Acta Polytechnica https://doi.org/10.14311/AP.2022.62.0050 Acta Polytechnica 62(1):50–55, 2022 © 2022 The Author(s). Licensed under a CC-BY 4.0 licence Published by the Czech Technical University in Prague GENERALIZED THREE-BODY HARMONIC OSCILLATOR SYSTEM: GROUND STATE Adrian M. Escobar-Ruiz∗, Fidel Montoya Universidad Autónoma Metropolitana, Departamento de Física, Av. San Rafael Atlixco 186, 09340 Ciudad de México, CDMX, México ∗ corresponding author: admau@xanum.uam.mx Abstract. In this work we report on a 3-body system in a d−dimensional space Rd with a quadratic harmonic potential in the relative distances rij = |ri − rj | between particles. Our study considers unequal masses, different spring constants and it is defined in the three-dimensional (sub)space of solutions characterized (globally) by zero total angular momentum. This system is exactly-solvable with hidden algebra sℓ4(R). It is shown that in some particular cases the system becomes maximally (minimally) superintegrable. We pay special attention to a physically relevant generalization of the model where eventually the integrability is lost. In particular, the ground state and the first excited state are determined within a perturbative framework. Keywords: Three-body system, exact-solvability, hidden algebra, integrability. 1. Introduction The two-body harmonic oscillator, i.e. two particles with masses m1 and m2 interacting via the transla- tional invariant potential V ∝ |ri − rj | 2, appears in all textbook in Classical Mechanics. In an arbitrary d−dimensional Euclidean space Rd this system ad- mits separation of variables in the center-of-mass and relative coordinates as well as exact solvability. The relevance of such a system is obvious: any scalar po- tential U = U(|ri − rj |) can be approximated by the two-body harmonic oscillator. In this case, the center- of-mass and relative coordinates are nothing but the normal coordinates. Therefore, in the n-body case of n > 2 particles interacting by a quadratic pairwise potential it is natural to ask the question about the existence of normal coordinates and the correspond- ing explicit exact solutions. Interestingly, even for the three-body case n = 3 a complete separation of variables can not be achieved in full generality. Starting in 1935, the quantum n−body problem in R3 was studied by Zernike and Brinkman [1] using the so-called hyperspherical-harmonic expansion. Two decades later, this method possessing an underlying group-theoretical nature was then reacquainted and refined in the papers by Delves [2] and Smith [3]. Nevertheless, in practice the success of the method is limited to the case of highly symmetric systems, namely identical particles with equal masses and equal spring constants. In a previous work [4], the most general quantum system of a three-body chain of harmonic oscillators, in Rd, was explored exhaustively. For arbitrary masses and spring constants this problem possesses spherical symmetry. It implies that the total angular momen- tum is a well-defined Observable which allows to re- duce effectively the number of degrees of freedom in the corresponding Schrödinger equation governing the states with zero angular momentum. In the sector of vanishing angular momentum, it turns out that this three-body quantum system is exactly solvable. The hidden algebra sℓ(4,R) responsible of the exact solv- ability was exhibited in [4] using the ρ-representation. In the present work we consider a physically relevant generalization of the model where eventually the inte- grability properties are lost. Again, in our analysis we assume a system of arbitrary masses and spring con- stants with the total angular momentum identically zero. In the current study we revisited the algebraic struc- ture and solvability of the quantum 3-body quantum oscillator system in the special set of coordinates ap- pearing in [5], [6]. Afterwards, a physically motivated generalization of the model is considered. The goal of the paper is two-fold. Firstly, in the (sub)-space of zero total angular momentum we will describe the reduced Hamiltonian operator which admits a hidden sℓ(4; R) algebraic structure, hence, allowing exact- analytical eigenfunctions. Especially, at any d ≥ 1 it is demonstrated the existence of an exactly-solvable model that solely depends on the moment of inertia of the system. This model, admits a quasi-exactly- solvable extension as well. Secondly, we explore a physically relevant general- ization of the model. Approximate solutions of the problem are presented just for the case of equal masses in the framework of standard perturbation theory and complemented by the variational method. The first excited state, thus the energy gap of the system, is briefly discussed. 50 https://doi.org/10.14311/AP.2022.62.0050 https://creativecommons.org/licenses/by/4.0/ https://www.cvut.cz/en vol. 62 no. 1/2022 Generalized three-body harmonic oscillator system: ground state 2. Generalities The quantum Hamiltonian in Rd (d > 1) for three non- relativistic spinless particles with masses m1,m2,m3 and translationally invariant potential is given by H = − 3∑ i=1 1 2 mi ∆(d)i + V (r12, r13, r23) , (1) (ℏ = 1) see e.g. [4, 5], where ∆(d)i stands for the individual Laplace operator of the ith mass with d−dimensional position vector ri, and rij = |ri − rj | , (2) (j = 1, 2, 3) is the relative mutual distance between the bodies i and j. The eigenfunctions of (1) which solely depend on the ρ-variables, ρij = r2ij , are governed by a three-dimensional reduced Hamiltonian [4] Hrad ≡ −∆rad + V (ρ) , (3) where ∆rad = 2 µ12 ρ12 ∂ 2 ρ12 + 2 µ13 ρ13 ∂ 2 ρ13 + 2 µ23 ρ23 ∂ 2 ρ23 + 2(ρ13 + ρ12 − ρ23) m1 ∂ρ13 , ρ12 + 2(ρ13 + ρ23 − ρ12) m3 ∂ρ13 , ρ23 + 2(ρ23 + ρ12 − ρ13) m2 ∂ρ23 , ρ12 + d µ12 ∂ρ12 + d µ13 ∂ρ13 + d µ23 ∂ρ23 , (4) c.f. [5], and µij = mi mj mi + mj , denotes a reduced mass. The operator (3) de- scribes three-dimensional (radial) dynamics in vari- ables ρ12,ρ13,ρ23. This operator Hrad is, in fact, equivalent to a Schrödinger operator, see [4]. We call it three-dimensional (radial) Hamiltonian. All the d−dependence in (3) occurs in the coefficients in front of the first derivatives. 2.1. Case of identical particles: τ -representation Now, let us consider the case of identical masses m1 = 1 ; m2 = 1 ; m3 = 1 , thus, µij = 12 , and the operator (4) is S3 permutationally-invariant in the ρ-variables. It sug- gests the change of variables ρ ↔ τ where τ1 = ρ12 + ρ13 + ρ23 , τ2 = ρ12 ρ13 + ρ12 ρ23 + ρ13 ρ23 , τ3 = ρ12 ρ13 ρ23 , (5) are nothing but the lowest elementary symmetric poly- nomials in ρ-coordinates. In these variables (5), the coefficients of the operator ∆rad are also polynomials, hence, this operator is algebraic in both representations. Explicitly, ∆rad = 6 τ1∂21 + 2τ1(7τ2 − τ 2 1 )∂ 2 2 + 2τ3(6τ2 − τ 2 1 )∂ 2 3 + 24 τ2∂21,2 + 36τ3∂ 2 1,3 + 2 [9τ3τ1 + 4τ2(τ2 − τ 2 1 )]∂ 2 2,3 + 6 d∂1 + 2 (2d + 1)τ1 ∂2 + 2 [(d + 4)τ2 − τ21 ] ∂3 (6) ∂i ≡ ∂τi, i = 1, 2, 3. 3. Laplace-Beltrami operator Now, as a result of calculations it is convenient to consider the following gauge factor Γ4 = (S2△) 2−d M I , (7) M = m1 + m2 + m3, where S 2 △ = 2ρ12 ρ13 + 2ρ12 ρ23 + 2ρ23 ρ13 − ρ212 − ρ213 − ρ223 16 , and I = m1m2 ρ12 + m1m3 ρ13 + m2m3 ρ23 M , possess a geometrical meaning. The term S2△ is the area (squared) of the triangle formed by the position vectors of the three bodies whilst the term I is the moment of inertia of the system with respect to its center of mass. The radial operator Hrad (3) is gauge- transformed to a truly Schrödinger operator [4], HLB ≡ Γ−1 Hrad Γ = −∆LB + V + V (eff) , (8) here ∆LB stands for the Laplace-Beltrami operator ∆LB (ρ) = √ | g | ∂µ 1√ | g | gµν∂ν , (ν,µ = 1, 2, 3) and ∂1 = ∂∂ρ12 ,∂2 = ∂ ∂ρ13 ,∂3 = ∂∂ρ23 . The corresponding co-metric in ∆LB (ρ) reads g µν =   2 µ12 ρ12 (ρ13 +ρ12 −ρ23 ) m1 (ρ23 +ρ12 −ρ13 ) m2 (ρ13 +ρ12 −ρ23 ) m1 2 µ13 ρ13 (ρ13 +ρ23 −ρ12 ) m3 (ρ23 +ρ12 −ρ13 ) m2 (ρ13 +ρ23 −ρ12 ) m3 2 µ23 ρ23   . Its determinant | g | ≡ Detgµν = 32 M2 m21m 2 2m 2 3 I S2△ , (9) admits factorization and is positive definite. The term V (eff) denotes an effective potential V (eff) = 3 8 1 I + (d − 2)(d − 4) 32 M I m1 m2 m3 S 2 △ , which depends on the two variables I and S2△ alone. Thus, the underlying geometry of the system emerges. The classical analogue of the quantum Hamilto- nian operator (8) describes an effective non-relativistic 51 Adrian M. Escobar-Ruiz, Fidel Montoya Acta Polytechnica Figure 1. 3-body chain of harmonic oscillators. classical particle in a three-dimensional curved space. Explicitly, the Hamiltonian function takes the form H(classical)LB = g µν Πµ Πν + V , (10) where Πµ , µ = 12, 23, 13 are the associated canon- ical conjugate momenta to the ρ-coordinates. The Hamilton-Jacobi equation, at vanishing potential V = 0 (free motion), is clearly integrable. However, a complete separation of variables is absent in the ρ-representation. The Poisson bracket between the ki- netic energy T = gµν Πµ Πν and the linear function in momentum variables L (c) 1 = (ρ13−ρ23)Π12+(ρ23−ρ12)Π13+(ρ12−ρ13)Π23 , is zero. 4. Three body harmonic oscillator system In the spectral problem with Hamiltonian (3) we take the harmonic potential V (HO)(ρ) = 2 ω2 [ ν12 ρ12 + ν13 ρ13 + ν23 ρ23 ] , (11) ω > 0 is frequency and ν12, ν13, ν23 > 0 are con- stants with dimension of mass. This problem can be solved exactly [4]. In particular, in ρ-space the reduced operator (3) possesses multivariate polyno- mial eigenfunctions, see below. We call the above potential V (HO)(ρ) the 3-body oscillator system. We mention that in the case d = 1 (3 particles on a line), the corresponding spectral problem was studied in the paper [7]. In the current report, we analyze the d−dimensional case with d > 1. In r-variables, ρ = r2, the potential (11) can be interpreted as a three-dimensional (an)isotropic one- body oscillator. It is displayed in Figure 1. The configuration space is a subspace of the cube R3+(ρ) in E3 ρ-space. The ρ-variables must obey the “tri- angle condition” S2△ ⩾ 0, namely the area of the triangle formed by the position vectors of the bodies is always positive. 4.1. Solution for the ground state In the harmonic potential (11), the ground state eigen- function reads Ψ(HO)0 = e −ω (a1 µ12 ρ12 + a2 µ13 ρ13 + a3 µ23 ρ23) , (12) where the parameters a1, a2, a3 ≥ 0 are introduced for convenience. They define the spring constants, see below. The associated ground state energy E0 = ω d (a1 + a2 + a3) , (13) is mass-independent. There exists the following alge- braic relations ν12 = a21 µ12 + a1 a2 µ12 µ13 m1 + a1 a3 µ12 µ23 m2 − a2 a3 µ13 µ23 m3 , ν13 = a22 µ13 + a1 a2 µ12 µ13 m1 + a2 a3 µ13 µ23 m3 − a1 a3 µ12 µ23 m2 , ν23 = a23 µ23 + a1 a3 µ12 µ23 m2 + a2 a3 µ13 µ23 m3 − a1 a2 µ12 µ13 m1 . 5. Lie algebraic structure Using the previous function Ψ(HO)0 (12) as a gauge factor, the transformed Hamiltonian Hrad (3) h(algebraic) ≡ ( Ψ(HO)0 )−1 [−∆rad + V − E0] Ψ(HO)0 (14) is an algebraic operator, i.e. the coefficient are poly- nomials in the ρ-variables. The E0 is taken from (13). In addition, this algebraic operator (14) is of Lie- algebraic nature. It admits a representation in terms of the generators J −i = ∂ ∂yi , J 0ij = yi ∂ ∂yj , J 0(N) = 3∑ i=1 yi ∂ ∂yi − N , J +i (N) = yi J 0(N) = yi   3∑ j=1 yj ∂ ∂yj − N   , (i,j = 1, 2, 3) of the algebra sℓ(4,R), see [8, 9] here N is a constant. The notation y1 = ρ12 , y2 = ρ13 , y3 = ρ23 , was employed for simplicity. If N is a non-negative integer, a finite-dimensional representation space takes place, VN = ⟨yn11 y n2 2 y n3 3 | 0 ≤ n1 + n2 + n3 ≤ N⟩ . (15) 52 vol. 62 no. 1/2022 Generalized three-body harmonic oscillator system: ground state 6. Relation with the Jacobi oscillator Now, we can indicate an emergent relation between the harmonic potential (11) and the Jacobi oscillator system H(Jacobi) ≡ 2∑ i=1 [ − ∂2 ∂zi∂zi + 4 Λi ω2 zi · zi ] , (16) where ω > 0, Λ1, Λ2 ≥ 0, and z1 = √ m1 m2 m1 + m2 (r1 − r2) z2 = √ (m1 + m2) m3 m1 + m2 + m3 ( r3 − m1 r1 + m2 r2 m1 + m2 ) are standard Jacobi variables, see e.g. [10]. This Hamiltonian describes two decoupled harmonic os- cillators in flat space, see [6]. Consequently, it is an exactly-solvable problem. The complete spectra and eigenfunctions can be calculated by pure algebraic means. The solutions of the Jacobi oscillator that solely depend on the Jacoby distances zi = |zi| are governed by the operator, H(Jacobi)rad = 2∑ i=1 [ − ∂2 ∂zi∂zi − (d − 1) zi ∂ ∂zi ] + 4 Λ1 ω2 z21 + 4 Λ2 ω 2 z22 . (17) In this case, the associated hidden algebra is given by sl ⊗ (2)2 which acts on the two-dimensional space (z1,z2). In particular, the eigenfunctions of H(Jacobi) (16) can be employed to construct approximate solutions for the n-body problem, for this discussion see [10]. Assuming any of the two conditions m2 m3 = ν12 ν13 ; m1 m2 = ν13 ν23 , in the harmonic oscillator potential V (HO) (11), we obtain U (HO) J ≡ 4 Λ1 ω 2 z21 + 4 Λ2 ω 2 z22 = 2 ω2 [ ν12 ρ12 + ν13 ρ13 + ν23 ρ23 ] = V (HO) (18) with Λ1 = Λ2 = m1 + m2 + m3 2 m1 m3 ν13 , hence, in this case the three-body oscillator poten- tial coincides with the two-body Jacobi oscillator potential. In fact, imposing the singly condition m2 ν13 = m3 ν12 the equality (18) is still valid but Λ1 ̸= Λ2 and the system is not maximally superinte- grable any more. 6.1. Identical particles: hyperradious A remarkable simplification occurs in the case of three identical particles with the same common spring con- stant, namely m1 = m2 = m3 = 1 , a1 = a2 = a3 ≡ a . (19) Thus, the potential (11) reduces to V (HO) = 3 2 a2 ω2 (ρ12 + ρ13 + ρ23) = 3 2 a2 ω2 τ1 . Consequently, the ground state solutions (12) and (13) read Ψ(3a)0 = e − ω2 a ( ρ12 + ρ13 + ρ23 ) = e− ω 2 a τ1 , (20) E0 = 3 ω da , (21) respectively. Moreover, from (6) it follows that in this case there exists an infinite family of eigenfunctions ΨN (τ1) = e− 1 2 a ω τ1 L (d−1) N (aω τ1) , with energy EN = 3 aω ( d + 2 N ) , N = 0, 1, 2, 3, . . ., that solely depend on the variable τ1, the so called hyperradious, here L (d−1) N (x) denotes the generalized Laguerre polynomial. These solutions are associated with a hidden sℓ(2,R) Lie-algebra. 6.2. Arbitrary masses: moment of inertia A generalization of the results presented in Section 6.1 can be derived from the decomposition of ∆rad (4) ∆rad = ∆I + ∆̃ , (22) where ∆I = ∆I (I) is an algebraic operator for arbi- trary d ≥ 1. It depends on the moment of inertial I only. Explicitly, we have ∆I = 2 I ∂2I,I + 2 d∂I . (23) The operator ∆̃ = ∆̃(I,q1,q2) depends on I and two more (arbitrary) variables q1, q2 for which the coordinate transformation {ρij } → {I,q1,q2} is in- vertible (not singular). Since such an operator ∆̃ annihilates any function F = F(I), i.e. ∆̃ F = 0, the splitting (22) indicates that for any potential of the form V = V (I) , (24) the eigenvalue problem for the operator Hrad = −∆rad + V is further reduced to a one-dimensional spectral problem, namely [ −∆I + V (I) ] ψ = E ψ , (25) which can be called the I−representation. In the case of equal masses m1 = m2 = m3 the co- ordinate I is proportional to the hyperspherical radius (hyperradious). Also, HI (25) is gauge-equivalent to a one-dimensional the Schrödinger operator. 53 Adrian M. Escobar-Ruiz, Fidel Montoya Acta Polytechnica Figure 2. Classical generalized three-body harmonic oscillator system: average Lyapunov exponent in the space of parameters (H, m1). The values m2 = m3 = 1, ω = 1, ν12 = ν13 = ν23 = 1 and R12 = R13 = R23 = 1 were used. 7. Generalized three body harmonic oscillator system Now, let us consider the following potential V (R) = 2 ω2 [ ν12 ( √ ρ12 − R12)2 + ν13 ( √ ρ13 − R13)2 + ν23 ( √ ρ23 − R23)2 ] , (26) where R12,R13,R23 ⩾ 0 denote the rest lengths of the system. At R12 = R13 = R23 = 0 we re- cover the exactly solvable 3-body oscillator system, V (R) → V (HO). The relevance of V (R) comes from the fact that any arbitrary potential V = V (rij ) can be approximated, near its equilibrium points, by this generalized 3-body harmonic potential. However, the existence of non-trivial exact solu- tions is far from being evident. Even for the most symmetric case of equal masses and equal spring con- stants, we were not able to find a hidden Lie algebra in the corresponding spectral problem (3). Moreover, at the classical level such a system is chaotic. This can be easily seen by computing the average Lya- punov exponent in the space of parameters (H, m1), see Figure 2, where H is the value of the classical Hamiltonian (energy) with potential V (R) (26). Also, for one-dimensional systems it is said (see [11]) that a classical orbit is PT -symmetric if the orbit re- mains unchanged upon replacing x(t) by −x∗(−t). There are several classes of complex PT -symmetric non-Hermitian quantum-mechanical Hamiltonians whose eigenvalues are real and with unitary time evo- lution [12, 13]. However, while the corresponding quantum three-body oscillator Hamiltonian is Her- mitian, it can still have interesting complex classical trajectories. 7.1. Identical particles In order to simplify the problem one can consider the simplest case of equal masses and equal spring constants (19) with ω = 1. Also, we will assume equal rest lengths R12 = R13 = R23 = R > 0 . Ε0PT [0.5, R] Ε0PT [1, R] Ε0PT [2, R] 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 R 5 10 15 20 Ε0[ a, R] Figure 3. Ground state energy of the generalized 3- body harmonic oscillator vs R for different values of the parameter a which defines the spring constant, see text. The solid lines correspond to the variational result whilst the dashed ones refer to the value calculated by perturbation theory up to first order. In this case, approximate solutions for the Schrödinger equation can be obtained using perturbation theory in powers of R. 7.1.1. Ground state Taking the R−dependent terms in (26) as a small perturbation, the first correction E1,0 to the ground state energy takes the form E1,0 = 3 a 2 π ( 3 π aR2 − 4 R √ 6 π a ) . The domain of validity of this perturbative approach is estimated by means of the variational method. The use of the simple trial function Ψtrial0 = e − ω2 a α ( ρ12 + ρ13 + ρ23 ) c.f. (20), where α is a variational parameter to be fixed by the procedure of minimization, leads to the results shown in Figure 3. 7.1.2. First excited state It is important to mention that for the 3-body har- monic oscillator (R = 0) the exact first excited state possesses a degeneracy equal to 3. For R > 0, the perturbation theory partially breaks this degeneracy. The energy of the approximate first excited state cal- culated by perturbation theory, up to first order, is displayed in Figure 4. 8. Conclusions In this report for a 3-body harmonic oscillator in Eu- clidean space Rd we consider the Schrödinger operator in ρ-variables ρij = r2ij , HLB = −∆LB(ρij ) + V (HO)(ρij ) + V (eff)(ρij ) , (27) where the kinetic energy corresponds to a 3- dimensional particle moving in a non-flat space. The 54 vol. 62 no. 1/2022 Generalized three-body harmonic oscillator system: ground state E1[0.5, R] E1[1, R] E1[2, R] 0.2 0.4 0.6 0.8 1.0 R 5 10 15 20 25 30 E1[a, R] Figure 4. First excited state of the generalized 3- body harmonic oscillator vs R. Schrödinger operator (27) governs the S-states solu- tions of the original three-body system (1), in par- ticular, it includes the ground state. It implies that the solutions of corresponding eigenvalue problem depend solely on three coordinates, contrary to the (3d)-dimensional Schrödinger equation. The reduced Hamiltonian HLB is an Hermitian operator, where the variational method can be more easily implemented (the energy functional is a 3-dimensional integral only). The classical analogue of (27) was presented as well. The operator (27) up to a gauge rotation is equivalent to an algebraic operator with hidden algebra sℓ(4,R), thus, becoming a Lie-algebraic operator. In the case of identical masses and equal frequencies the aforementioned model was generalized to a 3-body harmonic system with a non-zero rest length R > 0. In this case, no hidden algebra nor exact solutions seem to occur. An indication of the lost of integra- bility is the fact that the classical counterpart of this model exhibits chaotic motion. Using perturbation theory complemented by the variational method it was shown that the ground state energy vs R develops a global minimum, hence, defining a configuration of equilibrium. Acknowledgements AMER thanks the support through the Programa Especial de Apoyo a la Investigación 2021, UAM-I. FM and AMER are thankful to Mario Alan Quiroz for helping in the numerical computations. References [1] F. Zernike, H. C. Brinkman. Hypersphärische Funktionen und die in sphärischen Bereichen orthogonalen Polynome. Koninklijke Nederlandse Akademie van Wetenschappen 38:161–170, 1935. [2] L. M. Delves. Tertiary and general-order collisions (I). Nuclear Physics 9(3):391–399, 1958. https://doi.org/10.1016/0029-5582(58)90372-9. [3] F. Smith. A symmetric representation for three-body problems. I. Motion in a plane. Journal of Mathematical Physics 3(4):735, 1962. https://doi.org/10.1063/1.1724275. [4] A. Turbiner, W. Miller Jr, M. A. Escobar-Ruiz. Three-body problem in d-dimensional space: ground state, (quasi)-exact-solvability. Journal of Mathematical Physics 59(2):022108, 2018. https://doi.org/10.1063/1.4994397. [5] A. Turbiner, W. Miller Jr, M. A. Escobar-Ruiz. Three-body problem in 3D space: ground state, (quasi)-exact-solvability. Journal of Physics A: Mathematical and Theoretical 50(21):215201, 2017. https://doi.org/10.1088/1751-8121/aa6cc2. [6] W. Miller Jr, A. Turbiner, M. A. Escobar-Ruiz. The quantum n-body problem in dimension d ≥ n − 1: ground state. Journal of Physics A: Mathematical and Theoretical 51(20):205201, 2018. https://doi.org/10.1088/1751-8121/aabb10. [7] F. M. Fernández. Born-Oppenheimer approximation for a harmonic molecule. ArXiv:0810.2210v2. [8] A. V. Turbiner. Quasi-exactly-solvable problems and the sl(2, R) algebra. Communications in Mathematical Physics 118:467–474, 1988. https://doi.org/10.1007/BF01466727. [9] A. V. Turbiner. One-dimensional quasi-exactly-solvable Schrödinger equations. Physics Reports 642:1–71, 2016. https://doi.org/10.1016/j.physrep.2016.06.002. [10] L. M. Delves. Tertiary and general-order collisions (II). Nuclear Physics 20:275–308, 1960. https://doi.org/10.1016/0029-5582(60)90174-7. [11] C. M. Bender, J.-H. Chen, D. W. Darg, K. A. Milton. Classical Trajectories for Complex Hamiltonians. Journal of Physics A: Mathematical and General 39(16):4219–4238, 2006. https://doi.org/10.1088/0305-4470/39/16/009. [12] C. M. Bender, S. Boettcher. Real spectra in Non-Hermitian Hamiltonians having PT symmetry. Physical Review Letters 80(24):5243–5246, 1998. https://doi.org/10.1103/PhysRevLett.80.5243. [13] P. Dorey, C. Dunning, R. Tateo. Supersymmetry and the spontaneous breakdown of PT symmetry. Journal of Physics A: Mathematical and General 34(28):L391, 2001. https://doi.org/10.1088/0305-4470/34/28/102. 55 https://doi.org/10.1016/0029-5582(58)90372-9 https://doi.org/10.1063/1.1724275 https://doi.org/10.1063/1.4994397 https://doi.org/10.1088/1751-8121/aa6cc2 https://doi.org/10.1088/1751-8121/aabb10 http://arxiv.org/abs/0810.2210v2 https://doi.org/10.1007/BF01466727 https://doi.org/10.1016/j.physrep.2016.06.002 https://doi.org/10.1016/0029-5582(60)90174-7 https://doi.org/10.1088/0305-4470/39/16/009 https://doi.org/10.1103/PhysRevLett.80.5243 https://doi.org/10.1088/0305-4470/34/28/102 Acta Polytechnica 62(1):50–55, 2022 1 Introduction 2 Generalities 2.1 Case of identical particles: -representation 3 Laplace-Beltrami operator 4 Three body harmonic oscillator system 4.1 Solution for the ground state 5 Lie algebraic structure 6 Relation with the Jacobi oscillator 6.1 Identical particles: hyperradious 6.2 Arbitrary masses: moment of inertia 7 Generalized three body harmonic oscillator system 7.1 Identical particles 7.1.1 Ground state 7.1.2 First excited state 8 Conclusions Acknowledgements References