Topological defects in 1D potential and superlattices

We study the energy spectrum of 1D periodic potential with a topological defect, i.e. the defect that breaks globally the translational symmetry. The topological defect appears by matching two identical periodic potentials at a point where the derivative is different from zero. At the defect the derivative (the force) suffers a discontinuous jump. The amplitude of the jump characterizes the strength of the defect. We show that topological defects may give rise to two localized states within the energy gaps. This situation is different from point defects that lead to a single discrete level in every gap.