Volume dependence of charged-particle bound states

Simulating a quantum system in a finite volume is a powerful theoretical tool to extract information about it. The pioneering work of Lüscher has shown that the real-world properties of the system are encoded in how its discrete energy levels change with the size of the volume, and many aspects of this idea have already been studied. The approach is commonly used for example to analyze lattice calculations of atomic nuclei. We consider the finite-volume correction to the binding energy of two-body systems with a repulsive Coulomb interaction. The long-range nature of the Coulomb interaction constitutes an interesting challenge for the formalism, which normally assumes the presence of short (or finite) range interactions only. We investigate this problem in one and three-dimensional periodic boxes and show that the challenge can be overcome by truncating the Colomb potential at the box boundary. This approach then yields analytic expressions for the volume dependence of bound states in terms of Whittaker functions, which reduce to known results in the limit where the Coulomb interaction is switched off. We test our results against numerical calculations and show how the method can be used to extract asymptotic normalization coefficients for charged-particle bound states.