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Exchange Operators and Extended Heisenberg Algebra for the Three 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.
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