Bound states of an ultracold atom interacting with a set of stationary impurities

In this paper we analyze the properties of bound states of an atom interacting with a set of static impurities. We begin with the simplest system of a single atom interacting with two static impurities. We consider two types of atom-impurity interaction: (i) a zero-range potential represented by a regularized δ interaction and (ii) a more realistic polarization potential, representing the long-range part of the atom-ion interaction. For the former we obtain analytical results for energies of bound states. For the latter we perform numerical calculations based on the application of the finite-element method. We then discuss the case of a single atom interacting with a one-dimensional (1D) infinite chain of static ions. Such a setup resembles the Kronig-Penney model of a 1D crystalline solid, where the energy spectrum exhibits band-structure behavior. For this system we derive analytical results for the band structure of bound states, assuming a regularized δ interaction, and perform numerical calculations, considering the polarization potential to model an atom-impurity interaction. Both approaches agree quite well when the separation between impurities is much larger than the characteristic range of the interaction potential.