Existence of a ground state for the confined hydrogen atom in non-relativistic QED

We consider a system of a hydrogen atom interacting with the quantized electromagnetic field. Instead of fixing the nucleus, we assume that the system is confined by its center of mass. This model is used in theoretical physics to explain the Lamb-Dicke effect. After a brief review of the literature, we explain how to verify some properly chosen binding conditions which, by [25], lead to the existence of a ground state for our model, and for all values of the fine-structure constant.