  We investigate how confining a transverse spatial dimension influences the
few- and many-body properties of non-relativistic bosons with pointlike
interactions. Our main focus is on the dimensional crossover from three to two
dimensions, which is of relevance for ultracold atom experiments. Using
Functional Renormalization Group equations and T-matrix calculations we study
how the phase transition temperature changes as a function of the spatial
extent of the transverse dimension and relate the 3D and 2D s-wave scattering
lengths. The analysis reveals how the properties of the lower-dimensional
system are inherited from the higher-dimensional one during the renormalization
group evolution. We limit the discussion to confinements in a potential well
with periodic boundary conditions and argue why this qualitatively captures the
physics of other compactifications such as transverse harmonic confinement as
in cold atom experiments.
