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\begin{document}
\title{\Large\bf On the number of particles which a curved quantum
waveguide can bind}
\author{Pavel Exner$^{a,b}$ and Simeon A. Vugalter$^a$}
\date{}
\maketitle
\begin{center}
a) Nuclear Physics Institute, Academy of Sciences,
25068 \v{R}e\v{z} near Prague, \\
b) Doppler Institute, Czech Technical University, B\v rehov\'a 7,
11519 Prague, \\
{\em exner@ujf.cas.cz, vugalter@ujf.cas.cz}
\end{center}
\begin{abstract}
We discuss the discrete spectrum of $\,N\,$ particles in a curved
planar waveguide. If they are neutral fermions, the maximum number of
particles which the waveguide can bind is given by a one--particle
Birman--Schwinger bound in combination with the Pauli principle. On
the other hand, if they are charged, \eg, electrons in a bent quantum
wire, the Coulomb repulsion plays a crucial role. We prove a
sufficient condition under which the discrete spectrum of such a
system is empty.
\end{abstract}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Introduction}
A rapid progress of mesoscopic physics brought, in particular,
interesting new problems concerning relations between geometry and
spectral properties of quantum Hamiltonians. They involve models of
quantum wires, dots, and similar systems. While in reality these are
rather complicated systems composed of different semiconductor
materials, experience tells us that their basic features can be
explained using simple models in which electrons (regarded as free
particles with an effective mass) are supposed to be confined to an
appropriate spatial region, either by a potential or by a hard wall.
A brief description of this approximation with a guide to further
reading is given in Ref.~\cite{DE}.
In addition, such models apply not only to electrons in semiconductor
microstructures; a different example is represented by atoms trapped
in hollow optical fibers \cite{SMZ}.
It is natural that most theoretical results up to date refer to the
case of a single particle in the confinement. On the other hand, from
the practical point of view it is rather an exception than a rule
that an experimentalist is able to isolate a single electron or atom,
and therefore many--body problems in this setting are of interest.
For instance, two--dimensional quantum dots which can be regarded as
artificial atoms have been studied recently, usually in presence of a
magnetic field, either for a pair of electrons or in the semiclasical
situation when a Thomas--Fermi--type approach is applicable --- \cf
[3-6]
%\cite{Ha,ElS,PS,Y}
and references therein.
In these studies, however, geometry of the dot played a little role,
because the confinement was realized by a harmonic potential or a
circular hard wall. This is not the case for open systems modelling
quantum wires where a deformation of a straight channel is needed to
produce nontrivial spectral properties. In particular, a quantum
waveguide exhibit bound states if it is bent \cite{DE,ES,GJ},
protruded
%\cite{AS,BGRS,EV1}
[9-11]
or allowing a leak to another duct
%\cite{ESTV,EV2,EV3},
[12-14],
and the
discrete spectrum depends substantially on the shape of the channel.
With few exceptions such as Ref.~\cite{NTV}, however, the known
results refer to the one--particle case.
It is the aim of the present paper to initiate a rigorous
investigation of many--particle effects in quantum waveguides.
We are going to discuss here a system of $\,N\,$ particles in
a bent planar Dirichlet tube, \ie, a hard--wall channel, and ask
whether $\,N$--particle bound states exist for a given geometry.
After collecting the necessary preliminaries in the next section,
we shall derive first in Section~3 a simple bound for the neutral
case which follows from the Birman--Schwinger estimate of the
one--particle Hamiltonian in combination with the Pauli principle.
The main result of the paper is formulated and proved in Section~4.
It concerns the physically interesting case of charged particles; the
example we have in mind is, of course, electrons in a bent
semiconductor quantum wire. The electrostatic repulsion makes
spectral analysis of the corresponding Hamiltonian considerably more
complicated. Using variational technique borrowed from atomic
physics, we derive here a sufficient condition under which the
discrete spectrum is empty. The condition is satisfied for $\,N\,$
large enough and represents an implicit equation for the maximum
number of charged particles which a waveguide of a given curvature
and width can bind. Some other aspects of the result and open
questions are discussed briefly in the concluding section.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Preliminaries}
The waveguide in question will be modelled by a curved planar strip
$\,\Sigma\,$ in $\,\R^2\,$, of a constant width $\,d=2a\,$. It can be
obtained by transporting the perpendicular interval $\,[-a,a]\,$
along the curve $\,\Gamma\,$ which is the axis of $\,\Sigma\,$. Up
to Euclidean transformations, the strip is uniquely characterized by
its halfwidth $\,a\,$ and the (signed) curvature
$\,s\mapsto\gamma(s)\,$ of $\,\Gamma\,$, where $\,s\,$ denotes the
arc length. We adopt the regularity assumptions of
Refs.~\cite{DE,ES}:
\begin{description}
\item{\em (i)} $\;\Omega\,$ is not self--intersecting,
\item{\em (ii)} $\;a\|\gamma\|_{\infty}<1\,$,
\item{\em (iii)} $\;\gamma\,$ is piecewise $\,C^2\,$ with
$\,\gamma',\, \gamma''\,$ bounded,
\end{description}
and restrict our attention to the case when the tube is curved in a
bounded region only:
\begin{description}
\item{\em (iv)} there is $\,b>0\,$ such that $\,\gamma(s)=0\,$ for
$\,|s|>b\,$; without loss of generality we may assume that $\,2b>a\,$.
\end{description}
As usual we put $\,\hbar=2m=1\,$; then the one--particle Hamiltonian
of such a waveguide is the Dirichlet Laplacian
$\,-\Delta_D^{\Sigma}\,$ defined in the conventional way --- \cf
\cite{RS}, Sec.~XIII.15. Using the natural locally orthogonal
curvilinear coordinates $\,s,u\,$ in $\,\Sigma\,$ one can map
$\,-\Delta_D^{\Sigma}\,$ unitarily onto the operator
%
\begin{equation} \label{Hamiltonian 1}
H_1\,=\,-\partial_s\, (1+u\gamma)^{-2}\, \partial_s\,-\,
\partial_u^2 \,+\,V(s,u)
\end{equation}
%
on $\,L^2(\R\times(-a,a)\,)\,$ with the effective curvature--induced
potential
%
\begin{equation} \label{effective potential}
V(s,u)\,:=\,-\,{\gamma(s)^2\over 4(1+u\gamma(s))^2}\,+\,
{u\gamma''(s) \over 2(1+u\gamma(s))^3}\,-\,
{5 \over 4}\,{u\gamma'(s)^2 \over(1+u\gamma(s))^4}
\end{equation}
%
which is e.s.a. on the core $\,D(H)\,=\, \{\,\psi\,:\,\psi\in
C^{\infty},\; \psi(s,\pm a)=0,\; H\psi\in L^2\,\}\,$ --- \cf
Refs.~\cite{DE,ES} for more details.
If the waveguide contains $\,N\,$ particles, the state Hilbert space
is $\,L^2(\Sigma))^N$; the Pauli principle will be taken into account
later. We assume that each particle has the charge $\,e\,$; using the
same ``straightening" transformation we are then able to rewrite the
Hamiltonian as
%
\begin{eqnarray} \label{Hamiltonian N}
H_N \,\equiv\, H_N(\gamma,a,e) &\!=\!& \sum_{j=1}^N \left\lbrace\,
-\partial_{s_j}\, (1+u_j\gamma(s_j))^{-2}\, \partial_{s_j}\,-\,
\partial_{u_j}^2 \,+\,V(s_j,u_j) \,\right\rbrace
\nonumber \\ \nonumber \\
&\! +\!& e^2 \sum_{1\le j2b>a\,$ is a parameter to be specified later. By abuse
of notation, we use the symbols $\,v,\,g\,$ again both for these
functions and the corresponding operators of multiplication. It is
straightforward to evaluate $\,\left([H_N,v]\psi,v\psi\right)\,$ and
the analogous expression with $\,v\,$ replaced by $\,g\,$ for a
vector $\,\psi\in D(H_N)\,$; in both cases it is only the
longitudinal kinetic part in (\ref{Hamiltonian N}) which contributes.
This yields the identity
%
\begin{eqnarray*}
(H_N\psi,\psi) &\!=\!& (H_Nv\psi,v\psi)\,+\, (H_Ng\psi,g\psi) \\
&\!+\!& \sum_{j=1}^N \left\lbrace\, \left\|(1\!+\!u_j\gamma_j)^{-1}
v_j\psi \right\|^2+\, \left\|(1\!+\!u_j\gamma_j)^{-1} g_j\psi
\right\|^2 \,\right\rbrace\,,
\end{eqnarray*}
%
where we have used the shorthands $\,v_j:= {\partial v\over\partial
s_j}\,$, $\,g_j:= {\partial g\over\partial s_j}\,$, and $\,\gamma_j:=
\gamma(s_j)\,$. Notice further that the factors
$\,(1\!+\!u_j\gamma_j)^{-1}\,$ may be neglected, because $\,v_j\,
g_j\,$ are nonzero only if $\,s_j\ge \beta>2b\,$ in which case
$\,\gamma_j=0\,$. Furthermore, with the exception of the hyperplanes
where two or more coordinates coincide (which is a zero measure set)
the norm $\,\|s\|_{\infty}\,$ coincides with just one of the
coordinates $\,s_1,\dots,s_n\,$, and therefore
%
\begin{equation} \label{vg error}
\sum_{j=1}^N \left\lbrace\, \left\| v_j\psi \right\|^2+\, \left\|
g_j\psi \right\|^2 \,\right\rbrace\,\le\, \|\psi\|^2 \max_{1\le j\le
N} \left\lbrace\, \left\| v_j \right\|^2_{\infty}+\, \left\| g_j
\right\|^2_{\infty} \,\right\rbrace \,\le\, \beta^{-2}C_0
\|\psi\|^2\,,
\end{equation}
%
where $\,C_0:=\|v'\|_{\infty}^2\!+ \|g'\|_{\infty}^2\,$. We arrive at
the estimate
%
\begin{equation} \label{decomposition 1}
(H_N\psi,\psi)\,\ge\, L_1[v\psi]+ L_1[g\psi]
\end{equation}
%
with
%
\begin{equation} \label{L_1}
L_1[\phi]\,:=\, (H_N\phi,\phi)\,-\, {C_0\over\beta^2}\,
\|\phi\|^2_{\NN_\beta}\,,
\end{equation}
%
where the last index symbolizes the norm of the vector $\,\phi\,$
restricted to the subset $\,\NN_\beta:= \left\lbrace\,s:\;
\beta\le\|s\|_{\infty}\le\, {3\beta \over 2}\,\right\rbrace\,$ of the
configuration space.
Next one has to estimate separately the contributions from the inner
and outer parts. Let us begin with the exterior. We introduce the
following functions:
%
\begin{eqnarray*}
f_1(s) &\!=\!& v\left(2s_1\|s\|^{-1}_{\infty}\right)\,, \\
f_j(s) &\!=\!& v\left(2s_j\|s\|^{-1}_{\infty}\right)
\prod_{n=1}^{j-1} g\left(2s_n\|s\|^{-1}_{\infty}\right)\,, \qquad
j=2,\dots,N\!-\!1\, \\
f_N(s) &\!=\!& \prod_{n=1}^{N-1}
g\left(2s_n\|s\|^{-1}_{\infty}\right)\,.
\end{eqnarray*}
%
It is clear from the construction that
%
\begin{equation} \label{decomposition 2}
\sum_{j=1}^N f_j(s)^2= \,1\,.
\end{equation}
%
Moreover, the functions
%
$$
s_j\,\mapsto\, v(2s_j\|s\|_{\infty}),\, g(2s_j\|s\|_{\infty})
$$
%
have a non--zero derivative only if $\,|s_j|\ge\, {1\over
2}\|s\|_{\infty}^{-1}$. Hence on the support of $\,s\mapsto
v(\|s\|_{\infty}\beta^{-1})$ the derivative is non--zero if
$\,|s_j|\ge \,{1\over 2}\beta>b\,$. In other words, the function
$\,s\mapsto f_j(s)^2 v(\|s\|_{\infty}\beta^{-1})$ has zero derivative
in all the parts of the configuration space where at least one of the
electrons dwells in the curved part of the waveguide. Commuting the
(longitudinal kinetic part of) $\,H_N\,$ with $\,f_j\,$, we get in
the same way as above the identity
%
\begin{equation} \label{decomposition 3}
L_1[v\psi]\,=\, \sum_{j=1}^N \left\lbrace\, L_1[f_jv\psi] -
\|(\nabla_sf_j)v\psi\|^2 \,\right\rbrace\,,
\end{equation}
%
where $\,\nabla_s:= \left(\partial_{s_1},\dots, \partial_{s_1}
\right)\,$. Next we need a pointwise upper bound on $\,\sum_{j=1}^N
(\nabla_sf_j)^2$: denoting $\,\sigma_j:= 2s_j\|s\|_{\infty}$, we can
write
%
\begin{eqnarray*}
\sum_{j=1}^N |(\nabla_sf_j)(s)|^2 &\!=\!& {4\over\|s\|_{\infty}^2}\,
\bigg\lbrace\, v'(\sigma_1)^2 \\ \\
&& +g'(\sigma_1)^2 v(\sigma_2)^2 +g(\sigma_1)^2 v'(\sigma_2)^2
+ \cdots \\
&& +g'(\sigma_1)^2 g(\sigma_2)^2\!\dots g(\sigma_N)^2 +\cdots+
g(\sigma_1)^2 \!\dots g(\sigma_{N-1})^2 g'(\sigma_N)^2\,
\bigg\rbrace\,,
\end{eqnarray*}
%
which gives after a partial resummation
%
\begin{eqnarray*}
&\!=\!& {4\over\|s\|_{\infty}^2}\, \bigg\lbrace\, v'(\sigma_1)^2
+ g'(\sigma_1)^2 +g(\sigma_1)^2 g'(\sigma_2)^2 + \cdots \\
&& +g(\sigma_1)^2 \!\dots g(\sigma_{N-1})^2 g'(\sigma_N)^2\,
\bigg\rbrace \\ \\
&\!\le\!& {4\over\|s\|_{\infty}^2}\, \left\lbrace\, v'(\sigma_1)^2
+\sum_{j=1}^N g'(\sigma_j)^2 \,\right\rbrace\,\le\,
{4NC_0\over\|s\|_{\infty}^2} \;;
\end{eqnarray*}
%
recall that $\,C_0:=\|v'\|_{\infty}^2\!+ \|g'\|_{\infty}^2\,$.
Consequently,
%
\begin{eqnarray} \label{L_1 decomposition}
L_1[v\psi] &\!\ge \!& \sum_{j=1}^N L_1[f_j v\psi] \,-\, 4NC_0 \left\|
v\psi \|s\|_{\infty}^{-1} \right\|^2 \nonumber \\ \nonumber \\
&\! =\!& \sum_{j=1}^N\, \left\lbrace\, L_1[f_j v\psi] -4NC_0 \left\|
f_j v\psi \|s\|_{\infty}^{-1} \right\|^2\, \right\rbrace \nonumber \\
\nonumber \\
&\! =\!& \sum_{j=1}^N\, L_2[f_j v\psi]\,,
\end{eqnarray}
%
where
%
\begin{equation} \label{L_2}
L_2[\phi]\,:=\, L_1[\phi] \,-\,4NC_0 \left\| \phi \|s\|_{\infty}^{-1}
\right\|^2 \,.
\end{equation}
%
Hence we have to find a lower bound to $\,L_2(\psi_j)\,$ with
$\,\psi_j:= f_jv\psi)\,$. Since $\,s_j\ge\, {1\over 2}\|s\|_{\infty}
\ge\, {1\over 2}\beta >b\,$ holds on the support of $\,\psi_j\,$, we have
$\,V(s_j,u_j)=0\,$ there. This allows us to write
%
$$
(H_N\psi_j,\psi_j)\,=\, (H_{N-1}\psi_j,\psi_j)+\, \left\|
\partial_{s_j} \psi_j \right\|^2+\, \left\| \partial_{u_j} \psi_j
\right\|^2+ e^2\, \sum_{j\ne l=1}^N \left( |\vec r_j\!- \vec
r_l|^{-1} \psi_j, \psi_j \right)\,,
$$
%
where $\,H_{N-1}$ refers to the system with the $\,j$--th electron
excluded, and therefore
%
$$
(H_N\psi_j,\psi_j)\,\ge\, \left(\, \mu_{N-1}+\, \left( \pi\over 2a
\right)^2 \right) \|\psi_j\|^2 +\, e^2\, \sum_{j\ne l=1}^N \left(
|\vec r_j\!- \vec r_l|^{-1} \psi_j, \psi_j \right)\,.
$$
%
Since $\,|\vec r_j\!- \vec r_l|\le \sqrt{ (s_j\!-\!s_l)^2\! +4a^2}\,
\le\, 2\, \sqrt{\|s\|_{\infty}^2\!+a^2}\,$, we have
%
$$
(H_N\psi_j,\psi_j)\,\ge\, \left(\, \mu_{N-1}+\, \left( \pi\over 2a
\right)^2 \right) \|\psi_j\|^2 +\, {e^2(N\!-\!1)\over 2}\, \left(
(\|s\|^2\!+a^2)^{-1/2} \psi_j,\psi_j \right)\,.
$$
%
The sought lower bound then follows from (\ref{L_2}) and (\ref{L_1}):
%
\begin{eqnarray*}
L_2[\psi_j] &\! \ge \!& \left(\, \mu_{N-1}+\, \left( \pi\over 2a
\right)^2 \right) \|\psi_j\|^2 -\,4NC_0 \left\| \psi_j
\|s\|_{\infty}^{-1} \right\|^2\\ \\
&\! -\!& C_0\beta^{-2} \|\psi_j\|^2_{\NN_\beta} +\,
{e^2(N\!-\!1)\over 2}\, \left( (\|s\|^2\!+a^2)^{-1/2} \psi_j,\psi_j
\right)\;;
\end{eqnarray*}
%
recall that $\,\NN_\beta:= \left\lbrace\,s:\;
\beta\le\|s\|_{\infty}\le\, {3\beta \over 2}\,\right\rbrace\,$. The
second and the third term at the \rhs can be combined using
%
$$
4NC_0 \left\| \psi_j \|s\|_{\infty}^{-1}\right\|^2 +\,C_0 \beta^{-2}
\|\psi_j\|^2_{\NN_\beta} \,\le\, (4N\!+\!1)C_0 \left\| \psi_j
\|s\|_{\infty}^{-1}\right\|^2\,.
$$
%
Furthermore, $\,\|s\|_{\infty}\ge \beta >2b>a\,$ yields
$\,(\|s\|^2\!+a^2)^{1/2} \le \sqrt{2}\, \|s\|_{\infty}$ and
%
\begin{eqnarray} \label{L_2 bound}
L_2[\psi_j] &\!\ge\!& \left(\, \mu_{N-1}+\, \left( \pi\over 2a
\right)^2 \right) \|\psi_j\|^2 \nonumber \\ \nonumber \\
&\!+\!& \left(\, {e^2(N\!-\!1)\over 2\sqrt{2}} \,-\,
{C_0(4N\!+\!1)\over \beta}\,\right)\, \left\| \psi_j
\|s\|_{\infty}^{-1}\right\|^2 \,.
\end{eqnarray}
%
We are interested in the situation when the second term at the \rhs
is positive. This is achieved if
%
$$
{e^2(N\!-\!1)\over 2\sqrt{2}} \,>\, {C_0(4N\!+\!1)\over \beta}
$$
%
which is ensured if we choose $\,\beta\,$ in such a way that
%
\begin{equation} \label{beta bound}
\beta\,>\, {18\sqrt{2} C_0\over e^2}\;;
\end{equation}
%
recall that $\,N\ge 2\,$. Owing to the identity (\ref{L_1
decomposition}) we then have
%
\begin{equation} \label{outer bound}
L_1[v\psi]\,\ge\, \left(\, \mu_{N-1}+\, \left( \pi\over 2a
\right)^2 \right) \|v\psi\|^2\,,
\end{equation}
%
which means in view of (\ref{ess spectrum}) that the external part of
$\,\psi\,$ does not contribute to the discrete spectrum.
Let us turn now to the inner part. The corresponding quadratic form
in the decomposition (\ref{decomposition 1}) can be estimated with
the help of (\ref{Hamiltonian N}) and (\ref{L_1}) by
%
\begin{eqnarray} \label{inner bound}
L_1[g\psi] &\! \ge \!& \delta_+^{-2} \|\nabla_s g\psi\|^2+\,
\|\nabla_u g\psi\|^2 +\, \sum_{j=1}^N \left( V(s_j,u_j)g\psi, g\psi
\right) \nonumber \\ \nonumber \\
&\! +\!& e^2 \sum_{1\le2b>a\,$, so we arrive at the estimate
%
$$
|\vec r_j\!- \vec r_k| \le\, \sqrt{7} \beta\,,
$$
%
which yields
%
$$
\sum_{1\le0\,$ and the remaining terms in
(\ref{absence}) are independent od $\,e\,$ we see that $\,\sigma_{\rm
disc}(H_N)= \emptyset\,$ for any $\,N\ge 2\,$ provided $\,e\,$ is
large enough. Thus our result confirms the natural expectation that
for a given curved tube and sufficiently charged particles just
one--particle bound states can survive.
We have not addressed in this paper the question about the minimum
number of particles which a curved quantum waveguide can bind. The
gap between the trivial result which follows from the one--particle
theory \cite{DE,ES,GJ} and the condition (\ref{absence}) leaves a lot
of space for improvements. Moreover, it is a natural question whether
strongly curved tubes which can bind many particles allow for some
semiclassical description analogous to the case of the quantum dots
\cite{Y}. This is a task for a future work.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section*{Acknowledgments}
The research has been partially supported by the Grants
No.~202--0218, GACR, and ME099, Ministry of Education of the Czech
Republic.
\vspace{5mm}
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