Further developments for the auxiliary field method

TL;DR

The paper enhances the auxiliary field method for quantum eigenvalue problems, providing a simplified approach to approximate solutions for many-body systems with insights into its variational properties and critical coupling constants.

Contribution

It introduces a refined formulation linking mean momentum and distance, and explores the method's variational nature and relation to perturbation theory.

Findings

Eigenvalues approximated by kinetic and potential energy at mean values

Relation between mean momentum and distance derived from quantum numbers

Results on critical coupling constants for finite bound states

Abstract

The auxiliary field method is a technique to obtain approximate closed formulae for the solutions of both nonrelativistic and semirelativistic eigenequations in quantum mechanics. For a many-body Hamiltonian describing identical particles, it is shown that the approximate eigenvalues can be written as the sum of the kinetic operator evaluated at a mean momentum $p_0$ and of the potential energy computed at a mean distance $r_0$. The quantities $p_0$ and $r_0$ are linked by a simple relation depending on the quantum numbers of the state considered and are determined by an equation which is linked to the generalized virial theorem. The (anti)variational character of the method is discussed, as well as its connection with the perturbation theory. For a nonrelativistic kinematics, general results are obtained for the structure of critical coupling constants for potentials with a finite number of bound states.