# What is a singular continuous spectrum?

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?

• This paper has some examples and references. More via Google. Mar 23 '19 at 16:50
• "Can I say that all free states have absolutely continuous spectra, and all bound states have pure point spectra? So what lies in between?" There can be bound states in the continuum (although I don't think this is related to singular continuous spectrum). Mar 23 '19 at 18:09
• This already useful, so I thank you for that. Mar 23 '19 at 18:23
• Singular Continuous Measures in Scattering Theory offers physical interpretations. Mar 23 '19 at 22:00