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\begin{document}
\title{Absolute continuity in periodic thin tubes \\ and
strongly coupled leaky wires}
\author{Francois Bentosela$^{1,2}$, Pierre Duclos$^{1,3}$,
and Pavel Exner$^{4,5}$}
\date{}
\maketitle
\begin{quote}
\emph{1$\;$ Centre de Physique Th\'eorique, C.N.R.S., Luminy Case
907, \\ \phantom{aa} F-13288 Marseille Cedex 9, \\
2$\;$ Universit\'{e} de la Mediterran\'{e}e
(Aix--Marseille II), F-13288 \\ \phantom{aa} Marseille, \\
3$\;$ PhyMat, Universit{\'e} de
Toulon et du Var, BP 132, F-83957 \\ \phantom{aa} La Garde Cedex, France; \\
4$\;$ Department of Theoretical Physics, Nuclear Physics Institute, \\
\phantom{aa} Academy of Sciences, CZ-25068 \v{R}e\v{z} near Prague, \\
5$\;$ Doppler Institute, Czech Technical University, B\v{r}ehov{\'a}
7, \\ \phantom{aa} CZ-11519 Prague, Czechia; \\
\texttt{\phantom{aa} bento@cpt.univ-mrs.fr, duclos@univ-tln.fr, \\
\phantom{aa} exner@ujf.cas.cz}} \\[5mm]
%
{\small {\bf Abstract:} Using a perturbative argument, we show
that in any finite region containing the lowest transverse eigenmode,
the spectrum of a periodically
curved smooth Dirichlet tube in two or three dimensions is
absolutely continuous provided the tube is sufficiently thin. In a
similar way we demonstrate absolute continuity at the bottom of
the spectrum for generalized Schr\"odinger operators with a
sufficiently strongly attractive $\delta$ interaction supported by
a periodic curve in $\mathbb{R}^d,\: d=2,3$.}
\end{quote}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Introduction}
In models of periodically structured quantum systems, absolute
continuity of the spectrum is a crucial property. For usual
Schr\"odinger operators and many other PDE's with periodic
coefficients the problem is well understood -- see, e.g. \cite{Ku,
RS}. On the other hand, there are important classes of operators
which still pose open questions. An example is represented by
so-called \emph{quantum waveguides}, i.e. systems the Hamiltonian
of which is (a multiple of) the Laplacian in an infinitely long
tube-shaped region, usually with Dirichlet boundary conditions.
In the two-dimensional setting, where the region in question is a
periodically curved planar strip the absolute continuity has been
demonstrated recently in \cite{SV}. Unfortunately, the method used
in this work does not seem to generalize to other dimensions
including the physically interesting case of a periodic tube in
$\R^d,\, d=3$. This is why we present in this letter a simpler
result stating the absolute continuity at the bottom of the
spectrum for tubes which are thin enough. With physical
applications in mind we formulate it for $d=2,3$, but the argument
can be used in any dimension.
We also address an analogous problem concerning Schr\"odinger
operators in $L^2(\R^d)$, $d=2,3$, with an attractive $\delta$
interaction supported by a periodic curve; we prove absolute
continuity, again at the bottom of the spectrum, for a
sufficiently strong attraction. If $d=2$ the answer was known for
a family of curves periodic in two independent directions
\cite{BSS}, however, for a single infinite curve results have been
missing, to say nothing about the more singular case of dimension
$d=3$.
Our method is perturbative. We show that for a sufficiently thin
tube or strongly attractive $\delta$ interaction the Floquet
eigenvalues do not differ much from the Floquet eigenvalues of a suitable
comparison problem in one dimension which are known to be
nonconstant as functions of quasimomentum. A similar argument was
used recently for periodically perturbed magnetic channels
\cite{EJK}, and in a different context to demonstrate existence of
persistent currents in leaky quantum wire loops \cite{EY2}.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Thin curved tubes}
\setcounter{equation}{0}
Let $\Gamma:\: \R\to\R^d,\, d=2,3$, be a $C^4$ smooth curve
without self-intersections which is periodic, i.e. there are $L>0$
and a nonzero vector $b\in\R^d$ such that
% ------------- %
$$ %\begin{equation}
\Gamma(s+L) = b+\Gamma(s)\,, \quad\forall s\in\R\,.
$$ %\end{equation}
% ------------- %
With an abuse of notation we will use the same symbol $\Gamma$ for
the map and for the image $\Gamma(\R)$. Let further
$\Omega:= \{x\in\R^d:\, \mathrm{dist}\,(x,\Gamma)0$. If the latter is small enough, such a tube has
no self-intersections and we can parametrize it by natural
curvilinear coordinates. More specifically, take a ball
$\BB_a:=\{u\in\R^{d-1},\,|u|0$ there is $a_E>0$ such that the spectrum of
$-\Delta_D^{\Omega}$ is absolutely continuous in the interval
$[0,a^{-2}\kappa_1^2+E]$ for all $a0$
such that $|\epsilon_{1,n}(a,\theta) -a^{-2}\kappa_1^2
+\lambda_n(\theta)|\le c_E a$ holds for each of them according to
Lemma~\ref{tubepert}. Since the functions $\lambda_n(\cdot)$ are
non-constant by \cite[Sec.~XIII.16]{RS}, the conclusion follows.
\end{proof} \hspace{1em}
% ------------- %
\begin{remark} \label{small gamma}
{\rm A similar perturbative argument shows that the spectrum is
absolutely continuous at its bottom if $a$ is fixed and
$\|\gamma\|_\infty$ is sufficiently small.}
\end{remark}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Leaky quantum wires}
\setcounter{equation}{0}
Now we are going to consider the analogous problem for another
class of operators. Let $\Gamma$ be the same $C^4$-smooth periodic
curve in $\R^d$ and consider the generalized Schr\"odinger
operator given by the formal expression
% ------------- %
$$ %\begin{equation} \label{}
H_{\alpha,\Gamma} = -\Delta-\alpha \delta(x-\Gamma)\,.
$$ %\end{equation}
% ------------- %
Its meaning is different for different dimensions. If $d=2$ we
regard it as the unique self-adjoint operator associated with the
quadratic form
% ------------- %
$$ %\begin{equation} \label{}
q_{\alpha,\Gamma}[\psi] = \|\nabla\psi\|^2 - \alpha \int_\R
|\psi(\Gamma(s))|^2 ds\,, \quad \psi\in W^{2,1}(\R^2)\,,
$$ %\end{equation}
% ------------- %
which is closed and below bounded by \cite{BEKS}; we suppose that
the singular interaction is attractive, $\alpha>0$. The situation
is more complicated in the three-dimensional case when
$\mathrm{codim}\,\Gamma=2$. Following the construction given in
\cite{EK1} -- see also \cite{Po} for a more general background --
one starts from the family of curves $\phi_\Gamma
(\cdot,\rho,\vartheta_0)$, obtained by translating $\Gamma$
by $\rho(\cos(\vartheta_0),\sin(\vartheta_0))$, which are used
to determine the generalized
boundary values of a function $f\psi\in W_{\mathrm{loc}}^{2,2}(
\R^3 \setminus \Gamma)\cap L^{2}(\R^3)$ as the following limits
% ------------- %
\begin{eqnarray*}
L_0(\psi)(s) &\!:=\!& -\lim_{\rho \to 0}\: \frac{1}{\ln \rho }\,
\psi(\phi_\Gamma(s,\rho,\vartheta_0))\,, \\
L_1(\psi)(s) &\!:=\!& \lim_{\rho \to 0}\,
\left[\, \psi(\phi_\Gamma(s,\rho,\vartheta_0))
+L_0(\psi)(s)\ln \rho \,\right] \,.
\end{eqnarray*}
% ------------- %
We call $\Upsilon_\Gamma$ the family of those $\psi$ for which
these limits exist a.e. in $\R$, are independent of $\vartheta_{0}$,
and define a pair functions belonging to $L_{\mathrm{loc}}^2(\R)$.
The sought operator is then defined as
% ------------- %
\begin{eqnarray*}
H_{\alpha,\Gamma}\psi(x) &\!:=\!& -\Delta\psi(x)\,,\quad x\in
\R^3 \setminus\Gamma\,, \\
D(H_{\alpha,\Gamma}) &\!:=\!& \{f\in \Upsilon_\Gamma :\: 2\pi
\alpha L_0(\psi)(s)=L_1(\psi)(s)\,\}\,.
\end{eqnarray*}
% ------------- %
Being interested in strong coupling we suppose hereafter that the
coupling parameter $\alpha$ is negative though $H_{\alpha,\Gamma}$
is well defined for all $\alpha\in\R$; recall that the
two-dimensional $\delta$ interaction is always attractive
\cite{AGHH}.
In an important particular case when $\Gamma$ is a straight line
one uses separation of variables to show that the spectrum is
absolutely continuous and covers the interval $[\zeta(\alpha),
\infty)$, with the threshold given by the corresponding
$(d\!-\!1)$-dimensional $\delta$ interaction eigenvalue,
% ------------- %
$$ %\begin{equation} \label{}
\zeta(\alpha) := \left\lbrace \begin{array}{lcl} -{1\over
4} \alpha^2 \quad & \dots & \quad d=2 \\ -4\e^{2(-2\pi\alpha
+\psi(1))} \quad & \dots & \quad d=3 \end{array} \right.
$$ %\end{equation}
% ------------- %
where $-\psi(1)\approx 0.577$ is the Euler number. This also
illustrates that the strong coupling means $(-1)^d\alpha
\to\infty$ for $d=2,3$. Due to the injectivity and periodicity
assumptions the curve decomposes into a disjoint union of
translates of the period cell $\Gamma_\PP:= \Gamma \upharpoonright
[0,L)$. Since $H_{\alpha,\Gamma}$ now acts in the whole Euclidean
space, we need also a decomposition of the space $\R^d$ with the
period cell
% ------------- %
$$ %\begin{equation} \label{}
\PP := \left\lbrace \, \LL+tb:\: t\in[0,1)\, \right\rbrace\,,
$$ %\end{equation}
% ------------- %
where $\LL\subset\R^d$ is a affine space which is not colinear
with $b$. We denote by $b_\perp$ the component of $b$ in the
direction orthogonal to $\LL$; it follows that $b_\perp\ne0$. It
is important that the two decompositions are chosen in a
consistent way, $\Gamma_\PP=\PP\cap\Gamma$. We will assume in
addition that
\\ [.5em]
% ------------- %
(c) the restriction of $\Gamma_\PP$ to the interior of $\PP$ is
connected. \\ [.5em]
% ------------- %
It should be noted that the choice of a slab for $\PP$ is made
rather for convenience -- see Remarks~\ref{puzzle,crochet} below.
We start again from the Floquet decomposition with respect to the
Brillouin zone $\BB:=[-\pi|b_\perp|^{-1}, \pi|b_\perp|^{-1})$.
% ------------- %
\begin{lemma} \label{leakyfloq}
There is a unitary $\UU:\: L^2(\R^d) \to \int^\oplus_\BB
L^2(\PP)\, d\theta$ such that
% ------------- %
$$ %\begin{equation} \label{}
\UU H_{\alpha,\Gamma} \UU^{-1} = \int^\oplus_\BB
H_{\alpha,\Gamma}(\theta)\, d\theta\,,
$$ %\end{equation}
% ------------- %
where the fibre operator satisfies periodic b.c. in the direction
of $b$ acting as
% ------------- %
$$ %\begin{equation}
H_{a,\Gamma}(\theta) = (-i\nabla+\theta)^2
-\alpha\delta(x-\Gamma)\,,
$$ %\end{equation}
% ------------- %
and the interaction term in $L^2(\PP)$ is interpreted in the
above described sense, the quadratic form if $d=2$ and boundary
conditions if $d=3$.
\end{lemma}
% ------------- %
\begin{proof} See \cite{EY1} for $d=2$ and \cite{EK2} for $d=3$.
\end{proof} \vspace{1em}
It is easy to see that in distinction to the previous case the
essential spectrum is non-empty and equals
$\sigma_\mathrm{ess}(H_{a,\Gamma}(\theta))= [\theta^2,\infty)$; we
will be interested in the eigenvalues below its threshold. They
are again real-analytic functions:
% ------------- %
\begin{lemma} \label{leakyanal}
$\: \{ H_{\alpha,\Gamma}(\theta):\: \theta\in\BB\, \}$ is a type A
analytic family.
\end{lemma}
% ------------- %
\begin{proof} Similar to that of Lemma~\ref{tubeanal}.
\end{proof} \vspace{1em}
The role of the small tube width from the previous section is
played here by strong coupling. While the wave function may be
nonzero at large distances from $\Gamma$, it is localized in its
vicinity as $(-1)^d\alpha \to\infty$. Then one can choose a
tubular neighbourhood $\Omega$ of $\Gamma$ and estimate
the operator in question from both sides by imposing the Dirichlet
and Neumann condition at $\pd\Omega$. The exterior part does not
contribute to the negative spectrum, while the part in $\Omega$ can be
treated as in the previous section, with the additional $\delta$
interaction on the tube axis and different boundary conditions for
the lower bound. The argument is thus more complicated, however,
it was done in \cite{EY1} and \cite{EK2} with the following
result.
% ------------- %
\begin{lemma} \label{leakypert}
The number of isolated eigenvalues of $H_{\alpha,\Gamma}(\theta)$
exceeds any fixed $n\in\N$ as $(-1)^d\alpha \to\infty$. The $n$-th
eigenvalue behaves asymptotically as
% ------------- %
$$ %\begin{equation} \label{}
\epsilon_n(\alpha,\theta):= \zeta(\alpha)+ \lambda_n(\theta) +
\left\lbrace \begin{array}{l} \OO(\alpha^{-1}\ln\alpha) \\
\OO(\e^{\pi\alpha}) \end{array} \right\rbrace
$$ %\end{equation}
% ------------- %
in the strong coupling limit for $d=2,3$, respectively, uniformly
in $\theta$, i.e. the error terms is for a fixed $n$ bounded in
$\BB$. Here $\lambda_n(\theta)$ means again the $n$-th eigenvalue
of the operator $S(\theta)$ defined in the previous section.
\end{lemma}
% ------------- %
The main result of this section then reads as follows.
% ------------- %
\begin{theorem} \label{leakymain}
Under the stated assumptions, to any $E>0$ there exists an
$\alpha_E>0$ such that the spectrum of the operator
$H_{\alpha,\Gamma}$ is absolutely continuous in $(-\infty,
\zeta(\alpha)+E]$ as long as $(-1)^d\alpha >\alpha_E$.
\end{theorem}
% ------------- %
\begin{proof}
The argument is analogous to the one used for Theorem~\ref{tubemain}.
\end{proof} \hspace{1em}
% ------------- %
\begin{remarks} \label{puzzle,crochet}
{\rm (i) It is not always possible to choose $\PP$ is the
Cartesian-product form, as we did above, which would satisfy the
assumption (c). Counterexamples with a sufficiently entangled
periodic $\Gamma$ are easily found. However, if we choose instead
another period cell $\PP$ with a smooth boundary, which is not a
planar slab and for which the property (c) is valid, the argument
modifies easily and the claim of Theorem~\ref{leakymain} remains
true. \\ [.2em]
% ------------- %
(ii) In the case $d=2$ such a ``puzzle-like'' decomposition can be
always found. To see that, fold the plane into a cylinder of
radius $|b|/2\pi$ so $\Gamma$ becomes a loop which encircles the
cylindrical surface, dividing into two parts $\CC^\pm$ which are
disjoint apart of the common boundary; each of them is connected
because $\Gamma$ has by assumption no self-intersections. Choosing
a point at $\Gamma$, one can thus find two smooth semi-infinite
curves in $\CC^\pm$, even straight from some point on, which go
the two cylinder ``infinities'' without crossing $\Gamma$.
\\ [.2em]
% ------------- %
(iii) On the other hand, an analogous decomposition into
translates of a suitable $\PP$ satisfying the hypothesis (c) may
not exist if $d=3$. It depends on the topology of $\Gamma$, a
simple counterexample is given by a ``crotchet-shaped'' curve
which enters $\PP$ on its ``left side'' twice and leaves it once,
and vice versa on the right, without being topologically
equivalent to a line. We conjecture that the claim of
Theorem~\ref{leakymain} remains valid in such situations too,
however, a different method is required to demonstrate it. }
\end{remarks}
% ------------- %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection*{Acknowledgments}
P.E. is grateful for the hospitality in Centre de Physique
Th\'eorique, C.N.R.S., where this work was started, and all the
authors to Institut ``Simion Stoilow'' of the Romanian Academy,
where it was finished. The research has been partially supported
by GAAS under the contract A1048101.
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% -------------- %
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% ------------- %
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% -------------- %
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% ------------- %
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% ------------- %
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% -------------- %
\end{thebibliography}
\end{document}