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

Abstract

Three-body systems that are continuously squeezed from a three-dimensional space into a two-dimensional space are investigated. Such 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 d 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 three dimensions 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.

Efimov effect evaporation after confinement

“…The equivalence between the d-method, using d as a continuous parameter, and the more natural approach of confining the system by means of an external squeezing potential included explicitly in a 3D calculation, has been investigated in Refs. [10,11,12]. The main conclusion is that it is possible to establish a connection between a non-integer value of d and a specific strength of the squeezing potential.…”

“…1. If the interpretation made in [9,12] is correct, when close to the state disappearance, one of the heavy particles, particle 2 in Fig. 1, should be pretty far from the bound heavy-light system (particles 1 and 3).…”

“…The price to pay when using the d-formalism is that the calculation of a given observable, unavoidably measured in an integer-dimensional space, requires a translation of the d-dimensional wave function into the ordinary 3D space. This is done by considering the d-wave function as a wave function in the ordinary 3D space, but deformed along the squeezing direction [10,11,12].…”

“…The other infinitely many Efimov states disappear very fast around the two-body threshold. This fact was interpreted as a process where, when decreasing d, one of the heavy particles is becoming less and less bound with respect to the bound heavy-light two-body system, being eventually lost [9,12].…”

“…1, this is the fingerprint of the disappearance process anticipated in Refs. [9,12], namely, one of the heavy particles is moving far apart, being progressively less and less bound with respect to the bound heavy-light system, and lying eventually into the continuum.…”