Rearranging Schrödinger equation with Coulomb potential in Dirac notation

[Dmon1Unlimited]

I'm looking through my lecture notes, (studying relativistic corrections/perturbation theory using hydrogen), and I seem to have a mind block with one of the equations (the last one from the 3 in the middle).

I know that the kinetic energy and coulomb potential has been subbed in for the operator H0, but I don't understand the rearrangement. Simply multiply both sides by 2*m, then take the coulomb potential to the other side.

The coulomb potential (the e^2 term) on the left hand side is positive, moving it to the right would make it negative, so why is it positive?

 

[Simon Bridge]

I know that the kinetic energy and coulomb potential has been subbed in for the operator H0, but I don't understand the rearrangement. Simply multiply both sides by 2*m, then take the coulomb potential to the other side.

... I think I see what you mean.
$V(r)=e^2/4\pi\epsilon_0r$ would be a repulsive potential wouldn't it?
If the potential is supposed to be attractive, then they've made a typo in the initial substitution: there should be a minus sign in there.