0 D ec 1 99 7 Mapping of the B N - type Calogero - Sutherland - Moser system to decoupled Harmonic Oscillators

The BN -type Calogero-Sutherland-Moser system in one-dimension is shown to be equivalent to a set of decoupled oscillators by a similarity transformation. This result is used to show the connection of the AN and BN type models and explain the degeneracy structure of the later. We identify the commuting constants of motion and the generators of a linear W∞ algebra associated with the BN system. † E-mail: [email protected] Typeset using REVTEX 1 In the recent literature, there is an increasing interest in the quantum mechanical, exactly solvable N -particle systems in one-dimension [1-4]. All these models are characterised by non-trivial, long-range interactions between particles and Jastrow-type correlated manybody wave functions. Interestingly, they have been found relevant for the description of universal properties of various physical systems. These include spin chains [5], quantum Hall effect [6], universal conductance fluctuations in mesoscopic systems [7], two-dimensional gravity [8], gauge theories [9] and random matrices [2,10]. In a recent paper [11], the present authors have constructed a similarity transformation which maps the AN -type Calogero-Sutherland (CS) model [1,2], having pair-wise inversesquare and harmonic interactions, to decoupled harmonic oscillators. Starting from the symmetrised eigenfunctions of N free harmonic oscillators, the wave functions of the CS model can then be constructed explicitly. In general, the Hamiltonians which can be brought through a suitable transformation to the generalised form H̃ = ∑ i xi ∂ ∂xi + c+ F̂ can also be mapped to ∑ i xi ∂ ∂xi + c by a similarity transformation. The operator that accomplishes this is given by exp{−dF̂}; where, F̂ is any homogeneous function of ∂ ∂xi and xi with degree d, and c is a constant. For the normalizability of the wave functions, one needs to check that the action of exp{−dF̂} on an appropriate linear combination of the eigenstates of ∑ i xi ∂ ∂xi yields a polynomial solution. In this letter, the BN -type Calogero-Sutherland-Moser (CSM) model [4] is diagonalised using the above procedure. We (i) prove the equivalence of BN -type model to decoupled oscillators, (ii) construct the complete set of eigenfunctions for this model from those of harmonic oscillators, (iii) show the existence of N independent commuting constants of motion and a linear W∞ algebra. We also point out the connection of AN and BN -type models and explain the origin of degeneracies. The CSM Hamiltonian (in the units h̄ = m = ω = 1) is given by H = − 1 2 N