  We describe a non-perturbative method for computing the energy band
structures of one-dimensional models with general point potentials sitting at
equally spaced sites. This is done thanks to a Bethe ansatz approach and the
method is applicable even when periodicity is broken, that is when Bloch's
theorem is not valid any more. We derive the general equation governing the
energy spectrum and illustrate its use in various situations. In particular, we
get exact results for boundary effects. We also study non-perturbatively the
effects of impurities in such systems. Finally, we discuss the possibility of
including interactions between the particles of these systems.
