reducing their dimensionality has become a key concept in the contemporary study of ultracold atomic few- and many-body systems, as exotic low-dimensional quantum phases [@girardeau1960; @kinoshita2004; @paredes2004] such as the Tonks-Girardeau gas in one dimension (1D) or the Berezinsky-Kosterlitz-Thouless transition in two dimensions (2D) are available. Besides these intriguing phenomena, the reduction of dimensionality also allows for a novel mechanism to control the scattering physics of two-body interactions. This was first pointed out by Olshanii [@olshanii1998] who studied the influence of a tight cylindrical confinement on scattering events. In particular, a resonance appears when the $s$-wave scattering length becomes comparable with the length scale associated with the confining waveguide potential. This so called confinement-induced resonance (CIR) as a result can be controlled by tuning the trap frequency, which recently also led to the first experimental realization of a super Tonks-Girardeau gas [@haller2009; @haller2009]. A similar prediction of a CIR for spin-polarized fermions [@granger2004] was in the following also experimentally confirmed [@guenter2005; @moritz2005; @froehlich2011]. Except these experiments discovering CIRs in 1D and 2D, a recent experiment [@lamporesi2010] was carried out in mixed-dimensions.

However, the substantial theoretical effort prevail the experimental observations, while at the same time building up a comprehensive understanding 