# Algorithm for finding eigenfunctions

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,

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

• somehow I have difficulty parsing your formula for $\Omega$; could you write down explictly how $\Omega$ acts on a function in $L^2(\mathbb{R})$ ? – Carlo Beenakker Oct 12 '15 at 21:58
• I just did. Please let me know if you need more detail. – aidan.plenert.macdonald Oct 13 '15 at 23:17
• 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. – Francois Ziegler Oct 13 '15 at 23:44
• 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. – aidan.plenert.macdonald Oct 14 '15 at 0:03
• The $\partial \phi(a, b)$ is my a, b space measure. – aidan.plenert.macdonald Oct 14 '15 at 0:05