View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by CERN Document Server EXCHANGE OPERATORS AND EXTENDED HEISENBERG ALGEBRA FOR THE THREE-BODY CALOGERO-MARCHIORO-WOLFES PROBLEM 1 ; 2 C. QUESNE Physique NucleaireTheorique et Physique Mathematique, Universite Libre de Bruxel les, Campus de la Plaine CP229, BoulevardduTriomphe, B-1050 Brussels, Belgium Abstract The exchange op erator formalism previously intro duced for the Calogero problem is extended to the three-b o dy Calogero-Marchioro-Wolfes one. In the absence of oscillator p otential, the Hamiltonian of the latter is interpreted as a free particle Hamiltonian, expressed in terms of generalized momenta. In the presence of oscillator processed by the SLAC/DESY Libraries on 15 May 1995. 〉 p otential, it is regarded as a free mo di ed b oson Hamiltonian. The mo di ed b oson op erators are shown to b elong to a D -extended Heisenb erg algebra. A pro of of 6 complete integrability is also provided. PostScript 1 Directeur de recherches FNRS HEP-TH-9505071 2 E-mail: [email protected] e 1 1 Intro duction A long time ago, Calogero [1] solved the Schr}odinger equation for three particles in one dimension, interacting pairwise via harmonic and inverse square p otentials. Later, Wolfes [2] extended Calogero's metho d to the case where there is an additional three-b o dy p otential of a sp ecial form. The same problem in the absence of harmonic p otential was also studied by Calogero and Marchioro [3]. Other exactly solvable many-b o dy problems were then intro duced, analyzed from the viewp oint of classical or quantum integrability, and shown to b e related to ro ot systems of Lie algebras [4]. A breakthrough in the study of integrable mo dels o ccurred some three years ago when Brink et al [5] and Polychronakos [6] indep endently intro duced an exchange op erator for- malism, leading to covariant derivatives, known in the mathematical literature as Dunkl op erators [7], and to an S -extended Heisenb erg algebra [8, 9 ]. In terms of the latter, N the N -b o dy quantum-mechanical Calogero mo del can indeed b e interpreted as a mo del of free mo di ed oscillators. Such an approach emphasizes the relations b etween the Calogero problem and fractional statistics [10] and is connected with the spin generalization of the former (see e.g. Ref. [11 ] and references quoted therein). Recently, there has b een a renewed interest in the three-b o dy Calogero-Marchioro- Wolfes (CMW) problem [2, 3] and some other related three-particle problems including a three-b o dy p otential. Khare and Bhaduri [12] indeed showed that they can b e solved by using sup ersymmetric quantum-mechanical techniques. Their approach is based up on Calogero's original metho d [1], wherein after eliminating the centre-of-mass motion, the three-b o dy problem is mapp ed to that of a particle on a plane and the corresp onding Schr}odinger equation is separated into a radial and an angular equations. The purp ose of the present letter is to show that the exchange op erator formalism of Brink et al [5] and Polychronakos [6] can b e extended to the CMW problem, thereby leading to a further generalization of the Heisenb erg algebra. Our starting p oint will b e another sup ersymmetric approach to the three-b o dy problem, wherein the latter is mapp ed to that of a particle in three-dimensional space and use is made of the Andrianov et al generalization of sup ersymmetric quantum mechanics for multidimensional Hamiltonians [13]. 2 2 Sup ersymmetric Approach to the CMW Problem The three-particle Hamiltonian of the CMW problem is given by [2, 3 ] 3 3 3 X X X 1 1 2 2 2 H = @ + ! x + g +3f ; (2.1) i i 2 2 (x x ) (x + x 2x ) i j i j k i=1 i;j =1 i;j;k =1 i6=j i6=j 6=k 6=i where x , i = 1, 2, 3, denote the particle co ordinates, @ @=@x , and the inequalities i i i g>1=4, and f>1=4 are assumed to b e valid to prevent collapse. Let x x x , ij i j i 6= j , and y x + x 2x , i 6= j 6= k 6= i, where in the latter we suppressed index k as ij i j k it is entirely determined by i and j . Since for singular p otentials crossing is not allowed, in the case of distinguishable par- ticles the wave functions in di erent sectors of con guration space are disconnected. We shall therefore restrict the particle co ordinates to the ranges x >x >x if g 6=0, f =0, 1 2 3 x >x , x >x , jx x j<x x , jx x j <x x if g =0, f 6= 0, and x >x >x , 1 3 2 3 1 2 1 3 1 2 2 3 1 2 3 x x <x x if g 6=0, f 6= 0. In the case of indistinguishable particles, there is an 1 2 2 3 additional symmetry requirement, whichwe shall not review here as it was discussed in detail in Refs. [2] and [3]. For distinguishable particles, the unnormalized ground-state wave function of Hamilto- nian (2.1), corresp onding to the eigenvalue E = 3! (2 +1) if g 6=0;f =0; 0 = 3!(2 +1) if g =0;f 6=0; = 3!(2 +2 +1) if g 6=0;f 6=0; (2.2) is ! X 2 1 (x) = exp x jx x x j if g 6=0;f =0; ! 0 12 23 31 i 2 i ! X 2 1 = exp ! x jy y y j if g =0;f 6=0; 12 23 31 i 2 i ! X 2 1 = exp ! x jx x x j jy y y j if g 6=0;f 6=0; (2.3) 12 23 31 12 23 31 i 2 i 3 p p 1 1 where (1 + 1+4g), (1 + 1+4f) (implying g = ( 1), f = ( 1)). 2 2 In terms of the function (x)= ln (x), one can construct six di erential op erators 0 Q = @ + @ , i = 1, 2, 3, whose explicit form is given by i i i X 1 if g 6=0;f =0; Q = @ + !x i i i x ij j6=i 0 1 X X B C 1 1 B C = @ +!x if g =0;f 6=0; i i @ A y y ij jk j6=i j;k i6=j 6=k 6=i 0 1 X X X B C 1 1 1 B C = @ +!x if g 6=0;f 6=0: (2.4) i i @ A x y y ij ij jk j6=i j 6=i j;k i6=j 6=k 6=i (0) It can b e easily shown [13] that H E can b e regarded as the H comp onentofa 0 sup ersymmetric Hamiltonian 1 0 (0) H 0 0 0 C B (1) 0 H 0 0 C B ^ C (2.5) H = B (2) A @ 0 0 H 0 (3) 0 0 0 H with sup ercharge op erators 1 0 1 0 0 Q 0 0 0 0 0 0 1;0 + C B C B y 0 0 Q 0 Q 0 0 0 C B C B + + 2;1 0;1 ^ ^ ^ C ; (2.6) = B C ; Q = Q Q = B + A @ A @ 0 0 0 Q 0 Q 0 0 3;2 1;2 + 0 0 0 0 0 0 Q 0 2;3 + ^ ^ ^ i.e., H , Q , Q generate the sup ersymmetric quantum-mechanical algebra sqm(2), o h i h i o n o n n + + + + ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ =0; Q ;H = Q ;H =0: (2.7) = Q ; Q = H; Q ; Q Q ; Q + + + In (2.6), Q , Q , Q (resp. Q , Q , Q ) denote 3 1, 3 3, 1 3 (resp. 1 3, 3 3, 0;1 1;2 2;3 1;0 2;1 3;2 + + + + , Q ), , Q (resp. Q , P = Q 3 1) matrices, whose elements are Q , P = Q ij k ij k i i i ij i ij k k where is the antisymmetric tensor and there is a summation over dummy indices. ij k ^ The comp onents of H can b e expressed in terms of such matrices as (n) 0(n) 00(n) 0(n) + 00(n) + H = H + H ; H = Q Q ; H = Q Q ; (2.8) n1;n n;n1 n+1;n n;n+1 4 + + where n =0,1,2,3,andQ = Q =Q =Q =0. Apart from some addi- 1;0 0;1 3;4 4;3 + + (0) (3) tive constants, H = Q Q , and H = Q Q are given by (2.1) with g = ( 1), i i i i f = ( 1), and g = ( + 1), f = ( + 1) resp ectively, while the remaining two com- (1) (2) p onents H and H have the form of Schr}odinger op erators with matrix p otentials. The op erators (2.8) satisfy the intertwining relations 0(n+1) + + 00(n) 0(n+1) 00(n) H Q = Q H ; Q H = H Q ; (2.9) n;n+1 n;n+1 n+1;n n+1;n 0(n+1) 00(n) (n+1) (n) and similar relations with H and H replaced by H and H , resp ectively.