Non-relativistic bound states in a finite volume

TL;DR

This paper derives general formulas for the mass shifts of bound states with angular momentum in finite cubic volumes, aiding lattice simulations of various quantum systems and verifying results numerically.

Contribution

It provides new analytical expressions for mass shifts of bound states with angular momentum in finite volumes, applicable to lattice simulations of hadronic molecules and related systems.

Findings

Analytical formulas for mass shifts in finite volumes

Verification through numerical calculations

Relations between effective range, binding momentum, and wave function normalization

Abstract

We derive general results for the mass shift of bound states with angular momentum l >= 1 in a periodic cubic box in two and three spatial dimensions. Our results have applications to lattice simulations of hadronic molecules, halo nuclei, and Feshbach molecules. The sign of the mass shift can be related to the symmetry properties of the state under consideration. We verify our analytical results with explicit numerical calculations. Moreover, we comment on the relations connecting the effective range parameter, the binding momentum of a given state and the asymptotic normalization coefficient of the corresponding wave function. We give explicit expressions for this relation in the shallow binding limit.