Compact equations for the envelope theory

TL;DR

This paper derives simplified sets of equations for the envelope theory, enabling easier approximation of solutions for complex quantum many-body problems involving multiple particle types and dimensions.

Contribution

It introduces compact equations for the envelope theory applicable to systems with multiple particle types, extending previous formulations for identical particles.

Findings

Derivation of 3 equations for identical particles in D dimensions.

Extension to 7 equations for systems with two different particle types.

Provides a foundation for improving the envelope theory method.

Abstract

The envelope theory is a method to easily obtain approximate, but reliable, solutions for some quantum many-body problems. Quite general Hamiltonians can be considered for systems composed of an arbitrary number of different particles in $D$ dimensions. In the case of identical particles, a compact set of 3 equations can be written to find the eigensolutions. This set provides also a nice interpretation and a starting point to improve the method. It is shown here that a similar set of 7 equations can be determined for a system containing an arbitrary number of two different particles.