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A lot of papers define a 'diabolical point' as a "double semi-simple eigenvalue." I know a semi-simple eigenvalue is one which has algebraic multiplicity and geometric multiplicity to be equal. However, I could not find any definition of a double semi-simple eigenvalue.

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"Double" simply means a degenerate eigenvalue (repeated root of the characteristic equation), thus a "double semi-simple eigenvalue" is a once repeated eigenvalue (i.e., with algebraic multiplicity 2) that spans a 2D vector space (i.e., its geometric multiplicity is also 2).

You can check, e.g., the second section of this paper (e-print), or 4.1 of this (e-print), or section 9.2.4 of this book, or, apparently, chapter 5 of this book.

These points are relevant mostly because they are associated to systems at bifurcations, i.e., structurally unstable systems whose behavior can change qualitatively under small perturbations. Such sensitivity has been used in the construction of very sensitive sensors, and even more sensitive than the diabolical points are the even more degenerate "exceptional points" (where "not only do resonant frequencies coincide but their resonant modes do too") . Both situations are schematically illustrated for light propagation modes (see this article) in the following figure, which shows the modes split with growing perturbation intensity $\epsilon$:

Modes split with growing perturbation intensity epsilon

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  • $\begingroup$ Conical intersections are stable features in that a perturbation to the system will shift the intersection instead of breaking it. Is that also the case for the other degeneracies you describe? Or are those more fragile? If they are indeed stable, despite their higher order, can you comment on what features enable that? $\endgroup$ – Emilio Pisanty Dec 17 '17 at 16:15
  • $\begingroup$ @EmilioPisanty, I mean here parametric perturbations. The degeneracy is a single point in a 2D parameter space, so a perturbation doesn't break it indeed, but will take the system away from it. As for the "more degenerate", it was unclear, thanks for pointing out - I included now in the answer the quote from the linked Phys. Today article that describes it: "not only do resonant frequencies coincide but their resonant modes do too". $\endgroup$ – stafusa Dec 17 '17 at 16:33
  • $\begingroup$ @stafusa Exceptional points are nothing but eigenvalues with their geometric multiplicity being less than the algebraic multiplicity, right? $\endgroup$ – Chetan Waghela Dec 20 '17 at 8:33
  • $\begingroup$ @ChetanWaghela I don't know. I hadn't heard about them before reading the article I link to, so I have no deeper understanding of them. $\endgroup$ – stafusa Dec 20 '17 at 8:35
  • $\begingroup$ @stafusa Ok, no problem. $\endgroup$ – Chetan Waghela Dec 20 '17 at 8:43

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