding
gap in $S(\alpha',\lambda)$ with measure larger than
$|g|-12|\lambda(\alpha-\alpha')|^{1/2} $.
\bigskip\bigskip\noindent
{\bf 3. The Ten Martini Problem}
\medskip\noindent
It is a long standing conjecture that for all irrational $\alpha$'s and
all $\lambda$'s, $S(\alpha, \lambda)$ is a Cantor set;
namely, a closed, nowhere dense
set, which has no isolated points. The absence of isolated points for
$S(\alpha, \lambda)$ is known from general principles
about ergodic Jacobi matrices
[7]. Thus, the remaining
question is whether or not the spectrum is nowhere dense.
The Cantor structure was conjectured in a 1964 paper by Azbel [4].
In 1976, Hofstadter [13] found clear numerical evidence that for
$|\lambda|=2$,
$S(\alpha, \lambda)$ is a Cantor set of zero Lebesgue measure.
In 1981, during a talk at the AMS annual meeting, Mark Kac offered ten
Martinis to anyone who shows that $S(\alpha, \lambda)$
``has all its gaps there.''
This led Simon [19] to name it ``The Ten Martini Problem.'' In fact,
there are two versions of the problem: The regular form
%to which we shall refer (following Shubin [18]) as TMP,
is to prove that for all $\lambda$'s
and irrational $\alpha$'s, $S(\alpha, \lambda)$ is a Cantor set.
The strong (or dry) form
%to which we shall refer as STMP,
is to prove that all the gaps which are allowed
by the Gap Labeling Theorem (see [19] and references therein)
are really open. For simplicity, we shall only relate to the (weaker)
regular form.
Several partial results on the Ten Martini Problem have been obtained
so far. They establish the Cantor structure either in certain limits or
for special values of $\alpha$ and/or $\lambda$. In general, however,
the problem is still open.
The first rigorous result on the Cantor structure of
$S(\alpha, \lambda)$ has been given in 1982 by Bellissard and Simon [5].
They have shown that for a dense $G_\delta$ of pairs
$(\lambda,\alpha)\in R^2$ (and thus ``generically,'' in the commonly
used topological sense), $S(\alpha, \lambda)$ is a Cantor set.
The $\alpha$'s which
are relevant to their proof need to be very well approximated by
rationals, but are not really specified. Nor are the $\lambda$'s.
Sinai, in a 1987 paper [20], has proven that for
(Lebesgue) a.e.\ $\alpha$, and
sufficiently large (or small) $|\lambda|$, $S(\alpha, \lambda)$
is a Cantor set.
Sinai's proof is perturbative and quite complicated.
The relevant $\alpha$'s are those with typical diophantine
properties (namely, are not too well approximated by rationals), such that
they are explicitly given. The $\lambda$'s are not explicitly given, and
the required largeness (or smallness) of
$|\lambda|$ may depend on $\alpha$.
An alternative perturbative proof of Sinai's result was recently given by
Goldstein [8].
In 1989, Helffer and Sj\"ostrand [12] have
proven that $S(\alpha, \lambda)$ is a Cantor
set, for the special case $|\lambda|=2$,
and for a special class of irrationals
characterized through their continued fraction expansions as follows: Let
$$\alpha\,=\,[n_1,\,n_2,\,n_3,\,\dots\;]\,=\,
{\strut\displaystyle 1\over {\displaystyle n_1+{\strut 1\over
{\displaystyle n_2 +
{\strut 1\over {\displaystyle n_3+\;\cdots\;}}}}}}\;\;, \eqno (3.1) $$
then there exists $C$ such that if
$\; n_i>C\;\;\forall i\;$ then $S(\alpha,2)$ is a Cantor set (of zero
Lebesgue measure). Their analysis is based on a unique spectral
renormalization procedure, which exploits
elaborate semi-classical analysis,
and is based on ideas (and a heuristic analysis) of Wilkinson [26]. The
class of relevant $\alpha$'s is
nowhere dense and of zero Lebesgue measure.
In their 1990 paper, Choi, Elliot, and Yui [6]
have proven that $S(\alpha, \lambda)$
is a Cantor set for all $\lambda$'s,
and for (Liouville) $\alpha$'s obeying
the diophantine condition: $|\alpha - p/q| < C^{-q}$ for infinitely many
rationals $p/q$. $C$ is an explicitly given constant. This set of relevant
$\alpha$'s is of zero Lebesgue measure,
but it contains a dense $G_\delta$.
In particular, their result strengthens the Bellissard-Simon result. The
Choi-Elliot-Yui result is a simple consequence of their lower bound on
gap sizes for rational $\alpha$'s (Theorem 2.1),
along with the continuity
properties of $S(\alpha, \lambda)$ (Theorem 2.2).
More recently, Last [15] has shown that
for $|\lambda|=2$, and (Lebesgue)
a.e.\ $\alpha$, $S(\alpha, \lambda)$ is a (zero measure) Cantor set.
The Cantor structure, in this case, follows immediately from showing that
$S(\alpha, \lambda)$ has zero measure.
The relevant $\alpha$'s are explicitly given in terms of a
diophantine condition or, equivalently, as having unbounded continued
fraction expansions (see more details below). While this class of
$\alpha$'s is intuitively larger than the class of
$\alpha$'s discussed by Helffer and Sjostrand,
it should be pointed out that their result covers
some $\alpha$'s that are excluded from Last's result.
For an additional review of results on the Ten Martini Problem and
many references, we refer the reader to the recent paper by Shubin [18].
\bigskip\bigskip\noindent
{\bf 4. Lebesgue Measure of the Spectrum}
\medskip\noindent
In a 1980 paper [1], Aubry and Andre described numerical evidence
that for $p,q$ relatively prime:
$$\lim_{q\to\infty}|S(p/q,\lambda)|\;=\;|4-2|\lambda||\;, \eqno (4.1)$$
where $|\cdot|$ denotes Lebesgue measure.
This led to the conjecture that for every irrational $\alpha$:
$$|S(\alpha, \lambda)| = |4-2|\lambda||\;, \eqno (4.2)$$
which, in particular, agrees with
Hofstadter's 1976 numerical findings [13]
that $|S(\alpha,2)| = 0\;$ ($\alpha$ irrational).
It should be pointed out that the set $S(\alpha, \lambda)$
and its structure do
depend strongly on $\alpha$. Clear evidence for this can be found
in Guillement, Helffer, and Treton's 1989 paper [9], which contains
many numerically computed illustrations of Almost Mathieu spectra.
The universality of the measure, predicted by (4.2), is, therefore,
a rather remarkable property.
The first rigorous work on this issue was done by Thouless [21] in 1983.
He has shown that for every rational $p/q$:
$\;|S(p/q,\lambda)|\;\geq\;|4-2|\lambda||\;$, which, by the continuity
of $S(\alpha, \lambda)$, implies:
$$|S(\alpha, \lambda)|\;\geq\;|4-2|\lambda|| \eqno (4.3)$$
for any real $\alpha$.
In their 1990 paper [3], Avron, van Mouche, and Simon, have proven
that for $|\lambda|\leq 2$ and $p,q$ relatively
prime:
$$4-2|\lambda|\;\leq\;
|S({p/q},\lambda)|\;\leq\;4-2|\lambda| +4\pi{\left(|\lambda|\over
2\right)}^{q/ 2}\;, \eqno (4.4)$$
and, moreover,
$$|S_- ({p/q}, \lambda)|\;=\;4-2|\lambda|\;\;, \eqno (4.5)$$
where $\displaystyle{S_-(\alpha, \lambda) \;\equiv\;
\bigcap_\theta \sigma (\alpha, \lambda, \theta)}$.
(4.4) proves the numerical observation (4.1) of Aubry and Andre, for any
$|\lambda|\not= 2$. A 1993 observation of Last [14] has shown that
(4.4) combined with Theorem 2.2 implies the conjecture (4.2) for
$|\lambda|\not= 2$ and (Lebesgue)
a.e.\ $\alpha$. For $|\lambda| = 2$, however, (4.4) implies nothing, and
the vanishing of the measure in this case remained open.
The $|\lambda|=2$ case was recently tackled by Last and Wilkinson
[16] and Last [15], who have shown that
$${{2(\sqrt5+1)}\over q}\; < \;|S({p/q}, 2)|\; < \;{{8e}\over q}\;,
\eqno (4.6)$$
where $e\equiv\exp(1) =2.71.\,.\,.$. It is obtained as a $\lambda\to 2$
limit of the remarkable exact equality
(4.5) of Avron, van Mouche, and Simon.
(4.6) is strongly related to a conjecture of Thouless
[21,22,23,24]
that $\displaystyle{\lim_{q\to\infty}\, q|S({p/q}, 2)|\; = \;
{\rm Const} \; = \; 9.32.\,.\,.}$, and, in particular,
it allows to complete
the proof of the following theorem:
\bigskip\noindent
{\bf Theorem 4.1. } {\it If $\alpha$ is an irrational, for
which there is a
sequence of rationals $\{p_n/q_n\}$ obeying:
$$\lim_{n\to\infty}\;{q_n}^2\Bigl|\alpha-{p_n\over {q_n}}\Bigr|\;=\;
0 \; ,$$
then for every $\lambda\in R$:
$$|S(\alpha, \lambda)| = |4-2|\lambda|| \; .$$
\bigskip\medskip\noindent
Remark. } The set of irrationals characterized in
Theorem 4.1 is precisely
the set of irrationals having unbounded continued fraction expansions.
This set is known to have full Lebesgue measure [10].
\vfil\eject
\bigskip\noindent
{ \bf { }References}
\bigskip\noindent
\item{1.}\ref{Aubry, S., Andre, G.}{Analyticity breaking and Anderson
localization in incommensurate lattices}
{Ann.\ Israel Phys.\ Soc.}{3}{133-164}
{1980}
\bigskip\noindent
\item{2.}\ref{Avron, J., Simon, B.}
{Almost periodic Schr\"odinger operators.\
II.\ The integrated density of states}{Duke Math.\ J.}{50}{369-391}{1983}
\bigskip\noindent
\item{3.}\ref{Avron, J., van Mouche, P., Simon, B.}{On the measure of the
spectrum for the almost Mathieu operator}
{Commun.\ Math.\ Phys.}{132}{103-118}
{1990}
\bigskip\noindent
\item{4.}\ref{Azbel, M.Ya.}{Energy spectrum of a conduction
electron in a magnetic field}{Sov.\ Phys.\ JETP}{19}{634-645}{1964}
\bigskip\noindent
\item{5.}\ref{Bellissard, J., Simon, B.}{Cantor spectrum for the almost
Mathieu equation}{J.\ Funct.\ Anal.}{48}{408-419}{1982}
\bigskip\noindent
\item{6.}\ref{Choi, M.D., Elliott, G.A., Yui, N.}
{Gauss polynomials and the
rotation algebra}{Invent.\ Math.}{99}{225-246}{1990}
\bigskip\noindent
\item{7.} Cycon, H.L., Froese, R.G., Kirsch, W., Simon, B.:
Schr\"odinger operators. Berlin, Heidelberg, New York: Springer 1987
\bigskip\noindent
\item{8.} Goldstein, M.: Preprint (1992)
\bigskip\noindent
\item{9.}\ref{Guillement, J.P., Helffer, B., Treton, P.}
{Walk inside Hofstadter's butterfly}{J.\ Phys.\ France}{50}{2019-2058}
{1989}
\bigskip\noindent
\item{10.}Hardy, G.H., Wright, E.M.: An introduction to the theory of
numbers, Fifth ed. Oxford: Oxford University Press 1979
\bigskip\noindent
\item{11.}\ref{Harper, P.G.}{Single band motion of conduction electrons
in a uniform magnetic field}{Proc.\ Phys.\ Soc.\ London}
{A68}{874-892}{1955}
\bigskip\noindent
\item{12.}\ref{Helffer, B., Sj\"ostrand, J.}{Semi-classical analysis for
Harper's equation.\ III.\ Cantor structure of the spectrum}{M\'em.\ Soc.\
Math.\ France (N.S.)}{39}{1-139}{1989}
\bigskip\noindent
\item{13.}\ref{Hofstadter, D.R.}{Energy levels and wave functions
of Bloch electrons in a rational or irrational magnetic field}
{Phys.\ Rev.}{B14}{2239-2249}{1976}
\bigskip\noindent
\item{14.}\ref{Last, Y.}
{A relation between a.c.\ spectrum of ergodic Jacobi
matrices and the spectra of periodic approximants}
{Commun.\ Math.\ Phys.}{151}{183-192}{1993}
\bigskip\noindent
\item{15.}\ref{Last, Y.}
{Zero measure spectrum for the Almost Mathieu operator}
{Commun.\ Math.\ Phys.}{164}{421-432}{1994}
\bigskip\noindent
\item{16.}\ref{Last, Y., Wilkinson, M.}{A sum rule for the dispersion
relations of the rational Harper's equation}{J.\ Phys.}{A 25}{6123-6133}
{1992}
\bigskip\noindent
\item{17.}\ref{Rauh, A.}{Degeneracy of Landau levels in crystals}
{Phys.\ Status Solidi}{B 65}{K131-135}{1974}
\bigskip\noindent
\item{18.}\ref{Shubin, M.A.}{Discrete magnetic Laplacian}
{Commun.\ Math.\ Phys.}{164}{259-275}{1994}
\bigskip\noindent
\item{19.}\ref{Simon, B.}
{Almost periodic Schr\"odinger operators: A review}
{Adv.\ Appl.\ Math.}{3}{463-490}{1982}
\bigskip\noindent
\item{20.}\ref{Sinai, Ya.G.}{Anderson localization for one-dimensional
difference Schr\"odinger operator with quasiperiodic potential}
{J.\ Stat.\ Phys.}{46}{861-909}{1987}
\bigskip\noindent
\item{21.}\ref{Thouless, D.J.}{Bandwidth for a
quasiperiodic tight binding model}{Phys.\ Rev.\ B}
{28}{4272-4276}{1983}
\bigskip\noindent
\item{22.}\ref{Thouless, D.J.}{Scaling for the discrete Mathieu equation}
{Commun.\ Math.\ Phys.}{127}{187-193}{1990}
\bigskip\noindent
\item{23.}\ref{Thouless, D.J., Tan, Y.}{Total bandwidth for the Harper
equation. III. Corrections to scaling}{J.\ Phys.}{A 24}{4055-4066}{1991}
\bigskip\noindent
\item{24.}\ref{Thouless, D.J., Tan, Y.}{Scaling, localization and
bandwidths for equations with competing periods}
{Physica}{A177}{567-577}{1991}
\bigskip\noindent
\item{25.}\ref{van Mouche, P.M.H.}{The coexistence problem
for the discrete Mathieu operator}{Commun.\ Math.\ Phys.}{122}{23-34}
{1989}
\bigskip\noindent
\item{26.}\ref{Wilkinson, M.}{An exact renormalization group for
Bloch electrons in a magnetic field}{J.\ Phys.}{A 20}{4337-4354}
{1987}
\bigskip\noindent
\vfill\eject
\end