  Quantum electrodynamics (QED) provides a highly accurate description of
phenomena involving the interaction of atoms with light. We argue that the
quantum theory describing the interaction of cold atoms with a vibrating
membrane--quantum acoustodynamics (QAD)--shares many issues and features with
QED. Specifically, the adsorption of an atom on a vibrating membrane can be
viewed as the counterpart to QED radiative electron capture. A calculation of
the adsorption rate to lowest-order in the atom-phonon coupling is finite;
however, higher-order contributions suffer from an infrared problem mimicking
the case of radiative capture in QED. Terms in the perturbation series for the
adsorption rate diverge as a result of massless particles in the model
(flexural phonons of the membrane in QAD and photons in QED). We treat this
infrared problem in QAD explicitly to obtain finite results by regularizing
with a low-frequency cutoff that corresponds to the inverse size of the
membrane. Using a coherent state basis for the soft phonon final state, we then
sum the dominant contributions to derive a new formula for the multiphonon
adsorption rate of atoms on the membrane that gives results that are finite,
nonperturbative in the atom-phonon coupling, and consistent with the KLN
theorem. For micromembranes, we predict a reduction with increasing membrane
size for the low-energy adsorption rate. We discuss the relevance of this to
the adsorption of a cold gas of atomic hydrogen on suspended graphene.
