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I am now reading some mathematical note on Anderson localization. It mentioned two types of continuous spectrum. What are absolutely continuous spectrum and singularly continuous spectrum?

I only had the idea of point spectrum and continuous spectrum, or bound states and extended states.

So, what is this finer classification?

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It is a distinction corresponding to different types of spectral measures. The absolutely continuous spectrum corresponds to absolutely continuous measures, singular spectrum to continuous singular measures (both with respect to Lebesgue measure). Refer e.g. to Reed-Simon Chapter VII for a more detailed description.

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    $\begingroup$ My favourite quote from that reference: "[...]This decomposition will recur in a quantum-mechanical context where any state will be a sum of bound states, scattering states and states without physical meaning." :) $\endgroup$
    – Daniel
    Sep 25, 2014 at 15:22

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