Content-Type: multipart/mixed; boundary="-------------0209171917703" This is a multi-part message in MIME format. ---------------0209171917703 Content-Type: text/plain; name="02-387.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="02-387.keywords" Kronig-Penney model, Landauer resistivity, quasiperiodic potential ---------------0209171917703 Content-Type: application/x-tex; name="Landauer14.tex" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="Landauer14.tex" %#format LaTeX \ifx\documentclass\undefined \documentstyle[12pt]{article} \else \documentclass[12pt]{article} \fi %\documentclass[11pt,a4j, leqno]{article} % leqno: numbering for expressions is put on the left \sloppy %%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Adaptation of spaces in eqnarray \makeatletter \renewcommand{\theequation}{\thesection.\arabic{equation}} \@addtoreset{equation}{section} \def\eqnarray{% \stepcounter{equation}% \let\@currentlabel=\theequation \global\@eqnswtrue \global\@eqcnt\z@ \tabskip\@centering \let\\=\@eqncr $$\halign to \displaywidth\bgroup\@eqnsel\hskip\@centering $\displaystyle\tabskip\z@{##}$&\global\@eqcnt\@ne \hfil$\displaystyle{{}##{}}$\hfil &\global\@eqcnt\tw@$\displaystyle\tabskip\z@{##}$\hfil \tabskip\@centering&\llap{##}\tabskip\z@\cr} \makeatother %%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Taken From Mathsing \def\bbbr{{\rm I\!R}} %reelle Zahlen \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{$\sf\scriptscriptstyle Z\kern-0.2em Z$}}}} % \def\bbbq{{\mathchoice {\setbox0=\hbox{$\displaystyle\rm Q$}\hbox{\raise 0.15\ht0\hbox to0pt{\kern0.4\wd0\vrule height0.8\ht0\hss}\box0}} {\setbox0=\hbox{$\textstyle\rm Q$}\hbox{\raise 0.15\ht0\hbox to0pt{\kern0.4\wd0\vrule height0.8\ht0\hss}\box0}} {\setbox0=\hbox{$\scriptstyle\rm Q$}\hbox{\raise 0.15\ht0\hbox to0pt{\kern0.4\wd0\vrule height0.7\ht0\hss}\box0}} {\setbox0=\hbox{$\scriptscriptstyle\rm Q$}\hbox{\raise 0.15\ht0\hbox to0pt{\kern0.4\wd0\vrule height0.7\ht0\hss}\box0}}}} % \def\bbbc{{\mathchoice {\setbox0=\hbox{$\displaystyle \rm C$}\hbox{\raise 0.06\ht0\hbox to0pt{\kern0.4\wd0\vrule height0.9\ht0\hss}\box0}} {\setbox0=\hbox{$\textstyle\rm C$}\hbox{\raise 0.06\ht0\hbox to0pt{\kern0.4\wd0\vrule height0.9\ht0\hss}\box0}} {\setbox0=\hbox{$\scriptstyle\rm C$}\hbox{\raise 0.06\ht0\hbox to0pt{\kern0.4\wd0\vrule height0.8\ht0\hss}\box0}} {\setbox0=\hbox{$\scriptscriptstyle\rm C$}\hbox{\raise 0.06\ht0\hbox to0pt{\kern0.4\wd0\vrule height0.8\ht0\hss}\box0}}}} % \def\B{\bf B} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % eqnum \makeatletter \renewcommand{\theequation}{% \thesection.\arabic{equation}} \@addtoreset{equation}{section} \makeatother %%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}{Lemma}[section] \newtheorem{corollary}{Corollary}[section] \newtheorem{proposition}{Proposition}[section] \newtheorem{remark}{Remark}[section] \newtheorem{definition}{Definition}[section] \newsavebox{\toy} \savebox{\toy}{\framebox[0.65em]{\rule{0cm}{1ex}}} \newcommand{\QED}{\usebox{\toy}} \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 $0 \le c_j(\epsilon_F)2$: % 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. 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