  In this work, we discuss the resonance states of a quantum particle in a
periodic potential plus a static force. Originally this problem was formulated
for a crystal electron subject to a static electric field and it is nowadays
known as the Wannier-Stark problem. We describe a novel approach to the
Wannier-Stark problem developed in recent years. This approach allows to
compute the complex energy spectrum of a Wannier-Stark system as the poles of a
rigorously constructed scattering matrix and solves the Wannier-Stark problem
without any approximation. The suggested method is very efficient from the
numerical point of view and has proven to be a powerful analytic tool for
Wannier-Stark resonances appearing in different physical systems such as
optical lattices or semiconductor superlattices.
