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I have an $ L^2(\mathbb{R}) $ operator that looks like $$ \Omega = \int \partial\phi(a, b)\ \ |b, a\rangle \langle b, a |, $$ where $ \langle x | a, b \rangle = f_a(x - b) e^{x^2/2} $ and $ f_a \in L^2(\mathbb{R}) \ \ \forall a $

I want to get the eigenvectors of the operator. I was hoping for one of two things,

  • An algorithm I can use to approximate them via sampling the vectors and using the finite samples as simple discrete vectors and getting eigenvectors by a standard algorithm
  • Some theoretical means of reducing the problem (group theory?)

My concern is that if I sample, I might have some sort of convergence or error problem. And I wouldn't know where to look for further research in the issue.

I am not a mathematician, although I wish I were, so please forgive any miscommunication on my part. Thank you for any help.

Update:

The explicit formula for the operator is as follows,

$$ \langle x | \Omega | h \rangle = \int \partial\phi(a, b)\ \ \langle x |b, a\rangle \langle b, a | h \rangle \\ = \int \partial\phi(a, b)\ \ f_a(x - b) e^{x^2/2} \left[ \int dx \ \ \overline{f_a(x - b) e^{x^2/2}} h(x) \right] $$

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  • $\begingroup$ somehow I have difficulty parsing your formula for $\Omega$; could you write down explictly how $\Omega$ acts on a function in $L^2(\mathbb{R})$ ? $\endgroup$ – Carlo Beenakker Oct 12 '15 at 21:58
  • $\begingroup$ I just did. Please let me know if you need more detail. $\endgroup$ – aidan.plenert.macdonald Oct 13 '15 at 23:17
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    $\begingroup$ What is $\partial$? (Some sort of $d$?) What is $\int\partial\phi(a,b)\dots$? (Some sort of Stieltjes integral?) What is $a$? (Some sort of number?) How does $f_a$ depend on it? (Arbitrarily?) Etc. $\endgroup$ – Francois Ziegler Oct 13 '15 at 23:44
  • $\begingroup$ I edited it thinking of a different problem I am working on. If you saw the change ignore it. I changed it back. My general goal is to take a family of functions $ \{ f_a \} $, and take all translations (could be discrete if this is easier) of these functions and extract a "covariance" matrix of these functions localized in a gaussian window as vectors in a Hilbert space. I am doing Principle Component Analysis on these vectors, but first localizing them in space. Thus I can ask what the principle localized components are and use this to decompose other functions into orthogonal components. $\endgroup$ – aidan.plenert.macdonald Oct 14 '15 at 0:03
  • $\begingroup$ The $ \partial \phi(a, b) $ is my a, b space measure. $\endgroup$ – aidan.plenert.macdonald Oct 14 '15 at 0:05

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