Exact results in quantum many-body systems of interacting particles in many dimensions with SU(1,2)¯ (1̄,1̄) as the dynamical group

Abstract

We consider a class of system of N interacting particles in any dimension—the potential includes a quadratic pair potential and an arbitrary translation-invariant position-dependent potential that is homogeneous of degree −2. The group SU(1,2)¯ (1̄,1̄) is the dynamical group for the Hamiltonian. We illustrate the significance of the Casimir operator in relation to the separation of variables method; obtain a series of eigenfunctions that transform under the unitary irreducible representations of SU(1,2)¯ (1̄,1̄) labeled by the ground state energy; indicate the structure of arbitrary eigenfunctions; and specify when the complete energy spectrum is linear. We treat N-body examples which include two- and three-body forces. For N identical particles in one dimension interacting with a quadratic pair potential and an inverse square pair potential, we exhibit a series of eigenfunctions characterized by four quantum numbers. These eigenfunctions reduce to the complete set of eigenfunctions for five particles. We indicate how a complete set of eigenfunctions for N particles are obtained.

Affine Toda–Sutherland systems

“…Except for the systems based on the A (1) series, the small oscillations near the equilibrium do not have integer (times the coupling constant) eigenvalues other than 2, which is universal for all the potentials with quadratic plus inverse quadratic dependence on the coordinate q [23].…”

“…In all the other cases the only integer (times ω) eigenvalues of W is 2, which exists in all the cases based on any root system. In fact it is more universal and exists for all the potentials with quadratic (q 2 ) plus inverse quadratic dependence on the coordinate q [23,9] without any root or weight structure. This eigenvalue 2 gives rise to exact quantum eigenfunctions φ n (q) which is proportional to the Laguerre polynomial [23,9] in q 2 : 4) in which E 0 = (βh + r/2)ω is the ground state energy and h is the Coxeter number (2.12).…”

Relationship between the energy eigenstates of Calogero-Sutherland models with oscillator and coulomb-like potentials

“…Recall at this point that the many-body interaction of all the known higher dimensional CSM type models is homogeneous with degree −2. For example, the Calogero-Marchioro model [9], models with novel correlations [10], models with two-body interactions [11] and models considered in [12,13] have this property.…”

“…Similar to the one dimensional oscillator problem, we define the three operators for the D ′ dimensional many-body problem with oscillator potential as [12],…”

Quantum Calogero-Moser models: integrability for all root systems

“…As is emphasised by Perelomov [12] and Gambardella [30] the sl(2, R) algebra and the corresponding Laguerre wavefunctions are more universal than Calogero-Moser models.…”