Minimum Principle for Effective Potentials

Abstract

In the familiar Rayleigh-Ritz method, a finite-dimensional Hamiltonian matrix is constructed using a set of linearly independent trial functions. The eigenvalues of this matrix, suitably ordered, are guaranteed to lie above the corresponding true eigenvalues. An analogous situation is shown to hold in scattering theory for energies low enough so that only two-body channels are open. An effective Hamiltonian, which is a matrix in the open-channel subspace, is constructed variationally. Since the corresponding Schr\"odinger equation is of the two-body type, it may be solved, numerically if necessary. The eigenvalues of the reaction matrix, in a partial-wave representation, are then guaranteed to lie below the true eigenvalues. The essential point involved is that the error in the effective Hamiltonian can be shown to be a nonpositive operator, provided the trial function satisfies a certain natural constraint. Effects due to the identity of particles are easily accounted for. As an application, the three-body problem is considered in some detail, and the required constraint on the trial function is explicitly defined for this problem in terms of simple orthogonality conditions.