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quantum wires, Dirichlet Laplacian, eigenvalues, Lieb-Thirring estimates
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\begin{document}


\title{Lieb-Thirring Inequalities on Trapped Modes in Quantum Wires}


\author{Pavel Exner and Timo Weidl}

\begin{abstract}
We prove Lieb-Thirring type bounds on the discrete spectrum of Schr{\"o}dinger
operators on tube-like domains and give explicit estimates on the constants
involved.
\end{abstract}
\maketitle

\section{Introduction}

Let \( \omega  \) be a connected, bounded, and open set in \( \mathbb {R}^{d-1} \).
Consider the tube \( \Omega =\omega \times \mathbb {R}\subset \mathbb {R}^{d} \)
generated by the cross section \( \omega  \). Let \( -\Delta ^{\omega } \)
and \( -\Delta ^{\Omega } \) be the Dirichlet Laplacians on \( L^{2}(\omega ) \)
and \( L^{2}(\Omega ) \), respectively.

The spectrum of the operator \( -\Delta ^{\omega } \) is discrete. Let \( \{\mu _{k}\}_{k=1}^{\infty } \)
be the ordered sequence of its eigenvalues (counting multiplicities) and let
\( \{\psi _{k}\}_{k=1}^{\infty } \)be a corresponding ortho-normed eigenbasis
in \( L^{2}(\omega ) \). These give rise to generalized eigenfunctions
\[
\psi _{k}(x^{\prime })e^{i\xi x_{d}},\quad x=(x^{\prime },x_{d}),\, x^{\prime }=(x_{1},\dots ,x_{d-1}),\]
for the operator \( -\Delta ^{\Omega } \) on \( L^{2}(\Omega ) \). Hence,
by separation of variables, the spectrum of \( -\Delta ^{\Omega } \) is purely
absolutely continuous and consists of the eigenvalue branches \( [\mu _{k},\infty ) \).
In particular, the point \( \mu _{1}>0 \) is the lower bound of the spectrum
of \( -\Delta ^{\Omega } \). 

If we perturb the operator \( -\Delta ^{\Omega } \) by some real-valued attractive
potential \( -\alpha V\leq 0 \), the resulting Schr{\"o}dinger operator
\begin{equation}
\label{H}
H_{\alpha }^{\Omega }=-\Delta ^{\Omega }-\alpha V,\quad \alpha >0,
\end{equation}
with Dirichlet boundary conditions on \( L^{2}(\Omega ) \) will have spectrum
below the threshold \( \mu _{1} \). Under certain regularity and decay assumptions
on \( V \) this portion of the spectrum is discrete. Let \( \{\lambda _{n}(\alpha )\} \)
stand for the respective non-decreasing sequence of the eigenvalues of \( H_{\alpha }^{\Omega } \)
below \( \mu _{1} \) (counting multiplicities). In this paper we study Lieb-Thirring
type bounds on the spectral moments\footnote{
We use the notation \( x_{\pm }=(|x|\pm x)/2 \).
}
\[
S_{\gamma }^{\Omega }(\alpha )=tr(H_{\alpha }^{\Omega }-\mu _{1})_{-}^{\gamma }=\sum _{n}(\mu _{1}-\lambda _{n}(\alpha ))^{\gamma },\quad \gamma \geq 0,\]
for tube-like domains \( \Omega =\omega \times \mathbb {R} \). Such estimates
are of interest, since the operators \( H_{\alpha }^{\Omega } \) can model
various effects in quantum wires. 

Compared with the usual Schr{\"o}dinger operator on \( L^{2}(\mathbb {R}^{d}) \),
the spectral behaviour of \( H^{\Omega }_{\alpha } \) reflects the mixed dimensionality
of the tube \( \Omega  \). One the one hand, as for any open domain, for sufficiently
regular \( V \) the quantity \( S^{\Omega }_{\gamma }(\alpha ) \) shows the
\( d \)-dimensional large coupling Weyl asymptotics
\begin{equation}
\label{las1}
S_{\gamma }^{\Omega }(\alpha )=(1+o(1))S^{cl}_{\gamma ,d}(\alpha )\quad \mbox {as}\quad \alpha \to \infty ,
\end{equation}
where
\begin{equation}
\label{scl}
S_{\gamma ,d}^{cl}(\alpha )=\int \int (|\xi |^{2}-\alpha V(x))_{-}^{\gamma }\frac{dxd\xi }{(2\pi )^{d}}=\alpha ^{\gamma +\frac{d}{2}}L_{\gamma ,d}^{cl}\int V_{+}^{\gamma +\frac{d}{2}}dx
\end{equation}
and
\[
L_{\gamma ,d}^{cl}=\frac{\Gamma (\gamma +1)}{2^{d}\pi ^{d/2}\Gamma (\gamma +\frac{d}{2}+1)}.\]
 

On the other hand, in the weak coupling limit \( \alpha \to 0 \) the ground
state of \( H_{\alpha }^{\Omega } \) will extend longitudinally in the tube
and its respective asymptotics is similar to that of a one-dimensional Schr{\"o}dinger
operator. Namely, if \( V\geq 0 \) is continuous and compactly supported, then
for sufficiently small \( \alpha >0 \) the operator \( H_{\alpha } \) has
exactly one bound state \( \lambda _{1}(\alpha ) \) below \( \mu _{1} \) and
\begin{equation}
\label{las}
S_{1/2}^{\Omega }(\alpha )=\frac{\alpha }{2}\int _{\omega }\int _{-\infty }^{+\infty }V(x^{\prime },x_{d})|\psi _{1}(x^{\prime })|^{2}dx^{\prime }dx_{d}+O(\alpha ^{2})
\end{equation}
as \( \alpha \to 0+ \) \cite{Si,DE}. Similar weak coupling asymptotics are
valid also if the perturbation is of a geometric rather than potential type
\cite{BGRS,DE,EV}.

An appropriate estimate should reflect this behaviour. Accordingly we prove
a two-term Lieb-Thirring formula. The first term has the homogeneity of order
\( \alpha ^{\gamma +\frac{1}{2}} \) and stands for the one-dimensional effects.
The second one is a multiple of the phase space average \( S_{\gamma ,d}^{cl}(\alpha ) \)
and controls the large coupling behaviour. The sum of both gives a uniform bound
on \( S_{\gamma }^{\Omega }(\alpha ) \) for all \( \alpha \geq 0 \).

Because of the one-dimensional effects, these spectral estimates hold for \( \gamma \geq 1/2 \)
only. This agrees with the validity range of the usual Lieb-Thirring estimates
in dimension one (see section 2). In particular, if \( \gamma =1/2 \) then
the one-dimensional term captures the asymptotics (\ref{las}) and the constant
in front of it is almost sharp. Based on \cite{LW,HLW} we provide also reasonable
explicit estimates on the constants for the semiclassical term. 


\section{Lieb-Thirring bounds on \protect\( L^{2}(\mathbb {R}^{d})\protect \)}

In this section we collect auxiliary material on Lieb-Thirring estimates on
\( L^{2}(\mathbb {R}^{d}) \), which shall be of use below. Usually these inequalities
concern the operator
\[
H_{\alpha }=-\Delta -\alpha V\quad \mbox {on}\quad L^{2}(\mathbb {R}^{d}).\]
Assume that \( V\geq 0 \), \( \alpha \geq 0, \) and put
\[
S_{\gamma }(\alpha )=tr\, (H_{\alpha })_{-}^{\gamma }.\]
Then for \( \gamma \geq 1/2 \) if \( d=1 \), for \( \gamma >0 \) if \( d=2 \),
and for \( \gamma \geq 0 \) if \( d\geq 3 \), the inequality
\begin{equation}
\label{LTh}
S_{\gamma }(\alpha )\leq R(\gamma ,d)S_{\gamma }^{cl}(\alpha )
\end{equation}
holds true \cite{LT,C,L,R}. For all other pairs of \( \gamma  \) and \( d \)
the estimate (\ref{LTh}) fails.

The bound (\ref{LTh}) agrees with the asymptotics (\ref{las1}), which on its
turn implies \( R(\gamma ,d)\geq 1 \). We mention the following known estimates
for the constants \( R(\gamma ,d) \) in (\ref{scl})
\begin{eqnarray}
R(\gamma ,d)=1 & \quad \mbox {if}\quad  & \gamma \geq 3/2,\, d\in \mathbb {N},\label{R1} \\
R(\gamma ,d)\leq 2 & \quad \mbox {if}\quad  & 1\leq \gamma <3/2,\, d\in \mathbb {N},\label{R2} \\
R(\gamma ,d)\leq 2 & \quad \mbox {if}\quad  & 1/2\leq \gamma <1,\, d=1,\label{R3} \\
R(\gamma ,d)\leq 4 & \quad \mbox {if}\quad  & 1/2\leq \gamma <1,\, d\in \mathbb {N},\, d\geq 2,\label{R4} 
\end{eqnarray}
see \cite{LW,HLW,HLT}.

We point out that for \( \gamma =1/2 \), \( d=1 \) and non-negative integrable
potentials \( V \) actually a two-sided Lieb-Thirring bound \cite{W,HLT}
\begin{equation}
\label{d12}
S_{1/2}^{cl}(\alpha )\leq S_{1/2}(\alpha )\leq 2S_{1/2}^{cl}(\alpha ),\quad \alpha \geq 0,
\end{equation}
holds. Both constants are sharp and
\begin{eqnarray}
S_{1/2}(\alpha )=S_{1/2}^{cl}(\alpha )(1+o(\alpha )) & \quad \mbox {as}\quad  & \alpha \to \infty ,\label{sup} \\
S_{1/2}(\alpha )=2S_{1/2}^{cl}(\alpha )(1+o(\alpha )) & \quad \mbox {as}\quad  & \alpha \to 0.\label{slow} 
\end{eqnarray}


Along with (\ref{LTh}) we shall need the following generalization to Schr{\"o}dinger
operators with operator-valued potentials. Namely, let \( G \) be a separable
Hilbert space and let \( W \) be a function on \( \mathbb {R}^{d} \) which
takes a.e. non-negative compact operators on \( G \) as its values. We consider
eigenvalue moments of the Schr{\"o}dinger type operator
\[
Z=1_{G}\otimes (-\Delta )-W(x)\quad \mbox {on}\quad G\otimes L^{2}(\mathbb {R}^{d}).\]
Assume that \( tr_{G}W^{\gamma +\frac{d}{2}}(\cdot )\in L^{\gamma +\frac{d}{2}}(\mathbb {R}^{d}) \).
Then for \( \gamma \geq 1/2 \) if \( d=1 \), and for \( \gamma >0 \) if \( d\geq 2 \),
the following estimate holds true:
\begin{equation}
\label{es12}
tr_{G\times L^{2}(\mathbb {R}^{d})}\, Z_{-}^{\gamma }\leq r(\gamma ,d)L_{\gamma ,d}^{cl}\int _{\mathbb {R}^{d}}tr_{G}\, W^{\gamma +\frac{d}{2}}(x)dx.
\end{equation}
The constants \( r(\gamma ,d) \) satisfy the inequalities (\ref{R1})-(\ref{R4})
if one replaces \( R(\gamma ,d) \) by \( r(\gamma ,d) \). For the proof of
this generalization we refer to \cite{LW} for \( \gamma \geq 3/2 \) and to
\cite{HLW} for \( 1/2\leq \gamma <3/2 \).

While the validity of (\ref{LTh}) is settled, it seems not to be clear whether
(\ref{es12}) holds for \( \gamma =0 \) and \( d\geq 3 \).\footnote{
Recently D. Hundertmark has announced that (\ref{es12}) holds true for \( \gamma =0 \)
and \( d\geq 3 \).
} Moreover, it is obvious that \( R(\gamma ,d)\leq r(\gamma ,d) \). In all cases
where their sharp values are known, these constants coincide. However, to our
knowledge it is not clear whether \( R(\gamma ,d)=r(\gamma ,d) \) holds in
general.


\section{Statement of the Result}

The main result of this paper is the following:

\begin{thm*}
\label{tm: 1}Let us assume that \( \Omega =\omega \times \mathbb {R} \), \( V\geq 0 \),
and \( V\in L^{\gamma +\frac{d}{2}}(\Omega )\cap L^{\gamma +\frac{1}{2}}(\mathbb {R},dx_{d};L^{2}(\omega ,|\psi _{1}(x^{\prime })|^{2}dx^{\prime })) \).
Then for \( \gamma \geq 1/2 \), \( \alpha \geq 0 \), and all \( \epsilon >0 \)
the following estimate holds true
\begin{equation}
\label{gres}
S_{\gamma }^{\Omega }(\alpha )\leq c_{1}\alpha ^{\gamma +\frac{1}{2}}\int _{\mathbb {R}}\left( \int _{\omega }V(x^{\prime },x_{d})|\psi _{1}(x^{\prime })|^{2}dx^{\prime }\right) ^{\gamma +\frac{1}{2}}dx_{d}+c_{2}S_{\gamma ,d}^{cl}(\alpha ).
\end{equation}
The constants \( c_{1} \) and \( c_{2} \) satisfy the bounds
\begin{eqnarray*}
c_{1} & \leq  & (1+\epsilon )^{\gamma +\frac{1}{2}}r(\gamma ,1)L_{\gamma ,1}^{cl},\\
c_{2} & \leq  & (1+\epsilon ^{-1})^{\gamma +\frac{d}{2}}\left( \frac{\mu _{2}}{\mu _{2}-\mu _{1}}\right) ^{\frac{d-1}{2}}r(\gamma ,1)R(\gamma +\frac{1}{2},d-1),
\end{eqnarray*}
where \( R(\cdot ,\cdot ) \) are the sharp constant in (\ref{LTh}) and \( r(\cdot ,\cdot ) \)
are the sharp constants in (\ref{es12}).
\end{thm*}
\begin{rem*}
If \( \gamma =1/2 \) then (\ref{gres}) gives rise to
\begin{eqnarray*}
S_{1/2}^{\Omega }(\alpha ) & \leq  & (1+\epsilon )\frac{\alpha }{2}\int _{\Omega }V(x^{\prime },x_{d})|\psi _{1}(x^{\prime })|^{2}dx\\
 &  & +4(1+\epsilon ^{-1})^{\frac{d+1}{2}}\alpha ^{\frac{d+1}{2}}\left( \frac{\mu _{2}}{\mu _{2}-\mu _{1}}\right) ^{\frac{d-1}{2}}\int _{\Omega }V^{\frac{d+1}{2}}(x)dx.
\end{eqnarray*}
This formula allows us to extend the small coupling asymptotics
\[
S_{1/2}^{\Omega }(\alpha )=\frac{\alpha }{2}\int _{\Omega }V(x^{\prime },x_{d})|\psi _{1}(x^{\prime })|^{2}dx+o(\alpha )\quad \mbox {as}\quad \alpha \to 0\]
to all non-negative potentials \( V\in L^{\frac{d+1}{2}}(\Omega )\cap L^{1}(\Omega ,|\psi _{1}(x^{\prime })|^{2}dx^{\prime }dx_{d}) \).
The constant in the first term on the r.h.s. of the above estimate is sharp
up to the factor \( (1+\epsilon ) \). Minimizing in \( \epsilon =\epsilon (\alpha ,V) \)
one can extract a three-term formula with an additional intermediate homogeneity
in \( \alpha  \), such that the term serving for small perturbations appears
with the optimal constant.
\end{rem*}
In the remaining part of this paper we sketch the proof of the theorem. 


\section{Reduction in Dimension}

As the first step we apply (\ref{es12}) to reduce bounds for \( S_{\gamma }^{\Omega }(\alpha ) \)
to appropriate integrals of spectral estimates on the operator
\[
W^{\omega }(\alpha ;x_{d})=-\Delta ^{\omega }-\alpha V(\cdot ,x_{d})\quad \mbox {on}\quad L^{2}(\omega )\]
(satisfying Dirichlet boundary conditions) for fixed coordinate parameter \( x_{d}\in \mathbb {R} \).
Indeed, evaluating the quadratic form of \( H^{\Omega }_{\alpha } \) on the
form core \( C_{0}^{\infty }(\omega \times \mathbb {R}) \), Fubini's theorem
and a standard variational argument imply
\begin{eqnarray*}
S_{\gamma }^{\Omega }(\alpha ) & = & tr_{L^{2}(\Omega )}\, (-\Delta -\mu _{1}-\alpha V)_{-}^{\gamma }\\
 & \leq  & tr_{L^{2}(\omega )\otimes L^{2}(\mathbb {R})}\left( -\frac{d}{dx_{d}}\otimes 1_{L^{2}(\omega )}-(W^{\omega }(\alpha ;x_{d})-\mu _{1})_{-}\right) ^{\gamma }_{-}.
\end{eqnarray*}
The bound (\ref{es12}) with \( W(x_{d})=(W^{\omega }(\alpha ;x_{d})-\mu _{1})_{-} \),
\( d=1 \) and \( G=L^{2}(\omega ) \) gives now
\begin{equation}
\label{s12}
S^{\Omega }_{\gamma }(\alpha )\leq r(\gamma ,1)L_{\gamma ,1}^{cl}\int _{\mathbb {R}}tr\, (W^{\omega }(\alpha ;x_{d})-\mu _{1})^{\gamma +\frac{1}{2}}_{-}dx_{d}.
\end{equation}
 


\section{Spectral Estimates on the Cross Sections}

In the next step we have to find appropriate estimates on the quantities \( tr\, (W^{\omega }(\alpha ;x_{d})-\mu _{1})^{\gamma +\frac{1}{2}}_{-} \).
For that end we recall that \( \mu _{k} \) and \( \psi _{k} \) denote the
eigenvalues and eigenfunctions of the Dirichlet Laplacian \( -\Delta ^{\omega } \)
on \( L^{2}(\omega ) \). The lowest eigenvalue \( \mu _{1} \) is simple. Let
\( P \) be the projection on the linear span of \( \psi _{1} \) of rank one,
and let \( Q \) be the projection onto its orthogonal complement in \( L^{2}(\omega ) \).
Since \( V\geq 0 \), we have the variational estimate
\[
W^{\omega }(\alpha ;x_{d})\geq PW((1+\epsilon )\alpha ;x_{d})P+QW((1+\epsilon ^{-1})\alpha ;x_{d})Q.\]
Since \( P(-\Delta ^{\omega }-\mu _{1})P=0 \), one claims that
\[
W^{\omega }(\alpha ;x_{d})-\mu \geq \hat{W}^{\omega }((1+\epsilon )\alpha ;x_{d})+\tilde{W}^{\omega }((1+\epsilon ^{-1})\alpha ;x_{d})\]
for any \( \epsilon >0 \), where
\begin{eqnarray}
\hat{W}^{\omega }(\alpha ;x_{d}) & = & -\alpha PV(\cdot ,x_{d})P,\label{ha} \\
\tilde{W}^{\omega }(\alpha ;x_{d}) & = & Q(W(\alpha ;x_{d})-\mu _{1})Q.\label{ti} 
\end{eqnarray}
Obviously, the operators (\ref{ha}) and (\ref{ti}) reduce on the orthogonal
subspaces \( L^{2}(\omega )=PL^{2}(\omega )\oplus QL^{2}(\omega ) \) and
\begin{eqnarray*}
tr\, (W^{\omega }(\alpha ;x_{d})-\mu )^{\gamma +\frac{1}{2}}_{-} & \leq  & tr\, (\hat{W}^{\omega }((1+\epsilon )\alpha ;x_{d}))_{-}^{\gamma +\frac{1}{2}}\\
 &  & +tr\, (\tilde{W}^{\omega }((1+\epsilon ^{-1})\alpha ;x_{d}))_{-}^{\gamma +\frac{1}{2}}.
\end{eqnarray*}


Now we shall estimate the last two terms separately. Since \( P \) is a perturbation
of rank one and \( V\geq 0 \), by (\ref{ha}) it is clear that
\begin{equation}
\label{v4}
tr\, (\hat{W}^{\omega }((1+\epsilon )\alpha ;x_{d}))_{-}^{\gamma +\frac{1}{2}}=\alpha ^{\gamma +\frac{1}{2}}(1+\epsilon )^{\gamma +\frac{1}{2}}\left( \int _{\omega }V(x^{\prime },x_{d})|\psi _{1}(x^{\prime })|^{2}dx^{\prime }\right) ^{\gamma +\frac{1}{2}}.
\end{equation}
With regard to the second term we note, that because of \( \mu _{k}\geq \mu _{2}>\mu _{1}>0 \)
for \( k=2,3,\dots  \), the operator inequality
\begin{eqnarray*}
Q(-\Delta ^{\omega }-\mu _{1})Q & = & \sum _{k=2}^{\infty }(\mu _{k}-\mu _{1})\psi _{k}(\cdot ,\psi _{k})_{L^{2}(\omega )}\\
 & \geq  & \sum _{k=2}^{\infty }\frac{\mu _{2}-\mu _{1}}{\mu _{2}}\mu _{k}\psi _{k}(\cdot ,\psi _{k})_{L^{2}(\omega )}\\
 & = & \frac{\mu _{2}-\mu _{1}}{\mu _{2}}Q(-\Delta ^{\omega })Q
\end{eqnarray*}
holds true. Hence,
\begin{eqnarray*}
tr\, (\tilde{W}^{\omega }((1+\epsilon ^{-1})\alpha ;x_{d}))_{-}^{\gamma +\frac{1}{2}} & \leq  & tr\, (Q(-\tau \Delta ^{\omega }-\tau \rho V(\cdot ,x_{d}))Q)_{-}^{\gamma +\frac{1}{2}}\\
 & = & \tau ^{\gamma +\frac{1}{2}}tr\, (QW(\rho ;x_{d})Q)_{-}^{\gamma +\frac{1}{2}}\\
 & \leq  & \tau ^{\gamma +\frac{1}{2}}tr\, (W(\rho ;x_{d}))_{-}^{\gamma +\frac{1}{2}},
\end{eqnarray*}
where \( \tau =\mu _{2}^{-1}(\mu _{2}-\mu _{1}) \) and \( \rho =\tau ^{-1}\alpha (1+\epsilon ^{-1}) \).
Clearly, the negative eigenvalues of the operator \( W(\rho ;x_{d}) \) on \( L^{2}(\omega ) \)
with Dirichlet boundary conditions at \( \partial \omega  \) can be estimated
from below by the respective negative eigenvalues of
\[
H_{\rho }(x_{d})=-\Delta -\rho V(\cdot ,x_{d})\quad \mbox {on}\quad L^{2}(\mathbb {R}^{d-1}).\]
Now the usual Lieb-Thirring inequality (\ref{LTh}) for \( \gamma ^{\prime }=\gamma +\frac{1}{2} \)
and \( d^{\prime }=d-1 \) implies that
\begin{eqnarray*}
tr\, (\tilde{W}^{\omega }((1+\epsilon ^{-1})\alpha ;x_{d}))_{-}^{\gamma ^{\prime }} & \leq  & \tau ^{\gamma ^{\prime }}S_{\gamma ^{\prime }}(\rho )\\
 & \leq  & \tau ^{\gamma ^{\prime }}R(\gamma ^{\prime },d^{\prime })L_{\gamma ^{\prime },d^{\prime }}^{cl}S_{\gamma ^{\prime },d^{\prime }}^{cl}(\rho )\\
 & = & R(\gamma ^{\prime },d^{\prime })L_{\gamma ^{\prime },d^{\prime }}^{cl}\tau ^{\gamma ^{\prime }}\rho ^{\gamma +\frac{d}{2}}\int _{\omega }V^{\gamma +\frac{d}{2}}(x^{\prime },x_{d})dx^{\prime }.
\end{eqnarray*}
In the last expression we made use of the relation \( \gamma ^{\prime }+\frac{d^{\prime }}{2}=\gamma +\frac{d}{2} \). 

Now we insert the spectral estimates on (\ref{ha}) and (\ref{ti}) into (\ref{s12}).
In view of \( L_{\gamma ^{\prime },d^{\prime }}^{cl}L_{\gamma ,1}^{cl}=L_{\gamma ,d}^{cl} \)
this gives
\begin{eqnarray*}
S_{\gamma }^{\Omega }(\alpha ) & \leq  & \alpha ^{\gamma +\frac{1}{2}}(1+\epsilon )^{\gamma +\frac{1}{2}}r(\gamma ,1)L_{\gamma ,1}^{cl}\int _{\mathbb {R}}\left( \int _{\Omega }V(x)|\psi _{1}(x^{\prime })|^{2}dx\right) ^{\gamma +\frac{1}{2}}dx_{d}\\
 &  & +(1+\epsilon ^{-1})^{\gamma +\frac{d}{2}}\left( \frac{\mu _{2}}{\mu _{2}-\mu _{1}}\right) ^{\frac{d-1}{2}}r(\gamma ,1)R(\gamma ^{\prime },d^{\prime })S_{\gamma ,d}^{cl}(\alpha ).
\end{eqnarray*}
This completes the proof.


\section{Acknowledgements}

The work on this paper has been supported by the Royal Swedish Academy of Sciences
and the Academy of Sciences of the Czech Republic. Both authors are also grateful
to the ICMP 2000 for financial support.

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\address{{\footnotesize P. Exner: Dept. of Theoretical Physics, Nuclear Physics Institute,
Academy of Sciences, CZ-25068 \v{R}e\v{z} near Prague, Czech Republic, exner@ujf.cas.cz.
T. Weidl: Dept. of Mathematics, Royal Institute of Technology, S-10044 Stockholm,
Sweden, weidl@math.kth.se.}}

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