Equivalence of the Calogero-Sutherland Model to Free Harmonic Oscillators

A similarity transformation is constructed through which a system of particles interacting with inverse-square two-body and harmonic potentials in one dimension, can be mapped identically, to a set of free harmonic oscillators. This equivalence enables us to write down the complete set of eigenfunctions. We find the exact constants of motion and a linear W1+∞ algebra associated with this model. It is also demonstrated that a large class of models with long-range interactions, both in one and higher dimensions can be made equivalent to decoupled oscillators. PACS numbers: 03.65.Ge, 03.65.Fd Typeset using REVTEX 1 In recent times, the one-dimensional system of identical particles having pair-wise inversesquare and harmonic interactions [1], known in the literature as the Calogero-Sutherland (CS) model, has generated wide interest. This model and its generalizations to the periodic case [2] and the spin systems [3], have been found relevant for the description of various physical phenomena such as the universal conductance fluctuations in mesoscopic systems [4], quantum Hall effect [5], wave propagation in stratified fields [6], random matrix theory [2,7], fractional statistics [8], two-dimensional gravity [9] and gauge theories [10]. The quantum CS model is exactly solvable, with the energy eigenvalues and the level degeneracies matching identically with those of harmonic oscillators, apart from a coupling dependent shift of the ground-state energy. Since its inception, this remarkable manybody system has been studied quite extensively in the literature for a better understanding of the origin of solvability and the underlying symmetries [11,12]. However, the explicit construction of the complete set of eigenfunctions, including the degenerate ones, has yet to progress beyond a few particles [13]. The similarity of the spectra of the CS model and free harmonic oscillators suggests to the interesting possibility of a map between these two systems. In this paper, we provide a similarity transformation which realizes this goal by mapping the CS system of interacting particles to a set of free harmonic oscillators. The complete eigenstates of the CS model, including the degenerate ones, are explicitly constructed starting from the symmetrized form of the eigenstates of the harmonic oscillators. The following results are also obtained; (i) we exactly determine the N linearly independent, mutually commuting constants of motion, (ii) show the existence of a linear W1+∞ algebra as the infinite dimensional symmetry associated with the CS system and (iii) demonstrate that, a large class of CS type models in one and higher dimensions can also be solved in an analogous manner. The N particle CS Hamiltonian is given by (in the units h̄ = ω = m = 1) H = − 2 N