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Lettere al nuovo cimento della societa italiana di fisica/Lettere al nuovo cimento

Exact solution of the classical and quantal one-dimensional many-body problems with the two-body potential {ie383-01}

P. Calogero, O. Ragnisco, Carlo Marchioro

Published Jul 1, 1975• 145 citations
Publisher
DOI
10.1007/bf02742674

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The exact solution is presented of the scattering problem of three equal particles interacting in one-dimension via two- and/or three-body inverse-square potentials. Both the classical and the quantal problems are treated. It is shown that the outcome of this scattering problem is an extremely simple relation between initial and final momenta, the latter being univocally determined by the former even in the quantal case. The solvability of the problem, and the simple results just mentioned, are peculiar to the equal particle case.
The quantal problem of a particle interacting in one dimension with an external time-dependent quadratic potential and a constant inverse square potential is exactly solved. The solutions are found both in the Schrödinger representation, by using a generating function or a time-dependent raising operator, and in the Heisenberg picture. They depend only on the solution of the classical harmonic oscillator. The generalizations to the n-dimensional problem and to the problem of N particles in one dimension, interacting pairwise via a quadratic time-dependent potential and a constant inverse square potential, are finally sketched.
The exact solution for the ground state of the quantum one-dimensional NNN-boson problem with attractive $\ensuremath{\delta}$-function two-body potentials is compared with the (exact) self-consistent solution of the corresponding variational Hartree problem.