Perturbations

Abstract

The periodic Sturm-Liouville or Dirac operator on the whole real line has a purely absolutely continuous spectrum of band-gap structure; the regular end-point of the operator restricted to a half-line only introduces a single eigenvalue, if any, into each spectral gap. In applications, however, one does not always have exact periodicity of the coefficients, and the question arises how the spectral properties of the operator change if a non-periodic perturbation is added to the periodic background potential. In many ways this is analogous to the general question of the spectrum generated by a more or less localised potential added to a free Sturm-Liouville or Dirac operator, but here we take as an unperturbed reference a periodic operator, whose spectral properties are very well known by the results shown in the preceding chapters.