Selfadjointness of Coulomb Hamiltonian in $d\geq3$ dimensions

Question

I know that the Coulomb-Hamiltonian $H=-\Delta - |\cdot|^{-1}$ is self-adjoint with $\operatorname{dom}(H)=H^2(\mathbb R^3)$. This follows by the Kato-Rellich-Theorem.

Has the corresponding quadratic form a form domain of $Q(H) = H^1(\mathbb R^3)$?

Is it also true for dimensions which are bigger that $3$? How to see it?

Best wishes :)

Answer 1

This is related to a problem posed by Tosio Kato in 1953 and not resolved until 2002, a few years after Kato's death. Take a look at the references on this page to Kato's Conjecture:

     http://www.math.missouri.edu/~hofmann/

I believe that you are essentially correct because the form domain is related to the square root of the elliptic operator, and that is related to Kato's conjecture.

     http://en.wikipedia.org/wiki/Kato_conjecture

I don't think that the Coulombic singularity is a problem for what you are wanting to say.