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The Fučík spectrum seems to gain momentum among people working on spectral theory, with almost 300 articles published on this topic over the last 5 years, according to Google scholar. There exist several accounts and surveys about this topic, like this one by Schechter, but I am still unable to understand the deep significance of this notion. I must admit that I am a bit uneasy with the idea of introducing a completely new object whose characterization is a complete mess even in the case of small matrices.

It would be great if knowledgeable MO users could give me some hint.

  • Can the Fučík spectrum of an operator or a matrix be used to say something relevant about the operator/matrix itself?
  • Does the Fučík spectrum yield relevant information about the usual spectrum?
  • Does the Fučík spectrum appear in relevant applications?
  • [Very minimal requirement] Do there exist cases, apart from diagonal matrices, where the Fučík spectrum can be described in an easy way?
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About applications.

I know that the Fučík spectrum appears in some models of suspension bridges. See, for example, this article. You can also google the phrase like fucik spectrum bridges, and find another articles about this application (and, probably, others).

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  • $\begingroup$ Upvote, but I'm not completely sure that that paper by Drabek & co. is actually devoted to the Fucik spectrum. In my eyes, it's rather like a parameter-dependent semilinear beam equation that, when one focuses on its stationary version, can be seen as a very particular kind of secular equation for the Fucik spectrum. No special derivation for the stationary version is provided, though, and it is not even clear whether the beam equation is converging towards a stationary state. Given the lack of a Fucik pendant of the spectral theorem, I really don't see a clear connection. $\endgroup$
    – Delio Mugnolo
    Commented Nov 30, 2014 at 20:30
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I have just discovered an interesting relation between Fučík spectrum and so-called spectral minimal partitions in a 2005 article by Conti-Terracini-Verzini (Calc. Var. 22, 45–72 (2005)).

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