On-shell approximation for the s -wave scattering theory

Quantum fluctuations in atomic Josephson junctions: the role of dimensionality

“…Firstly, renormalizing the zero-point energy of the excitations is not enough to deal with the UV divergency, since a Γ(−2) appears. Under the minimal subtraction scheme we obtain a finite result [37] for the zero-point energy of the excitations, which now depends on an energy cut-off Λ. However this energy cut-off cancels out with the same energy cut-off which appears in the expression for the mean-field 2-dimensional coupling constant, giving a Λ−independent expression for the beyond-mean-field grand potential.…”

“…However, while in the D = 3, 2 cases the mean-field approach is an adequate approximation for highly dilute systems, or in other words when the gas parameter is close to zero; vice versa, in the D = 1 case the mean-field approximation is correct once one considers systems with the limit of the gas parameter that is going to −∞. This difference is related to the expression of the interaction parameter g 0 , which is qualitatively different in the various dimensions [37]. Indeed, while in D = 3 we have g 0 ∝ a s , in D = 2 we have g 0 ∝ ln −1 (κa s with a constant κ, and in D = 1, the relation is g 0 ∝ a −1 s .…”

“…Therefore, before computing the potential term, a discussion on the coupling is needed [37]. Since the coupling g k,r is different for each site, it is useful to define a coupling g r for the entire system…”