Does there always exist an equivalent geometric problem for a given differential equation?

Question

I recently came up with a method to show for any given differential equation whose solution is the power series there exists an equivalent geometric problem in Hilbert space. I was wondering if something like this already existed and could this expanded upon to include any function. Here's a link to my work:

https://www.dropbox.com/s/fjpuk316osxfo2p/intro2.pdf

Answer 1

What you are doing is trying to construct an orthonormal basis in $L^2$ space consisting of polynomials. Such bases are known as orthogonal polynomials, and if properly constructed any function in that $L^2$ can be expanded over them. However, the choice of $x^n$ as orthonormal vectors is not a very good one for convergence purposes. Your geometric operators are somewhat similar to creation and annihilation operators used in quantum mechanics. They represent action of some differential operators on the basis, and so can be used to solve the corresponding equations in terms of the basis.