A perturbative approach to the quantum elliptic Calogero-Sutherland m odel

arXiv:math-ph/0205042v1 29 May 2002

A perturbative approach to the quantum elliptic Calogero-Sutherland m odel

J.Fernandez N u~nez,W .G arc a Fuertes,A .M .Perelom ov

D epartam ento de F sica, Facultad de C iencias, U niversidad de O viedo,E-33007 O viedo,Spain

A bstract

W e solve perturbatively the quantum elliptic C alogero-Sutherland m odelin the regim e in w hich the quotient betw een the realand im aginary sem iperiods ofthe W eierstrass P function is sm all.

T he class ofquantum and classicalintegrable system s know n as C alogero-Sutherland m odels w ere rst introduced by these authors in the seventies,[1,2],and have since then attracted considerable interest, both for their intrinsic m athem aticalbeauty and depth and for the num erous applications found,w hich range from condensed m atterto supersym m etricYang-M illstheory and strings/M -theory,see forinstance [3]. T hese m odels, w hose integrability stem s from the fact that their H am iltonians coincide w ith the Laplace-B eltram i operators on som e sym m etric spaces, can be form ulated for an arbitrary num ber of particles. T here are ve possible interaction potentials: (a) V (q) = q²; (b) V (q) = sinh² q; (c) V (q)= sin² q;(d)V (q)= P (q),P being the elliptic W eierstrass function;and (e)V (q)= q²+ !²q². In allcases,the particle coordinates enter in these potentials in com binations w hich are given by the roots ofsom e sim ple Lie algebra,see [4]for details.

T he m ost generalam ong these system s is the elliptic one: allthe other potentials arise as suitable in nite lim its ofone ofboth sem iperiods ofthe P -function. N evertheless,to solve the quantum elliptic C alogero-Sutherland m odelis,even in the m ost sim ple cases, a di cult task. T he elliptic problem for only one particle and specialvalues ofthe coupling constant w as solved for Lam e m ore than one century ago in the course of his analysis of the stationary distribution of tem peratures on an ellipsoid [5],and later in greater generality by H erm ite [6]. A part from this,one ofthe m ost successfulresults obtained so far is the exact solution for the case ofthree particles given in [7,8]. H ow ever,the nalform ula for the eigenvalues is a very com plicated expression involving transcendentalfunctions,and it is therefore quite hard to grasp its content. In this letter, w e w ill show that in som e cases it is possible to take advantage of the solutions of the trigonom etric C alogero-Sutherland m odeldeveloped in [9,10,11] to give approxim ate solutions to the elliptic problem . T hese solutions are expressed by sim ple rational functions and the procedure for nding them is fairly elem entary.

W e begin by recalling som e basic facts taken from [9,10,11]. T he trigonom etric quantum C alogero- Sutherland m odel of A_n-type describes the m utual interaction of N = n + 1 particles m oving on the circle. T he coordinates ofthese particles are q_j,j = 1;:::;N ,and the Schrodinger equation reads

H^trig = E^trig( )

O n leave of absence from the Institute for T heoretical and E xperim ental P hysics, 117259, M oscow , R ussia. C urrent E -m ailaddress: perelom o@ dftuz.unizar.es

H^trig= 1

2 + ( 1)

XN

j< k

sin²(q_j q_k); = XN

j= 1

@²

@q_j²: (1)

T he quantum eigenstates depend on a n-tuple ofquantum num bers m = (m₁;m₂;:::;m_n) H^trig _m = E_m^trig( ) _m

E_m^trig( ) = 2( + ; + ); (2)

w here is the highest w eight ofthe representation ofA_n labelled by m ,i. e. = ^{P n}_{i= 1}m_{i i}w ith _ithe fundam entalw eights ofA_n,and is the standard W eylvector, = ¹₂^P _{2 R}+ ,w ith the sum extended over allthe positive roots ofAn. T he center-of-m ass-fram e eigenfunctions are ofthe form

m (q_i)=

n YN

j< k

sin(q_j q_k) o

P_m (z_i); (3)

w here the z_i;i= 1;2;:::;n,variables are the elem entary sym m etric functions ofx_j= e^2iqj,

z_p= XN

j₁< j₂< :::< jp

x_j1x_j2:::x_jp; (4)

and P_m are the generalized G egenbauerpolynom ialsrelated to A_n. Som e propertiesofthese polynom ials, as w ellas explicit exam ples, can be found in [9,10,11,12,13]. W e m ention, in particular, that each product ofthe form z_iP_m can be decom posed as a linear com bination ofG egenbauer polynom ials w hich m im ics the structure ofthe C lebsch-G ordan series for the irreducible representations ofSU (n). H ere w e only quote tw o ofthese recurrence relations w hich are specially relevant for w hat follow s:

z₁P_m = XN

j= 1

c_j;mP_{m +}

j

z_nP_m = XN

j= 1

~

c_j;mP_{m j}; (5)

in these form ulas _jis the n-tuple w hose i-th elem ent is _{i;j i;j 1} and c_j;m; ~c_j;m are som e coe cients w hich can be obtained by know n algorithm s, see [14, 15, 16]. N ote that as z_n = z₁^y both recurrence relations are sim ply related.

T he elliptic m odel related to A_n has the sam e structure. T he Schrodinger equation is H ^ell = E^ell( ) ,the H am iltonian being

H^ell= 1

2 + ( 1)

XN

j< k

P (q_j q_k;!1;!₂); (6)

w here P (z;!₁;!₂) is the W eierstrass elliptic function w ith sem iperiods chosen to be !₁ = ₂ and !₂ an im aginary num ber. T hisensuresthatthe P function w illtake realvalueson the realaxis. Forj!₂j !₁, the W eierstrass function can be expanded in the param eter g = e^4j!2j:

P (z;

2;ln g

4i)= sin ² z 1 3+ 8

X1

k= 1

kg^k

1 g^k(1 cos2kz); (7)

that is,P is represented by the explicit pow er series P (z;

2;ln g

4i)= sin² z 1 3+

X1

p= 1

g^pV_p(z); (8)

w ith

V_p(z)= 8 X

h2 Dp

h(1 cos2hz); (9)

D_p being the set ofnaturaldivisors ofp,i.e.,D_p= fh 2 N jp=h 2 N g. T herefore,

H^ell= H^trig 1

6 ( 1)N (N 1)+ ( 1)

X1

p= 1

g^p XN

j< k

V_p(q_j q_k) (10)

and a perturbative treatm ent of the elliptic problem becom es feasible. T he rst order term in that expansion is

( 1)g

XN

j< k

V₁(q_j q_k)= 8g ( 1) XN

j< k

(1 cos2(q_j q_k))= 4g ( 1)(N² z₁z_n); (11)

and thus, rst order perturbation theory gives E^ell_m ( ) = E_m^trig( ) 1

6 ( 1)N (N 1)+ ₁E_m ( )+ o(g²) (12)

1E_m ( ) = 4g ( 1)h _m jN² z₁z_nj _m i

h _m j _m i : (13)

T he recurrence relations for the trigonom etric case allow an easy evaluation ofthe energy correction:

it follow s from (5) that

z₁z_nP_m = a_m P_m + ; (14)

w here the dots stand for term s proportionalto polynom ials other than P_m and

a_m = XN

j= 1

~ c_j;m c_j;m

j: (15)

T he orthogonality properties ofthe system ofgeneralized G egenbauer polynom ials guarantee that these term s do not contribute to (13),and w e com e to the sim ple result

1E_m ( )= 4g ( 1) h

N² a

m

i

: (16)

W e can use the explicit expression ofthe coe ents c_j;m ; ~c_j;m given in [11,12]to w rite this correction for the A₁;A₂ and A₃ cases,i.e. for tw o,three and four particles,respectively.

A₁ case: H ere m = (m ); N = 2 and

c_1;m = 1 c_2;m = c_m( )

~

c_1;m = c_m( ) ~c_2;m = 1 (17)

w ith

c_m( )= m (m 1 + 2 )

(m + )(m 1 + ): (18)

T herefore

1E_m( )= 8g ( 1) 1 + ( 1)

(m + 1 + )(m 1 + ) : (19)

A₂ case: H ere m = (m ;n); N = 3 and

c_{1;(m ;n)}= 1 c_{2;(m ;n)}= c_m( ) c_{3;(m ;n)}= a_{m ;n}( )

~

c_{1;(m ;n)}= a_n;m( ) c~_{2;(m ;n)}= c_n( ) c~_{3;(m ;n)}= 1 (20)

w ith

a_{m ;n}( )= n(m + n + )(n 1 + 2 )(m + n 1 + 3 )

(n + )(n 1 + )(m + n + 2 )(m + n 1 + 2 ): (21) T herefore

1E_{m ;n}( ) = 24g ( 1)+ 8g ²( 1)²3 ²+ 3(m + n) + m²+ n²+ m n 3 (m + 1 + )(m 1 + )(n + 1 + ) 2 ²+ (3m + 3n + 1) + m²+ n²+ m n 1

(n 1 + )(m + n + 1 + 2 )(m + n 1 + 2 ): (22)

A₃ case: H ere m = (m ;l;n); N = 4 and

c_{1;(m ;l;n)}= 1 c_{2;(m ;l;n)}= c_m( ) c_{3;(m ;l;n)}= a_{m ;l}( ) c_{4;(m ;l;n)}= d_{m ;l;n}( )

~

c_{1;(m ;l;n)}= d_n;l;m( ) ~c_{2;(m ;l;n)}= a_n;l( ) ~c_{3;(m ;l;n)}= c_n( ) c~_{4;(m ;l;n)}= 1 (23) w ith

d_{m ;l;n}( )= n(l+ n + )(n 1 + 2 )(m + l+ n + 2 )(l+ n 1 + 3 )(m + l+ n 1 + 4 )

(n + )(n 1 + )(l+ n + 2 )(l+ n 1 + 2 )(m + l+ n + 3 )(m + l+ n 1 + 3 ): (24) T herefore

1E_{m ;l;n}( ) = 4g ( 1)

16 n(l+ 1)(l+ m + 1 + )(l+ 2 )(n 1 + 2 )(l+ m + 3 ) (l+ )(l+ 1 + )(n + )(n 1 + )(l+ m + 2 )(l+ m + 1 + 2 )

(n + 1)(l+ n + 1 + )(n + 2 )(l+ m + n + 1 + 2 )(l+ n + 3 )(l+ m + n + 4 ) (n + )(n + 1 + )(l+ n + 2 )(l+ n + 1 + 2 )(l+ m + n + 3 )(l+ m + n + 1 + 3 )

m (l+ m + )(m 1 + 2 )(l+ m + n + 2 )(l+ m 1 + 3 )(l+ m + n 1 + 4 ) (m + )(m 1 + )(l+ m + 2 )(l+ m 1 + 2 )(l+ m + n + 3 )(l+ m + n 1 + 3 )

l(m + 1)(l+ n + )(m + 2 )(l 1 + 2 )(l+ n 1 + 3 )

(l+ )(l 1 + )(m + )(m + 1 + )(l+ n + 2 )(l+ n 1 + 2 ) : (25) T his expression becom es particularly sim ple w hen only one quantum num ber is non-vanishing:

1E_{m ;0;0}( ) = 24g ( 1) h

2 + 4 ³+ (4m 2)²+ (m ² 2) (m 1 + )(1 + 2 )(m + 1 + 3 )

i

1E_0;l;0( ) = 16g ( 1) h

3 + 3 ³+ 4l²+ (l² 3) (1 + )(l 1 + )(l+ 1 + 3 )

i

1E_0;0;n( ) = 24g ( 1) h

2 + 4 ³+ (4n 2)²+ (n² 2) (n 1 + )(1 + 2 )(n + 1 + 3 )

i

: (26)

T he extension ofthe perturbative approach to higher orders is straightforw ard. T here is only a new ingredient: due to the contribution ofinterm ediate states,the norm s ofthe unperturbed eigenfunctions

enter explicitly in the corrections to the energies. Fortunately, these norm s are know n [16,17]. A part from this,the procedure to follow is analogous to that used in rst order and the keypoint is that the recurrence relations w illsave us from doing allthe di cult integrals. W e w illanalyse only the sim plest exam ple,that is,second order perturbation theory for the A₁ case.

T he second order contribution to the H am iltonian H ^ell(10) ( 1)g²

XN

j< k

V₂(q_j q_k) = 8 ( 1)g² XN

j< k

[3 cos2(q_j q_k) 2cos4(q_j q_k)]

= 4 ( 1)g²[3N ² z₁z_n 2(z²₁ 2z₂)(z²_n 2z_{n 1})] (27) for the A1case gives,taking z0= 1,

E^ell_m ( )= E_m^trig( ) 1

6 ( 1)N (N 1)+ ₁E_m( )+ ₂E_m( )+ o(g³) (28) w ith

2E_m( ) = 4g² ( 1)h _mj4 + 7z₁² 2z₁⁴j _mi h _mj _mi + 16g^{2 2}( 1)²

h _mj _mi X

n6= m

1 h _nj _ni

h _njz₁²j _mi² E^trig_m ( ) E_n^trig( )

: (29)

T he recurrence relations for the G egenbauer polynom ials related to A₁ (see (5) and (17)) also hold for the eigenfunctions _m ofthe trigonom etric problem ,i. e.,z_{1 m} = _{m + 1}+ c_{m m 1},from w hich and the H erm itian character ofz₁ (for A₁) it follow s that

h _mj _mi= c_mh _{m 1}j _{m 1}i; (30)

a very usefulrelation w hich allow s to express the equation (29) in term s ofthe coe cients c_m only:

2E_m ( ) = 4g² ( 1)[4 + 7(c_m + c_{m + 1}) 2(c_m + c_{m + 1})² 2c_{m + 2}c_{m + 1} 2c_mc_{m 1}] + 16g^{2 2}( 1)²

(

c_{m + 2}c_{m + 1}

E_m^trig( ) E^trig_{m + 2}( )+ c_m c_{m 1} E^trig_m ( ) E^trig_{m 2}( )

)

: (31)

Finally, using (18) and the expression E_m^trig = m²+ 2 m ² for the energy levels, (31) can be evaluated quite easily,obtaining the sim ple result

2E_m( )= 8g^{2 2}( 1)² (

3

2 (10m + 6) 5m²+ 8 [(m + )² 4][(m + )² 1]

)

+ 4g^{2 2}( 1)² m +

m (m 1)(m 1 + 2 )(m 2 + 2 ) (m 1 + )³(m 2 + )

(m + 1)(m + 2)(m + 2 )(m + 1 + 2 ) (m + 1 + )³(m + 2 + ) :

(32)

A cnow ledgm ents

W e are grateful to Prof. M . Lorente for interesting discussions. A .M .P. w ould like to express his gratitude to the D epartm ent ofPhysics ofthe U niversity ofO viedo for the hospitality during his stay as a V isiting Professor. T he w ork ofJ.F.N .and W .G .F.has been partially supported by the U niversity of O viedo,V icerrectorado de Investigacion,grant M B -02-514.

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