\magnification= \magstep1
\baselineskip= 11 pt
\par \noindent
c
\par \vskip 3 cm \noindent
\centerline{ $ { \bf STATISTICS \; IN \; SPACE \; DIMENSION \; TWO }$ }
\par \vskip 2 cm \noindent
\centerline{ Gianfausto Dell'Antonio }
\par \vskip 3 pt \noindent
\centerline {Dipartimento di Matematica, Univ. di Roma La Sapienza}
\par \vskip 2 pt \noindent
\centerline{ and}
\centerline{ Laboratorio interdisciplinare, SISSA, Trieste}
\par \vskip 9 pt \noindent
\centerline{ Rodolfo Figari}
\par \vskip 3 pt \noindent
\centerline {Dipartimento di Fisica, Univ. di Napoli}
\centerline{ and}
\centerline{ INFN Sezione di Napoli}
\par \vskip 9 pt \noindent
\centerline{ Alessandro Teta }
\par \vskip 3 pt \noindent
\centerline {Dipartimento di Matematica, Univ. di Roma La Sapienza}
\par \vskip 2 cm \noindent
\centerline { \it ABSTRACT \rm}
\par \vskip 4 pt \noindent
We construct as a selfadjoint operator the Schroedinger hamiltonian for a
system of $N$ identical particles on a plane, obeying the statistics defined
by a representation $\pi_1$ of the braid group. We use quadratic forms and
potential theory, and give details only for the free case; standard
arguments provide the extension of our approach to the case of potentials
which are small in the sense of forms with respect to the laplacian.
\par \noindent
We also comment on the relation between the analysis given here and other
approaches to the problem, and also on the connection with the description of a
quantum particle on a plane under the influence of a shielded magnetic field
(Aharanov-Bohm effect).
\vfill \eject
\par
\par \vskip 1 cm \noindent
0. $ { \bf INTRODUCTION } $
\par \vskip 3 pt
For quantum particles in space dimension two (as can be considered, at least
to first approximation, particles confined in a very thin layer) statistics
has a richer structure than in three
dimensions. Loosely speaking, this is due to the fact that on a plane
one can define a relative winding number for two impenetrable particles: given
a configuration of the particles on the plane, the relative winding number
counts the number of times one of the particles has gone around
the other before reaching again the initial configuration. It is a
topological invariant (given the impenetrability).
\par \noindent
This richer topological structure has an influence on the quantum mechanical
description of a collection of $N$ identical particles in a two-dimensional
world and in particular on the definition of the Schroedinger
hamiltonian as a self-adjoint operator in the suitable Hilbert space
(see e.g. [BCMS],[GS],[LM],[MS],[S.2],[W],[Wu]).
\par \noindent
Knot theory is
the natural tool to study the topology of the problem. In this paper we shall
advocate the point of view that the natural functional-analytic tool are
quadratic forms.
\par \vskip 3 pt \noindent
There is a connection between the analysis given here and a possible
description of a quantum particle on a plane under the influence of a
shielded magnetic field, as in the Aharanov-Bohm effect for
a particle
in $R^2$ in presence of a singular magnetic field produced by a finite number
of solenoids ([AB],[R],[S.1]). We shall touch briefly upon this relation in
Section 3.
\par \vskip 3 pt \noindent
Throughout this note,
we shall consider only the free case, paying
special attention to the description of statistics; standard arguments allow
to extend the results to the case in which there is a perturbation which is
small in the sense of forms. Singular perturbations, such as point interaction,
can be introduced but a detailed treatment requires further work
and will not be discussed here (see e.g. [MT]).
\par \vskip 5 pt \noindent
1. ${ \bf GENERAL \; FORMULATION }$
\par \vskip 3 pt \noindent
Consider
$N$ identical particles of mass $m$ in $R^2 $ and
denote by
$$
D_N \equiv \{ (R^2)^N \ni x \equiv \{ x^k, \; k=1, \ldots N \}: \; \exists
\; i < k \; \; for \; which \; \; x^{i} = x^k \}
$$
the ''coincidence set''.
\par \noindent
To simplify notations, we choose units in which $2m = \hbar =1.$
\par \noindent
Let $S_N$ be the permutation group of $N$ elements, which acts on $
(R^2)^N$ in an obvious way, leaving $D_N$ invariant.
\par \noindent
The particles are identical and impenetrable; therefore, as a
set, a natural candidate for configuration space is the quotient
$$
Y_N \equiv \left( (R^2)^N - D_N \right) / S_N \eqno 1.1
$$
The space $Y_N$ has a natural differential structure and also a
natural measure $\mu$ induced by the Lebesgue measure on $(R^2)^N.$
\par \noindent
The space $
Y_N $ is not simply connected; its first homology group is the ''braid group''
for N elements on $R^2,$ denoted $B_N $ (this is one way of stating in
mathematical terms the presence of a nontrivial relative winding number).
\par \noindent
We denote by $ \tilde Y_N $ be the universal covering space of
$ Y_N $ and denote by $\pi_0$ the representation of $B_N$ as the fundamental
group of $ \tilde Y_N;$ $ \pi_0 (g), \; g \in B_N $ is a map on
$\tilde Y_N $ which projects to the identity in $Y_N.$
\par \noindent
Let $ \pi_1 $ be
a finite-dimensional unitary representation of $B_N$ on a complex vector space
$V$ of dimension $d$ with scalar product $<,>.$
\par \noindent
Identical quantum particles with statistics given by the representation
$ \pi_1$ are described by (complex valued) functions $ f $ from $ \tilde Y_N $ to $V$ satisfying the
equivariance relation
$$
f (\pi_0 (g) \tilde y) = \pi_1 (g) f (\tilde y) \qquad g \in B_N \eqno 1.3
$$
One refers to these particles as \it plektons; \rm if $ V $ is one-dimensional
they are called \it anyons. \rm
\par \vskip 3 pt \noindent
We want to define a Schroedinger operator which describes the dynamics of
plektons.
\par \noindent
A possible line of approach is the
following [MS].
\par \noindent
A Hilbert space structure is introduced on functions which satisfy (1.3)
through the scalar product
$$
( f, g ) \equiv \int_{Y_N} < f (\tilde y), g (\tilde y)> d \mu (y)
\eqno 1.4
$$
where $ \tilde y $ is any point on the orbit of $ \pi_0$ through $y.$
Since $ \pi_1 (g)$ is unitary,
the integrand is constants on the orbits of $\pi_0$ and therefore (1.4) is
well defined.
\par \vskip 2 pt \noindent
We shall denote by ${\cal H }$ the closure with respect to (1.4) of
continuous $ V$-valued functions on $ \tilde Y_N$ with compact support
and satisfying (1.3).
\par \noindent
The next step is to introduce dynamics; this is done by choosing
a ''hamiltonian'', i.e. a self-adjoint operator $H$ on $ {\cal H }.$
This can be done by defining on smooth functions on $ \tilde Y_N $
an operator $\tilde H $ which commutes with the representation ${\cal R}$ of
$ B_N$ which is defined by
$$
({\cal R }(g) f) ( \pi_0 (g) \tilde y ) \equiv \pi_1 (g) f ( \tilde y) \eqno
1.5 $$
and then restricting $\tilde H $ to ${\cal H}.$
\par \noindent
Recall that locally one has
$$
\tilde Y_N \simeq Y_N \otimes W \eqno 1.6
$$
where $ W$ is a discrete set.
\par \noindent
One can then consider the special case in which
$ \tilde H $ acts locally as
$ (- \Delta + U) \otimes I $ where $ \Delta $
is the Laplacian on $Y_N)$ and the operator $U ( y) $
(the ''potential energy) acts multiplicatively on functions on $Y_N$.
\par \noindent
One can also envisage more general situations, for example
$\Delta$ could be a covariant second order differential operator.
We shall not consider these generalizations here.
\par \noindent
If one identifies $Y_N$ with a section of $ \tilde Y_N$ and one sets
$ \tilde U( \pi_0 (g) \tilde y) =
U(y)$ for every $ g \in B_N$ one can easily extend $U$ to be a multiplication
operator on smooth functions on $ \tilde Y_N; $ by construction it leaves
invariant the linear space of
functions satisfying (1.3) and therefore can be restricted to ${\cal H}.$
\par \noindent
If $U$ is sufficiently regular the resulting multiplication operator
is self-adjoint.
\par \vskip 3 pt \noindent
It is not entirely obvious how to define the differential part of the
operator in such a way that $H$ is selfadjoint.
\par \noindent
The reason is that $\tilde H $ was defined as acting as differential
operator on smooth functions on $\tilde Y_N$ but, unless the action of
$\pi_0$ on $W$ can be reduced to an
action on a finite set (as is the case for Bose and Fermi statistics)
it is not clear how to find a countably
additive measure which is invariant under $
\pi_0$ (so that (1.5) defines a unitary representation) and with respect
to which $\tilde H $ is self-adjoint.
\par \noindent
We shall overcome this difficulty by constructing a unitary map between
${\cal H}$ and a suitable Hilbert space $ {\cal H}'$
and then defining the Schroedinger operator on $ {\cal H}'.$
\par \noindent
Let
$$
\Omega_N \equiv
\{ x \in (R^2)^N \; |\; x^k_2 - x^{k+1}_2 < 0, \; k=1, \ldots N-1 \} \eqno
1.7
$$
where the index $2$ denotes the second coordinate with
respect to a fixed cartesian frame.
\par \noindent
Define
$$
\Sigma_N^{\pm} \equiv \cup_k \Sigma_N^{k,\pm}
$$
$$
\Sigma_N^{h,\pm} \equiv \{ x\in (R^2)^N: \; x^1_2 < \ldots
< x^h_2 = x^{h+1}_2 < \ldots < x^N_2, \;\; \pm (x^{h+1}_1 - x^h_1) > 0 \} \eqno 1.8
$$
$$
\Sigma_N^k \equiv \Sigma_N^{k,+} \cup \Sigma_N^{k,-}
\qquad \Sigma_N \equiv \Sigma_N^+ \cup \Sigma_N^-
$$
Then
$$
\Sigma_N^{+} \cap \Sigma_N^{ -} = \emptyset, \qquad \bar \Sigma_N^{+}
\cup \bar \Sigma_N^{-} \equiv \partial \Omega_N
$$
Moreover
$$
\bar \Sigma_N^{+} \cap \bar \Sigma_N^{-} = \bar \Omega_N \cap D_N
$$
Let $\phi $ be the product of the maps $ \phi_k $ which take each point on
$ \Sigma_N^{k,-} $ to that point on $ \Sigma_N^{k,+} $ which is obtained by
exchanging the numerical
values of $x^k_1 $ and $x^{k+1}_1.$
\par \noindent
Then as a set $ Y_N$ can be identified with
$ \Omega_N \cup \Sigma_N^+ $
and as a topological space $ Y_N$ can be identified with
$ \Omega_N \cup \left( \Sigma_N ) / \phi \right) $
\par \vskip 3 pt \noindent
\it REMARK \rm
\par
\it A simple example is the case $N=2.$ Using coordinates
$ x^1 + x^2 $ and $ x^1 - x^2 $ one readly sees that $ Y_2 $ is diffeomorphic
to the product of $R^2$ with a cone with
vertex in the origin, which can be obtained from the closed upper
half plane minus the origin identifying by reflection every point on $R^-$
with the corresponding one on $R^+.$ \rm
\par \vskip 3 pt \noindent
Denote by $g_k$ the element of the braid group which can be identified with
$ \phi_k \cdot R_{\pi,k}$ as a map of $ \Sigma_N^{k,+}$ onto itself,
where $R_{\pi,k}$ denotes rotation of ninety
degrees in the anticlockwise direction (i.e. going through $\Omega_N);$
these elements generate the entire group, so that the representation $ \pi_1$ is completely
determined by its restriction to the $ g_k.$
\par \noindent
Then there is a one-to-one correspondence between continuous functions $ f $ on
$\tilde Y_N$ which satisfy (1.3) and continuous functions $F$ on
$ \Omega_N \cup \Sigma_N $ which satisfy
the boundary condition
$$
F( x) = \pi_1 (g_k) F ( \phi_k (x)), \quad x \in \cup_k \Sigma_N^{k,-} \eqno 1.9
$$
It is obtained by identifying $ \Omega_N \cup \Sigma_N^+ $ with $Y_N,$ as
described above, and then $Y_N$ with a subset of $\tilde Y_N.$
Condition (1.9) guarantees that $F$ can be
extended uniquely to a function $f$ on $ \tilde Y_N$ which satisfies the
equivariance requirement (1.3).
\par \noindent
This correspondence is an isometry for the corresponding scalar products
and extends therefore to a unitary equivalence between
$ {\cal H }$ and the closure in $L^2(\Omega_N, V)$ (Lebesgue measure
is understood) of the continuous
compactly supported functions satisfying (1.9). Notice that the latter
concides with $L^2(\Omega_N, V)$ (in fact
the subset of functions which have support in $\Omega_N,$ and therefore
satisfy (1.9), is dense in $L^2(\Omega_N, V)). $
We shall therefore define the Schroedinger operator as a
self-adjoint operator on $L^2(\Omega_N, V).$
\par \noindent
Statistics, i.e. the representation $\pi_1$ of the braid group $B_N,$ will
then enter through the explicit structure of the operator (and in
particular through its domain). More
precisely, the free Schroedinger operator will be identified with
the self-adjoint operator whose quadratic form is the restriction
of the energy form [S.2]
$$
Q^{\pi_1} (f,f) \equiv \int_{ (R^2)^N} <\nabla f(x), \nabla f(x)> dx \eqno 1.10
$$
to the domain
$$
D(Q^{\pi_1}) = \{ f \in H^1(\Omega_N,V) |\; f \; satisifies \; (1.9) \}
$$
We have denoted by $H^s$ the Sobolev space of order $s$ (see e.g. [A])
\par \noindent
This is the \it definition \rm of statistics we shall advocate;
it has the advantage of allowing for a
simple treatment of the issue of essential self-adjiointness.
\par \noindent
A similar approach has been suggested by P.Stovicek in [S.2].
\par \vskip 3 pt \noindent
\it REMARK 1 \rm
\par
\it The distributions in the domain of the quadratic form (1.10)
are in general not continuous, but they have a trace on
$\Sigma_N $ which belongs to $H^{1/2} (\Sigma_N,V);$
condition (1.9) will be understood to hold in this slightly
more general sense. \rm
\par \vskip 3 pt \noindent
>From potential theory we know that
for the laplacian in $\Omega_N$ with smooth
boundary $ \bar \Sigma_N, $ boundary conditions which are local can be
described by double-layer potentials, i.e. by distributions of
''dipoles'' on the boundary.
\par \noindent
Also, as we will discuss in detail in the next Section, it is known that
there is a one-to one
correspondence (continuous in a suitable topology) between dipole charges
and the restriction to $\Sigma_N $ of the corresponding potential.
\par \vskip 3 pt \noindent
The explicit construction of the closed quadratic form which defines
the free laplacian for the given choice of statistics will be given in Section 2.
The form domain will be the disjoint union of $ H_0^1 (\Omega_N, V)$
and of a suitable Hilbert space of $V-$valued (dipole charge)
distributions on $\Sigma_N.$
\par \noindent
The dependence on the statistics (i.e. on the representation
$\pi_1$ of $B_N)$ will be through the expression of the quadratic form on
dipole distributions.
\par \vskip 3 pt \noindent
\it REMARK 2 \rm
\par
\it We have already remarked that $ \Sigma_N^k $
does not intersect the subspace $ \{ x\;|\; x^k = x^{k+1}\}.$ Therefore a
boundary condition is not placed
directly at coincidence points; since functions in $H^1 ((R^2)^N,V) $ are
not continuous in general, the introduction of such ''boundary condition''
must be understood in a more general
sense [GFT] and would correspond to the introduction of a ''point interaction''
among plektons. While there seems to be no particular difficulty in applying
the methods of [GFT] to
achieve this construction, we shall not do it here. \rm
\par \vskip 4 pt \noindent
In the remaining part of this Section we shall describe briefly the relation of
our approach to one which can be found in the literature for the study of
anyons (in this case the representation $\pi_1$ is abelian).
In this approach one selects an open dense subset $Z_N \subset(R^2)^N,$
chosen so that
its Riemann covering has $B_N$ as fundamental group.
This allows for the introduction of an equivariance condition
in the spirit of (1.3).
\par \noindent
Define
$$
\eta_N \equiv \cup_{k \lambda_0 $ every function
$ f \in H^1 (\bar \Omega, V) $ can be uniquely written
as
$$
f = f_0 ^{\lambda}+ \hat G_{\lambda} * \mu ,
\qquad f_0^{\lambda} \in H^1_0 (\bar \Omega, V), \quad \
\mu \in H^{1/2} (\Sigma, V) \eqno 2.7
$$
(notice that $\mu$ in (2.7) does not depend on $\lambda).$
\par \noindent
The advantage of this
decomposition lies in the fact that the two terms in the decomposition (2.7)
are orthogonal in the scalar product defined by the energy form.
This is due to the fact that
$ (-\Delta + \lambda I) (\hat G_{\lambda} * \mu) $ is a distribution
supported by $ \Sigma $ and is therefore orthogonal to smooth functions
vanishing in a
neighbourhood of $ \Sigma.$
\par \noindent
One has therefore for any such
function $g$
$$
\left( \nabla g, \nabla (\hat G_{\lambda} * \mu) \right) + \lambda
(g,(\hat G_{\lambda} * \mu)) =
(g, ( - \Delta + \lambda I) (\hat G_{\lambda}
* \mu)) = 0 \eqno 2.8
$$
where we have indicated by $(\cdot, \cdot)
$ the scalar product in $ L^2 (\Omega, V)$
\par \vskip 2 pt \noindent
The proof of (2.8) for any function in $H^{1}_0 (\Omega, V)$ is
then obtained by a standard approximation procedure.
\par \noindent
As a consequence, the quadratic form which represents the laplacian with the
given boundary conditions will be written as the sum of the energy form on
$ H^1 (\Omega, V)$ and of a bilinear form for functions on $\Sigma.$
\par \noindent
This will simplify the study of the resolvent and of the spectral properties.
\par \vskip 3 pt \noindent
Using (2.7) and Green's theorem one has
$$
(\nabla f, \nabla f) + \lambda (f,f) = (\nabla f_0^{\lambda}, \nabla
f_0^{\lambda}) + \lambda (f_0^{\lambda},f_0^{\lambda}) - {1\over 2}
\int_{\Sigma} \mu { \partial \over \partial n} G_{\lambda} * \mu \eqno 2.9
$$
It is known from potential theory that the last term is positive, and bounded if
$\mu \in H^{1/2} (\Sigma,V).$ This is seen explicitely by taking Fourier
transforms, upon which this term can be written as
$$
{ 1 \over 4 \pi} \sum_k
\int \sqrt { \sum_{h \not= k,\; k+1} q_h^2 + \sum_i p_i^2 +
\zeta_k^2 +
\lambda}\; \; | \mu_k (q^{(k)}, p, \zeta_k)|^2 dq^{(k)} dp \; d \zeta_k
+ N_{\lambda} \eqno 2.10
$$
where $N_{\lambda}$ is a ''nondiagonal'' term (containing the sum over
integrations on terms bilinear in $ \mu_k$,$\mu_h$ for $h \not= k)$
which can be made arbitrarly small by taking $\lambda$ large enough.
\par \noindent
In (2.10) the
$p_k 's$ are the variables conjugated to the $ x^k_1,$ the $ q_h $ are the
variables conjugated to the $ x^h_2$ and in the integral over $\Sigma^k,$ $
\zeta_k$ is conjugated to $ 1/2 \left( x^k_2 + x^{k+1}_2 \right) $ and $ q^{(k)}
$ denotes the collection of the coordinates $q_h$ with the exception of $q_k$
and $q_{k+1}.$
\par \noindent
>From (2.9) the form associated to the laplacian on $ H^1(\bar \Omega, V) $
without restrictions due to statistics is
$$
Q(f,f) = ( \nabla f_0^{\lambda}, \nabla f_0^{\lambda}) - {1\over 2} \int_{\Sigma}
\mu { \partial \over \partial n} G_{\lambda} * \mu -
\lambda (f,f) + \lambda (f_0^{\lambda},f_0^{\lambda})
\eqno 2.11
$$
defined on
$$
H^{1}_0 (\Omega, V) \times H^{1/2} (\Sigma^+, V)
$$
\par \vskip 4 pt
\noindent
The quadratic form for particles obeying the statistics defined by the
representation $\pi_1$ of the braid group $B_N$ is obtained restricting (2.11) to
functions which satisfy (1.9).
\par \noindent
Since by construction $f_0^{\lambda}$ vanishes on $ \Sigma$ the restriction is
entirely on $ \hat G_{\lambda} * \mu;$
in view of Proposition
2.1 it can be expressed as a relation in (2.12) between $ \mu^-$ and $
\mu^+,$ the restrictions of $ \mu $ to $ \Sigma^{\pm}.$
\par \noindent
In order to work out the details of this construction,
notice that if one sets, for functions $\nu $ defined on
$\Sigma^{\pm}$
$$
\nu_k^{\pm} \equiv \theta (\pm \xi^k_1 ) \nu_k \eqno 2.12
$$
(where $ \theta (x) $ is 1 for $ x > 0 $ and $0$ for $ x \leq 0) $
one obtains
$$
\Gamma_{\lambda} \mu = { I \over 2} \mu + R_{\lambda} \mu^+
+ R_{\lambda} \mu^- \eqno 2.13
$$
\par \vskip 3 pt \noindent
The boundary conditions which define the statistics have the form
$$
\left( \Gamma_{\lambda}^k \mu \right)^- = \pi_1 (g_k) \left(
\Gamma_{\lambda}^k \mu \right)^+ \eqno 2.14
$$
where $g_k$ is the element of $B_N$ which has been described in Section 1
and the $ \pm $ sign indicates restriction of $ \Gamma _{\lambda} \mu $ to $
\Sigma_k^{\pm}. $
\par \vskip 3 pt \noindent
We write (2.14) in the concise form
$$
\left( \Gamma_{\lambda} \mu \right) ^{-} = A _{\lambda} (\pi_1)
\left( \Gamma_{\lambda}
\mu \right)^{+}
$$
This \it defines \rm a linear map $ A_{\lambda}( \pi_1) $ on $ V^N, $ which
depends on the representation $ \pi_1.$
>From (2.13) one has
$$
{ 1 \over 2} \mu^- + R^{-,+}_{\lambda} \mu^+
+ R^ {-,-}_{\lambda} \mu^-
= A_{\lambda} (\pi_1)
\left[ { 1 \over 2} \mu^+ + R^{+,+}_{\lambda} \mu^+
+ R^{+,-}_{\lambda} \mu^- \right] \eqno 2.15
$$
where $ R ^{\pm,\pm}_{\lambda} (\xi, \eta) $
are integral operators with kernels given by $ R_{\lambda} (\xi, \eta) $ for $
\xi \in \Sigma^{\pm} $, $ \eta \in \Sigma^{\pm}.$
After rearrangment of terms (2.15) becomes
$$
{ 1 \over 2} \mu^-
+ R^{-,-}_{\lambda} \mu^- - A_{\lambda}( \pi_1) R^{+,-}_{\lambda} \mu^-
=
{ 1 \over 2} A_{\lambda} (\pi_1) \mu^+ + A_{\lambda}( \pi_1)
R^{+,+}_{\lambda} \mu^+ - R^{-,+}_{\lambda} \mu^+
\eqno 2.16
$$
For $ \lambda$ sufficiently large the $ H^{1/2} $ norms of $
R^{\tau,\nu}_{\lambda} \mu ^{\nu}, \; \; (\tau, \; \nu = \pm )$
become arbitrarly small with respect to the norm of $ \mu$
(the proof is identical to the one given above for the operator
$R^{\pm}_{\lambda}).$
\par \noindent
Therefore if one takes $ \lambda $ sufficiently large (depending
only on $N)$ one can write (2.16) in the form
$$
\mu^- = \left[ I
+ 2 R^{-,-}_{\lambda} - 2 A_{\lambda}( \pi_1) R^{+,-}_{\lambda}
\right]^{-1}
A _{\lambda}(\pi_1) \left[ I + 2 R^{+,+}_{\lambda} - 2
A_{\lambda}^{-1}(\pi_1) R^{-,+}_{\lambda} \right] \mu^+
\eqno 2.17
$$
which, in components, can be written as
$$
\mu^-_k = \sum_{h} T^{\lambda}_{k,h}(\pi_1)
\mu^+_h \eqno 2.18
$$
where the matrix $T^{\pi_1}$ defines a bounded invertible linear
map from $ H^{1/2}( \Sigma^+, V)$ to $ H^{1/2}( \Sigma^-, V)$
which depends on the representation $ \pi_1 $ chosen for the braid group.
\par \noindent
Equation (2.18), or rather its explicit form given through (2.17)
will be used to give the Dirichlet form $ Q^{\pi_1} $ which
describes the Laplacian for a system of $N$ identical impenetrable particles in
$R^2$ which satisfy the
statistic defined by the representation $\pi_1$ of the braid group $B_N.$
\par \noindent
Substituting (2.18) in (2.12) one has
\par \vskip 4 pt \noindent
\it PROPOSITION 2.2 \rm
\par
\it The quadratic form associated to the free hamiltonian for a system of N
identical impenetrable particles in $R^2$ described by the representation $
\pi_1 $ of the braid group is given by
$$
Q^{\pi_1} (f,f) = ( \nabla f_0, \nabla f_0) -
\lambda (f,f) + \lambda (f_0^{\lambda},f_0^{\lambda}) - {1\over 2}
\int_{\Sigma} \mu { \partial \over \partial n} G_{\lambda} * \mu \eqno 2.19
$$
$$
\mu^{-}_ k \equiv \sum_{h} T^{\lambda}_{k,h}(\pi_1) \mu^+_h \eqno 2.20
$$
where $ T^{\lambda}_{k,h}(\pi_1) $ is defined in (2.17),(2.18). \rm
\par \vskip 4 pt \noindent
The analysis given above can be easily extended to the study
of the Schroedinger equation with a potential $U$ which is relatively small
as form with respect to the laplacian.
\par \noindent
More singular potentials, and in particular
interactions supported by
$ \cup_k \left( \bar \Sigma_k^+ \cap \bar \Sigma_k^- \right) )$ (''zero
range interactions'') require more care. We shall not discuss them here, but
weremark that the characterization of the statistics in terms of a quadratic
form for the laplacian allows to introduce point interactions in a natural
manner [GFT].
\par \vskip 5 pt \noindent
We shall now briefly outline the domain of the hamiltonian and its action on the domain. Along similar lines one can give an expression for the resolvent. The
simplicity of the action of the operator on elements in its domain suggests
that one can obtain a rather explicit form for the resolvent. We shall not work
it out here, but in the next Section we shall consider more in detail the
analogous problem for a quantum particle in $R^2$ under the influence of the
magnetic field of $N$ solenoids.
\par \vskip 3 pt \noindent
Recall that, if $
Q(v,u)$ is a strictly positive bilinear form on a Hilbert space $ {\cal H} $,
the domain of the corresponding operator is the subset of those $u's$ in the
form domain for which $Q(v,u) $ is continuous in $v$ in the topology of ${\cal
H}.$
\par \noindent
In our case, considering a sequence of functions $v_n$ of the form
$$
v_n = \hat G_{\lambda} * \mu_n , \qquad \mu_n \in H^{1/2}(\Sigma,V) \eqno 2.21
$$
which converge in $L^2(\Omega,V)$ and requiring convergence of
$ Q^{\pi_1} (u,v_n)$ one finds the condition
$$
u = u_0 + \hat G_{\lambda} * \mu, \qquad \mu \in H^{3/2}(\Sigma,V) \eqno 2.22
$$
If the same procedure is applied to a sequence of functions of the form
$$
v_n = \hat G_{\lambda} * \mu + h_n, \qquad \mu \in H^{1/2} (\Sigma,V),
\qquad h_n \in H^1_0 (\Omega,V)
$$
which converge in $L^2(\Omega,V),$ and convergence of $ Q^{\pi_1} (u,v_n)$ is
required, one finds that $u_0$ must belong to $H^2(\bar \Omega,V)$
and that the normal derivative of $u$ at $\Sigma$ (which exists since $\mu
\in H^{3/2}(\Sigma,V))$ must satisfy (1.9) (this condition cancels boundary terms
which would appear in the integration by parts and for which the limit does not
exist in general).
\par \noindent
If these conditions on $u$ are satisfied, by Riesz's representation theorem
the quadratic form can be written, for every $v $ in the form domain
$$
Q^{\pi_1} (v,u) = (v,\xi_u) \eqno 2.23
$$
for some $\xi_u \in L^2(\Omega,V),$ and then $Hu \equiv \xi_u.$
\par \noindent
Setting $v = v_0 + \hat G_{\lambda} * \nu$ and performing the integrations by
parts one concludes
\par \vskip 4 pt \noindent
\it PROPOSITION 2.3 \rm
\par
\it The hamiltonian $H$ corresponding to the quadratic form $Q^{\pi_1} (v,u)$
is characterized, for $ \lambda $
sufficiently large, by
$$
D(H) \equiv
$$
$$
\{ u = u_0 + \hat G_{\lambda} * \mu,
\; \; \; u_0 \in H^2 (\bar \Omega,V) \cap H^1_0 (\Omega,V), \;\;
\mu \in H^{3/2}(\Sigma,V), \;\;
\mu \simeq (2.20) \;\; \partial_n u \simeq
(1.9) \}
$$
(we have used the symbol $\simeq$ to indicate that the corresponding identity
is satisfied).
\par \vskip 2 pt \noindent
If $ u \in D(H) $ one has \rm
$$
(H + \lambda ) u = ( - \Delta + \lambda ) u_0 \eqno 2.24
$$
\par \vskip 8 pt \noindent
We conclude this Section giving some details of the presentation of the
Schroedinger operator for $N$ identical quantum particles satisfying Fermi
statistics as an operator on a space of symmetric functions. It could be called
pictorially ''bosonization of free nonrelativistic fermions''
\par \noindent
We shall treat for simplicity only the case $N=2$.
\par \noindent
One starts
with the operator
$$
\Delta =
\Delta_1 + \Delta_2, \qquad \Delta_k \equiv \sum_{i=1,2} { \partial ^2 \over
\partial (x^k_i)^2 }
$$
defined on smooth functions on the open set
$$
\Omega
\equiv \{ x: \; x^2_2 > x^1_2 \}
$$
Introducing the new coordinates $ y,\; z $ defined by
$$
y \equiv x^2 -
x^1, \qquad z \equiv { 1 \over 2Ê} (x^1 + x^2)
$$
one has
$$
\Delta = \Delta_z
+ \Delta_y \quad on \quad \{ y,z: \; y_2 > 0 \}
$$
The operator $ \Delta_z$ is essentially
selfadjoint in the given domain and its closure is represented by the energy
form on $R^2.$ We shall then restrict attention to $ \Delta_y $, which is the
Laplacian defined on smooth functions on the upper half plane $ \{y :\; y_2 > 0 \}.$
\par \noindent
Here $ \bar \Sigma \equiv \{ y\; : \; y_2 = 0 \} $
and we are looking for an extension
with the property that every function in its form domain belongs to
$H^1 ( R^2 - \bar \Sigma)$ and is such that it has opposite traces on opposite
sides of $ \Sigma.$ We describe this by requiring
$$
f(- y_1,0_-) = -
f(y_1,0_+), \quad y_1 \in R^+ \eqno 2.25
$$
The isometry corresponding to bosonization
extends to a map from smooth functions $f$ in the open upper half plane, which
have a limit (in $ H^{1/2} (R)) $ when $ y_2 \to 0_+$ to smooth functions
$ \tilde f $ on $ R^2 - \Sigma $ which have a limit from both
sides and the two limits sum to zero:
$$
lim_{y_2 \to 0_+} \tilde f + lim_{y_2 \to 0_-} \tilde f = 0 \eqno 2.26
$$
The problem of finding the selfadjoint extension which corresponds to fermions
is now posed as the problem of finding the self-adjoint extension of $ \Delta^0$
which has in its domain functions which satisfy (2.25).
\par \noindent
Here we have denoted by $\Delta^0$ the Laplacian as symmetric operator
defined on smooth functions in $R^2$ with support which is compact and does not
intersect the ''horizontal'' axis $ \{y: \; y_2 = 0 \}.$
\par \vskip 3 pt \noindent
According to (2.7) a function in the form domain of any of these self-adjoint
extensions can be written
(for $\lambda$ sufficiently large) as
$$
u(x) = u^0_{\lambda} (x) + \hat G_{\lambda} * \mu (x), \quad x \in
R^2 - \bar \Sigma \eqno 2.27
$$
The symmetric function $ u^0 $ belongs to $ H^1( R^2 - \bar \Sigma)$
(and therefore vanishes on $ \bar \Sigma), $ while the function $
\mu \in H^{1/2}(R) $ is symmetric.
\par \noindent
In fact, every symmetric function in $ H^1 (R^2) $ can be represented (and
then uniquely) as in (2.27).
\par \noindent
As a special case of (2.19) the energy
form $ Q(u) $ reads on functions represented as in (2.27), with the required
symmetry
$$
Q(u) = \left[ ( \nabla u^0_{\lambda}, \nabla u^0_{\lambda}) -
\lambda (u,u) + \lambda (u^0_{\lambda}, u^0_{\lambda}) \right]
+ \Phi_{\lambda} (\mu) \eqno 2.28
$$
$$
\Phi_{\lambda} (\mu) \equiv { 1 \over 4 \pi} \int_R \sqrt { p^2 + \lambda }
\; | \hat \mu (p) |^2 dp \eqno 2.29
$$
\par \vskip 4 pt \noindent
The selfadjoint operator defined by the quadratic form $Q$ is unitarily
equivalent to the laplacian defined on antisymmetric functions on $R^2.$
\par \noindent
The unitary transformation is defined by
$$
f'(x) = f(x), \;\; x_2 > 0, \qquad f'(x) = - f(x), \;\; x_2 < 0
$$
Notice that, due to the symmetry requirement we have made on the function
$\mu,$ the image of $ \hat G * \mu $ is continuous at $\Sigma$ and in
fact a function in $ H^1 (R^2).$
\par \vskip 8 pt \noindent
3. $ { \bf THE \; AHARANOV-BOHM \; HAMILTONIAN } $
\par \vskip 3 pt \noindent
We exploit here the method of layer potentials to analyse the Aharanov-Bohm
hamiltonian describing the interaction of a charged quantum particle on a plane
$ \Pi $, interacting with $ M$ thin solenoids which are orthogonal to the plane.
We identify the plane with $ R^2$ and denote by $ y \equiv \{ y_1,\; y_2 \} $
the coordinates relative to a fixed reference frame.
\par \noindent
Let $ y^1,\ldots y^M $ the coordinates of the intersections of
the solenoids with $ \Pi$ and let $ \alpha_k$ be the magnetic flux associated
to the $ k^{th}$ solenoid.
\par \noindent
To simplify notation, we choose unit
for which $ \hbar = c = e = 2m =1 $ where $e$ is the charge of the particle and
$ m$ is its mass.
\par \noindent
We agree to choose an ordering (and a reference
frame) such that
$$
y^k_2 < y^{k+1}_2, \qquad 1 \leq k \leq M-1
$$
and for each point consider the half-line
$$
\gamma_k \equiv \{ x \in R^2| \; x_1 > y^k_1, \; x_2 = y_2^k \} \eqno 3.1
$$
A singular gauge transformation maps the formal hamiltonian for a particle
interacting with $M$ solenoids into the free laplacian in
$$
\Xi \equiv R^2 - \cup_k \gamma_k
$$
with the following boundary conditions on each $ \gamma_k:$
$$
u_k^- = e^{i \alpha_k} u_k^+, \qquad
\left( { \partial u \over \partial x_2 } \right)^-_k = e^{i \alpha_k}
\left( { \partial u \over \partial x_2 } \right)^+_k \eqno 3.2
$$
where $\alpha_k$ are the preassigned magnetic fluxes and for a function $ g(x) $
we have denoted by $ g_k^+$
the trace on $ \gamma_k $ from above and by
$ g_k^-$ the trace on $ \gamma_k $ from below (see e.g. [S.1],[R]).
\par \vskip 3 pt \noindent
In order to describe a self-adjoint realization
in $ L^2 (R^2) $ of this formal operator, we
introduce the quadratic form $F$
defined on complex-valued functions as follows
$$
D(F) \equiv \{ u \in L^2 (R^2) | \;
u \in H^1 (\Xi), \;\; u_k^- = e^{i\alpha_k} u_k^+ \} \eqno 3.3
$$
$$
F(u) \equiv \int_{\Xi} | \nabla u|^2 dx \eqno 3.4
$$
This form is obviously closed and positive,
and defines a positive self-adjoint operator $H$ which is by definition the
hamiltonian of a charged particle in $R^2$ interacting with $M$ solenoids
orthogonal to the plane.
\par \noindent
In the remaining part of
this section we
shall give a rather detailed study of this form and of the associated operator,
as well as a fairly explicit form of its resolvent.
\par \noindent
Our results
generalize those of Stovicek [S.1] where the same hamiltonian is analysed in the
case when the points $ y^k$ are colinear.
\par \noindent
The analysis parallels
to a large extent the one given in the previous section for the case of $N$
plektons, with some simplifying features which allow for a more explicit
description. \par \noindent One has then
\par \vskip 4 pt \noindent
\it PROPOSITION 3.1 \rm
\par
\it For $ \lambda $ sufficiently large one has
$$
D(F) \equiv \{ u \in L^2 (R^2)| \; u = v + \sum_k \hat G_{\lambda} *
\mu_k, \quad v \in H^1 (R^2), \;\; \mu_k \in H^{1/2} (\gamma_k),
$$
$$
{ i \mu_k \over 2} cotg { \alpha_k \over 2 } -
\sum_{ j \not= k } \left( \hat G_{\lambda} * \mu_j \right)_k =
v_k \} \eqno 3.5
$$
and, on its domain, the quadratic form $F$ is given by
$$
F(u) \equiv \int_{R^2} \left( | \nabla v |^2 + \lambda |v|^2 -
\lambda |u|^2 \right) dx
$$
$$
- \sum_k \int_{\gamma_k} \bar \mu_k (z)
{ \partial \over \partial n } \left( \hat G_{\lambda} * \mu_k \right)_k (z)
dz - \sum_{j \not= k } \int_{\gamma_k} \bar \mu_k (z) \left( { \partial
\over \partial n }\hat
G_{\lambda}* \mu_j \right)_k (z) dz \eqno 3.6
$$
where $n$ is the normal
at $ \gamma_k$ oriented to increasing values of $ y_2.$ \rm
\par \vskip 3 pt \noindent
\it Proof \rm
\par
Let $u$ belong to the domain of $F.$ Then $u \in H^1 (\Xi)$ and,
making use of the discontinuity properties of $ \hat G_{\lambda} *\mu_k $ at $
\gamma_k $ one computes
$$
u_k^- - e^{i \alpha_k} u_k^+ = v_k + \sum_{j \not= k} \left( \hat G_{\lambda}
* \mu_j \right)_k - { \mu_k \over 2} - e^{i \alpha_k} \left[ v_k +
\sum_{j \not=
k} \left( \hat G_{\lambda} * \mu_j \right)_k + { \mu_k \over 2 } \right]
$$
$$
= (1 - e^{i \alpha_k} ) \left[ v_k - {i \over 2} \mu_k cotg ({ \alpha_k \over
2}) + \sum_{j \not= k} \left( \hat G_{\lambda} * \mu_j \right)_k \right] = 0
$$
Conversely, given $ u \in D(F)$ define
$$
\mu_k \equiv u_k^+ - u_k^-, \qquad v \equiv u -
\sum_k \hat G_{\lambda} * \mu_k \eqno 3.7
$$
Then $ \mu_k \in H^{1/2} (\gamma_k) $ and moreover,
by direct computation, $ v_k^+ = v_k^-$ which implies $ v \in H^1 (R^2).$
\par \noindent
Still by direct inspection one verifies that the
relation between $ \mu_k$ and $ v_k $ given in (3.5) is satisfied.
\par \noindent
>From (3.4) and (3.7), integrating by
parts and using the discontinuity properties of
double layer potentials one
derives the expression for $F(u)$ given in Proposition 3.1. $ \qquad
\diamondsuit$ \par \vskip 5 pt
\noindent
\it REMARK 3.2 \rm
\par
\it In the case of only one solenoid the expression
for the form $F$ and for its domain $D(F)$ can be simplified.
\par \noindent
>From (3.5) one has (setting $ \mu_1 \equiv \mu, \ldots)$
$$
\mu = - 2i v \; tg ({\alpha \over 2})
$$
and, substituting in (3.6)
$$
D(F) = \{ u \in L^2(R^2)\; | \; u = v - 2 i \; tg ( {\alpha \over 2} ) \; \hat
G_{\lambda}* v, \;\;\; v \in H^1 (R^2) \}
$$
$$
F(u) =\int_{R^2} \left( | \nabla v |^2 + \lambda |v|^2 - \lambda |u|^2
\right) dx - 4 \; tg^2 \left({ \alpha \over 2}\right) \int_{\gamma} \bar v (z)
\left( { \partial \over \partial n} \hat G_{\lambda}
* v \right) (z) dz
$$
$$
= \int_{R^2} \left( | \nabla v |^2 + \lambda |v|^2 - \lambda |u|^2
\right) dx - 4 \; tg^2 \left({ \alpha \over 2} \right) \int _R \sqrt{ k^2 +
\lambda} | \tilde v (k) |^2 dk
$$
where $ \tilde v $ is the Fourier transform of $v.$
\par \vskip 3 pt \noindent
In the present context, in order to describe the operator and its
domain it turns out to be more convenient to introduce, in addition to the
double-layer potentials, also the single-layer potentials associated to charge
distributions $\sigma_k$ on $ \gamma_k $, defined by
$$
\left( \tilde G_{\lambda} * \sigma_k \right)
(x) \equiv \int_{\gamma_k} G_{\lambda} (x-z) \sigma_k (z) dz
$$
In this notation one has
\par \vskip 5 pt \noindent
\it PROPOSITION 3.3 \rm
\par
\it
For $ \lambda $ sufficiently large the domain of
the operator $H$ associated to the form $F(u)$ is given by
$$
D(H) = \{ u \in L^2 (R^2)\; | \; u = w + \sum_k \tilde G_{\lambda} * \sigma_k +
\sum_k \hat G_{\lambda} * \mu_k \} \eqno 3.8
$$
where
$$
w \in H^2 (R^2)\cap H^1 (\Xi ), \qquad
\sigma_k\in H^{1/2} (\gamma_k), \qquad \mu_k \in H^{3/2} (\gamma_k)
$$
$$
{ i \over 2
} \mu_k cotg \left({ \alpha_k \over 2} \right) - \sum_{j \not= k} \left( \hat
G_{\lambda} * \mu_j \right)_k - \sum_j \left( \tilde G_{\lambda} *
\sigma_j\right)_k = 0
$$
$$
\quad { i \over 2 } \sigma_k cotg { \alpha_k \over 2} -
\sum_{j \not= k} \left( { \partial \over \partial n} \tilde G_{\lambda}
* \sigma_j \right)_k - \sum_j \left( { \partial \over \partial n} \hat
G_{\lambda} \mu_j \right)_k = \left({ \partial w \over
\partial n }\right)_k
$$
(recall that elements in $ H^1 (\Xi )$ have zero trace on all $ \gamma_k).$
\par \noindent
On $D(H)$ the action of $H$ is given by \rm
$$
( H + \lambda) u = ( - \Delta + \lambda) w \eqno 3.9
$$
The proof follows the same scheme outlined in the previous section
and will be omitted.
\par \vskip 4 pt \noindent
\it REMARK 3.4 \rm
\par
\it One can rephrase Proposition 3.1, making use of both single and
double layer potentials, so that the condition on the domain coincides with the
first condition in Proposition 3.3. \rm
\par \vskip 4 pt \noindent
We conclude this section giving a rather explicit expression for the resolvent.
We point out that a good representation for the resolvent is the starting point
for a detailed analysis of the spectrum and for scattering theory. We plan to
come back to these problems making use of the formalism developped here.
\par \vskip 3 pt \noindent
For $ f \in L^2(R^2) $ we want to construct $ (H + \lambda \; I )^{-1} \; f $
for $\lambda $ sufficiently large.
\par \noindent
This is equivalent to solving the following boundary value problem
$$
( - \Delta + \lambda \; I ) u = f\qquad on \; \; R^2 - \cup_k \gamma_k \eqno
3.10
$$
$$
u^-_k = e^{i \alpha_k} u_k^+, \qquad
\left( { \partial u \over \partial n } \right)^-_k= e^{i \alpha_k} \left(
{ \partial u \over \partial n } \right)_k^+ \eqno 3.11
$$
Taking into account Proposition 3.2 it is natural to represent the solution of
(3.10), (3.11) as
$$
u = G_{\lambda} * f + \sum_k \tilde G_{\lambda} * \sigma_k + \sum_k \hat
G_{\lambda} * \mu_k \eqno 3.12
$$
\par \vskip 3 pt \noindent
\it REMARK 3.5 \rm
\par
\it Strictly speaking, this prescription is different from the one used in
Proposition 3.1, since the first term does not vanish on $ \cup_k \gamma_k.$
The representation we use now is more convenient in this context than the
previous one, and they only differ for a different definition of the charge
distributions. \rm
\par \vskip 4 pt \noindent
If one imposes the boundary conditions (3.11) in (3.12) one obtains a system of
equations for the charge distributions
$$
{ i \over 2} \mu_k cotg \left({ \alpha_k \over 2 } \right) - \sum _{j \not= k}
\left( \hat G_{\lambda} * \mu_j \right)_k - \sum_j \left( \tilde G_{\lambda} *
\sigma_j \right)_k = \left( G_{\lambda} * f \right)_k \eqno 3.13
$$
$$
-{ i \over 2} \sigma_k cotg \left({ \alpha_k \over 2 }\right) - \sum _{j \not= k}
\left( { \partial \over \partial n }\tilde G_{\lambda} * \sigma_j \right)_k -
\sum_j \left( { \partial \over \partial n }\hat G_{\lambda} * \mu_j \right)_k =
\left({ \partial G_{\lambda} * f \over \partial n } \right)_k \eqno 3.14
$$
We now prove
\par \vskip 4 pt \noindent
\it PROPOSITION 3.6 \rm
\par
\it For $ \lambda$ sufficiently large the system (3.13), (3.14) has a unique
solution \rm
$$
\mu_k \in H^{3/2} (\gamma_k), \qquad \sigma_k \in H^{1/2} (\gamma_k) \eqno 3.15
$$
\par \vskip 3 pt \noindent
\it Proof \rm
\par
Observe that, since $ G_{\lambda} * f \in H^2 (R^2),$ the r.h.sides of (3.13),
(3.14) have the regularity described in (3.8).
\par \noindent
We rewrite (3.13),(3.14) as
$$
\sum_j \Lambda^D _{k,j} ( \mu_j, \sigma_j ) + \sum_j \Lambda^{ND}_{k,j} (\mu_j,
\sigma_j) =
\left( ( G_{\lambda} * f )_k , ({\partial \over
\partial n} G_{\lambda} * f )_k \right) \eqno 3.16
$$
where
$$
\sum_j \Lambda^D_{k,j} (\mu_j, \sigma_j ) = \left( { i \over 2} \mu_k cotg
\left({ \alpha_k \over 2} \right) - ( \tilde G_{\lambda} * \sigma_k )_k, -{ i
\over 2} \sigma_k cotg \left({ \alpha_k \over 2 } \right)- ( \hat G_{\lambda} *
\mu_k )_k \right) \eqno 3.17
$$
$$
\sum_j \lambda^{ND}_{k,j} (\mu_j, \sigma_j ) =
$$
$$
\left( - \sum_{j \not= k} \left[ ( \hat G_{\lambda} * \mu_j )_k +
(\tilde G_{\lambda} * \sigma_j )_k \right] \;,\;
- \sum_{j \not= k} \left[ ( { \partial \over \partial
n } \tilde G_{\lambda} * \sigma_j )_k + ( { \partial \over
\partial n } \hat G_{\lambda} * \mu_j)_k \right] \right) \eqno 3.18
$$
We prove first that the operator $ \Lambda^D $ is a bijection, i.e. the
following system has a unique solution
$$
{ i \over 2} \mu_k cotg \left( { \alpha_k \over 2}\right) -
\left( \tilde G_{\lambda} *
\sigma_k \right)_k = \xi_k,
\; \; -{ i \over 2} \sigma_k cotg { \alpha_k \over 2} - \left( {\partial
\over \partial n } \hat G_{\lambda} * \mu_k \right)_k = \zeta_k \eqno 3.19
$$
where
$$
\xi_k \in H^{3/2} (\gamma_k), \qquad \zeta_k \in H^{1/2} (\gamma_k)
$$
Operating with
$ G_{\lambda} $ on the second equation and substituting in the first, and making
use of the identity
$$
- \left[ \tilde G_{\lambda} * ( { \partial \over
\partial n }\hat G_{\lambda} * \mu_k ) \right]_k = { \mu_k \over 4 } \eqno 3.20
$$
(see e.g. [K]) one finds as unique solution
$$
\mu_k = { - { i \over 2} \xi_ k cotg \left({ \alpha_k \over 2 } \right)+ (
\tilde G_{\lambda} * \zeta_k )_k \over
{ 1 \over 4} (1 + cotg^2 \left({ \alpha_k \over 2} \right)) } \in H^{3/2}
(\gamma_k) \eqno 3.21
$$
$$
\sigma_k = { { i \over 2} \zeta_ k cotg \left( { \alpha_k \over 2 } \right)
+ ( {
\partial \over \partial n }\hat G_{\lambda} * \xi_k)_k \over
{ 1 \over 4} (1 + cotg^2 \left({ \alpha_k \over 2}\right)) } \in H^{1/2}
(\gamma_k) \eqno 3.22
$$
The off-diagonal term $ \Lambda^{ND} $ is easily seen to be a bounded operator
in the spaces indicated, with norm decreasing to zero when $ \lambda \to
\infty.$
\par \noindent
This concludes the proof of Proposition 3.6. $ \qquad \diamondsuit$
\par \vskip 4 pt \noindent
\it REMARK 3.7 \rm
\par
\it In case $M=1$ the off diagonal operator is not present and the above
procedure leads to the explicit solutions of the equations for $ \mu$ and
$\sigma$ and thus to a completely explicit formula for the resolvent:
$$
(H + \lambda \; I)^{-1} f = G_{\lambda} * f + \tilde G_{\lambda} * \sigma + \hat
G_{\lambda} * \mu \eqno 3.23
$$
where $ \mu$ and $\sigma$ are given in (3.21) and (3.22).
\par \vskip 12 pt \noindent
\it ACKNOWLEDGMENTS \rm
\par
This work was completed while one of the authors (GFDA) was a visitor to the
Depts. of Mathematics and Physics of New York University. The warm ospitality of
these Institutions and a partial support are gratefully acknowledged.
\par
\vskip 1 cm
\noindent
\it REFERENCES \rm
\par \vskip 3 pt \noindent
[A] \it R.A.Adams \rm
\par
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\par \vskip 3 pt \noindent
[AB] \it Y.Aharanov, D.Bohm \rm
\par
Phys. Rev. ${ \bf 115 }$, 485-491 (1959)
\par \vskip 3 pt \noindent
[BCMS] \it G.Baker, G.Canright, S.Mulay, C.Sundberg \rm
\par
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[GFT] \it G.F.Dell'Antonio, R.Figari, A.Teta \rm
\par
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[GS] \it G.Goldin, D.H.Sharp \rm
\par
Phys. Rev. D ${ \bf 28}$,830-832 (1983)
\par \vskip 3 pt \noindent
[K] \it R.Kress \rm
\par
LINEAR INTEGRAL EQUATIONS, Springer Verlag Berlin Heidelberg 1989
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[LM] \it J.M.Leinas, J. Myrheim \rm
\par
Nuovo Cimento ${ \bf 37 B}$, 1-23, 1977
\par \vskip 3 pt \noindent
[MS] \it G.Mund, R.Schrader \rm
\par
"Advances in Dynamical Systems and Quantum
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\par
Capri 1993, S.Albeverio, R.Figari, E.Orlandi, A.Teta eds, World Scientific 1995
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[MT] \it C.Manuel, R.Tarrach\rm
\par
Phys.Letters B ${ \bf 268 }$,222-226 (1991)
\par \vskip 3 pt \noindent
[R] \it S.N.M.Ruijsenaars \rm
\par
Annals of Physics ${\bf 146}$, 1-34 (1983)
\par \vskip 3 pt \noindent
[S.1] \it P.Stovicek \rm
\par
Duke Mathematical Journal, ${ \bf 76}$, 303-332 (1994)
\par \vskip 3 pt \noindent
[S.2] \it P. Stovichek \rm
\par
Proceedings of the Workshop on Singular Schroedinger Operators
\par
Trieste 1994, Preprint ILAS/FM 16/1995
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[W] \it F.Wilczek \rm
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Phys. Rev. Letters ${ \bf 49}$, 957-1149 (1982)
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[Wu] \it Y.S.Wu \rm
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Phys. Rev. Letters ${ \bf 53}$,11-114 (1984)
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