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Nuclear Physics B

Optimized lower bounds in the three-body problem

Jean-Louis Basdevant, André Martin, J. M. Richard, Tai Tsun Wu

Published Mar 1, 1993• 22 citations
Publisher
DOI
10.1016/0550-3213(93)90240-p

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Generalizing a method used previously for three-body, four-body and very recently for five-body systems, we derive a lower bound for the ground state energy of an N-body Hamiltonian, with arbitrary N. Because the expression of the lower bound obtained in this way depends on a number of parameters, we obtain in fact a family of lower bounds, a lower bound for each set of values of these parameters. The best of these is of course obtained by maximizing over these parameters and is correspondingly named optimized lower bound. The set of values of the parameters corresponding to the optimized lower bound satisfy a number of relations, named universal dynamical constraints, which result from the application of a dynamical principle and are independent of the particular form of the potential. For N = 3, 4, 5, they can be worked out in the most general case. For N = 6 up, they can be worked out only for particular mass configurations. Furthermore, the optimized lower bound proves to be saturated in the harmonic oscillator case.
A method for systematically improving Hall–Post–Stenschke lower bounds to the bound state eigenvalues of three-body Schrödinger equations is given. The improved bounds are obtained by solving coupled one variable integral equations; the bounds get better as the number of coupled equations is increased. The method generates explicit wave functions which can be used to obtain complementary upper bounds via the Rayleigh–Ritz variational method. Either identical or nonidentical particles can be handled. The method is illustrated by calculations for three identical particles bound by Hooke’s law forces. A brief discussion of extensions to more than three particles is given.