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charge transport, Anderson localization,  
adiabatic theorem
---------------9908261934257
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\begin{document}
\setlength{\baselineskip}{15pt}
\title{\bf{Absence of transport under the slowly varying potential 
in disordered systems}
}
\author{
Fumihiko Nakano
\thanks{
present address:
Department of Physics, Princeton University, 
Princeton, NJ08544, USA, 
On leave of absence from:
Mathematical Institute, Tohoku University, 
Sendai, 980-77, Japan.} \
and \
Masahiro Kaminaga
\thanks{Multimedia Systems Research Department, 
Hitachi Ltd., Central Research Laboratory, 
1-280, Higashi-Koigakubo, Kokubunji-shi, Tokyo, 185-8601, Japan. }
}
\date{}
\maketitle

%%%%% DEFINITION %%%%%

\begin{abstract}
In the tight-binding 
random Hamiltonian on ${\bf Z}^d$, 
we consider the charge transport induced by 
the electric potential which varies sufficiently slowly in time, 
and prove that it is almost surely 
equal to zero at high disorder. 
%
In order to compute the charge transport, 
we adopt the adiabatic approximation 
and prove a weak form of adiabatic theorem 
while there is no spectral gap at the Fermi energy. 

\hspace*{1ex}

{\bf KEY WORDS:}
charge transport, Anderson localization, 
adiabatic theorem. 

\end{abstract}



\hspace*{1ex}

%%%%%% INTRO %%%%%
\section{INTRODUCTION}
\hspace*{3ex}
%
About 
four decades ago, 
Anderson$^{(4)}$ discussed that 
a certain disorder may cause materials to have insulating property. 
%
And recently, 
there is much progress 
toward the mathematical understanding of this phenomenon 
$^{(e.g., \,1, \,2, \,3, \,10, \,11, \,20, \,22, \,etc.)}$ such as 
derivation of the exponential decay of the eigenfunctions. 
%
In this paper, 
we consider the charge transport 
raised by the time-dependent flux$^{(6, 13)}$, 
and prove that this quantity is zero 
when the external field varies slowly in time. 

Our model is 
the standard, tight-binding, 
random Hamiltonian on $l^2 ({\bf Z}^d)$:
%
%%%%% Hamiltonian %%%%%
%
\begin{equation}
(H_{\omega} \varphi) (x) := 
\sum_{ |x-y|=1 } \varphi(y) 
+
\lambda V_{\omega}(x) \varphi(x), 
\quad
\varphi \in l^2({\bf Z}^d). 
\label{Hamiltonian}
\end{equation}
%
%
$\lambda >0$ is the coupling constant 
which represents the strength of disorder 
potential. 
%
$\{ V_{\omega} (x) \}_{ x\in {\bf Z}^d }$ is 
the independent, identically distributed 
random variables 
on a probability space 
$(\Omega, {\cal F}, {\bf P})$ 
such that the probability distribution of 
$V_{\omega}(0)$ has the density 
$r(v) dv$, 
where $r \in L^p({\bf R})$, 
$1 < p \leq +\infty$, 
and $r(v)$ is compactly supported. 
%
It follows that, 
for $a \in {\bf Z}^d$, 
there is a corresponding measure preserving map 
$T^a \; : \; \Omega \rightarrow \Omega$, 
such that
%
%%%%% Covariant %%%%%
%
\begin{equation}
U(a) H_{\omega} U(a)^* = H_{T^a \omega}, 
\label{covariant}
\end{equation}
%
%
where $U(a)$ is the translation operator on $l^2 ({\bf Z}^d)$ 
: $(U(a) \varphi)(x) := \varphi(x-a)$. 
% 
${\bf P}$ is ergodic 
with respect to $T$, and 
$\{ H_{\omega} \}_{\omega\in\Omega}$ 
is the ergodic family of bounded self-adjoint operators. 
%
Let $V(t)$ 
be a multiplication operator on $l^2 ({\bf Z}^d)$ :
$(V(t)\varphi) (x) := \exp[i\int_0^t g(x^{(1)},s) ds] \varphi(x)$, 
%
where $g (x^{(1)},s) : {\bf Z}\times [0,1] 
\rightarrow {\bf R}$ 
is bounded and periodic in $x^{(1)}$ 
with period $L\in {\bf N}$.  
%
$x^{(1)}$ is 
the first component of $x\in {\bf Z}^d$. 
%
We define 
the time-dependent random Hamiltonian: 
%
%%%%% tHamiltonian %%%%%
%
\begin{eqnarray}
H_{\omega}(t)&:=& V(t) H_{\omega} V^*(t), 
\quad t\in [0,1], \nonumber \\
{\rm or \; equivalently, } \nonumber \\
\quad
(H_{\omega}(t) \varphi)(x) &:=& \sum_{ |x-y|=1 }
\exp \Bigl[ i  \int_0^t  \bigl( g(x^{(1)},s) - g(y^{(1)},s) \bigr) ds \Bigr] \varphi(y)
\nonumber \\
&&\quad\quad\quad+
\lambda V_{\omega}(x) \varphi(x), 
\quad
\varphi \in l^2 ({\bf Z}^d).
\label{tHamiltonian}
\end{eqnarray}
% 
%
$H_{\omega}(t)$ describes 
non-interacting 
electrons in the random potential 
$\lambda V_{\omega}(x)$ 
under the time-dependent electric field 
potential $g(x^{(1)},t)
^{\; (9, \, Chapter \, 7)}$. 
%
This 
formulation was introduced by 
Avron-Seiler-Yaffe and Klein-Seiler
$^{(6, \, 13)}$ 
to study the quantum Hall effect. 

We shall 
prepare some notations which are necessary 
to define the charge transport. 
%
Let $U_{\omega}(t)$ be 
the unitary evolution operator of $H_{\omega}(t)$ 
which satisfies:
%
%%%%% Uomega %%%%%
%
\begin{eqnarray}
i\frac {d}{dt} U_{\omega}(t)\varphi &=&
H_{\omega}(t)U_{\omega}(t)\varphi, 
\quad
t\in (0,1), \nonumber\\
%
U_{\omega}(0)\varphi &=& \varphi, 
\quad
\varphi \in l^2 ({\bf Z}^d). 
\label{Uomega}
\end{eqnarray}
%
%
Let $\varepsilon_F \in {\bf R}$ 
be the Fermi energy 
which can be arbitrary in this paper, 
and 
$P_{\omega}(t) := \chi_{ (-\infty, \; \varepsilon_F] }
( H_{\omega}(t) )$ 
be the corresponding Fermi projection, 
where $\chi_C$ is the characteristic function 
of the set $C \subset {\bf R}$.  
%
For simplicity, 
we write $P_{\omega} := P_{\omega}(0)$. 
%
For an operator 
$A$ on $l^2 ({\bf Z}^d)$, 
we define ${\cal T}(A)$ as the trace per unit volume:
%
%%%%% traceper %%%%%
%
\begin{equation}
{\cal T}(A) := 
\lim_{ |\Lambda| \to \infty}
\frac{1}{|\Lambda|} {\rm trace} (\chi_{\Lambda} A \chi_{\Lambda}),
\label{traceper}
\end{equation}
%
%
as long as it exists, where 
$\Lambda:= 
\{
x \in {\bf Z}^d : 
|x^{(i)}| \leq M, 
i=1, \cdots, d
\}
\subset {\bf Z}^d$, $M\in{\bf N}$.  
%
In what follows, 
we shall 
define the quantity 
which we call the charge transport 
induced by slowly varying potential. 
%
This is 
based on 
what was considered in 
(refs. 6, 13). 
%
%
Firstly, 
we consider the charge transport 
from $t=0$ to $t=1$, 
which is defined 
as the thermal average of the current operator: 
$J_{\omega}(t) := U^*_{\omega}(t) 
i[H_{\omega}(t), x]U_{\omega}(t)$ 
at zero temperature, 
in the grand canonical ensemble: 
%
%%%%% transport %%%%%
%
\begin{equation}
j_{\omega} := 
\int_0^1 \; dt \; 
{\cal T}( U^*_{\omega}(t) i[H_{\omega}(t), \; x ] 
U_{\omega}(t) P_{\omega} ), 
\label{transport}
\end{equation}
%
where 
$[A, B]:= AB-BA$ is the commutator 
which is defined by those integral kernels 
(this will be mentioned in  Remark after Lemma 2.1).
%
We will confirm that 
$j_{\omega}$ is well-defined in Lemma 2.4. 
%
Secondly, 
we consider the limit 
under which the electric potential varies 
slowly in time. 
%
In order to do this, 
we rescale the time $s:= t/\tau$ ($\tau >0$) 
and replace $t$ by $s$. 
%
When $s$ varies 
from $s=0$ to $s=1$, 
the real time $t$ goes from $t=0$ to $t=\tau$, 
and the 
electric potential becomes $g(x^{(1)},s)/\tau$. 
%
The corresponding 
time evolution operator is the solution to:
%
%%%%% Uomegatau %%%%%
%
\begin{eqnarray}
i \frac{d}{ds} U_{\omega, \tau}(s) \varphi 
&=&
\tau H_{\omega}(s) U_{\omega, \tau}(s) \varphi, 
\quad
s\in (0,1), 
\nonumber\\
U_{\omega, \tau}(0) \varphi &=& \varphi, 
\quad
\varphi \in l^2 ({\bf Z}^d). 
\label{Uomegatau}
\end{eqnarray}
%
%
The charge transport   
from $t=0$ to $t=\tau$ becomes: 
%
%%%%% scaledtransport %%%%%
%
\begin{equation}
j_{\omega, \tau} := 
\int_0^1 \; ds \;
{\cal T} 
\left(
U^*_{\omega,\tau}(s) 
i\tau [H_{\omega}(s), \; x] 
U_{\omega,\tau}(s) 
P_{\omega} 
\right). 
\label{scaledtranport}
\end{equation}
%
We define 
$\tau \rightarrow \infty$ limit 
of $j_{\omega}$ 
as the charge transport 
induced by slowly varying 
electric potential.   
%
%%%%% conductivity %%%%%
%
\begin{equation}
\sigma_{\omega} 
:= \lim_{\tau \rightarrow \infty} 
j_{\omega,\tau}. 
\label{conductivity}
\end{equation}
%

{\bf {\it Remarks.}}

(1) 
A typical 
example of $g(x^{(1)},s)$ is 
the bounded periodic potential 
which is modulated in time: 
$g(x^{(1)},s) = h(x^{(1)}/L - t)$, 
where $h(x)$ is 
a function of period $1$. 
%
It was 
first introduced by 
Thouless and Niu$^{(14, \, 15, \, 16, \, 21)}$ 
in their study of 
quantized adiabatic charge transport. 
%
Niu$^{(14)}$ 
showed that, 
in some special choices of 
$h(x)$ and $V(x)\equiv 0$, 
there are nonzero transport 
(that is, $\sigma\ne 0$). 
%
Therefore, 
it seems to be plausible 
to take $g(x^{(1)},s)$ 
as an electric potential 
to study the transport properties. 

(2)
Since 
the electric potential becomes 
$g(x^{(1)},s)/\tau$ 
under the above scaling, 
$\tau\rightarrow\infty$ limit 
is equivalent to 
the small electric field limit. 
%
Thus, 
$\sigma_{\omega}$ can be regarded as 
the electrical conductivity 
under the bounded and time-dependent 
electric potential $g(x^{(1)},s)$. 

(3)
There have been 
some definitions of the electrical conductivity 
in the study of quantum Hall effect. 
%
The definition 
of the charge transport in (refs. 6, 13) 
is almost the same as that of ours 
except the following two points: 
(a) usual trace in (refs. 6, 13) is replaced by the 
trace per unit volume here.  
This replacement seems to be important to consider 
Anderson localization,  
(b) taking average with respect to flux is not necessary here.  
%

The definition of 
the electrical conductivity in (ref. 7, 19) is also 
similar to that of ours. 
%
The difference is:
(a) they considered 
time independent Hamiltonian 
with constant electric field 
and dissipative term,  
(b) they adopted 
Abel limit to compute the conductivity explicitly, 
while we will use adiabatic approximation here. 

\vspace*{1em}

The main result 
of this paper is, at high disorder, 
the charge transport 
defined in (\ref{conductivity}) 
indeed vanishes almost surely. 

\vspace*{1em}

%%%%% MAIN RESULT %%%%%
{\bf Theorem 1.1.}

{\it
For sufficiently large $\lambda >0$, 
$\sigma_{\omega} =0$,  
for  ${\bf P}$-a.e. $\omega$. 
}

\vspace*{1em}

{\bf {\it Remarks.} }

(1)
More precisely, 
$\lambda$ is taken large such that 

$\lambda > \lambda_0 (s):= 
2 \| r \|_p^{1/\sigma} (\frac{2\sigma}{\sigma-s} d)^{1/s}$, 
for some $0 < s < \sigma$, 
where $\| \cdot \|_p$ is the 
$L^p ({\bf R})$ norm, 
and 
$\sigma := 1-\frac{1}{p}$ ($\sigma=1$, if $p=+\infty$)$^{(2)}$.  
%
On the other hand, 
our proof does not apply to the weak disorder case 
(that is, $\lambda$ is small) 
even if $\epsilon_F$ lies 
in the localized states, 
because 
we can not control the tunneling 
to the delocalized states. 
%

(2)
The main 
tool we use to prove Theorem 1.1 is the 
exponential decay estimates of the 
fractional moment of Green's function$^{(1,\,2,\,3)}$. 
%
Thus, the above 
assertion also holds to many other Hamiltonians, 
such as that with long range hopping terms, 
or 
that with periodic background potential$^{(2)}$. 
%

(3)
It is 
desirable to consider the constant electric field. 
%
However, 
if we take $g(x^{(1)},s) :=x^{(1)}$, then 
(a) the proof of Lemma 2.2 in section 2 fails, 
(b) the adiabatic Hamiltonian 
defined in (\ref{adiabatic}) 
would be unbounded 
and 
analysis of it will not be easy. 
%

(4) 
Many people 
might think that our result is almost trivial 
because 
spectrum of $H_{\omega}$  
is all pure point 
and corresponding eigenfunctions 
decay exponentially.  
%
Our opposition 
to this opinion is: 
%
(a) 
they 
do not consider any external fields, 
%
(b) 
We only 
know that each eigenfunction 
decays exponentially 
at each rate. 
%
However, 
from the statistical mechanics 
point of view, 
we have to consider 
infinitely many states at the same time 
at least near the Fermi energy. 
%
And, 
del Rio, et. al.$^{(10)}$ suggest that 
the eigenfunctions of $H_{\omega}$ 
do not localize ``uniformly", 
%
(c) 
the eigenvalues of $H_{\omega}$ 
distribute densely in the spectrum$^{(20)}$ 
which implies there is no spectral gap, 
and there are infinitely many eigenvalues 
near the Fermi energy.  
%
Since there might be 
infinitely small excitation beyond the Fermi energy, 
our result seems 
not to be trivial. 

(5) 
By using Kubo formula, 
vanishing conductivity follows immediately from 
the exponential decay estimates of Green's function
$^{(2, \, 11, \, 19)}$. 
%
Thus, our result 
can be regarded as another presentation of the 
vanishment of the electrical conductivity 
except 
the electric field must be bounded. 

\vspace*{1em}

In fact, we expect that 
the excitation beyond the Fermi energy 
can be negligible under the weak electric field. 
%
In another words, 
we consider the adiabatic approximation 
of $j_{\omega,\tau}$ to prove Theorem 1.1. 
%
To this end, 
we consider the adiabatic evolution operator 
$U^A_{\omega,\tau}(s)$ 
which satisfies: 
%
%%%%% adiabatic %%%%%
%
\begin{eqnarray}
i\frac{d}{ds}U^A_{\omega,\tau}(s)\varphi
&=&
\tau H^A_{\omega}(s)U^A_{\omega,\tau}(s)\varphi, 
\quad
s\in(0,1), 
\nonumber \\
U^A_{\omega,\tau}(0)\varphi &=& \varphi, 
\quad
\varphi\in l^2({\bf Z}^d), 
\label{adiabatic}
\end{eqnarray}
%
%
where $H^A_{\omega}(s):=
H_{\omega}(s) + \frac{i}{\tau}
[\dot{P_{\omega}}(s), P_{\omega}(s)]$ 
(which is bounded and self-adjoint on $l^2 ({\bf Z}^d)$). 
%
$\dot{P_{\omega}}(s)$ is 
the derivative of $P_{\omega}(s)$ 
in the operator norm topology. 
%
In fact, 
$\dot{P_{\omega}}(s)=
iV(s)[g(x,s), P_{\omega}]V^*(s)$. 
%
Since 
$U^A_{\omega,\tau}(s)$ is expected to approach to  
$U_{\omega,\tau}(s)$ for sufficiently large $\tau 
^{\;(6, \, 12, \, 13)}$, 
we consider the adiabatic approximation of 
$j_{\omega,\tau}$ as: 
%
%%%%% jA %%%%%
%
\begin{equation}
j^A_{\omega,\tau}:=
\int_0^1 \,ds\,
{\cal T}
(U^{A*}_{\omega,\tau}(s)
i\tau [H_{\omega}^A(s), x]
U^{A}_{\omega,\tau}(s)
P_{\omega}).
\label{jA}
\end{equation}
%
%
Our theorem 
is the corollary 
of following two propositions. 
%

\vspace*{1em}

%%%%% Proposition 1.2 %%%%%
{\bf Proposition 1.2.}

{\it 
For sufficiently large $\lambda>0$, 
%
$
j_{\omega,\tau}=j^A_{\omega,\tau}+o(1), 
$ 
% 
as  $\tau \rightarrow \infty$, 
for ${\bf P}$-a.e. $\omega$.
}

\vspace*{1em}

%%%%% Proposition 1.3 %%%%%
{\bf Proposition 1.3.}

{\it 
For sufficiently large $\lambda>0$, 
%
%
$
j^A_{\omega,\tau}=0,
$ 
% 
for ${\bf P}$-a.e. $\omega$.
}

\vspace*{1em}

%
The rest 
of this paper 
is organized as follows. 
%
In section 2, 
we study some properties 
of ${\cal T}$, 
derive rapid decay of 
$U_{\omega,\tau}(s)$ and 
$U^A_{\omega,\tau}(s)$, 
and prove the well-definedness of $j_{\omega,\tau}$ 
and $j^A_{\omega,\tau}$. 
%
In section 3, 
we prove Proposition 1.2 and 1.3. 
%
Proposition 1.3 
follows easily from 
the intertwining property of 
$U^A_{\omega,\tau}(s)$$^{(6)}$ 
and 
the time reversal invariance of 
$H_{\omega}$. 
%
The proof 
of Proposition 1.2 needs 
a sort of adiabatic theorem. 
%
Adiabatic 
theorem is proved in 
(refs. 6, 12, 13) 
in an abstract setting 
when $\epsilon_F$ lies 
in the spectral gap. 
%
We follow 
their argument 
to prove our adiabatic theorem. 
%
Then 
we will have an additional term 
which originates from 
the fact that 
there is no spectral gap in our case. 
%
We 
estimate this term 
by exponential decay of 
the fractional moment of 
Green's function
$^{(1, \,2, \,3)}$. 
%
%
%%%%% SECTION 2 %%%%% Preliminaries %%%%%
%

\section{PRELIMINARIES}
\hspace*{3ex}

For an operator $A_{\omega}$ on $l^2 ({\bf Z}^d)$, 
we write 
$<x | \; A_{\omega} \; | y> := 
( \delta_x, \;A_{\omega} \delta_y )$, 
$x$, $y\in{\bf Z}^d$, 
%
where $( \cdot\; , \; \cdot )$ is the inner product on 
$l^2 ({\bf Z}^d)$, 
and $\delta_x (z) \in l^2 ({\bf Z}^d)$ is 
defined as: 
$\delta_x (z) = 1$ (if $z=x$), 
$\delta_x (z) = 0$ (otherwise). 
%
We call 
$<x | \; A_{\omega} \; |y>$ the integral kernel of $A_{\omega}$. 
%
%
We say 
an operator $A_{\omega}$ satisfies covariance relation 
((CR) in short), 
if $A_{\omega}$ obeys 
%
% 
%%%%% CR %%%%%
\begin{equation}
U(a) A_{\omega} U^*(a) = A_{T^a \omega}, 
\label{CR}
\end{equation}
%
%
whenever $a\in \{ x\in {\bf Z}^d : 
x^{(1)}=kL, \, k\in{\bf Z} \}$. 
%
$L$ was 
defined in section 1 to be 
the period of $g(x^{(1)},s)$. 
%
We introduce 
some classes of 
families of 
operators on $l^2 ({\bf Z}^d)$:
%
%
%%%%% %%%%%
\begin{eqnarray*}
{\cal S}_{\alpha}:=
\{
&A_{\omega}&{\rm  an \; operator \; on \; } l^2({\bf Z}^d):\\
%
&(1)& A_{\omega} \; {\rm satisfies \; (CR)}, \; 
\\
%
&(2)& \forall\beta>0, \; \exists C_{\alpha\beta}>0 
\; {\rm s.t.} \; 
{\bf E}|<x| \; A_{\omega} \; |y>|^{\alpha}
\leq C_{\alpha \beta}<x-y>^{-\beta}
\}, 
\\
%
{\cal S}:=&\cup_{\alpha>1}&{\cal S}_{\alpha}, 
\quad
\tilde{{\cal S}}:=\cap_{\alpha>1}{\cal S}_{\alpha},
\end{eqnarray*}
%
where 
$<x>:=(1 + |x|^2)^{1/2}$. 
%
${\bf E}$ stands 
for the expectation value 
with respect to ${\bf P}$. 
%
We set 
$
\| A \|_{{\cal S}_{\alpha}, \beta}
:= 
(
\sup_{x,y\in{\bf Z}^d}
<x-y>^{\beta}
{\bf E}
|<x| \; 
A_{\omega}
\; |y>|^{\alpha}
)^{1/\alpha}
$. 
%

%
%%%%% Lemma 2.1 %%%%%
%
%

\vspace*{1em}

{\bf Lemma 2.1.} 

{\it
The followings 
hold for ${\bf P}-a.e. \omega$. 

(1) 
Let $A_i \in {\cal S}_{\alpha_i}$, 
$i=1,\cdots, n$ such that 
$\sum_{i=1}^n \alpha_i^{-1} = 1$. 
%
%
Then, 
for $\beta>1$,  
%
%
%%%%% %%%%%
\begin{equation}
|{\cal T}(\prod_{i=1}^n A_i)|
\leq
C_n
\prod_{i=1}^n 
\| A_i \|_{{\cal S}_{\alpha_i}, \beta\alpha_i},
\label{Hoelder}
\end{equation}
%
%
where $C_n>0$ is an universal constant. 

(2) 
Let $A_i\in {\cal S}_{\alpha_i}$, 
$i=1$, $2$, such that 
$\sum_{i=1}^2 \alpha^{-1}_i=1$. 
Then, 
%
%
%%%%% EXCHANGE %%%%%
\begin{equation}
{\cal T}(A_1 A_2) = 
{\cal T}(A_2 A_1). 
\label{exchange}
\end{equation}
%
%
}

{\bf {\it Remark.} }

$A_{\omega}\in {\cal S}_{\alpha}$ 
is not necessarily bounded 
on $l^2({\bf Z}^d)$ 
for fixed $\omega\in \Omega$. 
%
In this case, 
we define the product of two operators 
$A$, $B\in {\cal S}_{\alpha}$ by their integral kernels:
%
%%%%% kernel %%%%%
%
\begin{equation}
<x | \; AB \; |y> := 
\sum_{z\in{\bf Z}^d}
<x | \; A \; |z> <z | \; B \; |y>, 
\label{kernel}
\end{equation}
%
%
whenever it exists and finite. 
%
In all cases in this paper, 
we can prove that for all $x$, $y \in {\bf Z}^d$, 
the above expression converges absolutely and 
$<x|\;AB\;|y>$ is finite for ${\bf P}$-a.e. $\omega$. 
%
Therefore, 
$A_{\omega}\in {\cal S}_{\alpha}$ 
always has a meaning 
as a function on $\Omega\times{\bf Z}^d\times{\bf Z}^d$. 
%

\vspace*{1em}

{\bf {\it proof.} }

(1)
%
By definition, 
%
%%%% %%%%%
\begin{eqnarray*}
<x| \; 
\prod_{i=1}^n A_i 
\; |x>
&=&
\sum_{z_1, z_2, \cdots, z_{n-1}\in{\bf Z}^d}
<x| \; A_1 \; |z_1>
<z_1| \; A_2 \; |z_2>
\\
%
&&\quad
\cdots
<z_{n-2}| \; A_{n-1} \; |z_{n-1}>
<z_{n-1}| \; A_n \; |x>
\end{eqnarray*}
%
%
We take 
expectation on both sides 
and use Fubini's Theorem and H\"{o}lder's inequality 
with respect to ${\bf P}$:
%
%
%%%%%% %%%%%
\begin{eqnarray*}
&&{\bf E}
|<x|  
\prod_{i=1}^n A_i 
\; |x>|
\\
%
&\leq&
{\bf E}
\sum_{z_1, z_2, \cdots, z_{n-1}\in{\bf Z}^d}
|<x| \; A_1 \; |z_1>|
|<z_1| \; A_2 \; |z_2>|
\\
%
&&\quad\quad\cdots
|<z_{n-2}| \; A_{n-1} \; |z_{n-1}>|
|<z_{n-1}| \; A_{n} \; |x>|
\\
%
&\leq&
\sum_{z_1, z_2, \cdots, z_{n-1}\in{\bf Z}^d}
({\bf E}|<x| \; A_1 \; |z_1>|^{\alpha_1})^{1/\alpha_1}
({\bf E}|<z_1| \; A_2 \; |z_2>|^{\alpha_2})^{1/\alpha_2}
\\
%
&&\quad\quad\cdots
({\bf E}|<z_{n-2}| \; A_{n-1} 
\; |z_{n-1}>|^{\alpha_{n-1}})^{1/\alpha_{n-1}}
({\bf E}|<z_{n-1}| \; A_n 
\; |x>|^{\alpha_n})^{1/\alpha_n}
\\
%
&\leq&
\sum_{z_1, z_2, \cdots, z_{n-1}\in{\bf Z}^d}
\| A_1 \|_{{\cal S}_{\alpha_1},\beta\alpha_1}
<x-z_1>^{-\beta}
\| A_2 \|_{{\cal S}_{\alpha_2},\beta\alpha_2}
<z_1-z_2>^{-\beta}
\\
%
&&\quad\cdots
\| A_{n-1} \|_{{\cal S}_{\alpha_{n-1}},\beta\alpha_{n-1}}
<z_{n-2}-z_{n-1}>^{-\beta}
\| A_n \|_{{\cal S}_{\alpha_n},\beta\alpha_n}
<z_{n-1}-x>^{-\beta}
\\
%
&\leq&
C_n
\prod_{i=1}^n
\| A_i \|_{{\cal S}_{\alpha_i},\beta \alpha_i}.
\end{eqnarray*}
%
%
Thus, 
it suffices to show 
the following equation 
which follows from Birkhoff's ergodic theorem:
%
%
%%%%% prodA %%%%%
\begin{equation}
{\cal T}
(\prod_{i=1}^n A_i)
=
\frac{1}{L}
\sum_{x^{(1)}=1}^{L}
{\bf E}
<x| \; 
\prod_{i=1}^n A_i 
\; |x>, 
\quad a.e. \; \omega, 
\label{prodA}
\end{equation}
%
%
where 
$x^{(i)}=0$, $i=2, \cdots, d$. 

(2)
We write 
$x\in {\bf Z}^d$ as 
$x=(p, q)$, 
$p\in {\bf Z}$, 
$q\in {\bf Z}^{d-1}$. 
%
From (\ref{prodA}), 
%
\begin{eqnarray}
{\cal T}(A_1 A_2)&=&
\frac{1}{L}
\sum_{p=1}^{L}
{\bf E}
\sum_{ k\in{\bf Z}, r\in {\bf Z}^{d-1} }
\sum_{q=1}^{L}
\nonumber
\\
%
&&<p, 0 \,|\, A_{1, \omega} \,|\, kL+q, r>
<kL+q, r \,|\, A_{2, \omega} \,|\, p, 0>,
\label{AA}
\end{eqnarray}
%
for a.e. $\omega$. 
%
Since 
$A_{1,\omega}$, $A_{2,\omega}$ 
satisfy (CR), we have 
%
%
\begin{eqnarray}
<p, 0 \,|\, A_{1, \omega} \,|\, kL+q, r> 
= 
<-kL+p, -r \,|\, A_{1, T^{-k\vec{L}-\vec{r}}\omega } \,|\, q, 0>, 
\label{covA} \\
<kL+q, r \,|\, A_{2, \omega} \,|\, p, 0> 
= 
< q, 0 \,|\, A_{2, T^{-k\vec{L}-\vec{r}}\omega } \,|\, -kL+p, -r>, 
\label{covAA}
\end{eqnarray}
%
where 
$\vec{L}:=(L, 0)\in {\bf Z}^d$, 
$\vec{r}:=(0, r)\in {\bf Z}^d$. 
%

We substitute 
(\ref{covA}) and (\ref{covAA}) into (\ref{AA}), and 
use Fubini's theorem (which is permitted by Lemma 2.1 (1)). 
%
Therefore, 
%
%
\begin{eqnarray*}
&{\cal T}&(A_1 A_2) \\
&=&
\frac{1}{L}
\sum_{p=1}^{L}
\sum_{ k\in {\bf Z}, r\in {\bf Z}^{d-1} }
\sum_{q=1}^{L}
{\bf E}
\\
%
&&\quad
<q, 0 \,|\, A_{2, T^{-k\vec{L}-\vec{r}}\omega } \,|\, -kL+p, -r>
<-kL+p, -r \,|\, A_{1, T^{-k\vec{L}-\vec{r}}\omega } \,|\, q, 0>
\\
&=&
\frac{1}{L}
\sum_{q=1}^{L}
{\bf E}
\sum_{ k\in {\bf Z}, r\in{\bf Z}^{d-1} }
\sum_{p=1}^{L}
<q, 0 \,|\, A_{2, \omega } \,|\, kL+p, r>
<kL+p, r \,|\, A_{1, \omega } \,|\, q, 0>
 \\
&=&
{\cal T}(A_2 A_1).
\end{eqnarray*}
%
%
%
$\Box$

\vspace*{1em}

The following 
lemma implies that electrons are 
still localized even under the electric field, 
and 
it is here that 
we have to restrict our analysis 
to bounded electric potential.
%
%
%
%%%%% Lemma 2.2 %%%%%
%

\vspace*{1em}

{\bf Lemma 2.2.}

{\it
For arbitrary 
$\alpha>0$, there exists a constant 
$C_{\alpha}>0$ such that 
%
%%%%% UAdecay %%%%%
%
\begin{eqnarray}
{\bf E}|<x| \; U_{\omega,\tau}(s) \; |y>|
\leq
C_{\alpha}<x-y>^{-\alpha},
\label{Udecay} \\
%
%
{\bf E}|<x| \; U^A_{\omega,\tau}(s) \; |y>|
\leq
C_{\alpha}<x-y>^{-\alpha},
\label{UAdecay}
\end{eqnarray}
%
%
for $\lambda>0$ sufficiently large, 
where $C_{\alpha}$ is independent of 
$x$, $y\in{\bf Z}^d$, 
$s\in[0,1]$, and $\tau>0$. 
}

\vspace*{1em}

{\bf {\it proof.} }

(1) 
At first, we prove (\ref{Udecay}), (\ref{UAdecay}) 
when $d=1$. 

Let 
$\bar{U}_{\omega,\tau}(s)$,  
$\bar{U}_{\omega,\tau}^A(s)$ 
be the unitary evolution operators 
of time-dependent Schr\"{o}dinger equations:
%
%
\begin{eqnarray}
i \frac{d}{ds}
\bar{U}_{\omega,\tau}(s)
\varphi
&=&
\tau
\big(H_{\omega} 
+ \frac{1}{\tau}g(x,s)\big)
\bar{U}_{\omega,\tau}(s)
\varphi, 
\nonumber\\
%
\bar{U}_{\omega,\tau}(0)\varphi
&=&
\varphi, 
%\quad
%s\in (0,1), 
%\quad
\label{Uelec}
\\
%%
i \frac{d}{ds}
\bar{U}_{\omega,\tau}^A(s)
\varphi
&=&
\tau
\Big(H_{\omega} 
+ 
\frac{1}{\tau}g(x,s) 
\nonumber \\
%
&&\qquad
+
\frac{1}{\tau}
[[P_{\omega}, g(x,s)], P_{\omega}]\Big)
\bar{U}_{\omega,\tau}^A(s)
\varphi, 
\nonumber\\
%
\bar{U}_{\omega,\tau}^A(0)\varphi
&=&
\varphi, 
\quad
s\in (0,1), 
\quad
\varphi\in l^2({\bf Z}). 
\label{UAelec}
\end{eqnarray}
%
%
It follows that$^{\,(9, \, Chapter\,7)}$
%
%
\[
U_{\omega,\tau}(s)
=
V(s) \bar{U}_{\omega,\tau}(s), 
\quad
U_{\omega,\tau}^A(s)
=
V(s) \bar{U}_{\omega,\tau}^A(s).
\]
%
%
Therefore 
(\ref{Udecay}), (\ref{UAdecay}) 
are equivalent to the rapid decay of 
$\bar{U}_{\omega,\tau}(s)$, 
$\bar{U}_{\omega,\tau}^A(s)$ repectively.
%
%
We will 
only show the decay of $\bar{U}_{\omega,\tau}^A(s)$
%
(that is, 
$
{\bf E}
|<x|
\bar{U}_{\omega,\tau}^A(s)
|y>|
\leq
C_{\alpha}
<x-y>^{-\alpha}
$), 
because
the proof of the decay of 
$\bar{U}_{\omega,\tau}(s)$ is similar.  
%

When $x=y$, 
this is obvious because 
$\bar{U}^A_{\omega,\tau}(s)$ is bounded on $l^2 ({\bf Z})$,  
so that 
we will show 
%
%
\[
{\bf E} |<x |\; \bar{U}^A_{\omega,\tau}(s) \;|y>|
\leq C_{\alpha}
|x-y|^{-\alpha},
\]
%
%
when $x \ne y$. 
%
 $\bar{U}^A_{\omega,\tau}(s)$ can be written 
by Dyson expansion which converges in operator norm:
\[
\bar{U}^A_{\omega,\tau}(s)
=
\sum_{j=0}^{\infty} T_j(s),
\]
where $T_0(s):=e^{-i\tau s H_{\omega}}$, 
and for $j\geq 1$, 
%
%
%%%%% Dyson %%%%%
%
\begin{eqnarray}
T_j(s)&:=&
(-i)^j
\int_{0\leq\sum_{i=1}^j s_i\leq s}
ds_1 ds_2 \cdots ds_j
\nonumber\\
%
&&
U^0_{\omega,\tau}(s-\sum_{i=1}^j s_i)
X_{\omega}(\sum_{i=1}^j s_i)
U^0_{\omega,\tau}(s_1)
X_{\omega}(\sum_{i=2}^j s_i)
U^0_{\omega,\tau}(s_2)
X_{\omega}(\sum_{i=3}^j s_i)
\nonumber\\
%
&&
\cdots
X_{\omega}(s_{j-1}+s_j)
U^0_{\omega,\tau}(s_{j-1})
X_{\omega}(s_j)
U^0_{\omega,\tau}(s_j).
\label{Dyson}
\end{eqnarray}
%
%
In (\ref{Dyson}), 
we wrote 
$U^0_{\omega,\tau}(s)
:=
e^{-i \tau s H_{\omega}}$, 
and 
$X_{\omega}(s)
:=
g(x,s)
+
[[P_{\omega},g(x,s)], P_{\omega}]$. 
%
We will show that, 
$\sum_{j=0}^{\infty}
{\bf E}|<x | \; T_j(s) \; |y>|$ 
converges and 
has the rapid decay 
for sufficiently small $s>0$, 
and 
that how small $s$ should be 
does not depend on $\alpha$.  
%
Then, 
the conclusion follows by the semigroup property 
of $\bar{U}^A_{\omega,\tau}(s)$ 
(
we have to 
consider ${\bar U}^A_{\omega,\tau}(t,s)$ 
which is defined 
similarly to (\ref{UAelec}) 
except 
${\bar U}^A_{\omega,\tau}(0)\varphi=\varphi$ 
is replaced by 
${\bar U}^A_{\omega,\tau}(t,t)\varphi=\varphi$ 
) 
and the following simple fact:
%
let $A_{\omega}$, $B_{\omega}$ 
be bounded operators on $l^2({\bf Z})$ 
which satisfy ($\alpha >2$) 
%
\begin{eqnarray*}
{\bf E} |<x|\,A_{\omega}\,|y>| \,
&\leq& 
<x-y>^{-\alpha}, 
\\
{\bf E} |<x|\,B_{\omega}\,|y>| \,
&\leq& 
<x-y>^{-\alpha}, 
\quad x,\,y\in {\bf Z}, 
\end{eqnarray*}
%
Then, 
the operator $C_{\omega} := A_{\omega}B_{\omega}$ 
satisfies 
%
\[
{\bf E} |<x|\,C_{\omega}\,|y>| 
\leq  C <x-y>^{-\alpha/2}, 
\]
%
for some constant $C>0$.
%

The integral kernel 
of $<x | \; T_j \; |y>$ can be written as:
%
%
%%%%% Tj %%%%%
%
\begin{eqnarray}
<&x& |\; T_j \;|y>=(-i)^j
\int_{0\leq\sum_{i=1}^j s_i\leq s}
ds_1 ds_2 \cdots ds_j
\nonumber\\
%
&&\sum_{z_1,\cdots,z_j}
<x| 
\; U^0_{\omega,\tau}(s-\sum_{i=1}^j s_i)
X_{\omega}(\sum_{i=1}^j s_i) 
\; |z_1>
<z_1 | \; 
U^0_{\omega,\tau}(s_1)X_{\omega}(\sum_{i=2}^j s_i) 
\; |z_2>
\nonumber\\
%
&&\quad\quad\cdots
<z_{j-1} | \; 
U^0_{\omega,\tau}(s_{j-1})
X_{\omega}(s_j)
\; |z_j>
<z_j | \; 
U^0_{\omega,\tau}(s_j)
\; |y>. 
\label{Tj}
\end{eqnarray}
%
%
In order to use 
Fubini's theorem, we take expectation 
at first and consider:
%
%
%%%%% Tjtilde %%%%%
%
\begin{eqnarray}
\tilde{T}_j:= 
&&\sum_{z_1,z_2,\cdots,z_j\in{\bf Z}}
{\bf E}
|<x| 
\; U^0_{\omega,\tau}(s-\sum_{i=1}^j s_i)
X_{\omega}(\sum_{i=1}^j s_i) 
\; |z_1>|
\nonumber\\
%
&&
|<z_1 | \; 
U^0_{\omega,\tau}(s_1)X_{\omega}(\sum_{i=2}^j s_i) 
\; |z_2>|
\nonumber\\
%
&&
\cdots
|<z_{j-1} | \; 
U^0_{\omega,\tau}(s_{j-1})
X_{\omega}(s_j)
\; |z_j>|
|<z_j | \; 
U^0_{\omega,\tau}(s_j)
\; |y>|.
\label{Tjtilde}
\end{eqnarray}
%
By H\"{o}lder's inequality, 
%
%
%%%%% %%%%%
%
\begin{eqnarray*}
&&\tilde{T}_j
\leq
\sum_{z_1,z_2,\cdots,z_j\in{\bf Z}}
({\bf E}|<x| \; 
U^0_{\omega,\tau}(s-\sum_{i=1}^j s_i)
X_{\omega}(\sum_{i=1}^j s_i)
\; |z_1>|^{j+1})^{\frac{1}{j+1}}
\nonumber\\
%
&&({\bf E}|<z_1| \; 
U^0_{\omega,\tau}(s_1)
X_{\omega}(\sum_{i=2}^j s_i)
\; |z_2>|^{j+1})^{\frac{1}{j+1}}
\cdots
\\
%
&&
({\bf E}|<z_{j-1}| \; 
U^0_{\omega,\tau}(s_{j-1})
X_{\omega}(s_j)
\; |z_j>|^{j+1})^{\frac{1}{j+1}}
%
({\bf E}|<z_{j}| \; 
U^0_{\omega,\tau}(s_{j})
\; |y>|^{j+1})^{\frac{1}{j+1}}.
\end{eqnarray*}
%
We use 
the results in (ref. 1, 2):
%
%%%%% U0decay and Pdecay %%%%%
%
\begin{eqnarray}
{\bf E}|<x|\; U^0_{\omega,\tau}(s) \;|y>|
\leq
C_1
e^{-\mu |x-y|}, 
\label{U0decay}
\\
%
{\bf E} |<x |\; P_{\omega} \; |y>|
\leq
C_1 e^{-\mu|x-y|}, 
\label{Pdecay}
\end{eqnarray}
%
%
for sufficiently large $\lambda>0$, 
where constants $C_1 >0$ and $\mu>0$ 
are independent of 
$x$, $y\in{\bf Z}$, 
$s\in[0, 1]$, 
and $\tau>0$. 
%
We used 
the fact that, in this case, 
$H_{\omega}$ has only pure point spectrum
$^{(9, \, 20)}$. 
%
%
By (\ref{U0decay}), 
(\ref{Pdecay}), H\"{o}lder's inequality, 
and the boundedness of $U^0_{\omega,\tau}(s)$ and $X_{\omega}(s)$, 
we can deduce that, 
when $\lambda>0$ is sufficiently large, 
%
%
\begin{equation}
{\bf E}
|<x| \; 
U^0_{\omega,\tau}(s)X_{\omega}(t)
\; |y>|
\leq
C_2 e^{-\mu |x-y|}
\label{UX}
\end{equation}
% 
%
for some constant $C_2>0$ 
which is also independent of 
$x$, $y\in{\bf Z}$, 
$s$, $t\in[0,1]$, and $\tau>0$. 
%
Moreover, 
since 
$U^0_{\omega,\tau}(s)$ and $X_{\omega}(t)$ 
are bounded, 
%
%
\[
{\bf E}
|<x| \; 
U^0_{\omega,\tau}(s-\sum_{i=1}^j s_i)
X_{\omega}(\sum_{i=1}^j s_i)
| \; z_1>|^{j+1}
\leq
C^j_3 C_2 e^{-\mu |x-z_1|},
\]
%
for some constant $C_3>0$. 
%
By estimating 
the other terms in the same way, 
%
%
%%%%% %%%%%
%
\begin{eqnarray*}
\tilde{T}_j
\leq
&\sum_{z_1,z_2,\cdots,z_j\in{\bf Z}}&
C_3^{\frac{j}{j+1}}C_2^{\frac{1}{j+1}}
e^{-\frac{\mu}{j+1}|x-z_1|}
%
C_3^{\frac{j}{j+1}}C_2^{\frac{1}{j+1}}
e^{-\frac{\mu}{j+1}|z_1-z_2|}
\nonumber\\
%
&&
\cdots
%
C_3^{\frac{j}{j+1}}C_2^{\frac{1}{j+1}}
e^{-\frac{\mu}{j+1}|z_{j-1}-z_j|}
%
C_1^{\frac{1}{j+1}}
e^{-\frac{\mu}{j+1}|z_{j}-y|}.
\end{eqnarray*}
%
%
We notice that, 
for $i=1$, $2$, $3$, 
$C_i^{\frac{1}{j+1}}\leq 1$, 
$C_i^{\frac{j}{j+1}}\leq 1$ 
if $C_i \leq 1$, 
and 
%
$C_i^{\frac{1}{j+1}}\leq C_i$, 
$C_i^{\frac{j}{j+1}}\leq C_i$ 
if $C_i > 1$. 
%
Therefore, 
by taking 
$C_4 := 
\max(1, C_1, C_2, C_3)$, 
%
%
%%%%% %%%%%
%
\begin{eqnarray*}
\tilde{T_j}
&\leq&
\sum_{z_1,z_2,\cdots,z_j\in{\bf Z}}
C_4^{2j+1}
e^{-\frac{\mu}{j+1}
(|x-z_1|+|z_1-z_2|+\cdots+|z_{j-1}-z_j|
+|z_j-y|)}
\nonumber\\
%
&\leq&
C_4^{2j+1}
\sum_{z_1,z_2,\cdots,z_j\in{\bf Z}}
e^{-(\frac{\mu-\epsilon}{j+1})|x-y|}
e^{-\frac{\epsilon}{j+1}
(|x-z_1|+|z_1-z_2|+\cdots+|z_{j-1}-z_j|
+|z_j-y|)}. 
\end{eqnarray*}
%
%
For $\epsilon>0$ small, 
$\sum_{x\in{\bf Z}}e^{-\epsilon |x|}
\leq \frac{4}{\epsilon}$. 
%
We ignore 
$e^{-\frac{\epsilon}{j+1}|z_j-y|}$, and 
take the sum in the order of 
$z_j$, $z_{j-1}$, $\cdots$, $z_2$, $z_1$. 
%
The conclusion is 
%
%
%%%%% %%%%%
\begin{eqnarray*}
\tilde{T}_j
&\leq&
C_4^{2j+1}
(\sum_{z\in{\bf Z}}e^{-\frac{\epsilon}{j+1}|z|})^j
e^{-\frac{\mu-\epsilon}{j+1}|x-y|}
\\
%
&\leq&
C_4^{2j+1}
\{
\frac{4}{\epsilon}(j+1)
\}^j
e^{-\frac{\mu-\epsilon}{j+1}|x-y|}.
\end{eqnarray*}
%
%
In general, 
for $\mu>0$, $\alpha>0$, 
$e^{-\mu x} \leq (\frac{\alpha}{\mu})^{\alpha}
e^{-\alpha}x^{-\alpha}$. 
%
Therefore, 
for arbitrary $\alpha>0$, 
%
%
%%%%% %%%%%
\[
\tilde{T}_j
\leq
C_4^{2j+1}
(\frac{4}{\epsilon}(j+1))^j
(\frac{j+1}{\mu-\epsilon}\alpha)^{\alpha}
e^{-\alpha}|x-y|^{-\alpha}.
\]
%
%
We return 
to (\ref{Tj}). 
%
We interchange 
${\bf E}$ with $\sum_{z_1,z_2,\cdots,z_j\in{\bf Z}}$ 
and compute
%
%
%%%%% %%%%%
\begin{eqnarray*}
{\bf E}|<x| & T_j & |y>|
\\
%
&\leq&
\int_{0\leq\sum_{i=1}^j s_i \leq s }
ds_1 ds_2 \cdots ds_j 
\;
C_4^{2j+1}
(\frac{4}{\epsilon}(j+1))^j
(\frac{j+1}{\mu-\epsilon}\alpha)^{\alpha}
e^{-\alpha}|x-y|^{-\alpha}
\\
%
&\leq&
\frac{s^j}{j!}
C_4^{2j+1}
(\frac{4}{\epsilon}(j+1))^j
(\frac{j+1}{\mu-\epsilon}\alpha)^{\alpha}
e^{-\alpha}|x-y|^{-\alpha}. 
\end{eqnarray*}
%
%
By taking 
%
$C_5:= 
(\frac{\alpha}{\mu-\epsilon})^{\alpha}
e^{-\alpha}C_4$, 
and 
$C_6:= 
4C_4^2/\epsilon$, 
%
%
%%%%% %%%%%
\[
{\bf E}|<x| \; T_j \; |y>|
\leq
\frac{s^j}{(j+1)!}
(j+1)^{\alpha}
C_6^j C_5 (j+1)^{j+1}
|x-y|^{-\alpha}.
\]
%
%
By Stirling formula, 
for some constants $C_7$, $C_8>0$, 
%
%
%%%%% %%%%%
\[
(C_7 n)^n 
\leq n!
\leq (C_8 n)^n.
\]
%
%
Therefore, 
%
%
%%%%% %%%%%
\begin{eqnarray*}
{\bf E}|<x| \; T_j \; |y>|
&\leq&
(j+1)^{\alpha} C_6^j s^j C_7^{-(j+1)} 
C_5 |x-y|^{-\alpha}
\\
%
&\leq&
(j+1)^{\alpha} 
(\frac{C_6}{C_7})^j \frac{C_5}{C_7} s^j |x-y|^{-\alpha}. 
\end{eqnarray*}
%
%
Hence, for sufficiently small $s>0$, 
there exists a constant $C_{\alpha}>0$ 
such that 
%
%
%%%%% %%%%%
\[
\sum_{j=1}^{\infty}
{\bf E}|<x| \; T_j \; |y>|
\leq
C_{\alpha}|x-y|^{-\alpha}. 
\]
%
%
By Fubini's theorem, 
%
%
%%%%% %%%%%
\begin{eqnarray*}
{\bf E}
|<x| \; \bar{U}^A_{\omega,\tau}(s) \; |y>|
%
&=&
{\bf E}
|<x| \; \sum_{j=0}^{\infty}T_j \; |y>|
\\
%
&\leq&
{\bf E}
\sum_{j=0}^{\infty}
|<x| \; T_j \; |y>|
\\
%
&=&
\sum_{j=0}^{\infty}
{\bf E}
|<x| \; T_j \; |y>|
\\
%
&\leq&
C_{\alpha}|x-y|^{-\alpha}.
\end{eqnarray*}
%

(2)
When $d>1$, 
we write $j$ as $j=dk + l$, 
$k\in {\bf N}$, 
$l=0, 1, \cdots, d-1$, 
and write the sum 
$\sum_{j=1}^{\infty}$ as 
$\sum_{j=1}^{\infty}
=
\sum_{j\equiv l, mod \, d}
\sum_{k=1}^{\infty}$.
%
Then, it is sufficient to use the following 
estimates instead of (\ref{UX}):
%
\begin{eqnarray}
&{\bf E}&
|<x|
\overbrace{
(U^0 X)(U^0 X) \cdots (U^0 X)
}^{d}
|y>|
\leq
Ce^{-\mu |x-y|}, 
\label{UXUX}
\\
%
&{\bf E}&
|<x|
\overbrace{
(U^0 X)(U^0 X) \cdots (U^0 X)
}^{l (<d)}
(U_0 X U_0 X)
|y>|
\leq
Ce^{-\mu |x-y|}, 
\label{UXUXU}
%
\end{eqnarray}
%
%
(we omit the $s_i$-dependence). 
%
The rest is similar. 
%
$\Box$

\vspace*{1em}

%
%
%%%%% Lemma 2.3 %%%%%
{\bf Lemma 2.3.}

{\it 
For sufficiently large $\lambda>0$, 
$U_{\omega,\tau}(s)$, 
$U^A_{\omega,\tau}(s)$ and 
$P_{\omega}(s)
\in \tilde{{\cal S}}$. 
}

\vspace*{1em}

{\bf {\it proof.} }

It is 
straightforward to see 
$U_{\omega,\tau}(s)$, $U^A_{\omega,\tau}(s)$ 
and $P_{\omega}(s)$ satisfy (CR). 
%
Then, 
$P_{\omega}\in\tilde{ {\cal S} }$ 
follows from (\ref{Pdecay}) 
and boundedness of $P_{\omega}$ on $l^2({\bf Z}^d)$. 
%
$U_{\omega,\tau}(s)$, 
$U^A_{\omega,\tau}(s) \in \tilde{{\cal S}}$ 
follows from 
Lemma 2.2 and their unitarity.
%
$\Box$

\vspace*{1em}

%
%
%%%%% Lemma 2.4 %%%%%

{\bf Lemma 2.4.}

{\it 
$j_{\omega,\tau}$, $j^A_{\omega,\tau}$ 
is well-defined for ${\bf P}$-a.e. $\omega$. 
}

\vspace*{1em}

{\bf {\it proof.} }

It follows easily  
from Lemma 2.1(1) and 2.3.
%
$\Box$

\vspace*{1em}

The following 
lemma is not necessary to prove our theorem. 
%
However, 
it gives us another point of view of the charge transport. 
%

\vspace*{1em}

%
%
%%%%% Lemma 2.5 %%%%%
{\bf Lemma 2.5.}

{\it
\begin{equation}
j_{\omega} = 
{\cal T}\left(
(U^*_{\omega}(1) x U_{\omega}(1) - x)P_{\omega}
\right), 
\;
 for \;
{\bf P}\!-\!a.e. \; \omega.
\label{new}
\end{equation}
%
%
}

\vspace*{1em}

{\bf {\it Remarks.} }

(1) 
In the RHS of 
(\ref{new}), 
$(U^*_{\omega}(1) x U_{\omega}(1) - x)$ 
is at first defined as a form on 
$C_0:= \{\varphi\in l^2({\bf Z}^d)
\; : \;
\varphi(x)=0$, 
 for  sufficiently  large  $|x| \,\}$, 
and then extended to an operator. 

(2) 
We can consider 
$(U^*_{\omega}(1) x U_{\omega}(1) - x)$ 
stands for the displacement of electrons 
from $t=0$ to $t=1$.
% 
Therefore, 
multiplying $P_{\omega}$ by this operator, 
and taking trace per unit volume 
corresponds to the measurement of the  
displacement of electrons below the Fermi energy. 

\vspace*{1em}

{\bf {\it proof.} }

At first, 
we note that 
%
%%%%% %%%%%
%
\begin{eqnarray*}
{\rm the \, RHS \, of} \, (\ref{new}) 
&=&
{\cal T}\left(
\int_0^1 \,dt\,
U^*_{\omega}(t)
i[H_{\omega}(t),x]
U_{\omega}(t)
P_{\omega}
\right) \\
&=&
\lim_{|\Lambda| \to \infty}
\frac{1}{|\Lambda|}
{\rm trace}
\left(
\chi_{\Lambda} 
\int_0^1 \,dt\,
U^*_{\omega}(t)
i[H_{\omega}(t),x]
U_{\omega}(t)
P_{\omega}
\chi_{\Lambda}
\right). 
\end{eqnarray*}
%
%
Thus, 
it suffices to show that 
we can interchange 
$\int_0^1 \,dt$ with 
$\lim_{|\Lambda | \to \infty}$ 
and $\int_0^1\,dt\,$ with trace. 
%
By Lemma 2.1 (2), 2.3,  
%
%%%%% %%%%%
%
\begin{eqnarray*}
{\cal T}
(
U^*_{\omega}(t)
i[&H_{\omega}(t)&,x]
U_{\omega}(t)P_{\omega}
)
\\
&=&
{\cal T}
\left(
P_{\omega}
U^*_{\omega}(t)
i[H_{\omega}(t),x]
U_{\omega}(t)P_{\omega}
\right)
\\
&=&
\lim_{|\Lambda|\rightarrow\infty}
\frac{1}{|\Lambda|}
{\rm trace}
\left(
\chi_{\Lambda}
P_{\omega}
U^*_{\omega}(t)
i[H_{\omega}(t),x]
U_{\omega}(t)P_{\omega}
\chi_{\Lambda}
\right), 
\end{eqnarray*}
%
%
for ${\bf P}$-a.e. $\omega$. 
%
$A_{\omega}:=
U^*_{\omega}(t)
i[H_{\omega}(t),x]
U_{\omega}(t)$ 
is bounded on $l^2 ({\bf Z}^d)$. 
%
Then, 
by decomposing $A_{\omega}$ 
into the linear combination of four unitrary 
operators$^{(\,17, \, Chapter\,6)}$, 
we can assume $A_{\omega}$ is unitary 
(we should note it still satisfies (CR)). 
%
By using 
$|{\rm trace}(AB)| \leq 
{\rm trace}(A A^*)^{1/2}
{\rm trace}(B^*B)^{1/2}$ 
(where $A$, $B$ belong to Hilbert-Schmidt class), 
%
%%%%% %%%%%
%
\begin{eqnarray*}
|\frac{1}{|\Lambda|}
&{\rm trace}&(\chi_{\Lambda}
P_{\omega}A_{\omega}P_{\omega}
\chi_{\Lambda})|
\\
&\leq &
\frac{1}{|\Lambda|}
{\rm trace}
(\chi_{\Lambda} P_{\omega}A_{\omega}A^*_{\omega}P_{\omega}
\chi_{\Lambda})^{1/2}
{\rm trace}
(\chi_{\Lambda}
P_{\omega}P_{\omega}\chi_{\Lambda})^{1/2}
\\
&=&
\frac{1}{|\Lambda|}
|{\rm trace}(\chi_{\Lambda}
P_{\omega}\chi_{\Lambda})|
\end{eqnarray*}
%
%
%
Therefore, 
%
%%%%% trace %%%%%
%
\begin{eqnarray}
|\frac{1}{|\Lambda|}
{\rm trace}(\chi_{\Lambda}
&P_{\omega}&U^*_{\omega}(t)
i[H_{\omega},x]U_{\omega}(t)P_{\omega}
\chi_{\Lambda})|
\nonumber\\
%
&\leq&
\| U^*_{\omega}(t)
i[H_{\omega},x]
U_{\omega}(t)\|_{op}
\frac{1}{|\Lambda|}
{\rm trace}(\chi_{\Lambda} P_{\omega} \chi_{\Lambda})
\nonumber\\
%
&\leq&
C\frac{1}{|\Lambda|}
{\rm trace}
( \chi_{\Lambda} P_{\omega} \chi_{\Lambda} ), 
\label{trace}
\end{eqnarray}
%
%
where the constant $C>0$ 
can be taken independent of $t\in[0,1]$. 
%
$\|\cdot\|_{op}$ is 
the operator norm on $l^2({\bf Z}^d)$. 
%
Since 
the RHS of (\ref{trace}) converges to 
$C\,{\cal T}(P_{\omega})$ for ${\bf P}$-a.e. $\omega$, 
the LHS of (\ref{trace})
 is uniformly bounded w.r.t. $t\in[0,1]$ 
for $|\Lambda|$ sufficiently large. 
%
Then, 
by the dominated convergence theorem, 
we can interchange 
$\int_0^1\,dt$ with 
$\displaystyle \lim_{|\Lambda| \rightarrow \infty}$. 
%
%
Next, 
%
%%%%% %%%%%
%
\begin{eqnarray*}
\int_0^1\,dt\,
&{\rm trace}&
(\chi_{\Lambda}
P_{\omega}
U^*_{\omega}(t)
i[H_{\omega}(t),x]
U_{\omega}(t)P_{\omega}
\chi_{\Lambda})
\\
%
&=&
\int_0^1 \,dt\,
\sum_{n=1}^N
(\varphi_n, 
(\chi_{\Lambda}
P_{\omega}
U^*_{\omega}(t)
i[H_{\omega}(t),x]
U_{\omega}(t)P_{\omega}
\chi_{\Lambda})\varphi_n), 
\end{eqnarray*}
%
%
where $\{\varphi_n(x)\}_{n=1}^{N}$ 
is the complete orthonormal basis of 
$l^2(\Lambda)$.  
%
Since 
$\sum_{n=1}^N$ is a finite sum, 
we can interchange 
$\int_0^1 dt$ with trace. 
$\Box$
%
%
%
%
%

%%%%%% SECTION 3 %%%%% Proof of Theorem %%%%%
\section{PROOF OF THEOREM 1.1.}
\hspace*{3ex}

%
We defer 
the proof of Proposition 1.2 
which is a little complicated, 
and prove Proposition 1.3 first.  
%

\vspace*{1em}

{\bf {\it proof of Proposition 1.3.} }

We decompose 
the integrand of (\ref{jA}) = I + II, 
where
%
%%%%% %%%%%
%
\begin{eqnarray*}
{\rm I}&=&i\tau
{\cal T}
(U^{A*}_{\omega,\tau}(s)
[H_{\omega}(s), x]U^A_{\omega,\tau}(s)P_{\omega}), 
\\
{\rm II}&=&-
{\cal T}
(U^{A*}_{\omega,\tau}(s)
[[\dot{P_{\omega}}(s),P_{\omega}(s)], x]
U^A_{\omega,\tau}(s)P_{\omega}). 
\end{eqnarray*}
%
%
Since 
$U^A_{\omega,\tau}(s)$ satisfies 
the following intertwining property$^{(6)}$:
%
%%%%% intertwine %%%%%
%
\begin{equation}
U^A_{\omega,\tau}(s)P_{\omega}
=
P_{\omega}(s)U^A_{\omega,\tau}(s),
\label{intertwine}
\end{equation}
%
%
We can simplify I:
%
%%%%% %%%%%
%
\begin{eqnarray*}
{\rm I} &=&
i\tau
{\cal T}
(U^{A*}_{\omega,\tau}(s)
[H_{\omega}(s), x]
P_{\omega}(s)U^{A}_{\omega,\tau}(s)) \\
%
&=&
i\tau
{\cal T}
(P_{\omega}(s)[H_{\omega}(s), x]
P_{\omega}(s)) \\
%
&=&
i\tau
{\cal T}
(V(s)P_{\omega}[H_{\omega},x]P_{\omega}V^*(s)) \\
%
&=&
i\tau
{\cal T}
(P_{\omega}[H_{\omega},x]P_{\omega}). 
\end{eqnarray*}
%
%
In the second 
and fourth equality, we used Lemma 2.1 (2). 
%
Thus, by Proposition 2 in (ref. 6), 
${\rm I}=0$ for ${\bf P}$-a.e. $\omega$. 
%
As for ${\rm II}$, 
by using intertwining property (\ref{intertwine}) 
and Lemma 2.1(2), 
%
%%%%% %%%%%
%
\begin{eqnarray*}
{\rm II}
&=&
-{\cal T}
(U^{A*}_{\omega,\tau}(s)
[[\dot{P_{\omega}}(s),P_{\omega}(s)],x]P_{\omega}(s)
U^A_{\omega,\tau}(s)) \\
%
&=&
-{\cal T}
(P_{\omega}(s)
[[\dot{P_{\omega}}(s),P_{\omega}(s)],x]P_{\omega}(s))\\
%
&=&
i{\cal T}
(P_{\omega}
[[[P_{\omega},g(x,s)],P_{\omega}],x]
P_{\omega})\\
%
&=&
i{\cal T}
(P_{\omega}
[[P_{\omega},g(x,s)], [P_{\omega},x]]
P_{\omega})\\
%
&=&
-2 {\rm Im }
{\cal T}
(P_{\omega}
[[P_{\omega},g(x,s)],[P_{\omega},x]]
P_{\omega}),
\end{eqnarray*}
%
%
Therefore 
${\rm II}=0$, because $<x | P_{\omega} | y>$ 
is real due to the time-reversal invariance. 
%
$\Box$

\vspace*{1em}

We shall 
compare $U_{\omega,\tau}(s)$ 
with $U^A_{\omega,\tau}(s)$, 
and prove a kind of adiabatic theorem (Lemma 3.2) 
which is necessary to prove Proposition 1.2. 
% 
Let 
$\Omega_{\omega,\tau}(s)
:=
U^{A*}_{\omega,\tau}(s)U_{\omega,\tau}(s)$. 
%
Then, 
it solves the following integral equation.
%
%
%%%%% integral equation %%%%%
\begin{equation}
\Omega_{\omega,\tau}(s)=I-
\int_0^s dt \;
U^{A*}_{\omega,\tau}(t)X_{\omega}(t)U^A_{\omega,\tau}(t)
\Omega_{\omega,\tau}(t), 
\label{integralequation}
\end{equation}
%
%
where $X_{\omega}(t):=[\dot{P}_{\omega}(t),P_{\omega}(t)]$
(it is different 
from what is defined in the proof of 
Lemma 2.2), 
and $I$ is the identity operator.  
%
We shall 
estimate the second term in the RHS of 
(\ref{integralequation}), 
and will show that 
it becomes small in certain sense 
when $\tau$ is large, 
basically 
along the argument in (refs. 6, 13). 
%
At first 
we introduce  
%
%%%%% Xtilde %%%%%
\begin{equation}
\tilde{X}_{\omega}(t):=
-\frac{1}{2\pi i}
\int_{\Gamma_{\delta}}dz
R_{\omega}(z,t)X_{\omega}(t)R_{\omega}(z,t), 
\label{Xtilde}
\end{equation}
%
%
where $R_{\omega}(z,t):=(H_{\omega}(t)-z)^{-1}$. 
%
$\Gamma_{\delta}:= 
\Gamma\setminus\tilde{\Gamma}_{\delta}$, 
where 
$\Gamma\subset{\bf C}$ is a 
rectangle whose vertices are located at 
%
$\epsilon_F+ia$, 
%
$-N+ia$, 
%
$-N-ia$, 
%
and 
$\epsilon_F-ia$ 
($N$ is taken sufficiently large 
such that $\inf\sigma(H_{\omega}) > -N$). 
%
And 
$\tilde{\Gamma}_{\delta}:= 
\{
\epsilon_F+ic : 
-\delta\leq c \leq \delta \}$, 
for some $\delta\in (0, a)$. 
%
Then, by direct computation, we have 
%
%
%%%%% %%%%%
\begin{eqnarray*}
[H_{\omega}(t), \tilde{X}_{\omega}(t)]
&=&
[X_{\omega}(t), \; P_{\omega}(t)]
\\
%
&&
+\frac{1}{2\pi i}
[
X_{\omega}(t), \; 
\int_{\tilde{\Gamma}_{\delta}}dz
R_{\omega}(z,t)]
, 
\end{eqnarray*}
%
where 
$\int_{ \tilde{\Gamma}_{\delta} } 
dz R_{\omega}(z,t)$ is defined to be 
%
%
\[
\int_{ \tilde{\Gamma}_{\delta} } 
dz R_{\omega}(z,t)
:=
{\rm s}\!-\!\lim_{\eta \downarrow 0}
(\int_{\epsilon_F-i\delta}^{\epsilon_F-i\eta}
+
\int^{\epsilon_F+i\delta}_{\epsilon_F+i\eta})
R_{\omega}(z,t) dz.
\]
%
%
We used the fact
$^{\;(9, \, Chapter \, 9)}$:
${\bf P}\{\omega\in\Omega$: $\epsilon_F$ is an eigenvalue of 
$H_{\omega}\}=0$. 
%
We define:
%
%
%%%%% S %%%%%
\begin{eqnarray}
S_{\omega}(t)&:=&
\frac{1}{2\pi i}
[
X_{\omega}(t),
\;
T_{\omega}(t)],
\label{S}
\\
T_{\omega}(t)&:=&
 \int_{\tilde{\Gamma}_{\delta}}dz
R_{\omega}(z,t).
\label{T}
\end{eqnarray}
%
%
Let $Q_{\omega}(t):= I-P_{\omega}(t)$, 
and $Q_{\omega}:=Q_{\omega}(0)$. 
%
The second term 
of (\ref{integralequation}) 
multiplied by $Q_{\omega}$ from the left 
is computed as in the same way in (ref. 6, Lemma 2.5). 
%
The result is:
%
%
%%%%% compute %%%%%
\begin{eqnarray}
Q_{\omega}
\int_0^s &dt&
U^{A*}_{\omega,\tau}(t) X_{\omega}(t) U^A_{\omega,\tau}(t) 
\Omega_{\omega,\tau}(t) 
\nonumber\\
%
&=&
-\frac{i}{\tau}Q_{\omega}
\Bigl\{
[U^{A*}_{\omega,\tau}(t)
\tilde{X}_{\omega}(t)
U^A_{\omega,\tau}(t)P_{\omega}\Omega_{\omega,\tau}(t)
]^{t=s}_{t=0}
\nonumber\\
%
&-&
\int_0^s dt \;
U^{A*}_{\omega,\tau}(t)
\tilde{X}_{\omega}(t)
U^A_{\omega,\tau}(t)P_{\omega}
\dot{\Omega}_{\omega,\tau}(t)
\nonumber\\
&-&
\int_0^s dt \;
U^{A*}_{\omega,\tau}(t)
(\dot{\tilde{X}}_{\omega}(t) - 
[\dot{P}_{\omega}(t), \tilde{X}_{\omega}(t)])
U^A_{\omega,\tau}(t)P_{\omega}\Omega_{\omega,\tau}(t)
\Bigr\}
\nonumber\\
%
&-&
Q_{\omega}\int_0^s dt \;
U^{A*}_{\omega,\tau}(t)S_{\omega}(t)U^A_{\omega,\tau}(t)
\Omega_{\omega,\tau}(t).
\label{compute}
\end{eqnarray}
%
%
We estimate 
the last term in the RHS of (\ref{compute}) 
which is the additional term 
we mentioned at the end of section 1. 
%

\vspace*{1em}

%
%%%%% Lemma 3.1 %%%%%
{\bf Lemma 3.1.}

{\it
$T_{\omega}(t)
\in
{\cal S}$.  
Moreover, 
for $\alpha$, $\beta$ which satisfy 
$1\leq \alpha <2$, $\alpha-1 < \beta < 1 - \frac1p$, 
there exists a constant $C_{\alpha \beta}>0$ 
such that 
for sufficiently large $\lambda>0$, 
%
%
%%%%% estimateS %%%%%
\begin{equation}
{\bf E}
|<x| \; 
T_{\omega}(t)
\; |y>|^{\alpha}
\leq
C_{\alpha \beta}\delta^{\beta} e^{-\mu|x-y|}.
\label{estimateS}
\end{equation}
%
%
}

{\bf {\it proof.} }

Let 
$z=\epsilon_F + i\eta$, 
$\eta\in(-\delta,\delta)$. 
%
%
%%%%% Jensen %%%%%
\begin{eqnarray}
{\bf E}
|<x &|& 
T_{\omega}(t)
\; |y>|^{\alpha}
%
\leq
(2\delta)^{\alpha-1}
{\bf E}
\int_{\tilde{\Gamma}_{\delta}}dz
|<x| \; 
R_{\omega}(z,t)
\; |y>|^{\alpha}
\nonumber\\
%
&=&
(2\delta)^{\alpha-1}
{\bf E}
\int_{-\delta}^{\delta}d\eta
|<x| \; 
R_{\omega}(\epsilon_F+i\eta, t)
\; |y>|^{\alpha}
\nonumber\\
%
&\leq&
(2\delta)^{\alpha-1}
{\bf E}
\int_{-\delta}^{\delta}d\eta
\eta^{-(\alpha-\beta)}
|<x| \; 
R_{\omega}(\epsilon_F+i\eta, t)
\; |y>|^{\beta}, 
\label{Jensen}
\end{eqnarray}
%
%
where $0<\beta<1 - \frac1p$. 
In the first 
inequality, 
we used the Jensen's inequality. 
%
If $\alpha-\beta<1$, 
we use the exponential decay 
estimates of Green's function 
derived in (ref. 3):
%
%
%%%%% Greendecay %%%%%
%
\begin{equation}
{\bf E}
|<x| \; 
R_{\omega}(\epsilon_F + i\eta, t)
| \; y>|^{\beta}
\leq
Ce^{-\mu|x-y|},
\label{Greendecay}
\end{equation}
%
%
for $0<\beta<1 - \frac1p$ and 
$\lambda>0$ is sufficiently large.  
%
Hence we obtain
%
%
%%%%% %%%%%
\[
{\rm the \; RHS \; of} \; (\ref{Jensen})
\leq
C\delta^{\beta}
e^{-\mu|x-y|}.
\]
%
%
$\Box$

\vspace*{1em}

%
By using 
above estimates, we prove 
$\Omega_{\omega,\tau}(s)$ is 
close to identity 
in certain sense when $\tau$ is large 
(adiabatic theorem). 
%
Usually, 
adiabatic theorem is stated as$^{(6, 12, 13)}$ 
%
%
%%%%% %%%%%
\[
U_{\tau}(s)=U^A_{\tau}(s)+O(\tau^{-1}), 
\quad 
as \; \tau \rightarrow\infty,
\]
%
in the operator norm. 
%
However, 
since there is no spectral gap 
in the cases 
where Anderson localization takes place, 
our adiabatic theorem presented below is rather weak. 
%

%
%%%%% Lemma 3.2 %%%%%
{\bf Lemma 3.2.}

{\it
$\Omega_{\omega,\tau}(s)$ has 
the following form:
%
\begin{eqnarray}
\Omega_{\omega,\tau}(s)=I+
\frac{1}{\tau}
\left\{
A_1(s) + \int_0^s dt \, A_2(t)
\right\}
+ 
\int_0^s dt \, B(t), 
\label{Omega}
\\
A_1 = \prod_{i=1}^l A^{(i)}_1, 
\quad
A_2(t) = \prod_{i=1}^m A_2^{(i)}(t), 
\quad
B(t) = \prod_{i=1}^n B^{(i)}(t), 
\end{eqnarray}
%
where, 

(1) 
$A_1^{(i)}$, $A_2^{(i)}(t) \in {\tilde {\cal S}}$ 
and 
$\| A_1^{(i)}\|_{ {\cal S}_{\alpha}, \beta}$, 
$\| A_2^{(i)}(t)\|_{ {\cal S}_{\alpha}, \beta}$ 
are all independent of $\tau>0$, $t\in[0,1]$. 

(2)
There is an $i_0 \leq n$ such that, 
$B^{(i)}(t)\in{\tilde {\cal S}}$ 
for $i\ne i_0$, 
where 
$\| B^{(i)}(t)\|_{ {\cal S}_{\alpha}, \beta}$ 
are all independent of $\tau>0$, $t\in[0,1]$. 
%
And $B^{(i_0)}(t) = T_{\omega}(t)$ 
which is defined in (\ref{T}) 
and satisfies (\ref{estimateS}). 
}


\vspace*{1em}

{\bf {\it Remark.} }

We can not deduce 
the adiabatic theorem 
in the operator norm from (\ref{Omega}). 
%
What 
Lemma 3.2 tells us is that 
%
%
\begin{eqnarray}
\lim_{\tau\to\infty}
{\bf E}
(\varphi, 
\bigl(U_{\omega,\tau}(s)-U_{\omega,\tau}^A(s)\bigr)
\psi)=0, 
\label{adia}\\
%
\lim_{\tau \to \infty}
{\bf E}
\|
(U_{\omega,\tau}(s) - U^A_{\omega, \tau}(s))
\varphi
\|_2^2
=0, 
\label{adiaa}
\end{eqnarray}
%
%
for $\varphi$, $\psi \in l^2({\bf Z}^d)$. 
%
$\| \cdot \|_2$ 
is the $l^2({\bf Z}^d)$-norm. 
%
The rate of 
convergence is $\tau^{-\alpha}$, 
where 
$\alpha < \left( 1 + \frac{2}{\beta}(d+1) \right)^{-1}$, 
if 
$\lambda > \lambda_0 (\beta)$, 
$0<\beta< 1-\frac1p$. 
%
Thus 
(\ref{adia}), (\ref{adiaa}) can be regarded as a weak form of the 
adiabatic theorem. 
%
On the other hand, 
we should note that the adiabatic theorem 
in the operator norm is proved in (ref. 5) 
without gap condition in general setting, 
provided Range$P(s)$ is finite dimensional. 


\vspace*{1em}

{\bf {\it proof.} }

It suffices 
to estimate the RHS of (\ref{compute}), 
since we can estimate the second term of 
(\ref{integralequation}) multiplied by $P_{\omega}$ 
from the left 
in the same manner. 
%
We will 
show 
the first term of (\ref{compute}) 
is the component of $A_1$ in (\ref{Omega}),  
and 
the second and third terms are   
the components of $A_2(t)$. 
%
In order 
to do this, 
we have to check 
the operators appearing in the first, second and 
third terms in (\ref{compute}) 
satisfy (1) in the statement of Lemma 3.2. 
%
We have 
only to check 
$\dot{\Omega}_{\omega,\tau}(t)$ and 
$\dot{\tilde{X}}_{\omega}(t)$, 
because 
for the rest, 
it is the direct conclusion of Lemma 2.3. 
%
By 
(\ref{integralequation}), 
$\dot{\Omega}_{\omega,\tau}(t)$ 
can be written as 
%
%
%%%%% %%%%%
\[
\dot{\Omega}_{\omega,\tau}(s)=
-U^{A*}_{\omega,\tau}(s)X_{\omega}(s)U^A_{\omega,\tau}(s)
\Omega_{\omega,\tau}(s).
\]
%
%
By Lemma 2.3, the RHS of this equation obviously 
satisfies (1) in the statement of Lemma 3.2. 
%
As for 
$\dot{\tilde{X}}_{\omega}(s)$, 
it has the following form,  
%
%
%%%%% %%%%%
\begin{eqnarray*}
\dot{\tilde{X}}_{\omega}(s)
&=&
-\frac{1}{2\pi i}
\int_{\Gamma_{\delta}}dz \,
(\dot{R}_{\omega}(z,s)X_{\omega}(s)R_{\omega}(z,s)
\\
%
&&\;+
R_{\omega}(z,s)\dot{X}_{\omega}(s)R_{\omega}(z,s)
+
R_{\omega}(z,s)X_{\omega}(s)\dot{R}_{\omega}(z,s)). 
\end{eqnarray*}
%
%
We note that 
$\Gamma_{\delta}$ does not contain the 
spectrum of $H_{\omega}$. 
%
Thus  
$R_{\omega}(z,s)$ satisfies 
the exponential decay estimate 
which can be shown by Combes-Thomas argument
$^{(1, 8)}$ for $z\in \Gamma_{\delta}$:
%
%
%%%%% %%%%%
\[
|<x| \; 
R_{\omega}(z,s)
\; |y>|
\leq 
Ce^{-\mu|x-y|}.
\]
%
%
Then, 
the conclusion follows from the explicit form 

$\dot{R}_{\omega}(z,s)
=
iV(s) [g(x,s), R_{\omega}(z,0)] V^* (s)$, 
$\dot{P}_{\omega}(s)
=
i V(s) [g(x,s), P_{\omega}] V^* (s)$, 
the boundedness of 
$R_{\omega}(z,0)$ and $P_{\omega}$, 
and the Jensen's inequality. 
%

That 
the last term of (\ref{compute}) 
satisfies (2) in the statement 
of Lemma 3.2 follows easily from 
Lemma 2.3, 3.1. 
$\Box$

\vspace*{1em}


\vspace*{1em}

We are 
at the stage of the proof of Proposition 1.2. 

{\bf {\it proof of Proposition 1.2.} }

We shall  
compare the following two quantities:
%
%
%%%%% %%%%%
\begin{eqnarray*}
j_{\omega,\tau}&:=&
i\tau
\int_0^1 ds 
{\cal T}
(U^*_{\omega,\tau}(s)
[H_{\omega}(s),x]
U_{\omega,\tau}(s)
P_{\omega}),
\\
%
j_{\omega,\tau}^A&:=&
i\tau
\int_0^1 ds 
{\cal T}
(U^{A*}_{\omega,\tau}(s)
[H^A_{\omega}(s),x]
U^A_{\omega,\tau}(s)
P_{\omega}).
\end{eqnarray*}
%
%
We compute
%
%
%%%%% %%%%%
\begin{eqnarray*}
j_{\omega,\tau}=
i\tau
\int_0^1 &ds& 
{\cal T}
([U^*_{\omega,\tau}(s)
H_{\omega}(s),x]
U_{\omega,\tau}(s)
P_{\omega})
\\
%
&-&
i\tau
\int_0^1 ds 
{\cal T}
([U^*_{\omega,\tau}(s),x]
H_{\omega}(s)
U_{\omega,\tau}(s)
P_{\omega}).
\end{eqnarray*}
%
%
Since 
$[U^*_{\omega,\tau}(s)H_{\omega}(s), x]$, 
$[U^*_{\omega,\tau}(s), x]\in\tilde{{\cal S}}$, 
the above two terms are both 
well-defined by Lemma 2.1 (1). 
%
Here, we defined 
the product of operators by their integral kernels, 
since $[U^*_{\omega,\tau}(s)H_{\omega}(s), x]$ and 
$[U^*_{\omega,\tau}(s), x]$ 
could be unbounded 
(Remark after Lemma 2.1). 
%
We 
substitute the Schr\"{o}dinger equation 
(\ref{Uomega}):
%
%
%%%%% %%%%%
\begin{eqnarray*}
j_{\omega,\tau}=i\tau
\int_0^1 &ds& 
{\cal T}(
[-\frac{i}{\tau}\frac{d}{ds}U^*_{\omega,\tau}(s), x]
U_{\omega,\tau}(s)P_{\omega})
\\
%
&&\quad
-i\tau
\int_0^1 ds 
{\cal T}(
[U^*_{\omega,\tau}(s), x]
\frac{i}{\tau}\frac{d}{ds}U_{\omega,\tau}(s)P_{\omega})
\\
%
=
\int_0^1 &ds& 
{\cal T}\left(
\frac{d}{ds}(
[U^*_{\omega,\tau}(s), x]U_{\omega,\tau}(s)P_{\omega})
\right). 
\end{eqnarray*}
%
%
Because 
$\frac{d}{ds}(
[U^*_{\omega,\tau}(s), x]U_{\omega,\tau}(s)P_{\omega})$ 
is the sum of two operators, 
and each operators are products of operators 
which belong to $\tilde{{\cal S}}$, 
we can 
interchange $\int_0^1 ds$ with $\int_{\Omega}d{\bf P}$. 
%
Therefore, 
%
%
%%%%% %%%%%
\[
j_{\omega,\tau}=
{\cal T}
([U^*_{\omega,\tau}(1), x]U_{\omega,\tau}(1)P_{\omega}). 
\]
%
%
And in the same manner, 
%
%
%%%%% %%%%%
\[
j_{\omega,\tau}^A=
{\cal T}
([U^{A*}_{\omega,\tau}(1), x]U^A_{\omega,\tau}(1)P_{\omega}). 
\]
%
%
To compare 
$j_{\omega,\tau}^A$ with $j_{\omega,\tau}$, 
we substitute 
$U^A_{\omega,\tau}(s)=U_{\omega,\tau}(s)
\Omega^*_{\omega,\tau}(s)$ 
into the above equation. 
%
%
%%%%% %%%%%
\begin{eqnarray*}
j_{\omega,\tau}^A&=&
{\cal T}([\Omega_{\omega,\tau}(1)U^{*}_{\omega,\tau}(1), x]
U_{\omega,\tau}(1)\Omega^*_{\omega,\tau}(1)P_{\omega})
\\
%
&=&
{\cal T}(\Omega_{\omega,\tau}(1)
[U^*_{\omega,\tau}(1), x]U_{\omega,\tau}(1)
\Omega^*_{\omega,\tau}(1)P_{\omega})
%
+
{\cal T}([\Omega_{\omega,\tau}(1), x]
\Omega^*_{\omega,\tau}(1)P_{\omega})
\\
%
&=:&C+D.
\end{eqnarray*}
%
We will 
show that 
$C=j_{\omega,\tau}+o(1)$ 
and 
$D=o(1)$ 
as 
$\tau\rightarrow\infty$. 
%
This 
concludes the proof. 
%
%
By subtracting 
the following two equations, 
%
%
%%%%% %%%%%
\begin{eqnarray*}
P_{\omega}\Omega_{\omega,\tau}(s)
-
P_{\omega}\Omega_{\omega,\tau}(s)P_{\omega}
=
P_{\omega}\Omega_{\omega,\tau}(s)Q_{\omega}, 
\\
%
\Omega_{\omega,\tau}(s)P_{\omega}
-
P_{\omega}\Omega_{\omega,\tau}(s)P_{\omega}
=
Q_{\omega}\Omega_{\omega,\tau}(s)P_{\omega},
\end{eqnarray*}
%
%
we obtain 
%
%
%%%%% %%%%%
\begin{eqnarray*}
P_{\omega}\Omega_{\omega,\tau}(s)
-
\Omega_{\omega,\tau}(s)P_{\omega}
&=&
P_{\omega}\Omega_{\omega,\tau}(s)Q_{\omega}
-
Q_{\omega}\Omega_{\omega,\tau}(s)P_{\omega}
\\
%
&=:&W(s),
\\
%
{\rm where} \quad
W(s)&:=&
P_{\omega}
\left[
\frac{1}{\tau}
\left\{
A_1(s) + \int_0^s dt A_2 (t)
\right\}
+
\int_0^s dt B(t)
\right]
Q_{\omega}
\\
%
&-&
Q_{\omega}
\left[
\frac{1}{\tau}
\left\{
A_1(s) + \int_0^s dt A_2 (t)
\right\}
+
\int_0^s dt B(t)
\right]
P_{\omega}.
\end{eqnarray*}
%
%
Then, 
%
%
%%%%% %%%%%
\begin{eqnarray}
&&{\cal T}(
\Omega_{\omega,\tau}(1)
[U^*_{\omega,\tau}(1), x]U_{\omega,\tau}(1)
(\Omega^*_{\omega,\tau}(1)P_{\omega}
-
P_{\omega}\Omega^*_{\omega,\tau}(1)))
\nonumber\\
%
&&\quad=
\frac{1}{\tau}
{\cal T}(
\Omega_{\omega,\tau}(1)
[U^*_{\omega,\tau}(1), x]U_{\omega,\tau}(1)
(Q_{\omega}A_1^*(1)P_{\omega}
-
P_{\omega}A_1^*(1)Q_{\omega}))
\nonumber\\
%
&&\quad\quad+
\frac{1}{\tau}
\int_0^1 ds 
{\cal T}(
\Omega_{\omega,\tau}(1)
[U^*_{\omega,\tau}(1), x]U_{\omega,\tau}(1)
(Q_{\omega}A_2^*(s)P_{\omega}
-
P_{\omega}A_2^*(s)Q_{\omega}))
\nonumber\\
%
&&\quad\quad+
\int_0^1 ds 
{\cal T}(
\Omega_{\omega,\tau}(1)
[U^*_{\omega,\tau}(1), x]U_{\omega,\tau}(1)
(Q_{\omega}B^*(s)P_{\omega}
-
P_{\omega}B^*(s)Q_{\omega}))
\nonumber\\
%
&&\quad:=
E_1 + E_2 + F. 
\label{and}
\end{eqnarray}
%
%
By Lemma 2.1(1), 3.2, 
there exist constants $\beta>0$, $C_{\beta}>0$ such that 
$F$ in (\ref{and}) satisfies 
%
$
|F| \leq C_{\beta} \delta^{\beta}.
$
%
Therefore, 
for arbitrary small $\epsilon>0$, 
we have $|F| < \epsilon$ 
by taking $\delta>0$ 
sufficiently small. 
%
After fixing 
such $\delta>0$, 
we let $\tau>0$ sufficiently large 
and obtain $|E_1|$, $|E_2| < \epsilon$ 
(by using Lemma 2.1(1), 3.2 again). 
%
Hence 
%
%
%%%%% %%%%%
\[
{\cal T}(
\Omega_{\omega,\tau}(1)
[U^*_{\omega,\tau}(1), x]U_{\omega,\tau}(1)
(\Omega^*_{\omega,\tau}(1)P_{\omega}
-
P_{\omega}\Omega^*_{\omega,\tau}(1)))
=
o(1), 
\]
%
%
and therefore, 
%
%
\[
C=
{\cal T}(
\Omega_{\omega,\tau}(1)
[U^*_{\omega,\tau}(1), x]
U_{\omega,\tau}(1)P_{\omega}
\Omega^*_{\omega,\tau}(1))
+
o(1), 
\]
%
%
as $\tau\rightarrow\infty$. 
%
By Lemma 2.1 (2), it follows that 
%
%
%%%%% %%%%%
\begin{eqnarray*}
C&=&
{\cal T}(
P_{\omega}\Omega^*_{\omega,\tau}(1)
\Omega_{\omega,\tau}(1)
[U^*_{\omega,\tau}(1), x]
U_{\omega,\tau}(1)P_{\omega})
+
o(1)
\\
%
&=&
{\cal T}(
P_{\omega}[U^*_{\omega,\tau}(1), x]
U_{\omega,\tau}(1)P_{\omega})
+
o(1)
\\
&=&
j_{\omega,\tau}+o(1). 
\end{eqnarray*}
%
%
It remains to show 
$D=o(1)$. 
%
In fact, by Lemma 3.2   
%
%
%%%%% %%%%%
\begin{eqnarray*}
D&=&
{\cal T}
([\Omega_{\omega,\tau}(1), x]
\Omega^*_{\omega,\tau}(1)P_{\omega})
\\
%
&=&
\frac{1}{\tau}{\cal T}
([A_1(1), x]
\Omega^*_{\omega,\tau}(1)P_{\omega})
%
+
\frac{1}{\tau}
\int_0^1 ds
{\cal T}
([A_2 (s), x]
\Omega^*_{\omega,\tau}(1)P_{\omega})
\\
%
&&\quad+
\int_0^1 ds
{\cal T}
([B(s), x]
\Omega^*_{\omega,\tau}(1)P_{\omega}).
\end{eqnarray*}
%
%
In the same manner 
as in $C$, 
we obtain $D=o(1)$. 
$\Box$
%
%

\vspace*{1em}

%
%
%%%%% Acknowledgement %%%%%

\vspace*{1em}

{\bf ACKNOWLEDGEMENTS}

It is our pleasure to thank 
Michael Aizenman, 
Akira Iwatsuka, 
Tohru Koma, Shu Nakamura, and Hermann Schultz-Baldes 
for their useful discussions and suggestions. 
We would like to thank the referee for important critical comments. 
%
One of the authors (F.N.) is partially supported by the 
Japan Society for the Promotion of Science. 

\vspace*{1em}

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\end{document}



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