Let $H = H_0 + V$ be the Hamiltonian of the single electron where $H_0 = - \Delta, V = - \frac{\gamma}{|x|}$. Now one defines the dilation group $U(s) \psi(x) = e^{-ns/2} \psi(e^{-s}x), s \in \mathbb R$. My questions:
1) What is the motivation for the exact definition of $U$?
2) How does one compute the infitesimal generator of $U$? This should be $D = 1/2(x\cdot p + p \cdot x)$.
3) Why does the action of $U$ on $V$ equal $e^{-s}V$, that is, why is $U(-s)VU(s) = e^{-s}V$?
4) The virial theorem now asserts that an eigenfunction $\psi$ of $H$ satisfies $-\langle \psi, H_0, \psi \rangle = \langle \psi, V \psi\rangle$. What does change if $V$ satisfies $U(-s)VU(s) = e^{-\alpha s}V$ for some $\alpha \in \mathbb R \setminus \{0\}?$