Content-Type: multipart/mixed; boundary="-------------0501050959475" This is a multi-part message in MIME format. ---------------0501050959475 Content-Type: text/plain; name="05-3.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="05-3.keywords" periodic decorated graphs, gaguge-periodic Schrodinger operator ---------------0501050959475 Content-Type: application/x-tex; name="Perr.tex" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="Perr.tex" \documentclass[12pt]{article} \usepackage{amsmath,amsthm,amsfonts} %\date{\Large\today} \renewcommand{\baselinestretch}{1.1} \usepackage{cite} \usepackage{mathptm} \usepackage{times} \theoremstyle{plain} \newtheorem{prop}{Proposition} \newtheorem{lemma}[prop]{Lemma} \newtheorem{corol}[prop]{Corollary} \newtheorem{theorem}[prop]{Theorem} \theoremstyle{remark} \newtheorem{rem}[prop]{\bf Remark} \newtheorem{example}[prop]{\bf Example} \renewcommand{\proofname}{{\bf Proof}} \renewcommand{\Tilde}{\widetilde} \numberwithin{equation}{section} \sloppy %\raggedbottom \hfuzz=2pt \parindent=0pt \parskip=6pt \textwidth=160mm \textheight=240mm \oddsidemargin=-0.4mm \topmargin=-5.4mm \newcommand{\spec}[1]{\mathop{\mathrm{spec}}(#1)} \newcommand{\res}[1]{\mathop{\mathrm{res}}(#1)} \newcommand{\resm}[1]{\mathop{\mathrm{res}^-}(#1)} \DeclareMathOperator{\Ran}{Ran} \DeclareMathOperator{\supess}{sup\,\,ess} \DeclareMathOperator{\supp}{supp} %\documentstyle[12pt,amsfonts,eucal]{article} % %\oddsidemargin 0in %\topmargin -0.5in %\textwidth 16.5truecm %\textheight 23truecm %%\pagestyle{empty} %%\parindent=45pt \newcommand{\bra}{\langle} \newcommand{\ket}{\rangle} \newcommand{\CC}{{\mathbb C}} \newcommand{\Cr}{{C^*_{red}\,(\Gamma,{\cal K})}} \newcommand{\dd}{\partial} \newcommand{\DD}{{\cal D}} \newcommand{\EE}{{\cal E}} \newcommand{\FF}{{\cal F}} \newcommand{\GG}{{\cal G}} \newcommand{\Gr}{{\rm Gr}\,} \newcommand{\HP}{{\mathbb H}} \newcommand{\HH}{{\cal H}} \newcommand{\jj}{$j=1,\,\ldots,\,n$} \newcommand{\KK}{{\rm K}} \newcommand{\Kk}{{\cal K}} \newcommand{\Kr}{{\rm Ker}} \newcommand{\LL}{{\bf L}} \newcommand{\NN}{{\mathbb N}} \newcommand{\N}{{\cal N}} \newcommand{\Q}{{\cal Q}} \newcommand{\RR}{{\mathbb R}} \newcommand{\Rs}{{\mathbb R}^+} %\newcommand{\SS}{{\mathbb S}} \renewcommand{\sp}{{\rm spec}} \newcommand{\TT}{{\mathbb T}} \newcommand{\Tr}{{\rm Tr}} \newcommand{\tQ}{{\tilde Q}} \newcommand{\Wr}{{W^*(\Gamma,{\cal F})}} \newcommand{\zep}{z_{\epsilon}} \newcommand{\ZZ}{{\mathbb Z}} \newcommand{\vr}{{\vec r}} \newcommand{\ve}{{\vec e}}\newcommand{\vx}{{\vec x}} \newcommand{\vy}{{\vec y}} \newcommand{\va}{{\vec a}} \newcommand{\vb}{{\vec b}} \newcommand{\ee}{{\bf e}} \newcommand{\ba}{{\bf a}} \newcommand{\bb}{{\bf b}} \newcommand{\Ker}{{\rm Ker}} %\newcommand{\Ran}{{\rm Ran}} \newcommand{\halm}{{\vrule height7pt width4pt depth0pt}} \renewcommand{\Im}{{\rm Im}} \renewcommand{\Re}{{\rm Re}} \renewcommand{\epsilon}{{\varepsilon}} \renewcommand{\phi}{{\varphi}} \renewcommand{\hat}{\widehat} \renewcommand{\tilde}{\widetilde} \renewcommand{\det}{{\rm det}} \renewcommand{\baselinestretch}{1.5} \begin{document} { \raggedright {\LARGE\bf Spectral properties of gauge-periodic Schr\"odinger operators on graphs decorated by Riemannian manifolds} } \medskip \leftskip 20mm {\Large Jochen Br\"uning $^1$, Vladimir Geyler $^{2,\,*}$, Igor Lobanov~$^{1,2}$.} \medskip \leftskip 35mm $^1$ Institut f\"ur Mathematik, Humboldt-Universit\"at zu Berlin, Rudower Chaussee 25, 12489 Berlin, Germany $^2$ Mathematical Faculty, Mordovian State University, 430000 Saransk, Russia $^*$ Corresponding author; e-mail addresses: geyler@mrsu.ru, geyler@mail.ru \bigskip \leftskip 20mm \textbf{\large Abstract.} { Gauge-periodic Schr\"odinger operators on periodic graphs decorated by compact Riemannian manifolds are considered. Results concerning the dependence of the spectral structure of such operators on properties of the periodicity group are announced. } \leftskip 0mm \large \bigskip Spectral properties of the Laplace--Beltrami operators on periodic Riemannian manifolds attract noticeable interest recently (see, e.g. \cite{1} and the references therein). In particular, this interest is motivated by transport properties of the carbon nanostructures composed from fullerenes and nanotubes \cite{2}, \cite{2a}; the geometry of such structures is simulated adequately by means of so-called hybrid manifolds, i.e. by means of the topological spaces glued from manifolds of distinct dimensions. We consider here a particular case of hybrid manifolds, so-called periodic decorated graphs, and announce some results concerning the spectrum of the periodic Schr\"odinger operators on such spaces. Note that the class of periodic decorated graphs includes periodic metric graphs (quantum networks) \cite{3} as a very special particular case. Below $\Gamma$ denotes an oriented graph with vertex set $V$ and edge set $E$. For every $v\in V$ we fix a compact Riemannian manifold $M_v$ with dimension less or equal to 3, and for every $e\in E$ we fix a closed interval $M_e=[a_1(e),a_2(e)]\subset \mathbb{R}$, где $a_1(e)\le a_2(e)$. By definition, a {\it decorated graph} is a topological space $X$ obtained from the disjoint union $Y$ of the manifolds $M_v$ and the intervals $M_e$ by gluing each of the point $a_1(e)$ (respectively, $a_2(e)$) to the manifold $M_v$ where $v$ is the initial (respectively, terminal) vertex of the edge $e$. Note that if all the manifolds $M_v$ ($v\in V$) are zero-dimensional and all the intervals $M_e$ ($e\in E$) are nondegenerate, then $X$ is just a {\it metric graph}. There is a natural measure on $X$ such that $L^2(X)$ is isomorphic to the direct sum $\bigoplus_{\xi\in\Xi} L^2(M_\xi)$ where $\Xi$ denotes the disjoint union of the sets $V$ and $E$. For every $\xi\in\Xi$ we fix a Schr\"odinger operator $H_\xi$ on $L^2(M_\xi)$ defined by a second order differential expression with sufficiently smooth coefficients on $M_\xi$; if $\xi\in E$ we impose some self-adjoint boundary conditions on the boundary of the interval $M_\xi$. Then we can define a Schr\"odinger operator on $X$ in the following way. Let $S_\xi$ be the restriction of $H_\xi$ to the set of all the functions from the domain ${\mathcal D}(H_\xi)$ vanishing at the gluing points, and let $S$ be the direct sum of all the operators $S_\xi$, $\xi\in\Xi$. Every self-adjoint extension of $S$ is called a \emph{Schr\"odinger operator on $X$}. It is convenient to describe all the Schr\"odniger operators generated by $S$ with the help of the Krein resolvent formula. For this purpose we denote by $H^0$ the direct sum of $H_\xi$ ($\xi\in\Xi$) and introduce for every $\xi$ an auxiliary Hilbert space ${\mathcal G}_\xi$ with the dimension equal to the deficiency index of the operator $S_\xi$. Let $\gamma_\xi$ and $Q_\xi$ be the $\Gamma$-field and $\mathcal{Q}$-function respectively for the pair $(H_\xi,S_\xi)$ \cite{4}; in particular, $\gamma_\xi(z)$ (respectively $Q_\xi(z)$) is a bounded linear operator from ${\mathcal G}_\xi$ into $L^2(M_\xi)$ (respectively, from ${\mathcal G}_\xi$ into ${\mathcal G}_\xi$) for every $z\in{\mathbb C}\setminus{\rm spec}(H^0)$. Let now ${\mathcal G}=\bigoplus_{\xi\in\Xi}{\mathcal G}_\xi$, $\gamma=\bigoplus_{\xi\in\Xi}\gamma_\xi$, $Q=\bigoplus_{\xi\in\Xi}Q_\xi$. Then the following formula gives a bijection between the resolvents of all the self-adjoint extensions $H$ of the operator $S$ and self-adjoint (generally speaking, multi-valued) operators $\Lambda$ in ${\mathcal G}$: $$ (H-z)^{-1}=(H^0-z)^{-1}-\gamma(z)[Q(z)-\Lambda]^{-1}\gamma^*(\bar z). $$ We will denote by $H_\Lambda$ the extension $H$ corresponding to $\Lambda$. The extensions $H_\Lambda$ and $H^0$ of $S$ are {\it disjoint} (i.e., ${\mathcal D}(H_\Lambda)\cap{\mathcal D}(H^0)={\mathcal D}(S)$) if and only if the operator $\Lambda$ is single-valued. To simplify the statements below, we suppose further that $H_\Lambda$ and $H^0$ are disjoint. In essence, this supposition does not restrict generality since the set of all the Lagrangian subspaces in the space ${\mathcal G}\times{\mathcal G}$ equipped with the standard skew-Hermitian product coincides with the set of the graphs of all $\Lambda$ \cite{5}; it is clear that the graphs of all single-valued $\Lambda$ are points of general position in this set. We consider periodic decorated graphs and gauge-periodic operators $H_\Lambda$ on the graphs. With this in mind we assume that the graph $\Gamma$ is equipped with a free action of a discrete group $G$, in particular, $G$ acts on $\Xi$. We will suppose that $M_{g\xi}=M_\xi$ ($g\in G$, $\xi\in\Xi$) and that the gluing points are invariant with respect to action of $G$ on $Y$; as a result we obtain a natural action of $G$ on $X$. Let now $\sigma$ be a 2- cocycle on $G$ with values in ${\bf U}(1)$. We say that a closed linear operator $A$ in $L^2(X)$ is {\it gauge-periodic}, if the resolvent of $A$ is $(G,\bar\sigma)$-invariant. If, in addition, $\sigma=1$, then $A$ is called {\it periodic}. We assume further that $H_{g\xi}=H_\xi$ $\forall g\in G$. Under this assumption, it is easy to prove that {\it a Schr\"odinger operator $H_\Lambda$ is gauge-periodic if and only if $\Lambda$ is $(G,\bar\sigma)$-invariant}. Below we consider only $(G,\bar\sigma)$-invariant operators $\Lambda$. Further we suppose that the action of $G$ on $\Gamma$ has finite number of orbits; denote by $F$ a fundamental set of the action of $G$ on $\Xi$, i.e. $F$ is a full system of distinct representatives for the orbits of this action. Denote by $C^*_r(G,\sigma)$ the reduced twisted group $C^*$-algebra for $G$ with the multiplier $\sigma$ \cite{6}. We denote by ${\mathcal H}_f$ (respectively, by ${\mathcal G}_f$) the space $\bigoplus_{\xi\in F}L^2(M_\xi)$ (respectively, the space $\bigoplus_{\xi\in F}{\mathcal G}_\xi$). We denote by ${\mathcal K}({\mathcal H})$ the space of all compact linear operators in a Hilbert space ${\mathcal H}$, the standard trace in $C^*_r(G,\sigma)\otimes{\mathcal K}({\mathcal H})$ will be denoted by ${\rm Tr}$. Since ${\mathcal G}_f$ is finite-dimensional, the following theorem considerably simplifies the study of the spectrum of a gauge-periodic operator $H_\Lambda$; in this theorem and further, we identify the space $L^2(X)$ (respectively, ${\mathcal G}$) with $l^2(G,{\mathcal H}_f)$ (respectively $l^2(G,{\mathcal G}_f)$). {\bf Theorem~1.} {\it If $\Lambda$ belongs to the $C^*$-algebra $C^*_r(G,\sigma)\otimes{\mathcal K}({\mathcal G}_f)$, then the resolvent of the operator $H_\Lambda$ belongs to $C^*_r(G,\sigma)\otimes{\mathcal K}({\mathcal H}_f)$. } The condition of Theorem~1 is satisfied in the following very important particular case: $\Lambda$ is a bounded linear operator in ${\mathcal G}$ and the standard matrix representation of $\Lambda$ has only finite numbers non-vanishing diagonals. Moreover, let $G$ be equipped with a left-invariant metric $\rho$. Then using some methods from \cite{7} we can prove the following theorem. {\bf Теорема~2.} {\it Let $\Lambda$ be a bounded operator such that for some constants $C, \delta>0$ all the matrix elements $\Lambda(\alpha,\beta)$ satisfy the condition $$ ||\Lambda(\alpha,\beta)||\leq C \eta(\rho(\alpha,\beta))^{-1-\delta}, $$ where $\eta(R)$ is the numbers of points in any ball of radius $R$. Then $\Lambda\in C^*_r(G,\sigma)\otimes{\mathcal K}({\mathcal G}_f)$. } \bigskip Further we assume that $H_\Lambda$ is semi-bounded below. Fix a real number $E'<\inf{\rm spec}\,(H^\Lambda)$ and define for $E\notin {\rm spec}\,(H^\Lambda)$ $$ N(E):=\left\{\begin{array}{ll} {\rm Tr} P_{[E',E]}, & E\geq E';\\ 0, & E0$. Then the spectrum of $H_\Lambda$ has a band structure and has no singular continuous component. } {\bf Remark}. Simple examples show that under the condition of Theorem~3, the operator $H_\Lambda$ can have eigenvalues in the spectrum. There are examples of periodic Schr\"odinger operators $H$ on periodic decorated graphs $X$ with ${\rm dim}\,X\ge2$ such that $H$ have infinite number of gaps in their spectrum (see \cite{11}); hence, the Bethe--Sommerfeld conjecture \cite{12} is not valid for such operators. The next theorem shows that really this conjecture is not valid for a wide class of periodic decorated graphs. {\bf Theorem~4.} {\it Let the action of $G$ on the set of vertices $V$ be transitive. Suppose that every operator $H_v$ $($in the considered case, $H_v$ is independent of $v$$)$ has infinite number of eigenvalues with multiplicity greater than the degree of $v$ $($this degree is also independent of $v$$)$. Suppose in addition that all the intervals degenerate into a point. Then under generic position of the gluing points (both in the sense of the Riemann--Lebesgue measure and the Baire category) $H_\Lambda$ has infinite number of gaps in its spectrum. } \bigskip \noindent{\bf Acknowledgements.} The paper is partially supported by grants of RFBR (\#02-01-00804), DFG, and INTAS. \bigskip \begin{thebibliography}{99} \bibitem{1} O.~Post. Periodic manifolds with spectral gaps. {\it J. Diff. Eqs.} {\bf 187} (2003), 23--45. \bibitem{2} H.~Aoki, M.~Koshino, D.~Takeda, H.~Morise, K.~Kuroki. Electronic structure of periodic curved surfaces: Topological band structure. {\it Phys. Rev. B.} {\bf 65} (2002), 035102.1--035102.8. \bibitem{2a} A.~R.~Sabirov, I.~V.~Stankevich, L.~A.~Chernozatonskii. Hybrids of Carbyne and Fullerene (in Russian). {\it Pis'ma ZhETF}. {\bf 79} (2004), 153--157; English transl.: {\it JETP Letters}. {\bf79} (2004), 121--125. \bibitem{3} P.~Kuchment. Quantum graphs: I. 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