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Canadian Journal of Physics

Collective phenomena in the one-dimensional three-particle model of high-energy potential scattering

V. I. Inozemtsev

Published Jul 1, 1979• 0 citations
Publisher

Abstract

Within the framework of an exactly solvable model of three particles interacting through zero-range forces it is shown that the high-energy scattering with high momentum transfer is due to collective phenomena. The spectrum of particles scattered through 180° is compared with the results of impulse approximation.
DOI
10.1139/p79-134

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The exactly solvable Luttinger model for a system of one-dimensional, massless, two-component fermions interacting via general two-body forces is used as a test case for a systematically generated sequence of decoupling approximations commonly referred to as random-phase approximations (RPA) of which the simple RPA is the lowest. It is found, irrespective of both the nature and strength of the interaction, that the exact (collective) spectrum, as well as the exact correlation energy, can already be obtained in the simple RPA, and furthermore, that the next higher RPA gives a vanishing correction to these quantities. It is also shown, quite in contradistinction to the exact result, that the free-gas momentum distribution n‾k{\overline{n}}_{k}nk​ persists in the second RPA as well as in the first, thereby eliminating the higher RPA as a possible approach to the calculation of n‾k{\overline{n}}_{k}nk​. Thus, in the context of this particular model, although the first few orders of RPA are highly effective for calculating the excitation spectrum and correlation energy, they are sensibly powerless in their ability to predict any variation in structure beyond that of the free-gas distribution, a result which most probably holds as well in the realistic three-dimensional case.