Multiphase semiclassical approximation of an electron in a one-dimensional crystalline lattice ⋆ I. Homogeneous problems

We present a computational approach for the WKB approximation of the wave function of an electron moving in a periodic one-dimensional crystal lattice. We derive a non-strictly hyperbolic system for the phase and the intensity where the flux functions originate from the Bloch spectrum of the Schrödinger operator. Relying on the framework of the multibranch entropy solutions introduced by Brenier and Corrias, we compute efficiently multiphase solutions using well adapted and simple numerical schemes. In this first part we present computational results for vanishing exterior potentials which demonstrate the effectiveness of the proposed method.