%Paper: hep-th/9306094
%From: mash@phys.unit.no
%Date: Sun, 20 Jun 1993 19:18:21 +0200

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\begin{document}
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\mbox{ }\hfill{\normalsize ITP-93-32E}\\
\mbox{ }\hfill{\normalsize hep-th/9306094}\\
\mbox{ }\hfill{\normalsize May 1993}\\
%
\begin{center}
{\Large \bf Exact Multiplicities in the Three-Anyon Spectrum}\\
\vspace{1cm}
{\large Stefan V.~Mashkevich\footnote{email: gezin@gluk.apc.org}}\\[.5cm]
{\large \it
N.N.Bogolyubov Institute for Theoretical Physics, \\ 252143 Kiev,
Ukraine}
\end{center}
\vspace{.5cm}
\begin{abstract}
Using the symmetry properties of the three-anyon spectrum, we obtain
exactly the multiplicities of states with given energy and angular
momentum. The results are shown to be in agreement with the proper
quantum mechanical and semiclassical considerations, and the
unexplained
points are indicated.
\end{abstract}
\newpage
%


     It is well known that the quantum mechanical spectrum of $N$
non-interacting bosons or  fermions  can  be  obtained  given  just
the
single-particle spectrum, but for anyons this does not hold  because
the $N$-anyon problem is essentially many-particle.  For  arbitrary
$N$
there are two classes of exact solutions \cite{Chou,Poly,Dunn,IJMP}
but
their relative
number decreases rapidly with $N$ increasing. The only case  in
which
one can proceed with the exact analysis more or less  far  ahead  is
that of $N=3.$ In the previous work \cite{PL} we have shown that it
is
possible to calculate exactly all the degeneracies in  the
three-anyon
spectrum using certain symmetry properties of the  latter.  Here  we
will carry out an analogous  calculation,  taking  into  account  in
addition the angular momentum. Our present consideration will  allow
us to shed at least some light on the problem of quantum  mechanical
description of anyonic spectra, which at  the  moment  is  far  from
being closed.

     As it was done earlier, we consider the problem of  three
non-interacting anyons in a harmonic potential, with the  particle
mass and the frequency set to unity. The Hamiltonian is
$\hat{H}=\sum_{j=1}^{3}\hat{H}_{j}$ with the one-particle Hamiltonian
$\hat{H}_{j}=\frac{1}{2}(-\Delta_{j}+{\bf r}_{j}^{2}).$
The single-particle state is uniquely determined by the two quantum
numbers -- energy $E$ which may equal $1,2,\dots,$ and angular
momentum
$L=-(E-1),-(E-3),\dots,E-3,E-1.$ Formally, the number of
single-particle
states with energy $E$ and momentum $L$ is given by
\be
g_{1}(E,L) = \left\{ \begin{array}{cl}
                r(E-L,2) & $ if $ |L|\le E-1 \\ \\
                0 & \mbox{ otherwise}
                \end{array}
             \right.
\label{single}
\ee
where $r(a,b)$ is the remainder of the division of $a$ by $b$. (Here
and
further $E$ is implicit to be a positive integer and $L$ an integer.)
In
what follows it will be also convenient to use, along with $E$ and
$L$,
the numbers ${\ell}^{+}=(E+L-1)/2$ and ${\ell}^{-}=(E-L-1)/2$.
Obviously,
any pair of non-negative integers $({\ell}^{+},{\ell}^{-})$
corresponds
to exactly one state.

     To calculate the multiplicities  $\tilde{g}_{3B}(E,L)$ and
$\tilde{g}_{3F}(E,L)$ in the relative motion spectrum of three
bosons  and  three  fermions,
respectively, we start from the multiplicities  $g_{3}(E,L)$
corresponding to Boltzmann statistics, then come to  $g_{3B}(E,L)$
and $g_{3F}(E,L)$ for the full spectrum, and afterwards  separate
the
center-of-mass motion.

     The calculation is simplified by using
\be
{\cal L}^{\pm}=\frac{\ds1}{\ds2}(E\pm L-3);
\label{lplus}
\ee
since ${\cal L}^{\pm}={\ell}^{\pm}_{1}+{\ell}^{\pm}_{2}+{\ell}^{\pm}
_{3}$ with the ${\ell}^{\pm}_{i}$'s as defined earlier, the
Boltzmann degeneracy is $g_{3}(E,L)=S_{3}({\cal L}^{+})S_{3}({\cal
L}^{-})$
where $S_{3}(N)$ is the number
of ordered triples of integers the sum of which is  $N$.
One has $S_{3}(N)=\frac{\ds1}{\ds2}(N+1)(N+2)$ and consequently
\be
%% FOLLOWING LINE CANNOT BE BROKEN BEFORE 70 CHAR
g_{3}(E,L)=\frac{1}{64}\left[(E^{2}-1)^{2}-2(E^{2}+1)L^{2}+L^{4}\right]. .
\ee
Now, in Boltzmann count one bosonic state is taken six times if  all
the particles occupy different single-particle states,  three  times
if two are in the same state but the third is in another one and one
time if all are in one and the same state. Therefore
\be
%% FOLLOWING LINE CANNOT BE BROKEN BEFORE 70 CHAR
g_{3B}(E,L)=\frac{1}{6}\left[g_{3}(E,L)+3d_{3}(E,L)+2t_{3}(E,L)\right].
\ee
where  $d_{3}(E,L)$ is the number of states with two  particles
in  the same state (the third one may be  or  not  be  in  that
state)  and  $t_{3}(E,L)$  is the number of states with all  three
particles  in  the same state. For fermions, correspondingly,
\be
%% FOLLOWING LINE CANNOT BE BROKEN BEFORE 70 CHAR
g_{3F}(E,L)=\frac{1}{6}\left[g_{3}(E,L)-3d_{3}(E,L)+2t_{3}(E,L)\right].
\ee
By virtue of the aforesaid,
\be
d_{3}(E,L)=Q_{3}({\cal L}^{+})Q_{3}({\cal L}^{-})
\ee
with $Q_{3}(N)$ the number of ordered pairs $(\ell_{1},\ell_{3})$
such that
$2\ell_{1}+\ell_{3}=N\;\;(\ell_{2}=\ell_{1}),$
\be
Q_{3}(N)=\left[\frac{N}{2}\right]+1\equiv\frac{1}{2}[N-r(N,2)]+1,
\ee
where $[n]$ stands for the entire part of $n$. Finally,
\be
t_{3}(E,L)=d(E,3)d(L,3),
\ee
where we have introduced the "multiplicity function"
\be
d(a,b)=
  \left\{ \begin{array}{cc}
     1 & $ if $ r(a,b)=0 \\ \\
     0 & \mbox{ otherwise}
     \end{array}
  \right.  .
\ee
Explicit expressions for $g_{3B}$ and $g_{3F}$ read
$$
%% FOLLOWING LINE CANNOT BE BROKEN BEFORE 70 CHAR
g_{3B}(E,L)=\frac{1}{384}\left[(E^{2}-L^{2})^{2}+10E^{2}-14L^{2}+24E+13\right]
$$
\be
+\frac{1}{16}[2{\cal M}^{+}{\cal M}^{-}+{\cal M}^{+}(L-E-1)-{\cal
M}^{-}(L+E+1)]+\frac{1}{3}d(E,3)d(L,3),
\label{g3b}
\ee
$$
%% FOLLOWING LINE CANNOT BE BROKEN BEFORE 70 CHAR
g_{3F}(E,L)=\frac{1}{384}\left[(E^{2}-L^{2})^{2}-14E^{2}+10L^{2}-24E-11\right]
$$
\be
-\frac{1}{16}[2{\cal M}^{+}{\cal M}^{-}+{\cal M}^{+}(L-E-1)-{\cal
M}^{-}(L+E+1)]+\frac{1}{3}d(E,3)d(L,3),
\label{g3f}
\ee
where ${\cal M}^{\pm}=r({\cal L}^{\pm},2)$ , ${\cal L}^{\pm}$ are
defined by
(\ref{lplus}), and $E$ and $L$ should be of different parity.

     In order to separate the center-of-mass motion, note that
\be
g_{3S}(E,L)=\sum_{e=1}^{E}\sum_{l=-e+1}^{e-1}\tilde{g}_{3S}(E-e,L-l)
\label{g3s}
\ee
for  $S = B,F$  ( $e$  and  $l$  are the center-of-mass energy and
angular momentum, respectively). Eq.(\ref{g3s}) implies
\be
\tilde{g}_{3S}(E,L)=g_{3S}(E+1,L)+g_{3S}(E-1,L)
-g_{3S}(E,L-1)-g_{3S}(E,L+1).
\label{gt3s}
\ee
Substituting (\ref{g3b}) and (\ref{g3f}), one gets
$$
\tilde{g}_{3(B/F)}(E,L)=\frac{1}{24}(E^{2}-L^{2})+\frac{1}{3}\left[
d(E+1,3)d(L,3)+d(E-1,3)d(L,3)\right.
$$
\be
%% FOLLOWING LINE CANNOT BE BROKEN BEFORE 70 CHAR
\left.-d(E,3)d(L-1,3)-d(E,3)d(L+1,3)\right]\pm\frac{1}{8}r(E-L,4)r(E+L,4),
\label{grel}
\ee
the upper/lower sign referring to  $B/F$ ; here,  $E$  and  $L$  have
 to
be of the same parity. In the table below, the values  of
$\tilde{g}_{3B}(E,L)$ and $\tilde{g}_{3F}(E,L)$  for all  $E\le 12$
are listed.

  Now, our main goal is to investigate how  the  spectrum
interpolates between the bosonic and fermionic  ones.  Anyonic wave
functions satisfy the interchange conditions  ${\cal P}_{jk}\Psi=
\exp(i\pi\delta)\Psi$ for each pair $j,k$ , where  ${\cal P}_{jk}$
is the operator of anticlockwise interchange
of particles  $j$  and  $k$  and  $\delta$  is the statistical
parameter; there is a continuous transition from bosons to
fermions as  $\delta$  goes  from 0 to 1 . Having turned an
$N$-anyon system by one complete  revolution
in an anticlockwise direction, one will have the wave function
multiplied by  $\exp[i\pi N(N-1)\delta]$ . Consequently, the
possible  values  of angular momentum for anyons are
$L = \frac{N(N-1)}{2}\delta+$integer. In  our case  $N = 3$
one gets  $L = 3\delta+$integer . Thus, a state with angular
momentum  $L$  at Bose statistics interpolates to  that  with
angular momentum  $L+3$  at Fermi statistics. As for energy,
it is known \cite{Spor,Murt,Illu} that possible values of
the difference  $E_{Fermi}-E_{Bose}$ are $+3,+1, -1,-3.$
In the first and the last cases the  $\delta$  dependence  of  $E$
is linear and the states may be found exactly (in  Ref.\cite{PL}  we
called them "good" states), in the other two ones ("bad" states)
this  dependence is non-linear and  not  found  exactly  (see
Ref.\cite{Amel} for an interesting hypothesis concerning the
mentioned dependence). To summarize, all bosonic states with the
quantum  numbers   $(E,L)$ fall into four classes according to
the  quantum  numbers  of  fermionic states to which they
interpolate;  those  can equal $(E+3,L+3),(E+1,L+3),(E-1,L+3),$ or
$(E-3,L+3).$ (It is implicit of course that in the subspace
of states with the same  $(E,L)$  one chooses the "correct" ones
in the same way as in perturbation theory with degenerate states;
it can always be done since  $E$  and  $L$  themselves are
good quantum numbers for anyons.)

     We will use  $\tilde{r}^{n}(E,L)$ to denote the number of
states which come from  $(E,L)$  at Bose statistics to  $(E+n,L+3)$
at Fermi  statistics. (Since it has been shown by numerical
calculations  \cite{Spor,Murt}  that  the energies of different
"bad" states interpolating  between  the  same bosonic and
fermionic ones are in general different  for  fractional
$\delta$ , one has for anyons, strictly speaking, "numbers" of
such  states rather than "multiplicities".) As  $n$ can take four
values, one needs four equations for each $(E,L)$ to determine those
numbers. The "law of conservation of states" at bosonic and
fermionic  points reads
\be
\tilde{r}^{+3}(E,L)+\tilde{r}^{+1}(E,L)+\tilde{r}^{-1}(E,L)+
\tilde{r}^{-3}(E,L)=\tilde{g}_{3B}(E,L),
\label{boson}
\ee
$$
\tilde{r}^{+3}(E-3,L-3)+\tilde{r}^{+1}(E-1,L-3)
$$
\be
+\tilde{r}^{-1}(E+1,L-3)+\tilde{r}^{-3}(E+3,L-3)
=\tilde{g}_{3F}(E,L),
\label{fermion}
\ee
respectively. Two more equations follow from the known symmetry
properties of the spectrum. First, in perturbation theory it is easy
to
establish \cite{SenN,SpoP} that at Fermi statistics, two states
with opposite angular momenta have opposite values of slopes, that
is,
derivatives  $(dE/d\delta)_{\delta=1}.$ Since the linear behavior
of  the  "good"  states  is exact, this implies the equality
\be
\tilde{r}^{+3}(E-3,L-3)=\tilde{r}^{-3}(E+3,-L-3)
\label{pert}
\ee
(see Fig.1). Second, there is the supersymmetry property. Namely,
it was pointed out by Sen \cite{SenP} that there exists an
operator  $\hat{Q}$  which annihilates some of the "good"
states but acting on  a  "bad"  state with statistical
parameter  $\delta$  always produces another  "bad"  state
with same energy  and  with  statistical  parameter  $1+\delta.$
It is straightforward to show that  $[\hat{L},\hat{Q}]=2\hat{Q}$
so that a state coming from  $(E,L)$  at  $\delta=0$ to $(E+1,L+3)$
at  $\delta=1$ turns  under  $\hat{Q}$ into that coming from
$(E,L+2)$  at $\delta=1$ to $(E+1,L+5)$ at $\delta=2.$
Now, parity transformation turns the latter into a state
coming from  $(E+1,-L-5)$  at  $\delta=0$  to $(E,-L-2)$
at $\delta = 1$  (Fig.2).  Therefore the last equation is
\be
\tilde{r}^{+1}(E,L)=\tilde{r}^{-1}(E+1,-L-5).
\label{super}
\ee
The four equations (\ref{boson})-(\ref{super}) and the expression
(\ref{grel}) for $\tilde{g}_{3B}$ and $\tilde{g}_{3F}$
would suffice to calculate  $\tilde{r}^{n}(E,L).$ It  seems,
however, hardly possible to obtain closed-form  expressions
for  them  in  a straightforward manner as it is not clear
in which form they  should be searched for. Instead, it is easy
to solve (\ref{boson})-(\ref{super}) numerically for low-lying
levels and it turns out to be possible to pick up the regularity
in the numbers.

    The results for  $\tilde{r}^{n}(E,L)$ for  $E\le 12$  are
summarized in Tab.2. The general tendency is clear and can be
said to coincide
with what one could more or less expect, but to move further it is
useful  to
involve the concept of towers \cite{Spor,SpoP,SenP}.  Again following
Ref.\cite{SenP},  we recall that there exists an operator
$\hat{K}_{-}$  such
that $[\hat{H},\hat{K}_{-}]=-2\hat{K}_{-}$ and
$[\hat{L},\hat{K}_{-}]=0$ ---
therefore it either annihilates a  common  eigenstate of $\hat{H}$
and
$\hat{L}$ or lowers  its  energy  by  two  units  without changing
its angular
momentum. This  means  in  turn  that  all the states fall into
"towers"
descending along which is realized by $\hat{K}_{-}$ until one reaches
a bottom
state for which $\hat{K}_{-}\Psi = 0.$ Therefore  any $(E,L)$ state
either
is  a  bottom  state  or  has  its  correspondent $(E-2,L)$ state
obtained from
it by action of $\hat{K}_{-}$ .  Denoting  by $\tilde{b}^{n}(E,L)$
the number
of bottom states with the  same characteristics as in
$\tilde{r}^{n}(E,L)$ ,
we have
\be
\tilde{b}^{n}(E,L)=\tilde{r}^{n}(E,L)-\tilde{r}^{n}(E-2,L),
\label{btor}
\ee and correspondingly
\be \tilde{r}^{n}(E,L)=\sum_{k=0}^{[E/2]}\tilde{b}^{n}(E-2k,L).
\label{rtob} \ee
Thus, instead of counting (and finding) all states it is sufficient
to count
(and find) only bottom states. The values of  $\tilde{b}^{+3}(E,L)$
and
$\tilde{b}^{+1}(E,L)$ are displayed in Tabs.3-4.

     Here at last the regularities are obvious. The  formal
expressions
read
$$
\tilde{b}^{+3}(E,L)=\left[\frac{E+1}{6}\right]-
\left[\frac{E+L}{4}\right]+\left[\frac{L}{2}\right]+1-d(E,6)
$$
\be
\mbox{for}\;\; \frac{E-2}{3}\le L\le E-2 ;
\label{bp3}
\ee
$$
\tilde{b}^{+1}(E,L)=\left[\frac{E^{*}}{6}\right]+
\left[\frac{L-L^{*}}{4}+\frac{1-r(E,2)}{2}\right]
+\frac{r(E^{*},6)}{4}
$$
$$
+\left[1-2r(E,2)\right]\frac{d(E^{*}-2,6)[r(L-L^{*},4)-1]}{2}
$$
\be
\mbox{for}\;\; \frac{-E-2}{3}\le L\le E-6 ,
\label{bp1}
\ee
where
\be
E^{*}=E-3r(E,2) , \;\;
L^{*}=2\left[\frac{E^{*}}{6}\right]-2+r(E,2) .
\label{eandl}
\ee
{}From these, one obtains the formulas for $\tilde{r}^{n}(E,L)$
immediately  by applying (\ref{rtob}):
\be
\tilde{r}^{+3}(E,L)=
       \left\{ \begin{array}{l}
         {\displaystyle
         \frac{L^{2}+6L+5}{12}+\frac{1-r(L,2)}{4}+\frac{d(L,3)}{3}}
\\  \\
         \hspace{5cm}$for $0\le L\le \frac{E-2}{3}, \\  \\
         {\displaystyle
         \frac{-E^{2}+6EL-5L^{2}+12E-12L}{48}+
         \frac{d(L,3)-d(E,3)}{3}} \\ \\
         +\left[{\textstyle 1-2r(E,2)}\right]
         {\displaystyle
         \frac{d(E-L-2,4)}{4}} \\  \\
         \hspace{5cm}$for $\frac{E-2}{3}\le L \le E-2;
       \end{array}\right.
\label{rp3}
\ee
\be
\tilde{r}^{+1}(E,L)=
         \left\{ \begin{array}{l}
         {\displaystyle
         \frac{E^{2}+6EL+9L^{2}+16E+48L+48}{48}+\frac{r(E-L,4)}{8}
         +\frac{d(E-1,3)}{3}}  \\  \\
         \hspace{6cm}$for $\frac{-E-2}{3}\le L <-1, \\ \\
         {\displaystyle
         \frac{E^{2}+6EL-15L^{2}+16E-72L-96}{48}+\frac{r(E-L,4)}{8}
         +\frac{d(E-1,3)}{3}}  \\  \\
         \hspace{6cm}$for $-1\le L \le\frac{E-8}{3}, \\  \\
         {\displaystyle
         \frac{E^{2}-2EL+L^{2}-4E+4L}{16}+\frac{r(E-L,4)}{8}} \\ \\
         \hspace{6cm}$for $\frac{E-8}{3}<L\le E -6.
       \end{array}\right.
\label{rp1}
\ee

Formulas (\ref{bp3}),(\ref{bp1}), (\ref{rp3}),(\ref{rp1}) are exact
and are
the main  result  of this paper. In these formulas,  $E$  and $L$
should  be  of  the  same parity, i.e.  $r(E-L,2) = 0$ ;
otherwise, as well as if  $L$ does  not fall in any of the ranges
specified in the formulas, the corresponding multiplicities
vanish. Formulas for  $\tilde{b}^{-1},\tilde{r}^{-1}$ and
$\tilde{b}^{-3},\tilde{r}^{-3}$ follow immediately from
the obtained ones upon applying (\ref{pert}) and (\ref{super}).

     Finally, summation over  $L$  yields the total numbers of
states with given energy and slope:
\be
\tilde{r}^{+3}(E)=\frac{1}{216}\left\{E^{3}+9E^{2}+
\left[42-27r(E,2)-24d(E,3)\right]E \\
+\left[f_{r(E^{*},6)}-81r(E,2)\right]\right\},
\ee
\be
\tilde{r}^{+1}(E)=\frac{1}{216}\left\{2E^{3}+3E^{2}+
\left[24d(E-1,3)-18\right]E \\
+\left[4r(E^{*},6)-27r(E,2)\right]\right\},
\ee
where  $f_{0}=0,f_{2}=88,f_{4}=56,$ and  $E^{*}$ has been defined
by (\ref{eandl}); these are exactly the expressions obtained in
Ref.\cite{PL} (where  slightly different notations were used).

     Let us now discuss the obtained results. For clarity, the
plots of  $\tilde{r}^{n}(E,L)$  for  $E = 100$  are displayed
in Fig.3. We  have  learned that for a given  $E$ , the states
of each of the four  classes  exist for not all values of  $L$
allowed for that  $E$ . Tab.5 shows the
lowest and the highest values of  $L$ for  which  the
corresponding states exist, as well as the values for which
the numbers of such states are maximal.


     Since all the states are solutions of the Schr\"{o}dinger
equation, all these values should in principle be deduced
directly  from  this equation. Producing them in this way is an
interesting and apparently difficult problem, and perhaps
at least some points of its  solution, if available, could
be used also to understand  the  structure of the  $N$-anyon
spectrum. What we can do at the moment is to explain
why for  $L=E-2$  and  $E-4$ , as the table shows,  there
exist  only the (+3) states. In the context of our previous
considerations, this follows immediately from (\ref{pert}) and
(\ref{super}) by noticing that $\tilde{r}^{+1}(E,E-k)
=\tilde{r}^{-1}(E+1,-E+k-5)$ does not vanish only for  $k\ge6$,
because  there are no $(E,L)$ states with  $L<-E+2$ , and
analogously for  $\tilde{r}^{-1}$ and $\tilde{r}^{-3}$ .
However, a purely quantum mechanical proof of this fact
can be given \cite{IJMP,UNP}, which we will describe here.

     Choosing as independent coordinates the one of the center
of mass $Z=(z_{1}+z_{2}+z_{3})/\sqrt{3}\;\;(z_{j}=x_{j}+iy_{j}$
is the complex coordinate of $j$-th particle) and two relative
ones  $z_{12}=(z_{1}-z_{2})/\sqrt{2}\;,\;z_{23}=(z_{2}-z_{3})/
\sqrt{2}$ and searching for the  wave  function
in the form  $\Psi=\Psi_{cm}\tilde{\chi}\exp\left[-\frac{1}{2}
(z_{1}z_{1}^{*}+z_{2}z_{2}^{*}+z_{3}z_{3}^{*})\right]$ , one
has for the relative Hamiltonian  $\tilde{H}=\tilde{H}_{(1)}
+\tilde{H}_{(2)}$  where $\tilde{H}_{(1)}=z_{12}\partial_{12}+
%% FOLLOWING LINE CANNOT BE BROKEN BEFORE 70 CHAR
z_{12}^{*}\partial_{12}^{*}+z_{23}\partial_{23}+z_{23}^{*}\partial_{23}^{*}
+2\;,\;\tilde{H}_{(2)}=-2(\partial_{12}\partial_{12}^{*}+
%% FOLLOWING LINE CANNOT BE BROKEN BEFORE 70 CHAR
\partial_{23}\partial_{23}^{*})\;,\;\partial_{jk}\equiv\partial/\partial z_{jk}
\;\;\;(\;\tilde{H}$ acts on $\tilde{\chi}$),
and for the relative angular momentum
$\tilde{L}=z_{12}\partial_{12}-
%% FOLLOWING LINE CANNOT BE BROKEN BEFORE 70 CHAR
z_{12}^{*}\partial_{12}^{*}+z_{23}\partial_{23}-z_{23}^{*}\partial_{23}^{*}.$
The function $\tilde{\chi}$ is to be searched  for
as a linear combination of functions of the form
\be
\left|l_{12}\;\bar{l}_{12}\;;\;l_{23}\;\bar{l}_{23}\;;\;
l_{31}\;\bar{l}_{31}\;\right\rangle_{+} \\
\equiv\left\{(z_{12})^{l_{12}}(z_{12}^{*})^{\bar{l}_{12}}
(z_{23})^{l_{23}}(z_{23}^{*})^{\bar{l}_{23}}
(z_{31})^{l_{31}}(z_{31}^{*})^{\bar{l}_{31}}\right\}_{+}
\label{func}
\ee
($z_{31}=-z_{12}-z_{23}$) where  $\{...\}_{+}$  means symmetrization
over  $z_{j}$'s   and the equalities  $l_{jk}-\bar{l}_{jk}=\delta
$+integer  should fulfil.  The  function (\ref{func})
then satisfies the anyonic interchange conditions
${\cal P}_{jk}\tilde{\chi}=\exp(i\pi\delta)\tilde{\chi},$
and the coefficients in the linear combination should
be chosen so that  $\tilde{\chi}$ be non-singular and satisfy
the equation $\tilde{H}\tilde{\chi}=E\tilde{\chi}$. Obviously,
$$
\tilde{L}\left|l_{12}\;\bar{l}_{12}\;;\;l_{23}\;\bar{l}_{23}\;;\;
l_{31}\;\bar{l}_{31}\;\right\rangle_{+}
$$
\be
=\left(l_{12}-\bar{l}_{12}+l_{23}-\bar{l}_{23}+
l_{31}-\bar{l}_{31}\right)
\left|l_{12}\;\bar{l}_{12}\;;\;l_{23}\;\bar{l}_{23}\;;\;
l_{31}\;\bar{l}_{31}\;\right\rangle_{+} ,
\ee
$$
\tilde{H}_{(1)}
\left|l_{12}\;\bar{l}_{12}\;;\;l_{23}\;\bar{l}_{23}\;;\;
l_{31}\;\bar{l}_{31}\;\right\rangle_{+}
$$
\be
=\left(l_{12}+\bar{l}_{12}+l_{23}+\bar{l}_{23}+
l_{31}+\bar{l}_{31}+2\right)
\left|l_{12}\;\bar{l}_{12}\;;\;l_{23}\;\bar{l}_{23}\;;\;
l_{31}\;\bar{l}_{31}\;\right\rangle_{+} ,
\ee
and $\tilde{H}_{(1)}\left|l_{12}\;\bar{l}_{12}\;;\;l_{23}\;
\bar{l}_{23}\;;\;l_{31}\;\bar{l}_{31}\;\right\rangle_{+} $ consists
of pieces of the form \[l_{jk}\bar{l}_{mn}\left|\dots l_{jk}-1\dots
\bar{l}_{mn}-1\dots\right\rangle_{+}.\]
The states which at  $\delta=0$
have the maximal angular momentum  $L = E-2$
are those with all  $\bar{l}_{jk}$ vanishing, i.e. of
the form  $\left|l\;0 ; m\;0 ; n\;0 \right\rangle_{+}$
(with integral  $l,m,n$);
the corresponding anyonic states  $\left|l+\delta\;0 ; m+\delta\; 0 ;
n+\delta\; 0 \right\rangle$ always are stationary because
$\tilde{H}_{(2)}$ annihilates  them,  and  their
energy is  $E=E_{Bose}+3\delta\;$(where  $E_{Bose}=l+m+n+2$).
Further, to construct a state which has $L=E-4$ at $\delta=0$
one should start from
$\left|l+\delta\; 1 ; m+\delta\;0 ; n+\delta\; 0 \right\rangle_{+}
\equiv\left|1\right\rangle$.
Acting on this with  $\tilde{H}_{(2)}$  yields
a sum of the terms of the form  $\left|p+\delta\; 0 ; q+\delta\; 0 ;
r+\delta\;0\right\rangle_{+}$, which are annihilated by
$\tilde{H}_{(2)}$ .
If all of them are non-singular,  then  a stationary
state can be built. (One has  $\tilde{H}_{(1)}\left|1\right\rangle=
E\left|1\right\rangle\;,\;
\tilde{H}_{(2)}\left|1\right\rangle=\left|2\right\rangle\;,\;
\tilde{H}_{(1)}\left|2\right\rangle=(E-2)\left|2\right\rangle\;,\;$
and $\tilde{H}_{(2)}\left|2\right\rangle=0$, consequently
$\tilde{H}\left[
\left|1\right\rangle+{\ds\frac{1}{2}}\left|2\right\rangle\right]=
%% FOLLOWING LINE CANNOT BE BROKEN BEFORE 70 CHAR
E\left[\left|1\right\rangle+{\ds\frac{1}{2}}\left|2\right\rangle\right]$.)
Now, the only potentially  singular  term in  $\left|2\right\rangle$
could be
$\left|-1+\delta\; 0 ; q+\delta\; 0 ; r+\delta\;0\right\rangle_{+}$
with  non-negative integral  $q$ and $r$  .  However,  such
function, which equals  $(z_{12}z_{23}z_{31})^{\delta}
\left| -1\; 0 ; q\; 0 ; r\;0 \right\rangle$  , can easily be shown to
be
non-singular: the singularities cancel out due to symmetrization
\cite{IJMP,UNP}. This completes the proof of the fact that
all bosonic states with $E=L-4$ , as well as those with $E=L-2$,
have their correspondent anyonic states with the slope +3.

     At present it is unknown how the other features of the
obtained results could be derived from  such  quantum
mechanical  considerations. However, the qualitative
picture can be justified using semiclassical arguments. In
the two-anyon problem,  there  is  only  one relative
coordinate  $z_{12}$ , the change of which is  governed  by  the
oscillator equation. The corresponding  classical
trajectories   are  ellipses,  and  the  anyonic  interchange
conditions  demand  that  the relative angular momentum be
$L=\delta+2l$ ,  where  the  integer $l$  determines the sign  of
$L$  at $\delta=0$ . Thus, as one starts   increasing $\delta$
from zero, one should expect linear increase of the energy of
those states in which the relative vector,  in  classical
terms, rotates anticlockwise $(l>0)$  or does not rotate
$(l=0)$ , and linear decrease for those in which it rotates
clockwise. This is indeed the case: The semiclassical
description of the two-anyon spectrum  turns out to yield the
exact values of levels \cite{Bhad,Illu}. For $N$ anyons,
there are $\frac{N(N-1)}{2}$ relative vectors, and the analogous
consideration would yield linear dependences with the  slopes
$E(1)-E(0)=\frac{N(N-1)}{2}-2s$ with $s$ the number of the
relative  vectors  rotating  clockwise. The set of the
slopes is correct \cite{Chin}, which encourages us  to use this
picture for a qualitative  analysis  (although  we  do  not
argue that it is good in other senses, for example that it
gives the correct multiplicities). So, for a state to  have
the  slope  +3  , neither  of  the  relative  vectors  should
rotate  clockwise.  Certainly it cannot be so if  $L<0$ .  For
$L=0$  a (+3)  state  can  be  only  realized  as  a  radial
excitation  of  the   bosonic   ground   state;   indeed,
$\tilde{r}^{+3}(E,0)=1$ (for even $E$ ). Further,  if  we
choose a state with a given $L$ "at random", then the more is
$L$ ,  the  more  vectors  "in average" rotate  anticlockwise.
Therefore the states with  $L$ close to maximal  $L=E-2$
have at most the slope +3 , and  with  $L$ decreasing,  states
with successively decreasing  slopes  emerge,  pass the
point  of their maximal number, and vanish, just as
one observes in Fig.3. Apparently the picture is analogous for
arbitrary $N$.

     To summarize, we have determined exactly the multiplicities  of
states with given energies and angular momenta in  the  spectrum  of
three non-interacting anyons in a harmonic well and shown  that  the
results are in conformity with certain quantum mechanical and
semiclassical considerations. Generally speaking, these  results
should follow directly from the Schr\"{o}dinger equation, but the
concrete procedure of deriving them this way is still to be found.

     I thank G.M.Zinovjev for his attention to my work and  constant
support, and Diptiman Sen for stimulating discussions. This work was
supported, in part, by a  Soros  Foundation  Grant  awarded  by  the
American Physical Society.

\newpage

\begin{center}
\begin{tabular}{|rr|c|c|c|rr|c|c|c|rr|c|c|c|rr|c|c|}
\cline{1-4}\cline{6-9}\cline{11-14}\cline{16-19}
$E$ & $L$ & $\tilde{g}_{3B}$ & $\tilde{g}_{3F}$ & \hspace{-.25cm} &
$E$ & $L$ & $\tilde{g}_{3B}$ & $\tilde{g}_{3F}$ & \hspace{-.25cm} &
$E$ & $L$ & $\tilde{g}_{3B}$ & $\tilde{g}_{3F}$ & \hspace{-.25cm} &
$E$ & $L$ & $\tilde{g}_{3B}$ & $\tilde{g}_{3F}$    \\
\cline{1-4}\cline{6-9}\cline{11-14}\cline{16-19}
2 & 0 & 1 & 0 & \hspace{-.25cm} & 7 & 1 & 2 & 2 &
\hspace{-.25cm} & 9 & -5 & 2 & 2 & \hspace{-.25cm} & 11 & -1 & 5 & 5
\\
\cline{1-4}
3 & 1 & 0 & 0 & \hspace{-.25cm} & 7 & -1 & 2 & 2 &
\hspace{-.25cm} & 9 & -7 & 1 & 1 & \hspace{-.25cm} & 11 & -3 & 5 & 5
\\
\cline{11-14}
3 & -1 & 0 & 0 & \hspace{-.25cm} & 7 & -3 & 2 & 2 &
\hspace{-.25cm} & 10 & 8 & 2 & 1 & \hspace{-.25cm} & 11 & -5 & 4 & 4
\\
\cline{1-4}
4 & 2 & 1 & 0 & \hspace{-.25cm} & 7 & -5 & 1 & 1 &
\hspace{-.25cm} & 10 & 6 & 3 & 3 & \hspace{-.25cm} & 11 & -7 & 3 & 3
\\
\cline{6-9}
4 & 0 & 1 & 1 & \hspace{-.25cm} & 8 & 6 & 2 & 1 &
\hspace{-.25cm} & 10 & 4 & 4 & 3 & \hspace{-.25cm} & 11 & -9 & 2 & 2
\\
\cline{16-19}
4 & -2 & 1 & 0 & \hspace{-.25cm} & 8 & 4 & 2 & 2 &
\hspace{-.25cm} & 10 & 2 & 4 & 4 & \hspace{-.25cm} & 12 & 10 & 2 & 1
\\
\cline{1-4}
5 & 3 & 1 & 1 & \hspace{-.25cm} & 8 & 2 & 3 & 2 &
\hspace{-.25cm} & 10 & 0 & 5 & 4 & \hspace{-.25cm} & 12 & 8 & 3 & 3
\\
5 & 1 & 1 & 1 & \hspace{-.25cm} & 8 & 0 & 3 & 3 &
\hspace{-.25cm} & 10 & -2 & 4 & 4 & \hspace{-.25cm} & 12 & 6 & 5 & 4
\\
5 & -1 & 1 & 1 & \hspace{-.25cm} & 8 & -2 & 3 & 2 &
\hspace{-.25cm} & 10 & -4 & 4 & 3 & \hspace{-.25cm} &
 12 & 4 & 5 & 5  \\
5 & -3 & 1 & 1 & \hspace{-.25cm} & 8 & -4 & 2 & 2 &
\hspace{-.25cm} & 10 & -6 & 3 & 3 & \hspace{-.25cm} & 12 & 2 & 6 & 5
\\
\cline{1-4}
6 & 4 & 1 & 0 & \hspace{-.25cm} & 8 & -6 & 2 & 1 &
\hspace{-.25cm} & 10 & -8 & 2 & 1 & \hspace{-.25cm} & 12 & 0 & 6 & 6
\\
\cline{6-9}\cline{11-14}
6 & 2 & 1 & 1 & \hspace{-.25cm} & 9 & 7 & 1 & 1 &
\hspace{-.25cm} & 11 & 9 & 2 & 2 & \hspace{-.25cm} & 12 & -2 & 6 & 5
\\
6 & 0 & 2 & 1 & \hspace{-.25cm} & 9 & 5 & 2 & 2 &
\hspace{-.25cm} & 11 & 7 & 3 & 3 & \hspace{-.25cm} & 12 & -4 & 5 & 5
\\
6 & -2 & 1 & 1 & \hspace{-.25cm} & 9 & 3 & 3 & 3 &
\hspace{-.25cm} & 11 & 5 & 4 & 4 & \hspace{-.25cm} & 12 & -6 & 5 & 4
\\
6 & -4 & 1 & 0 & \hspace{-.25cm} & 9 & 1 & 3 & 3 &
\hspace{-.25cm} & 11 & 3 & 5 & 5 & \hspace{-.25cm} & 12 & -8 & 3 & 3
\\
\cline{1-4}
7 & 5 & 1 & 1 & \hspace{-.25cm} & 9 & -1 & 3 & 3 &
\hspace{-.25cm} & 11 & 1 & 5 & 5 & \hspace{-.25cm} & 12 & -10 & 2 & 1
 \\
7 & 3 & 2 & 2 & \hspace{-.25cm} & 9 & -3 & 3 & 3 &
\hspace{-.25cm} & & & & & \hspace{-.25cm} & & & & \\
\cline{1-4}\cline{6-9}\cline{11-14}\cline{16-19}
\end{tabular}
\end{center}

Tab.1. The values of $\tilde{g}_{3B}(E,L)$ and $\tilde{g}_{3F}(E,L).$

\newpage

\begin{center}
\begin{tabular}{|rr|c|c|c|c|c|rr|c|c|c|c|c|rr|c|c|c|c|}
\cline{1-6}\cline{8-13}\cline{15-20}
 $E$ & $L$ & \multicolumn{4}{c|}{$n$} &  & $E$ & $L$ &
\multicolumn{4}{c|}{$n$}
 &  & $E$ & $L$ & \multicolumn{4}{c|}{$n$} \\
\cline{3-6}\cline{10-13}\cline{17-20}
   &  & 3 & 1 & -1 & -3 & & & & 3 & 1 & -1 & -3 &  &  &  & 3 & 1 & -1
& -3 \\
\cline{1-6}\cline{8-13}\cline{15-20}
 2  & 0 & 1 &  &  &  &  & 8 & 4 & 2 &   &   &   &  & 10 &-8 &  &  & 1
& 1  \\
\cline{1-6}\cline{15-20}
 3  & 1 &   &  &  &  &  & 8 & 2 & 2 & 1 &   &   &  & 11 & 9 & 2&  &
&    \\
 3  &-1 &   &  &  &  &  & 8 & 0 & 1 & 2 &   &   &  & 11 & 7 & 3&  &
&    \\
\cline{1-6}
 4  & 2 & 1 &  &  &  &  & 8 &-2 &   & 2 & 1 &   &  & 11 & 5 & 3& 1&
&    \\
 4  & 0 & 1 &  &  &  &  & 8 &-4 &   &   & 2 &   &  & 11 & 3 & 3& 2&
&    \\
 4  &-2 &   & 1&  &  &  & 8 &-6 &   &   & 1 & 1 &  & 11 & 1 & 1& 4&
&    \\
\cline{1-6}\cline{8-13}
 5  & 3 & 1 &  &  &  &  & 9 & 7 & 1 &   &   &   &  & 11 &-1 &  & 4& 1
&    \\
 5  & 1 & 1 &  &  &  &  & 9 & 5 & 2 &   &   &   &  & 11 &-3 &  & 2& 3
&    \\
 5  &-1 &   & 1&  &  &  & 9 & 3 & 2 & 1 &   &   &  & 11 &-5 &  &  & 4
&    \\
 5  &-3 &   &  & 1&  &  & 9 & 1 & 1 & 2 &   &   &  & 11 &-7 &  &  & 2
& 1  \\
\cline{1-6}
 6  & 4 & 1 &  &  &  &  & 9 &-1 &   & 3 &   &   &  & 11 &-9 &  &  & 1
& 1  \\
\cline{15-20}
 6  & 2 & 1 &  &  &  &  & 9 &-3 &   & 1 & 2 &   &  & 12 &10 & 2&  &
&    \\
 6  & 0 & 1 & 1&  &  &  & 9 &-5 &   &   & 2 &   &  & 12 & 8 & 3&  &
&    \\
 6  &-2 &   & 1&  &  &  & 9 &-7 &   &   & 1 &   &  & 12 & 6 & 4& 1&
&    \\
\cline{8-13}
 6  &-4 &   &  & 1&  &  &10 & 8 & 2 &   &   &   &  & 12 & 4 & 3& 2&
&    \\
\cline{1-6}
 7  & 5 & 1 &  &  &  &  &10 & 6 & 3 &   &   &   &  & 12 & 2 & 2& 4&
&    \\
 7  & 3 & 2 &  &  &  &  &10 & 4 & 3 & 1 &   &   &  & 12 & 0 & 1& 5&
&    \\
 7  & 1 & 1 & 1&  &  &  &10 & 2 & 2 & 2 &   &   &  & 12 &-2 &  & 4& 2
&    \\
 7  &-1 &   & 2&  &  &  &10 & 0 & 1 & 4 &   &   &  & 12 &-4 &  & 1& 4
&    \\
 7  &-3 &   & 1& 1&  &  &10 &-2 &   & 3 & 1 &   &  & 12 &-6 &  &  & 4
& 1  \\
 7  &-5 &   &  & 1&  &  &10 &-4 &   & 1 & 3 &   &  & 12 &-8 &  &  & 2
& 1  \\
\cline{1-6}
 8  & 6 & 2 &  &  &  &  &10 &-6 &   &   & 2 & 1 &  & 12 &10 &  &  & 1
& 1  \\
\cline{1-6}\cline{8-13}\cline{15-20}
\end{tabular}
\end{center}

Tab.2. The values of  $\tilde{r}^{n}(E,L).$ Zeros are not
written down for clarity.

\newpage

\begin{center}
\begin{tabular}{||c|r|cccccccccc||}
\cline{1-12}
\multicolumn{2}{||c|}{} &
\multicolumn{10}{c||}{$E$}  \\
\cline{3-12}
\multicolumn{2}{||c|}{} & 2 & 4 & 6 & 8 & 10 & 12 & 14 & 16 & 18 & 20
 \\
\cline{1-12}
  & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
  & 2 &  & 1 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\
  & 4 &  &  & 1 & 1 & 1 & 0 & 1 & 0 & 0 & 0  \\
  & 6 &  &  &  & 2 & 1 & 1 & 1 & 1 & 0 & 1  \\
$L$ & 8 &  &  &  &  & 2 & 1 & 2 & 1 & 1 & 1  \\
  & 10 &  &  &  &  &  & 2 & 2 & 2 & 1 & 2 \\
  & 12 &  &  &  &  &  &  & 3 & 2 & 2 & 2 \\
  & 14 &  &  &  &  &  &  &  & 3 & 2 & 3  \\
  & 16 &  &  &  &  &  &  &  &  & 3 & 3  \\
  & 18 &  &  &  &  &  &  &  &  &  & 4 \\
\cline{1-12}
\end{tabular}

\begin{tabular}{||c|r|cccccccccc||}
\cline{1-12}
\multicolumn{2}{||c|}{} &
\multicolumn{10}{c||}{$E$}  \\
\cline{3-12}
\multicolumn{2}{||c|}{} & 3 & 5 & 7 & 9 & 11 & 13 & 15 & 17 & 19 & 21
 \\
\cline{1-12}
  & 1 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
  & 3 &  & 1 & 1 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\
  & 5 &  &  & 1 & 1 & 1 & 1 & 0 & 1 & 0 & 0  \\
  & 7 &  &  &  & 1 & 2 & 1 & 1 & 1 & 1 & 0  \\
$L$ & 9 &  &  &  &  & 2 & 2 & 1 & 2 & 1 & 1  \\
  & 11 &  &  &  &  &  & 2 & 2 & 2 & 2 & 1 \\
  & 13 &  &  &  &  &  &  & 2 & 3 & 2 & 2 \\
  & 15 &  &  &  &  &  &  &  & 3 & 3 & 2  \\
  & 17 &  &  &  &  &  &  &  &  & 3 & 3  \\
  & 19 &  &  &  &  &  &  &  &  &  & 3 \\
\cline{1-12}
\end{tabular}

Tab.3. The values of  $\tilde{b}^{+3}(E,L)$ .
\end{center}

\newpage

\begin{center}
\begin{tabular}{||c|r|cccccccccc||}
\cline{1-12}
\multicolumn{2}{||c|}{} &
\multicolumn{10}{c||}{$E$}  \\
\cline{3-12}
\multicolumn{2}{||c|}{} & 2 & 4 & 6 & 8 & 10 & 12 & 14 & 16 & 18 & 20
 \\
\cline{1-12}
  & -6& 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 1 \\
  & -4& 0 & 0 & 0 & 0 & 1 & 0 & 1 & 1 & 1 & 1 \\
  & -2&   & 1& 0 & 1 & 1 & 1 & 1 & 2 & 1 & 2  \\
  & 0 &   &  & 1& 1 & 2 & 1 & 2 & 2 & 2 & 2  \\
  & 2 &   &  &  & 1 & 1 & 2 & 2 & 3 & 2 & 2  \\
$L$ & 4  &  &  &  &  & 1 & 1 & 2 & 2 & 3 & 3  \\
  &  6 &   &  &  &  &  & 1 & 1 & 2 & 2 & 3 \\
  &  8 &   &  &  &  &  &  & 1 & 1 & 2 & 2 \\
  & 10 &   &  &  &  &  &  &  & 1 & 1 & 2  \\
  & 12 &   &  &  &  &  &  &  &  & 1 & 1  \\
  & 14 &   &  &  &  &  &  &  &  &  & 1 \\
  & 16 &   &  &  &  &  &  &  &  &  &   \\
\cline{1-12}
\end{tabular}

\begin{tabular}{||c|r|cccccccccc||}
\cline{1-12}
\multicolumn{2}{||c|}{} &
\multicolumn{10}{c||}{$E$}  \\
\cline{3-12}
\multicolumn{2}{||c|}{} & 3 & 5 & 7 & 9 & 11 & 13 & 15 & 17 & 19 & 21
 \\
\cline{1-12}
  & -7& 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\
  & -5& 0& 0 & 0 & 0 & 0 & 1 & 0 & 1 & 1 & 1 \\
  & -3& 0& 0& 1 & 0 & 1 & 1 & 1 & 1 & 2 & 1  \\
  & -1&  & 1& 1& 1 & 1 & 2 & 1 & 2 & 2 & 2  \\
  &  1&  &  & 1& 1 & 2 & 2 & 2 & 2 & 3 & 2  \\
$L$ & 3 &  &  &  & 1& 1 & 2 & 2 & 3 & 3 & 3  \\
  &  5 &  &  &  &  & 1& 1 & 2 & 2 & 3 & 3 \\
  &  7 &  &  &  &  &  & 1& 1 & 2 & 2 & 3 \\
  &  9 &  &  &  &  &  &  & 1& 1 & 2 & 2  \\
  & 11 &  &  &  &  &  &  &  & 1& 1 & 2  \\
  & 13 &  &  &  &  &  &  &  &  & 1& 1 \\
  & 15 &  &  &  &  &  &  &  &  &  & 1 \\
\cline{1-12}
\end{tabular}

Tab.4. The values of  $\tilde{b}^{+1}(E,L)$ .
\end{center}

\newpage

\begin{center}
\begin{tabular}{|c|c|c|c|}
\hline
    & Low & Max. & High \\ \hline
$ +3$ & 0 & ${\ds\frac{3E-6}{5}}$ & $E-2$ \\ \hline
$ +1$ & ${\ds\frac{-E-2}{3}}$ & ${\ds\frac{E-12}{5}}$ & $E-6$ \\
\hline
$ -1$ & $-E+2$ & ${\ds\frac{-E-12}{5}}$ & ${\ds\frac{E-14}{3}}$ \\
\hline
$ -3$ & $-E+2$ & ${\ds\frac{-3E-6}{5}}$ & $-6$ \\ \hline
\end{tabular}
\end{center}

Tab.5. The lowest and highest values of $L$ for which the
states belonging to $\tilde{r}^{n}(E,L)$ exist, and the
value for which the number of such states is maximal.

\newpage

\begin{thebibliography}{99}
\bibitem{Chou}C.Chou,  {\em Phys.Rev.} D {\bf 44}, 2533 (1991);
Erratum:
Phys.Rev. D 45, 1433 (1992).
\bibitem{Poly}A.P.Polychronakos,  {\em Phys.Lett.} {\bf B264}, 362
(1991).
\bibitem{Dunn} G.Dunne, A.Lerda, S.Sciuto, C.A.Trugenberger,  {\em
Nucl.Phys.}
{\bf B370}, 601 (1992).
\bibitem{IJMP}S.V.Mashkevich,  {\em Int.J.Mod.Phys.} {\bf A7}, 7931
(1992).
\bibitem{PL}S.V.Mashkevich,  {\em Phys.Lett.} {\bf B295}, 233 (1992).
\bibitem{Spor}M.Sporre, J.J.M.Verbaarschot, I.Zahed,  {\em
Phys.Rev.Lett.}
{\bf 67}, 1813 (1991).
\bibitem{Murt}M.V.N.Murthy, J.Law, M.Brack, R.K.Bhaduri,
{\em Phys.Rev.Lett.}  67, 1817 (1991);  {\em Phys.Rev.} B {\bf 45},
4289 (1992).
\bibitem{Illu}F.Illuminati, F.Ravndal, J.Aa.Ruud,  {\em Phys.Lett.}
{\bf A161}, 323 (1992).
\bibitem{Amel}C.Chou, L.Hua, G.Amelino-Camelia,  {\em Phys.Lett.}
{\bf B286}, 329 (1992);
see also  G.Amelino-Camelia,  {\em Phys.Lett.} {\bf B299}, 83 (1993).
\bibitem{SenN}D.Sen,  {\em Nucl.Phys.} {\bf B360}, 397 (1991).
\bibitem{SpoP}M.Sporre, J.J.M.Verbaarschot, I.Zahed,  preprint
SUNY-NTG-91/40
(1991).
\bibitem{SenP}D.Sen,  {\em Phys.Rev.Lett.} {\bf 68}, 2977 (1992).
\bibitem{UNP}S.V.Mashkevich  (unpublished).
\bibitem{Bhad}R.K.Bhaduri, R.S.Bhalerao, A.Khare, J.Law,
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{\em Phys.Rev.Lett.} {\bf 66}, 523 (1991).
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(1992).
\end{thebibliography}

\newpage

{\large Figure Captions}

     Fig.1. Two "good" states with opposite angular momenta
            in the fermion limit.

     Fig.2. Two "bad" states related by (supersymmetry+parity)
            transformation.

     Fig.3. The plots of  $\tilde{r}^{n}(E,L)$  for  $E = 100$ .

\end{document}

%%!!!!!
%%% Postscript figure #1 follows

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2450 3400 M 6450 3400 D S
2450 3400 M 2450 3310 D S
2321 3278 M 2304 3273 D 2293 3256 D 2287 3227 D 2287 3210 D 2293 3181
D 2304 3164 D 2321 3158 D 2333 3158 D 2350 3164 D 2361 3181 D 2367
3210 D 2367 3227 D 2361 3256 D 2350 3273 D 2333 3278 D 2321 3278 D
2310 3273 D 2304 3267 D 2299 3256 D 2293 3227 D 2293 3210 D 2299 3181
D 2304 3170 D 2310 3164 D 2321 3158 D S
2333 3158 M 2344 3164 D 2350 3170 D 2356 3181 D 2361 3210 D 2361 3227
D 2356 3256 D 2350 3267 D 2344 3273 D 2333 3278 D S
2413 3170 M 2407 3164 D 2413 3158 D 2419 3164 D 2413 3170 D S
2493 3278 M 2476 3273 D 2464 3256 D 2459 3227 D 2459 3210 D 2464 3181
D 2476 3164 D 2493 3158 D 2504 3158 D 2521 3164 D 2533 3181 D 2539
3210 D 2539 3227 D 2533 3256 D 2521 3273 D 2504 3278 D 2493 3278 D
2481 3273 D 2476 3267 D 2470 3256 D 2464 3227 D 2464 3210 D 2470 3181
D 2476 3170 D 2481 3164 D 2493 3158 D S
2504 3158 M 2516 3164 D 2521 3170 D 2527 3181 D 2533 3210 D 2533 3227
D 2527 3256 D 2521 3267 D 2516 3273 D 2504 3278 D S
4450 3400 M 4450 3310 D S
4321 3278 M 4304 3273 D 4293 3256 D 4287 3227 D 4287 3210 D 4293 3181
D 4304 3164 D 4321 3158 D 4333 3158 D 4350 3164 D 4361 3181 D 4367
3210 D 4367 3227 D 4361 3256 D 4350 3273 D 4333 3278 D 4321 3278 D
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showpage
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%%!!!!!
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% Copyright (C) 1988 Golden Software, Inc.
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1625 5000 M 1540 5000 D S
1206 5035 M 1216 5040 D 1232 5057 D 1232 4944 D S
1227 5051 M 1227 4944 D S
1206 4944 M 1254 4944 D S
1302 5035 M 1307 5030 D 1302 5024 D 1297 5030 D 1297 5035 D 1302 5046
D 1307 5051 D 1323 5057 D 1345 5057 D 1361 5051 D 1366 5046 D 1372
5035 D 1372 5024 D 1366 5014 D 1350 5003 D 1323 4992 D 1313 4987 D
1302 4976 D 1297 4960 D 1297 4944 D S
1345 5057 M 1356 5051 D 1361 5046 D 1366 5035 D 1366 5024 D 1361 5014
D 1345 5003 D 1323 4992 D S
1297 4955 M 1302 4960 D 1313 4960 D 1340 4949 D 1356 4949 D 1366 4955
D 1372 4960 D S
1313 4960 M 1340 4944 D 1361 4944 D 1366 4949 D 1372 4960 D 1372 4971
D S
1436 5057 M 1420 5051 D 1409 5035 D 1404 5008 D 1404 4992 D 1409 4965
D 1420 4949 D 1436 4944 D 1447 4944 D 1463 4949 D 1473 4965 D 1479
4992 D 1479 5008 D 1473 5035 D 1463 5051 D 1447 5057 D 1436 5057 D
1425 5051 D 1420 5046 D 1415 5035 D 1409 5008 D 1409 4992 D 1415 4965
D 1420 4955 D 1425 4949 D 1436 4944 D S
1447 4944 M 1457 4949 D 1463 4955 D 1468 4965 D 1473 4992 D 1473 5008
D 1468 5035 D 1463 5046 D 1457 5051 D 1447 5057 D S
1625 5167 M 1569 5167 D S
1625 5333 M 1569 5333 D S
1625 5500 M 1540 5500 D S
1206 5535 M 1216 5540 D 1232 5556 D 1232 5444 D S
1227 5551 M 1227 5444 D S
1206 5444 M 1254 5444 D S
1307 5556 M 1297 5502 D 1307 5513 D 1323 5519 D 1340 5519 D 1356 5513
D 1366 5502 D 1372 5486 D 1372 5476 D 1366 5460 D 1356 5449 D 1340
5444 D 1323 5444 D 1307 5449 D 1302 5454 D 1297 5465 D 1297 5470 D
1302 5476 D 1307 5470 D 1302 5465 D S
1340 5519 M 1350 5513 D 1361 5502 D 1366 5486 D 1366 5476 D 1361 5460
D 1350 5449 D 1340 5444 D S
1307 5556 M 1361 5556 D S
1307 5551 M 1334 5551 D 1361 5556 D S
1436 5556 M 1420 5551 D 1409 5535 D 1404 5508 D 1404 5492 D 1409 5465
D 1420 5449 D 1436 5444 D 1447 5444 D 1463 5449 D 1473 5465 D 1479
5492 D 1479 5508 D 1473 5535 D 1463 5551 D 1447 5556 D 1436 5556 D
1425 5551 D 1420 5545 D 1415 5535 D 1409 5508 D 1409 5492 D 1415 5465
D 1420 5454 D 1425 5449 D 1436 5444 D S
1447 5444 M 1457 5449 D 1463 5454 D 1468 5465 D 1473 5492 D 1473 5508
D 1468 5535 D 1463 5545 D 1457 5551 D 1447 5556 D S
1625 5667 M 1569 5667 D S
1625 5833 M 1569 5833 D S
1625 6000 M 1540 6000 D S
1206 6035 M 1216 6040 D 1232 6056 D 1232 5944 D S
1227 6051 M 1227 5944 D S
1206 5944 M 1254 5944 D S
1323 6056 M 1307 6051 D 1302 6040 D 1302 6024 D 1307 6013 D 1323 6008
D 1345 6008 D 1361 6013 D 1366 6024 D 1366 6040 D 1361 6051 D 1345
6056 D 1323 6056 D 1313 6051 D 1307 6040 D 1307 6024 D 1313 6013 D
1323 6008 D S
1345 6008 M 1356 6013 D 1361 6024 D 1361 6040 D 1356 6051 D 1345 6056
D S
1323 6008 M 1307 6003 D 1302 5997 D 1297 5987 D 1297 5965 D 1302 5954
D 1307 5949 D 1323 5944 D 1345 5944 D 1361 5949 D 1366 5954 D 1372
5965 D 1372 5987 D 1366 5997 D 1361 6003 D 1345 6008 D S
1323 6008 M 1313 6003 D 1307 5997 D 1302 5987 D 1302 5965 D 1307 5954
D 1313 5949 D 1323 5944 D S
1345 5944 M 1356 5949 D 1361 5954 D 1366 5965 D 1366 5987 D 1361 5997
D 1356 6003 D 1345 6008 D S
1436 6056 M 1420 6051 D 1409 6035 D 1404 6008 D 1404 5992 D 1409 5965
D 1420 5949 D 1436 5944 D 1447 5944 D 1463 5949 D 1473 5965 D 1479
5992 D 1479 6008 D 1473 6035 D 1463 6051 D 1447 6056 D 1436 6056 D
1425 6051 D 1420 6046 D 1415 6035 D 1409 6008 D 1409 5992 D 1415 5965
D 1420 5954 D 1425 5949 D 1436 5944 D S
1447 5944 M 1457 5949 D 1463 5954 D 1468 5965 D 1473 5992 D 1473 6008
D 1468 6035 D 1463 6046 D 1457 6051 D 1447 6056 D S
1625 6166 M 1569 6166 D S
1625 6333 M 1569 6333 D S
1625 6500 M 1540 6500 D S
1195 6535 M 1200 6530 D 1195 6524 D 1190 6530 D 1190 6535 D 1195 6546
D 1200 6551 D 1216 6557 D 1238 6557 D 1254 6551 D 1259 6546 D 1265
6535 D 1265 6524 D 1259 6514 D 1243 6503 D 1216 6492 D 1206 6487 D
1195 6476 D 1190 6460 D 1190 6444 D S
1238 6557 M 1248 6551 D 1254 6546 D 1259 6535 D 1259 6524 D 1254 6514
D 1238 6503 D 1216 6492 D S
1190 6455 M 1195 6460 D 1206 6460 D 1232 6449 D 1248 6449 D 1259 6455
D 1265 6460 D S
1206 6460 M 1232 6444 D 1254 6444 D 1259 6449 D 1265 6460 D 1265 6471
D S
1313 6535 M 1323 6540 D 1340 6557 D 1340 6444 D S
1334 6551 M 1334 6444 D S
1313 6444 M 1361 6444 D S
1436 6557 M 1420 6551 D 1409 6535 D 1404 6508 D 1404 6492 D 1409 6465
D 1420 6449 D 1436 6444 D 1447 6444 D 1463 6449 D 1473 6465 D 1479
6492 D 1479 6508 D 1473 6535 D 1463 6551 D 1447 6557 D 1436 6557 D
1425 6551 D 1420 6546 D 1415 6535 D 1409 6508 D 1409 6492 D 1415 6465
D 1420 6455 D 1425 6449 D 1436 6444 D S
1447 6444 M 1457 6449 D 1463 6455 D 1468 6465 D 1473 6492 D 1473 6508
D 1468 6535 D 1463 6546 D 1457 6551 D 1447 6557 D S
1625 6667 M 1569 6667 D S
1625 6833 M 1569 6833 D S
1625 7000 M 1540 7000 D S
1195 7035 M 1200 7029 D 1195 7024 D 1190 7029 D 1190 7035 D 1195 7045
D 1200 7051 D 1216 7056 D 1238 7056 D 1254 7051 D 1259 7045 D 1265
7035 D 1265 7024 D 1259 7013 D 1243 7002 D 1216 6992 D 1206 6986 D
1195 6976 D 1190 6960 D 1190 6944 D S
1238 7056 M 1248 7051 D 1254 7045 D 1259 7035 D 1259 7024 D 1254 7013
D 1238 7002 D 1216 6992 D S
1190 6954 M 1195 6960 D 1206 6960 D 1232 6949 D 1248 6949 D 1259 6954
D 1265 6960 D S
1206 6960 M 1232 6944 D 1254 6944 D 1259 6949 D 1265 6960 D 1265 6970
D S
1345 7045 M 1345 6944 D S
1350 7056 M 1350 6944 D S
1350 7056 M 1291 6976 D 1377 6976 D S
1329 6944 M 1366 6944 D S
1436 7056 M 1420 7051 D 1409 7035 D 1404 7008 D 1404 6992 D 1409 6965
D 1420 6949 D 1436 6944 D 1447 6944 D 1463 6949 D 1473 6965 D 1479
6992 D 1479 7008 D 1473 7035 D 1463 7051 D 1447 7056 D 1436 7056 D
1425 7051 D 1420 7045 D 1415 7035 D 1409 7008 D 1409 6992 D 1415 6965
D 1420 6954 D 1425 6949 D 1436 6944 D S
1447 6944 M 1457 6949 D 1463 6954 D 1468 6965 D 1473 6992 D 1473 7008
D 1468 7035 D 1463 7045 D 1457 7051 D 1447 7056 D S
1625 7167 M 1569 7167 D S
1625 7333 M 1569 7333 D S
1625 7500 M 1540 7500 D S
1195 7535 M 1200 7529 D 1195 7524 D 1190 7529 D 1190 7535 D 1195 7546
D 1200 7551 D 1216 7556 D 1238 7556 D 1254 7551 D 1259 7546 D 1265
7535 D 1265 7524 D 1259 7513 D 1243 7503 D 1216 7492 D 1206 7487 D
1195 7476 D 1190 7460 D 1190 7444 D S
1238 7556 M 1248 7551 D 1254 7546 D 1259 7535 D 1259 7524 D 1254 7513
D 1238 7503 D 1216 7492 D S
1190 7454 M 1195 7460 D 1206 7460 D 1232 7449 D 1248 7449 D 1259 7454
D 1265 7460 D S
1206 7460 M 1232 7444 D 1254 7444 D 1259 7449 D 1265 7460 D 1265 7471
D S
1297 7556 M 1297 7524 D S
1297 7535 M 1302 7546 D 1313 7556 D 1323 7556 D 1350 7540 D 1361 7540
D 1366 7546 D 1372 7556 D S
1302 7546 M 1313 7551 D 1323 7551 D 1350 7540 D S
1372 7556 M 1372 7540 D 1366 7524 D 1345 7497 D 1340 7487 D 1334 7471
D 1334 7444 D S
1366 7524 M 1340 7497 D 1334 7487 D 1329 7471 D 1329 7444 D S
1436 7556 M 1420 7551 D 1409 7535 D 1404 7508 D 1404 7492 D 1409 7465
D 1420 7449 D 1436 7444 D 1447 7444 D 1463 7449 D 1473 7465 D 1479
7492 D 1479 7508 D 1473 7535 D 1463 7551 D 1447 7556 D 1436 7556 D
1425 7551 D 1420 7546 D 1415 7535 D 1409 7508 D 1409 7492 D 1415 7465
D 1420 7454 D 1425 7449 D 1436 7444 D S
1447 7444 M 1457 7449 D 1463 7454 D 1468 7465 D 1473 7492 D 1473 7508
D 1468 7535 D 1463 7546 D 1457 7551 D 1447 7556 D S
1625 7666 M 1569 7666 D S
1625 7833 M 1569 7833 D S
1625 8000 M 1540 8000 D S
1195 8035 M 1200 8030 D 1195 8024 D 1190 8030 D 1190 8035 D 1195 8046
D 1200 8051 D 1216 8057 D 1238 8057 D 1254 8051 D 1259 8040 D 1259
8024 D 1254 8014 D 1238 8008 D 1222 8008 D S
1238 8057 M 1248 8051 D 1254 8040 D 1254 8024 D 1248 8014 D 1238 8008
D 1248 8003 D 1259 7992 D 1265 7982 D 1265 7965 D 1259 7955 D 1254
7949 D 1238 7944 D 1216 7944 D 1200 7949 D 1195 7955 D 1190 7965 D
1190 7971 D 1195 7976 D 1200 7971 D 1195 7965 D S
1254 7998 M 1259 7982 D 1259 7965 D 1254 7955 D 1248 7949 D 1238 7944
D S
1329 8057 M 1313 8051 D 1302 8035 D 1297 8008 D 1297 7992 D 1302 7965
D 1313 7949 D 1329 7944 D 1340 7944 D 1356 7949 D 1366 7965 D 1372
7992 D 1372 8008 D 1366 8035 D 1356 8051 D 1340 8057 D 1329 8057 D
1318 8051 D 1313 8046 D 1307 8035 D 1302 8008 D 1302 7992 D 1307 7965
D 1313 7955 D 1318 7949 D 1329 7944 D S
1340 7944 M 1350 7949 D 1356 7955 D 1361 7965 D 1366 7992 D 1366 8008
D 1361 8035 D 1356 8046 D 1350 8051 D 1340 8057 D S
1436 8057 M 1420 8051 D 1409 8035 D 1404 8008 D 1404 7992 D 1409 7965
D 1420 7949 D 1436 7944 D 1447 7944 D 1463 7949 D 1473 7965 D 1479
7992 D 1479 8008 D 1473 8035 D 1463 8051 D 1447 8057 D 1436 8057 D
1425 8051 D 1420 8046 D 1415 8035 D 1409 8008 D 1409 7992 D 1415 7965
D 1420 7955 D 1425 7949 D 1436 7944 D S
1447 7944 M 1457 7949 D 1463 7955 D 1468 7965 D 1473 7992 D 1473 8008
D 1468 8035 D 1463 8046 D 1457 8051 D 1447 8057 D S
1625 8167 M 1569 8167 D S
1625 8333 M 1569 8333 D S
1625 8500 M 1540 8500 D S
1195 8535 M 1200 8529 D 1195 8524 D 1190 8529 D 1190 8535 D 1195 8545
D 1200 8551 D 1216 8556 D 1238 8556 D 1254 8551 D 1259 8540 D 1259
8524 D 1254 8513 D 1238 8508 D 1222 8508 D S
1238 8556 M 1248 8551 D 1254 8540 D 1254 8524 D 1248 8513 D 1238 8508
D 1248 8502 D 1259 8492 D 1265 8481 D 1265 8465 D 1259 8454 D 1254
8449 D 1238 8444 D 1216 8444 D 1200 8449 D 1195 8454 D 1190 8465 D
1190 8470 D 1195 8476 D 1200 8470 D 1195 8465 D S
1254 8497 M 1259 8481 D 1259 8465 D 1254 8454 D 1248 8449 D 1238 8444
D S
1302 8535 M 1307 8529 D 1302 8524 D 1297 8529 D 1297 8535 D 1302 8545
D 1307 8551 D 1323 8556 D 1345 8556 D 1361 8551 D 1366 8540 D 1366
8524 D 1361 8513 D 1345 8508 D 1329 8508 D S
1345 8556 M 1356 8551 D 1361 8540 D 1361 8524 D 1356 8513 D 1345 8508
D 1356 8502 D 1366 8492 D 1372 8481 D 1372 8465 D 1366 8454 D 1361
8449 D 1345 8444 D 1323 8444 D 1307 8449 D 1302 8454 D 1297 8465 D
1297 8470 D 1302 8476 D 1307 8470 D 1302 8465 D S
1361 8497 M 1366 8481 D 1366 8465 D 1361 8454 D 1356 8449 D 1345 8444
D S
1436 8556 M 1420 8551 D 1409 8535 D 1404 8508 D 1404 8492 D 1409 8465
D 1420 8449 D 1436 8444 D 1447 8444 D 1463 8449 D 1473 8465 D 1479
8492 D 1479 8508 D 1473 8535 D 1463 8551 D 1447 8556 D 1436 8556 D
1425 8551 D 1420 8545 D 1415 8535 D 1409 8508 D 1409 8492 D 1415 8465
D 1420 8454 D 1425 8449 D 1436 8444 D S
1447 8444 M 1457 8449 D 1463 8454 D 1468 8465 D 1473 8492 D 1473 8508
D 1468 8535 D 1463 8545 D 1457 8551 D 1447 8556 D S
1625 8667 M 1569 8667 D S
1625 8833 M 1569 8833 D S
1625 9000 M 1540 9000 D S
1195 9035 M 1200 9029 D 1195 9024 D 1190 9029 D 1190 9035 D 1195 9046
D 1200 9051 D 1216 9056 D 1238 9056 D 1254 9051 D 1259 9040 D 1259
9024 D 1254 9013 D 1238 9008 D 1222 9008 D S
1238 9056 M 1248 9051 D 1254 9040 D 1254 9024 D 1248 9013 D 1238 9008
D 1248 9003 D 1259 8992 D 1265 8981 D 1265 8965 D 1259 8954 D 1254
8949 D 1238 8944 D 1216 8944 D 1200 8949 D 1195 8954 D 1190 8965 D
1190 8971 D 1195 8976 D 1200 8971 D 1195 8965 D S
1254 8997 M 1259 8981 D 1259 8965 D 1254 8954 D 1248 8949 D 1238 8944
D S
1361 9040 M 1356 9035 D 1361 9029 D 1366 9035 D 1366 9040 D 1361 9051
D 1350 9056 D 1334 9056 D 1318 9051 D 1307 9040 D 1302 9029 D 1297
9008 D 1297 8976 D 1302 8960 D 1313 8949 D 1329 8944 D 1340 8944 D
1356 8949 D 1366 8960 D 1372 8976 D 1372 8981 D 1366 8997 D 1356 9008
D 1340 9013 D 1334 9013 D 1318 9008 D 1307 8997 D 1302 8981 D S
1334 9056 M 1323 9051 D 1313 9040 D 1307 9029 D 1302 9008 D 1302 8976
D 1307 8960 D 1318 8949 D 1329 8944 D S
1340 8944 M 1350 8949 D 1361 8960 D 1366 8976 D 1366 8981 D 1361 8997
D 1350 9008 D 1340 9013 D S
1436 9056 M 1420 9051 D 1409 9035 D 1404 9008 D 1404 8992 D 1409 8965
D 1420 8949 D 1436 8944 D 1447 8944 D 1463 8949 D 1473 8965 D 1479
8992 D 1479 9008 D 1473 9035 D 1463 9051 D 1447 9056 D 1436 9056 D
1425 9051 D 1420 9046 D 1415 9035 D 1409 9008 D 1409 8992 D 1415 8965
D 1420 8954 D 1425 8949 D 1436 8944 D S
1447 8944 M 1457 8949 D 1463 8954 D 1468 8965 D 1473 8992 D 1473 9008
D 1468 9035 D 1463 9046 D 1457 9051 D 1447 9056 D S
6080 3284 M 6035 3550 D 5990 3800 D 5945 4034 D 5900 4267 D 5855 4467
D 5810 4667 D 5765 4850 D 5720 5017 D 5675 5167 D 5630 5317 D 5585
5433 D 5540 5550 D 5495 5650 D 5450 5733 D 5405 5800 D 5360 5867 D
5315 5900 D 5270 5933 D 5225 5950 D 5180 5950 D 5135 5933 D 5090 5917
D 5045 5867 D 5000 5817 D 4955 5750 D 4910 5667 D 4865 5567 D 4820
5467 D 4775 5333 D 4730 5200 D 4685 5050 D 4640 4883 D 4595 4700 D
4550 4517 D 4505 4334 D 4460 4167 D 4415 4017 D 4370 3867 D 4325 3734
D 4280 3617 D 4235 3500 D 4190 3400 D 4145 3317 D 4100 3233 D 4055
3167 D 4010 3117 D 3965 3067 D 3920 3033 D 3875 3016 D 3830 3000 D
3785 3000 D 3740 3000 D 3695 3000 D 3650 3000 D 3605 3000 D 3560 3000
D 3515 3000 D 3470 3000 D 3425 3000 D 3380 3000 D 3335 3000 D 3290
3000 D 3245 3000 D 3200 3000 D 3155 3000 D 3110 3000 D 3065 3000 D
3020 3000 D 2975 3000 D 2930 3000 D 2885 3000 D 2840 3000 D 2795 3000
D 2750 3000 D 2705 3000 D 2660 3000 D 2615 3000 D 2570 3000 D 2525
3000 D 2480 3000 D 2435 3000 D 2390 3000 D 2345 3000 D 2300 3000 D
2255 3000 D 2210 3000 D 2165 3000 D 2120 3000 D 2075 3000 D 2030 3000
D 1985 3000 D 1940 3000 D 1895 3000 D 1850 3000 D 1805 3000 D 1760
3000 D 1715 3000 D 1670 3000 D S
6080 3000 M 6035 3000 D 5990 3016 D 5945 3033 D 5900 3067 D 5855 3100
D 5810 3150 D 5765 3200 D 5720 3267 D 5675 3333 D 5630 3417 D 5585
3500 D 5540 3600 D 5495 3700 D 5450 3817 D 5405 3933 D 5360 4067 D
5315 4200 D 5270 4350 D 5225 4500 D 5180 4667 D 5135 4833 D 5090 5017
D 5045 5200 D 5000 5400 D 4955 5600 D 4910 5817 D 4865 6033 D 4820
6267 D 4775 6500 D 4730 6750 D 4685 7000 D 4640 7267 D 4595 7533 D
4550 7817 D 4505 8050 D 4460 8250 D 4415 8400 D 4370 8517 D 4325 8583
D 4280 8617 D 4235 8600 D 4190 8550 D 4145 8450 D 4100 8317 D 4055
8133 D 4010 7917 D 3965 7650 D 3920 7350 D 3875 7000 D 3830 6617 D
3785 6200 D 3740 5817 D 3695 5450 D 3650 5117 D 3605 4800 D 3560 4517
D 3515 4250 D 3470 4017 D 3425 3800 D 3380 3617 D 3335 3450 D 3290
3317 D 3245 3200 D 3200 3117 D 3155 3050 D 3110 3016 D 3065 3000 D
3020 3000 D 2975 3000 D 2930 3000 D 2885 3000 D 2840 3000 D 2795 3000
D 2750 3000 D 2705 3000 D 2660 3000 D 2615 3000 D 2570 3000 D 2525
3000 D 2480 3000 D 2435 3000 D 2390 3000 D 2345 3000 D 2300 3000 D
2255 3000 D 2210 3000 D 2165 3000 D 2120 3000 D 2075 3000 D 2030 3000
D 1985 3000 D 1940 3000 D 1895 3000 D 1850 3000 D 1805 3000 D 1760
3000 D 1715 3000 D 1670 3000 D S
6080 3000 M 6035 3000 D 5990 3000 D 5945 3000 D 5900 3000 D 5855 3000
D 5810 3000 D 5765 3000 D 5720 3000 D 5675 3000 D 5630 3000 D 5585
3000 D 5540 3000 D 5495 3000 D 5450 3000 D 5405 3000 D 5360 3000 D
5315 3000 D 5270 3000 D 5225 3000 D 5180 3000 D 5135 3000 D 5090 3000
D 5045 3000 D 5000 3000 D 4955 3000 D 4910 3000 D 4865 3000 D 4820
3000 D 4775 3000 D 4730 3000 D 4685 3000 D 4640 3000 D 4595 3000 D
4550 3000 D 4505 3016 D 4460 3067 D 4415 3134 D 4370 3233 D 4325 3350
D 4280 3500 D 4235 3667 D 4190 3867 D 4145 4083 D 4100 4334 D 4055
4600 D 4010 4900 D 3965 5217 D 3920 5567 D 3875 5933 D 3830 6333 D
3785 6734 D 3740 7100 D 3695 7417 D 3650 7700 D 3605 7934 D 3560 8133
D 3515 8283 D 3470 8400 D 3425 8467 D 3380 8500 D 3335 8483 D 3290
8433 D 3245 8333 D 3200 8200 D 3155 8017 D 3110 7800 D 3065 7533 D
3020 7267 D 2975 7000 D 2930 6750 D 2885 6500 D 2840 6267 D 2795 6033
D 2750 5817 D 2705 5600 D 2660 5400 D 2615 5200 D 2570 5017 D 2525
4833 D 2480 4667 D 2435 4500 D 2390 4350 D 2345 4200 D 2300 4067 D
2255 3933 D 2210 3817 D 2165 3700 D 2120 3600 D 2075 3500 D 2030 3417
D 1985 3333 D 1940 3267 D 1895 3200 D 1850 3150 D 1805 3100 D 1760
3067 D 1715 3033 D 1670 3016 D S
6080 3000 M 6035 3000 D 5990 3000 D 5945 3000 D 5900 3000 D 5855 3000
D 5810 3000 D 5765 3000 D 5720 3000 D 5675 3000 D 5630 3000 D 5585
3000 D 5540 3000 D 5495 3000 D 5450 3000 D 5405 3000 D 5360 3000 D
5315 3000 D 5270 3000 D 5225 3000 D 5180 3000 D 5135 3000 D 5090 3000
D 5045 3000 D 5000 3000 D 4955 3000 D 4910 3000 D 4865 3000 D 4820
3000 D 4775 3000 D 4730 3000 D 4685 3000 D 4640 3000 D 4595 3000 D
4550 3000 D 4505 3000 D 4460 3000 D 4415 3000 D 4370 3000 D 4325 3000
D 4280 3000 D 4235 3000 D 4190 3000 D 4145 3000 D 4100 3000 D 4055
3000 D 4010 3000 D 3965 3000 D 3920 3000 D 3875 3000 D 3830 3000 D
3785 3000 D 3740 3016 D 3695 3033 D 3650 3067 D 3605 3117 D 3560 3167
D 3515 3233 D 3470 3317 D 3425 3400 D 3380 3500 D 3335 3617 D 3290
3734 D 3245 3867 D 3200 4017 D 3155 4167 D 3110 4334 D 3065 4517 D
3020 4683 D 2975 4833 D 2930 4983 D 2885 5100 D 2840 5217 D 2795 5317
D 2750 5400 D 2705 5467 D 2660 5534 D 2615 5567 D 2570 5600 D 2525
5617 D 2480 5617 D 2435 5600 D 2390 5583 D 2345 5534 D 2300 5483 D
2255 5417 D 2210 5333 D 2165 5234 D 2120 5133 D 2075 5000 D 2030 4867
D 1985 4717 D 1940 4550 D 1895 4367 D 1850 4184 D 1805 3967 D 1760
3750 D 1715 3517 D 1670 3267 D S
3517 2165 M 3517 2007 D S
3524 2165 M 3524 2007 D S
3569 2120 M 3569 2060 D S
3494 2165 M 3614 2165 D 3614 2120 D 3607 2165 D S
3524 2089 M 3569 2089 D S
3494 2007 M 3547 2007 D S
3667 2165 M 3659 2157 D 3667 2149 D 3674 2157 D 3667 2165 D S
3667 2112 M 3667 2007 D S
3674 2112 M 3674 2007 D S
3644 2112 M 3674 2112 D S
3644 2007 M 3697 2007 D S
3772 2112 M 3757 2105 D 3749 2097 D 3742 2082 D 3742 2067 D 3749 2052
D 3757 2044 D 3772 2037 D 3787 2037 D 3802 2044 D 3809 2052 D 3817
2067 D 3817 2082 D 3809 2097 D 3802 2105 D 3787 2112 D 3772 2112 D S
3757 2105 M 3749 2089 D 3749 2060 D 3757 2044 D S
3802 2044 M 3809 2060 D 3809 2089 D 3802 2105 D S
3809 2097 M 3817 2105 D 3832 2112 D 3832 2105 D 3817 2105 D S
3749 2052 M 3742 2044 D 3734 2030 D 3734 2022 D 3742 2007 D 3764 1999
D 3802 1999 D 3824 1992 D 3832 1985 D S
3734 2022 M 3742 2015 D 3764 2007 D 3802 2007 D 3824 1999 D 3832 1985
D 3832 1977 D 3824 1962 D 3802 1955 D 3757 1955 D 3734 1962 D 3727
1977 D 3727 1985 D 3734 1999 D 3757 2007 D S
3892 2022 M 3884 2015 D 3892 2007 D 3899 2015 D 3892 2022 D S
3959 2135 M 3967 2127 D 3959 2120 D 3952 2127 D 3952 2135 D 3959 2149
D 3967 2157 D 3989 2165 D 4019 2165 D 4042 2157 D 4049 2142 D 4049
2120 D 4042 2105 D 4019 2097 D 3997 2097 D S
4019 2165 M 4034 2157 D 4042 2142 D 4042 2120 D 4034 2105 D 4019 2097
D 4034 2089 D 4049 2075 D 4057 2060 D 4057 2037 D 4049 2022 D 4042
2015 D 4019 2007 D 3989 2007 D 3967 2015 D 3959 2022 D 3952 2037 D
3952 2044 D 3959 2052 D 3967 2044 D 3959 2037 D S
4042 2082 M 4049 2060 D 4049 2037 D 4042 2022 D 4034 2015 D 4019 2007
D S
4117 2022 M 4109 2015 D 4117 2007 D 4124 2015 D 4117 2022 D S
5800 5334 M 5806 5344 D 5816 5355 D 5833 5355 D 5838 5350 D 5838 5339
D 5833 5318 D 5822 5280 D S
5827 5355 M 5833 5350 D 5833 5339 D 5827 5318 D 5816 5280 D S
5833 5318 M 5843 5339 D 5854 5350 D 5865 5355 D 5875 5355 D 5881 5350
D 5881 5344 D 5875 5339 D 5870 5344 D 5875 5350 D S
5928 5429 M 5928 5371 D S
5899 5400 M 5956 5400 D S
5982 5425 M 5985 5422 D 5982 5419 D 5979 5422 D 5979 5425 D 5982 5432
D 5985 5435 D 5995 5438 D 6008 5438 D 6018 5435 D 6021 5429 D 6021
5419 D 6018 5413 D 6008 5409 D 5998 5409 D S
6008 5438 M 6014 5435 D 6018 5429 D 6018 5419 D 6014 5413 D 6008 5409
D 6014 5406 D 6021 5400 D 6024 5393 D 6024 5384 D 6021 5377 D 6018
5374 D 6008 5371 D 5995 5371 D 5985 5374 D 5982 5377 D 5979 5384 D
5979 5387 D 5982 5390 D 5985 5387 D 5982 5384 D S
6018 5403 M 6021 5393 D 6021 5384 D 6018 5377 D 6014 5374 D 6008 5371
D S
6093 5414 M 6082 5403 D 6072 5387 D 6061 5366 D 6056 5339 D 6056 5318
D 6061 5291 D 6072 5269 D 6082 5253 D 6093 5243 D S
6082 5403 M 6072 5382 D 6066 5366 D 6061 5339 D 6061 5318 D 6066 5291
D 6072 5275 D 6082 5253 D S
6157 5393 M 6125 5280 D S
6163 5393 M 6131 5280 D S
6184 5360 M 6174 5318 D S
6141 5393 M 6222 5393 D 6216 5360 D 6216 5393 D S
6147 5339 M 6179 5339 D S
6109 5280 M 6190 5280 D 6200 5307 D 6184 5280 D S
6249 5280 M 6243 5285 D 6249 5291 D 6254 5285 D 6254 5280 D 6249 5269
D 6238 5259 D S
6340 5393 M 6307 5280 D S
6345 5393 M 6313 5280 D S
6324 5393 M 6361 5393 D S
6291 5280 M 6372 5280 D 6382 5312 D 6366 5280 D S
6414 5414 M 6425 5403 D 6436 5387 D 6446 5366 D 6452 5339 D 6452 5318
D 6446 5291 D 6436 5269 D 6425 5253 D 6414 5243 D S
6425 5403 M 6436 5382 D 6441 5366 D 6446 5339 D 6446 5318 D 6441 5291
D 6436 5275 D 6425 5253 D S
4563 8371 M 4568 8382 D 4579 8393 D 4595 8393 D 4600 8387 D 4600 8376
D 4595 8355 D 4584 8318 D S
4590 8393 M 4595 8387 D 4595 8376 D 4590 8355 D 4579 8318 D S
4595 8355 M 4606 8376 D 4616 8387 D 4627 8393 D 4638 8393 D 4643 8387
D 4643 8382 D 4638 8376 D 4633 8382 D 4638 8387 D S
4690 8466 M 4690 8408 D S
4661 8437 M 4719 8437 D S
4751 8463 M 4758 8466 D 4767 8476 D 4767 8408 D S
4764 8473 M 4764 8408 D S
4751 8408 M 4780 8408 D S
4856 8451 M 4845 8441 D 4834 8425 D 4824 8403 D 4818 8376 D 4818 8355
D 4824 8328 D 4834 8307 D 4845 8291 D 4856 8280 D S
4845 8441 M 4834 8419 D 4829 8403 D 4824 8376 D 4824 8355 D 4829 8328
D 4834 8312 D 4845 8291 D S
4920 8430 M 4888 8318 D S
4925 8430 M 4893 8318 D S
4947 8398 M 4936 8355 D S
4904 8430 M 4984 8430 D 4979 8398 D 4979 8430 D S
4909 8376 M 4941 8376 D S
4872 8318 M 4952 8318 D 4963 8344 D 4947 8318 D S
5011 8318 M 5006 8323 D 5011 8328 D 5016 8323 D 5016 8318 D 5011 8307
D 5000 8296 D S
5102 8430 M 5070 8318 D S
5107 8430 M 5075 8318 D S
5086 8430 M 5124 8430 D S
5054 8318 M 5134 8318 D 5145 8350 D 5129 8318 D S
5177 8451 M 5187 8441 D 5198 8425 D 5209 8403 D 5214 8376 D 5214 8355
D 5209 8328 D 5198 8307 D 5187 8291 D 5177 8280 D S
5187 8441 M 5198 8419 D 5203 8403 D 5209 8376 D 5209 8355 D 5203 8328
D 5198 8312 D 5187 8291 D S
2418 8371 M 2423 8382 D 2434 8393 D 2450 8393 D 2455 8387 D 2455 8376
D 2450 8355 D 2439 8318 D S
2445 8393 M 2450 8387 D 2450 8376 D 2445 8355 D 2434 8318 D S
2450 8355 M 2461 8376 D 2471 8387 D 2482 8393 D 2493 8393 D 2498 8387
D 2498 8382 D 2493 8376 D 2488 8382 D 2493 8387 D S
2516 8437 M 2574 8437 D S
2606 8463 M 2613 8466 D 2622 8476 D 2622 8408 D S
2619 8473 M 2619 8408 D S
2606 8408 M 2635 8408 D S
2711 8451 M 2700 8441 D 2689 8425 D 2679 8403 D 2673 8376 D 2673 8355
D 2679 8328 D 2689 8307 D 2700 8291 D 2711 8280 D S
2700 8441 M 2689 8419 D 2684 8403 D 2679 8376 D 2679 8355 D 2684 8328
D 2689 8312 D 2700 8291 D S
2775 8430 M 2743 8318 D S
2780 8430 M 2748 8318 D S
2802 8398 M 2791 8355 D S
2759 8430 M 2839 8430 D 2834 8398 D 2834 8430 D S
2764 8376 M 2796 8376 D S
2727 8318 M 2807 8318 D 2818 8344 D 2802 8318 D S
2866 8318 M 2861 8323 D 2866 8328 D 2871 8323 D 2871 8318 D 2866 8307
D 2855 8296 D S
2957 8430 M 2925 8318 D S
2962 8430 M 2930 8318 D S
2941 8430 M 2979 8430 D S
2909 8318 M 2989 8318 D 3000 8350 D 2984 8318 D S
3032 8451 M 3042 8441 D 3053 8425 D 3064 8403 D 3069 8376 D 3069 8355
D 3064 8328 D 3053 8307 D 3042 8291 D 3032 8280 D S
3042 8441 M 3053 8419 D 3058 8403 D 3064 8376 D 3064 8355 D 3058 8328
D 3053 8312 D 3042 8291 D S
1855 5784 M 1861 5794 D 1871 5805 D 1888 5805 D 1893 5800 D 1893 5789
D 1888 5768 D 1877 5730 D S
1882 5805 M 1888 5800 D 1888 5789 D 1882 5768 D 1871 5730 D S
1888 5768 M 1898 5789 D 1909 5800 D 1920 5805 D 1930 5805 D 1936 5800
D 1936 5794 D 1930 5789 D 1925 5794 D 1930 5800 D S
1954 5850 M 2011 5850 D S
2037 5875 M 2040 5872 D 2037 5869 D 2034 5872 D 2034 5875 D 2037 5882
D 2040 5885 D 2050 5888 D 2063 5888 D 2073 5885 D 2076 5879 D 2076
5869 D 2073 5863 D 2063 5859 D 2053 5859 D S
2063 5888 M 2069 5885 D 2073 5879 D 2073 5869 D 2069 5863 D 2063 5859
D 2069 5856 D 2076 5850 D 2079 5843 D 2079 5834 D 2076 5827 D 2073
5824 D 2063 5821 D 2050 5821 D 2040 5824 D 2037 5827 D 2034 5834 D
2034 5837 D 2037 5840 D 2040 5837 D 2037 5834 D S
2073 5853 M 2076 5843 D 2076 5834 D 2073 5827 D 2069 5824 D 2063 5821
D S
2148 5864 M 2137 5853 D 2127 5837 D 2116 5816 D 2111 5789 D 2111 5768
D 2116 5741 D 2127 5719 D 2137 5703 D 2148 5693 D S
2137 5853 M 2127 5832 D 2121 5816 D 2116 5789 D 2116 5768 D 2121 5741
D 2127 5725 D 2137 5703 D S
2212 5843 M 2180 5730 D S
2218 5843 M 2186 5730 D S
2239 5810 M 2229 5768 D S
2196 5843 M 2277 5843 D 2271 5810 D 2271 5843 D S
2202 5789 M 2234 5789 D S
2164 5730 M 2245 5730 D 2255 5757 D 2239 5730 D S
2304 5730 M 2298 5735 D 2304 5741 D 2309 5735 D 2309 5730 D 2304 5719
D 2293 5709 D S
2395 5843 M 2362 5730 D S
2400 5843 M 2368 5730 D S
2379 5843 M 2416 5843 D S
2346 5730 M 2427 5730 D 2437 5762 D 2421 5730 D S
2469 5864 M 2480 5853 D 2491 5837 D 2501 5816 D 2507 5789 D 2507 5768
D 2501 5741 D 2491 5719 D 2480 5703 D 2469 5693 D S
2480 5853 M 2491 5832 D 2496 5816 D 2501 5789 D 2501 5768 D 2496 5741
D 2491 5725 D 2480 5703 D S
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