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Fm(+)2024 944 y(1)p 2024 988 49 4 v 2024 1080 a(4)2099 928 y Fd(X)2103 1113 y Fl(i)p Fc(6)p Fk(=)p Fl(j)2246 944 y Fg(\027)2294 959 y Fl(ij)2354 944 y Fm(\()p Fg(\027)2440 959 y Fl(ij)2523 944 y Fe(\000)23 b Fm(1\))p 2246 988 464 4 v 2271 1089 a(\()p Fg(x)2364 1104 y Fl(i)2415 1089 y Fe(\000)g Fg(x)2570 1104 y Fl(j)2607 1089 y Fm(\))2644 1041 y Fk(2)2719 1011 y Fm(\()2800 944 y(1)p 2767 988 114 4 v 2767 1080 a Fg(m)2852 1095 y Fl(i)2913 1011 y Fm(+)3058 944 y(1)p 3021 988 122 4 v 3021 1080 a Fg(m)3106 1095 y Fl(j)3153 1011 y Fm(\)+)1047 1360 y(+)1133 1292 y(1)p 1133 1336 49 4 v 1133 1428 a(2)1343 1277 y Fd(X)1208 1461 y Fl(i)p Fc(6)p Fk(=)p Fl(j;i)p Fc(6)p Fk(=)p Fl(k)r(;j)t Fc(6)p Fk(=)p Fl(k)1950 1292 y Fg(\027)1998 1307 y Fl(ij)2058 1292 y Fg(\027)2106 1307 y Fl(j)t(k)p 1624 1336 883 4 v 1624 1428 a Fg(m)1709 1443 y Fl(j)1746 1428 y Fm(\()p Fg(x)1839 1443 y Fl(j)1898 1428 y Fe(\000)g Fg(x)2053 1443 y Fl(i)2081 1428 y Fm(\)\()p Fg(x)2212 1443 y Fl(j)2271 1428 y Fe(\000)g Fg(x)2426 1443 y Fl(k)2469 1428 y Fm(\))2517 1360 y Fg(:)898 b Fm(\(1\))24 1607 y(The)33 b(ground)g(state)g(w)m(a)m (v)m(e)h(function)e(is)g(of)g(the)h(Calogero)f(t)m(yp)s(e:)515 1892 y(\011)591 1907 y Fk(0)630 1892 y Fm(\()p Fg(x)723 1907 y Fk(1)763 1892 y Fg(;)17 b(:::;)g(x)987 1907 y Fl(n)1034 1892 y Fm(\))27 b(=)1206 1809 y Fd(Y)1203 1991 y Fl(i)28 b Fm(0)p Fg(;)1268 1974 y Fm(\()p Fg(ii)p Fm(\))196 b Fg(\027)1654 1989 y Fk(1)1721 1974 y Fm(=)28 b Fe(\000)p Fm(\()1950 1906 y Fg(N)2028 1921 y Fk(2)p 1950 1950 118 4 v 1950 2042 a Fg(N)2028 2057 y Fk(1)2077 1974 y Fm(\))g Fg(<)g Fm(0)p Fg(:)24 2226 y Fm(Their)k(ph)m(ysical)h(implications)c (are)j(summarized)g(in)f(T)-8 b(able)33 b(1.)170 2496 y(T)-8 b(able)32 b(1.)p 24 2606 4247 4 v 162 2690 a Fg(\027)210 2705 y Fk(1)471 2690 y Fg(\024)224 b(\025)621 b(\017)1468 2705 y Fk(0)2903 2690 y Fm(Commen)m(ts)p 24 2850 V 115 2895 a Fl(N)171 2904 y Fa(2)206 2895 y Fc(\000)p Fk(1)p 115 2911 181 4 v 115 2969 a Fl(N)171 2978 y Fa(1)206 2969 y Fc(\000)p Fk(1)437 2935 y Fm(+1)99 b Fg(\025)28 b(>)f Fm(0)338 b(2)p Fg(N)1363 2950 y Fk(1)1403 2935 y Fg(N)1481 2950 y Fk(2)1548 2935 y Fg(>)27 b Fm(0)480 b(Ph)m(ysical)32 b(solution,)g(no)g(three-b)s(o)s(dy)h(in)m(teraction.) 73 3183 y Fe(\000)p Fm(\()198 3144 y Fl(N)254 3153 y Fa(2)p 199 3160 91 4 v 199 3218 a Fl(N)255 3227 y Fa(1)299 3183 y Fm(\))100 b(+1)179 b Fe(\000)383 b Fg(N)1279 3198 y Fk(1)1340 3183 y Fm(+)22 b Fg(N)1516 3198 y Fk(2)1583 3183 y Fg(>)28 b Fm(0)360 b(Ph)m(ysical)32 b(solution,)g(with)g(a)g (three-b)s(o)s(dy)h(in)m(teraction.)115 3393 y Fl(N)171 3402 y Fa(2)206 3393 y Fc(\000)p Fk(1)p 115 3409 181 4 v 115 3466 a Fl(N)171 3475 y Fa(1)206 3466 y Fc(\000)p Fk(1)442 3432 y Fm(-)f(1)185 b Fe(\000)626 b Fm(0)545 b(Unph)m(ysical)33 b(solution,)e(with)h(a)h(three-b)s(o)s(dy)g(in)m (teraction.)73 3680 y Fe(\000)p Fm(\()198 3641 y Fl(N)254 3650 y Fa(2)p 199 3657 91 4 v 199 3715 a Fl(N)255 3724 y Fa(1)299 3680 y Fm(\))122 b(-1)e Fg(\025)28 b(<)f Fm(0)100 b Fg(N)1076 3695 y Fk(1)1137 3680 y Fm(+)22 b Fg(N)1313 3695 y Fk(2)1375 3680 y Fe(\000)h Fm(2)p Fg(N)1602 3695 y Fk(1)1641 3680 y Fg(N)1719 3695 y Fk(2)1786 3680 y Fg(<)28 b Fm(0)183 b(Unph)m(ysical)33 b(solution,)e(no)i(three-b)s(o)s (dy)g(in)m(teraction.)p 24 3848 4247 4 v 24 4023 a(A)f(few)h(remarks)g (are)g(in)f(order.)24 4223 y Ff(R)-5 b(emark)34 b(1)p Fm(.)43 b(Notice)32 b(that)h(solution)e(\()p Fg(i)p Fm(\))h(requires)i Fg(N)1978 4238 y Fk(1)p Fl(;)p Fk(2)2100 4223 y Fe(\025)28 b Fm(2.)24 4422 y Ff(R)-5 b(emark)26 b(2)p Fm(.)41 b(If)24 b(the)h(generalized)f(strong-w)m(eak)h(dualit)m(y)e(condition)g(is)h (imp)s(osed,)h(i.e.)41 b Fg(\027)3223 4437 y Fl(I)3263 4422 y Fg(\027)3311 4437 y Fl(J)3388 4422 y Fm(=)27 b(1,)24 4621 y(\()p Fg(I)35 b Fe(6)p Fm(=)28 b Fg(J)36 b Fm(=)28 b(1)p Fg(;)17 b Fm(2)p Fg(;)g(:::F)d Fm(\),)31 b(then)j(it)d(follo)m (ws)h(that)g(it)f(can)i(b)s(e)g(satis\014ed)g(for)f Fg(F)41 b Fm(=)28 b(2)k(only)-8 b(.)24 4820 y Ff(R)j(emark)42 b(3)p Fm(.)69 b(In)42 b(Refs.[7,9,10])f(w)m(e)h(sho)m(w)m(ed)i(that)d (there)h(existed)g(a)f(critical)e(p)s(oin)m(t)h Fg(\017)3264 4835 y Fk(0)p Fl(R)3400 4820 y Fm(=)i(0)24 5020 y(at)k(whic)m(h)i(the)f (system)h(describ)s(ed)g(b)m(y)1592 4994 y(~)1566 5020 y Fg(H)8 b Fm(\()p Fg(!)t Fm(\))1796 5035 y Fl(R)1900 5020 y Fm(collapsed)46 b(completely)-8 b(,)49 b(i.e.)86 b(the)48 b(relativ)m(e)1746 5269 y(12)p eop %%Page: 13 13 13 12 bop 24 159 a Fm(momen)m(ta,)51 b(the)f(relativ)m(e)e(energy)h (and)g(the)h(relativ)m(e)e(co)s(ordinates)g(w)m(ere)i(all)d(zero)i(at)f (this)24 358 y(critical)32 b(p)s(oin)m(t.)51 b(The)36 b(ground)f(state)h(w)m(as)g(a)f(square-in)m(tegrable)g(function)f(only) h(for)g Fg(\017)3258 373 y Fk(0)p Fl(R)3383 358 y Fg(>)d Fm(0.)24 557 y(This)26 b(is)g(the)h(reason)f(wh)m(y)i(w)m(e)f(ascrib)s (e)f(the)h(term)f('unph)m(ysical')g(to)g(the)h(last)e(solutions)h(in)f (T)-8 b(able)24 756 y(1.)24 1160 y Fh(5)161 b(Conclusion)24 1529 y Fm(In)40 b(this)g(Letter)h(w)m(e)h(ha)m(v)m(e)f(studied)g(the)g 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b(W)-8 b(e)40 b(ha)m(v)m(e)i(paid) e(sp)s(ecial)24 3123 y(atten)m(tion)e(to)i(the)f(SU\(1,1\))h(structure) g(of)f(the)h(mo)s(del.)62 b(W)-8 b(e)40 b(ha)m(v)m(e)h(found)e(certain) g(relations)24 3322 y(b)s(et)m(w)m(een)31 b(SU\(1,1\))d(generators)h (that)g(are)g(univ)m(ersal)g(for)f(all)f(c)m(hoices)i(of)g(masses)g (and)g(coupling)24 3521 y(constan)m(ts.)80 b(W)-8 b(e)45 b(particularly)d(sho)m(w)k(ho)m(w)f(to)f(obtain)f(SU\(1,1\))h (generators)h(of)f(the)h(whole)24 3720 y(system)39 b(from)d(SU\(1,1\))i (generators)g(of)g(arbitrary)f(subsystem.)61 b(Moreo)m(v)m(er,)41 b(the)d(same)g(rela-)24 3920 y(tions)k(are)h(v)-5 b(alid)42 b(for)g(an)h(arbitrary)g(n)m(um)m(b)s(er)g(of)g(dimensions)f(and)h(for) g(all)e(p)s(oten)m(tials)h(that)24 4119 y(b)s(eha)m(v)m(e)37 b(as)g(a)f(kinetic)g(energy)h(term)f(under)h(the)f(dilatation)d (represen)m(ted)39 b(b)m(y)e(the)g(generator)24 4318 y Fg(T)81 4333 y Fk(0)120 4318 y Fm(.)46 b(There)34 b(is)f(only)f(one)i (di\013erence)g(b)s(et)m(w)m(een)h(one)f(and)f(higher)g(dimensions.)44 b(In)34 b(the)f(case)h(of)24 4518 y(one)j(dimension,)h(one)f(can)h (exclude)g(the)g(three-b)s(o)s(dy)g(in)m(teraction)e(b)s(et)m(w)m(een)k (particles)c(from)24 4717 y(the)42 b(b)s(eginning,)g(while)f(there)h (is)g(no)f(kno)m(wn)i(w)m(a)m(y)g(ho)m(w)f(to)g(do)f(this)h(in)f (dimensions)f(higher)24 4916 y(than)e(one.)61 b(Our)39 b(results)g(can)f(also)g(b)s(e)h(extended)h(to)e(other)h(systems)g (with)g(the)f(underlying)1746 5269 y(13)p eop %%Page: 14 14 14 13 bop 24 159 a Fm(conformal)30 b(or)i(sup)s(erconformal)g(symmetry) h([12].)170 628 y Fn(Ac)m(kno)m(wledgmen)m(t)24 827 y Fm(This)27 b(w)m(ork)h(w)m(as)g(supp)s(orted)g(b)m(y)g(the)f(Ministry)g (of)f(Science)i(and)f(T)-8 b(ec)m(hnology)28 b(of)e(the)i(Republic)24 1026 y(of)k(Croatia)f(under)i(con)m(tracts)h(No.)43 b(0098003)31 b(and)i(No.)43 b(0119261.)1746 5269 y(14)p eop %%Page: 15 15 15 14 bop 24 159 a Fh(References)72 498 y Fm([1])49 b(F.)32 b(Calogero,)g(J.)h(Math.)g(Ph)m(ys.)h Fn(10)f Fm(\(1969\))e(2191,)h (2197;)224 697 y(F.)g(Calogero,)g(J.)h(Math.)g(Ph)m(ys.)h Fn(12)f Fm(\(1971\))e(419)224 896 y(B.)i(Sutherland,)g(J.)f(Math.)h(Ph) m(ys.)i Fn(12)d Fm(\(1971\))g(246.)72 1179 y([2])49 b(M.)33 b(Lassigg,)f(Ph)m(ys.)j(Rev.)e(Lett.)g Fn(77)f Fm(\(1996\))g(526;)224 1378 y(N.F.Johnson,)h(L.Quiroga,)f(Ph)m(ys.)i(Rev.)f(Lett.)g Fn(74)g Fm(\(1995\))e(4277;)224 1577 y(H.)i(Azuma,)f(S.)h(Iso,)g(Ph)m (ys.)i(Lett.)d(B)h Fn(331)g Fm(\(1994\))e(107;)224 1776 y(N.)i(Ka)m(w)m(ak)-5 b(ami,S.K.Y)d(ang,)31 b(Ph)m(ys.)k(Rev.)e(Lett.)g Fn(67)f Fm(\(1991\))g(2493;)224 1976 y(G.W.)h(Gibb)s(ons,)e(P)-8 b(.K.)33 b(T)-8 b(o)m(wnsend,)35 b(Ph)m(ys.)g(Lett.)d(B)h Fn(454)g Fm(\(1999\))e(187;)224 2175 y(D.Birmingham,)e(K.S.)k(Gupta,)f (S.)h(Sen,)g(Ph)m(ys.)i(Lett.)d(B)h Fn(505)g Fm(\(2001\))e(191;)224 2374 y(H.)i(Ujino,)e(M.)i(W)-8 b(adati,)32 b(J.)h(Ph)m(ys.)h(So)s(c.)f (Jpn.)g Fn(65)g Fm(\(1996\))e(653;)224 2573 y(L.)i(Lap)s(oin)m(te,)f (L.)g(Vinet,)g(Comm.)g(Math.)h(Ph)m(ys.)h Fn(178)f Fm(\(1996\))e(425.) 72 2856 y([3])49 b(D.)32 b(Bernard,)h(Y.-S.W)-8 b(u,)33 b(ArXiv)g(cond-mat/9404025;)224 3055 y(M.V.N.Murth)m(y)-8 b(,)35 b(R.)d(Shank)-5 b(ar,)33 b(Ph)m(ys.)i(Rev.)e(Lett.)g Fn(73)f Fm(\(1994\))g(1574.)72 3337 y([4])49 b(F.D.M.)32 b(Haldane,)h(Ph)m(ys.)h(Rev.)f(Lett.)g Fn(67)p Fm(,)g(937)f(\(1991\);) 224 3537 y(Y.S.W)-8 b(u,)34 b Ff(ibid.)d Fn(73)p Fm(,)i(922)f (\(1994\).)72 3819 y([5])49 b(A.P)-8 b(.)33 b(P)m(olyc)m(hronak)m(os,)h (Ph)m(ys.)h(Rev.)e(Lett.)g Fn(69)f Fm(\(1992\))g(703;)224 4018 y(L.)h(Brink,)f(T.H.)h(Hansson,)h(M.)f(V)-8 b(asiliev,)31 b(Ph)m(ys.)j(Lett.)f(B)g Fn(286)f Fm(\(1992\))g(109;)224 4217 y(L.)g(Brink,)g(T.H.)h(Hansson,)g(S.E.)g(Konstein,)f(M.)h(V)-8 b(asiliev,)31 b(Nucl.)h(Ph)m(ys.)i(B)e Fn(384)g Fm(\(1993\))224 4417 y(591.)72 4699 y([6])49 b(P)-8 b(.)33 b(F)-8 b(orrester,)33 b(J.)g(Ph)m(ys.)h(A:)f(Math.)g(Gen.)f Fn(25)h Fm(\(1992\))e(L607)h(;) 224 4898 y(C.)h(F)-8 b(urtlehner,)33 b(S.)f(Ouvry)-8 b(,)34 b(Mo)s(d.Ph)m(ys.Lett.)h(B)p Fn(9)d Fm(\(1995\))g(503;)1746 5269 y(15)p eop %%Page: 16 16 16 15 bop 224 159 a Fm(D.)32 b(Sen,)i(Nucl.)e(Ph)m(ys.)i(B)f Fn(479)g Fm(\(1996\))e(554;)224 358 y(A.)i(Dasniers)f(de)h(V)-8 b(eigy)g(,)32 b(Nucl.)h(Ph)m(ys.)h(B)f Fn(483)g Fm(\(1997\))e(580;)224 557 y(S.)i(Maskhevic)m(h,)i(Ph)m(ys.)f(Lett.)f(A)g Fn(233)f Fm(\(1997\))g(30.)72 839 y([7])49 b(S.)33 b(Meljanac,)f(M.)h(Milek)m(o) m(vi)m(\023)-46 b(c)33 b(,)f(A.Samsaro)m(v,)h(Ph)m(ys.)h(Lett.)f(B)p Fn(573)g Fm(\(2003\))e(202.)72 1122 y([8])49 b(V.)33 b(de)g(Alfaro,)e(S.)i(F)-8 b(ubini,)31 b(G.)h(F)-8 b(urlan,)31 b(Nuo)m(v)m(o)j(Cimen)m(to)e(A)g Fn(34)h Fm(\(1976\))e(569;)224 1321 y(A.M.)23 b(P)m(erelomo)m(v,)h Ff(Gener)-5 b(alize)g(d)24 b(Coher)-5 b(ent)25 b(States)g(and)g(Their)f(Applic)-5 b(ations)21 b Fm(\(Springer,)224 1520 y(1986\).)72 1802 y([9])49 b(S.)41 b(Meljanac,)h(M.)f(Milek)m(o)m(vi)m(\023)-46 b(c)40 b(,)i(A.Samsaro)m(v,)h("A)d(m)m(ultisp)s(ecies)f(mo)s(del)g(in)h (arbitrary)224 2002 y(dimensions")32 b(\()g(submitted)g(to)g(Ph)m (ys.Rev.)k(B\).)24 2284 y([10])48 b(S.)33 b(Meljanac,)f(M.)h(Milek)m(o) m(vi)m(\023)-46 b(c)33 b(,)f(M.)h(Sto)5 b(ji)m(\023)-46 b(c)65 b(,)32 b(Eur.)h(Ph)m(ys.)i(J.)e(C)g Fn(60)f Fm(\(2002\))g(331;) 224 2483 y(V.)h(Bardek,)g(L.)f(Jonk)m(e,)h(S.)g(Meljanac,)f(M.)h(Milek) m(o)m(vi)m(\023)-46 b(c,)32 b(Ph)m(ys.Lett.)i(B)e Fn(531)g Fm(\(2002\))g(311.)24 2765 y([11])48 b(I.)33 b(Andri)m(\023)-46 b(c)32 b(,)h(D.)f(Jurman,)g(Ph)m(ys.Lett.)j(A)e Fn(313)f Fm(\(2003\))g(252.)24 3048 y([12])48 b(P)-8 b(.K.Ghosh,)33 b(J.Ph)m(ys.A)i Fn(34)p Fm(,)e(\(2001\))e(5583.)1746 5269 y(16)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF