Content-Type: multipart/mixed; boundary="-------------0008062137716" This is a multi-part message in MIME format. ---------------0008062137716 Content-Type: text/plain; name="00-311.comments" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="00-311.comments" 9 Pages, 3 figures ---------------0008062137716 Content-Type: text/plain; name="00-311.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="00-311.keywords" Spectral Theory, Graph Theory ---------------0008062137716 Content-Type: application/x-tex; name="SpectralGaps.tex" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="SpectralGaps.tex" % ----------------------------- % AMS-LaTeX Paper ****************** % J.H. Schenker and M. Aizenman: % ``Spectral Gaps and Graph Decoration'' % **** ----------------------- \documentclass[reqno]{amsart} \usepackage{amsfonts,amsmath} % amsmath,times,psfig} \date{Aug. 4, 2000} \input epsf %preferred figures program \newcommand{\eq}[1]{eq.~(\ref{#1})} \def\abs#1{\left\vert #1 \right\vert} \def\ip#1{\left < #1 \right >} \def\c{\mathrm{Const.}} \def\Chi {{\cal X}} \def\d {{\rm d}} \def\deg {{\rm deg}} \def\dist{\operatorname{dist}} %%% distance \def\dim{\operatorname{dim}} %%% dimension \def\E{{\mathbb E}} %%% for "expectation value" (in math mode) \def\half {{1 \over 2}} \def\norm#1{\left\| #1 \right\|} \def\Z{{\mathbb Z}} %%% for Z (in math mode) \def\N{{\mathbb N}} %%% for N (in math mode) \def\Pr{\operatorname{Prob}} %%% appears in many equations Prob \def\R{{\mathbb R}} %%% for R (in math mode) \def\C{{\mathbb C}} %%% Complex (in math mode) \def\T{{\mathbb T}} \def\H{{\mathcal H}} \def\K{{\mathcal K}} \def\B{\mathcal B} \def\F{\mathcal F} \def\O{{\rm O}} \def\i{{\rm i}} \def\1{{\bf 1}} % THEOREMS ------------------------------------------------------- \newtheorem{thm}{Theorem}[section] \newtheorem{cor}[thm]{Corollary} \newtheorem{lem}[thm]{Lemma} \newtheorem{prop}[thm]{Proposition} \theoremstyle{definition} \newtheorem{defn}[thm]{Definition} \theoremstyle{remark} \newtheorem{rem}[thm]{Remark} \numberwithin{equation}{section} % MATH ----------------------------------------------------------- \newcommand{\set}[1]{\left\{#1\right\}} \newcommand{\eps}{\varepsilon} \newcommand{\To}{\longrightarrow} \def\blackbox{{\vrule height 1.3ex width 1.0ex depth -.2ex} \hskip 1.5truecm} \newenvironment{proof_of}[1]{ \noindent{\bf Proof of #1: } }{ \hfill \blackbox \bigskip} % ---------------------------------------------------------------- \begin{document} \bibliographystyle{mabib} \title[Spectral Gaps and Graph Decoration]{The Creation of Spectral Gaps by Graph Decoration} \author[J.H. Schenker]{Jeffrey H. Schenker$^{(a)}$} \address{Princeton University, Departments of Mathematics$^{(a)}$ and Physics$^{(b)}$, Princeton, NJ 08544, USA.} \email{schenker@princeton.edu, aizenman@princeton.edu} \author[M. Aizenman]{Michael Aizenman$^{(a,b)}$} % ---------------------------------------------------------------- \begin{abstract} We present a mechanism for the creation of gaps in the spectra of self-adjoint operators defined over a Hilbert space of functions on a graph, which is based on the process of graph decoration. The resulting Hamiltonians can be viewed as associated with discrete models exhibiting a repeated local structure and a certain bottleneck in the hopping amplitudes. \end{abstract} \maketitle % ---------------------------------------------------------------- \section{Introduction} Energy spectra characterized by the presence of bands and gaps are familiar from the Bloch theory of periodic systems. In this note, we present another mechanism for the creation of spectral gaps which does not rely on translation invariance. The band-gap spectral structure plays an important role in the theory of the solid state~\cite{Kittel}, as well as in the properties of dialectric and acustic media~\cite{FiKl}. Of particular interest are also situations in which localized states are injected into existing gaps (see ref.~\cite{DH} for a mathematical discussion with further references). These applied models are mentioned only as distant analogies; the topic we discuss pertains to spectral properties of Hamiltonians of discrete models, whose ``hopping terms'' can be viewed as associated with graphs exhibiting a repeated local structure and a certain bottleneck in hopping amplitudes. To present the principle described herein it is convenient to introduce the notion of ``graph decoration''. Given two graphs $\Gamma$ and $G$, we may ``decorate'' $\Gamma$ with $G$ by ``gluing'' a copy of $G$ to each vertex $v$ of $\Gamma$ in such a way that $v$ is identified with the appropriate copy of some distinguished vertex $\O_G \in G$ (see \S\ref{sec:decoration} for a formal definition, and figures \ref{fig:example1} and \ref{fig:example2} for typical examples). Given self-adjoint operators $H_o$ on $\ell^2(\Gamma)$ and $A$ on $\ell^2(G)$ there is a natural way to define an operator extension $H$ of $H_o$ and $A$ to $\ell^2(\Gamma \triangleleft G)$ where $\Gamma \triangleleft G$ denotes the decorated graph just described. In the absence of certain degeneracy, there is a simple relation between the spectra of $H$ and $H_o$, denoted here by $\sigma(H_{...})$, which allows us to conclude that intervals around certain energies $\eps_j$ are excluded from the spectrum of $H$. Specifically, there is a function $\gamma$ such that \begin{equation} \sigma (H) \ = \ \gamma^{-1} \left ( \sigma (H_o) \right ) \; , \label{eq:spectralsplitting} \end{equation} and $\gamma$ is of the form \begin{equation} \gamma(E) \ = \ E + c + \sum_j w_j \ {1 \over \eps_j - E} \; , \end{equation} where $w_j > 0$ and $\eps_j,c \in \R$ (see figure \ref{fig:typicalgamma}). In fact, we shall see that $\eps_j$ are exactly the eigenvalues of the operator $\widehat P A \widehat P $ where $\widehat P$ is the projection onto the subspace of functions in $\ell^{2}(G)$ which vanish at $\O_{G}$. Thus, we have the appealing picture in which the eigenenergies of the decorated graph are repelled by resonances with the ``inner spectrum'' of the decoration. \begin{figure}[tbp] \centering \epsfxsize=3in \epsffile{gamma.eps} \caption{\label{fig:typicalgamma} A schematic depiction of the spectral relation $\sigma(H) = \gamma^{-1}(\sigma(H_{o}) )$. } \end{figure} In \S\ref{sec:decoration} we define graph decoration and describe the operator extension mentioned above. In \S\ref{sec:evaluation} we present our main result (Prop. \ref{prop:main}) which describes the spectral relationship presented above in full detail. Finally, in \S\ref{sec:examples} we provide several examples and applications of Prop. \ref{prop:main}. \section{Graph decoration}\label{sec:decoration} In this section, we suppose that we are given a graph $\Gamma$ and a self adjoint operator $H_o$ on $\ell^2(\Gamma)$, the space of square summable functions on the vertices of $\Gamma$ (see below). Our goal is to describe a certain class of graph extensions of $\Gamma$ and a corresponding class of operator extensions of $H_o$. Recall that a graph $G$ is described by specifying two sets: (1) $V(G)$ whose elements are called vertices, and (2) $E(G)$ a set of (unordered) pairs of vertices called edges. The edges play a secondary role in our discussion, for we are mainly concerned with the Hilbert space of square summable functions mapping $V(G) \rightarrow \C$, which we denote $\ell^2(\Gamma)$. The situation of interest is when $E(G)$ is defined so that a given operator $A$ on $\ell^2(G)$ is {\it compatible} with $G$, by which we mean that the off diagonal matrix elements $\ip{x|A|y}$ vanish whenever $\{x,y\} \not \in E(G)$. \footnote{We use the Dirac bra-ket notation for matrix elements in the standard basis $ \left | x \right > \ = \ \delta_x $, with $\delta_x (y) $ the Kronecker function.} Correspondingly, our notation generally identifies a graph $G$ with its vertex set: by $x \in G$ we indicate $x \in V(G)$ and we shall say that a graph is countable (finite) if $V(G)$ is countable (finite). The graph extensions of $\Gamma$ shall be obtained by ``gluing'' copies of a second graph $G$ to each vertex of $\Gamma$. The extended graph may be visualized as a field in which are tethered many identical kites (see figures \ref{fig:example1} and \ref{fig:example2}). Formally given any graph $G$ with a distinguished vertex $\O_G$ we define the {\it decoration of~ $\Gamma$ by $(G,\O_G)$}, denoted $\Gamma \triangleleft G$, to be the following graph: \begin{enumerate} \item $V \left ( \Gamma \triangleleft G \right ) = V(\Gamma) \times V(G)$ . \item $E \left ( \Gamma \triangleleft G \right ) = E_{\rm field} \cup E_{\rm kite}$ , where: \begin{enumerate} \item $E_{\rm field}= \left \{ \ \{(x,\O_G),(y,\O_G) \} \ | \ \{x,y\} \in E(\Gamma) \ \right \}$ . \item $E_{\rm kite} = \left \{ \ \{(x,h),(x,g)\} \ | \ x \in V(\Gamma) \mbox{ and } \{h,g\} \in E(G) \ \right \}$ . \end{enumerate} \end{enumerate} We think of the space $\ell^2(\Gamma \triangleleft G)$ as the tensor product $\ell^2(\Gamma) \otimes \ell^2(G)$, which is natural since the vertex set of $\Gamma \triangleleft G$ is $V(\Gamma) \times V(G)$. The subspace of functions which are supported on $\Gamma_o = \{(x,\O_g): x \in \Gamma\}$ is naturally identified with $\ell^{2}(\Gamma)$. We denote by $P$ the orthogonal projection onto this space. Let $A$ be a self adjoint operator on $\ell^2(G)$. A natural extension of $H_{o}$ to $\ell^2(\Gamma) \otimes \ell^2(G)$, incorporating $A$, is \begin{equation}\label{eq:defnofH} H \ := \ P H_o P + \1 \otimes A \;. \end{equation} The above operator is appropriate to the geometry of graph decoration, for if $H_o$ and $A$ are compatible with $\Gamma$ and $G$ respectively, then $H$ is compatible with $\Gamma \triangleleft G$. An example of an operator of the form described in eq. \eqref{eq:defnofH} is provided by the discrete Laplacian. On any graph $H$ the discrete Laplacian, $\Delta_H$, is defined by \begin{equation} [\Delta_H \psi](x) \ := \ \sum_{y:\, \{x,y\} \in E(H)} \psi(y) - \psi(x) \; . \end{equation} For decorated graphs, if we take $H_o = -\Delta_\Gamma$ and $A = - \Delta_G$ then the operator defined by \eqref{eq:defnofH} is $H = -\Delta_{\Gamma \triangleleft G}$. \section{A resolvent evaluation principle}\label{sec:evaluation} We now focus on the case $|G| < \infty$ and present our main result.\footnote{This result may be easily extended to the case $|G|= \infty$ provided the spectrum of $A$ is discrete.} \begin{prop}\label{prop:main} Let $H$ be a bounded \footnote{More generally, $H$ may be unbounded provided the set of functions with finite support forms a core for $H$.} self adjoint operator of the form described in eq. \eqref{eq:defnofH} with $G$ a finite graph. If $\left |\O_G \right > $ is a cyclic vector for $A$, then \begin{equation}\label{eq:spectralmap} \sigma(H) \ = \ \gamma^{-1}\left (\sigma(H_o) \right )\; , \end{equation} where $\gamma$ is a function of the form \begin{equation}\label{eq:formofgamma} \gamma(E) \ = \ E \ + \ c \ + \sum_{j=1}^{n} w_j {1 \over \eps_j - E } \; , \end{equation} with $c,\eps_j \in \R$, and $w_j > 0$. Furthermore, whether or not $\left |\O_G \right>$ is cyclic, there is a function $\gamma$ of the form \eqref{eq:formofgamma} such that for each $x \in \Gamma$ and $z \in \C \setminus \R$ \begin{equation}\label{eq:greenrelation} \ip{x,\O_G| (H-z)^{-1} |x,\O_G} \ = \ \ip{x|(H_o-\gamma(z))^{-1}|x} \; , \end{equation} and the spectral measure, $\widetilde \mu_x$, for $H$ associated to $\left |x,\O_G \right >$ is related to the spectral measure, $\mu_x$, for $H_o$ associated to $\left |x \right >$ by \begin{equation}\label{eq:spectralmeasures} \widetilde \mu_x(dE) \ = \ {1 \over \gamma'(E)} \ \mu_x(d \gamma(E)) \; . \end{equation} Thus \begin{equation}\label{eq:spectralsubset} \gamma^{-1}(\sigma(H_o)) \ \subseteq \ \sigma(H) \; . \end{equation} \end{prop} {\noindent \bf Remarks:} \begin{itemize} \item Recall that given a self-adjoint operator $K$ on a Hilbert space $\mathcal K$, the spectral measure associated to a vector $v \in {\mathcal K}$ is defined via the functional calculus and the Riesz-Markov theoerem as the unique regular Borel measure, $\nu$, such that \begin{equation} \int f(E) \nu(dE) \ = \ \ip{v,f(K)v} \; , \end{equation} for each $f \in C_o(\R)$, the family of continuous functions on $\R$ which vanish at infinity. \item Eq. \eqref{eq:spectralmeasures} is a formal expression which indicates the following identity for the expectations of a function $f \in C_o(\R)$: \begin{equation} \int \widetilde \mu_x(dE) \, f(E) \ = \ \int \mu_x(d \eps) \, \sum_{E \in \gamma^{-1}(\eps)} {1 \over \gamma'(E)} f(E) \; . \end{equation} \end{itemize} \begin{proof_of}{Proposition \ref{prop:main}} The heart of Prop.~\ref{prop:main} is the relation \eqref{eq:greenrelation}, so let us begin with a derivation of this equation. Fix $x \in \Gamma$ and $z \in \C \setminus \R$. Recall that the Green function, \begin{equation} G(y,u) \ := \ \ip{x,\O_G | (H-z)^{-1} |y,u} \; , \ (y,u) \in \Gamma \triangleleft G \; , \end{equation} is the unique square summable solution to the equation \begin{equation}\label{eq:greeneq} (H - z )G = \left | x, \O_G \right > \; . \end{equation} A natural guess is that the solution factors: \begin{equation}\label{eq:greenansatz} G(y,u) \ = \ g(y) \, h(u) \; . \end{equation} With this ansatz, eq. \eqref{eq:greeneq} yields for $g$ and $h$: \begin{subequations} \begin{equation} g(y) \, [ (A - z)h](u) \ = \ 0 \; , \ u \neq \O_G \; , \label{eq:g} \end{equation} and \begin{equation} [H_o g](y) \, h(\O_g) \ + \ g(y) \, [(A - z)h](\O_g) \ = \ \left | x \right > \; . \label{eq:h} \end{equation} \end{subequations} It is now an easy exercise to solve these equations using the Green functions for $H_o$ and $A$: eq. \eqref{eq:g} gives $g$ as a function of $h$, \begin{equation} g(y) \ = \ {1 \over h(\O_G)} \, \ip{x| \left (H_o + { [(A - z)h](\O_g) \over h(\O_g) } \right )^{-1} |y} \; ; \end{equation} while eq. \eqref{eq:h} shows that $h$ is a multiple of the Green function for $A$, \begin{equation} h(u) \ = \ C \, \ip{\O_G | (A-z)^{-1}|u} \; , \end{equation} with $C$ an arbitrary factor which drops out of the resulting solution: \begin{equation} G(y,u)\ \ = \ \ip{x| \left (H_o + { 1 \over \ip{\O_G | (A-z)^{-1}|\O_G} } \right )^{-1} |y} \, {\ip{\O_G | (A-z)^{-1}|u} \over \ip{\O_G | (A-z)^{-1}|\O_G}} \; . \end{equation} Setting $u = \O_G$ and $y = x$ in this expression yields \eqref{eq:greenrelation} with \begin{equation} \gamma(z) \ := \ {-1 \over \ip{\O_G | (A - z)^{-1} |\O_G} } \; . \end{equation} Because $\dim \ell^2(G)$ is finite, $\gamma$ is a rational function with finitely many simple real poles (which occur at the zeros of $\ip{\O_G | (A- E)^{-1} | \O_G}$). Hence, the partial fraction expansion (alternatively, the represenation theory of Herglotz functions) shows that $\gamma$ is of the form displayed in eq. \eqref{eq:formofgamma}: \begin{equation} \gamma(z) \ = \ z \ - \ c \ + \ \sum_{j=1}^{n -1} w_j {1 \over \eps_j - z} \; , \end{equation} with $c, \eps_j \in \R$ and $w_j > 0$. (The coefficient of $z$ is one, since $z / \gamma(z) \rightarrow 1$ as $z \rightarrow \infty$.) We now turn to the verification of the relation between the spectral measures expressed in \eqref{eq:spectralmeasures}. First consider the situation when $\Gamma$ is finite. For a self-adjoint operator $B$ on a finite dimensional vector space there is a useful formula for the spectral measure, $\nu_\psi$, associated to a vector $\psi$: \begin{equation} \nu_\psi(dE) \ = \ \delta \left ( {-1 \over \ip{\psi, (B-E)^{-1} \psi}} \right )dE \; , \label{eq:deltaequation} \end{equation} where $\delta$ is the Dirac-delta ``function.'' (This formula offers a simple route to the ``spectral averaging'' principle discussed in ref.~\cite{SW}; its derivation is an instructive exercise which we leave to the reader.) Coupled with eq.~\eqref{eq:greenrelation}, \eqref{eq:deltaequation} easily yields: \begin{multline} \widetilde \mu_x(dE) \ = \ \delta \left ( {-1 \over \ip{x,\O_G| (H - E)^{-1}|x,\O_G}} \right ) dE \\ = \ \delta \left ( {-1 \over \ip{x| (H_o - \gamma(E))^{-1}|x}} \right ) {1 \over \gamma'(E)} d\gamma(E) \ = \ {1 \over \gamma'(E)} \mu_x(d\gamma(E)) \; . \end{multline} When $\Gamma$ is infinite we must turn to a more abstract derivation of eq. \eqref{eq:spectralmeasures}. Writing each side of eq. \eqref{eq:greenrelation} as a spectral integral we find that \begin{equation} \int {1 \over E - z} d \widetilde \mu_x (E) \ = \ \int {1 \over E - \gamma(z)} d\mu_x(E) \; . \end{equation} Expanding the right side of this equality with partial fractions yields \begin{equation} \int {1 \over E - z} d \widetilde \mu_x (E) \ = \ \int \sum_{\lambda \in \gamma^{-1}(E)} {1 \over \gamma'(\lambda)} {1 \over \lambda - z} d\mu_x(E) \; . \label{eq:rescase} \end{equation} This equation can be viewed as a special case of: \begin{equation} \int f(E) \, d \widetilde \mu_x(E) \ = \ \int \sum_{\lambda \in \gamma^{-1}(E)}{1 \over \gamma'(\lambda)} f(\lambda) \, d \mu_x(E) \; , \label{eq:general} \end{equation} and indeed \eqref{eq:rescase} implies \eqref{eq:general}, for all $f \in \C_o(\R)$, since by the Stone-Weierstrass Theorem the set of finite sums of the form $\sum_j c_j {1 \over E - z_j}$ is dense in $ \C_o(\R)$. As mentioned previously, \eqref{eq:general} is the statement claimed in \eqref{eq:spectralmeasures}. Finally, the spectral inclusion $\gamma^{-1}(\sigma(H_o)) \subseteq \sigma(H)$ follows since \begin{equation}\label{eq:unionofsupports} \gamma^{-1}(\sigma(H_o)) \ = \ \bigcup_{x \in \Gamma} \mbox{supp.}(\widetilde \mu_x) \; , \end{equation} which may be verified using \eqref{eq:spectralmeasures}. If $\{\left | x,\O_G \right >\}$ is a cyclic family for $H$ then eq. \eqref{eq:unionofsupports} shows further that $\gamma^{-1}(\sigma(H_o)) = \sigma(H)$. In case $\left | \O_G \right >$ is a cyclic vector for $A$, this family is easily seen to be cyclic for $H$. This completes the proof of the proposition. \end{proof_of} We conclude this section with several remarks regarding proposition \ref{prop:main} and its proof: \begin{enumerate} \item Eigenfunctions and generalized eigenfunctions of $H$ factor in the same way as the Green function (see eq. \eqref{eq:greenansatz}). That is \begin{equation} \Psi(y,u) \ = \ \psi(y) \, \phi(u) \end{equation} satisfies $H\Psi = E \Psi$ provided \begin{subequations} \begin{equation} \phi(u) \ = \ {\ip{\O_G|(A-E)^{-1}|u} \over \ip{\O_G|(A-E)^{-1} |\O_G} } \; , \end{equation} and \begin{equation} H_o \psi = \gamma(E) \psi \; . \end{equation} \end{subequations} \item The relationship between the spectrum of $H_o$ and $H$ is a stronger relationship than the simple inclusion $\gamma^{-1}(\sigma(H_o)) \subset \sigma(H)$: {\em the spectral type is preserved under the map $\gamma^{-1}$}. So, a bound state for $H_o$ gives rise to $n$ bound states for $H$. Similarly, a band of absolutely continuous spectrum for $H_o$ gives rise to $n$ bands of absolutely continuous spectrum for $H$. If $H_o$ possesses singular continuous spectrum, then such spectrum also occurs in the spectrum of $H$. \item Generically, $\left |\O_G \right >$ is a cyclic vector for $A$, and $\sigma(H) \ = \ \gamma^{-1}(\sigma(H_o))$. However, even when $\left |\O_G \right >$ is not cyclic, we may still determine the spectrum of $H$. We need only decompose the space $\ell^2(G)$ as a direct sum $V \oplus V^\bot$ with each summand invariant under $A$ and $\left |\O_G \right >$ cyclic for $A|_V$. Then, \begin{equation} \sigma(H) = \gamma^{-1}(\sigma(H_o)) \cup \sigma(A|_{V^\bot}) \; , \end{equation} which may be verified by noting that \begin{equation} H \cong \begin{pmatrix} H_o + \1 \otimes A|_V & 0 \\ 0 & \1 \otimes A|_{V^\bot} \end{pmatrix} \; . \end{equation} Note that the eigenvalues in $\sigma(A|_{V^\bot})$ occur with multiplicity a multiple of $|\Gamma|$. \item The poles $\eps_j$ of $\gamma$ are eigenvalues of $\widehat A = \widehat P A \widehat P$, where $\widehat P$ is the projection of $\ell^2(G)$ onto the space of functions which vanish at $\O_G$. To see this, recall that $\eps_j$ satisfy $\ip{\O_G | (A-\eps_j)^{-1} |\O_G} = 0$. Thus the Green functions $\psi_{j}(y) \ = \ \ip{\O_G | (A - \eps_j)^{-1} |y}$ themselves are eigenfunctions: $\widehat A \psi_j = \eps_j \psi_j$. Conversely, if $\left | \O_G \right > $ is cyclic for $A$ then every eigenvalue of $\widehat A$ is a pole of $\gamma$. \item The coefficients $c$ and $w_j$ in the partial fraction expansion of $\gamma$ (eq. \eqref{eq:formofgamma}) satisfy \begin{subequations} \begin{equation} c \ = \ -\ip{\O_G \, | \, A \, | \, \O_G} \; , \end{equation} \begin{equation}\label{eq:uncertainty} \sum_{j=1}^n w_j \ = \ \ip{\O_G \, | \, A^2 \, | \, \O_G} - \ip{\O_G \, | \, A \, | \, \O_G}^2 \; , \end{equation} and \begin{equation} w_j \ = \ \lim_{E \rightarrow \eps_j} (\eps_j - E) \gamma(E) \ = \ {1 \over \ip{\O_G|(A-\eps_j)^{-2}|\O_G}} \; . \end{equation} \end{subequations} The first two equalities may be verified by expanding $\gamma$ in a Laurent series around $0$. \end{enumerate} \section{Examples and Applications}\label{sec:examples} \subsection{Splitting the spectrum of the Laplacian on $\Z^d$}\label{sec:example1} For a simple example of the phenomenon, consider the operator $H = -\Delta_{\Z^d \triangleleft G}$ where $G$ is the graph consisting of two vertices $V(G) = \{\O_G, 1_G\}$ and a single edge $E(G) = \{\{\O_G,1_G\}\}$ (see figure \ref{fig:example1}). As described at the end of \S \ref{sec:decoration}, $H$ is of the form \eqref{eq:defnofH} with $H_o = -\Delta_{\Z^d}$ and $A=-\Delta_G$. \begin{figure} [tbp] \centering \epsfysize=1.9in \epsffile{example1.eps} \caption{\label{fig:example1} A portion of the graph discussed in \S\ref{sec:example1} in the case $d=2$. % A situation in which $\left |\O_G % \right >$ is cyclic (sec \ref{sec:example1}). } \end{figure} The spectrum of $H_{o}=-\Delta_{\Z^d}$ is \begin{equation} \sigma(H_{o}) = [0,4 d] \end{equation} and all the associated spectral measures are purely absolutely continuous. In this case, the function $\gamma$ is easy to calculate: \begin{equation} \gamma(E) \ = \ - \left [ {1-E \over (1-E)^2 -1 }\right ]^{-1} \ = \ E \ - \ 1 \ + \ {1 \over 1 - E} \; , \end{equation} and the vector $\left |\O_G \right >$ is cyclic. Thus \begin{equation} \begin{split} \sigma(H) \ =& \ \left \{ E \mid 0 \ \le \ E \ - \ 1 \ + \ {1 \over 1 - E} \ \le \ 4 d \right \} \\ =& \ \left [ 0, 1 + 2d - \sqrt{1 + 4d^2 } \right ] \cup \left [ 2, 1 + 2d + \sqrt{1 + 4d^2} \right ] \; , \end{split} \end{equation} and the spectral measures are purely absolutely continuous. \subsection{An example in which $\left | \O_G \right >$ is not cyclic.} \label{sec:example2} Consider now the discrete Laplacian on the graph $\Z^d \triangleleft G$ where $G$ is the fully connected graph with three vertices $V(G) = \{\O_G,1_G,2_G\}$ (see figure \ref{fig:example2}). \begin{figure} [bpt] \centering \epsfysize=1.9in \epsffile{example2.eps} \caption{\label{fig:example2}A situation in which $\left |\O_G \right >$ is not cyclic (\S \ref{sec:example2}).} \end{figure} The involution $R$ on $\ell^2(G)$ obtained by interchanging $\left | 1_G \right >$ and $\left | 2_G \right >$ commutes with $\Delta_G$. Hence $\Delta_G$ leaves invariant the subspaces $V_{+(-)}$ of functions which are symmetric (anti-symmetric) with respect to this involution. A non-normalized basis of simultaneous eigenfunctions for $R$ and $(-\Delta_G ) $ consists of: $\left |\O_G \right > + \left |1_G \right > + \left |2_G \right > \in V_+$ with eigenvalue $0$ , $2 \left |\O_G \right > - \left |1_G \right > - \left |2_G \right > \in V_+ $ with eigenvalue $3$ , and $ \left |1_G \right > - \left |2_G \right > \in V_- $ with eigenvalue $3$. Using this basis, it is easy to see that $\left |\O_G \right >$ is a cyclic vector for the restriction of $\Delta_G$ to $V_+$. Furthermore, we can calculate $\gamma$: \begin{equation} \gamma(E) \ = \ {- 1 \over {2 \over 3} {1 \over 3 - E} + {1 \over 3}{1 \over 0 - E}} \ = \ E - 2 + {2 \over 1 - E} \; , \end{equation} and note that $\sigma(-\Delta_G|_{V_-}) = \{3\}$. Thus, \begin{equation} \begin{split} \sigma(-\Delta_{\Gamma \triangleleft G}) \ =& \ \{3\} \cup \left \{ E \mid 0 \le {E - 2 + {2 \over 1 - E} } \le 4 d \right \} \\ =& \left [ 0, \eps^- \right ] \cup \left [ 3, \eps^+ \right ] \; , \end{split} \end{equation} where $\eps^{\pm}$ are the solutions to \begin{equation} E - 2 + {2 \over 1 - E} \ = \ 4d \; , \end{equation} with $\eps^- < 1$ and $\eps^+ > 1$. The spectrum is purely absolutely continuous except for the presence of an infinitely degenerate eigenvalue at $E=3$. \subsection{Persistence of band edge localization} The operator $H$ may include disorder, in the form of a random potential at the sites of $\Gamma_{o}$. It is generally expected that in such a situation the spectrum of $H_{o}$ will exhibit Anderson localization ({\it i.e.}, dense pure point spectrum) at all the spectral edges. (This has been rigorously shown to be true in various situations~ \cite{FiKl,BCH,KSS,Klp,ASFH}). Let us note that the mechanism of gap creation via graph decorations preserves such band edge localization, even if the randomness is not introduced at all the sites of the decorated graph. \newpage % \bibliography{resolvent} \begin{thebibliography}{1} \bibitem{Kittel} C.~Kittel, {\em Quantum Theory of Solids}. \newblock (Wiley: New York, London, Sydney, 1963). \bibitem{FiKl} A.~Figotin and A.~Klein, ``Midgap defect modes in dielectric and acoustic media,'' {\em SIAM J. Appl. Math}, {\bf 58}, 1748, (1998). \bibitem{DH} P.~A. Deift and R.~Hempel, ``On the existence of eigenvalues of the {S}chr{\"o}dinger operator {$H-\lambda W$} in a gap of {$\sigma(H)$},'' {\em Comm. Math. Phys.}, {\bf 103}, 461, (1986). \bibitem{SW} B.~Simon and T.~Wolff, ``Singular continuous spectrum under rank one perturbations and localization for random {H}amiltonians,'' {\em Comm. Pure Appl. Math.}, {\bf 39}, 75, (1986). \bibitem{BCH} J.~M.Barbaroux, J.-M. Combes, and P.~D. Hislop, ``Localization near band edges for random {S}chr{\"o}dinger operators,'' {\em Helv. Phys. Acta}, {\bf 70}, 16, (1997). \bibitem{KSS} W.~Kirsch, P.~Stollman, and G.~Stolz, ``Localization for random perturbations of periodic {S}chr{\"o}dinger operators,'' {\em Rand. Op. Stoch. Eq.}, {\bf 6}, 241, (1998). \bibitem{Klp} F.~Klopp, ``Internal {L}ifshits tails for random perturbations of periodic {S}chr\"odinger operators.,'' {\em Duke Math. J.}, {\bf 98}, 335, (1999). \bibitem{ASFH} M.~Aizenman, J.~H.Schenker, R.~M. Friedrich, and D.~Hundertmark, ``Finite-volume fractional-moment criteria for {A}nderson localization,'' to appear in {\em Comm. Math. Phys.} \newblock {http://xxx.lanl.gov/abs/math-ph/9910022}. \end{thebibliography} \end{document} % ---------------------------------------------------------------- ---------------0008062137716 Content-Type: application/postscript; name="example1.eps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="example1.eps" %!PS-Adobe-3.0 EPSF-3.0 %%BoundingBox: 0 0 361 253 %%LanguageLevel: 1 %%Creator: Corel PHOTO-PAINT 9 %%Title: example1.eps %%CreationDate: Tue Aug 01 17:30:41 2000 %%DocumentProcessColors: Black %%DocumentSuppliedResources: (atend) %%EndComments %%BeginProlog /AutoFlatness false def /AutoSteps 0 def /CMYKMarks true def /UseLevel 1 def %Build: Corel PHOTO-PAINT 9 Version 9.337 %Color profile: Generic offset separations profile %%BeginResource: procset wCorel9Dict 9.0 0 /wCorel9Dict 300 dict def wCorel9Dict begin % Copyright (c)1992-1999 Corel Corporation % All rights reserved. v9.0 r0.5 /bd{bind def}bind def/ld{load def}bd/xd{exch def}bd/_ null def/rp{{pop}repeat} bd/@cp/closepath ld/@gs/gsave ld/@gr/grestore ld/@np/newpath ld/Tl/translate ld /$sv 0 def/@sv{/$sv save def}bd/@rs{$sv restore}bd/spg/showpage ld/showpage{} bd currentscreen/@dsp xd/$dsp/@dsp def/$dsa xd/$dsf xd/$sdf false def/$SDF false def/$Scra 0 def/SetScr/setscreen ld/@ss{2 index 0 eq{$dsf 3 1 roll 4 -1 roll pop}if exch $Scra add exch load SetScr}bd/SepMode_5 where{pop}{/SepMode_5 0 def}ifelse/CurrentInkName_5 where{pop}{/CurrentInkName_5(Composite)def} ifelse/$ink_5 where{pop}{/$ink_5 -1 def}ifelse/$c 0 def/$m 0 def/$y 0 def/$k 0 def/$t 1 def/$n _ def/$o 0 def/$fil 0 def/$C 0 def/$M 0 def/$Y 0 def/$K 0 def /$T 1 def/$N _ def/$O 0 def/$PF false def/s1c 0 def/s1m 0 def/s1y 0 def/s1k 0 def/s1t 0 def/s1n _ def/$bkg false def/SK 0 def/SM 0 def/SY 0 def/SC 0 def/$op false def matrix currentmatrix/$ctm xd/$ptm matrix def/$ttm matrix def/$stm matrix def/$ffpnt true def/CorelDrawReencodeVect[16#0/grave 16#5/breve 16#6/dotaccent 16#8/ring 16#A/hungarumlaut 16#B/ogonek 16#C/caron 16#D/dotlessi 16#27/quotesingle 16#60/grave 16#7C/bar 16#82/quotesinglbase/florin/quotedblbase/ellipsis/dagger/daggerdbl 16#88/circumflex/perthousand/Scaron/guilsinglleft/OE 16#91/quoteleft/quoteright/quotedblleft/quotedblright/bullet/endash/emdash 16#98/tilde/trademark/scaron/guilsinglright/oe 16#9F/Ydieresis 16#A1/exclamdown/cent/sterling/currency/yen/brokenbar/section 16#a8/dieresis/copyright/ordfeminine/guillemotleft/logicalnot/minus/registered/macron 16#b0/degree/plusminus/twosuperior/threesuperior/acute/mu/paragraph/periodcentered 16#b8/cedilla/onesuperior/ordmasculine/guillemotright/onequarter/onehalf/threequarters/questiondown 16#c0/Agrave/Aacute/Acircumflex/Atilde/Adieresis/Aring/AE/Ccedilla 16#c8/Egrave/Eacute/Ecircumflex/Edieresis/Igrave/Iacute/Icircumflex/Idieresis 16#d0/Eth/Ntilde/Ograve/Oacute/Ocircumflex/Otilde/Odieresis/multiply 16#d8/Oslash/Ugrave/Uacute/Ucircumflex/Udieresis/Yacute/Thorn/germandbls 16#e0/agrave/aacute/acircumflex/atilde/adieresis/aring/ae/ccedilla 16#e8/egrave/eacute/ecircumflex/edieresis/igrave/iacute/icircumflex/idieresis 16#f0/eth/ntilde/ograve/oacute/ocircumflex/otilde/odieresis/divide 16#f8/oslash/ugrave/uacute/ucircumflex/udieresis/yacute/thorn/ydieresis]def /L2?/languagelevel where{pop languagelevel 2 ge}{false}ifelse def/Comp?{ /LumSepsDict where{pop false}{/AldusSepsDict where{pop false}{1 0 0 0 @gs setcmykcolor currentcmykcolor @gr add add add 0 ne 0 1 0 0 @gs setcmykcolor currentcmykcolor @gr add add add 0 ne 0 0 1 0 @gs setcmykcolor currentcmykcolor @gr add add add 0 ne 0 0 0 1 @gs setcmykcolor currentcmykcolor @gr add add add 0 ne and and and}ifelse}ifelse}bd/@PL{/LV where{pop LV 2 ge L2? not and{@np /Courier findfont 12 scalefont setfont 72 144 m (The PostScript level set in the Corel application is higher than)show 72 132 m (the PostScript level of this device. 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ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff %%PageTrailer @rs @rs %%Trailer @EndSysCorelDict end %%DocumentSuppliedResources: procset wCorel9Dict 9.0 0 %%EOF ---------------0008062137716 Content-Type: application/postscript; name="gamma.eps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="gamma.eps" %!PS-Adobe-3.0 EPSF-3.0 %%BoundingBox: 0 0 361 361 %%LanguageLevel: 1 %%Creator: Corel PHOTO-PAINT 9 %%Title: gamma.ep$ %%CreationDate: Thu Aug 03 18:33:40 2000 %%DocumentProcessColors: Black %%DocumentSuppliedResources: (atend) %%EndComments %%BeginProlog /AutoFlatness false def /AutoSteps 0 def /CMYKMarks true def /UseLevel 1 def %Build: Corel PHOTO-PAINT 9 Version 9.337 %Color profile: Generic offset separations profile %%BeginResource: procset wCorel9Dict 9.0 0 /wCorel9Dict 300 dict def wCorel9Dict begin % Copyright (c)1992-1999 Corel Corporation % All rights reserved. v9.0 r0.5 /bd{bind def}bind def/ld{load def}bd/xd{exch def}bd/_ null def/rp{{pop}repeat} bd/@cp/closepath ld/@gs/gsave ld/@gr/grestore ld/@np/newpath ld/Tl/translate ld /$sv 0 def/@sv{/$sv save def}bd/@rs{$sv restore}bd/spg/showpage ld/showpage{} bd currentscreen/@dsp xd/$dsp/@dsp def/$dsa xd/$dsf xd/$sdf false def/$SDF false def/$Scra 0 def/SetScr/setscreen ld/@ss{2 index 0 eq{$dsf 3 1 roll 4 -1 roll pop}if exch $Scra add exch load SetScr}bd/SepMode_5 where{pop}{/SepMode_5 0 def}ifelse/CurrentInkName_5 where{pop}{/CurrentInkName_5(Composite)def} ifelse/$ink_5 where{pop}{/$ink_5 -1 def}ifelse/$c 0 def/$m 0 def/$y 0 def/$k 0 def/$t 1 def/$n _ def/$o 0 def/$fil 0 def/$C 0 def/$M 0 def/$Y 0 def/$K 0 def /$T 1 def/$N _ def/$O 0 def/$PF false def/s1c 0 def/s1m 0 def/s1y 0 def/s1k 0 def/s1t 0 def/s1n _ def/$bkg false def/SK 0 def/SM 0 def/SY 0 def/SC 0 def/$op false def matrix currentmatrix/$ctm xd/$ptm matrix def/$ttm matrix def/$stm matrix def/$ffpnt true def/CorelDrawReencodeVect[16#0/grave 16#5/breve 16#6/dotaccent 16#8/ring 16#A/hungarumlaut 16#B/ogonek 16#C/caron 16#D/dotlessi 16#27/quotesingle 16#60/grave 16#7C/bar 16#82/quotesinglbase/florin/quotedblbase/ellipsis/dagger/daggerdbl 16#88/circumflex/perthousand/Scaron/guilsinglleft/OE 16#91/quoteleft/quoteright/quotedblleft/quotedblright/bullet/endash/emdash 16#98/tilde/trademark/scaron/guilsinglright/oe 16#9F/Ydieresis 16#A1/exclamdown/cent/sterling/currency/yen/brokenbar/section 16#a8/dieresis/copyright/ordfeminine/guillemotleft/logicalnot/minus/registered/macron 16#b0/degree/plusminus/twosuperior/threesuperior/acute/mu/paragraph/periodcentered 16#b8/cedilla/onesuperior/ordmasculine/guillemotright/onequarter/onehalf/threequarters/questiondown 16#c0/Agrave/Aacute/Acircumflex/Atilde/Adieresis/Aring/AE/Ccedilla 16#c8/Egrave/Eacute/Ecircumflex/Edieresis/Igrave/Iacute/Icircumflex/Idieresis 16#d0/Eth/Ntilde/Ograve/Oacute/Ocircumflex/Otilde/Odieresis/multiply 16#d8/Oslash/Ugrave/Uacute/Ucircumflex/Udieresis/Yacute/Thorn/germandbls 16#e0/agrave/aacute/acircumflex/atilde/adieresis/aring/ae/ccedilla 16#e8/egrave/eacute/ecircumflex/edieresis/igrave/iacute/icircumflex/idieresis 16#f0/eth/ntilde/ograve/oacute/ocircumflex/otilde/odieresis/divide 16#f8/oslash/ugrave/uacute/ucircumflex/udieresis/yacute/thorn/ydieresis]def /L2?/languagelevel where{pop languagelevel 2 ge}{false}ifelse def/Comp?{ /LumSepsDict where{pop false}{/AldusSepsDict where{pop false}{1 0 0 0 @gs setcmykcolor currentcmykcolor @gr add add add 0 ne 0 1 0 0 @gs setcmykcolor currentcmykcolor @gr add add add 0 ne 0 0 1 0 @gs setcmykcolor currentcmykcolor @gr add add add 0 ne 0 0 0 1 @gs setcmykcolor currentcmykcolor @gr add add add 0 ne and and and}ifelse}ifelse}bd/@PL{/LV where{pop LV 2 ge L2? not and{@np /Courier findfont 12 scalefont setfont 72 144 m (The PostScript level set in the Corel application is higher than)show 72 132 m (the PostScript level of this device. 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bind loop setflat}bd/@stroke/stroke ld/stroke{currentflat{{@stroke}stopped {@ifl}{exit}ifelse}bind loop setflat}bd}if L2?{/@ssa{true setstrokeadjust}bd}{ /@ssa{}bd}ifelse/d/setdash ld/j/setlinejoin ld/J/setlinecap ld/M/setmiterlimit ld/w/setlinewidth ld/O{/$o xd}bd/R{/$O xd}bd/W/eoclip ld/c/curveto ld/C/c ld/l /lineto ld/L/l ld/rl/rlineto ld/m/moveto ld/n/newpath ld/N/newpath ld/P{11 rp} bd/u{}bd/U{}bd/A{pop}bd/q/@gs ld/Q/@gr ld/&{}bd/@j{@sv @np}bd/@J{@rs}bd/g{1 exch sub/$k xd/$c 0 def/$m 0 def/$y 0 def/$t 1 def/$n _ def/$fil 0 def}bd/G{1 sub neg/$K xd _ 1 0 0 0/$C xd/$M xd/$Y xd/$T xd/$N xd}bd/k{1 index type /stringtype eq{/$t xd/$n xd}{/$t 0 def/$n _ def}ifelse/$k xd/$y xd/$m xd/$c xd /$fil 0 def}bd/K{1 index type/stringtype eq{/$T xd/$N xd}{/$T 0 def/$N _ def} ifelse/$K xd/$Y xd/$M xd/$C xd}bd/x/k ld/X/K ld/sf{1 index type/stringtype eq{ /s1t xd/s1n xd}{/s1t 0 def/s1n _ def}ifelse/s1k xd/s1y xd/s1m xd/s1c xd}bd/i{ dup 0 ne{setflat}{pop}ifelse}bd/v{4 -2 roll 2 copy 6 -2 roll c}bd/V/v ld/y{2 copy c}bd/Y/y ld/@w{matrix rotate/$ptm xd matrix scale $ptm dup concatmatrix /$ptm xd 1 eq{$ptm exch dup concatmatrix/$ptm xd}if 1 w}bd/@g{1 eq dup/$sdf xd {/$scp xd/$sca xd/$scf xd}if}bd/@G{1 eq dup/$SDF xd{/$SCP xd/$SCA xd/$SCF xd} if}bd/@D{2 index 0 eq{$dsf 3 1 roll 4 -1 roll pop}if 3 copy exch $Scra add exch load SetScr/$dsp xd/$dsa xd/$dsf xd}bd/$ngx{$SDF{$SCF SepMode_5 0 eq{$SCA} {$dsa}ifelse $SCP @ss}if}bd/p{/$pm xd 7 rp/$pyf xd/$pxf xd/$pn xd/$fil 1 def} bd/@MN{2 copy le{pop}{exch pop}ifelse}bd/@MX{2 copy ge{pop}{exch pop}ifelse}bd /InRange{3 -1 roll @MN @MX}bd/@sqr{dup 0 rl dup 0 exch rl neg 0 rl @cp}bd /currentscale{1 0 dtransform matrix defaultmatrix idtransform dup mul exch dup mul add sqrt 0 1 dtransform matrix defaultmatrix idtransform dup mul exch dup mul add sqrt}bd/@unscale{}bd/wDstChck{2 1 roll dup 3 -1 roll eq{1 add}if}bd /@dot{dup mul exch dup mul add 1 exch sub}bd/@lin{exch pop abs 1 exch sub}bd /cmyk2rgb{3{dup 5 -1 roll add 1 exch sub dup 0 lt{pop 0}if 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fffffffffffffffffffff3dfffffffffffffffff3fffffffffffffffffffffffff39ffffffffffffffffffffff fffffffffffffffffffff3dfffffffffffffffff3fffffffffffffffffffffffff39ffffffffffffffffffffff fffffffffffffffffffff3dfffffffffffffffff3fffffffffffffffffffffffff39ffffffffffffffffffffff fffffffffffffffffffff3dfffffffffffffffff3fffffffffffffffffffffffff39ffffffffffffffffffffff fffffffffffffffffffff3dfffffffffffffffff3ffffffffffffffffffffffffff9ffffffffffffffffffffff ffffffffffffffffffffffdfffffffffffffffff3ffffffffffffffffffffffffff9ffffffffffffffffffffff %%PageTrailer @rs @rs %%Trailer @EndSysCorelDict end %%DocumentSuppliedResources: procset wCorel9Dict 9.0 0 %%EOF ---------------0008062137716--