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periodic decorated graphs, gaguge-periodic Schrodinger operator
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\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
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\end{document}
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