Scattering in periodic systems: from resonances to band structure

We report a study of the spectrum of scattering resonances in spatially ext ended open quantum systems. We consider the Schrodinger equation with a pot ential composed of n copies of a unit cell periodically arranged in a one-d imensional space and vanishing at large distances. We develop an asymptotic theory which explains the structure of the resonance spectrum. We also sho w that, in the limit of a large system, the leading scattering resonances c onverge to the allowed energy bands of the infinitely extended periodic sys tem.