Three identical bosons: Properties in non-integer dimensions and in external fields

TL;DR

This paper explores the properties of three identical bosons as they transition from three-dimensional to two-dimensional space using a continuous dimension parameter, simplifying calculations of confined three-body systems.

Contribution

It introduces a novel approach using a continuous dimension parameter to study three-body boson systems under confinement, offering a simpler alternative to external potential methods.

Findings

Established a translation between dimension parameter and external confinement.

Demonstrated the equivalence of the $d$-method and external potential approach.

Provided insights into the evolution of three-body states during confinement.

Abstract

Three-body systems that are continuously squeezed from a three-dimensional (3D) space into a two-dimensional (2D) space are investigated. Such a squeezing can be obtained by means of an external confining potential acting along a single axis. However, this procedure can be numerically demanding, or even undoable, especially for large squeezed scenarios. An alternative is provided by use of the dimension $d$ as a parameter that changes continuously within the range $2\leq d \leq 3$. The simplicity of the $d$-calculations is exploited to investigate the evolution of three-body states after progressive confinement. The case of three identical spinless bosons with relative $s$-waves in 3D, and a harmonic oscillator squeezing potential is considered. We compare results from the two methods and provide a translation between them, relating dimension, squeezing length, and wave functions from both methods. All calculations are then possible entirely within the simpler $d$-method, but simultaneously providing the equivalent geometry with the external potential.