\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
$$
\eta_N \equiv \cup_{k<h}   \zeta_N^{k,h}, \qquad \zeta_N^{k,h} \equiv
\{ x| x^k_2 - x^h_2 =0, \; x^k_1 < x^h_1 \}  \eqno 1.11
$$
and set 
$$ 
Z_N \equiv (R^2)^N - \bar \eta_N
$$ 
Denote by $ \tilde Z_N$  its Riemann covering. Then one can 
verify that $ \tilde Y_N$ is diffeomorphic to the quotient of 
$ \tilde  Z_N$ by $S_N$ and in  this way
the braid group $B_N$  is identified with the fundamental group of  
$ \tilde Z_N.$ 
\par \noindent
We shall denote by $g \to \pi_2(g)$ this identification.
\par \noindent
While this presentation of $\tilde Y_N$ is not 'canonical'' 
(it depends on the arbitrary choice of $\eta_N)$ it leads to a 
concrete construction of the Schroedinger operator. 
\par \noindent
One considers the Hilbert space completion of functions on 
$ \tilde Z _N$ which satisfy the  equivariance requirement 
$$
\phi (\pi_2(g) \tilde z) = \pi_1 (g) \phi (\tilde z), 
\qquad \tilde z \in \tilde Z_N  \eqno 1.12
$$
To set a unitary correspondence with the Hilbert space $ {\cal H}$ 
one must now find a bijection between functions on $ \tilde Y_N   $ 
and a linear subset of functions on $ \tilde Z_N.$ 
\par \noindent
Once this is done, the definition  of 
a second order differential operator presents no problems. 
The proof that its restriction to functions satisfying (1.12) defines a
self-adjoint operator is less simple. This is clear from the fact that
$Z_N$  is  $(R^2)^N$  with a subset of codimension one  removed.  
The definiton of $ \Delta$ as a selfadjoint operator on $\tilde Z_N$   requires
the specification of boundary conditions on $ \eta_N.$
\par \vskip 2 pt \noindent
To construct a bijection, one can extend to $ Z_N $ and then 
to $\tilde Z_N$ the functions defined in $\Omega_N$ and for which (1.9) is
satisfied.  
\par  \noindent
Any choice of extension is in principle legitimate, under some unitarity
conditions.  One can for example extend every function from $ \Omega_N$ to a
dense subset of $Z_N$ by setting $ f( \sigma (x) ) = f(x) $ for every element
$ \sigma$ of the permutation group. In this way one obtains functions which have
discontinuities on $ \eta_N$ and the definiton of the laplacian requires 
boundary conditions on subsets of $ \eta_N.$  
\par \noindent 
Notice that this is the   way
in which  Bose  statistics is usually introduced, with the representation
$\pi_1$  chosen to be $ \pi_1 (g) = 1 \;\;\forall g \in B_N.$  
\par \noindent
To give an example, we shall discuss at the end of the next Section the
presentation  of Fermi statistics which one obtains if one chooses the extension
given by    
$$  f(\sigma (x)) = f(x) 
\qquad \forall \; \sigma \in S_N  \qquad x \in \Omega_N \eqno 1.13  
$$ 
The choice (1.13) provides a
presentation in which the Hilbert space is made of symmetric functions, 
but there are functions in the form domain of the Schroedinger operator which
are not in $H^1((R^2)^N).$ 
\par \vskip 3 pt \noindent
In the abelian case (anyons) there is a natural construction which 
associates to continuous functions on $ \tilde Y_N$ continuous functions on $
\tilde Z_N .$ \par \noindent
Let $w (x)$ be a  differentiable function on $ Z_N $ which extends by equivariance (for the abelian  representation $\pi_1 )$ to the covering space $ \tilde Z_N $
as a differentiable function and satisfies
$$
w (\pi_2 (g) \tilde x) = (\omega (g) )^{-1} w (\tilde x),  \quad \tilde x \in  \tilde Z_N  \eqno 1.14
$$
where we have set
$$
S^1 \ni \omega (g) \equiv \pi_1 (g), \quad g \in B_N 
$$
to stress the fact that $ \pi_1$ is  abelian, and therefore $ \pi_1(g)$ is a 
phase factor, in view of unitarity (if the representation is not irreducible, 
one can consider separately the irreducible components).
\par \noindent
A function $w$ with the property (1.14) can be constructed by
exploiting the identification of $R^2$  with the complex plane and choosing 
for $w$ a suitable meromorphic function.
\par \noindent
Then the map  
$$ 
W \; : \; f \to  f \; w \eqno 1.15
$$
takes continuous functions on $ \tilde Z_N$ which satisfy (1.12) to
continuous functions on  $\tilde Z_N $ which transform under the identity 
representation of the fundamental group (and correspond therefore to continuous
functions on $ Z_N).$  
\par \noindent
Since $ |w(x)|=1 $ the map (1.15) provides an isometry.
\par \noindent
In deriving the  transformation properties of  $  f \cdot w $  essential use is
made of the fact that the representation $ \pi_1 $ of $B_N$ is abelian. A direct
extension of this construction to the nonabelian case does not seem to be
possible. 
\par \vskip 2 pt \noindent 
One can now choose the natural bijection
between continuous symmetric  functions on $ (R^2)^N$ and their restriction to
the closure of $ \Omega_N.$ 
\par \noindent
This leads to consider the linear subspace of  functions $ f $ on $ Z_N$ 
which are of the form  
$$
f(x) = F(x) w(x), \qquad F(x) = F(\sigma (x) )  \eqno 1.16
$$ 
where  $\sigma$ is any element of the permutation group $S_N.$
\par \noindent
Notice that for this class of functions, the restriction to 
$ \bar \Omega_N $ satisfies
$$
f(\pi_0 (x)) = \omega ( g_k )^{1/2} f(x) =  \omega (g_k^{1/2} ) f(x), \qquad x
\in \Sigma _N^{k,+} 
$$
(also here we have used the fact that the representation is abelian) 
and therefore the statistics defined by this choice of extension corresponds, in
the notations we have used previously, to the representation $ g_k \to \pi_1
(g_k) ^{1/2}.$ 
\par \noindent
One can now proceed to define the Schroedinger operator on the linear space 
of functions on $Z_N$ which satisfy (1.16).
\par \vskip 3 pt  \noindent
A possible approach is to define it through its quadratic form, and describe 
the  domain as the direct sum of $H^1_0 (Z_N)$ and of potentials of charges
concentrated at $ \eta_N.$ This approach has the advantage that an explicit
description can be given of  domains of selfadjointness. 
\par  \noindent 
Moreover this construction  closely parallels the description of a 
quantum particle in the field generated by several solenoids perpendicular to a
fixed plane to which the particle is restricted (Aharanov-Bohm effect);  we
shall briefly discuss this in Section 3. 
\par \noindent
As we shall see there, in the Aharanov-Bohm problem the role of the set $ Z_N$
is  taken by  
$$
R^2 - \gamma \qquad \gamma \equiv  \cup_{j=1,\ldots N}  \{ x \in R^2 \; | \; 
x_2 - y_2^{j} \in R^+ \}   
$$
where $y^{j}$ are the positions of the $N$ solenoids (more precisely, the
intersections of the $N$ solenoids with the plane).     
\par \vskip 3 pt \noindent  
For example, if only one solenoid were present at the point $y,$ 
the self-adjoint Schroedinger operator corresponding to  the presence of the
solenoid is defined by  requiring that the functions in its form domain 
satisfy at  $\gamma$ the boundary condition  
$$
f^- = e^{i e \alpha/ \hbar  c} f^+ \eqno 1.17
$$
where $e$ is the electric charge of the particle, $ \alpha $ is the ''strength''
of the solenoid (the total flux through any domain which contains $y$ in its
interior), $ f^{\pm} $ are the traces of $f$ on the two sides of $\gamma.$  
\par \noindent 
This is a ''local condition'', and is treated by the introduction
of a dipole density on $ \gamma,$ as we will explain in Section 3. 
\par \noindent
In the case of a particle interacting with $N$ solenoids this procedure allows
to give a rather explicit characterization of the domain of the hamiltonian and
of the resolvent.  
\par \noindent 
We point out here that the use of boundary
conditions to study the dynamics of a quantum particle in a plane interactiong
with $N$ point solenoids has been  suggested by P.Stovichek in [S.1] for 
the case
when the $N$ points $y_k$ are colinear.  
\par \vskip 3 pt \noindent 
An alternative way to define the Schroedinger operator starting from the
condition (1.16) on functions in its domain proceeds as follows. 
\par \noindent
Let $ \Delta_0 $ be the restriction of $ \Delta$ on $ C^2 (\tilde Z_N) $ to
functions which have support disjoint from the image of $\bar \eta_N$ in $\tilde
Z_N.$  
\par \noindent 
The map $W$ takes these functions to functions which are
twice differentiable on $Z_N$ and have support disjoint from $\bar \eta_N.$ We
denote by ${ \bf F}_N $ this class of functions. 
\par \noindent 
Under $W$ the
image of the operator $ - \Delta $ is  
$$ 
K \equiv - \Delta + 2 \alpha \cdot \nabla + \beta,
\quad \alpha (x) \equiv w^{-1} \nabla w, \quad \beta (x) = 
w^{-1} \Delta w   \eqno 1.19
$$
a symmetric operator defined on ${ \bf F}_N, $  a dense subset of $
L^2 ( (R^2)^N,  dx) $  
\par \noindent 
Notice that the coefficients $\alpha (x) $ and $\beta (x)$
are singular on $ \eta_N$ so that the hamiltonian is now a 
singular perturbation of the laplacian. 
\par \noindent
The proof that  $K$ defined in (1.19) is essentially selfadjoint on 
${ \bf F}_N$ presents now serious difficulties, and to our knowledge it has
been carried out in detail only in the case when there is an additional  hard
core potential, and the particles are confined to a compact subset of $R^2$  so
that the resulting operator has a compact resolvent and therefore pure point
spectrum [BCMS].
\par \vskip 8 pt \noindent 
2. $ { \bf  QUADRATIC \; FORMS \; FOR \; PLEKTONS } $
\par \vskip 3 pt \noindent
In this section we shall give the construction of the quadratic form 
corresponding to the free Schroedinger operator for plektons. 
Addition of a potential which is relatively small in form requires no
essential  additional work.
\par \noindent
Consider $N$ plektons in $R^2.$ As configuration space we choose $ \Omega_N $
as described in (1.7), Section 1.
\par \noindent
The form domain will be a subset of $ H^1 ( \bar \Omega_N, V)$, the collection 
of 
distributions with support in the closure of $ \Omega_N$ and with 
square-integrable gradient; we denote by $\| \cdot \|_1$ the corresponding norm.
\par \vskip 3 pt \noindent
\it From now on we shall omit the index $N$ in $\Omega_N $ and   
$ \Sigma_N $ (for their definition see (1.8)).\rm
\par \vskip 3 pt \noindent
Any function $f$ in  $ H^1 (\bar \Omega, V) $ has a trace $F_f$ on 
$ \Sigma $ and if
$ f_1 $ and $ f_2 $ have the same trace, then 
$ f_1 - f_2 \in  H^1_0 ( \bar \Omega, V) $
\par \noindent
Moreover (see e.g.[A]) $ F_f \in H^{ 1/2} (\Sigma, V); $ we denote by $\| \cdot
\|_{1/2}$ the norm for this space. 
\par \noindent
Conversely, for each $ k=1, \ldots N $ let $ \mu_k $ be a $ V -$ 
valued function
on  $ \Sigma^k $ of class $ H^{ 1/2}$ and consider the $V-$valued 
function  defined in $
\Omega $ as the double-layer potential due to the distributions $\mu_k$ (of
''dipole charges'').
$$
(\hat G_{\lambda} * \mu) (\xi) \equiv \sum_k \int \hat G_{\lambda} (\xi-y)
\mu_k (y) \delta (y^k_2 - y^{k+1}_2) d^{2N }y \eqno 2.1
$$
where $ \hat G_{\lambda} $ is the double-layer Green function with
parameter $\lambda,$ defined by
$$
\hat G_{\lambda} (x-y )|_{y \in \Sigma^k} \equiv { \partial G_{\lambda} 
\over \partial n(y)} (x-y)
$$
where $ n(y)$ is the normal to $ \Sigma^k$ at $y$ directed toward $\Omega$ and
$$
G_{\lambda} (x-y) \equiv ( - \Delta + \lambda I)^{-1} (x-y)  \eqno 2.2
$$
\par \noindent
Due to the properties of the double-layer potential (see e.g. [K])
the function $ \hat G _{\lambda}* \mu $ belongs 
to $ H^1 (\bar \Omega,V); $ we denote by
$   \Gamma_{\lambda}  \mu $  its trace on $
\Sigma, $ a function in $ H^{1/2} (\Sigma,V)$,
$$
\Gamma _{\lambda} \mu \equiv  \left( \hat G _{\lambda}* \mu
\right)|_{\Sigma }
$$
and by $ \Gamma_{\lambda}^k \mu $  the restriction of 
$ \Gamma_{\lambda}  \mu $ to $\Sigma^k.$ 
\par \noindent
By construction $ \Gamma_{\lambda} $ is an integral operator with kernel
$$
\Gamma_{\lambda} (\xi, \eta) \equiv \hat G_{\lambda} (\xi - \eta), \qquad \xi,
\; \eta  \in \Sigma \eqno 2.3
$$
\par \vskip 3 pt \noindent
We prove now 
\par \vskip 5 pt \noindent
\it PROPOSITION 2.1 \rm
\par
\it  For $ \lambda $ sufficiently large (depending only on $N)$ 
the  operator $ \Gamma_{\lambda} $ is bounded symmetric and invertible on 
$ H^{1/2} (\Sigma,V).$  \rm
\par \vskip 3 pt \noindent
\it Proof \rm
\par
By the properties of the double-layer potential one has
$$
\Gamma_{\lambda}  \mu = { I \over 2} \mu + R_{\lambda}  \mu   \eqno 2.4 
$$
where $ R_{\lambda} $ is an integral operator with matrix kernel  
$R_{\lambda}^{k,h}, \; \; k, h  =  1, \ldots N  $ given by
$$
R_{\lambda}^{k,k} (\xi, \eta) =  0 \quad  \xi, \; \eta \in \Sigma^k  
$$
$$
R_{\lambda}^{k,h} (\xi, \eta) = {\partial \over \partial n_k (\xi)} {\partial
\over \partial n_h (\eta) } G_{\lambda} (\xi - \eta)  \quad  \xi \in \Sigma^k, \;
\; \eta \in \Sigma^h, \; \; h \not= k. 
$$ 
\par \noindent
If 
$$ 
\mu,\;  \mu' \in H^{1/2} (\Sigma,V) \quad \mu_{\Sigma^k} \equiv \mu_k 
$$
one has, denoting by $ \left( \cdot, \cdot \right)_{\Sigma}$ the scalar product
for $L^2 (\Sigma,V) $ (with
respect to Lebesgue measure) 
$$
\left( \mu', R_{\lambda} \mu \right)_{\Sigma} = \sum_{k \not= h} \int_{\Sigma^k}
d \xi \int_{\Sigma_h} d \eta \; \bar \mu'_k (\xi) \mu_h (\eta) {\partial
\over \partial n_k (\xi) } { \partial \over \partial n_h (\eta) }  G_{\lambda}
(\xi - \eta)  \eqno 2.5 
$$ 
Using the definition of $G_{\lambda}(\xi - \eta)$ and Schwartz's
inequality the $k,h$ term in the sum (2.5) can be estimated to be smaller in
absolute value than  $$  c_{\lambda} \| \mu'_h \|_{1/2} \| \mu_k \|_{1/2}
$$ 
where 
$$
lim_{\lambda \to \infty} c_{\lambda} = 0
$$ 
so that
$$
\| R_{\lambda} \| \leq (N-1) c_{\lambda}  \eqno 2.6
$$
Therefore for $ \lambda $ sufficiently large  $\Gamma_{\lambda} $ is 
(bounded and) invertible. $\qquad \diamondsuit $
\par \vskip 5 pt\noindent
It follows from Proposition 2.1 that one can find $\lambda_0 (N) $ such that for $ \lambda > \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
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\par \vskip 3 pt \noindent
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\par
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\par
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\par 
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\par
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\par
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\par
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\par
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\par 
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\par
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\par
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\par 
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[S.2] \it P. Stovichek \rm
\par   
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\par
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\par
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\par
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