Sturm-Liouville operators with periodically modulated parameters have spectral density that is continuous and positive everywhere on the real line under assumptions on the monodromy matrix at spectral parameter zero.
Elaydi, An introduction to difference equations , third ed., Undergraduate Texts in Mathematics, Springer- Verlag New Y ork
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Sturm-Liouville operators with periodically modulated parameters. Part I: Regular case
Sturm-Liouville operators with periodically modulated parameters have spectral density that is continuous and positive everywhere on the real line under assumptions on the monodromy matrix at spectral parameter zero.