PDE with distributional coefficient

Question

I'm a undergraduate major in physics and when learning mathematical aspects of quantum physics, I run into for example this problem:

for a $1$-dim system with $\delta$-potential $$-\frac{d^2}{dx^2}\phi + \delta (x) \phi=E\phi$$ with certain asymptotic conditions, in physics books this kind of PDEs is usually solved by ad hoc tricks, and the method of dealing with elliptic PDE doesn't work for the ill-defined multiplication.

So my question is:

Is there a general theory for PDE with distributional coefficient?

Answer 1

Standard methods of dealing with elliptic PDE do work: In one dimension, multiplication by $\delta(x)$ is a compact operator from $H^1$ to $H^{-1}$. So the usual variational approach works, among other things.