  The role of shape resonances and many-body effects on universal quantum
sticking of ultra cold atoms onto solid surfaces is examined analytically and
computationally using an exactly solvable representation of the Dyson equation.
We derive the self-energy renormalization of the the transition amplitude
between an ultra cold scattering atom and the bound states on the surface in
order to elucidate the role of virtual phonon exchanges in the limiting
behavior of the sticking probability. We demonstrate that, to first order in
the interactions for finite ranged atom-surface potentials, virtual phonons can
only rescale the strength of the atom-surface coupling and do not rescale the
range of the coupling. Thus, universal sticking behaviour at ultra-low energies
is to be expected for all finite ranged potentials. We demonstrate that the
onset of the universal sticking behavior depends greatly on the position of the
shape resonance of the renormalized potential and for sufficiently low energy
shape resonances, deviations from the universal $s(E)\propto\sqrt{E}$ can occur
near these energies. We believe that this accounts for many of the low energy
sticking trends observed in the scattering of sub-millikelvin H atoms from
superfluid $^4$He films.
