Spectral classification in a periodic structure1

Abstract

P=− d 2 dx2+ V, where V is a periodic real function. Essentially we show that the spectra of these operators can be determined by a smooth function α, the phase of the eigenvalues of a matrix M (E, x, V) expressing the states of a particle in a L− periodical structure. The recurrent spectrum and the transient spectrum of P are also calculated. Such structures are met meet, for example, in solid state physics in the study of a linear molecule formed of regularly spaced atoms.