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Physical Review Letters

Exact Ground State of Some One-Dimensional N -Body Systems with Inverse ("Coulomb-Like") and Inverse-Square ("Centrifugal") Pair Potentials

Francesco Calogero, Carlo Marchioro

Published Jul 12, 1971• 8 citations
Publisher

Abstract

We exhibit the exact ground state and the corresponding binding energy of the system composed of $N$ (nonidentical) quantum-mechanical particles interacting in one dimension via the potential ${V}_{\mathrm{ij}}({x}_{i}\ensuremath{-}{x}_{j})=g{({x}_{i}\ensuremath{-}{x}_{j})}^{\ensuremath{-}2}+({f}_{i}\ensuremath{-}{f}_{j}){({x}_{i}\ensuremath{-}{x}_{j})}^{\ensuremath{-}1}$. We discuss some special choices of the constants ${f}_{i}$ and $g$ that yield examples of possible physical interest.

DOI
10.1103/physrevlett.27.86

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We obtain the exact ground state for the Calogero-Sutherland problem in arbitrary dimensions. In the special case of two dimensions, we show that the problem is connected to the random matrix problem for complex matrices, provided the strength of the inverse-square interaction g = 2. In the thermodynamic limit, we obtain the ground state energy and the pair-correlation function and show that in this case there is no long-range order. Calogero-Sutherland type models are a class of exactly solvable models. Starting from the inception [1], till today, these models continue to be of interest, especially due to their exact solvability [2]. Moreover, these models are related to (1 + 1)-dimensional conformal field theory, random matrices and a host of other things [3]. Some variants of the models have attracted attention in very recent past [3–5]. The original model of Calogero [1] described N particles in one dimension, interacting through an inverse square law and a two-body harmonic interaction. The latter interaction was added to the effect of complete discretization of the spectrum. This model gave in to an analytic treatment yielding an exact solution. Sutherland [6] considered a variant of this problem where the harmonic interaction was replaced by a harmonic well containing the N particles. It was shown that one can exactly solve for the ground state of the system. The norm of the ground state wave-function was shown to be of a form that coincides with the joint probability density function of the eigenvalues of random matrices from an orthogonal, unitary or symplectic Gaussian ensemble, provided, the