  The spectrum and properties of quantum bound states is strongly dependent on
the dimensionality of space. How this comes about and how one may theoretically
and experimentally study the interpolation between different dimensions is a
topic of great interest in different fields of physics. In this paper we study
weakly bound states of non-relativistic two and three boson systems when
passing continuously from a three (3D) to a two-dimensional (2D) regime within
a 'squeezed dimension' model. We use periodic boundary conditions to derive a
surprisingly simple form of the three-boson Schr{\"o}dinger equation in
momentum space that we solve numerically. Our results show a distinct
dimensional crossover as three-boson states will either disappear into the
continuum or merge with a 2D counterpart, and also a series of sharp
transitions in the ratios of three-body and two-body energies from being purely
2D to purely 3D.
