Superconformal quantum mechanics via Wigner Heisenberg algebra

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DOI:10.1142/S0217732310033475 Vol. 25, No. 29 (2010) 2507–2521

c

World Scientific Publishing Company

SUPERCONFORMAL QUANTUM MECHANICS VIA WIGNER HEISENBERG ALGEBRA

H. L. CARRION∗,†_{and R. DE LIMA RODRIGUES}∗,‡ ∗_{Centro Brasileiro de Pesquisas F´ısicas, Rua Dr. Xavier Sigaud,}

150 CEP 22290-180, Rio de Janeiro-RJ, Brazil

†_{Escola de Ciˆ}_{encias e Tecnologia, Universidade Federal do Rio Grande do Norte,}

Av. Hermes da Fonseca, 1111 Tirol, 59020-315, Natal, RN, Brazil

‡_{Unidade Acadˆ}_{emica de Educa¸}_c˜_{ao, Universidade Federal de Campina Grande,}

Cuit´e – PB, CEP 58.175-000, Brazil

†_{hectors@ect.ufrn.br} ‡_{rafaelr@cbpf.br} ‡_{rafael@df.ufcg.edu.br}

Received 14 August 2009 Revised 27 March 2010

We show the natural relation between the Wigner Hamiltonian and the conformal Hamiltonian. A model in (super)conformal quantum mechanics with (super)conformal symmetry in the Wigner–Heisenberg algebra picture [x, px] = i(1 + cP) (P being the

parity operator) is presented. In this context, the energy spectrum, the Casimir operator, raising and lowering operators are defined.

Keywords: Superconformal; conformal Hamiltonian; Wigner–Heisenberg algebra. PACS Nos.: 03.65.Fd, 03.65.Ge, 12.60.Jv

1. Introduction

The problem of unifying quantum mechanics and gravity is one of the greatest un-solved problems in physics. In this context, the quantum mechanical black holes provide an arena in which quantum mechanics and gravity meet head on. The conformally-invariant quantum mechanical model was investigated firstly by Alfaro et al.1_{and an extended superconformal quantum mechanics has recently been}

dis-covered as a superconformal structure in multi-black-hole quantum mechanics (see Refs.2–6and references therein).

As an application of the simplest graded Lie algebras of the supersymmetric fields theories, the supersymmetry (SUSY) in Quantum Mechanics (QM) embodies the essential features of a theory of supersymmetry, i.e. a symmetry that gen-erates transformations between bosons and fermions or rather between bosonic and fermionic sectors associated with a SUSY Hamiltonian. SUSY QM is defined

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by Crombrugghe and Rittenberg7 _{as a graded Lie algebra satisfied by the charge}

operators Qi (i = 1, 2, . . . , N ) and the SUSY Hamiltonian H.

Also, some superconformal models are sigma models that describe the propa-gation of a nonrelativistic spinning particle in a curved background.10_{It was}

con-jectured by Gibbons and Townsend that large-n limit of an N = 4 superconformal extension of the n-particle Calogero model11 _{might provide a description of the}

extreme Reissner–Nordstr¨om black hole near the horizon.12 _{Calogero model}11

de-scribes the system of n bosonic particles interacting through the inverse square and harmonic potential, it is completely integrable at classical and quantum levels. The possibility that the n-particle Calogero model might be relevant for a microscopic description of the extreme Reissner–Nordstr¨om black hole, has also been discussed in Ref.13.

Also, the superconformal mechanics, black holes and nonlinear realizations have been investigated by Azc´arraga et al.14_{In addition, the relation between the}

super-conformal mechanics and the nonlinear supersymmetry can be found in Ref. 15. In 1950, Wigner16 proposed the interesting question: “Do the equations of mo-tion determine the quantum-mechanical commutamo-tion relamo-tions?” and he found as an answer a generalized quantum rule for the one-dimensional harmonic oscillator. In the following year, Yang17 _{found the coordinate representation for the linear}

momentum operator which realizes this aforementioned generalized quantum rule. Yang’s wave mechanical description was further studied by Ohnuki et al.18,19 and Mukunda et al.20 _{Jayaraman–Rodrigues have identified the free parameter of the}

Celka–Hussin’s model with that of the Wigner parameter of a related super-realized general 3D Wigner oscillator system satisfying a generalized (super) quantum com-mutation relation of the σ3-deformed Heisenberg algebra.21,22

The deformed Wigner–Heisenberg (WH) oscillator algebra has been investi-gated in the context of the generalized statistics (introduced in physics in the form of parastatistics as an extension of the Bose and Fermi statistics).23,24_{On the other}

hand, the elements of the conformal group can be represented in terms of ladder operators of deformed quantum oscillators25 _{and the WH algebra has also been}

investigated in connection with noncommutative geometry.26–28 _{Also, recently the}

second author has deduced the energy eigenvalues and eigenfunctions of the hy-drogen atom, which are mapped onto the super-Wigner oscillator by using the Kustaanheimo–Stiefel transformation via WH algebra in nonrelativistic quantum mechanics.29

Conformal sigma models may have applications in the context of AdS/CFT correspondence with AdS2× M background.30Also, another interesting application

of the superconformal symmetry is the treatment of the Dirac oscillator31 _{in the}

context of superconformal quantum mechanics.32,33

Here we show that the interesting new structures in conformal quantum mechan-ics in the WH picture. The generalized Hamiltonian (21), its energy spectrum, the Casimir operator, raising and lowering operators using the WH algebra have been

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investigated. Also, we present an example in superconformal quantum mechanics with WH algebra structure. In the same way, we study the energy spectrum and construct the supersymmetric Casimir operator.

This paper is organized as follows. In Sec. 2 we start by summarizing the essen-tial features of the formulation of P-deformed Wigner oscillator algebra. In Sec. 3 we present the conformal quantum mechanics based on the WH-algebra and discuss the eigenvalue problem by defining the Casimir operator and the ladder operators. In Sec. 4 we present the superconformal Hamiltonian based on the WH algebra pic-ture and define the supercharged operator in the Yang representation. We compute the superconformal algebra and discuss the eigenvalue problem in the supersym-metric context. As in the bosonic case, in the supersymsupersym-metric case we defined the raising and lowering operators and then we discuss the eigenvalue problem for the well-defined superconformal Hamiltonian. In Figs.1and2the supersymmetric potentials for each specific parity operator eigenvalue are plotted.

Next we consider the following convention: [A, B]_± = AB ± BA, for the com-mutation and anticomcom-mutation relations.

2. The P -Deformed Heisenberg Algebra

In order to make the paper self-contained we present a short discussion of the P-deformed Heisenberg algebra. The Wigner Hamiltonian expressed in the sym-metrized bilinear form in terms of the mutually adjoint operators ˆa±_{, is defined}

by ˆ HW = 1 2(ˆp 2 x+ ˆx2) = 1 2[ˆa−, ˆa +_] +, (1) where ˆ a± =√1 2(±iˆpx− ˆx) , pˆx= −i d dx, ~= 1 . (2)

Wigner showed that the Heisenberg’s equations of motion

[ ˆHW, ˆa±]− = ± ˙ˆa±, (3)

do not necessarily entail in the usual quantum rule [ˆa−_{, ˆ}_a+_]

−= 1 ⇒ [ˆx, ˆpx]−= i , (4)

but a more general quantum rule17–20_{given by}

[ˆa−_{, ˆ}_a+_]

− = 1 + c ˆR ⇒ [ˆx, ˆpx]−= i(1 + c ˆR) , (5)

where c is a real constant, called Wigner parameter, related to the ground state energy E_W(0) ≥ 0 by virtue of the positive semi-definite form of ˆHW

|c| = 2EW(0)− 1 . (6)

Note that the case c = 0 corresponds to the usual oscillator with E(0) ₌ 1 2, ~ =

ω = m = 1.

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The basic (anti-) commutation relations (1) and (3), together with the derived relation (5) constitute P-deformed Heisenberg algebra or in short the WH alge-bra. The WH algebra can also be obtained by the requirement that ˆx satisfies the classical equation of motion (¨x + ˆˆ x = 0).

Note that ˆR is an abstract operator satisfying the properties

[ ˆR, â±]+= 0 ⇒ [ ˆR, ˆHW]₋= 0 ; Rˆ† = ˆR−1= ˆR , Rˆ2= 1 . (7) Also, we have ˆ HW = â+â−+1 2(1 + c ˆR) = â−_ˆ_a+ −1 2(1 + c ˆR) . (8)

In the mechanical representation first investigated by Yang,17 _{R is realized by the}_ˆ

parity operator P:

P_{|xi = | − xi ⇒ [P, x]}+ = 0 , [P, px]+= 0 ,

P†_{= P}−1_{= P ,} _P2_{= 1 .}

(9) Indeed, Yang17_{has found the coordinate representation for the momentum operator}

pxas given by ˆ px→ px= p + i c 2xP, x → x ,ˆ p = −i d dx, (10) ˆ a±_{→ a}± c/2= 1 √ 2 ±_dxd ∓_2xc P_{− x} . (11)

Yang’s wave mechanical description was further investigated in Refs. 18–20. The P-deformed Heisenberg algebra is based on the following general quantum rule17

[ˆa−, ˆa+]₋ = 1 + cP ⇒ [ˆx, ˆpx]₋ = i(1 + cP) , (12)

where

[P, ˆa±_]

+= 0 ⇒ [P, ˆHW]−= 0 . (13)

When we replace Eq. (10) into Eq. (1) for the Wigner Hamiltonian, we obtain ˆ H_{W −}=1 2 p2+ x2+ 1 4x2(c 2_{+ 2c)}_, ₍₁₄₎

where the parity operator has been taken the value −1. If we consider the following Hamiltonian

˜ L0= p2 2 + g 2x2 + x2 2 , (15)

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and choose c2 _{+ 2c = 4g in (}₁₄_{), then the Wigner Hamiltonian is equal to the}

Hamiltonian (15) in the coordinate representation ˜L0= ˆHW−, for the case P → −1. The energy spectrum of the Hamiltonian ˜L0 (or ˆHW −) is well defined. The

eigenstates of ˜L0form an infinite tower above the ground state, in integer steps.

On the other hand, it is important to comment that there exists a hidden su-persymmetry in the WH-algebra structure. If we define as the bosonic generators { ˆHW, (â+)2, (â−)2} and the fermionic ones as {â+, â−}, then they close the osp(1/2)

superalgebra.8,9

3. Conformal Symmetry in the Wigner Heisenberg Picture

In this section we study the Hamiltonian with conformal symmetry in the WH-algebra picture. This is a modified version of the usual conformal Hamiltonian with a standard canonical quantum rule [ˆx, ˆpx]₋= i. Let us define the new

Hamil-tonian H = p 2 x 2 + g 2x2, (16)

where g is a coupling constant and pxis defined in (10) with new quantum rule (12).

This Hamiltonian (16) commutes with the parity operator P. Observe that, when the constant c vanishes, the Hamiltonian in Eq. (16) becomes the usual Fubini– Furlan conformal Hamiltonian.1,3

Next let us introduce the operators

D = pxx + xpx 2 , K = x 2 2 . (17)

We can demonstrate that these three operators satisfy the following SL(2, R) algebra

[H, D]₋= −2iH , [H, K]₋= −iD , [K, D]₋= 2iK , (18) where D is known to be the generator of dilatation (it generates the re-scalings x → γx, px→ px/γ) and K is the generator of special conformal transformations.

Since D and K do not commute with the conformal Hamiltonian H, they do not generate symmetries in the usual sense of relating the degenerate states. Rather they can be used to relate states with different eigenvalues of H (16).

It is possible to show in any quantum mechanical systems with operators obeying the SL(2, R) algebra (18), that if |χi is a state of energy E, then eıαD_{|χi is a state}

for the energy e2α_{E. Thus, if there is a state of nonzero energy, then the spectrum}

is continuous. Note that this result gives the spectrum of conformal Hamiltonian H (16) to be continuous and then its eigenstates are not normalizable.3

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Let us consider the following linear combinations L+= 1 2(H − K + iD) , L₋= 1 2(H − K − iD) , L0= 1 2(H + K) . (19)

The generators (19) satisfy the SL(2, R) algebra (Virasoro form), with the following commutation relations

[L₋, L+]−= 2L0, [L0, L+]− = L+, [L0, L−]−= −L−. (20)

With definition (17)–(19), we have L0= p 2 x 4 + g 4x2 + x2 4 . (21)

This is the new version for Eq. (2.6) of Ref.2. The potential energy for this operator (21) has an absolute minimum, it then has a discrete spectrum with normalized eigenstates. If L0 satisfies the following eigenvalue equation

L0|ni = ǫn|ni , (22)

and using the algebra (20) we see that L₋ and L+ form the raising and lowering

operators for the Hamiltonian L0, so that

ǫn= ǫ0+ n , n = 0, 1, 2, . . . . (23)

In other words, the eigenvalues of L0form an infinite tower above “the ground

state”, in integer steps.

The SL(2, R) Casimir operator is given by

L2= L0(L0− 1) − L+L₋, (24)

where

[L2, L+]− = [L2, L−]−= [L2, L0]−= 0 . (25)

This Casimir operator in terms of the generators of the conformal group becomes L2= HK + KH

2 −

D2

4 . (26)

Using the WH-algebra, the action of this Casimir operator on the eigenket |ni and defining L2_{= l}

0(l0− 1), we obtain

L2₌ 1

16[4g − 3 − c(2P − c)] (27)

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or L2₌            1 16[4g − 3 − 2c + c 2_{] ,} _P_{→ +1 ,} 1 16[4g − 3 + 2c + c 2_{] ,} _P_{→ −1} (28) with ℓ0=                  1 ±qg +1 4(1 − c)2 2 , P→ 1 , 1 ±qg +1 4(1 + c)2 2 , P→ −1 (29)

for the “ground state” |0i. One can see that the coupling constant defined in (16) must satisfy (g > −1

4(1 ± c)

2_{). Note that the lowest eigenfunction of L}

0is found by

the annihilation condition L₋|0, ℓ0i = 0.

When the constant c vanishes, L0 → ˜L0, the Casimir operator becomes L2 = 1

16(4g −3), so that the ground state has the eigenvalue ℓ0= 1 2(1±

q g +1

4), which is

a well-known result for the usual conformal quantum mechanics with the canonical commutation relation [x, p]₋= i.

Note that ǫ0= ℓ0or ǫ0= 1 − ℓ0 and

L+|n, ℓ0i =p(ǫn+ ℓ0)(ǫn− ℓ0+ 1)|n + 1, ℓ0i ,

L₋|n, ℓ0i =p(ǫn− ℓ0)(ǫn+ ℓ0− 1)|n − 1, ℓ0i .

(30) From Eqs. (12) and (19), we can express L+ and L− in terms of the operators

ˆ

a+and ˆa₋ as shown below

L+= − 1 2ˆa 2 ++ g 4x2, (31) L₋ = −1₂ˆa2₋+ g 4x2. (32)

4. The Superconformal Quantum Mechanics

The superconformal quantum mechanics has been examined in Refs.3–10. Another application for these models is in the study of the radial motion of test particle near the horizon of extremal Reissner–Nordstr¨om black holes.10,12 _{Also, another}

interesting application of the superconformal symmetry is the treatment of the Dirac oscillator, in the context of the superconformal quantum mechanics.31–33

In this section we introduce the explicit supersymmetry for the conformal Hamil-tonian in the WH-algebra picture. Let us consider the supersymmetric generaliza-tion of H, Eq. (16), given by

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H =1₂{Qc, Q†c} , (33)

where the new supercharge operators are given in terms of the momentum Yang representation Qc = −ipx+ √_g x Ψ†_, Q† c = Ψ ipx+ √_g x , (34)

with Ψ and Ψ† _{being Grassmannian operators so that its anticommutator is}

{Ψ, Ψ†_{} = ΨΨ}†_{+ Ψ}†_{Ψ = 1.}

Explicitly the superconformal Hamiltonian becomes H = 1₂ 1p2x+ 1_{g + √gB(1 − cP)} x2 , (35) where B = [Ψ†_{, Ψ]}

−, so that the parity operator is conserved, i.e. [H, P]− = 0.

When one introduces the following operators S = xΨ†_,

S† _{= Ψx ,} (36)

it can be shown that these operators together with the conformal quantum me-chanics operators D and K

D = 1 2(xpx+ pxx) , K = 1 2x 2_, (37)

satisfy the deformed superalgebra osp(2|2) (actually, this superalgebra is osp(2|2) when we fix P = 1 or P = −1), viz.,

[H, D]− = −2ıH , [H, K]−= −ıD , [K, D]₋ = 2ıK , [Qc, Q†c]+= 2H , [Qc, S†]+= −ıD − 1 2B(1 + cP) + √_{g ,} [Q† c, S]+= ıD − 1 2B(1 + cP) + √_{g ,}

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[Q† c, D]−= −ıQ†c, [Q† c, K]−= S†, [Q† c, B]−= 2Q†c, [Qc, K]₋= −S , [Qc, B]−= −2Qc, [Qc, D]−= −ıQc, [H, S]−= Qc, [H, S†]₋ = −Q†c, [B, S†_] −= −2S†, [B, S]−= 2S , [D, S]₋= −ıS , [D, S†_] −= −ıS†, [S†_{, S]}₊_{= 2K ,} (38)

where, H, D, K, B are bosonic operators and Qc, Q†c, S, S† are fermionic ones.

The supersymmetric extension of the Hamiltonian L0 (presented in the previous

section) is

H0=

1

2(H + K) , [H0, P]− = 0 . (39)

In general, superconformal quantum mechanics has interesting applications in supersymmetric black holes, for example in the problem of a quantum test particle moving in the black hole geometry.3

Continuing with our discussion, now we are able to build up the ladder operators for the spectral resolution of the super-Hamiltonian H0. Note that defining

M = Qc− S , (40) Q = Qc+ S , (41) h ≡ 1₂[M, M†_] + = 2H0+ 1 2(1 + cP)B − √_{g ,} ˜ h ≡ 1₂[Q, Q†_] += 2H0− 1 2(1 + cP)B + √g , (42) we obtain [h, Q†_] −= −2Q†, [h, Q]₋= 2Q , (43)

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and

[˜h, M†_]

− = 2M†,

[˜h, M ]₋ = −2M .

(44) Thus we can see that Q and Q†_{, M and M}† _{are ladder operators associated}

to the SUSY Hamiltonian operators h and ˜h, respectively. We remark that since h, ˜h and H0 commute with each other, they form a set of mutually commuting

operators.

The super-Hamiltonian ˜h is, with the exception of the constant 2mω, the ex-tension of the ones for the radial equation with conformal invariance in a Dirac oscillator.32

The supersymmetric generalizations of the ladder operators defined in (19) in the bosonic case are given by

L− ≡ − 1 4[M, Q†]+= − 1 2(H − K − ıD) , L+ ≡ −1 4[M†, Q]+= − 1 2(H − K + ıD) (45)

with the following commutation relations of the so(2, 1) algebra [L+, H0]−= −L+,

[L−, H0]−= L−,

[L+, L−]−= −2H0.

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The supersymmetric Casimir operator is given by

L2= H0(H0− 1) − L+L₋. (47)

Thus, the energy spectrum becomes

H0|m, si = ζm|m, si ,

L2_{|m, si = s(s − 1)|m, si ,} (48)

with ζm= ζ0+ m.

Since the eigenket |m, si is normalized, from (47) and (48) we obtain s = s0= ζ0.

In this case, we may write the analogous representations to the ladder operators given by Eq. (30)

L+|m, si = αm+1|m + 1, si ,

L−|m, si = βm−1|m − 1, si ,

(49) where αm+1and βm−1 have similar forms to the bosonic case, given in Eq. (30).

To obtain a super-realization of the conformal quantum mechanics, we intro-duce, in addition to the usual bosonic coordinates (x, −idxd), the fermionic ones

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b∓_{(= (b}±₎†_{) that commute with the bosonic set and are represented in terms of the}

usual Pauli matrices σi, (i = 1, 2, 3) by the combinations

b−_{= σ} += 1 2(σ1+ iσ2) = 0 1 0 0 (b−₎2_{= 0 ,} ₍₅₀₎ b+= σ₋= 1 2(σ1− iσ2) = 0 0 1 0 (b+)2= 0 , (51) [b−, b+]+= 1 (52)

so that the fermion number operator is given by Nf = b+b−= σ−σ+= 1 2(1 + σ3) = 0 0 0 1 . (53)

Observe that we are using the convention that the number operator Nf has the

fermion number nf = 0 and is associated to the bosonic state and its eigenstate is

given by 1₀.

Similarly, the eigenstates of Nf with the fermion number nf = 1 is associated

to the fermionic state, and is given by 0₁. Using Eq. (48) we obtain

L2= 1 16[(4g − 3 − c(2P − c))l2×2+ 4 √ gB(1 − cP)] , (54) where l_2×2 = 1 0 0 1 , B = −1 0 0 1 = −σ3 (55)

and the parity operator must be substituted by its eigenvalues on the eigenstates |m, si. In the case P → +1, one gets

L2= 1 16 4g − 3 − 2c + c2_{− 4}√_{g(1 − c)} ₀ 0 4g − 3 − 2c + c2_{+ 4√g(1 − c)} ! , (56) with the following eigenvalues

s(s − 1) = ₁₆1 (4g − 3 − 2c + c2− 4√g(1 − c)) , (57) s′(s′− 1) = ₁₆1 (4g − 3 − 2c + c2+ 4√g(1 − c)) . (58) From Eqs. (39) and (55) we obtain the SUSY conformal Hamiltonian in the coordinate representation H01= 1 4   p2_{+ x}2₊c2+2c+4g−4√g(1−c) 4x2 0 0 p2_{+ x}2₊c2_{+2c+4g+4g√g(1−c)} 4x2  , (59)

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V01(+) 1 2 3 4 5 6 –2 –1 0 1 2 x

Fig. 1. The bosonic part of the supersymmetric potential V01, for P = +1, c = 2, g = 1. with the following superconformal potential

V01(x) ≡ V01(+) 0 0 V01(−) ! = 1 4 x2+c 2_{+ 2c + 4g} 4x2 1_2×2_{− σ}3 √_{g(1 − c)} x2 ! . (60)

In Fig.1we plot V01(+), the bosonic part of this supersymmetric potential. In

the case P → −1 we get L2= 1 16 4g − 3 + 2c + c2_{− 4}√_{g(1 + c)} ₀ 0 4g − 3 + 2c + c2_{+ 4√g(1 + c)} ! . (61) Also, in this case

H02= 1 4   p2_{+ x}2₊c2−2c+4g−4√g(1+c) 4x2 0 0 p2+ x2+c2−2c+4g+4_4x2√g(1+c)   (62)

with superconformal potential V02(x) = 1 4 x2+c 2_{− 2c + 4g} 4x2 ! 1_2×2_{− σ}3 √_{g(1 + c)} x2 ! . (63)

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V02(+) 0.6 0.8 1 1.2 1.4 1.6 –2 –1 0 1 2 x

Fig. 2. The bosonic part of the supersymmetric potential V02, for P = −1, c = 2, g = 1. In Fig.2 we plot V02(+), the bosonic part of this supersymmetric potential of

the supersymmetric Hamiltonian of the Calogero interaction’s type and it is also an extension of the super-Hamiltonian for the radial equation in a Dirac oscillator.33

Finally, observing the supersymmetric Hamiltonians H01and H02we recall that

the parameter c is real and g must be positive. With appropriate choice of these parameters we can recover various cases, for example the oscillator type super-Hamiltonian (without the presence of the proportional term 1/x2_{) although the}

bosonic part without supersymmetry has this term (g 6= 0). In this case the super-symmetric potential will not have a singularity at the origin of coordinates.

5. Conclusion

In this work, we have analyzed the connection between the conformal quantum me-chanics1,3 _{and the Wigner–Heisenberg (WH) algebra.}17–19,21 _{With an appropriate}

relationship between the coupling constant g and real constant c one can identify the Wigner Hamiltonian with the simple Calogero Hamiltonian.

The important result is that the introduction of the WH algebra in the con-formal quantum mechanics is still consistent with the concon-formal symmetry, and a realization of superconformal quantum mechanics in terms of deformed WH algebra is discussed. The spectra for the Casimir operator and the Hamiltonian L0depend

on the parity operator. The ladder operators also depend on the parity operator. It is shown, for example, that the eigenvalues of Calogero-type Hamiltonian (21) depend on the Wigner parameter c and the eigenvalues of the parity operator P . When c = 0, we obtain the usual conformal Hamiltonian structure.

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We also investigated the supersymmetrization of this model, in that case we obtain a new spectrum for the supersymmetric Hamiltonian of the Calogero inter-action’s type.

In this case the spectra for the Casimir operator and the super-Hamiltonian depend also on the parity operator. Therefore, we have found a new realization of supersymmetric Calogero-type model on the quantum mechanics context in terms of deformed WH algebra.

Let us point out that the same analysis implemented in this work can be con-sidered for the Dirac oscillator,31 _{getting a generalization of the works,}32,34 _and

for a multispecies Calogero model with inverse-square two-body and three-body interactions.35

Acknowledgments

The authors would like to acknowledge S. Alves for hospitality at CCP-CBPF of Rio de Janeiro-RJ, Brazil, where the part of this work was carried out. The authors are partially supported by CNPq-CLAF, and are also grateful to J. A. Helayel Neto, Harold Blas and F. Toppan for interesting discussions and suggestions. R.L.R. is also grateful for interesting discussions with Jambunatha Jayaraman (in memory), whose advises and encouragement were fundamental.

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