In discussions about the spectrum of hyperbolic surfaces, people seem to be interested in 'eigenvalues embedded in the continuous spectrum'. I am wondering which definition of the term 'continuous spectrum' is usually meant in this context? The reason I am asking this is because people seem to use different definitions for the term 'continuous spectrum'. Let me list three definitions I know about.

Let $\mathcal{H}$ be a Hilbert space and $T:\mathcal{H}\supset \mathcal{D}(T)\rightarrow\mathcal{H}$ be a densely defined, self-adjoint, and closed operator. Then people refer to the following sets as continuous spectrum:

\begin{align} \sigma_c(T):&=\lbrace{\lambda\in\mathbb{K}: Im(T-\lambda)\neq \overline{Im(T-\lambda)}\rbrace}\\ \sigma_c(T)':&=\lbrace{\lambda\in\mathbb{K}: ker(T-\lambda)=\lbrace{0\rbrace},\,\, Im(T-\lambda)\neq \mathcal{H},\,\,\overline{Im(T-\lambda)}=\mathcal{H}\rbrace}\\ \sigma_c(T)'':&=\sigma_{ess}(T)-\sigma_{p}(T), \end{align} where $\sigma_{ess}(T)$ is the essential spectrum and $\sigma_{p}(T)$ the point spectrum (=set of all eigenvalues).

Moreover, there is also the notion of continuous spectrum coming from the spectral measure associated to $T$. I must admit that I do not understand that definition completely, because I do not know used to work with the spectral measure. I think it corresponds to the union of the 'absolutely continuous spectrum' and the 'singular spectrum', as defined on Wikipedia: https://en.wikipedia.org/wiki/Decomposition_of_spectrum_(functional_analysis)

I have two questions, which causes me confusion:

1)Which of the above definitions are meant in the context stated at the beginning above?

2) How is the last notion of 'continuous spectrum' related to the other definitions? Is it maybe possible that the continuous spectrum coming from the spectral measure is equal to one of the above ones?

I would very appreciate your help!

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    $\begingroup$ I suspect that "continuous spectrum" is often used, informally, as "all spectrum that is not eigenvalues". Very rarely precise definitions are given. That's only a suspicion of mine, arising from attending talks in PDEs. $\endgroup$ – Giuseppe Negro Jun 17 '18 at 14:28
  • $\begingroup$ Thank you very much for your comment. Maybe you are right! Do you know what is meant by embedded eigenvalue then? $\endgroup$ – Oliver Watt Jun 17 '18 at 14:32
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    $\begingroup$ Again, I suspect that it refers to an eigenvalue surrounded by an interval of continuous spectrum. $\endgroup$ – Giuseppe Negro Jun 17 '18 at 15:59
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    $\begingroup$ @GiuseppeNegro, continuous spectrum is a technical term. See, for example, Chapter 7 of Reed and Simon's functional analysis text. I also like Teschl's quantum mechanics book as a reference for this material. $\endgroup$ – fourierwho Jun 17 '18 at 18:02

The usual definition in terms of the spectral measure $E$ is that $\lambda$ is in the continuous spectrum of $T$ if $E(-\infty,\mu]$ is strongly continuous in $\mu$ at $\lambda$, which excludes only the case where $E\{\lambda\}\ne 0$. $E\{\lambda\}$ is the orthogonal projection onto the eigenspace of $T$ with eigenvalue $\lambda$. This follows the decomposition of a Borel measure on $\mathbb{R}$ into a discrete and continuous measure.

The singular part can be further decomposed into absolutely continuous and singular continuous components, and these are sometimes used in studying the spectral measure, especially in Quantum scattering theory.

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  • $\begingroup$ Thank you for your answer. Do you know whether your definition of continuous spectrum corresponds to the set $\sigma_c(T)$ above? $\endgroup$ – Oliver Watt Jun 18 '18 at 5:21
  • $\begingroup$ @OliverWatt : It would correspond to $\sigma_c(T)'$ in your post. $\endgroup$ – DisintegratingByParts Jun 18 '18 at 6:18

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