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I read some answers about this and the wikipedia page that basically always say that a spectrum can be decomposed into:

$$\mu = \mu_{ac} + \mu_{sc} + \mu_{pp}, $$

where $\mu_{ac}$ is absolutely continuous, $\mu_{pp}$ is a pure point spectrum, and $\mu_{sc}$ is "singular continuous".

What does it mean, physically, to have a singular continuous spectrum? Are there examples in physics?

Can I say that all free states have absolutely continuous spectra, and all bound states have pure point spectra? So what lies in between? Localised states?

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