Self-adjointness domain of P_r^2

[Clausius]

hi everybody, i'd like to discuss with you a problem occurred to me in the study of a central symmetry field in non rel. qm.
at a certain point, i get the linear operator $$P_{r}^{2}=\frac{1}{r}\frac{\partial^{2}}{\partial r^{2}}r$$, which is the square of radial momentum, and i want to determine the domain in which it is self-adjoint, i.e. the maximal subset $$D(P_{r}^{2})\in L^{2}(\mathbb{R}_{+},r^{2}\ \mathrm{d}r)$$ such that $$D(P_r^2)=D(P_r^2\dagger)$$ and the two coincide in the domain above.

 

[dextercioby]

Goerg Teschl (professor at TU in Vienna) wrote a book based on some lecture notes available somewhere on the internet for free. I think a link to these notes is right here on PF in the <Reference> session. You may search for the latest version of his notes. He has an extended coverage of the self-adjointess problem for different hamiltonians.

I think the $p_{r}^{2}$ operator has the same domain of self-adjointness as the radial part of the H-atom operator. But you may check on that.

 

[Landau]

Since it's Gerald Teschl, you may have trouble searching for Goerg Teschl, so here's the link ;)