Scattering in periodic systems: from resonances to band structure

We report a study of the spectrum of scattering resonances in spatially extended open quantum systems. We consider the Schrödinger equation with a potential composed of n copies of a unit cell periodically arranged in a one-dimensional space and vanishing at large distances. We develop an asymptotic theory which explains the structure of the resonance spectrum. We also show that, in the limit of a large system, the leading scattering resonances converge to the allowed energy bands of the infinitely extended periodic system.