Many-body forces with the envelope theory

TL;DR

This paper extends the envelope theory to efficiently approximate solutions for many-body systems with complex forces, providing analytical bounds and numerical methods, with applications to attractive potentials and semiclassical interpretation.

Contribution

It introduces a novel adaptation of the envelope theory to handle specific many-body forces, offering analytical bounds and practical numerical solutions.

Findings

Analytical lower and upper bounds for eigenvalues in certain cases

Numerical approximations for complex many-body forces

Critical coupling constants for attractive potentials

Abstract

Many-body forces are sometimes a relevant ingredient in various fields, such as atomic, nuclear or hadronic physics. Their precise structure is generally difficult to uncover. So, phenomenological effective forces are often used in practice. Nevertheless, they are always very heavy to treat numerically. The envelope theory, also known as the auxiliary field method, is a very efficient technique to obtain approximate, but reliable, solutions of many-body systems interacting via one- or two-body forces. It is adapted here to allow the treatment of a special form of many-body forces. In the most favourable cases, the approximate eigenvalues are analytical lower or upper bounds. Otherwise, numerical approximation can always be computed. Two examples of many-body forces are presented, and the critical coupling constants for generic attractive many-body potentials are computed. Finally, a semiclassical interpretation is given for the generic formula of the eigenvalues.