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recent developments in semiconductor technology ( e.g. mbe and electron lithography ) opened the possibility to create totally confined electron systems , the so called artificial atoms or quantum dots @xcite . this is one of the examples of present - day - physics interest in confined finite systems , among which one can mention also the atomic and electronic traps@xcite , and the condensation of confined bosons@xcite . quantum dots exhibit very interesting properties like the possibility of varying their parameters ( number of electrons , applied fields , dot s geometry , temperature ) over a wide range , the observation of conductance oscillations@xcite and kohn s theorem@xcite , etc . theoretically , these systems have been studied mainly with the help of numerical methods . however , analytic approaches have proven to work extremely well providing , at the same time , a qualitative understanding of the quantum dynamics . among these approaches one can mention the semiclassical quantisation@xcite , regularised perturbation theory@xcite , pade approximant techniques@xcite , and the @xmath4-expansion@xcite . in the present paper , we continue the analytic - qualitative line of research and apply the @xmath4-expansion ( @xmath5 is the absolute value of the angular momentum ) to compute the energy levels of two and three anyons in a model parabolic dot . the particles interact through a coulombic ( @xmath2 ) repulsive potential . a magnetic field is applied perpendicularly to the plane of motion . numerical results for the two - anyon system were obtained by myrheim et al@xcite . exact analytic solutions at particular values of the coupling constants were found in@xcite . bohr - sommerfeld quantisation was applied to this system at low magnetic fields@xcite . we shall see that our method provides extremely accurate solutions for states with angular momentum @xmath6 . a picture for the `` geometry '' of the states ( the spatial distribution of probability ) is obtained also . in the three - anyon problem , however , to our knowledge there are no numerical or approximate calculations . we start from the hamiltonian of @xmath7 anyons moving in a two dimensional quantum dot in the presence of a perpendicular homogenous magnetic field . in the bosonic gauge , it is given by the expression @xcite    @xmath8    in which the vector potential is taken in the symmetric gauge ,    @xmath9    @xmath10 is the statistical vector potential ,    @xmath11    @xmath12 is the unit vector perpendicular to the plane of motion of the anyons , @xmath13 is the anyon s charge , @xmath14 is the anyonic parameter , @xmath15 is the frequency of the parabolic potential needed to confine the anyons in the dot and @xmath16 is the dielectric constant of the medium . a dimensionless hamiltonian is obtained by means of the change of variables @xmath17    @xmath18    where @xmath19 is the cyclotronic frequency , @xmath20 , and @xmath21 is the square root of the ratio between the coulombic and oscillator characteristic energies . the problem has one exactly solvable limit : a low - density limit , which we call the wigner limit , reached when @xmath22 . in the @xmath23 ( oscillator ) limit , the two - anyon problem is exactly solvable @xcite , whereas the three - anyon system has an infinite family of exact linear states @xcite . in real semiconductors , @xmath24 .    introducing jacobi coordinates , @xmath25 the centre - of - mass and relative motions are separated    @xmath26    where    @xmath27    is the centre of mass hamiltonian and    @xmath28    is the hamiltonian of the relative motion . we introduced the following notation : @xmath29 , and @xmath30 . we will obtain approximate expressions for the energy eigenvalues of @xmath31 for @xmath32 and @xmath33 as a function of @xmath34 by means of the @xmath0-expansion . in the two - anyon problem , we have only one jacobi coordinate , @xmath35 , and the hamiltonian of the internal motion reads    @xmath36    notice that @xmath37 . after the scaling transformation @xmath38 @xmath39 , we get    @xmath40 = \frac { ( \tilde\nu+1)^2}{\rho _ 1 ^ 2}+\frac 14 \rho _ 1 ^ 2 + \frac { \tilde \beta ^3}{\rho _ 1}\nonumber\\ & - & \frac 1{j^2 } \left ( \frac{\partial ^2}{\partial \rho _ 1 ^ 2}+ \frac 1{\rho _ 1}\frac \partial { \partial \rho _ 1 } \right ) ,   \end{aligned}\ ] ]    where @xmath41 . we have  renormalised " the coupling constant , @xmath42 , and the statistical parameter , @xmath43 , in order to take account of the coulomb repulsion and the statistical interaction in a nonperturbative way when taking the formal limit @xmath44 . we shall look for symmetric eigenfunctions of @xmath45 , i.e. @xmath46 shall be even .    in the @xmath47 limit , the only term surviving in the hamiltonian is the effective ( classical ) potential energy    @xmath48    minimising @xmath49 , we obtain the leading contribution to the energy , @xmath50 , where the radius of the `` bohr orbit '' is obtained from    @xmath51    substituting @xmath52 in the r.h.s . of ( 11 ) and expanding , we get    @xmath53    where the operator coefficients are given by    @xmath54    similar series are written for the wave function and the scaled energy , that is ,    @xmath55=\sum \limits_{i=0}^ { \infty } \frac{\epsilon_i}{|j|^{i/2}}.\end{aligned}\ ] ]    inserting ( 19 ) , ( 20 ) and the series expansion for the hamiltonian into the schdinger equation , we may compute the coefficients @xmath56 and @xmath57 in a systematic way . up to second order , for example , the system is described in terms of small oscillations around the equilibrium orbit , i.e. the wave function is    @xmath58    where @xmath59 is the angle associated to the vector @xmath60 , the @xmath61 are two - dimensional harmonic oscillator radial states with frequency @xmath62 , and the first two coefficients for the energy are    @xmath63    afterward , we may take account of anharmonicities . the results for the next two coefficients are the following    @xmath64 -\frac { ( 2n+1)}{2 \omega_1 ^ 5 \rho_{01}^4}\left [ 87n^2 \right . \nonumber\\ & + & 87n+86 + \frac { 12(\tilde\nu+1)^2 \left ( 87n^2 + 87n+86\right ) } { \rho_{01}^4}\nonumber\\ & + & \left.\frac { 4(\tilde\nu+1)^4 \left ( 713n^2 + 713n+709\right ) } { \rho_{01}^8 } \right]+ \frac { 9(2n+1)}{2 \omega_1 ^ 7 \rho_{01}^4 } \nonumber\\ & \times & \left ( 25n^2 + 25n+19 \right ) \left ( 1+\frac { 6(\tilde\nu+1)^2}{\rho_{01}^4}\right ) \nonumber\\ & \times & \left ( 1+\frac { 4(\tilde\nu+1)^2}{\rho_{01}^4}\right)^2   - \frac { 15(2n+1)}{8 \omega_1 ^ 9 \rho_{01}^4}\left ( 47n^2 + 47n+31 \right ) \nonumber\\ & \times & \left ( 1+\frac { 4(\tilde\nu+1)^2}{\rho_{01}^4}\right)^4.\end{aligned}\ ] ]    notice that in both wigner ( @xmath65 ) and oscillator ( @xmath66 ) limits the corrections @xmath67 and @xmath68 go to zero . the expressions found in @xcite are reproduced if we take @xmath69 . we show in fig . 1 the relative weight of @xmath68 in @xmath70 for the first states with @xmath71 and 6 . the parameter @xmath72 was fixed to 1/2 ( semions ) . the relative contribution of @xmath68 is never greater than @xmath73 or @xmath74 for @xmath71 or 6 respectively . this shows that the @xmath0-series is extremely well behaved . a comparison with the exact solutions found in @xcite is carried on in fig . 2 , where the relative difference @xmath75 is plotted against @xmath72 . the state with @xmath76 , @xmath77 is shown . it may be easily verified that @xmath78 , @xmath79 . the comparison shows that the relative error of our estimate is not greater than @xmath80 . the internal hamiltonian of the system of three anyons in jacobi coordinates @xmath81 and @xmath82 is written as    @xmath83 + \frac 14 \sum\limits_{i=1}^2\rho _ i^2 \nonumber\\ & + & 9\nu ^2\frac{\left ( \rho _ 1 ^ 2 + \rho _ 2 ^ 2\right ) ^2 + 4\left ( \rho _ 1 ^ 2\rho _ 2 ^ 2-\left ( \vec   \rho _ 1\cdot \vec \rho _ 2\right ) ^2\right ) } { \rho _ 1 ^ 2\left|   \vec \rho _ 1+\sqrt{3}\vec \rho _ 2\right| ^2\left| \vec \rho _ 1- \sqrt{3}\vec \rho_2\right| ^2}+ \nonumber\\ & & \beta ^3\left [ \frac 1{\rho _ 1}+\frac 2{\left| \vec \rho _ 1 + \sqrt{3}\vec \rho _ 2\right| } + \frac 2{\left| \vec \rho _ 1-\sqrt{3 } \vec \rho _ 2\right|}\right].\end{aligned}\ ] ]      @xmath85   = \frac 14\left ( \frac 1{\rho _ 1 ^ 2}+\frac 1{\rho _ 2 ^ 2}\right ) + \frac 14\left ( \rho _ 1 ^ 2+\rho _ 2 ^ 2\right ) \nonumber\\ & + & \tilde \beta ^3\left [   \frac 1{\rho _ 1}+\frac 2{\left| \vec \rho _ 1+\sqrt{3}\vec \rho _ 2\right| } + \frac 2{\left| \vec \rho _ 1-\sqrt{3}\vec \rho _ 2\right| } \right ] \nonumber \\ & + & 4\tilde\nu \frac{\rho _ 1 ^ 2\left ( 2 - 3\cos ^2\theta \right ) + 3\rho _ 2 ^ 2\left ( 2-\cos ^2\theta \right ) } { \left| \vec \rho _ 1+\sqrt{3 } \vec \rho_2\right| ^2\left| \vec \rho _ 1-\sqrt{3}\vec \rho _ 2 \right| ^2}+\frac { \tilde\nu}{\rho _ 1 ^ 2 } \nonumber\\ & + & 9\tilde\nu ^2\frac{\left (   \rho _ 1 ^ 2+\rho_2 ^ 2\right ) ^2 + 4\rho _ 1 ^ 2\rho _ 2 ^ 2\sin ^2\theta } { \rho _ 1 ^ 2\left| \vec \rho_1+\sqrt{3}\vec \rho _ 2\right| ^2\left|   \vec \rho _ 1-\sqrt{3}\vec \rho_2\right| ^2 } +   \frac 1j \left [ \imath\left ( \frac 1{\rho _ 1 ^ 2 }   - \frac 1{\rho _ 2 ^ 2}\right ) \frac \partial { \partial \theta } \right . \nonumber\\ & + & \tilde\nu \left ( 12\imath \frac{\rho _ 1\rho _ 2\sin 2\theta } { \left| \vec \rho _ 1+\sqrt{3}\vec \rho _ 2\right| ^2\left|   \vec \rho _ 1-\sqrt{3}\vec \rho_2\right| ^2}\left ( \rho _ 1 \frac \partial { \partial \rho _ 2}-\rho _ 2\frac \partial   { \partial \rho _ 1}\right ) \right.\nonumber\\ & + & \imath \frac 2{\rho _ 1 ^ 2}\frac \partial { \partial \theta } - \left . \left . 8\imath \frac{\rho _ 1 ^ 2\left ( 1 - 3\cos ^2\theta   \right)+3\rho _ 2 ^ 2\left ( 1+\cos ^2\theta \right ) } { \left|   \vec \rho _ 1+\sqrt{3}\vec \rho _ 2\right| ^2\left| \vec \rho _ 1- \sqrt{3}\vec \rho _ 2\right| ^2}\frac \partial { \partial \theta } \right)\right ] \nonumber\\ & + & \frac 1{j^2}\left [ -\left ( \frac{\partial ^2 } { \partial \rho _ 1 ^ 2}+\frac 1{\rho _ 1 } \frac \partial { \partial \rho _ 1}+\frac{\partial ^2 } { \partial \rho_2 ^ 2}+\frac 1{\rho _ 2 } \frac \partial { \partial \rho _ 2}+\right . \nonumber\\ & & \left . \left ( \frac 1{\rho_1 ^ 2}+\frac 1{\rho _ 2 ^ 2}\right )   \frac{\partial ^2}{\partial \theta ^2}\right)\right ] , \end{aligned}\ ] ]      the minimum of the classical potential entering ( 27 ) is reached in the configuration of an equilateral triangle ( @xmath89 , @xmath90 ) . we choose , for example , @xmath89 , @xmath91 . @xmath92 is obtained as the solution of the equation        @xmath97 , \\    h_3 & = & -\frac 1{\rho _ { 01}}\left ( \frac{\partial}{\partial y_1}+ \frac{\partial}{\partial y_2}\right ) + \frac 2{\rho _ { 01}^3}\left ( y_1+y_2\right ) \nonumber\\ & + & \frac{3\imath~{\rm sign}(j)}{\rho _ { 01}^4}\left ( y_1 ^ 2-y_2 ^ 2\right ) \frac \partial { \partial z}-\frac 1{\rho _ { 01}^5 } \left(y_1 ^ 3+y_2 ^ 3\right ) \nonumber\\ & - & \frac 1{64\rho _ { 01}}\left ( 1-\frac 1{\rho _ { 01}^4}\right ) \left ( 19y_1 ^ 3 + 3y_1 ^ 2y_2 + 33y_1y_2 ^ 2 + 9y_2 ^ 2 \right . \nonumber\\ & - & \left . 9\rho _ { 01}^2y_1z^2 + 21\rho_{01}^2y_2z^2\right )   - \frac { 9\tilde \nu^2}{64 \rho_{01}^5}\left ( y_2 ^ 3 - 5y_1 ^ 3 -21y_1 ^ 2 y_2 \right.\nonumber\\ & - & \left . 39 y_1 y_2 ^ 2 + 7\rho_{01}^2 y_1 z^2 - 11\rho_{01}^2 y_2 z^2   \right ) \nonumber\\ & + & \tilde\nu\left [ \frac { 3\imath~{\rm sign}(j)}{2 \rho_{01}^4 }   \left \ { \left ( 4y_1 ^ 2 - 2y_2 ^ 2 - 2y_1y_2-\rho_{01}^2z^2 \right )   \frac { \partial}{\partial z } \right.\right.\nonumber\\ & - & \left . \rho_{01}^2 z\left ( y_1 \frac{\partial } { \partial y_2}+y_2 \frac{\partial}{\partial y_1 } \right ) + 2\rho_{01}^2 y_2 z \frac{\partial}{\partial y_2 } \right\ }   \nonumber\\ & - & \frac{3}{32 \rho_{01}^5 } \left ( 21 y_1 ^ 3 + 15 y_2 ^ 3 + 5 y_1 ^ 2 y_2 + 23 y_1 y_2 ^ 2 \right.\nonumber\\ & + & \left.\left . \rho_{01}^2 y_1 z^2 - 13 \rho_{01}^2y_2 z^2\right ) \right ] ,   \\    h_4 & = & \frac 1{\rho _ { 01}^2}\left ( y_1\frac{\partial } { \partial y_1}+y_2\frac{\partial}{\partial y_2}\right ) - \frac 3{\rho _ { 01}^4}\left(y_1 ^ 2+y_2 ^ 2\right ) \frac{\partial ^2 } { \partial z^2 } \nonumber\\ & - & \frac{4\imath~{\rm sign}(j)}{\rho _ { 01}^5}\left (   y_1 ^ 3-y_2 ^ 3\right ) \frac \partial { \partial z}+\frac 5{4\rho _ { 01}^6 } \left ( y_1 ^ 4+y_2 ^ 4\right ) \nonumber\\ & + & \frac 1{256\rho _ { 01}^2}\left ( 1-\frac 1{\rho _ { 01}^4}\right )   \left ( \frac{329}4y_1 ^ 4 - 25y_1 ^ 3y_2 \right.\nonumber\\ & + & \frac{123}2y_1 ^ 2y_2 ^ 2 + 135y_1y_2 ^ 3 + \frac 94y_2 ^ 4-\frac{99}2\rho _ { 01}^2y_1 ^ 2z^2 \nonumber \\ & + & 27\rho _ { 01}^2y_1y_2z^2 + \left . \frac{141}2\rho _ { 01}^2y_2 ^ 2z^2+\frac{41}4\rho _ { 01}^4z^4 \right ) \nonumber\\ & + & \tilde\nu \left [ -\frac { \imath~{\rm sign}(j)}{8 \rho_{01}^5 } \left\ { 6\left(11y_1 ^ 3-y_1 ^ 2y_2 - 7y_1y_2 ^ 2 - 3y_2 ^ 3 \right.\right.\right . \nonumber\\ & - & \left . 3 \rho_{01}^2y_1z^2-\rho_{01}^2 y_2z^2\right)\frac{\partial}{\partial z } \nonumber\\ & + & \rho_{01}^2 z\left(9y_1 ^ 2 - 3y_2 ^ 2 - 18y_1y_2-\rho_{01}^2z^2   \right)\frac{\partial}{\partial y_1 } \nonumber\\ & - & \left . \rho_{01}^2 z \left(9y_1 ^ 2 - 27y_2 ^ 2 + 6y_1y_2-\rho_{01}^2z^2 \right ) \frac{\partial}{\partial y_2 } \right\ } \nonumber\\ & + & \frac{3}{512 \rho_{01}^6 } \left ( 503y_1 ^ 4 - 28y_1 ^ 3y_2 + 266y_1 ^ 2y_2 ^ 2 + 612y_1y_2 ^ 3 \right . \nonumber\\ & + & 183y_2 ^ 4 - 58 \rho_{01}^2y_1 ^ 2z^2 + 20 \rho_{01}^2y_1y_2z^2 - 154 \rho_{01}^2y_2 ^ 2z^2 \nonumber\\ & - & \left.\left . 41 \rho_{01}^4z^4 \right ) \right ]   + \frac{3\tilde\nu ^2}{1024\rho _ { 01}^6}\left ( 549y_1 ^ 4 - 411y_2 ^ 4- 84y_1 ^ 3y_2 \right.\nonumber\\ & + & 2718y_1 ^ 2y_2 ^ 2 + 1836y_1y_2 ^ 3 - 750\rho_{01}^2 y_1 ^ 2z^2 + 1266\rho_{01}^2y_2 ^ 2z^2 \nonumber\\ & + & 60\rho_{01}^2y_1y_2z^2 + \left . 37\rho_{01}^4z^4\right ) .   \end{aligned}\ ] ]                                these are the `` linear '' three - anyon states . we stress that they are obtained as harmonic excitations against the equilateral triangle configuration , and are not necessarily related to a cigar - like shape of the wave function ( @xmath122 ) as described in @xcite . the set of numbers \{@xmath123 } compatible with the symmetry constraints ( the wave function shall be symmetric ) are obtained upon comparison with harmonic - oscillator wave functions at @xmath69 , @xmath124 . details may be found in @xcite . an additional requirement is that the state at @xmath124 should be a linear state . for example , the lowest linear states are the following : @xmath125 ( the g.s . , starting from @xmath126 at the bosonic end ) , @xmath127 , and @xmath128 ( starting from @xmath129 ) , @xmath130 and @xmath131 ( starting from @xmath132 ) , @xmath133 , @xmath134 , and @xmath135 ( starting from @xmath136 ) , etc . the lowest state with @xmath137 is @xmath138 , which starts at @xmath139 . of course , states with small values of @xmath46 can not be described within our method .    in what follows , we restrict the analysis to levels with quantum numbers @xmath140 . this leaves only the linear anyonic states with @xmath141 , where @xmath142 is an integer @xcite . the geometry of the state is an equilateral triangle . it can be seen from ( 29 ) that the side of the triangle increases with @xmath72 when @xmath143 , and decreases when @xmath137 . thus , the coulomb repulsion is much more stronger for @xmath137 states , and the ordering of levels may dramatically change as @xmath34 is increased . on the other hand , for @xmath65 the side grows like @xmath144 and becomes independent of the anyonic parameter , as one expects . a strong coupling expansion @xcite shows that the leading contribution to the energy ( potential energy ) is @xmath145 , the next corrections ( quantum fluctuations ) are @xmath146 , the angular momentum and the statistical parameter enter the second order corrections , which are @xmath147 . we show in fig . 3 the relative weight of @xmath67 in @xmath70 for three semions in states with @xmath162 and @xmath76 . the numbers are similar to those appearing in the two - anyon problem . thus , we expect a similar accuracy to this order , i.e. one part in @xmath163 or better .    in fig . 4 , the levels with @xmath164 are drawn . @xmath34 is increased from 0.5 to 8 . notice that the coulomb effects are stronger for the state with negative @xmath1 , and that the levels become flatter ( as a function of @xmath72 ) as @xmath34 rises .    in conclusion , the energy levels of two and three anyons in a model parabolic dot were computed by means of the @xmath0- expansion . the qualitative picture emerging from the @xmath0- expansion is that of a rigid structure ( an orbit in the two - anyon system , an equilateral triangle for three anyons ) against which harmonic and anharmonic oscillations are developed . the coulomb repulsion is much stronger for negative- @xmath1 states . comparison with exact particular solutions for two anyons shows excellent agreement . 9 electronic mail : rperez@cidet.icmf.inf.cu electronic mail : agonzale@perseus.unalmed.edu.co n. f. johnson , j. phys . : condens . matter * 7 * , 965 ( 1995 ) . j. h. jefferson and w. husler , cond - mat/9705012 . s. becker et al . , * 66 * , 4902 ( 1995 ) ; h. j. kluge , nucl . . res . * b 98 * , 500 ( 1995 ) . c. c. bradley , c. a. sackett , j. j. tollet and r. g. hulet , phys . lett . * 75 * , 1687 ( 1995 ) ; k. b. davis , m. o. mewes , m. r. andrews et al , phys . lett . * 75 * , 3969 ( 1995 ) . m. a. kastner , rev . * 64 * , 849 ( 1992 ) . b. meurer , d. heitmann and k. ploog , phys . lett . * 68 * , 1371 ( 1992 ) . p. a. maksym , phys . rev . * b 53 * , 10871 ( 1996 ) . a. matulis and f. m. peeters , j. phys . : condens . matter * 6 * , 7751 ( 1994 ) . a. gonzalez , i. mikhailov , cond - mat/9706013 ; int . * b 11 * ( 1997 ) , to appear . a. gonzalez , j. phys . : condens . matter * 9 * , 4643 ( 1997 ) . a. gonzalez , b. partoens , f. m. peeters , cond - mat/9707258 ; phys . * 56 * ( 1997 ) , to appear . a. belov , yu . lozovik , a. gonzalez , phys . lett . * a 142 * , 389 ( 1989 ) . l. quiroga , a. camacho , a. gonzalez , j. phys : condens . matter * 7 * , 7517 ( 1995 ) . a. gonzalez , r. prez and p. fileviez , j. phys . : condens . matter * 9 * , 8465 ( 1997 ) . j. myrheim , e. halvorsen and a. vercin , phys . * b 278 * , 171 ( 1992 ) . a. vercin , phys . lett . * b 260 * , 120 ( 1991 ) . l. salasnich , mod . * b 11 * , 269 ( 1997 ) . j. myrheim , anyons , in course on geometric phases , ictp , trieste , 1993 . j. m. leinaas and j. myrheim , nuovo cimento * 37 b * , 1 ( 1977 ) . y. s. wu , phys . 53 * , 111 ( 1984 ) ; m. d. jhonson and g. s. canright , phys . rev . * d 41 * , 6870 ( 1990 ) ; a. p. polychronakos , phys . * b 264 * , 362 ( 1991 ) . m. sporre , j. verbaarschot and i. zahed , nucl . phys . * b 389 * , 645 ( 1993 ) . a. gonzalez , l. quiroga and b. rodriguez , few - body systems * 21 * , 47 ( 1996 ) .
the energy levels of two and three anyons in a two - dimensional parabolic quantum dot and a perpendicular magnetic field are computed as power series in @xmath0 , where @xmath1 is the angular momentum . 
 the particles interact repulsively through a coulombic ( @xmath2 ) potential . in the two - anyon problem , the reached accuracy is better than one part in @xmath3 . for three anyons , 
 we study the combined effects of anyon statistics and coulomb repulsion in the `` linear '' anyonic states .