3 T he structure of the recurrence relations

arXiv:math-ph/0305012v1 7 May 2003

Som e resultson the eigenfunctionsofthe quantum trigonom etric Calogero-Sutherland m odelrelated to the Lie algebra D

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

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

zM ax P lanck Institut fur M athem atik, D -53111 B onn,G erm any

A bstract

W e express the H am iltonian ofthe quantum trigonom etric C alogero-Sutherland m odelrelated to the Lie algebra D₄in term sofa setofW eyl-invariantvariables,nam ely,the charactersofthe fundam ental representationsofthe Lie algebra.T hisparam etrization allow susto solveforthe energy eigenfunctions ofthe theory and to study propertiesofthe system oforthogonalpolynom ialsassociated to them such as recurrence relations and generating functions.

1 Introduction

Integrable m odels play a prom inent role in theoretical physics. T he reason is not only the direct phe- nom enologicalinterest ofsom e ofthem ,but also the fact that they often provide som e deep insights into the m athem aticalstructure ofthe theories in w hich they arise. Som etim es,they even revealunexpected relationsam ong di erentphysicalorm athem aticaltheories. In classicalm echanics,integrability notonly show s up itselfin som e ofthe m ost im portant and tim e-honored problem s,such as the K eplerian m otion or the Lagrange or K ovalevskaya top. Itappearsalso in a plethora ofnew hypothetical,highly nontrivial sytem s discovered m ainly during the three last decades of the past century (see [1, 2]for com prehen- sives review s). A m ong these,the so-called C alogero-Sutherland m odels form a distinguised class. T he rst analysis of a system of this kind w as perform ed by C alogero [3] w ho studied, from the quantum standpoint, the dynam ics on the in nite line of a set of particles interacting pairw ise by rational plus quadratic potentials,and found that the problem w as exactly solvable. Soon afterw ards,Sutherland [4]

arrived to sim ilar resultsforthe quantum problem on the circle,thistim e w ith trigonom etric interaction, and M oser [5] show ed that the classical version of both m odels enjoyed integrability in the Liouville sense. T he identi cation of the generalscope of these discoveries com es w ith the w ork of O lshanetsky and Perelom ov [6,7],w ho realized that it w as possible to associate m odels of this kind to allthe root sytem s of the sim ple Lie algebras, and that allthese m odels w ere integrable, both in the classicaland in the quantum fram ew ork [8, 9]. N ow adays, there is a w idespread interest in this type of integrable system s,and m any m athem aticaland physicalapplications for them have been found,see for instance [10].

T he eigenfunctions ofthe C alogero-Sutherland H am iltonian associated to the rootsystem ofa sim ple Lie algebra L are proportional to som e polynom ials w hich form a com plete orthogonal system in the quantum H ilbert space. For the specials values = 1, w here g = ( 1) are the coupling constants, they coincide w ith the irreducible characters of L. For L = A_n, these polynom ials provide naturalgeneralizations to n variables of the classical orthogonalpolynom ials in one indeterm inate. In

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@ m pim -bonn.m pg.de

particular,for the case w ith a trigonom etric potential,one obtains a generalized system ofG egenbauer polynom ials. A s it w as show n in the papers [11,12,13],these generalized G egenbauer polynom ials obey a setofrecurrence relationsw hich constitute a -deform ation ofthe C lebsch-G ordan seriesofthe algebra.

T he nding ofthese recurrence relationsopened the w ay to obtain m any concrete resultson the system of polynom ials,as for exam ple explicit expressions,ladder operators or generating functions [14,15]. T he recurrence relations are also the key ingredient to form ulate a perturbative approach to m ost general am ong the C alogero-Sutherland m odels,that involving the W eierstrass }-function as potential[16].

T he aim of this paper is to extend som e of the results w hich have been obtained for A_n to the polynom ials related to other sim ple algebras. W e think that it is a good idea to begin w ith a concrete case. W e choose to w ork in the rstplace the problem associated to D₄ because ofthe triality sym m etry exhibited by this algebra, w hich w illhelp us in sim plifying of the treatm ent. T he organization of the paper is as follow s. In Sect. 2,w e explain how to express the C alogero-Sutherland H am iltonian in term s ofthe fundam entalcharacters ofthe algebra and how to solve the Schrodinger equation. T hen,in Sect.

3,w e obtain the m ain recurrence relations am ong the polynom ials and use them to give algorithm s to calculate som e subsets ofthem . Sect. 4 is devoted to nd the generating functions for som e classes of charactersand m onom ialsfunctionsofD₄. M ore recurrrence relationsand som e otherrelevantresultsare included in Sect. 5,and nally,in Sect. 6,w e give som e briefconclusions. A lso,w e o er tw o appendices.

In A ppendix A ,for the convenience ofthe reader,w e collect som e ofthe basic facts about D4 w hich w e use in the m ain text. In A ppendix B w e list som e polynom ials,characters and m onom ialfunctions.

2 T he eigenvalue problem

T he H am iltonian operator for the trigonom etric C alogero-Sutherland m odelrelated to the root system ofa sim ple Lie algebra ofrank r has the form

H = 1 2(p;p)+

X

2 R⁺

( 1)sin²( ;q); (1)

w here q = (q1;:::;q_r),p = (p1;:::;p_r),( ; ) is the usualeuclidean inner product in R^r,R⁺ is the set of positive roots ofthe algebra,and are constants such that = ifjj jj= jj jj. In particular,for the case ofthe algebra D₄(see A ppendix A ),this leads to the follow ing Schrodinger equation:

H = E ( ) ;

H = 1

2 + ( 1)

0

@ X4

j< k

sin ²(q_j q_k)+

X4

j< k

sin ²(q_j+ q_k) 1

A ; =

X4

j= 1

@²

@q_j²: (2) T he q coordinates are assum ed to take values in the [0; ] interval, and therefore the equation can be interpreted as describing the dynam ics ofa system offour particles m oving on the circle. Let us notice that there is not translationalinvariance. W e recapitulate som e im portant facts about this m odelw hich follow from the generalstructure ofthe quantum C alogero-Sutherland m odels related to Lie algebras [9].

T he ground state energy and the (non-norm alized) w avefunction are E0( ) = 2( ; ) ²= 28 ²;

0(q) = 8

<

: Y4

j< k

sin(q_j q_k)sin(q_j+ q_k) 9

=

; ; (3)

w here is the standard W eylvector, = ¹₂^P _{2 R}⁺ ,w ith the sum extended over allthe positive roots ofD4. T he excited states depend on a four-tuple ofquantum num bers m = (m1;m2;m3;m4)

H _m = E_m ( ) _m ;

E_m ( ) = 2( + ; + ); (4) w here is the highest w eight ofthe irreducible representation ofD4labelled by m ,i.e., = ^{P 4}_{i= 1}m_{i i}, and _iare the fundam entalw eights ofD4. B y substitution in (4) of

m (q)= ₀(q) _m (q); (5)

w e are led to the eigenvalue problem

m = "_m( ) _m (6)

w ith

= 1 2 +

X4

j< k

ctg(q_j q_k) @

@q_j

@

@q_k

!

+ ctg(q_j+ q_k) @

@q_j+ @

@q_k

! !

(7) and

"_m ( )= E_m ( ) E0( )= 2( ; + 2 ): (8) Introducing the inverse C artan m atrix A_jk¹= ( _j; _k),w e can give a m ore explicit expression for "_m ( ):

"_m ( ) = 2 X4

j;k= 1

A_jk¹m_jm_k+ 4 X4

j;k= 1

A_jk¹m_j= 2(m²₁+ m²₃+ m²₄)+ 4m²₂+ 2(m1m₃+ m1m₄+ m3m₄) + 4m2(m1+ m3+ m4)+ 12 (m 1+ m3+ m4)+ 20 m 2: (9) T he m ain problem is to solve equation (6). A s it has been show n for the case of the algebra A_n [11,12,13],the best w ay to do that is to use a set of independent variables w hich are invariant under the W eylsym m etry ofthe H am iltonian,nam ely,the characters ofthe four fundam entalrepresentations ofthe algebra D4. U nfortunately,the expression ofthese characters in term s of the q-variables (w hich play the role ofcoordinates on the m axim altorus ofD4) is not very sim ple. D enoting the character of the irreducible representation ofm axim alw eight _jas z_j,w e nd

z₁ = X4

j= 1

x_j+ X4

j= 1

x_j¹

z2 = X4

i< j

x_ix_j+ X4

i< j

(x_ix_j) ¹+ X4

i;j

x_i¹x_j

z3 = x X4

i= 1

1 x_i+ 1

x X4

i= 1

x_i;

z4 = x + 1 x+ 1

x X4

i< j

x_ix_j

w here x_j= e^2iqj,and x = p

x₁x₂x₃x₄. T hese expressions m ake the direct change ofvariables from q_ito z_k quite cum bersom e. W e refrain from trying that approach,and choose an indirect route w hich has the added advantage of being also applicable to other algebras in w hich the expressions for the characters are even m ore involved. W e can infer from (7) the structure of w hen w ritten in the z-variables:

= X4

j;k= 1

a_jk(z_i)@_zj@_zk + X4

j= 1

h

b⁽⁰⁾_j (z_i)+ b⁽¹⁾_j (z_i) i

@_zj: (10)

O n the other hand,as it is w ellknow n [17], the _m are polynom ials w hich, w ith som e precise partial ordering for the m onom ials to be described later,start as follow s:

m (z_i)= P_m (z_i)= z₁^m1z^m₂²z₃^m3z₄^m4+ : (11) T herefore,m aking use of(9),w e conclude that

a_jk(z_i) = 2A_jk¹z_jz_k+ low er order term s;

b⁽_j^r)(z_i) = c⁽_j^r)z_j+ d⁽_j^r); r = 0;1: (12) N ow ,to obtain the fullexpressions for these coe cients,w e rely on the fact that,for = 1,the P _m polynom ialgives the character ofthe irreducible representation ofD4 w ith m axim alw eight^{P 4}_{i= 1}m_{i i}, w hile for = 0 the sam e polynom ial is the corresponding sym m etric m onom ial function [9]. B oth, characters and m onom ialfunctions,can be com puted by using the inform ation available in the literature (see,for instance,the \R eference C hapter" of[18]). In fact,the follow ing short list ofpolynom ials

P_2;0;0;0⁽¹⁾ (z) = z²₁ z2 1;

P_1;1;0;0⁽¹⁾ (z) = z₁z₂ z₃z₄; P_1;0;1;0⁽¹⁾ (z) = z1z3 z4; P_0;2;0;0⁽¹⁾ (z) = z²₂ z1z3z4+ z2; P_2;0;0;0⁽⁰⁾ (z) = z²₁ 2z2

is allw e need to obtain . B y substituting these polynom ials in (6) and using (9),(10),(12) and the triality sym m etry (that here im plies that the nalexpression for should be invariant under perm uta- tions ofthe indices 1,3,4),w e get enough sim ple linear algebraic equations to x allthe coe cients. W e give here only the nalresult:

1

2 = z²₁ 2z2 8 @_z²₁+ h

2z₂² 4 z₁²+ z²₃+ z₄² 2z1z3z4+ 8z2

i

@_z²₂ + z₃² 2z2 8 @_z²₃ + z²₄ 2z2 8 @_z²₄+ (2z1z2 6z3z4 8z1)@_z1@_z2 + (z1z3 8z4)@_z1@_z3

+ (z1z₄ 8z3)@_z1@_z4+ (2z2z₃ 6z1z₄ 8z3)@_z2@_z3 + (2z2z₄ 6z1z₃ 8z4)@_z2@_z4 + (z₃z₄ 8z₁)@_z3@_z4+ (6 + 1)z₁@_z1 + [2(5 + 1)z₂+ 8( 1)]@_z2 + (6 + 1)z₃@_z3

+ (6 + 1)z4@_z4: (13)

O nce the explicit expression for the operator in the z variables is given,the Schrodinger equation can be solved iteratively. B y direct application of to z^m z^m₁¹z₂^m2z₂^m3z^m₄⁴,w e nd

z^m = "_m ( )z^m X4

i= 1

aⁱ_m z^{m i} X

j2 I

b^j_m z^{m ( 2+ j)} X

ij2 T

c^ij_m z^{m ( 2+ i+ j)} 2a²_m

X

ij2 T

z^{m (2 2+ i+ j)} d_m z^{m ( 1+ 2 2+ 3+ 4)} 4

X

j2 I

a^j_m z^{m ( 1+ 2 2+ 3+ 4+ j)}; (14)

w here the sets ofindices are I = f1;3;4g and T = f13;14;34g,and aⁱ_m = 4m_i(m_i 1); b^j_m = 12m₂m_j;

c^ij_m = 16m_im_j; d_m ( )= 16m2

0

@2 m2 +

X

j2 I

m_j 1 A :

A llm onom ials in z^m take the form z^m w ith as a positive root. T hus,the polynom ialP_m has the form

P_m (z)=

X

2 Q⁺(m )

c z^m ; (15)

w here w e choose the norm alization c₀= 1 and,ifQ⁺ is the cone ofpositive roots,

Q⁺(m )= 2 Q⁺ jz^m is w ellde ned if z1z2z3z4= 0 : (16) T he above-m entioned partialordering of m onom ials is given sim ply by the height of ,i. e. z^{m 1} >

z^{m 2} ifht( ₁)< ht( ₂). From (14),the coe cients c obey the iterative form ula

c = N

"_m ( ) "_m ( ) (17)

w ith

N =

X4

i= 1

aⁱ_{m ( i)}c _i+ X

j2 I

b^j

m ( 2 j)c _{( 2+ j)}+ X

ij2 T

c^ij

m ( 2 i j)c _{( 2+ i+ j)}

+ 2

X

ij2 T

a²

m ( 2 2 i j)c _{(2 2+ i+ j)}+ d_{m ( 1 2 2 3 4)}c _{( 1+ 2 2+ 3+ 4)}

+ 4

X

j2 I

a^j

m ( 1 2 2 3 4 j)c _{( 1+ 2 2+ 3+ 4+ j)}:

A long w ith the explicitexpressionsfortherootsgiven in A ppendix A ,itissuitable forthe im plem entation on a sym bolic com puterprogram . A listofpolynom ialsobtained through the use ofthisform ula iso ered in A ppendix B .

3 T he structure of the recurrence relations

A s itis w ellknow n,allthe system s oforthogonalpolynom ials in one indeterm inate z,such that P_m(z)=

z^m + satisfy a recursive form ula zP_m(z) = a_m P_{m + 1}(z)+ b_m P_m(z)+ c_m P_m 1(z). In particular,the orthogonalpolynom ials associated to the trigonom etric C alogero-Sutherland m odelfor the case of tw o particles and Lie algebra A1are the classicalG egenbauer polynom ials,w hose recursive form ula is know n to be

z P_m(z)= P_{m + 1}(z)+ m (m 1 + 2 )

(m 1 + )(m + )P_{m 1}(z):

T his form ula is rem iniscent of the C lebsch-G ordan series for A1. In fact, for = 1 it reduces exactly to this C lebsch-G ordan series: the polynom ials are the characters ofA1and the coe cents are equalto one. Im m ediately the question arises about the existence ofanalogous recurrence relations,i.e.,w ith the structure of -deform ations ofthe corresponding C lebsch-G ordan series,for the polynom ials related to C alogero-Sutherland m odels associated to other sim ple Lie algebras. A s it w as show n in [11],the answ er turns out to be in the a rm ative for all root system s, but to obtain the expressions for the deform ed coe cients it is necessary to proceed through a case-by-case analysis. O nce the coe cients are know n, m any applications are possible. T he aim of this section is to x the structure of the basic recurrence relations for the case ofD4 and to give a sim ple illustration oftheir use.

W e w ant to study the form ulas for z_iP_m (z),i= 1;2;3;4. T herefore,as P_m⁽¹⁾(z) = z_ifor m_j= (_ji), and the recursive form ulas are deform ations ofthe C lebsch-G ordan series,w e need to know the w eights ofthe irreducible representations w hose integraldom inant w eights are 1, 2, 3and 4. For the case of

1, 3 and 4,these representations have dim ension eight. O n the other hand,ifw e act on the highest w eight w ith the W eylgroup in the w ay explained in the A ppendix A ,w e obtain eight di erent w eights.

T hus,these representations include only one orbitofthe W eylgroup and w e are done. For the case of 2, the representation has dim ension 28 and the orbit ofthe W eylgroup containing 2has only 24 elem ents.

B ut 2 = ⁺₁₂, the highest root, and thus this representation is the adjoint one and includes a second orbit: the C artan subalgebra,w ith four elem ents ofw eight zero. Let us sum m arize.

W eights in z1: 1; (1 2); (2 3 4); (3 4):

W eights in z2: 2; (2 2_j); (22 1 3 4); (2+ _{i j k}); (_i+ _j

k); (_{2 1 3 4}); 0,w ith i;j;k 2 I:

W eights in z3: 3; (3 2); (2 1 4); (1 4):

W eights in z4: 4; (4 2); (2 1 3); (1 3):

W ith these w eights,the structure ofthe recurrence relations results to be as follow s:

z1P_m1;m2;m3;m4(z) = P_{m1+ 1;m2;m3;m4}(z)+ a¹_m ( )P_{m1 1;m2;m3;m4}(z)+ b¹_m ( )P_{m1+ 1;m2 1;m3;m4}(z) + c¹_m ( )P_{m1 1;m2+ 1;m3;m4}(z)+ d¹_m ( )P_{m1;m2+ 1;m3 1;m4 1}(z)

+ e¹

m ( )P_{m1;m2 1;m3+ 1;m4+ 1}(z)+ f_m¹( )P_{m1;m2;m3+ 1;m4 1}(z) + g_m¹( )P_{m1;m2;m3 1;m4+ 1}(z)

z2P_m1;m2;m3;m4(z) = P_{m1;m2+ 1;m3;m4}(z)+ A_m ( )P_{m1;m2 1;m3;m4}(z)+ B_m ( )¹ P_{m1 2;m2 1;m3;m4}(z) + B_m ( )³ P_{m1;m2 1;m3 2;m4}(z)+ B_m( )⁴ P_{m1;m2 1;m3;m4 2}(z)

+ C_m ( ) P_{m1 1;m2 2;m3 1;m4 1}(z)+ D_m ( )¹ P_{m1 1;m2 1;m3 1;m4 1}(z) + D_m ( )³ P_m

1 1;m2 1;m3 1;m4 1(z)+ D_m ( )⁴ P_m

1 1;m2 1;m3 1;m4 1(z) + E_m ( )¹ P_m

1 1;m2;m3 1;m4 1(z)+ E_m ( )³ P_m

1 1;m2;m3 1;m4 1(z) + E_m ( )⁴ P_{m1 1;m2;m3 1;m4 1}(z)+ F_m ( ) P_{m1 1;m2 1;m3 1;m4 1}(z) + G_m ( )P_m1;m2;m3;m4(z);

z3P_m1;m2;m3;m4(z) = P_{m1;m2;m3+ 1;m4}(z)+ a³_m ( )P_{m1;m2;m3 1;m4}(z)+ b³_m ( )P_{m1;m2 1;m3+ 1;m4}(z) + c³

m ( )P_{m1;m2+ 1;m3 1;m4}(z)+ d³_m ( )P_{m1 1;m2+ 1;m3;m4 1}(z) + e³_m ( )P_{m1+ 1;m2 1;m3;m4+ 1}(z)+ f_m³( )P_{m1+ 1;m2;m3;m4 1}(z) + g_m³( )P_{m1 1;m2;m3;m4+ 1}(z);

z4P_m1;m2;m3;m4(z) = P_{m1;m2;m3;m4+ 1}(z)+ a⁴_m ( )P_{m1;m2;m3;m4 1}(z)+ b⁴_m ( )P_{m1;m2 1;m3;m4+ 1}(z) + c⁴_m ( )P_{m1;m2+ 1;m3;m4 1}(z)+ d⁴_m ( )P_{m1 1;m2+ 1;m3 1;m4}(z)

+ e⁴_m ( )P_{m1+ 1;m2 1;m3+ 1;m4}(z)+ f_m⁴( )P_{m1 1;m2;m3+ 1;m4}(z) + g_m⁴( )P_{m1+ 1;m2;m3 1;m4}(z);

w hereB_m ( )¹ P_m

1 2;m2 1;m3;m4(z)m eansB_m ( )¹⁺ P_m

1+ 2;m2 1;m3;m4(z)+ B_m ( )¹ P_m

1 2;m2+ 1;m3;m4(z), etc, and it is understood that all polynom ials involving negative quantum num bers are zero. T he re- currence relations re ect triality in the fact that not allthe coe cients appearing in these form ulas are independent. T here are coincidences upon perm utations ofthe quantum num bers,for instance

a¹_m

1;m2;m3;m4 = a³_m3;m2;m1;m4 = a⁴_m4;m2;m3;m1; (18)

and sim ilarly for b^j_m ;c^j_m ;d^j_m ;e^j_m ;f_m^j;g_m^j. In the sam e fashion,w e have also B_m¹

1;m2;m3;m4 = B_m³_3;m2;m1;m4 = B⁴_m4;m2;m3;m1 (19) and sim ilarly for D^j_m ;E_m^j .

A san exam ple,letusconsidera sim plecasein w hich only oneofthequantum num bersisnonvanishing, nam ely,

z1P_{m ;0;0;0}(z)= Pm + 1;0;0;0(z)+ a_m( )P_{m 1;0;0;0}(z)+ c_m( )P_{m 1;1;0;0}(z); (20) w here w e w rite a_m( )= a¹_{m ;0;0;0}( ) and c_m( )= c¹_{m ;0;0;0}( ). U sing form ulae

P_{m ;0;0;0}(z) = z₁^m m (m 1) 4 ²+ 4(m 2) + (m 1)(m 2)

(m 1 + )(m 1 + 3 )(m 2 + ) z₁^{m 2} m (m 1)

m 1 + z₁^{m 2}z₂+ P_{m ;1;0;0}(z) = z₁^m z2+ 4 ( 1)(m 2 + 2 )

(m + 1 + 5 )(m + 2 )(m 1 + )z^m₁ + ; w e obtain the coe cients in (20)

a_m( ) = m (m + 2 )(m 1 + 4 )(m 1 + 6 ) (m 1 + )(m 1 + 3 )(m + 3 )(m + 5 ); c_m( ) = m (m 1 + 2 )

(m + )(m 1 + ):

A s a byproduct oftriality,w e can also w rite other tw o recurrence relations w ith the sam e coe cients:

z3P_{0;0;m ;0}(z) = P0;0;m + 1;0(z)+ a_m( )P_{0;0;m 1;0}(z)+ c_m( )P_{0;1;m 1;0}(z)

z4P_0;0;0;m(z) = P0;0;0;m + 1(z)+ a_m( )P_{0;0;0;m 1}(z)+ c_m( )P_{0;1;0;m 1}(z): (21) T he rst of these recurrence relations can be used to devise an algorithm for the calculation ofthe polynom ials of the form P_{m ;0;0;0}(z) and P_{m ;1;0;0}(z). B y m ultiplying (20) by the di erential operator

"_{m 1;1;0;0}( ),the term involving P_{m 1;1;0;0}cancels. U sing the explicit expressions (9),(13),w e nd Pm + 1;0;0;0 = 1

4(m + )[ ;z1]P_{m ;0;0;0}(z) 1 + 4

2(m + )z1P_{m ;0;0;0}(z) + m (m + 2 )(m 1 + 4 )(m 1 + 6 )

(m 1 + )(m 1 + 3 )(m + )(m + 3 )P_{m 1;0;0;0}(z);

w here,from (13),

[ ;z1] = 4 z²₁ 2z2 8 @_z1+ 2(z1z3 8z4)@_z3 + 2(z1z4 8z3)@_z4 + 4(z1z2 3z3z4 4z1)@_z2 + 2(6 + 1)z1:

w here,from (13),

[ ;z₁] = 4 z²₁ 2z2 8 @_z1+ 2(z1z₃ 8z4)@_z3 + 2(z1z₄ 8z3)@_z4

+ 4(z2z2 3z3z4 4z1)@_z2 + 2(6 + 1)z1:

O nce the polynom ials P_{m ;0;0;0}(z) are know n, the recurrence relation (20) provides a form ula for each P_{m ;1;0;0}(z):

c_{m + 1}( )P_{m ;1;0;0}(z)= z1Pm + 1;0;0;0(z) Pm + 2;0;0;0(z) a_{m + 1}( )P_{m ;0;0;0}(z): (22)

4 Som e generating functions

W e present in this section the generating functions for som e characters and sym m etric m onom ialfunc- tions. Let us consider rst the case of the m onom ial functions w ith only one non-vanishing quantum num ber in the form P_{m ;0;0;0}⁽⁰⁾ (z). T he generating function for this subset is

F0(t;z)=

X1

m = 0

t^m P_{m ;0;0;0}⁽⁰⁾ (z): (23)

In term s ofthe x variables,the generalexpression for these m onom ialfunctions is

P_{m ;0;0;0}⁽⁰⁾ (x)=

X4

j= 1

x^m_j + x_j^m ; (24)

and,in particular,w e de ne P_0;0;0;0⁽⁰⁾ (z)= 8. In these variables,the com putation ofF0(t;x) only requires to sum the geom etric series:

F0(t;x)=

X4

j= 1

0

@ 1

1 tx_j+ 1 1 _x^t

j

1

A : (25)

T he change to the originalz variables can be done by the inspection ofthe coe cients ofthe pow ers of t in both the num erator and denom inator ofthis rationalexpression,w ith the result

F0(t;z)= N0(t;z)

D (t;z); (26)

w here

N0(t;z) = 8 7z1t+ 6z2t² 5(z3z4 z1)t³+ 4(z₃²+ z²₄ 2z2 2)t⁴ 3(z3z4 z1)t⁵ + 2z2t⁶ z1t⁷;

D (t;z) = 1 z1t+ z2t² (z3z4 z1)t³+ (z₃²+ z²₄ 2z2 2)t⁴ (z3z4 z1)t⁵

+ z₂t⁶ z₁t⁷+ t⁸: (27)

T here is an alternative approach. A s the m onom ialfunctions are eigenfunctions of ⁽⁰⁾w ith eigenvalues

"_{m ;0;0;0}(0)= 2m²,w e have

1 2

(0)F₀(t;z)=

X1

m = 0

m²t^m P_{m ;0;0;0}⁽⁰⁾ (z);

and,therefore,w e can w rite a di erentialequation for F0(t;z):

1 2

(0) (t@_t)² F0(t;z)= 0; F0(0;z)= 8: (28) O ne can verify by substitution that (26) satis es this equation. W hen F0(t;z) is know n,w e can easily obtain the generating function

G₀(t;z)=

X1

m = 0

t^m P_{m ;1;0;0}⁽⁰⁾ (z) (29)

by only recalling (20),w hich for = 0 is sim ply

z1P_{m ;0;0;0}⁽⁰⁾ (z)= Pm + 1;0;0;0⁽⁰⁾ (z)+ P_m⁽⁰⁾_1;0;0;0(z)+ P_m⁽⁰⁾_1;1;0;0(z): (30)

T his gives

G₀(t;z)= M ₀(t;z)

D (t;z) (31)

w ith

M0(t;z) = z2 4 + (6z1 3z3z4)t

+ ( 8 2z₁² 10z2 z₂²+ 4z²₃+ 2z1z3z4+ 4z₄²)t² + (10z1+ 5z1z2 3z1z₃² 4z3z4+ z2z3z4 3z1z²₄)t³ + (8z2 4z₁²+ 2z₂² z₂z₃²+ 4z1z₃z₄ z₂z²₄)t⁴

+ ( 6z1 6z1z₂ z₃z₄+ z2z₃z₄)t⁵+ (8 + 6z₁²+ 2z2 z²₂)t⁶ + ( 10z1+ z1z2)t⁷+ (4 z2)t⁸:

T he com putation ofthe generating functions for the characters P_{m ;0;0;0}⁽¹⁾ and P_{m ;1;0;0}⁽¹⁾ goes through sim ilar argum ents. In this case,the eigenvalues are "_{m ;0;0;0}(1)= 2m²+ 12m . H ence,

F1(t;z)=

X1

m = 0

t^mP_{m ;0;0;0}⁽¹⁾ (z); P_0;0;0;0⁽¹⁾ (z) 1 (32) is the solution ofthe equation

1 2

(1) (t@_t)² 6t@_t F1(t;z)= 0; F1(0;z)= 1: (33) T he W eylcharacter form ula im plies that the denom inator ofF1(t;z) should be the sam e D (t;z) found before. T hus,w e try an A nsatz

F₁(t;z)= N₁(t;z)

D (t;z) (34)

and obtain the sim ple answ er

N₁(t;z)= 1 t²: (35)

A pplying the recurrence relation (20)) w e obtain the generating function G1(t;z) for the characters P_{m ;1;0;0}⁽¹⁾ :

G1(t;z)= 1 D (t;z)

n

z2 z3z4t+ (z₃²+ z²₄ 2z2 1)t² (z3z4 z1)t³+ z2t⁴ z1t⁵+ t⁶ o

: (36)

5 M ore recurrence relations and other results

In this Section,w e give the rem aining recurrence relations involving the product ofa fundam entalchar- acter tim es a polynom ialw ith only one non-vanishing quantum num ber. W e also com m ent the existence of som e peculiar values for for w hich the polynom ials associated to som e special excited states are proportionalto integer pow ers ofthe fundam entalstate w avefunction.

To obtain the m entioned recurrence relations,it is necessary to com pute the coe cients ofa lim ited num ber of term s of the polynom ials involved. O nce the form of these term s is know n, w e can obtain the coe cients in the recurrence relations solving a system oflinear algebraic equations. W e do not give here the fullexpressions for the coe cients of the required term s,because som e of them are too long, and only list them :

P_{1;0;m ;0}(z) = z1z₃^m + A z₃^{m 1}z4+ ;

P_{0;m ;0;0}(z) = z₂^m + B z^m₂ ¹+ C z₂^{m 2}+ D z1z₂^{m 2}z3z4+ E z1z₂^{m 3}z3z4

+ F (z₁²z^m₂ ²+ z₂^{m 2}z₃²+ z₂^{m 2}z₄²)+ ; P_{1;m ;0;0}(z) = z1z₂^m + G z1z^m₂ ¹z4+ H z^m₂ ¹z3z4+ ; P_{0;m ;1;1}(z) = z₂^mz₃z₄+ Iz1z₂^m + ;

P_{m ;0;0;0}(z) = z₁^m + J z₁^{m 2}+ K z^m₁ ²z₂+ ;

P_{m ;1;0;0}(z) = z₁^mz₂+ L z₁^{m 2}z₂+ N z₁^{m 1}z₃z₄+ M z^m₁ + ; P_{m ;0;1;1}(z) = z₁^mz₃z₄+ N z₁^{m 1}z₂+ O z^{m + 1}₁ + ;

P_{1;m ;1;1}(z) = z₁z₂^mz₃z₄+ P z₂^m + Q z1z₂^{m 1}z₃z₄+ R (z²₁z₂^m + z₂^mz²₃+ z₂^mz₄²)+ S z₂^{m + 1}+ ; P_{2;m ;0;0}(z) = z₁²z₂^m + T z^m₂ + U z1z₂^{m 1}z3z4+ W z₂^{m + 1}+ :

T he use of the quantities denoted A to W in the previous form ulas in the generalstructure of the recurrence relations give the follow ing results:

Form ulae oftype z1P_{0;0;m ;0}(z):

z1P_{0;0;m ;0}(z) = P_{1;0;m ;0}(z)+ b_m( )P_{0;0;m 1;1}(z) z1P_0;0;0;m(z) = P_1;0;0;m(z)+ b_m( )P_0;0;1;m 1(z) z3P_{m ;0;0;0}(z) = P_{m ;0;1;0}(z)+ b_m( )P_{m 1;0;0;1}(z) z₃P_0;0;0;m(z) = P_0;0;1;m(z)+ b_m( )P_1;0;0;m 1(z) z₄P_{m ;0;0;0}(z) = P_{m ;0;0;1}(z)+ b_m( )P_{m 1;0;1;0}(z) z4P_{0;0;m ;0}(z) = P_{0;0;m ;1}(z)+ b_m( )P_{1;0;m 1;0}(z)

w ith

b_m( )= m (m 1 + 4 ) (m 1 + )(m + 3 ): Form ulae oftype z1P_{0;m ;0;0}(z):

z1P_{0;m ;0;0}(z) = P_{1;m ;0;0}(z)+ d_m( )P_{1;m 1;0;0}(z)+ e_m( )P_{0;m 1;1;1}(z) z3P_{0;m ;0;0}(z) = P_{0;m ;1;0}(z)+ d_m( )P_{0;m 1;1;0}(z)+ e_m( )P_{1;m 1;0;1}(z) z₄P_{0;m ;0;0}(z) = P_{0;m ;0;1}(z)+ d_m( )P_{0;m 1;0;1}(z)+ e_m( )P_{1;m 1;1;0}(z)

w ith

d_m( ) = 2m (m + )(m 1 + 3 )(m 1 + 4 )(2m 1 + 6 ) (m 1 + )(m 1 + 2 )(m + 3 )(2m 1 + 5 )(2m + 5 ); e_m( ) = m (m 1 + 3 )

(m 1 + )(m + 2 ):

Form ulae oftype z2P_{m ;0;0;0}(z):

z₂P_{m ;0;0;0}(z) = P_{m ;1;0;0}(z)+ f_m( )P_{m 2;1;0;0}(z)+ g_m( )P_{m 1;0;1;1}(z)+ h_m( )P_{m ;0;0;0}(z) z₂P_{0;0;m ;0}(z) = P_{0;1;m ;0}(z)+ f_m( )P_{0;1;m 2;0}(z)+ g_m( )P_{1;0;m 1;1}(z)+ h_m( )P_{0;0;m ;0}(z) z2P_0;0;0;m(z) = P_0;1;0;m(z)+ f_m( )P_0;1;0;m 2(z)+ g_m( )P_{1;0;1;m 1}(z)+ h_m( )P_0;0;0;m(z) w ith

f_m( ) = m (m 1)(m 2 + 2 )(m + 2 )(m 1 + 4 )(m 1 + 5 ) (m 2 + )(m 1 + )²(m 1 + 3 )(m + 3 )(m + 4 ); g_m( ) = m (m 1 + 3 )

(m 1 + )(m + 2 );

h_m( ) = 4 3³+ 5 ²+ (6m 1) + (m² 1) (m 1 + )(1 + 3 )(m + 1 + 5 ) :

Form ula for z2P_{0;m ;0;0}(z):

z2P_{0;m ;0;0}(z) = P0;m + 1;0;0(z)+ k_m( )P_{0;m 1;0;0}(z)+ p_m( )P_{1;m 1;1;1}(z)+ q_m( )P_{1;m 2;1;1}(z) + r_m( )

h

P_{2;m 1;0;0}(z)+ P_{0;m 1;2;0}(z)+ P_{0;m 1;0;2}(z) i

+ s_m( )P_{0;m ;0;0}(z) w ith

k_m( ) = 4m (m + )²(m + 2 )(m 1 + 3 )(m 1 + 4 )²(2m 1 + 4 )(m 1 + 5 )(2m 1 + 6 ) (m 1 + )(m 1 + 2 )²(m + 3 )²(m + 4 )(2m 2 + 5 )(2m 1 + 5 )²(2m + 5 ) ; p_m( ) = m (m 1 + 2 )

(m 1 + )(m + );

q_m( ) = 2m (m 1)(m + )²(m 2 + 2 )(m 1 + 3 )³(2m 1 + 6 ) (m 2 + )(m 1 + )²(m 1 + 2 )²(m + 2 )²(2m 1 + 5 )(2m + 5 ); r_m( ) = m (m + )(m 1 + 3 )(m 1 + 4 )

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

s_m( ) = 4t_m( )

( + 1)(m 1 + )(m + 1 + 4 )(2m 1 + 5 )(2m + 1 + 5 );

t_m( ) = ( 1 + 5m² 4m⁴)+ (2 + 25m 7m² 40m³+ 2m⁴) + (20 35m 123m²+ 20m³) ²; + ( 22 115m + 63m²)³+ ( 19 + 65m )⁴+ 20 ⁵:

Finally, w e m ention that for = ¹₂(n 1), n 2 N , the polynom ials associated to the dom inant w eight w hich is n tim es the W eylvector are proportionalto a pow er ofthe ground state w avefunction, nam ely

P

1 2(n 1)

n = ( 1)ⁿ2¹²ⁿ 8

<

: Y4

j< k

sin(q_j q_k)sin(q_j+ q_k) 9

=

;

n

:

T his form ula can be veri ed quite easily by direct application of ^{12(n 1)}in the form (7) to the right- hand side: one nds that the Schrodinger equation (6) w ith the appropriate eigenvalue is satis ed. T he m ost convenient w ay to x the proportionality constant is by perform ing an analytic continuation to com plex q_iand considering the region x_i2 R and x₁ x₂ x₃ x₄ 0. T hen,the polynom ials are dom inated by the leading order term ,P

1 2(n 1)

n ’ z₁ⁿz₂ⁿz₃ⁿzⁿ₄,and,on the other hand,using the form ulas for the fundam entalcharacters displayed in Section 2. one nds z1z₂z₃z₄ ’ x³₁x²₂x₃ and ^Q4_{j< k}sin(q_j q_k)sin(q_j+ q_k)’ 2¹²x³₁x²₂x3. T his gives the proportionality constant w ritten above.

6 C onclusions

In this paper, w e have show n how to solve the Schrodinger equation for the trigonom etric C alogero- Sutherland m odel related to the Lie algebra D₄ and w e have explored som e properties of the energy eigenfunctions. T he m ain point is that the use ofa W eyl-invariant set ofvariables,the characters ofthe fundam entalrepresentations, leads to a form ulation of the Schrodinger equation by m eans of a second order di erentialoperator w hich is sim ple enough to m ake feasible a recursive m ethod for the treatm ent ofthe spectralproblem . T he eigenfunctions provide a com plete system oforthogonalpolynom ials in four variables,and these polynom ials obey recurrence relations w hich are extensions ofthe C lebsch-G ordan seriesofthe algebra. T he structure ofsom e ofthese recurrence relationshasbeen xed and,forparticular cases,the coe cients involved have been com puted. A lso,som e generating functions for the polynom ials w ith param eter = 1 and = 0 have been obtained. T hese generating functions can give som e hints about the form ofthe generating function for general ,see [20].

A cknow ledgem ents

A .M .P.w ould like to express his gratitude to the M ax Planck Institut fur M athem atik for hospitality.

T he w ork ofJ.F.N .and W .G .F.hasbeen partially supported by the U niversity ofO viedo,V icerrectorado de Investigacion,grant M B -03-514-1.

A ppendix A :Sum m ary of results on the Lie algebra D

In this appendix,w e review som e standard facts about the root and w eight system s ofthe Lie algebra D4 that the reader could nd usefulto follow the m ain text. M ore extensive and sound treatm ents of these topics can be found in m any excellent textbooks,see for instance [18],[19].

T he m ost convenient explicit representation ofD4 is D4=

(

m b

c m^t

!

j m ;b;c real4 4 m atrices and b^t= b; c^t= c )

: T his gives dim D4= 28. O ne can choose the follow ing linear basis:

M _jk = E_j;k E_{4+j;4+ k}; j;k = 1;2;3;4

B_jk = E_{j;4+ k} E_{k;4+ j}; j;k = 1;2;3;4; j < k C_jk = E_4+j;k E_4+k;j; j;k = 1;2;3;4; j < k w ith (E_i;j)_kl= _{ik jl}. T he C artan subalgebra is

H = (

h = X4

i= 1

c_iM _ii j c_i2 R )

and this con rm s that the rank ofD4 is four. T he m atrix com m utators [h;M _jk] = (c_j c_k)M _jk;

[h;B_jk] = (c_j+ c_k)B_jk; [h;C_jk] = (c_j+ c_k)C_jk allow us to classify the 24 roots in tw o groups

jk(h) = c_j c_k; j 6= k;

jk(h) = (c_j+ c_k); j < k: