We will start by introducing a real space model of a scalar electromagnetic field coupled to a continuum collection of two level atoms. From this we will obtain a pair of nonlocal partial differential equations describing the energy eigenstates that have at most one photon present in the field. The rest of the talk will discuss spectral results in two different types of atomic distributions.
1. Compactly supported densities: In this setting the atoms are contained in a finite region in space. We will state necessary and sufficient conditions for the existence of eigenstates, as well as an upper bound on the number of such states.
2. Periodic densities: In this setting the atoms exhibit the symmetries of a lattice. We will present a decomposition of the continuous spectrum into spectral bands and state a corresponding structure theorem.
This work is joint with Erik Hiltunen, John Schotland, and Michael Weinstein.