Self-energy of a Cold Atom Interacting with an Elastic Membrane

The interaction of an atom with an elastic membrane is studied using Feynman-Dyson perturbation theory. The self-energy $\Sigma(E)$ of an atom with incident energy $E$ is calculated analytically to second-order in the atom-membrane interaction. We explicitly show that while the first-order contribution to the self-energy is well-behaved, the second-order contribution is divergent in the limit of infinite membrane size, and we identify the various divergent contributions. These results are discussed in the context of the ``quantum sticking'' and scattering of cold atoms from two dimensional materials such as graphene and monolayer transition metal dichalcogenides.