  Results from four different approximations to the phonon-assisted quantum
adsorption rate for cold atoms on a 2D material are compared and contrasted:
(1) a loop expansion (LE) based on the atom-phonon coupling, (2) non-crossing
approximation (NCA), (3) independent boson model approximation (IBMA), and (4)
a leading-order soft-phonon resummation method (SPR). We conclude that, of the
four approximations considered, only the SPR method gives a divergence-free
result in the large membrane regime at finite temperature. The other three
methods give an adsorption rate that diverges in the limit of an infinite
surface.
