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,

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