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\begin{center}
{\ten Centre de Physique Th\'eorique\footnote{Unit\'e Propre de
Recherche 7061} - CNRS - Luminy, Case 907}
{\ten F-13288 Marseille Cedex 9 - France }
\vspace{1 cm}
{\twelve CURVATURE--INDUCED BOUND STATES IN QUANTUM WAVEGUIDES
IN TWO AND THREE DIMENSIONS} \\
\vspace{0.3 cm}
\setcounter{footnote}{0}
\renewcommand{\thefootnote}{\arabic{footnote}}
{\bf Pierre DUCLOS}\footnote{and PHYMAT, Universit\'e de Toulon et du Var,
83130 Lagarde, France \\ {\em duclos@naxos.unice.fr}} {\bf and
Pavel EXNER}\footnote{Nuclear Physics Institute, AS CR,
25068 \v Re\v z near Prague \\
and Doppler Institute, Czech Technical University,
B\v rehov{\'a} 7, 11519 Prague, Czech Republic \\
{\em exner@ujf.cas.cz}}
\vspace{1.5 cm}
{\bf Abstract}
\end{center}
Dirichlet Laplacian on curved tubes of a constant cross section in
two and three dimensions is investigated. It is shown that if the
tube is non--straight and its curvature vanishes asymptotically,
there is always a bound state below the bottom of the essential
spectrum. An upper bound to the number of these bound states in thin
tubes is derived. Furthermore, if the tube is only slightly bent,
there is just one bound state; we derive its behaviour with respect
to the bending angle. Finally, perturbation theory of these
eigenvalues in any thin tube with respect to the tube radius is
constructed and some open questions are formulated.
\vspace{2 cm}
\noindent October 1994
\noindent CPT-94/P.3023
\bigskip
\noindent anonymous ftp or gopher: cpt.univ-mrs.fr
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%END OF THE DEFINITION
\section{Introduction}
\subsection{What are the quantum waveguides?}
A rapid development of microelectronics has stimulated in recent years
a remarkable progress in semiconductor physics. Many new ideas have
appeared and the experimental framework has widened considerably. At
the same time this progress has brought interesting mathematical
problems but this fact escaped the general attention. The present paper
is devoted to some of them.
The most spectacular manifestation of the mentioned achievements
concern fabrication techniques of microscopic structures of a pure
semiconductor material -- see, \eg, \cite{Sa} and the papers quoted
below. Let us recall some of them:
\begin{description}
\item{(i)} thin films produced on an insulating surface by one of the
epitaxial techniques; their thickness can be as small as $\,2\,nm\,$
which means that the transversal cross section of such a layer contains
several atoms only,
\item{(ii)} sandwiches obtained by combining films of different
semiconductor materials; a typical example is the combination of GaAs
and AlGaAs,
\item{(iii)} heterostructures, \ie, sandwiches with a variating layer
thickness -- see, \eg, \cite{DB},
\item{(iv)} {\em "quantum wires"} obtained from semiconductor films by
suitable masking and ion bombardment -- see \cite{TDP}; in this way one
can create on the substrate various graphs,
\item{(v)} similarly one can produce many--probe junctions like the
structures investigated in \cite{TBV,Ba} \etc
\end{description}
One often uses the term {\em mesoscopic physics} for experiments with
such structures and other similarly small objects (like, for instance,
metallic graphs prepared by ion lithography). The goal is to stress
that the systems involved are large enough to be shaped by the
experimentalist and at the same time so small that quantum effects like
interference are manifested on them. Among the experimental
achievements made in this field, recall the manifestation of
Aharonov--Bohm effect in heterostructures \cite{DMB} and metallic rings
\cite{BLD,UHL,WWU}; some other experiments will be mentioned later.
Recall also that the quantum interference in microstructures controlled
by an external field is of a considerable interest: there are several
proposals of interference transistors \cite{DB,ES3,SMRH} which surpass
by their size, speed and switching voltage the commonly used MOSFETs.
Let us summarize characteristic properties of the mentioned
semiconductor microstructures: they are
\begin{description}
\item{(a)} small size, typically from tens to hundreds of $\,nm\,$,
\item{(b)} high purity; the electron mean free path can be a few
$\,\mu m\,$ or even larger,
\item{(c)} crystallic structure,
\item{(d)} the wavefunctions are usually suppressed at the boundaries
between different semiconductor materials -- see, \eg, \cite{HH}.
\end{description}
Behaviour of an electron in such a structure is, of course, governed by
the many--body Schr\"odinger equation describing its interaction with
the lattice atoms including possible impurities. The mentioned
properties allow us, however, to adopt several fundamentally
simplifying assumptions.
It is obvious from (a) and (b) that using a sufficiently pure material
one can achieve that the mean free path is two or three orders of
magnitude greater than the size of the structure; hence the electron
motion can be assumed in a reasonable approximation as undisturbed by
the scattering on impurities. Experimentalists are used to speak about
a ballistic regime, and they are able to prove it, for example, by
measuring interference effects which are excluded under other
circumstances.
The property (c) allows us to make the most important simplification.
It is well known that a particle in a crystallic lattice moves as {\em
free} with some effective mass $\,m^*\,$; from the mathematical point
of view it follows from spectral properties of Schr\"odinger operators
with periodic potentials -- see \cite{RS}, Sec.XIII.16. The effective
mass changes, of course, along the spectrum but one can regard it as a
constant when we restrict our attention to the physically interesting
part of the valence band. Recall that its value may differ
substantially from the true electron mass, for instance, one has $\,m^*
= 0.067\,m_e\,$ for GaAs.
In combination with (d) we see that the electron motion inside the
microstructure can be modelled by a free (spinless) particle living in
the corresponding spatial region with the {\em Dirichlet condition} on its
boundary; an interaction term must be added only if the whole structure
is placed into an external field.
Investigation of quantum motion in a fixed subset of the configuration
space represented until recently rather a textbook illustration, and
nearly nothing was known concerning complicated regions like curved or
branched tubes, layers or sandwiches. The new developments in
semiconductor physics discussed above offer a strong motivation for
studying of such systems; in view of the natural analogy and also for
the sake of brevity we shall refer to them as to {\bf quantum
waveguides}.
Introducing such a notion, one is naturally compelled to ask about the
relations to the classical theory of acoustic and electromagnetic
waveguides (the literature devoted to these problems is rather
extensive; let us name \cite{CJ,Go,KLR,KRS,KRT}, \cite{Marc,Me} as a few
examples related in a way to the subject of the present paper). The
basic equations in the two cases, \ie, the Schr\"odinger and wave
equation are, of course, different but as far as one is concerned with
the stationary problems only, both of them reduce to the Helmholz
equation in the appropriate spatial region. The difference is then in
the physical meaning of the spectral parameter, and possibly also in
the boundary conditions which are Neumann for the acoustic and EM--mode
of the electromagnetic waveguides (in the quantum case, one can
encounter the Neuman conditions when considering ballistic motion of
electrons in metallic microstructures).
This means that some results of the classical theory can be easily
adapted to the quantum case. On the other hand, quantum mechanics and
the classical theory of waves use rather different mathematical methods
so a mutual inspiration could be useful on both sides. For instance,
the results discussed below can be used to establish existence of
trapping modes in bent electromagnetic waveguides with perfectly
conducting walls which was not suspected in the classical theory
\cite{ES2}.
\subsection{Subject of the paper and survey of results}
Let us specify first the scope of the paper. The quantum waveguide
theory, though it exists up to now in a very rudimentary form only,
offers a lot of interesting problems. In this paper, we are going to
concentrate to those related to the {\em curvature--induced discrete
spectrum} of the corresponding Hamiltonian {\em in bent tubes}.
As we have mentioned, the existence of square--integrable solutions to
the wave equation remain unnoticed in the classical theory of
waveguides. In quantum mechanics, several authors
\cite{dC1,dC2,dC3,Pi,JK,KJ,Ma} presented independently formal
considerations (aimed mostly at the quantization problem for motion on
curves and surfaces) which contained implicitly the existence of bound
states. One can find also other indications for existence of the
effect, for example, the numerical solution to Schr\"odinger equation
on the oval--shaped ring \cite{SRS} which demonstrates that the
wavefunction is localized to the regions of maximum curvature.
The effect was clearly recognized for the first time in the paper \cite
{ES1} where the Birman--Schwinger technique in combination with the
minimax principle was used to demonstrate that the Dirichlet Laplacian
on a smooth curved planar strip which is thin enough and whose
curvature decays sufficiently rapidly has at least one isolated
eigenvalue below the bottom of the essential spectrum. The analogous
result was proved in \cite{E2} for curved tubes in $\,\R^3\,$; the decay
assumption has been later substantially weakened \cite{E3}, and some
lower bounds to the bound--state energies were derived \cite{AE}.
Another considerable progress was achieved in \cite{GJ} (see also
\cite{DJ}) where a suitably chosen trial function was used to
demonstrate that
any bending pushes the threshold of the spectrum below the lowest
transverse--mode energy; this implies the existence of bound states
in curved tubes of any thickness provided the curvature is zero
outside a bounded region, or at least vanishes asymptotically.
Though all these results apply to smooth tubes of a constant cross
section, analogous results are valid also in similar geometries. An
example was given in \cite{ESS1} showing that bound states can exist
in a sharply broken L--shaped strip; the computed bound--state energy
has been verified experimentally in a flat electromagnetic waveguide
in \cite{CLMM1} (an early mathematical treatment of this effect can
be found in \cite{He}). Other examples of this type are given, \eg,
in \cite{ABGM,CLMM2,SRW}. As for physical consequences of
curvature--induced bound states, in addition to the trapping modes
already mentioned, \eg, the existence of edge--confined currents in
thin semiconductor films was conjectured \cite{ESS2} and a model of
resonance scattering in finite--length quantum wires was analyzed
\cite{E1}.
The present paper is devoted to a review and a further development of
ideas formulated in the above mentioned papers. In the next two
sections we discuss bent tubes in $\,\R^2\,$ and $\,\R^3\,$,
respectively. Following the idea of \cite{GJ}, we prove the existence
of bound states in a curved tube under not very strong assumptions
about the regularity of the boundary and the decay of the curvature.
Using a trick known from Schr\"odinger operator theory \cite{Kl,New,Se}
we are able to derive here also an upper bound to the number of bound
states in sufficiently thin tubes in terms of the tube curvature.
In the three--dimensional case one has to know also the torsion of
the generating curve to characterize the tube. For simplicity, we
consider mostly cylindrical tubes but we specify also a more general
class to which the results extend easily. Apart from this, the
torsion is much less important for the existence of bound states;
all we need it that it satisfies some weak regularity assumptions.
It is certainly not easy to compute the bound--state energies
directly, and therefore perturbation theory is of natural interest.
In Sections~4 and 5 we treat the eigenvalues in bent tubes perturbatively
with respect to two different parameters. First we study mildly curved
tubes replacing the curvature $\,\gamma\,$ by $\,\gamma_\beta:=
\beta\gamma(\cdot)\,$. The parameter $\,\beta\,$ is proportional to
the tube bending angle. If it is small enough, the corresponding
Dirichlet Laplacian has just one simple eigenvalue; we compute the
gap between it and the threshold of the continuous spectrum in terms
of $\,\beta\,$.
Another natural perturbative parameter is the tube radius $\,a\,$. In
Section~5 we compare the tube Hamiltonian $\,H\,$ to a simple
operator in which the transverse and longitudinal variables are
decoupled, and the discrete spectrum is given explicitly in terms of
the discrete spectra of the ``component'' operators. We find that for
thin enough tubes there is a bijective correspondence between this
spectrum and the discrete spectrum of $\,H\,$, and show that an
asymptotic expansion in terms of $\,a\,$ exists up to any prescribed
order; the perturbation series for the eigenvalues is computed. We
also show that the mode--coupling terms in the Hamiltonian begin to
contribute at the fourth order in $\,a\,$ only, and discuss an
alternative in which $\,H\,$ is compared to a suitable operator
family.
We conclude the paper by showing several open problems and directions
in which the results obtained here could be further extended.
Physics of quantum waveguides attracts nowadays a lot of attention;
recently a wave of papers has appeared containing results on
different levels of rigor, often numerical, concerning the
curvature--induced bound states \cite{ABGM,WSM1,WS} as well as those
coming from bulges and crossings \cite{SRW,WSMK,IN,NI},
geometrically induced resonances and the associated transport problem
\cite{VOK1,VOK2,WS} \etc
\section{A strip in the plane}
\subsection{Preliminaries}
Consider a curved strip $\,\Omega\,$ in $\,\R^2\,$ of a constant width
$\,d=2a\,$, and choose its axis as the reference curve $\,\Gamma\,$. Up
to Euclidean transformations, $\,\Omega\,$ is uniquely characterized by
$\,a\,$ and the function $\,s\mapsto\gamma(s)\,$, where $\,s\,$ is the
arc length of $\,\Gamma\,$ and $\,\gamma\,$ is the (signed) curvature.
We assume that
\begin{description}
\item{\em (r1)} $\;\gamma\in L^1_{loc}(\R)\,$,
\item{\em (r2)} $\;a\Vert\gamma\Vert_{\infty}<1\,$,
\item{\em (r3)} $\;\Omega\,$ is not self--intersecting.
\end{description}
Sometimes it is useful to consider strips with a large winding angle,
then the last assumption can be avoided by a slight generalization:
instead of a plane we consider a strip on a multi--sheeted Riemannian
surface.
We are interested in the Dirichlet Laplacian (\cf \cite{RS},
Sec.XIII.15) $\;-\Delta_D^{\Omega}\,$ on the strip, excluding the
trivial case of a straight $\,\Omega\,$. As in \cite{ES1}, one can
introduce locally orthogonal coordinates $\,s,u\;$; the operator
$\;-\Delta_D^{\Omega}\,$ is then unitarily equivalent to
\begin{equation} \label{Hamiltonian2w}
\tilde H\,=\,-g^{-1/2}\,\partial_s\, g^{-1/2} \partial_s\,-\,
g^{-1/2}\,\partial_u\, g^{1/2} \partial_u
\end{equation}
on $\,L^2(\R\times(-a,a),\, g^{1/2} ds\,du\,)\,$ with the Dirichlet
condition at $\,u=\pm a\,$, where $\,\partial_s:=\,
\partial/\partial s\,$, {\em etc.,} and $\, g^{1/2}(s,u):=
1+u\gamma(s)\,$ is the corresponding Jacobian. If $\,\gamma\,$ is not
smooth, the operator $\,\tilde H\,$ has to be understood in the form
sense. Under a stronger regularity assumption, namely
\begin{description}
\item{\em (r4)} $\;\gamma\,$ is piecewise $\,C^2\,$ with
$\,\dot\gamma,\, \ddot\gamma\,$ bounded
\end{description}
we can pass to another unitarily equivalent operator,
\begin{equation} \label{Hamiltonian2}
H\,=\,-\partial_s\, (1+u\gamma)^{-2}\, \partial_s\,-\,
\partial_u^2 \,+\,V(s,u)
\end{equation}
on $\,L^2(\R\times(-a,a)\,)\,$ with
\begin{equation} \label{effective potential2}
V(s,u)\,=\,-\,\frac{\gamma^2}{4(1+u\gamma)^2}\,+\,
\frac{u\ddot{\gamma}}{2(1+u\gamma)^3}\,-\,
\frac{5}{4}\,\frac{u^2\dot\gamma^2}{(1+u\gamma)^4}
\end{equation}
which is \esa on $\,D(H)\,=\, \{\,\psi\,:\,\psi\in C^{\infty},\;
\psi(s,-a)=\psi(s,a)=0,\; H\psi\in L^2\,\}\,$.
The above listed regularity assumptions cover various classes of
operators. We shall concentrate on the case where $\,\Omega\,$ is
curved substantially within a bounded region only, \ie, the curvature
obeys suitable decay assumptions, \eg,
\begin{description}
\item{{\em (d1)}} $\;\gamma, \dot\gamma\,\,
\ddot\gamma\in\,L^{\infty}_{\eps}(\R)\,$,
\item{{\em (d2)}} $\;\gamma, \dot\gamma\,\in\,L^2(\R,|s|\,ds)\,$,
and $\,\ddot{\gamma}\in L^1(\R,|s|\,ds)\;$;
\end{description}
recall that $\,L^{\infty}_{\eps}(\R)\,$ is the set of $\,L^{\infty}\,$
functions with the property that to any $\,\eps>0\,$ there is a compact
$\,K_{\eps}\,$ such that $\,\Vert f\,\restr\, K_{\eps}\Vert_{\infty}< \eps\,$.
Since the functions involved are bounded, {\em (d2)} strengthens {\em
(d1)}. The latter requires $\,\lim_{|s|\to\infty} \gamma(s)=0\,$, while
{\em (d2)} corresponds for a purely powerlike asymptotics of $\,\gamma\,$
to an $\,\OO(\vert s\vert^{-1-\eps})\,$ decay;
an easy integration shows that the borderline refers in the last
case to a logarithmic--spiral asymptotics of $\,\Gamma\,$.
\subsection{Existence of bound states}
The most general existence result is a consequence of the following
theorem the basic idea of which belongs to Goldstone and Jaffe
\cite{GJ}.
\vspace{3mm}
\noindent
{\bf 2.1 Theorem.} {\em Assume (r1)--(r3). If the strip is not
straight, i.e., $\,\gamma\neq 0\,$, then
$\,\inf\,\sigma(-\Delta_D^{\Omega})<
\kappa_1^2\,$, where $\,\kappa_1\,=\,\pi/2a\,$. In particular, if
$\,\inf\,\sigma_{ess}(-\Delta_D^{\Omega})=\kappa_1^2\,$, then
$\;-\Delta_D^{\Omega}\,$ has at least one isolated eigenvalue of
finite multiplicity.}
\vspace{3mm}
\noindent
{\bf Proof.} The argument is based on a variational estimate. Denote
the norm in $\,L^2(\R\times(-a,a),\, g^{1/2} ds\,du\,)\,$ as
$\,\Vert\cdot\Vert_g\;$; then it follows from (\ref{Hamiltonian2w})
that
\begin{equation} \label{form Hamiltonian2}
q[\Phi]\,:=\, \Vert\tilde H^{1/2}\Phi\Vert_g^2 -\,\kappa_1^2
\Vert\Phi\Vert_g^2\,=\, \Vert
g^{-1/4}\partial_s\Phi\Vert^2 +\,\Vert g^{1/4}\partial_u\Phi\Vert^2
-\,\kappa_1^2 \Vert g^{1/4}\Phi\Vert^2
\end{equation}
holds for any $\,\Phi\,$ in $\,Q(\tilde H)\,$, the quadratic form
domain of $\,\tilde H\,$, where $\,\Vert\cdot\Vert:=
\Vert\cdot\Vert_1\,$. In particular, choosing $\,\Phi(s,u):=
\varphi(s)\chi_1(u)\,$, where $\,\chi_1(u):= \sqrt{2\over d}\,
\cos(\kappa_1 u)\,$ is the lowest transverse--mode function, we find
$$
q[\Phi]\,=\, \Vert\langle g^{-1/4}\rangle \dot \varphi
\Vert^2_{L^2(\R)}\,,
$$
where $\,\langle\cdot\rangle\,$ means the expectation with respect to
$\,\chi_1\,$. It is clear that it is sufficient to find $\,\Phi\in D(\tilde
H)\,$ for which $\,q[\Phi]\,$ is negative.
Let $\,K:=[-c,c]\,$ for some $\,c>0\,$ and choose the function
$\,\varphi\,$ from the Schwartz space $\,\SS(\R)\,$ in such a way that
$\,\varphi(s)=1\,$ on $\,K\,$. Furthermore, we define the family
$\,\{\varphi_{\lambda}:\, \lambda\in\R\,\}\,$ by a scaling exterior
to $\,K\,$:
\begin{equation} \label{scaling}
\varphi_{\lambda}(s)\,:=\; \left\lbrace \begin{array}{lll} \varphi(s) \quad &
\dots \quad & |s|\leq c \\ \\ \varphi(\pm c+\lambda(s\mp c)) \quad &
\dots \quad & \pm s>c \end{array} \right.
\end{equation}
It is easy to check that
\begin{equation} \label{exterior kinetic energy 2}
q[\Phi_{\lambda}]\,\leq\, \Vert\langle
g^{-1/2}\rangle\Vert^2_{\infty} \Vert
\dot \varphi_{\lambda}\Vert^2 \,\leq\,
{\lambda\over 1-a\Vert\gamma\Vert_{\infty}}\, \Vert
\dot \varphi\Vert^2
\end{equation}
holds for $\,\Phi_{\lambda}:\; \Phi_{\lambda}(s,u)= \varphi_{\lambda}(s)
\chi_1(u)\,$, so $\,q[\Phi_{\lambda}]\,$ can be made arbitrarily
small by choosing $\,\lambda\,$ small enough.
Next we pick $\,j\in C_0^{\infty}(K\times (-a,a))\,$ and set
$\,\phi:= j^2(\tilde H-\kappa_1^2)\Phi_{\lambda}\,$. For non--zero
$\,\gamma,\,j\,$ this vector is non--zero; one can compute explicitly
$$
\phi(s,u)\,=\, -\left(\, j^2\, {\gamma\over 1+u\gamma}\,
\partial_u\chi_1 \right)(s,u)\,=\,
j^2(s,u)\,\sqrt{2\over d}\; {\kappa_1\gamma(s)\over
1+u\gamma(s)}\, \sin(\kappa_1u)\,.
$$
Recall that $\,\gamma\,$ is non--zero as an element of $\,L^1_{loc}\,$
if there is a compact set (without loss of generality, it may be
identified with $\,K\,$) such that $\,\int_K \vert \gamma(s)\vert\, ds
\ne 0\,$. Then $\,\gamma\ne 0\,$ also in $\,L^2_{loc}(\R)\,$ because
$\,\int_K \vert \gamma(s)\vert\, ds \le \vert K\vert^{1/2}
\left( \int_K \vert \gamma(s)\vert^2\, ds \right)^{1/2}\,$, and
therefore $\,\phi\ne 0\,$ as an element of $\,L^2(\R\times (-a,a))\,$.
Since both $\,\Phi_{\lambda}\,$ and $\,\phi\,$ belong to $\,D(\tilde
H)\,$, we have
$$
q[\Phi_{\lambda}+\eps\phi]\,=\, q[\Phi_{\lambda}]\,+\, 2\eps \Vert
j(\tilde H-\kappa_1^2)\Phi_\lambda\Vert^2 +\, \eps^2 (\phi,(\tilde
H-\kappa_1^2)\phi) \,,
$$
where the second term on the right side does not depend on $\,\lambda\,$
because the scaling acts out of the support of the
localization function $\,j\,$. For all sufficiently small negative
$\,\eps\,$ the sum of the last two terms is negative, and we can
choose $\,\lambda\,$ so that $\,q[\Phi_{\lambda}+\eps\phi]<0\,$.
\quad \QED
\vspace{3mm}
This result implies that the discrete spectrum of
$\;-\Delta_D^{\Omega}\,$ is non--empty provided the bending does not
push the essential--spectrum threshold down; this happens typically
if the curvature decays to zero at infinity. In particular,
we have the following result.
\vspace{3mm}
\noindent
{\bf 2.2 Corollary.} {\em Let a curved strip satisfy the assumptions
(r1)--(r3). Any of the conditions
\begin{description}
\item{(a)} $\;\gamma\,$ has a compact support,
\item{(b)} $\;$ (r4) and (d1)
\end{description}
is sufficient for $\;-\Delta_D^{\Omega}\,$ to have at least one bound
state of energy below $\,\kappa_1^2\,$.}
\vspace{3mm}
\noindent
{\bf Proof.} In the first case we can choose $\,s_0\,$ in such a way
that the cuts of the strip at $\,s=\pm s_0\,$ are outside of the
curved part. We impose there the Neumann condition; this turns
$\,\tilde H\,$ into the orthogonal sum $\,\tilde H_N= \tilde H_-
\oplus \tilde H_0 \oplus \tilde H_+\,$. The spectrum of the middle
part is purely discrete while the tail operators have a purely continuous
spectrum, so $\,\sigma_{ess}(\tilde H_N)= \sigma_{ess}(\tilde H_-
\oplus \tilde H_+)= \sigma(\tilde H_-\oplus \tilde H_+)=
[\kappa_1^2,\infty)\,$. On the other hand, we have $\,\tilde H_N\leq
\tilde H\;$ (\cf \cite{RS}, Sec.XIII.15), so the result follows by
the minimax principle.
Under the strengthened regularity assumption {\em (r4),} we may pass
to the unitarily equivalent operator (\ref{Hamiltonian2}). This makes
it possible to use known results of the theory of Schr\"odinger
operators, since the latter can be estimated by
\begin{equation} \label{estimate2}
H_{\pm}\,:=\,-\,(1\mp a\Vert\gamma\Vert_{\infty})^{-2}
\partial_s^2\,-\,\partial_u^2\,+\,V
\end{equation}
or more roughly by
$$
\hat H_{\pm}\,:=\,-\,(1\mp a\Vert\gamma\Vert_{\infty})^{-2}
\partial_s^2\,-\,\partial_u^2\,+\,\hat V_{\pm}(s)
$$
with $\,\hat V_{\pm}\,:=\,\pm\,\sup\lbrace \pm V(s,\cdot):\:
u\in(-a,a)\,\rbrace\,$. The
sought result follows from the last mentioned estimate, since
$\;V_{\pm}\in L^{\infty}_{\eps}\,$ due to {\em (d1),} so
$\,\inf \sigma_{ess} (\hat H_{\pm})\,=\,\kappa_1^2\,$ and the same
is, of course, true for $\,H\,$. \quad \QED
\vspace{3mm}
The part (a) corresponds to the situation treated in \cite{GJ}. As an
element of $\,L^2(\R\times (-a,a))\,$, the trial function used in the proof of
the theorem is $\,(s,u) \mapsto \sqrt{1+u\gamma}\,(\Phi_{\lambda}+
\eps\phi)\,$. It is clearly better than the one of Lemma~3.6 in
\cite{E3} because it yields the existence of eigenvalues for {\em
any} strip width.
\subsection{Thin strips: number of bound states}
Minimax estimates of the above mentioned type are useful in many
ways. As the first illustration, we shall use them to derive an upper
bound to the number of bound states supported by a curved strip
which is thin enough.
\vspace{3mm}
\noindent
{\bf 2.3 Proposition.} {\em Assume (r1--4) and (d2), and furthermore, let}
$$
a\,<\, \frac{c_2}{1+c_2\Vert\gamma\Vert_{\infty}}\,,
$$
{\em where}
$$
c_2\,:=\, \min\,\left\{\, \frac{\pi}{\Vert\gamma\Vert_{\infty}}\,,\,
\left(\, \frac{\pi^2}{2\Vert\ddot\gamma\Vert_{\infty}}\,\right)^{1/3}
\,,\, \left(\, \frac{\pi}{\sqrt{5}\Vert\dot\gamma\Vert_{\infty}}
\,\right)^{1/2}\, \right\}\,,
$$
{\em then the number $\,N(H)\,$ of isolated eigenvalues (counting
multiplicity) obeys}
\begin{equation} \label{bound-state number}
N(H)\,\leq\, 1\,+\, \frac{1}{8}\,\left(\,
\frac{1+a\Vert\gamma\Vert_{\infty}}{1-a\Vert\gamma\Vert_{\infty}}
\,\right)^2 \: \frac{\int_{\R^2}\, \gamma(s)^2 \vert s-t\,\vert
\gamma(t)^2\, ds\,dt}{\int_{\R}\, \gamma(s)^2\, ds}\,.
\end{equation}
\vspace{3mm}
\noindent
{\bf Proof.} We estimate $\,H\,$ from below by
$$
H\,\geq\,H_-\,:=\,\overline{\left(1+a\Vert\gamma\Vert_{\infty}
\right)^{-2} \,h_-\otimes I \,+\, I\otimes G}
$$
with
\begin{eqnarray*}
h_- \!&:=&\! -\,\frac{d^2}{ds^2}\,+\, \left(1+a\Vert\gamma\Vert_{\infty}
\right)^2 V_-(s)\,, \\ \\
V_-(s) \!&:=&\! -\,\frac{\gamma^2}{4(1-a\Vert\gamma\Vert_{\infty})^2}\,-\,
\frac{a\Vert\ddot{\gamma}\Vert_{\infty}}{2(1-a\Vert\gamma
\Vert_{\infty})^3}\,-\,
\frac{5}{4}\,\frac{a^2\Vert\dot\gamma^2\Vert_{\infty}}
{(1-a\Vert\gamma\Vert_{\infty})^4}
\end{eqnarray*}
and $\,G\,$ denoting $\,-\partial_u^2\,$ on $\,L^2(-a,a)\,$ with Dirichlet
b.c. In view of the assumption, the modulus of each term of
$\,V_-(s)\,$ is $\;\leq\,\pi^2/4a^2\,$, and therefore only the lowest
eigenvalue of $\,G\,$ can contribute to $\,\sigma_{d}(H_-)\,$. Hence
we have $\,N(H)\leq N(H_-)\leq N(h_-)\,$. Moreover, the two last
terms in the potential $\,(1+a\Vert\gamma\Vert_{\infty})^2\,
V_-(s)\,$ are $\,s$--independent so $\,N(h_-)=N(\tilde h_-)\,$, where
$$
\tilde h_-\,:=\, -\,\partial_s^2\,-\, \frac{1}{4}\,\left(\,
\frac{1+a\Vert\gamma\Vert_{\infty}}{1-a\Vert\gamma\Vert_{\infty}}
\,\right)^2\, \gamma(s)^2\,,
$$
and the known bound \cite{Kl,New,Se} yields the result. \quad \QED
\vspace{3mm}
In particular, we get a sufficient condition under which a thin strip
has just one bound state; roughly speaking, this happens if
$\,\Omega\,$ has one simple bend, \ie, $\,\gamma\,$ is
sign--preserving and has one well distinguished peak -- \cf\cite{E2}.
In Chapter~4 below we shall see that the same is true for any
$\,\Omega\,$ which satisfies the assumptions of Corollary~2.2(b)
provided it is only slightly bent.
\section{A tube in $\,\R^3\,$}
Now we apply the same ideas to a three--dimensional quantum waveguide.
\subsection{"Straightening" of a curved cylindrical tube}
Let $\,\Gamma\in C^k(\R,\R^3)\,,\,k\geq 2\,$, be a curve and
$\,(T,B,N)\,$ the corresponding Frenet triad frame. Given $\,a>0\,$,
we denote $\,B_a\,:=\,\{\,x\in\R^2 \,:\,\vert x\vert0\,$. Since the
integral of the curvature over an interval $\,[s_1,s_2]\,$ is the
angle between the tangent vectors to $\,\Gamma_{\beta}\,$ at the
corresponding points, it is clear that the parameter $\,\beta\,$
controls bending of the tube.
We have seen that in the three--dimensional case it is the curvature
and not torsion which is responsible for the existence of non--empty
discrete spectrum; hence we limit ourselves to families of tubes with
a scaled $\,\gamma\,$ and $\,\tau\,$ fixed. One should pay attention
to the nature of the straightening which corresponds to the limit
$\,\beta\to 0\,$. As an illustration, suppose that $\,\gamma\,$ has a
compact support; then the length of the curved part remains preserved
and the curvature radius at each point grows proportionally to
$\,\beta^{-1}$. We have the following general result.
\vspace{3mm}
\noindent
{\bf 4.1 Theorem.} {\em Assume that the hypotheses (r1)--(r4) and
(d2) (or (d2') in the three--dimensional case) are valid. Then for
all $\,\beta\,$ small enough, $\,-\Delta_D^{\Omega}\,$ corresponding
to the tube generated by $\,\Gamma_{\beta}\,$ has exactly one
isolated eigenvalue $\,e(\beta)\,$. Moreover,
\begin{description}
\item{(a)} in the two--dimensional case, its asymptotic behaviour
is given by
\begin{eqnarray} \label{angle asymptotics 2}
\lefteqn{\sqrt{\kappa_1^2-e(\beta)} \,=\, {\beta^2\over
8}\,\biggl\lbrace\, \Vert\gamma\Vert^2} \nonumber \\ \\ &&
+\,{1\over 2}\, \sum_{N=2}^{\infty}\, (\chi_N,u\chi_1)^2 \varrho_N\,
\int_{\R^2} \dot\gamma(s)\, e^{-\varrho_N|s-s'|} \dot\gamma(s')\,
ds\,ds'\, \biggr\rbrace\,+\, \OO(\beta^3)\,, \nonumber
\end{eqnarray}
where $\,\varrho_N:= \kappa_1 \sqrt{N^2-1}\;$; the sum runs in fact
over even $\,N\,$ only.
\item{(b)} Similarly, for a three--dimensional tube we have
\begin{eqnarray} \label{angle asymptotics 3}
\lefteqn{\sqrt{\kappa_0^2-e(\beta)} \,=\, {\beta^2\over 8}\,
\Vert\gamma\Vert^2+\, {\beta^2\over 16}\,
\sum_{N=1}^{\infty}\, \varrho_N\, \int_{B_a\times B_a}
\, d\omega\,d\omega'\chi_0(\omega)\chi_0(\omega')
\chi_N(\omega)\chi_N(\omega')} \nonumber \\ \\ &&
\times \int_{\R^2}\, ds\,ds' h_s(s,r,\theta)\,
e^{-\varrho_N|s-s'|} h_s(s',r',\theta')\,+\, \OO(\beta^3)\,,
\phantom{AAAAAAAAAAAA} \nonumber
\end{eqnarray}
where $\,h_s= r\,\gamma\tau\,\sin(\theta-\alpha)+
r\,\dot\gamma\,\cos(\theta-\alpha)\,$ was introduced in Section~3.2,
$\,d\omega:= r\,dr\,d\theta\,$ and
$\,\varrho_N:= \sqrt{\kappa_N^2-\kappa_0^2}\,$ with $\,\kappa_N^2=
j_{n,m}^2 a^{-2}$ are the transverse mode energies (properly
ordered).
\end{description} }
\noindent
The formulae (\ref{angle asymptotics 2}), (\ref{angle asymptotics 3})
have a transparent structure. The first term comes from the
attractive potential $\,-\,{1\over 4}\,\gamma^2\,$ which dominates
the picture for thin tubes, while the rest expresses the contribution
of higher transverse modes (see also Remark~4.3 below).
\subsection{Weakly coupled Schr\"odinger operators in a cylinder}
To prove the above theorem, we need an auxiliary result which is of
interest independently of the present context.
Let $\,M\,$ be an open connected set in $\,\R^{\nu-1},\:\nu\geq 2\,$,
with $\,\overline M\,$ compact, and consider the operator
$$
H_{\lambda}\,=\,-\,\Delta_D\,+\,\lambda V
$$
on $\,L^2(\R)\otimes L^2(M)\,$, where $\,-\Delta_D\,$ is the Dirichlet
Laplacian on $\,\R\times M\,$ which is closure of $\,(-d^2/dx^2)\otimes
I_{\nu-1}\,+\,I_1\otimes(-\Delta_D^M)\,$, and the potential $\,V\,$ is
supposed to be {\em measurable} and {\em bounded}. The last
assumption is made with our aim in mind; it is not difficult to
extend the argument presented below to potentials with local
singularities.
The Dirichlet Laplacian $\,-\Delta_D^M\,$ on $\,L^2(M)\,$ has a
purely discrete spectrum consisting of eigenvalues $\,0 < \kappa_0^2
< \dots < \kappa_N^2 < \kappa_{N+1}^2 < \dots\;$; the corresponding
normalized eigenvectors will be denoted as $\,\chi_{N_j}\,,
\;N=0,1,\dots\,$ and $\,j=1,\dots,d_N\,$. It is known that the
lowest eigenvalue is simple, $\,d_0=1\,$, and $\,\chi_0\,$ can be
chosen positive (\cite{RS}, App. to Sec.XIII.12). Due to the
boundedness, the functions
$$
V_{mn}\,:=\,\int_M\, \overline{\chi}_m(y) V(\cdot,y) \chi_n(y)\,dy
$$
are again measurable and bounded for any $\,m,n\,$. In analogy with
$\,V_{00}\,$, we define $\,\vert V\vert_{00}\,$ as the ground--state
matrix element of the function $\,\vert V(\cdot,\cdot)\vert\,$.
\vspace{3mm}
\noindent
{\bf 4.2 Theorem.} {\em Assume that $\,V\,$ is non--zero and $\,\vert
V\vert_{00}\in L^1(\R,\vert x\vert\,dx)\;$, then $\;H_{\lambda}\;$
has for small $\,\lambda\,$ at most one simple eigenvalue
$\,e(\lambda)\,<\,\kappa_0^2\,$, and this happens iff $\;\int_{\R}
\lambda V_{00}(x)\,dx\leq 0\,$. Moreover, if this condition holds the
following expansion is valid}
\begin{eqnarray} \label{expansion}
\sqrt{\kappa_0^2-e(\lambda)}\, & = &
\,-\,\frac{\lambda}{2}\,\int_{\R} V_{00}(x)\,dx\,-\,
\frac{\lambda^2}{4}\,\Biggl\lbrace\,
\int_{\R^2} V_{00}(x)\vert x-x'\vert V_{00}(x')\,dx\,dx' \nonumber \\
\nonumber \\
&& -\,\sum_{N=1}^{\infty} \sum_{j=1}^{d_N}\,\int_{\R^2}\,
V_{0N_j}(x)\,\frac{e^{-\sqrt{\kappa_N^2-\kappa_0^2}\vert x-x'\vert}}
{\sqrt{\kappa_N^2-\kappa_0^2}}\,V_{N_j0}(x')\,dx\,dx'\,\Biggr\rbrace\,
\nonumber \\ \nonumber \\ && +\,\OO(\lambda^3)\,.
\end{eqnarray}
\vspace{3mm}
\noindent
{\bf Proof.} According to Birman--Schwinger principle, $\,e(\lambda)\,$
is a bound state of $\,H_{\lambda}\,$ {\em iff} $\,\lambda
K_{\alpha}\,$ has eigenvalue $\,-1\,$ for $\,\alpha^2\,=\,e(\lambda)\,$
where $\,K_{\alpha}\,:=\, \vert V\vert^{1/2} (-\Delta_D-\alpha^2)^{-1}
V^{1/2}\,$ and $\,V^{1/2}\,:=\,\vert V\vert^{1/2}{\rm sgn\,}V\,$. Since
$\,-\Delta_D^M\,$ is reduced by the projections $\,P_N\,$ to its
eigenspaces, $\,-\Delta_D\,$ is reduced by $\,I_1\otimes P_N\,$ and the
same is true for its resolvent whose kernel equals
\begin{equation} \label{resolvent}
R_0(\alpha; x,y; x',y')\,=\,\sum_{N=0}^{\infty}\sum_{j=1}^{d_N}
\chi_{N_j}(y)\, \frac{e^{-k_N(\alpha)\vert x-x'\vert}}{2k_N(\alpha)}\,
\overline{\chi}_{N_j}(y')
\end{equation}
where we have denoted
$\,k_N(\alpha)\,:=\,\sqrt{\kappa_N^2-\alpha^2}\,$. The first term in
this expansion has a singularity at $\,\alpha = \kappa_0\,$ which we
want to single out. This can be done by taking
$\,K_{\alpha}\,=\,M_{\alpha}+L_{\alpha}\,$, where
$$
M_{\alpha}(x,y; x',y')\,=\,
\vert V(x,y)\vert^{1/2} \chi_0(y)\, \frac{e^{-k_0(\alpha)\vert
x-x'\vert}-1}{2k_0(\alpha)}\, \chi_0(y')V(x',y')^{1/2}
$$
as in \cite{ES1}, which would require $\,V_{00}\in L^1(\R,\vert
x\vert^2\,dx)\,$. To get a better result, we replace following
\cite{BGS,E3} the above decomposition for
$\,\alpha<\kappa_0\,$ by $\,K_{\alpha}\,=\,Q_{\alpha}
+P_{\alpha}\,$ with
$$
Q_{\alpha}(x,y; x',y')\,=\,
\frac{e^{-k_0(\alpha)\vert x\vert}}{2k_0(\alpha)}\,
\vert V(x,y)\vert^{1/2} \chi_0(y)\, e^{-k_0(\alpha)\vert x'\vert}\,
\chi_0(y')V(x',y')^{1/2}
$$
and $\,P_{\alpha}\,:=\,A_{\alpha} + \vert V\vert^{1/2}B_{\alpha}
V^{1/2}\,$, where
\begin{eqnarray}
\lefteqn{A_{\alpha}(x,y; x',y')\,} \nonumber \\ &&
=\,\frac{1}{k_0(\alpha)}\,
\vert V(x,y)\vert^{1/2} \chi_0(y)\, \left\lbrack
e^{-k_0(\alpha)\vert x\vert_>} \sinh k_0(\alpha)\vert x\vert_<
\right\rbrack\, \chi_0(y')V(x',y')^{1/2}
\end{eqnarray}
and the kernel of $\,B_{\alpha}\,$ is given by (\ref{resolvent}) with
the summation running over $\,N=1,2,\dots\;$; here we have introduced
$$
\vert x\vert_<\,:=\, \max \lbrace\,0\,, \min(\vert x\vert, \vert
x'\vert)\,{\rm sgn\,}(xx')\,\rbrace
$$
and $\,\vert x\vert_>\,:=\,\max\{\vert x\vert, \vert x'\vert\}\,$. It is
easy to see that for any $\,\alpha_0 < \kappa_1\,$, the norm $\,\Vert
B_{\alpha}\Vert\,$ has in $\,\lbrack 0,\alpha_0\,\rbrack\,$ a bound
independent of $\,\alpha\,$. The norm of $\,A_{\alpha}\,$ can be estimated
as follows:
\begin{eqnarray*}
\Vert A_{\alpha}\Vert^2 &\!\le\!& \Vert A_{\alpha}\Vert_{HS}^2
\\ \\ &\!\le\!& \int_{\R\times M}\, \int_{\R\times M}\,
\vert V(x,y)\chi_0(y)\vert^2
\vert x\vert_<^2 \vert V(x',y')\chi_0(y')\vert^2 \,dx\,dy\,dx'\,dy'
\\ \\ &\!\le\!& \left( \int_{\R}\,\vert x\vert \,
\vert V\vert_{00}(x)\,dx\,\right)^2\,,
\end{eqnarray*}
where we have employed the inequalities $\,e^{-z'}\sinh z\,\leq\, z\,$
for $\,z'\geq z\,$ and $\,\vert x\vert_<^2\,\leq\,\vert xx'\vert\,$,
respectively.
Moreover, by dominated convergence $\,A_{\alpha}\,\rightarrow\,
A_{\kappa_0}\,$ as $\,\alpha\rightarrow\kappa_0\,$ where the last
operator has the kernel $\,A_{\kappa_0}(x,y; x',y') \,=\, \vert
V(x,y)\vert^{1/2} \chi_0(y) \vert x\vert_< \chi_0(y') V(x',y')^{1/2}\,$,
so it is again bounded. Together we see that $\,\Vert
P_{\alpha}\Vert\,$ has in $\,\lbrack 0,\alpha_0\rbrack\,$ a bound
independent of $\,\alpha\,$. Then $\,\Vert\lambda
P_{\alpha}\Vert\,<\,1\,$ holds for small enough $\,\lambda\,$ and we
may write
$$
(I+\lambda K_{\alpha})^{-1}\,=\,
\lbrack I+\lambda(I+\lambda P_{\alpha})^{-1} Q_{\alpha}\rbrack^{-1}
(I+\lambda P_{\alpha})^{-1}\,,
$$
hence $\,\lambda K_{\alpha}\,$ has eigenvalue $\,-1\,$ {\em iff} the
same is true for $\,\lambda(I+\lambda P_{\alpha})^{-1}Q_{\alpha}\,$.
Since $\,Q_{\alpha}\,$ is a rank--one operator, $\,\lambda(I+\lambda
P_{\alpha})^{-1} Q_{\alpha}\,=\, (\psi,\cdot)\varphi\,$ with
$\,\psi\,:=\,\frac{\lambda}{2k_0(\alpha)}\,
e^{-k_0(\alpha)\vert\cdot\vert} V^{1/2}\chi_0\,$ and $\,\varphi\,:=\,
(I+\lambda P_{\alpha})^{-1} e^{-k_0(\alpha)\vert\cdot\vert}
\vert V\vert^{1/2}\chi_0\,$, it has just one eigenvalue $\,\xi(\lambda)
\,=\,(\psi,\varphi)\,$. Putting it equal to $\,-1\,$ we get for
$\,k_0(\alpha) \,:=\, z\,$ the following equation
\begin{equation} \label{implicit}
z\,=\,G(\lambda,z)
\end{equation}
where
\begin{equation} \label{function G}
G(\lambda,z)\,=\,-\,\frac{\lambda}{2} \int_{\R\times M}\,
e^{-z\vert x\vert} V(x,y)^{1/2}\chi_0(y)
\lbrack (I+\lambda P_{\alpha(z)})^{-1} e^{-z\vert\cdot\vert}
\vert V\vert^{1/2}\chi_0 \rbrack(x,y)\,dx\,dy\,.
\end{equation}
If $\,V\,$ decreases exponentially, $\,G\,$ is analytic around
$\,(0,0)\,$ and the assertion follows by implicit--function
theorem; it gives also a prescription how to compute other terms
in the expansion (\ref{expansion}).
In the general case, more caution is needed. First of all, writing
$\,z(\lambda) \,=\, a\lambda+b\lambda^2+r(\lambda)\,$ one finds
easily that (\ref{expansion}) solves again the equation
(\ref{implicit}), however, now we know only that the remainder is
$\,\OO(\lambda^3)\,$. Hence it is sufficient to check that there is at
most one solution; this will be true if we prove it for a non--positive
$\,V\,$ and $\,\lambda>0\,$. Then $\,V_{00}\,$ is also non--positive,
and excluding the trivial case $\,V=0\,$ we see that there is a
positive $\,C_1\,$ such that $\,\vert z^{-1}\vert\,\leq\, C_1
\lambda^{-1}\,$ for small $\,\lambda\,$ and any solution $\,z\,$ of
the equation (\ref{implicit}). The operator--valued function $\,z
\mapsto P_{\alpha(z)}\,$ is real--analytic for $\,z>0\,$ and has a
bounded limit as $\,z\to 0+\,$, hence Cauchy integral formula gives
$$
\left\Vert\,\frac{dP_{\alpha(z)}}{dz}\,\right\Vert\,<\:
C_2\vert z^{-1}\vert
$$
for some $\,C_2\,$ and small $\,\vert z\vert\,$. An explicit
calculation shows that
$$
\left( e^{-z\vert\cdot\vert} \vert V\vert^{1/2}\chi_0,\,
\left\lbrack\,2\vert\cdot\vert\,+\,\frac{dP_{\alpha(z)}}{dz}\,
\right\rbrack e^{-z\vert\cdot\vert} \vert V\vert^{1/2}\chi_0 \right)
$$
remains bounded as $\,z\to 0+\,$ since $\,V_{00}\in L^2(\R,
\vert x\vert dx)\,$ by assumption. Then $\,\left\vert\frac
{\partial G}{\partial z}\right\vert\,$ can be estimated by the sum of
two terms; the one containing $\,\frac{dP_{\alpha(z)}}{dz}\,$ can be
handled as in the Schr\"odinger--operator case \cite{Si} being
therefore $\;\leq C_3\lambda\,$ for some $\,C_3\,$. On the other
hand, the operator $\,B_z\,$ of multiplication by
$\,xe^{-z\vert x\vert}\,$ fulfils $\,\Vert B_z\Vert\,\leq\,
e^{-1}\vert z\vert^{-1}\,$ and the same argument applies; hence there
is $\,C_4>0\,$ such that
$$
\left\vert\frac{\partial G}{\partial z}\right\vert\,\leq\:C_4\lambda
$$
holds for $\,\vert z^{-1}\vert\,\leq\, C_1\lambda^{-1}\,$ and all
sufficiently small $\,\lambda\,$. Any two solutions $\,z_1,\,z_2\,$
of (\ref{implicit}) have to fulfil
$$
z_2-z_1\,=\, \int_{z_1}^{z_2}\,\frac{\partial G}{\partial z}\, dz
$$
so the uniqueness is ensured for $\,\lambda\,0\;.
\end{eqnarray*}
\subsection{Proof of Theorem 4.1}
Consider first the two--dimensional case. We use the bounds
(\ref{estimate2}), where
$$
H_{\pm}\,:=\, -\,(1\mp a\beta\Vert\gamma\Vert_{\infty})^{-2}
\partial_s^2 -\,\partial_u^2 +\,V_{\beta}(s,u)\,,
$$
where $\,V_{\beta}\,$ is the potential (\ref{effective potential2})
referring to $\,\gamma_{\beta}\,$. By Theorem~4.2 each of the
operators $\,H_{\pm}\,$ has for small $\,\beta\,$ just one eigenvalue
$\,e_{\pm}(\beta)\;$; the minimax principle tells us then that the
same is true for $\,H\,$ and $\,e_+(\beta)\geq e(\beta) \geq
e_-(\beta)\,$.
Theorem~4.2 gives us an expansion of $\,e_{\pm}(\beta)\;$; if we
rescale the longitudinal variable to $\,s_{\pm}:= (1\mp
a\beta\Vert\gamma\Vert_{\infty})s\,$ and change appropriately the
integration variables, we obtain
\begin{eqnarray*}
\lefteqn{\sqrt{\kappa_1^2-e_{\pm}(\beta)^2} \,=\, -\,\frac{1}{2}\,(1\mp
a\beta\Vert\gamma\Vert_{\infty})^{-1}\, \int_{\R}
(V_{\beta})_{11}(s)\,ds} \\ \\ &&
-\,\frac{1}{4}\,(1\mp a\beta\Vert\gamma\Vert_{\infty})^{-3}\,
\Biggl\lbrace\, \int_{\R^2} (V_{\beta})_{11}(s)\vert s-s'\vert
(V_{\beta})_{11}(s')\,ds\,ds' \\ \\ &&
-\,\sum_{N=2}^{\infty} \,\int_{\R^2}\,
(V_{\beta})_{1N}(s)\,\frac{e^{-\sqrt{\kappa_N^2-\kappa_1^2}\vert
s-s'\vert}} {\sqrt{\kappa_N^2-\kappa_1^2}}\,(V_{\beta})_{N1}(s')
\,ds\,ds'\,\Biggr\rbrace\, \\ \\ && +\,\OO(\beta^3)\,.
\end{eqnarray*}
Since $\,V_{\beta}\,$ itself expands in terms of $\,\beta\,$, we may
neglect the scaling factors $\,1\mp a\beta\Vert\gamma
\Vert_{\infty}\,$ when computing the leading term.
Let us write the right side as $\,I_1+\sum_{N=1}^{\infty}I_{2,N}
+\cdots\,$. The first term is through an easy integration by parts
equal to
\begin{eqnarray*}
I_1 &\!:=\!& {\beta^2\over 8}\, \int_{-a}^a\,du\, \int_{\R}\,ds\,
\chi_1(u)^2 \left\lbrack {\gamma^2\over (1+u\beta\gamma)^2}\,-\,
{u^2\dot\gamma^2\over (1+u\beta\gamma)^4} \right\rbrack \\ \\
&\!=\!& {\beta^2\over 8}\, \left\{ \Vert\gamma\Vert^2-\, \Vert
u\chi_1\Vert^2 \Vert\dot\gamma\Vert^2 \right\}\, +\,\OO(\beta^3)\;;
\end{eqnarray*}
we have used the fact that $\,\dot\gamma(s) \to 0\,$ as
$\,|s|\to\infty\,$ due to {\em (d2)}. What is important is that the
integration by parts cancels the term linear in $\,\beta\,$. Hence we
have to compute $\,\sum_{N=1}^{\infty}I_{2,N}\,$ because it contains
also contributions of order $\,\beta^2\,$ coming from the second term
in (\ref{effective potential2}). We have
\begin{eqnarray*}
I_{2,1} &\!=\!& -\,{1\over 4}\, \int_{\R^2}\, ds\,ds'\, \int_{-a}^a\,
\int_{-a}^a\, du\,du' \chi_1(u)^2 \chi_1(u')^2\, V_{\beta}(s,u)\,
|s-s'|\, V_{\beta}(s') \\ \\
&\!=\!& -\,{\beta^2\over 16}\, \int_{-a}^a\,
\int_{-a}^a\, du\,du'\,uu'\, \chi_1(u)^2 \chi_1(u')^2 \int_{\R^2}
ds\,ds'\, \ddot\gamma(s)\, |s-s'|\, \ddot\gamma(s')\, +\,\OO(\beta^3)\,.
\end{eqnarray*}
By a double integration by parts, the inner integral equals $\,-2\,
\int_{\R} \dot\gamma(s)^2 ds\,$, so
$$
I_{2,1}\,=\, {\beta^2\over 8}\, (\chi_1,u\chi_1)^2 \Vert\dot\gamma
\Vert^2 +\, \OO(\beta^3)\,.
$$
The leading part in the remaining terms is
\begin{eqnarray*}
I_{2,N} &\!=\!& {\beta^2\over 16}\, \int_{-a}^a\,
\int_{-a}^a\, du\,du'\,uu'\, \chi_1(u)\chi_N(u)
\chi_1(u')\chi_N(u')\, \\ \\ & \times &
\int_{\R^2}\, ds\,ds'\, \ddot\gamma(s)\,
{e^{-\varrho_N|s-s'|}\over\varrho_N}\, \ddot\gamma(s')\,
+\,\OO(\beta^3)\,,
\end{eqnarray*}
where the inner integral can be by a repeated integration by parts
rewritten as
$$
2\,\int_{\R}\, \dot\gamma(s)^2 ds\,+\, \varrho_N\, \int_{\R^2}\,
ds\,ds'\, \dot\gamma(s)\,e^{-\varrho_N|s-s'|}\, \dot\gamma(s')\,.
$$
Putting all the contributions together and using the Parseval
identity, $\,\Vert u\chi_1\Vert^2= \sum_{N=1}^{\infty}
(\chi_N,u\chi_1)^2\,$, we find that the terms containing
$\,\Vert\dot\gamma\Vert^2\,$ cancel and we arrive at the relation
(\ref{angle asymptotics 2}).
The proof for the three--dimensional case is completely analogous.
The transverse eigenfunctions are now numbered conventionall starting
{}from $\,N=0\,$, and one has to replace $\,1+u\beta\gamma\,$ by
$\,h^{\beta}:= 1+r\gamma_{\beta}\cos(\theta-\alpha)\,$, \ie,
$\,u\dot\gamma,\,u\ddot\gamma\,$ by $\,h_s,\, h_{ss}\,$, respectively.
If the torsion is non--zero, of course, one cannot in general
separate the integrations completely, and the Parseval identity in
the last step has to be applied to the function $\,(s,r,\theta)
\mapsto \chi_0(r,\theta)h_s(s,r,\theta)\,$ of $\,L^2(\R\times B_a)\,$.
\quad \QED
\vspace{3mm}
\noindent
{\bf 4.3 Remark.} Mimicking the argument from the last part of the
proof of Theorem~4.2, we check easily that the terms on the right
sides of (\ref{angle asymptotics 2}), (\ref{angle asymptotics 3})
which express the contribution of higher transverse modes are
positive. They are small in thin tubes when the exponential kernel is
decaying fast. In fact, we have the estimate
\begin{eqnarray*}
\sum_{N=2}^{\infty} I_{2,N} &\!=\!& {\beta^2\over 8}\,
\sum_{N=2}^{\infty} \varrho_N^2\, (\chi_n,u\chi_1)^2\, \int_{\R}\,
|\hat{\dot\gamma}(k)|^2 {dk \over k^2+\varrho_N^2} \\ \\
&\!\leq\!& {\beta^2\over 8}\, \sum_{N=1}^{\infty} (\chi_n,u\chi_1)^2\,
\Vert\hat{\dot\gamma}\Vert^2 \,=\, {\beta^2\over 8}\,
\Vert\dot\gamma\Vert^2\, \Vert u\chi_1\Vert^2
\end{eqnarray*}
in two dimensions, and similarly
$$
\sum_{N=1}^{\infty} I_{2,N}\,\leq\, {\beta^2\over 8}\,
\Vert \chi_0 h_s\Vert^2
$$
for a three--dimensional tube, so these terms are of order
$\,\OO(a^2)\,$ as $\,a\to 0\,$.
\section{Perturbation theory: thin tubes}
Now we are going to return to the situation where the tube axis is
any curve which satisfies our assumptions about regularity and decay
of the curvature, and to study the behaviour of the bound state
eigenvalues in the limit $\,a\to 0\,$.
We shall treat the two-- and three--dimensional cases on the same
footing; the corresponding Schr\"odinger operator is denoted as
$\,-\partial_s b\partial_s- \Delta_{B_a}+V\,$ acting on
$\,L^2(\R,ds)\otimes L^2(B_a,du)\,$ where $\,B_a\,$ is the ball of
radius $\,a\,$ in the transverse variable $\,u\,$ in $\,\R\,$ or
$\,\R^2\,$, respectively, according to the dimension, and $\,du:=
r\,dr\,d\theta\,$ in the latter case. The functions $\,b\,$ and
$\,V\,$ are defined by (\ref{Hamiltonian2}) and (\ref{effective
potential2}) for the two--dimensional case and by
(\ref{Hamiltonian3}) and (\ref{effective potential3}) for dimension
three. We adopt the following conventions to unify the notations
between the two cases:
\begin{eqnarray} \label{common notation}
h(s,u) &\!:=\!& h(s,r,\theta) \,:=\, 1+r\,\gamma(s)\, \cos(\theta-
\alpha(s))\,, \nonumber \\ \\
b &\!:=\!& h^{-2},\quad V\,:=\, -\,\frac{\gamma^2}{4h^2}\,+\,
\frac{1}{2}\, \frac{h_{ss}}{h^3}\,-\, \frac{5}{4}\,
\frac{h_s^2}{h^4}\,, \nonumber
\end{eqnarray}
where $\,u=\,\left(r,\,\frac{\pi}{2}\mp\frac{\pi}{2}\right)\,$ and $\,\tau=
\alpha=0\,$ in the two--dimensional situation. It is convenient to
replace the Hamiltonian by the following unitarily equivalent operator
acting on $\,L^2(\R\times B_1)\,$:
\begin{equation} \label{scaled Hamiltonian}
H\,:=\, -a^{-2} \Delta_{B_1}+ T\,, \quad\; T\,:=\, T(a)\,:=\,
-\partial_s b(s,au)\partial_s +V(s,au)\,,
\end{equation}
which is obtained by the scaling $\,(s,u) \to(s,au)\;$; we shall also
denote by $\,b\,$ and $\,V\,$ the functions $\,(s,u)\mapsto
b(s,au)\,$ and $\,(s,u)\mapsto V(s,au)\,$, respectively.
As in the previous chapter, an important ingredient here is the
transverse--mode decomposition. If $\,\chi_j\,$ denotes the $\,j$--th
eigenfunction of $\,-\Delta_{B_1}\,$ with eigenvalue $\,E_j\,$, we
introduce the embedding $\,J_j:\, L^2(\R,ds) \to L^2(\R,ds)\otimes
\{\chi_j\} \subset\HH\,$ and its adjoint $\,J_j^*\;$; the latter can
be regarded as the projection onto the $\,j$--th mode $\,\chi_j\,$ of
$\,-\Delta_{B_1}\,$. This allows us to express $\,H\,$ as an infinite
matrix differential operator, $\,(H_{j,k})_{j,k=1}^{\infty}\,$ with
\begin{equation} \label{matrix elements}
H_{j,k}:= J_i^* HJ_k\,=\, a^{-2}E_j\delta_{j,k}+T_{j,k}\,,\quad\;
T_{j,k}\,:=\, -\partial_s b_{j,k}\partial_s +V_{j,k}\,,
\end{equation}
where $\,b_{j,k}:= \int_{B_1} b(\cdot,au)
\,\overline\chi_j(u)\chi_k(u)\, du\,$ and $\,V_{j,k}:= \int_{B_1}
V(\cdot,au)\, \overline\chi_j(u)\chi_k(u) \,du\;$; we set
$\,H_j:=H_{j,j}\,,\; T_j:=T_{j,j}\,$ \etc In the most part of the
following analysis, we shall consider the simplest comparison
operator
\begin{equation} \label{comparison}
H^0\,:=\, -a^{-2}\Delta_{B_1}+ T^0\otimes I\,, \quad\; T^0\,:=\,
-\partial_s^2+V^0\,:=\, -\partial_s^2\,-\,\frac{1}{4}\,\gamma^2\,.
\end{equation}
Suppose that $\,\gamma\,$ is such that $\,\sigma_{ess}(T^0)=\R_+\,$
and $\,\sigma_d(T^0)\neq\emptyset\,$, and let $\,\lambda\,$ be one of
the negative eigenvalues of $\,T^0\,$. Obviously, $\,E^0:= a^{-2}E_1+
\lambda\,$ is then a discrete eigenvalue of $\,H^0\;$; we want to
show that it gives rise to an eigenvalue of $\,H\,$ for a small
enough $\,a\,$ when the perturbation $\,H-H^0$ is turned on, and to
derive the corresponding perturbation expansion.
\subsection{The main theorem}
To state the main result of this chapter, we introduce the following
notation for the perturbation:
\begin{eqnarray} \label{perturbation}
W \,:=\, H-H^0 &\!=\!& -\partial_s(b(s,au)\!-\!1)\partial_s +V(s,au)+\,
\frac{1}{4}\,\gamma(s)^2\,, \nonumber \\ \\
W_{j,k} \,:=\, J_j^* WJ_k &\!=\!& -\partial_s(b_{j,k}-\delta_{j,k})
\partial_s+ V_{j,k}+\, \frac{1}{4}\,\gamma^2 \delta_{j,k}\,.
\nonumber
\end{eqnarray}
We shall also use the formal expansion of the function $\,W_{j,k}\,$
into power series in $\,a\,$,
\begin{equation} \label{formal expansion}
W_{j,k}\,=\, \sum_{l=1}^{\infty}\, W_{j,k}^{(l)}\, a^l
\end{equation}
which is actually convergent in the topology of $\,\BB(\HH^2(\R),
L^2(\R))\,$ for all $\,a\,$ small enough (see Lemma~5.2 below).
Notice that $\,W_{j,k}\,$ as well as $\,W_{j,k}^{(l)}\,$ are second
order differential operators in the variable $\,s\,$ exclusively, in
particular,
\begin{equation} \label{formal expension 2}
W_{j,k}^{(l)} \,=\, -\partial_s b_{j,k}^{(l)} \partial_s\,
+V_{j,k}^{(l)}
\end{equation}
with
$$
b_{j,k}^{(l)} \,:=\, (l\!+\!1)\, \gamma^l\, \int_{B_1}\, (-r\,
\cos(\theta\!-\!\alpha))^l\, \overline\chi_j(u) \chi_k(u)\,du
$$
and
\begin{eqnarray*}
\lefteqn{V_{j,k}^{(l)} \,:=\, \frac{1}{4}(l\!+\!1)\,
(-\gamma)^{l-2}\, \int_{B_1}\, (r\, \cos(\theta\!-\!\alpha))^{l-2} r^2
} \\ \\ &&
\times\, \biggl\lbrace\, \left\lbrack -\gamma^4-l\gamma
(\ddot\gamma-\gamma\tau^2) \right\rbrack\, \cos^2(\theta\!-\!\alpha)
\\ \\ &&
-\,\left\lbrack l\gamma(2\dot\gamma\tau+ \gamma\dot\tau)\,+\,
{5\over 3}\, l(l\!-\!1)\gamma\dot\gamma\tau \right\rbrack\,
\sin(\theta\!-\!\alpha) \cos(\theta\!-\!\alpha)
\\ \\ &&
-\, {5\over 6}\, l(l\!-\!1)\gamma^2\tau^2
\sin^2(\theta\!-\!\alpha) \biggr\rbrace\,
\overline\chi_j(u) \chi_k(u)\,du \;;
\end{eqnarray*}
recall that with our convention $\,\int_{B_1} f(s,r,\theta)\,du=
\int_{B_1} f(s,r,\theta)\,r\,dr\,d\theta\,$ for the
three--dimensional waveguide and
$$
\int_{B_1} f(s,r,\theta)\,du\,=\, \int_0^1 f\left(s,r,0\right)\,du\,
+\,\int_{-1}^0 f\left(s,r,\pi\right)\,du
$$
in dimension two. Now we are ready to state the main result of this
chapter.
\vspace{3mm}
\noindent
{\bf 5.1 Theorem.} {\em Assume (r1--4) and (d1), then
\begin{description}
\item{(a)} each discrete eigenvalue $\,\lambda\,$ of
$\,T^0\,$ gives rise for $\;a\;$ small enough to an eigenvalue
$\,E\,$ of $\,H\,$ of multiplicity one, which is given by an
absolutely convergent series,
$$
E\,=\, a^{-2}E_1\,+\,\lambda\,+\, \sum_{m=1}^{\infty} e_m\,,
$$
where the coefficients $\,e_m\,$ are of order $\,\OO(a^m)\,$ as
$\;a\;$ goes to zero; the expressions for them are given by
(\ref{coefficients}) below;
\item{(b)} conversely, for any discrete eigenvalue $\,E\,$ of
$\,H\,$ there is a unique discrete eigenvalue $\,\lambda\,$ of
$\,T^0\,$ such that
$$
E\,-\, a^{-2}E_1\,-\,\lambda\,=\, \OO(a)\;;
$$
\item{(c)} each coefficient $\,e_m\,$ is a $\,C^{\infty}\,$
function of the variable $\;a\;$ in the interval $\,[0,a_0)\,$ for
some positive $\,a_0\,$,
\item{(d)} the coefficient $\,e_1\,$ is analytic in $\;a\;$ around
$\,a=0\,$ and can be expressed as
$$
e_1\,=\, \sum_{l=1}^{\infty} e_{1,l}\,a^l\,, \quad\;
e_{1,l}\,:=\, (\varphi, W_1^{(l)}\varphi)\,,
$$
where $\,\varphi\,$ is the eigenvector of $\,T^0\,$ associated with
$\,\lambda\;$; recall that $\,W_1^{(l)}:= W_{1,1}^{(l)}\,$.
\end{description} }
\subsection{Proof of Theorem 5.1}
We begin with the analyticity properties of $\,W\,$ in the variable
$\,a\,$.
\vspace{3mm}
\noindent
{\bf 5.2 Lemma.} {\em The operators $\,W\,$ form a type $\,A\,$
analytic family of self--adjoint operators with respect to $\;a\;$
around $\;a=0\;$; its domain is $\,\HH^2(\R)\otimes L^2(B_1)\,$. }
\vspace{3mm}
\noindent
{\bf Proof.} Since $\,b\,$ and $\,V\,$ are analytic in $\,r\,$ around
the point $\,r=0\,$, the same is true for $\,a\mapsto b(s,au)\,$ and
$\,a\mapsto V(s,au)\,$ regarded as operator--valued functions of the
variable $\,a\;$; in addition, $\,b(s,au)\,$ is strictly positive for
all $\,s\,$ and $\,u\,$, and any $\,a\,$ small enough. Hence
$\,T\,$, and therefore also $\,W=T-T(0)\,$ is a type $\,A\,$ analytic
family \cite[Chap.VII]{Ka} with
the domain $\,\HH^2(\R)\otimes L^2(B_1)\,$ for small
$\,a\,$. \quad \QED
\vspace{3mm}
In view of the assumptions we know that $\,\sigma_{ess}(T^0)=\R_+\,$,
the operator $\,T^0\,$ has at least one discrete eigenvalue and all
its discrete eigenvalues are simple (see, \eg, \cite{BGS}). Taking
further into account that the spectrum of $\,T^0\,$ is bounded from
below by $\,-\,\frac{1}{4}\,\Vert\gamma\Vert^2_{\infty}\,$, we see
that the discrete spectrum of $\,H^0\,$ is non--empty and simple for
$\,a\,$ small enough.
Following the standard perturbation--theory procedure, we shall
consider a contour $\,\Gamma\,$ enclosing a single eigenvalue $\,E^0=
a^{-2}E_1+\lambda\,$ and show that the perturbation $\,W\,$ is
uniformly small on $\,\Gamma\,$. To be more specific, we choose
$\,\Gamma\,$ so that $\,\Gamma-a^{-2}E_1\,$ is a circle centered at
$\,\lambda\,$ in the resolvent set of $\,T^0\,$. Let $\,a_1\,$ be the
radius of the circle in which $\,W\,$ is analytic; furthermore, we
denote by $\,R^0,\, R\,$ the resolvents of $\,H^0,\,H\,$,
respectively, and by $\,\hat R^0\,$ the reduced resolvent with
respect to $\,E^0\,$.
\vspace{3mm}
\noindent
{\bf 5.3 Lemma.} {\em There are constants $\,c^{(1)},\,
c^{(2)}_{\Gamma}\,$ such that for all $\,a3\,$, a
curved Dirichlet layer in $\,\R^n\,$ and similar structures. No
rigorous result of that type is known to us though some formal
calculations have been made \cite{dC1,dC2,To}. The first named case
seems to be of a rather mathematical interest while the other
provides, in addition, for $\,n=3\,$ a rough model of a thin
semiconductor film as discussed in the introduction. The problem is,
however, more complicated now; the mentioned formal results suggest
that the leading term in the effective potential analogous to
(\ref{effective potential2}) and (\ref{effective potential3}) can be
zero in a curved part of the layer if the generating surface is
locally spherical.
As mentioned in Section~2.3, a thin curved strip supports typically
one bound state if the curvature has one well distinguished peak. If
there are more clearly separated bends one conjectures that a bound
state is associated with each of them and the wavefunctions are
localized in the vicinity of the bends. One may attempt to estimate
the shift of the eigenvalues comparing to corresponding one--bend
tubes in analogy with the tunneling analysis of multiwell
Schr\"odinger operators \cite{BCD}, \cite{CDS,HeSj,Hel}.
An entirely new class of effects opens when we replace the
assumptions about the decay of curvature by a periodicity hypothesis.
In this case one expects that a thin tube would have a band--type
spectrum but it remains to be proven. Further extensions include
quasiperiodically curved tubes, tubes with a random curvature \etc
However, the continuous spectrum is important also for the tubes
considered here --- as a part of the scattering analysis.
The effective attractive potential responsible for existence of the
bound states generates at the same time resonances below the
thresholds of the higher transversal modes as can be shown for a
two--dimensional curved strip by combination of complex scaling and
perturbation techniques \cite{DES1,DES2}. In these papers also the
Fermi--rule contribution to the resonance width was computed and
shown to be exponentially small in the limit $\,d\to 0\,$. One can
conjecture that the same is true for the full resonance width and
that this result extends to curved three-dimensional tubes.
\subsection*{Acknowledgments}
Most of the results reviewed here was obtained during the visits of
P.D. to the Laboratory of Theoretical Physics, JINR Dubna and Nuclear
Physics Institute AS CR, \v Re\v z, and P.E. to the Universit\'e de
Toulon et du Var and C.P.T. C.N.R.S., Marseille; the authors express
their gratitude to the hosts. The work was partially supported by the
AS CR Grant No.148409.
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\end{document}