Confinement of two-body systems and calculations in d dimensions

Abstract

A continuous transition for a system moving in a three-dimensional (3D) space to moving in a lower-dimensional space, 2D or 1D, can be made by means of an external squeezing potential. A squeeze along one direction gives rise to a 3D to 2D transition, whereas a simultaneous squeeze along two directions produces a 3D to 1D transition, without going through an intermediate 2D configuration. In the same way, for a system moving in a 2D space, a squeezing potential along one direction produces a 2D to 1D transition. In this work we investigate the equivalence between this kind of confinement procedure and calculations without an external field, but where the dimension d is taken as a parameter that changes continuously from d = 3 to d = 1. The practical case of an external harmonic oscillator squeezing potential acting on a two-body system is investigated in details. For the three transitions considered, 3D → 2D, 2D → 1D, and 3D → 1D, a universal connection between the harmonic oscillator parameter and the dimension d is found. This relation is well established for infinitely large 3D scattering lengths of the two-body potential for 3D → 2D and 3D → 1D transitions, and for infinitely large 2D scattering length for the 2D → 1D case. For finite scattering lengths size corrections must be applied. The traditional wave functions for external squeezing potentials are shown to be uniquely related with the wave functions for specific non-integer dimension parameters, d.

Three identical bosons: Properties in noninteger dimensions and in external fields

“…One advantage is that the two masses only enter in the relative motion as the reduced mass, and only as a factor in the overall scale parameter. This is reported in previous papers using both momentum-space coordinates [6,7] and ordinary space coordinates [8][9][10].…”

“…Recently, a d-dependent formulation has been presented and applied to two-body systems [9,10]. The basic assumption is to use d as a parameter that can take noninteger values, in such a way that the external squeezing potential does not appear at all but, instead, is substituted by the correspondingly modified Schrödinger equation depending on d and the particle number, N [11,12].…”

“…The required numerical effort is similar to a standard calculation for an integer dimension, but where the external potential has disappeared. In these works, [9,10], the equivalence between the d-dependent method and the more direct procedure working in three dimensions, including explicitly the external squeezing potential, was investigated. In the case of a squeezing harmonic oscillator potential a connection between the oscillator length and the equivalent dimension d was found.…”

“…In this work we extend the method presented in Refs. [9,10] to three-body systems. First, we describe in Sec.…”

Noninteger Dimensional Spaces and the Inverse Square Law

“…The three-dimensionality of the common intuition in terms of length, breadth, and height is a consequence of our experience, so it is imperative to put it to test. We know that under certain conditions, it becomes necessary to go beyond three to noninteger dimensions, which are used in various branches of science to explain confinement in certain systems [3], the emergence of scale invariant and fractal phenomena such as turbulence [4], human physiology, medicine, and neuroscience [5] [6], and engineered networks [7]. The most commonly used measures are the box-counting, information, packing, and the Hausdorff dimensions [8].…”