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Lifshitz tail, random Schrodinger operator, magnetic Schrodinger operator, integrated density of states
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%%%%%%%%%%%%  Title  %%%%%%%%%%%%%%%%

\title{Lifshitz tail for 2D discrete Schr\"odinger operator \\
with random magnetic field}
\author{Shu Nakamura \\ 
{\normalsize Graduate School of Mathematical Sciences, University of Tokyo}
\\ {\normalsize 3-8-1, Komaba, Meguro-ku, Tokyo, Japan 153-8914}
\\ {\normalsize E-mail: {\tt shu@ms.u-tokyo.ac.jp}}}
\date{}

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%     Paper from here
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\begin{document}
\maketitle

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%   SECTION 1  INTRODUCTION  %%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\abstract{
Lifshitz tail for 2 dimensional discrete Schr\"odinger operator with Anderson-type 
random magnetic field is proved. We first prove local energy estimates for
deterministic discrete magnetic Schr\"odinger operators, and then follow the large
deviation argument of Simon~\cite{S}.
}

\section{Introduction}

We first define our Hamiltonian. Let $\faces$ be the set of unit squares with the
vertices in $\ze^2$,  and let $\edges$ be the set of edges, i.e., 
\begin{align*}
\faces &= \bigbra{ [x_1,x_1+1]\times[x_2,x_2+1] \; \big| \; x_1,x_2\in\ze}, \\
\edges &= \bigbra{(x,y)\;\big|\; x,y\in\ze^2, |x-y|=1}.
\end{align*}
For $e=(x,y)\in\edges$, we write $\overline{e}= (y,x)$. For $f\in \faces$, we 
denote the boundary of $f$ by $\pa f\subset \edges$, i.e., if
$f=[x_1,x_1+1]\times[x_2,x_2+1]$  then 
\[
\pa f = \bigbra{ (x,x+\d_1), (x+\d_1,x+\d_1+\d_2), (x+\d_1+\d_2, x+\d_2), (x+\d_2,x)
} , 
\]
where $\d_1=(1,0)$ and $\d_2=(0,1)\in \ze_2$. 

Let $A(e)$ be a function of $e\in \edges$ with values in $\re/(2\pi\ze)$ such that 
\[
A(\overline{e})= -A(e), \qquad e\in\edges. 
\]
The discrete magnetic Schr\"odinger operator $H$ on $\ze^2$ 
with a vector potential $A$ is defined by 
\[
H\g(x) = \sum_{|x-y|=1} \bigpare{\g(x)-e^{iA((x,y))}\g(y)}, \qquad x\in\ze^2, 
\]
for $\g\in \ell^2(\ze^2)$. $H$ is a bounded self-adjoint operator on $\ell^2(\ze^2)$, 
and $0\leq H\leq 8$. For a given vector potential $A$, the magnetic filed is defined by 
\[
B(f) = \sum_{e\in\pa f} A(e), \qquad f\in \faces,
\]
which is a function on $\faces$ with values in $\re/(2\pi\ze)$. It is well-known that
the spectral properties of $H$ depends only on $B$, not $A$. Namely, if $A$ and
$\tilde{A}$ induces the same magnetic field $B$, then the corresponding
Schr\"odinger operators $H$ and $\tilde H$ are unitarily equivalent by a gauge
transform (i.e., a unimodular multiplication operator on $\ze^2$). 

We suppose that $B=B_w(f)$, $w\in\Omega$, is an identically distributed independent
random variables (i.i.d.) with distribution $d\m$ (on $\re/(2\pi\ze)$). 
We denote the probability space by $\Omega$. 
Our principal example is the uniform distribution on $\re/(2\pi\ze)$. 
Then  $H$ is an ergodic operator, and the spectrum is independent of $w$ 
almost surely (a.s.). See, e.g., \cite{CL}. 
Moreover, the density of states: 
\[
k(E) = \lim_{L\to\infty} \frac{1}{|\Lambda_L|} \#\bra{\text{eigenvalues of
}H^{\Lambda_L}\leq E}
\]
exists a.s., where 
\[
\Lambda_L=\bigbra{x=(x_1,x_2)\in\ze^2\;\big|\; |x_1|,|x_2|\leq L}, 
\]
$|\Lambda_L| = (2L+1)^2$, and $H^{\Lambda_L}$ is the Hamiltonian restricted on 
$\ell^2(\Lambda_L)$. 
We suppose the distribution $d\m$ satisfies the following assumptions: 

\begin{ass} (1) $d\m$ is not point measure at $0$, i.e., $B_w$ is not identically
zero. \newline
(2) There is  $C$ and $a>0$ such that 
\[
d\m([-\e,\e]) \geq C \e^a, \qquad 0\leq \e\leq \pi, 
\]
where we identify $\re/(2\pi\ze)$ with $[-\pi,\pi)$. 
\end{ass}

\begin{thm} 
Let $k(E)$ be the density of states for $H=H_w$, and suppose Assumption~A. Then 
\[
\lim_{E\dto 0} \log(-\log k(E))/\log E = -1. 
\]
\end{thm}

\begin{rem}
(1) This is a natural analogue of the Lifshitz singularity of the density of states for
Schr\"odinger operator with random potential (see, e.g., \cite{S}, \cite{CL}, \cite{K}
and references therein). Lifshitz tail for magnetic Schr\"odinger is recently studied
by  Ueki \cite{U}, where the random magnetic field is supposed to be Gaussian random
field.  Note that the space dimension $d$ is 2 in our setting and the right
hand side of the statement is $-d/2$. \newline
(2) Under our assumption, $\s(H_w)=[0,8]$ a.s., and hence Theorem~1 describe the
behavior of $k(E)$ at the lower edge of the spectrum. Similar result holds at 
$8=\sup\s(H)$ with little modifications. 
\end{rem}

In Section~2, we prove a simple local energy estimate for discrete magnetic 
Schr\"odinger operators. It is an analogue of the Avron-Herbst-Simon estimate 
\cite{AHS} for continuous magnetic Schr\"odinger operators, and we also show that
it is optimal. In the rest of the paper, we mimic the argument of Simon~\cite{S} to
prove the Lifshitz tail. We define the Dirichlet-Neumann decoupling in Section~3 and
prove Theorem~1 in Section~4. In Appendix~A, we discuss a generalization of the
local energy estimate of Section~2. It is not necessary in this paper, but is
interesting in  itself and maybe useful in the analysis of discrete magnetic
Schr\"odinger operators.  In Appendix~B, we give a proof of the spectral property of an
example of Section~2. 

\noindent\textbf{Notations:} 
We denote the inner product of $\ell^2(\ze^2)$ by 
\[
\ip{\f}{\g} = \sum_{x\in\ze^2} \overline{\f(x)} \g(x), 
\qquad \f,\g\in\ell^2(\ze^2).
\]
$\Pro(\cdot)$ and $\Exp(\cdot)$ denote the  probability and the expectation
with respect to $w\in\Omega$, respectively. For $e\in\edges$, $i(e)$ and
$t(e)$ denote the initial point and the terminal point of $e$, respectively, i.e., 
\[
i(e) =x, \quad t(e)=y, \qquad \text{if $e=(x,y)$}.
\]
We write $\ze_+= \{0,1,2\dots\}$. 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%   SECTION 2 ENERGY ESTIMATES   %%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Local energy estimates for discrete magnetic Schr\"odinger operators}

In this section, we consider deterministic magnetic Schr\"odinger operators on
$\ze^2$.  For a given magnetic field $B$, we set 
\[
W_B(x) = \sum_{x\in f} (1-\cos(B(f)/4)), \qquad x\in\ze^2.
\]
Here we identify $\re/(2\pi\ze)$ with $[-\pi,\pi)$, and hence 
$B(f)/4\in [-\pi/4,\pi/4)$. Thus, in particular, 
\[
0\leq W_B(x)\leq 4(1-1/\sqrt{2}), \qquad x\in\ze^2.
\]


\begin{thm}
Let $H$ and $W_B$ as above. Then $H\geq W_B$, i.e., 
\[
\ip{\g}{H\g} \geq \ip{\g}{W_B\g} = \sum_{x\in\ze^2} W_B(x) |\g(x)|^2, 
\qquad \g\in\ell^2(\ze^2).
\]
\end{thm}

\begin{rem} 
The motivation of this estimate comes from recent works by Higuchi and 
Shirai \cite{HS}. (See also Sunada~\cite{Su} for related results on generalized
Harper operators.) 
\end{rem}

\begin{proof}
We note 
\begin{align*}
\ip{\g}{H\g} &= \sum_{|x-y|=1} \overline{\g(x)} \Bigpare{\g(x)-e^{iA((x,y))}\g(y)} \\
&= \sum_{|x-y|=1} \overline{\g(y)} \Bigpare{\g(y)-e^{-iA((x,y))}\g(x)} 
\end{align*}
since $A((y,x))=-A((x,y))$. We take the average of these expressions, and obtain
\begin{align*}
\ip{\g}{H\g} &= \frac12 \sum_{|x-y|=1} \Bigabs{\g(x)-e^{iA((x,y))} \g(y)}^2 \\
&= \frac12 \sum_{e\in\edges} \Bigabs{\g(i(e))-e^{iA(e)} \g(t(e))}^2.
\end{align*}
Since each $e\in\edges$ is an element of $\pa f$ for only one $f\in\faces$, we may
write
\[
\ip{\g}{H\g} = \frac12 \sum_{f\in\faces} \sum_{e\in\pa f} 
\Bigabs{\g(i(e))-e^{iA(e)} \g(t(e))}^2.
\]

\begin{lem}
\[
\sum_{e\in\pa f} \Bigabs{\g(i(e))-e^{iA(e)} \g(t(e))}^2
\geq 2\Bigpare{1-\cos\Bigpare{\frac{B(f)}{4}}} \sum_{x\in f} |\g(x)|^2.
\]
\end{lem}
Now Theorem~2 follows immediately. 
\end{proof}

\begin{proof}[Proof of Lemma~3]
Let 
\[
f=[x_1,x_1+1]\times [x_2,x_2+1], \qquad x=(x_1,x_2)\in\ze^2, 
\]
and we write 
\begin{align*}
&u_1=\g(x), \quad u_2=\g(x+\d_1), \quad 
u_3=\g(x+\d_1+\d_2), \quad u_4=\g(x+\d_2), \\
&\th_1=A((x,x+\d_1)),\quad \th_2 =
A((x+\d_1,x+\d_1+\d_2)), \\
&\th_3= A((x+\d_1+\d_2,x+\d_2)),\quad \th_4=A((x+\d_2,x)).
\end{align*}
Then we have 
\begin{align*}
&\sum_{e\in\pa f} \Bigabs{\g(i(e))-e^{iA(e)} \g(t(e))}^2 \\
&= (\overline{u_1}, \overline{u_2}, \overline{u_3}, \overline{u_4}) 
\begin{pmatrix} 
2 & -e^{i\th_1} & 0 & -e^{-i\th_4} \\
-e^{-i\th_1} & 2 & -e^{i\th_2} & 0 \\
0 & -e^{-i\th_2} & 2 & -e^{i\th_3} \\
-e^{i\th_4} & 0 & -e^{-i\th_3} & 2 \end{pmatrix}
\begin{pmatrix} u_1 \\ u_2 \\ u_3 \\ u_4 \end{pmatrix} 
\equiv \mathbf{u}^* h_\th \mathbf{u}. 
\end{align*}
We set  $\a_1=\th_1$, $\a_2=\th_1+\th_2$, $\a_3 = \th_1+\th_2+\th_3$, and 
\[
A= \begin{pmatrix} 
1 & & & \\  & e^{i\a_1} & & \\ & & e^{i\a_2} & \\ & & & e^{i\a_3}
\end{pmatrix}.
\]
Then we have 
\[
A h_\th A^*= \begin{pmatrix}
2 & -1 & 0 & -e^{-iB} \\ -1 & 2 & -1 & 0 \\
0 & -1 & 2 & -1 \\ -e^{iB} & 0 & -1 & 2 
\end{pmatrix} 
\equiv A_B
\]
since $B=B(f)=\th_1+\th_2+\th_3+\th_4$. Note that $A_B$ is the Hamiltonian 
of the free discrete Schr\"odinger operator on the closed chain $\{ 0,1,2,3\}$ 
with the periodic boundary condition with an additional phase $e^{iB}$. Thus the
eigenvectors are given by 
\[
\mathbf{v}^j = (1,e^{i\m_j},e^{2i\m_j},e^{3i\m_j}), \qquad \m_j=(B+2\pi j)/4, 
\qquad j=0,1,2,3,
\]
and the eigenvalues are $\l_j=2(1-\cos\m_j)$. 
In particular, the lowest eigenvalue is $\l_0=2(1-\cos(B/4))$ since $B\in[-\pi,\pi)$. 
This implies $A_B\geq 2(1-\cos(B/4))$, and hence $h_\th\geq 2(1-\cos(B/4))$
to conclude the assertion. 
\end{proof}

\begin{example}
Let $b\in(-\pi,\pi)$ and let 
\[
B(f)=\begin{cases} b, \quad &\text{if $x_1+x_2$ is even}, \\
-b, & \text{if $x_1+x_2$ is odd},  \end{cases}
\]
where $f=[x_1,x_1+1]\times[x_2,x_2+1]$. 
In this case, $W_B(x)=4(1-\cos(b/4))$ for any $x\in\ze^2$. 
$H$ is solvable and we can show 
\[
\s(H)= [4(1-\cos(b/4)), 4(1+\cos(b/4))].
\]
This example shows that Theorem~1 is optimal. See Appendix~B for the computation
of this example. 
\end{example}

We will use the following (almost trivial) analogue of the Kato inequality:

\begin{lem}
\[
\ip{\g}{H\g} \geq \ip{|\g|\;}{\;H_0 |\g|}, \qquad \g\in\ell^2(\ze^2),
\]
where $H_0$ denotes the free discrete Schr\"odinger operator on $\ze^2$, 
i.e., $H$ with $A\equiv 0$. 
\end{lem}

\begin{proof}
We have 
\begin{align*}
\ip{\g}{H\g} &= \frac12 \sum_e \Bigabs{\g(i(e))-e^{iA(e)}\g(t(e))}^2 \\
&\geq \frac12 \sum_e \bigabs{ |\g(i(e))|-|\g(t(e))|}^2 = \ip{|\g|\;}{\;H_0|\g|}. 
\end{align*}
\end{proof}

This implies, for example, 
\begin{align*}
\ip{\g}{H\g} &\geq \frac12 \ip{|\g|\;}{\,H_0\, |\g|}+\frac12 \ip{\g}{W_B\g}   \\
&= \frac12 \ip{|\g|\;}{(H_0+W_B)|\g|}.
\end{align*}
By the min-max principle,  we obtain the following lemma. 

\begin{lem}
\[
\inf\s(H) \geq \frac12 \inf\s(H_0+W_B).
\]
\end{lem}

Note that $H_0+W_B$ is a usual discrete Schr\"odinger operator. We will use 
(a modified version of) Lemma~5 in the proof of Theorem~1. 


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%   SECTION 3 D-N DECOUPLING    %%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Dirichlet-Neumann bracketing}

In this section, we generalize the Dirichlet and Neumann decoupling of 
Simon \cite[Section 2]{S} to discrete magnetic Schr\"odinger operators. 
Let $\ze^2= \sum_\a S_\a$ be a disjoint decomposition of $\ze^2$, and let 
\[
\Sigma= \bigset{e\in \edges}{e\not\subset S_\a \text{ for any } S_\a}
\]
be the boundary set of the decomposition.  Then for a given magnetic Schr\"odinger
operator $H$, we construct operators $H^{\Sigma;N}$ and $H^{\Sigma;D}$ such that
\[
H^{\Sigma;N} \leq H \leq H^{\Sigma;D}
\]
and they commute with the direct decomposition:\ 
$\ell^2(\ze^2)= \bigoplus_\a \ell^2(S_\a)$, i.e., 
they act on each $\ell^2(S_\a)$. 

For each $e\in\edges$, we set 
\[
L_e\g(x) = \begin{cases}
\frac12\Bigpare{\g(x)- e^{iA(e)} \g(t(e))}, \quad &\text{if $x=i(e)$}, \\
\frac12\Bigpare{\g(x)- e^{-iA(e)} \g(i(e))}, &\text{if $x=t(e)$},  \\
0, &\text{otherwise}. 
\end{cases}
\]
$L_e$ corresponds to $2\times 2$-matrix 
\[
\frac12 \begin{pmatrix}1&-e^{iA(\e)} \\ -e^{-iA(e)} &  1 \end{pmatrix}
\qquad \text{ on } \ell^2(\{i(e), t(e)\}), 
\]
and hence $L_e$ has eigenvalues 1 and 0. In particular, $L_e\geq 0$. We note 
\[
H = \sum_{e\in\edges} L_e.
\]
On the other hand, we also set
\[
M_e\g(x) = \begin{cases}
\frac12\Bigpare{\g(x)+ e^{iA(e)} \g(t(e))}, \quad &\text{if $x=i(e)$}, \\
\frac12\Bigpare{\g(x)+ e^{-iA(e)} \g(i(e))}, &\text{if $x=t(e)$},  \\
0, &\text{otherwise}. 
\end{cases}
\]
Then, similarly, we learn $M_e\geq 0$. Now we set
\begin{align*}
H^{\Sigma;N} &= H- \sum_{e\in\Sigma}L_e = \sum_{e\notin \Sigma} L_e, \\
H^{\Sigma;D} &= H+ \sum_{e\in\Sigma} M_e. 
\end{align*}
Then, clearly, we have
\[
0\leq H^{\Sigma;N} \leq H \leq H^{\Sigma;D}, 
\]
and $H^{\Sigma;\#}$ has no off-diagonal elements corresponding to $e\in \Sigma$, 
where $\#=N$ or $D$. Hence $H^{\Sigma;\#}$ acts on each component of the direct
decomposition: $\ell^2(\ze^2)= \bigoplus_\a \ell^2(S_\a)$. 

For $L\in \ze_+$, we set
\[
S_\a^{(L)}=\bigset{x\in\ze^2}{L\a_j\leq x_j<L(\a_j+1), j=1,2} 
\qquad \text{for } \a\in\ze^2. 
\]
Let $H^{L;\#}$ be the operator $H^{\Sigma;\#}$ restricted to $\ell^2(S_0^{(L)})$. 
As in \cite[Section 2]{S}, we set 
\[
k_L^\# (E) = L^{-2}\Exp\bigbrac{\#\{\text{eigenvalues of } H^{L;\#}\leq E\}}
\]
with $\#=N$ or $D$. Then we have 
\[
k_L^D(E) \leq k(E) \leq k_L^N(E), \qquad E\in\re,
\]
for each $L\in\ze_+$. We will estimate $k_D^N(E)$ and $k_L^N(E)$ to
obtain  lower and upper bounds of $k(E)$, respectively. 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%   SECTION 4 PROOF OF THEOREM 1   %%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Proof of Theorem 1}

The proof of Theorem~1 is similar to \cite{S}, and we mainly discuss the necessary
modifications. 

\subsection{Lower bound}
Recall that the lowest eigenvalue of $H_0^{L;D}$ is given by 
$e_0^{L;D}=2(1-\cos(\pi/L))$ (\cite[Theorem~2.4]{S}). For $E>0$, we 
set $L\in\ze_+$ such that 
\[
\sqrt{4\pi^2/E} < L \leq \sqrt{4\pi^2/E}+1
\]
so that $e_0^{L;D}\leq 2\pi^2/L^2 < E/2$. If we suppose 
\[
|B(f)| \leq E/8L \qquad \text{for any } f\subset S_0^{(L)}, 
\]
then we can find a vector potential $A$ such that 
\[
|A(e)|  \leq E/8 \qquad\text{for any } e\subset S_0^{(L)}.
\]
This implies $\bignorm{H_0^{L;D}-H^{L;D}}\leq E/2$, and hence the lowest eigenvalue
of $H^{L;D}$ is smaller than $E$. Thus we have
\begin{align*}
k_L^D(E) &\geq 
L^{-2} \Pro\bigpare{|B(f)|\leq E/8L\text{ for any }f\subset S_0^{(L)}} \\
&= L^{-2} \Pro\bigpare{|B(f)|\leq E/8L}^{L^2} 
\geq L^{-2} C^{L^2} (c E^{3/2})^{aL^2} \\
& = \exp\bigbrac{-2\log L +L^2 (\log C + a \log c + (3a/2) \log E) }
\end{align*}
This implies 
\[
\liminf_{E\to 0} \log(-\log k_L^D(E))/\log E \geq -1
\]
since $L^2 \sim 4\pi^2 E^{-1}$ as $E\to 0$,  and hence
\[
\liminf_{E\to 0} \log(-\log k(E))/\log E \geq -1.
\]

\subsection{Upper bound}
We first note an analogue of Lemma~5. 

\begin{lem} Let 
\[
W_B^L(x) = \sum_{x\in f\subset S_0^{(L)}} \Bigpare{1-\cos\Bigpare{\frac{B(f)}{4}}}.
\]
Then 
\[
\inf \s(H^{L,N}) \geq \frac12 \inf\s(H_0^{L,N}+W_B^L).
\]
\end{lem}

The proof is almost the same, and we omit it. 

We fix $\e_0$ and $f_0>0$ such that 
\[
\Pro(1-\cos (B(f)/4) \geq \e_0) = f_0 
\]
Now the main lemmas of \cite{S}, i.e., Theorems~4.1 and 4.2 are rewritten as follows
in our setting: 

\begin{prop} 
There exist $L_0$ and $\a_0$ such that if $L\geq L_0$ and if 
\[
L^{-2}\#\bigset{x\in S_0^{(L)}}{W_B^L(x)\geq \e_0}\geq f_0/2, 
\]
then
\[
\inf \s(H_0^{L;N}+W_B^L)\geq \a_0 L^{-2}.
\]
\end{prop}

\begin{prop}
\[
\Pro\Bigpare{L^{-2} \#\Bigset{f\in\faces}{f\subset S_0^{(L)}, 
1-\cos\Bigpare{\frac{B(f)}{4}} \geq \e_0} <\frac12 f_0} 
\leq \exp\Bigpare{-\frac12 f_0^2 L^2}.
\]
\end{prop}

Proposition~8 implies 
\[
\Pro\Bigpare{L^{-2} \#\Bigset{x\in S_0^{(L)}}{
W_B^L(x)\geq \e_0} <\frac12 f_0} 
\leq \exp\Bigpare{-\frac12 f_0^2 L^2}.
\]
by the definition of $W_B^L$. Combining this with Proposition~7, we observe 
\[
\Pro\Bigpare{\inf\s(H_0^{L;N}+W_B^L)< \a_0 L^{-2}} \leq
\exp\Bigpare{-\frac12 f_0^2 L^2}
\]
if $L>L_0$. Hence we have
\begin{align*}
k_L^N(E) &\leq \Pro(\inf\s(H^{L,N})\leq E) \\
&\leq \Pro\Bigpare{\inf\s(H_0^{L;N}+W_B^L)\leq 2E}
\leq \exp\Bigpare{-\frac12 f_0^2 L^2}
\end{align*}
if we choose $L$ so that $2E< \a_0 L^{-2}$, i.e., if 
\[
L < \sqrt{\a_0/2E}.
\]
We set $L$ be the largest integer satisfying the above condition, so that 
$L\sim \sqrt{\a_0/2} \cdot E^{-1/2}$ as $E\to 0$. Then we learn 
\[
k_L^N(E) \leq \exp\Bigpare{-\frac12 f_0^2 
\Bigpare{\sqrt{\a_0/2}\cdot E^{-1/2}-1}^2}  
\sim \exp\Bigpare{-\frac{\a_0}{4} f_0^2 E^{-1}}
\]
as $E\to 0$, and we conclude
\[
\limsup_{E\to 0} \log (-\log k(E)) / \log E \leq -1. 
\]

%%%%%%%%%%%%%%  APPENDIX A  %%%%%%%%%%%%%

\section*{Appendix A\ \  Generalization of local energy estimate}

Let $m\in \mathbb{N}$ and let $\faces_m$ be the set of squares of size $m$ with the
vertices at $\ze^2$, i.e., 
\[
\faces_m = \bigset{[x_1,x_1+m]\times[x_2,x_2+m]}{x_1,x_2\in \ze}.
\]
Let $A(e)$ be a given vector potential. Then for $f\in\faces_m$, we denote the 
magnetic flux through $f$ by 
\[
B(f)= \sum_{e\in \pa f} A(e), 
\]
where $\pa f$ is the boundary of $f$. Here we again identify $\re/(2\pi \ze)$ with 
$[-\pi,\pi)$ and we always suppose $B(f)\in [-\pi,\pi)$. Then we set 
\[
W_B^{(m)}(x) = \frac 1m \sum_{x\in f\in\faces_m} 
\Bigpare{1-\cos\Bigpare{\frac{B(f)}{4m}}}.
\]
\begin{thm}
Under the above notations, $H\geq W_B^{(m)}$ for any $m\in\mathbb{N}$.
\end{thm}

\begin{proof}
We note that each $e\in\edges$ is an element of $\pa f$ for $m$ different
$f\in\faces_m$. Hence, as in the proof of Theorem~2, we have 
\[
\ip{\g}{H\g} = \frac{1}{2m} \sum_{f\in\faces_m} 
\Bigabs{\g(i(e))-e^{iA(e)} \g(t(e))}^2.
\]
Then we use the following lemma:

\begin{lem}
For each $f\in \faces_m$, 
\[
\sum_{e\in\pa f} \Bigabs{\g(i(e))-e^{iA(e)}\g(t(e))}^2 
\geq 2\Bigpare{1-\cos\Bigpare{\frac{B(f)}{4m}}} \sum_{x\in f}|\g(x)|^2.
\]
\end{lem}

The lemma is proved similarly as Lemma~3, since the left hand side of the above
formula is the energy function for the free discrete Schr\"odinger operator on the
closed chain of size $4m$\/ with the periodic boundary condition with the phase
$e^{iB(f)}$. Now the theorem follows immediately. 
\end{proof}

Theorem~9 may imply better estimate if $B(f)$ does not change the sign frequently. 
Let us consider the constant magnetic case to observe this. 

\begin{example}
Suppose $B(f)=b$ for all $f\in\faces$. Then for $f\in\faces_m$, 
$B(f)= m^2b$ (modulo $2\pi\ze$).  We choose $m\in\mathbb{N}$ so that 
\[
\frac{\pi}{(m+1)^2} < |b| \leq \frac{\pi}{m^2}
\]
when $|b|$ is small enough. Then $m\sim |\pi/b|^{1/2}$ and 
$|m^2b|=\pi+O(|b|^{1/2})$ when $b\sim 0$. By simple computations, we learn
\[
H\geq W_B^{m}(x)= \frac{\pi}{4} |b| + O(|b|^{3/2})
\]
as $b\to 0$. Probably this lower bound is not optimal, but it is better than
the estimate which follows from Theorem~2 for small $b$. 
\end{example}

%%%%%%%%%%  APPENDIX B  %%%%%%%%%%%%%%

\section*{Appendix B \ \ Spectrum of Example 1}

We set $\th= b/2$, 
\[
A((x,x+\d_1))= \begin{cases} \th, \quad &\text{if $x_1+x_2$ is even}, \\
-\th, & \text{if $x_1+x_2$ is odd},  \end{cases}
\]
and $A((x,x+\d_2))=0$ for any $x\in\ze^2$. 
Then $A$ defines a magnetic Schr\"odinger operator of Example~1. 
Note that this operator is invariant under the following change of coordinates:
\[
T_1\g(x)= \g(x+\d_1+\d_2), \qquad 
T_2\g(x)= \g(x+\d_1-\d_2)
\]
where $\g\in\ell^2(\ze^2)$. Hence we can apply the Floquet-Bloch theory. 
Namely, we compute the (generalized) eigenfunction and eigenvalues of $H$ 
under conditions 
\[
T_1\g = e^{i\a}\g, \qquad T_2\g= e^{i\b}\g
\]
with fixed $\a,\b\in\re/(2\pi\ze)$. These lead to a system of equations:
\[
\left\{ 
\begin{split}
H\g(0) &= 4\g(0) - \bigbra{e^{i\th}+e^{i(\th-\a-\b)} + e^{-i\a} + e^{-i\b}} \g(\d_1), \\
H\g(\d_1) &= -\bigbra{e^{-i\th}+e^{i(-\th+\a+\b)} + e^{i\a} + e^{i\b}} \g(0) + 4\g(\d_1).
\end{split}
\right.
\]
Thus the eigenvalues are the characteristic roots of the matrix 
$\begin{pmatrix} 4 & \l \\ \overline{\l} & 4 \end{pmatrix}$ with 
$\l= -\bra{e^{i\th}+e^{i(\th-\a-\b)} + e^{-i\a} + e^{-i\b}}$. They are $4\pm |\l|$, and by
simple computations we obtain
\[
|\l|^2 = 4\bigbra{1+\cos\a\cos\b+\cos\th(\cos\a+\cos\b)}.
\]
It is then easy to see 
\[
\sup_{\a,\b} |\l|^2 = 4 (2+2\cos\th) = (4\cos(b/4))^2, 
\]
and the range of $|\l|$ (as a function of $\a$ and $\b$)  is 
$[0,4\cos(b/4)]$. Thus the spectrum of $H$ is given by
\[
\s(H)= \bigset{4\pm|\l|}{\a,\b\in\re/(2\pi\ze)}
= [4-4\cos(b/4), 4+4\cos(b/4)]
\]
and it is absolutely continuous. 


%%%%%%%%%%%%%% REFERENCES %%%%%%%%%%%%%%%%%%%%

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