The spectrum of periodic generalized diffusion operators (Q1101894)

The spectrum of a periodic generalized diffusion operator on the real line is expressed either as \(\cup^{\infty}_{n=0}[\mu_ n^{(2)},\mu^{(1)}_{n+1}]\) in terms of an infinite sequence \[ - \infty < \mu_ 0^{(2)} < \mu_ 1^{(1)}\leq \mu_ 1^{(2)} < ... < \mu_ n^{(1)}\leq \mu_ n^{(2)} < ... \uparrow \infty, \] or as \(\cup^{N-1}_{n=0}[\mu_ n^{(2)},\mu^{(1)}_{n+1}]\) in terms of a finite sequence \(-\infty <\mu_ 0^{(2)}<\mu_ 1^{(1)}\leq \mu_ 1^{(2)}<...\leq \mu^{(2)}_{N-1}<\mu_ N^{(1)}<\infty\), according as the support of associated measure intersects a bounded interval with an infinite set or with a finite set. The spectrum consists only of the continuous spectrum and the point spectrum is empty as long as the operators are treated on the real line. If we deal with the operators on the half line, both the continuous spectrum and the point spectrum are nonempty in most cases. The problem how boundary conditions affect the spectrum is also studied.