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I'm trying to study the spectrum operator via Floer Theory. The idea is to consider the Rayleigh quotient as an operator on the Hilbert space $W^{1,2}$ and to study it's negative gradient flow.

Any idea of how to recover the well-known results about its eigenvalues and eigenfunctions? Or do you have any concrete example of such a method?

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    $\begingroup$ Can you give an example of what you're trying to do? Which "well-known results" do you mean exactly? $\endgroup$
    – Manuel Bärenz
    Commented Apr 12, 2016 at 11:16
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    $\begingroup$ I suggest that you first try this for a real symmetric $N\times N$ matrix. In any case. if the eigenvalues are simple, what you will discover is a perfect complex computing the homology of $\mathbb{RP}^N$. In any case, this is a classical example, good as an exercise. $\endgroup$
    – Liviu Nicolaescu
    Commented Apr 12, 2016 at 11:48
  • $\begingroup$ I wish I could find again that it possesses an infinite countable number of eigenvalues , that the first on is simple,etc.... I need to study the negative gradient flow associated to thé Rayleigh quotient. I read many things about Floer theory but I am very new with it, and I must admit I haven't manager yet to make everything clear. I wanted to start with an easy example, yet it still is challenging for me. THANK you Liviu Nicolaescu, I'm going to try this. $\endgroup$
    – D.M.
    Commented Apr 13, 2016 at 15:38

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