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Kronig-Penney model, Landauer resistivity, quasiperiodic potential
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
% Adaptation of spaces in eqnarray
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%    Taken From Mathsing
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\def\bbbn{{\rm I\!N}} %natuerliche Zahlen
\def\bbbp{{\rm I\!P}}
\def\bbbe{{\rm I\!E}}
\def\bbbz{{\mathchoice {\hbox{$\sf\textstyle Z\kern-0.4em Z$}}
{\hbox{$\sf\textstyle Z\kern-0.4em Z$}}
{\hbox{$\sf\scriptstyle Z\kern-0.3em Z$}}
{\hbox{$\sf\scriptscriptstyle Z\kern-0.2em Z$}}}}
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%
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0.06\ht0\hbox to0pt{\kern0.4\wd0\vrule height0.9\ht0\hss}\box0}}
{\setbox0=\hbox{$\textstyle\rm C$}\hbox{\raise
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 \thesection.\arabic{equation}}
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\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}{Lemma}[section]
\newtheorem{corollary}{Corollary}[section]
\newtheorem{proposition}{Proposition}[section]
\newtheorem{remark}{Remark}[section]
\newtheorem{definition}{Definition}[section]
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\def\nlni{\par\ifvmode\removelastskip\fi\vskip\baselineskip\noindent}
\newenvironment{proof}{\nlni\begingroup\it Proof.\rm}{
\endgroup\vskip\baselineskip}
%
\begin{document}
% DOUBLE SPACED
\setlength{\baselineskip}{16pt}
%
\title{
The Landauer resistivity on quantum wires
}
\author{Masahiro Kaminaga
\thanks{
Department of Information Sciences,
Tokyo Denki University,
Hatoyama-machi, Hiki-gun, Saitama 350-03, JAPAN, 
e-mail: kaminaga@j.dendai.ac.jp. } 
and  
Fumihiko Nakano
\thanks{
Mathematical Institute, Tohoku University,
Sendai, 980-8578, JAPAN, 
e-mail: nakano@math.tohoku.ac.jp }
}

\date{}
\maketitle
%%%%%%% ABSTRACT %%%%%%%%%%%%%
\begin{abstract}
We study 
the Landauer resistivity of the Kronig-Penney model which has various behavior 
depending on the potential and the Fermi energy. 
% 
In the case of the Sturmian quasiperiodic potential, 
we discuss examples in which 
$\liminf$ of it is zero. 
\end{abstract}

Key Words : 
Kronig-Penney model; Landauer resistivity; 
quasiperiodic potential.

%%%%%%%%%%%%%%%%%%%% 
% Introduction
\section{Introduction}

The subject 
of this paper is the Landauer resistivity 
of the Kronig-Penney model defined by  
%
\begin{eqnarray*}
H:=
-\frac {d^2}{dx^2}
+
\sum_{j \in \bbbz}V(j) \delta(x-j), 
\quad
\mbox{on }
L^2 (\bbbr). 
\end{eqnarray*}
%
More precisely, 
$H$ is the Laplacian 
with boundary conditions on integer points: 
%
\begin{eqnarray*}
H:=
-\frac {d^2}{dx^2}
\quad
\mbox{on }
{\cal D}, 
\end{eqnarray*}
%
where 
%
\begin{eqnarray*}
{\cal D}
:=
\{ 
\psi \in H^1 (\bbbr) \cap H^2(\bbbr\setminus\bbbz)
:
\psi'(j+)-\psi'(j-)=V(j)\psi(j), 
\;
j \in \bbbz
\}.
\end{eqnarray*}
%
$H^p (\Omega)$ 
is the Sobolev space of order $p$ 
on $\Omega$ 
and 
$V(j) \in \bbbr$, $j \in \bbbz$. 
%
$H$ is 
self-adjoint 
\cite{Ge-Ki, Al-Ge-Hoe-Hol}  
and 
can be regarded as a model 
describing non-interacting electrons on the quantum wire. 
%
We would 
like to consider the Landauer resistivity of $H$  
which, however, 
is defined on Hamiltonians 
with compactly supported potentials. 
%
Thus 
we first consider the truncated Hamiltonian 
so that it has $n$ $\delta$-barriers.
%
\begin{eqnarray*}
H_n :=
-\frac {d^2}{dx^2}
+
\sum_{j=1}^n V(j) \delta(x-j), 
\quad
\mbox{on }
L^2 (\bbbr). 
\end{eqnarray*}
%
In other words, 
$H_n$ is the Laplacian on the domain 
%
\begin{eqnarray*}
{\cal D}_n
:=
\{
\psi \in H^1(\bbbr) \cap H^2 (\bbbr\setminus Y_n)
:
\psi'(j+)-\psi'(j-)=V(j)\psi(j), 
\;
j \in Y_n
\},
\end{eqnarray*}
%
where 
$Y_n 
:=
\{ 1,2, \cdots, n \}$. 
%
We fix 
the Fermi energy $\epsilon_F > 0$ 
arbitrary 
and set 
$k=\sqrt{\epsilon_F}$. 
%
The Jost solution $\psi$ 
of the equation 
$H_n \psi = \epsilon_F \psi$ 
is defined 
so that it satisfies following condition. 
% Jost solution
\begin{eqnarray}
\psi(x)
=
\cases{
ce^{ikx} + de^{-ikx}, 
& 
(if $x<1$), $\quad c, d \in \bbbc$, 
\cr
%
e^{ikx}, 
&
(if $x>n$). 
\cr
}
\label{Jost solution}
\end{eqnarray}
%
We do not 
consider the case where $\epsilon_F \leq 0$, 
for the Jost solution can not be defined. 
%
$c=c(n, \epsilon_F)$, $d=d(n, \epsilon_F)$ 
are determined by condition (\ref{Jost solution}) 
and 
the transmission probability 
$\tau (n, \epsilon_F)$ 
and 
the Landauer resistivity 
$\rho_L (n, \epsilon_F)$ 
are defined by 
\cite{Butt} 
%
\begin{eqnarray*}
\tau (n, \epsilon_F)
:=
\frac {1}{|c(n, \epsilon_F)|^2}, 
\quad
\rho_L (n, \epsilon_F)
:=
\frac {1-\tau (n, \epsilon_F)}{\tau(n, \epsilon_F)}. 
\end{eqnarray*}
%
We 
derive an explicit representation 
of $\rho_L(n, \epsilon_F)$ 
in terms of the transfer matrix of $H$ 
in the next section. 
%
It turns out 
that $0 < |c(n, \epsilon_F)| < \infty$ 
and hence 
$\tau (n, \epsilon_F)$ 
and 
$\rho_L (n, \epsilon_F)$ 
are always well-defined. 
%
If 
$\lim_{n \to \infty}
\rho_L (n, \epsilon_F)$ 
exists, 
it may be reasonable to regard it as the 
electrical resistivity of $H$ 
corresponding to the Fermi energy $\epsilon_F > 0$. 
%
This motivates us 
to study the behavior of 
$\rho_L (n, \epsilon_F)$ 
as $n$ tends to infinity,  
which is the purpose of this paper. 
%
Some of them 
are review of known facts, 
while that in the case of quasiperiodic potential is new.  

%%%%
In section 2, 
we 
introduce the transfer matrix and 
compute the transmission probability 
$\tau (n, \epsilon_F)$ 
and the Landauer resistivity 
$\rho_L (n, \epsilon_F)$ explicitly. 
%

%%%%%
In sections 3, 4, 5, 
we review spectral properties of $H$ 
and 
study the behavior of $\rho_L (n, \epsilon_F)$ 
as $n$ tends to infinity, 
when 
$V$ is periodic, random, and Sturmian quasiperiodic 
respectively. 
%
Section 6 is a summary 
of results. 

% Preliminaries
\section{Preliminaries}

First of all, 
we introduce the transfer matrix of $H$. 
%
The solution 
$\psi$ to the equation 
$H\psi = k^2 \psi$ 
($k > 0$, $k^2 = \epsilon_F$) 
has the following form. 
%
\begin{eqnarray}
\psi(x) 
=
C_j e^{ik(x-j)} + D_j e^{-ik(x-j)}, 
\quad
x \in (j, j+1), 
\;
C_j, D_j \in \bbbc. 
\label{solution}
\end{eqnarray}
%
By the boundary condition, 
$^t(C_j, D_j)$ 
satisfies
%
\begin{eqnarray*}
\left(
\begin{array}{c}
C_j \\
D_j 
\end{array}
\right)
=
T(j, k^2)
\left(
\begin{array}{c}
C_{j-1} \\
D_{j-1} 
\end{array}
\right), 
\quad
j \in \bbbz,
\end{eqnarray*}
%
where 
the (elementary) transfer matrix 
$T(j, k^2)$ 
is given by
%
\begin{eqnarray*}
T(j,k^2)
:=
\left(
\begin{array}{cc}
\left(
1+\frac {V(j)}{2ik}
\right) 
e^{ik} 
&
\frac {V(j)}{2ik} e^{-ik}
\\
%
-\frac {V(j)}{2ik} e^{ik}
&
\left(
1-\frac {V(j)}{2ik}
\right) 
e^{-ik} 
\end{array}
\right). 
\end{eqnarray*}
%
Let 
${\cal G}$ be a subgroup of 
$SL(2, \bbbc)$ 
%
\begin{eqnarray}
{\cal G}
:=
\left
\{
\left(
\begin{array}{cc}
\alpha & \beta \\
\overline{\beta} & \overline{\alpha}
\end{array}
\right)
: 
\;
\alpha, \beta \in \bbbc, 
\quad
|\alpha|^2 - |\beta|^2 = 1 
\right\}. 
\label{ge}
\end{eqnarray}
%
Then 
$T(j, k^2) \in {\cal G}$ 
and we can write 
%
\begin{eqnarray*}
W(n, k^2)
&:=&T(n,k^2)T(n-1,k^2)\cdots T(1,k^2)
\\
%
&=&
\left(
\begin{array}{cc}
a(n, k^2) & b(n, k^2) 
\\
\overline{b(n, k^2)} & \overline{a(n, k^2)}
\end{array}
\right), 
\end{eqnarray*}
%
where 
$a(n, k^2), b(n, k^2) \in \bbbc, 
\;
|a(n, k^2)|^2 - |b(n, k^2)|^2 = 1$. 
%
According to (\ref{Jost solution}), 
we need to derive 
$c=c(n, k^2)$, 
$d=d(n, k^2) \in \bbbc$ 
such that  
%
\begin{eqnarray*}
\left(
\begin{array}{c}
e^{ikn} \\
0
\end{array}
\right)
=
W(n, k^2) 
\left(
\begin{array}{c}
c 
\\
d
\end{array}
\right). 
\end{eqnarray*}
%
Direct computation gives 
%
\begin{eqnarray}
\tau (n, k^2)
=
\frac {1}{|a(n, k^2)|^2}, 
\quad
%
\rho_L(n, k^2)
=
|b(n, k^2)|^2.
\label{explicit form of rho}
\end{eqnarray}
%
Hence 
the Landauer resistivity 
$\rho_L (n, k^2)$ 
is equal to square of the absolute value of 
$(1,2)$-element of $W(n,k^2)$. 
%
We note 
$\rho_L(n, k^2)$ 
is related to the Hilbert-Schmidt norm of $W(n,k^2)$. 
%
\begin{eqnarray}
\rho_L(n,k^2)
=
\frac {\|W(n,k^2)\|^2_{HS}-2}{4}, 
\label{write rho by HS}
\end{eqnarray}
%
where 
$\| A \|_{HS}
=
\sqrt{\mbox{tr }(A^*A)}$ 
is the Hilbert-Schmidt norm. 
%
$\rho(n,k^2)$ 
is also related to the generalized eigenfunctions of $H$. 
%
In fact, 
$b(n,k^2)=
\frac 12
\left(
\psi(j) + \frac {\psi'(j+)}{ik}
\right)$ 
where $\psi$ is the solution to 
$H \psi = k^2 \psi$ 
with the initial condition 
$\psi(0) = -\frac {\psi'(0+)}{ik} = 1$. 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% periodic potential 
\section{Periodic Potential}

In the rest of this paper, 
we compute $\rho_L(n, \epsilon_F)$ 
and study its behavior as $n$ tends to infinity. 
%
In this section, 
we consider the case in which 
$V$ ($\not\equiv 0$) is periodic 
with period $L$: 
%
$V(j) = V(j+L), 
\;
j \in \bbbz.
$
%
By 
the direct integral decomposition 
and analytic perturbation theory, 
the spectrum of $H$ 
is seen to be absolutely continuous
\cite{Al-Ge-Hoe-Hol} 
%
which has the following representation. 
%
\begin{eqnarray*}
\sigma(H)
=
\{ E \in \bbbr : |D(E)| \leq 2 \}, 
\end{eqnarray*}
%
where 
%
$D(E) 
:=
\mbox{tr }W(L,E)$
is the discriminant. 
%
In section 2, 
we defined $T(j,E)$ 
only when $E>0$. 
%
When $E<0$, 
the branch of $k=\sqrt{E}$ is taken 
so that 
$\Im \sqrt{E} \geq 0$. 
%
When $E=0$, 
we take different basis 
of the solution to $H \psi = E \psi$. 
%
The behavior 
of $\rho_L (n, k^2)$ as $n \to \infty$ 
is summarized as follows. 
%%%%%%%%%%%%%%%%%%%%
% periodic case
\begin{theorem}
{\bf (periodic case)}

(1) 
If $\epsilon_F \in \rho(H)$, 
then 
$\rho_L (n, \epsilon_F)$ diverges exponentially. 
%

(2) 
If $|D(\epsilon_F)| \leq 2$, 
then 
for except at most countable set $A(\subset \bbbr)$, 
we have 

(i) 
if $|D(\epsilon_F)|=2$, 
then  
$\rho_L(n, \epsilon_F)$ 
diverges like $O(n^2)$ 
as 
$n \to \infty$. 

(ii) 
if $|D(\epsilon_F)| < 2$, 
then 
$\rho_L (n, \epsilon_F) = O(1)$ 
as $n \to \infty$ 
and does not converge. 
%
Moreover, 
for a.e. $\epsilon_F$ in $\sigma(H)$, 
$\overline{
\{\rho_L(n, \epsilon_F); n \in \bbbn\}}
=
\cup_{j=1}^L
[c_j(\epsilon_F), d_j(\epsilon_F)]$ 
for some $0 \le c_j(\epsilon_F)<d_j(\epsilon_F)$, 
$j=1, \cdots, L$. 
%
Otherwise 
$\{\rho_L(n,\epsilon_F);n\in\bbbn\}$ 
consists of finitely many points. 

(3) 
if $\epsilon_F \in A$, 
then 
$\{\rho_L(n,\epsilon_F);n\in\bbbn\}$ 
consists of finitely many points. 
\label{periodic case}
\end{theorem}
%%%%%%%%%%%%%%%%%%%%
%
\begin{remark}
The statement of 
Theorem \ref{periodic case} 
becomes clear in many specific cases. 
%
When 
$V(j) = V$ ($\ne 0$) is constant  
for instance, 
$A = \emptyset$. 
% 
$|D(\epsilon_F)|=2$ 
if and only if 
$\epsilon_F \in \partial \sigma (H)$,  
and 
$|D(\epsilon_F)|<2$ 
if and only if 
$\epsilon_F \in \sigma (H)^{\circ}$ 
($\sigma (H)^{\circ}$ 
is the set of interior points of $\sigma(H)$).  
%
In the case of (2)(ii), 
$c(\epsilon_F)=0$ 
and 
$d(\epsilon_F)
=
V^2/\epsilon_F (4-D(\epsilon_F))$. 
\end{remark}
%
\begin{remark}
The dichotomy 
in (2)(ii) in Theorem \ref{periodic case} originates in that 
$\theta/\pi \in \bbbq$ or $\in \bbbq^c$, 
where 
$2 \cos \theta = D(\epsilon_F)$, $\theta \in (0, \pi)$. 
%
In these cases, 
the corresponding generalized eigenfunction is periodic 
or quasiperiodic respectively. 
\end{remark}
%
%%%%%%%%%%%%%%%%%%%%
\begin{proof}
%
We first consider the points 
where $n=jL$, $j \in \bbbn$. 
%
Then 
the problem is reduced to the computation of  
$(1,2)$-element of 
$W(L,E)^j$ ($n=jL$). 
%
Let 
$\lambda$, $\lambda^{-1}$ ($\in \bbbc$) 
be eigenvalues of the matrix $W(L,E)$. 
%
We have 
the following expressions for 
$W(L, E)^j$. 
%
If $|\mbox{tr }W(L,E)| \ne 2$, 
then 
$\lambda \ne \lambda^{-1}$ 
and 
%
\begin{eqnarray}
W(L,E)^j 
=
\frac {\lambda^j - \lambda^{-j}}{\lambda-\lambda^{-1}}
W(L,E)
-
\frac {\lambda^{j-1} - \lambda^{-j+1}}{\lambda-\lambda^{-1}}I, 
\label{power-one}
\end{eqnarray}
%
where $I$  
is the $2 \times 2$-identity matrix. 
%
If $\mbox{tr }W(L,E) = \pm 2$, 
then 
$W(L,E)$ has a multiple eigenvalue $\lambda = \pm 1$ respectively and
%
\begin{eqnarray}
W(L,E)^j 
=
j \lambda^{j-1} W(L,E) - (j-1) \lambda^j I.
\label{power-two} 
\end{eqnarray}
%
By (\ref{power-one}), (\ref{power-two}), 
we compute $\rho_L(jL,\epsilon_F)$. 

Case (a) 
$|D(\epsilon_F)|>2$: 
%
Let 
$\lambda \in \bbbr$ be the eigenvalue of $W(L,E)$ 
with $|\lambda|>1$. 
%
Then 
$\lambda = (\mbox{sgn }\lambda) e^{\theta}$ 
for some $\theta >0$ 
and by (\ref{explicit form of rho}), 
%
\begin{eqnarray}
\rho_L (jL, \epsilon_F)
=
\left(
\frac {\sinh j \theta}{\sinh \theta}
\right)^2
|b(L,\epsilon_F)|^2, 
\quad
j \in \bbbz. 
\label{sinh}
\end{eqnarray}
%
We note, 
since $W(L,E) \in {\cal G}$ and $|\mbox{tr }W(L,E)|>2$, 
$b(L,\epsilon_F)\ne 0$. 

Case (b) 
$|D(\epsilon_F)|=2$:
%
$\lambda= \pm 1$ 
and 
%
\begin{eqnarray}
\rho_L (jL, \epsilon_F)=j^2 |b(L,\epsilon_F)|^2. 
\label{multiple roots}
\end{eqnarray}
%

Case (c) 
$|D(\epsilon_F)|<2$: 
%
$\lambda = e^{i\theta}$ 
for some 
$\theta \in (0, \pi)$  
and 
%
\begin{eqnarray}
\rho_L (jL, \epsilon_F)
=
\left(
\frac {\sin j \theta}{\sin \theta}
\right)^2
|b(L,\epsilon_F)|^2, 
\quad
j \in \bbbz. 
\label{sin}
\end{eqnarray}

When 
$|D(\epsilon_F)|>2$, 
(\ref{sinh}) implies 
$b(jL,\epsilon_F)$ diverges 
exponentially as $j \to \infty$. 
%
For intermediate points, 
that is, 
points of the form: 
$n = jL + l$, 
($1 \leq l \leq L-1$), 
we can see 
$\| W(n,\epsilon_F) \|_{op}$ 
diverges exponentially 
($\| \cdot \|_{op}$ is the operator norm),  
%
or alternatively we can use positivity of the Lyapunov exponent discussed in later sections. 

When 
$|D(\epsilon_F)| \leq 2$, 
it is possible that $b(L,\epsilon_F)=0$ 
%
(a simple example is: 
$L=2$, $V(1)+V(2)=0$, and 
$\epsilon_F=(n\pi)^2$, $n \in \bbbn$). 
%
This leads us to set 
%
\begin{eqnarray*}
A:=
\{ \epsilon_F \in \sigma (H) : 
b(L, \epsilon_F)=0 \}, 
\end{eqnarray*}
%
which consists of at most countably many isolated points. 
%
Since 
the statement of theorem is obvious for 
$\epsilon_F \in A$, 
let $\epsilon_F \notin A$. 
%
When 
$|D(\epsilon_F)| = 2$, 
(\ref{multiple roots}) implies 
$\rho_L(jL, \epsilon_F)=j^2 |b(L, \epsilon_F)|^2$ 
which diverges in the order of $n^2$. 
%
Moreover, 
Lemma \ref{linear function}
given below shows   
$b(jL+l, \epsilon_F)=\alpha_l \, j + \beta_l$ 
for some 
$\alpha_l$, $\beta_l \in \bbbc$, 
and 
$\alpha_l \ne 0$ 
unless 
$b(L, \epsilon_F)=0$. 
%
When 
$|D(\epsilon_F)|<2$, 
 direct computation gives 
$b(jL+l, \epsilon_F)
=
\gamma_l \sin j \theta + \delta_l \cos j \theta$ 
for some $\gamma_l$, $\delta_l \in \bbbc$, 
$\gamma_l \delta_l \ne 0$. 

Now the statement of 
Theorem \ref{periodic case} 
follows immediately from these considerations.
%
\QED
\end{proof}
%%%%%%%%%%%%%%%%%%%%
% lemma
%%%%%%%%%%%%%%%%%%%%
\begin{lemma}

Suppose 
$\epsilon_F \notin A$ 
and 
$|D(\epsilon_F)|=2$. 
%
Then 
$b(jL+l)=\alpha_l j + \beta_l$ 
for some $\alpha_l$, $\beta_l \in \bbbc$, 
$\alpha_l \ne 0$. 
\label{linear function}
\end{lemma}
%%%%%%%%%%%%%%%%%%%%
%
\begin{proof}
We suppose 
$\lambda=1$. 
%
The proof 
for $\lambda = -1$ 
is similar. 
%
Letting 
%
$B(l, \epsilon_F)
:=
T(l,\epsilon_F)
T(l-1,\epsilon_F)
\cdots
T(1,\epsilon_F) 
\;
(1 \leq l \leq L-1)
$, 
%
we have
%
\begin{eqnarray*}
b(jL+l,\epsilon_F)
=
j\left(
BW(1,2) - b(l,\epsilon_F)
\right) + b(l,\epsilon_F), 
\end{eqnarray*}
%
where 
$BW(1,2)$ is the $(1,2)$-element of the matrix 
$B(l,\epsilon_F)W(L,\epsilon_F)$.  
%
Hence 
$b(jL+l,\epsilon_F)= \alpha_l j + \beta_l$ 
for some $\alpha_l$, $\beta_l \in \bbbc$. 
%
Next 
we suppose $BW(1,2)=b(l,\epsilon_F)$ 
and would like to deduce $b(L,\epsilon_F)=0$. 
%
Since 
$W(L,\epsilon_F)$, $B(l,\epsilon_F) \in {\cal G}$, 
we can write 
%
\begin{eqnarray*}
W(L,\epsilon_F)
&=&
\left(
\begin{array}{cc}
\alpha_w, & \beta_w, \\
\overline{\beta_w} & \overline{\alpha_w}
\end{array}
\right), 
\quad
\alpha_w, \beta_w \in \bbbc, \;
|\alpha_w|^2 - |\beta_w|^2 = 1, 
\\
%
B(l,\epsilon_F)
&=&
\left(
\begin{array}{cc}
\gamma_l & \delta_l \\
\overline{\delta_l} & \overline{\gamma_l}
\end{array}
\right), 
\quad
\gamma_l, \delta_l \in \bbbc, 
\;
|\gamma_l|^2 - |\delta|^2 = 1. 
\end{eqnarray*}
%
Since $\lambda=1$ and $\epsilon_F \notin A$,  
$\alpha_w = 1 + ia$ 
for some $a \in \bbbr$, $a \ne 0$.  
%
$BW(1,2) = b(l,\epsilon_F)$ 
means 
$
\gamma_l \beta_w + \delta_l \overline{\alpha_w}
=
\delta_l 
$ 
and thus 
%
\[
\delta_l
=
\frac {\gamma_l \beta_w}{1- \overline{\alpha_w}}
=
\frac {\gamma_l \beta_w}{ia}. 
\]
%
Substituting it to the equation 
$| \gamma_l |^2 - | \delta_l |^2 =1$, 
we must have 
%
\[
|\gamma_l |^2 
\left(
1 - \frac {|\beta_w|^2}{a^2}
\right)
=1. 
\]
%
However, 
$|\beta_w|^2 = |\alpha_w|^2 - 1 = a^2$ 
implies 
LHS $= 0$. 
\QED
\end{proof}
%
\begin{remark}
If  
$\epsilon_F \in {\cal E} :=
\{ (n \pi )^2 : n \in \bbbn\}$ 
in which case 
$| D( \epsilon_F) | = 2$, 
the computation becomes easier. 
%
In fact, 
$\{ T(j, \epsilon_F) \}_{j \in \bbbz}$ 
commutes each other so that we have  
$b(n, \epsilon_F) = 
\frac {1}{2 i \sqrt{\epsilon_F}} 
\sum_{j=1}^n V(j)$. 
%
Hence 
$\lim_{n \to \infty}
\frac {\rho_L (n, \epsilon_F)}{n^2}
=
\frac {1}{4 \epsilon_F} 
\left(
\frac 1L
\sum_{j=1}^L V(j)
\right)^2$. 
%
We note, 
according 
to the relation found in \cite{BFLT}, 
$\sigma (H) \cap {\cal E}$ is related to that of the free 
Laplacian on $l^2 (\bbbz)$ \cite{KPI}, 
while 
the computation above implies 
the behavior of wave functions are not. 
\label{exceptional points}
\end{remark}
%


% Random potential
\section{Random Potential} 

Let 
$\{ V(j) \}_{j \in \bbbz}
=\{ V_{\omega}(j) \}_{j \in \bbbz}$ 
be the independent, identically distributed 
random variables 
on a probability space 
$(\Omega, {\cal F}, {\bf P})$. 
%
We assume 
the distribution of $V_{\omega}(0)$ 
has the density function 
which is bounded and compactly supported 
contained in the positive real line. 
%
That is, 
there exists constants 
$c_1$, $c_2$ such that 
%
$0 < c_1 \leq V_{\omega}(0) \leq c_2 < \infty, 
\;
a.s$.
%
Then 
\cite{Ki-Ma}
$\sigma (H) = \Sigma$, $a.s.$ where 
%
\begin{eqnarray*}
\Sigma 
= 
\left
\{
 E \in (0, \infty) : 
\left|
2 \cos \sqrt{E} 
+ 
V_{\inf} \frac {\sin \sqrt{E}}{\sqrt{E}}
\right|
\leq 2
\right\}, 
\;
V_{\inf} 
:=
\mbox{ess-}\inf_{\omega\in \Omega}V_{\omega}(0). 
\end{eqnarray*}
%
Moreover, 
the spectrum of $H$ on ${\cal E}^c$ 
is almost surely pure point with exponentially decaying eigenfunctions (Anderson localization). 
%
The positivity 
of the Lyapunov exponent 
for $\epsilon_F \notin  {\cal E}$ 
guaranteed by Furstenberg's theorem \cite{Bo-La} 
and 
Remark \ref{exceptional points} 
for $\epsilon_F \in {\cal E}$ give the following result. 
%
%%%%%%%%%%%%%%%%%%%%
\begin{theorem}
{\bf (random case)}

(1) If $\epsilon_F \notin {\cal E}$,  
%
then  
$\rho_L(n, \epsilon_F)$ 
diverges exponentially ${\bf P}$-a.s.

(2) If $\epsilon_F \in {\cal E}$, then 
$\lim_{n \to \infty} 
\frac {\rho_L (n, \epsilon_F)}{n^2}
=
\frac {1}{4 \epsilon_F} 
\left(
{\bf E} V_{\omega}(0) 
\right)^2$,  
${\bf P}$-a.s. 
where 
${\bf E}$ stands for taking expectations. 
\label{random case}
\end{theorem}
%%%%%%%%%%%%%%%%%%%%
%
\begin{remark}
In 
\cite{SB1, SB2, Absence I, Absence II}, 
they considered the charge transport 
on the multidimensional Anderson model  
and showed  
it is zero almost surely 
which is consistent with the theorem above. 
%
\end{remark}

% quasiperiodic potential
\section{Quasiperiodic Potential}

In this section, 
we consider the following quasiperiodic potential 
%
\begin{eqnarray*}
V(j)=V_{\theta}(j)
:=
\lambda_1 \chi_A (\Phi(\alpha j) + \theta) 
+
\lambda_2 \chi_{A^c} (\Phi(\alpha j) + \theta), 
\quad
j \in \bbbz, 
\end{eqnarray*}
%
where 
$\lambda_1$, $\lambda_2 \in \bbbr$, 
$\lambda_1 \ne \lambda_2$, 
$A=[1-\alpha,1) \subset \bbbr/\bbbz$, 
$\alpha \in (0,1) \cap \bbbq^c$, 
and 
$\theta \in \bbbr/\bbbz$. 
%
$\Phi : \bbbr \to \bbbr/\bbbz$ 
is the canonical projection. 
%
The spectrum of $H$ is purely singular continuous 
and is a Cantor set 
({\it i.e.,} nowhere dense closed set 
without isolated points) 
for $(\lambda_1, \lambda_2)$-a.e. in $\bbbr^2$ 
and 
for any $\theta \in \bbbr/\bbbz$ 
\cite{Su1, Su2, Be-Io-Sc-Te, Damanik-Lenz, KPII}. 
%
Moreover, 
the spectral measure is absolutely continuous w.r.t. 
${\cal H}^{\beta}$:  
the Hausdorff measure of dimension $\beta$ 
for some $\beta > 0$ 
(which follows from arguments in 
\cite{Ji-La1, Ji-La2, Damanik-Killip-Lenz}).
%
To state the results below, 
we consider the continued fraction expansion of $\alpha$. 
%
%%%%% FRACTION EXPANSION %%%%%
\begin{displaymath}
\alpha
=[0, a_1(\alpha), a_2(\alpha), \cdots]
:=\displaystyle{\frac{1}
{a_{1}(\alpha)+\displaystyle{\frac{1}
{a_{2}(\alpha)+\displaystyle{{}_{\ddots}}}}}}, 
\quad
a_n (\alpha) \in \bbbn. 
\end{displaymath}
%%%%%%%%%%%%%%%%%%%%%%
%
The associated 
rational approximation $p_n/q_n$ satisfies 
\cite{Lang} 
%
%%%%% ZENKASIKI OF P AND Q %%%%%
\begin{eqnarray}
&& p_{n+1}=a_{n+1}(\alpha)p_{n}+p_{n-1}, 
\label{p} 
\\
&& q_{n+1}=a_{n+1}(\alpha)q_{n}+q_{n-1}, 
\quad n \ge 0, 
\label{q}
\end{eqnarray}
%%%%%%%%%%%%%%%%%%%%%%
%
with  
$p_0 = 0$, $q_0=1$, 
$p_{-1}=1$, $q_{-1}=0$. 
%
We say 
$\alpha$ is a bounded density number  
if 
$\limsup_{n \to \infty}
\frac 1n 
\sum_{j=1}^n a_j (\alpha) < \infty$ 
(a typical example is the golden number :
$\alpha = (-1+\sqrt{5})/2$). 
%
A simple guess 
by the fact $\sigma (H) = \partial \sigma (H)$ 
and argument in Theorem \ref{periodic case} 
leads us to a speculation that 
$\rho_L (n, k^2)$ 
would grow polynomially. 
%
However, 
the situation may be more complicated 
as will be discussed later. 
%
\begin{theorem}
{\bf (quasiperiodic case)}

(1) 
If $\epsilon_F \in \rho(H)$, 
then $\rho_L(n, \epsilon_F)$ 
diverges exponentially. 

(2) 
(\cite{Su-Ko, Io-Te, Io-Ra-Te, Damanik-Lenz II}) 
If $\epsilon_F \in \sigma(H)$ and 
if $\alpha$ is a bounded density number, 
then 
$\rho_L(n, \epsilon_F)$ grows at most polynomial order. 

(3) 
If $\epsilon_F \in {\cal E}$, then 
$\lim_{n \to \infty}
\frac {\rho_L (n, \epsilon_F)}{n^2}
=
\frac {1}{4 \epsilon_F}
\left(
\alpha \lambda_1 + (1-\alpha) \lambda_2
\right)^2$. 
\label{quasiperiodic case}
\end{theorem}
%



%%%%%%%%%%%%%%%%%%%%
\begin{remark}

(1) 
This result 
contrasts with those 
in previous sections. 
%
The behavior 
of $\rho(n)$ is known to be complicated 
\cite{Su-Ko}. 

(2) 
Spectral properties 
of $H$ is related to the behavior of 
$\mbox{tr }W(q_n,E) $ 
as $n$ tends to $\infty$. 
%
In fact, 
$\sigma(H)$ coincides with the set 
where the sequence 
$\{ \mbox{tr }W(q_n,E) \}_{n=1}^{\infty}
(=
\{ 2 \Re a (q_n,E) \}_{n=1}^{\infty})$ 
is bounded \cite{Be-Io-Sc-Te, KPI}. 
%
On the other hand, 
the behavior of 
$\{ \rho_L (q_n, \epsilon_F) \}_{n=1}^{\infty}$ 
is related to that of 
$\{ \Im a(q_n, \epsilon_F) \}_{n=1}^{\infty}$. 
%
Therefore 
we may say that 
the Landauer resistivity 
reflects some aspects of the systems 
which is different from 
spectral properties. 

(3) 
If $V$ 
is of almost Mathieu type 
(that is, 
$V_{\theta}(j) 
=
\lambda \cos (2\pi(\alpha j + \theta))$, 
$\lambda \in \bbbr$, 
$\alpha \notin \bbbq$, 
$\theta \in \bbbr/\bbbz$), 
by Herman's theorem \cite{Her, CFKS}, 
the Lyapunov exponent is positive 
if $\epsilon_F \in B:=
\{ E \in \bbbr : 
| \lambda \sin \sqrt{E}/\sqrt{E} | > 2 \}$. 
%
Hence 
$\rho_L(n, \epsilon_F)$ diverges exponentially. 
%
On the other hand, 
if $\alpha$ is a Liouville number, 
$\sigma (H) \cap B$ (if it is nonempty) 
is shown to be singular continuous 
for a.e. $\lambda$ \cite{Av-Si}. 
%
This fact 
contrasts with Theorem 
\ref{quasiperiodic case}. 
%
\label{remark for quasiperiodic case}
\end{remark}
%%%%%%%%%%%%%%%%%%%%
%
The proof 
of (1) in Theorem \ref{quasiperiodic case} 
follows from positivity of the Lyapunov exponent 
due to the Combes-Thomas argument \cite{CT} 
which is roughly given by 
the distance between $\epsilon_F$ and $\sigma (H)$. 
%
The proof 
of (3) in Theorem \ref{quasiperiodic case} 
follows from Remark \ref{exceptional points}. 

We study  
some properties of $\rho_L(n, \epsilon_F)$ further 
assuming some properties on the 
behavior of tr $W(q_n, \epsilon_F)$. 
%
We first assume 
%
\begin{eqnarray}
\lim_{n \to \infty}
a_n (\alpha) = \infty, 
\label{infinity}
\end{eqnarray}
%
in which case 
the quasiperiodic potential $V(j)$ is 
`` close to periodic" 
so that we expect the argument in 
Theorem \ref{periodic case} would be useful. 

%%%%%
\begin{proposition}

Let 
$\theta = 0$, 
$\epsilon_F \in \sigma (H)$ 
and assume (\ref{infinity}). 
%

(1) 
If 
$\limsup_{n \to \infty}
|\mbox{tr }W(q_n,\epsilon_F)| > 2$, 
then 
$\limsup_{n \to \infty}\rho_L (n, \epsilon_F)=\infty$. 

(2) 
If 
$\liminf_{n \to \infty}
| \mbox{tr }W(q_n, \epsilon_F)|<2$, 
and 
$\{ \rho_L(q_n,\epsilon_F)\}_{n=1}^{\infty}$ 
is bounded, 
then 
$\liminf_{n \to \infty}
\rho_L (n, \epsilon_F)=0$. 
%
\label{strange behavior of rho}
\end{proposition}
%%%%%
%
To prove Proposition \ref{strange behavior 
of rho}, 
we prepare 
%%%%%
\begin{lemma}

%
Let 
$\lambda$, $\lambda^{-1} \in \bbbc$ 
($|\lambda| \ge 1$) 
be eigenvalues of $W(q_n, \epsilon_F)$. 
%

(1) 
If 
$|\mbox{tr }W(q_n,\epsilon_F)|<2$, 
then 
%
\begin{eqnarray*}
b(kq_n, \epsilon_F)
=
\frac {\sin k \theta}{\sin \theta}
b(q_n, \epsilon_F), 
\quad
1 \leq k \leq a_{n+1}(\alpha), 
\end{eqnarray*}
% 
where $\theta \in \bbbr$ 
is determined by 
$\lambda = e^{i\theta}$. 

(2) 
If 
$|\mbox{tr }W(q_n,\epsilon_F)|>2$, 
then 
%
\begin{eqnarray*}
b(kq_n, \epsilon_F)
=
\frac {\sinh k \theta}{\sinh \theta}
b(q_n, \epsilon_F), 
\quad
1 \leq k \leq a_{n+1}(\alpha), 
\end{eqnarray*}
% 
where 
$\theta >0$ is determined by 
$\lambda = (\mbox{sgn } \lambda)e^{\theta}$. 
%
\label{An explicit form of b(n)}
\end{lemma}
%
\begin{proof}
Since 
%
$V(q_n + k) = V(k), 
\;
(1 \leq k \leq q_{n+1}-2, 
\;
n \geq 1)$
%
\cite{Be-Io-Sc-Te}, 
we have 
%
$W(kq_n, \epsilon_F) 
=
W(q_n, \epsilon_F)^k, 
\;
k = 1,2,\cdots, a_{n+1}$.
%
Hence
the proof reduces to the computation 
of some powers of matrices 
which is done in the proof of 
Theorem \ref{periodic case}. 
%
\QED
\end{proof}
%%%%%%%%%%%%%%%%%%%%
{\it 
Proof of 
Proposition \ref{strange behavior of rho}}

(1) 
By assumption, 
there exists $\delta>0$ and a subsequence $n' = n(k)$ such that 
$|\mbox{tr }W(q_n, \epsilon_F) |> 2 + \delta$. 
%
We rewrite $n'$ as $n$. 
%
Let 
$\lambda(n)$ ($|\lambda(n)|>1$) be an eigenvalue of 
$W(q_n, \epsilon_F)$. 
%
We consider 
the triangulation of $W(q_n, \epsilon_F)$ 
by an unitary matrix $U(n)$ 
%
\begin{eqnarray*}
V(n, \epsilon_F)
:=
U^*(n) W(q_n, \epsilon_F) U(n) 
=
\left(
\begin{array}{cc}
\lambda(n) & c(n) \\
0 & \lambda(n)^{-1} 
\end{array}
\right), 
\quad
c(n) \in \bbbc.
\end{eqnarray*}
%
Then
%
\begin{eqnarray*}
V(n, \epsilon_F)^* V(n, \epsilon_F)
&=&
\left(
\begin{array}{cc}
|\lambda(n)|^2 & \overline{\lambda(n)}c(n) \\
\overline{c(n)}\lambda(n) & 
|\lambda(n)|^{-2}+|c(n)|^2
\end{array}
\right)
\end{eqnarray*}
%
and hence
%
\begin{eqnarray*}
\| W(q_n, \epsilon_F) \|^2_{HS}
%
& \geq &
|\lambda(n)|^2 + |\lambda(n)|^{-2}
%
=
(\mbox{tr }W(q_n, \epsilon_F))^2 -2. 
\end{eqnarray*}
%
By (\ref{write rho by HS}), 
%
$\rho_L(q_n, \epsilon_F)
=
\frac 14 
( \| W(q_n, \epsilon_F) \|_{HS}^2 - 2 ) > \delta_1$ 
%
for some $\delta_1>0$. 
%
By Lemma 
\ref{An explicit form of b(n)}(2), 
%
\begin{eqnarray*}
\rho_L(k q_n, \epsilon_F)
&=&
\left(
\frac {\sinh k \theta(n)}{\sinh \theta(n)}
\right)^2
\rho_L(q_n, \epsilon_F)
%
>
\left(
\frac {\sinh k \theta(n)}{\sinh \theta(n)}
\right)^2
\delta_1, 
\end{eqnarray*} 
%
($1 \leq k \leq a_{n+1}(\alpha)$). 
$\theta(n) >0$ 
is determined by 
$\lambda(n) = (\mbox{sgn }\lambda(n)) e^{\theta(n)}$.
%
Since 
$\left|
\lambda(n) + \lambda(n)^{-1}
\right|
=
|\mbox{tr }W(q_n,\epsilon_F)|
> 2 + \delta $, 
%
$|\lambda(n)| -1$ 
is bounded from below and thus 
$\theta(n) > \delta_2 > 0$ 
for some $\delta_2 > 0$. 
%
On the other hand, since 
$\epsilon_F \in \sigma(H)$, 
$|\mbox{tr }W(q_n, \epsilon_F) |$ 
is bounded (Remark \ref{remark for quasiperiodic case} (2)). 
%
Therefore 
$|\theta(n)|$ 
is bounded 
so that 
%
$| \sinh \theta(n) | < C_1$  
%
for some constant $C_1 > 0$. 
%
Hence 
%
\begin{eqnarray*}
\rho_L(a_{n+1}(\alpha) q_n, \epsilon_F)
>
\frac {\delta_1}{C_1^2}
\sinh^2 (a_{n+1}(\alpha) \delta_2).
\end{eqnarray*}
%
Since 
$\lim_{n \to \infty}a_n(\alpha) = \infty$, 
the result follows. 

(2) 
%
Let 
$\theta(n) \in (0, \pi)$ 
determined by 
$\lambda(n) = e^{i\theta(n)}$.
%
There exists 
$\theta \in [0, \pi]$
and a subsequence 
$\theta(n')$
such that 
$\theta(n') \to \theta$ as 
$n \to \infty$. 
%
Due to the assumption  
$\liminf_{n \to \infty}|\mbox{tr }W(q_n,\epsilon_F)|<2$, 
we can assume 
$\theta \in [\delta_1, \pi - \delta_1]$
for some $\delta_1 > 0$. 
%
We rewrite 
$\theta(n)$ instead of $\theta(n')$. 
%
Let 
$r_k / s_k$, $r_k \in \bbbn$, $s_k \in \bbbn$ 
be the Diophantine approximation of 
$\theta/\pi$.
%
Then we have \cite{Lang}
%
\begin{eqnarray*}
\frac {\theta}{\pi}
\in
\left(
\frac {r_k - \frac {1}{s_{k+1}}}{s_k}, 
\frac {r_k + \frac {1}{s_{k+1}}}{s_k}
\right). 
\end{eqnarray*}
%
We fix 
$k \in \bbbn$ arbitrary. 
%
Since 
$\theta(n) \to \theta$, 
there exists 
$N=N(k) \in \bbbn$ 
such that if 
$n \geq N$, we have 
%
\begin{eqnarray*}
\frac {\theta(n)}{\pi}
\in
\left(
\frac {r_k - \frac {1}{s_{k+1}}}{s_k}, 
\frac {r_k + \frac {1}{s_{k+1}}}{s_k}
\right).
\end{eqnarray*}
%
If  
$\theta/\pi \in \bbbq$, 
then 
$\theta/\pi=p/q$ 
for some $p$, $q \in \bbbn$  
and 
%
\begin{eqnarray*}
\frac {\theta(n)}{\pi}
\in
\left(
\frac {p - \epsilon'}{q}, 
\frac {p + \epsilon'}{q}
\right), 
\end{eqnarray*}
%
where 
$\epsilon'>0$ can be taken arbitrary small, 
when $n$ is large enough. 
%
Then the rest 
of this proof also works. 
%
Now we have 
%
\begin{eqnarray*}
\frac {s_k \theta(n)}{\pi}
\in
\left(
r_k - \frac {1}{s_{k+1}}, 
r_k + \frac {1}{s_{k+1}}
\right). 
\end{eqnarray*}
%
We fix 
$\epsilon >0$ arbitrary small. 
%
Since 
$\lim_{k \to \infty}s_k = \infty$, 
by taking 
$k$ sufficiently large, we have 
%
$\sin s_k \theta(n) 
\in
(-\epsilon, \epsilon), 
\;
n \geq N(k)$. 
%
Since 
$\theta(n)$ 
converges to 
$\theta \in [\delta_1, \pi - \delta_1]$, 
%
$| \sin \theta(n) | > \delta_2 > 0$   
%
for some $\delta_2 > 0$. 
%
By Lemma \ref{An explicit form of b(n)}(1), 
%
\[
\rho_L(l q_n, \epsilon_F)
=
\left|
\frac {\sin l \theta(n)}{\sin \theta(n)}
\right|^2
|b(n, \epsilon_F)|^2,
\]
%
for 
$1 \leq l \leq a_{n+1}(\alpha)$. 
%
By assumption, 
$|b(n,\epsilon_F)| < C_2$ 
for some $C_2>0$.  
%
Thus if we could let 
$l=s_k$, then 
%
\begin{eqnarray*}
\rho_L(s_k q_n, \epsilon_F)
\leq
\frac {C_2^2}{\delta^2_2}
|\sin s_k \theta(n)|^2
\leq
\frac {C_2^2\epsilon^2}{\delta^2_2}. 
\end{eqnarray*}
%
However, 
because 
$\lim_{n \to \infty}
a_n(\alpha) = \infty$, 
$a_{n+1}(\alpha) \geq s_k$ 
for sufficiently large $n$. 
\QED 
%%%%%
%

In what follows, 
we construct examples which satisfy the assumption of 
Proposition \ref{strange behavior of rho} (2). 
%
Let 
$M(0, E) :=T(0,E)$, 
$M(n,E) :=W(q_n, E)$, 
$r(n,E) :=\mbox{tr }M(n,E)$. 
%
$M(n,E)$ 
is known to satisfy the following recursive equation
\cite{Be-Io-Sc-Te}. 
%
\[
M(n+1,E) = M(n-1,E) M(n,E)^{a_{n+1}}, 
\quad
n \ge 1.
\]
%
Since 
$M(j, E) \in {\cal G}$, we can write 
%
\begin{eqnarray*}
M(n, E) = 
\left(
\begin{array}{cc}
\alpha_n & \beta_n \\
\overline{\beta_n} & \overline{\alpha_n} 
\end{array}
\right), 
\quad
\alpha_n, \, \beta_n \in \bbbc, 
\quad
|\alpha_n|^2 - |\beta_n|^2 = 1.
\end{eqnarray*}
%
The following lemma 
is Lemma 4.1 in \cite{KPII} 
which 
we also present here for the sake of completeness. 
%
%%%%%
\begin{lemma}
Fix $E \in \bbbr$. 
%
Suppose $a_k \in \bbbn$, $k = 1, \cdots, n$, 
$C>0$, $R \in \bbbn$, $\delta > 0$ 
are given which satisfy 

(i) 
$|\beta_k| \le C \prod_{l=1}^k 
\left(
1 + \frac {1}{2^l} 
\right)$, 
$k = 1, \cdots, n$. 

(ii) 
$|r(k, E) | < 2 - 2 \delta_k$, 
$R \le k \le n$, 
$\delta_k = \delta (\frac 12 + \frac {1}{2^k})$. 

Then 
we can choose $a_{n+1} \in \bbbn$ such that 
$M(n+1, E) := M(n-1,E) M(n,E)^{a_{n+1}}$ 
satisfies 

(1) 
$|\beta_{n+1}| \le C \prod_{l=1}^{n+1} 
\left( 1 + \frac {1}{2^l} \right)$, 

(2) 
$|r(n+1, E)| < 2 - 2 \delta_{n+1}$. 
%

Moreover, we have 
$\lim_{n \to \infty} a_n = \infty$. 
\label{Construction of discrete example}
\end{lemma}
%
\begin{proof}
Take 
$\theta_n \in (0, \pi)$ 
such that 
$2 \cos \theta_n = r(n,E)$. 
%
Pick 
$a_{n+1} \in \bbbn$ 
and let 
$M(n+1, E) := M(n-1,E) M(n,E)^{a_{n+1}}$. 
%
Then 
%
\begin{eqnarray*}
M(n+1, E) 
&=&
\frac {\sin a_{n+1} \theta_n}{\sin \theta_n}
M(n-1, E) M(n, E) 
\\
%
&&\qquad
-
\frac {\sin (a_{n+1}-1) \theta_n}{\sin \theta_n}
M(n-1, E).
\end{eqnarray*}
%
Set 
$C_k = C \prod_{l=1}^k 
\left( 1 + \frac {1}{2^l} \right)$. 
%
It is easy to see  
%
\begin{eqnarray*}
|\beta_{n+1}| 
&\le&
C_n 
\{ (2 \sqrt{1 + C_n^2} + 1) | \sin a_{n+1} \theta_n| 
+ |\sin \theta_n| \} 
\frac {1}{| \sin \theta_n |},
\\
%
| \Re \alpha_{n+1} | 
&\le&
\left|\frac {\sin a_{n+1} \theta_n} {\sin \theta_n} \right|
(2 + 2 C_n^2) + 1 - \delta_{n-1}.
\end{eqnarray*}
%
Take 
$\epsilon_n'$, $\epsilon_n'' > 0$ 
such that 
%
\begin{eqnarray*}
&&(2 \sqrt{1 + C_n^2} + 1 ) \epsilon'_n < 
|\sin \theta_n| \frac {1}{2^{n+1}},
\\
%
&&(2 + 2 C_n^2)
\frac {\epsilon''_n} {| \sin \theta_n |} 
< 
\frac {\delta}{2^{n+1}}.
\end{eqnarray*}
%
Set 
$\epsilon_n = \min \{ \epsilon_n', \epsilon_n'' \}$ 
and take 
$a_{n+1} \in \bbbn$, $a_{n+1} \ge n+1$  
such that 
$| \sin a_{n+1} \theta_n | < \epsilon_n$. 
%
Then 
$\beta_{n+1}$ and $r(n+1, E)$ satisfy (1) and (2) 
in the statement of Lemma \ref{Construction of discrete example} 
respectively. 
\QED
\end{proof}
%
%%%%%
\begin{remark}
An 
example which satisfies the hypothesis of 
Lemma \ref{Construction of discrete example} 
is: 
$\lambda_2 = 0$, 
$R=0$, 
and 
$\epsilon_F \in 
\{ k^2 \notin {\cal E} : 
|
2 \cos a_1 k + \frac {\lambda_1}{k} \sin a_1 k 
| < 2 \}$. 
\end{remark}
%
Take 
$\epsilon_F > 0$ 
which satisfies the hypothesis of Lemma 
\ref{Construction of discrete example} 
and 
let $\alpha = [0, a_1, a_2, \cdots] 
\in \bbbq^c \cap (0, 1)$ 
be corresponding irrational number associated with 
$\{ a_n \}_{n \in \bbbn}$ 
given by Lemma \ref{Construction of discrete example}. 
%
Then 
$|\mbox{tr }W(q_n, \epsilon_F) | < 2 - \delta$, 
$\rho_L (q_n, \epsilon_F) \le 
C^2 \prod_{n=1}^{\infty} 
\left(
1 + \frac {1}{2^n} 
\right)^2 
< \infty$, 
%
and thus 
$\alpha$, $\epsilon_F$ satisfy the assumption of 
Proposition \ref{strange behavior of rho} (2). 
%
Therefore 
$\liminf_{n \to \infty} \rho_L (n, \epsilon_F) = 0$ 
%
(in this case, 
$\rho_L (n, \epsilon_F)$ 
has at most polynomial growth even if 
$\alpha$ 
is not a bounded density number). 
%
This example tells us, if 
$\lim_{n \to \infty} a_n (\alpha) = \infty$, 
we can not expect to have lower bound of 
$\rho_L (n, \epsilon_F)$. 
%
We note, 
however, that 
$\sum_{n=1}^l \rho_L (n, \epsilon_F)
\ge C l^{\gamma}$ 
for some $\gamma > 0$ 
is shown 
\cite{Ji-La2, Damanik, Damanik-Killip-Lenz} 
%
when $q_n \le C^n$ for some $C>0$,  
where 
$\gamma>0$ depends on 
$\lambda_1$, $\lambda_2$, $\epsilon_F$, 
and $C>0$. 


%%%%%%%%%%%%%%%%%%
\section{Concluding Remarks}

We studied and reviewed 
the behavior of the Landauer resistivity
as sample size tends to infinity 
when the potential is 
periodic, random, and quasiperiodic.  
%
The results are summarized as follows. 

(1) 
When 
$\epsilon_F \in \rho(H)$, 
the Landauer resistivity 
diverges exponentially 
due to positivity of the Lyapunov exponent. 
%
This implies 
the conductivity vanishes in such cases, 
which is consistent with 
well-known result of 
the band theory in solid state physics. 
%
On the other hand, 
if $\epsilon_F \in \sigma(H)$, 
situation becomes different. 
%

(2) 
in the case of periodic potential 
where $\sigma(H)$ is absolutely continuous, 
the Landauer resistivity 
is bounded but does not converge if 
$\epsilon_F \in \sigma(H)$ 
unless 
it diverges like $n^2$. 
%
In this case 
$\overline{
\{ \rho_L (n, \epsilon_F) : n \in \bbbn \}}$ 
is a closed interval or finitely many discrete points 
which is not stable 
under 
the small variation of $\epsilon_F$, 
reflecting the behavior of corresponding Bloch waves. 

(3) 
in the case of random potential 
where 
$\sigma(H) \setminus {\cal E}$ 
is pure point with exponentially decaying eigenfunctions, 
the Landauer resistivity 
diverges exponentially, 
which implies zero conductivity. 
%
This fact 
is consistent with 
calculations by Kubo-type formula 
done in Anderson model
\cite{SB1, SB2, Absence I, Absence II}. 

(4) 
in the case of quasiperiodic potential 
where 
$\sigma(H)$ 
is generically purely singular continuous, 
it is known 
that the Landauer resistivity 
grows at most polynomial order. 
%
We found examples where 
$\liminf_{n \to \infty} \rho_L (n, \epsilon_F)$ 
is equal to zero. 

\vspace*{1em}
%
%%%%%
\noindent {\bf Acknowledgement }
One of authors (M.K) would like to thank 
Professor Sadao Sato for fruitful discussions. 

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\end{document}




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