Hyperspherical explicitly correlated Gaussian approach for few-body systems with finite angular momentum

TL;DR

This paper introduces a hyperspherical explicitly correlated Gaussian method for solving few-body quantum systems with finite angular momentum, providing an efficient computational approach for four-body problems relevant to cold atom physics.

Contribution

The paper develops a new theoretical framework using explicitly correlated Gaussian basis functions within the hyperspherical approach for systems with finite angular momentum L=1.

Findings

Efficient calculation of effective potentials and coupling matrix elements.

Application to four-body bound and scattering problems.

Demonstrated relevance to cold atom physics.

Abstract

Within the hyperspherical framework, the solution of the time-independent Schroedinger equation for a n-particle system is divided into two steps, the solution of a Schroedinger like equation in the hyperangular degrees of freedom and the solution of a set of coupled Schroedinger like hyperradial equations. The solutions to the former provide effective potentials and coupling matrix elements that enter into the latter set of equations. This paper develops a theoretical framework to determine the effective potentials, as well as the associated coupling matrix elements, for few-body systems with finite angular momentum L=1 and negative and positive parity. The hyperangular channel functions are expanded in terms of explicitly correlated Gaussian basis functions and relatively compact expressions for the matrix elements are derived. The developed formalism is applicable to any n; however, for n greater or equal to 6, the computational demands are likely beyond present-day computational capabilities. A number of calculations relevant to cold atom physics are presented, demonstrating that the developed approach provides a computationally efficient means to solving four-body bound and scattering problems with finite angular momentum on powerful desktop computers. Details regarding the implementation are discussed.