Shifting the energy reference level

Question

In non-relativistic QM, does it make a difference if an energy shift is applied to the systems's Lagrangian or Hamiltonian?

Answer 1

Because it detracts from the potential, as the shift in $L$ increases it tends to increase $\langle T \rangle$ to the point that a molecule first implodes ($E=2 \langle T \rangle$) and then -- discounting nuclear fusion -- explodes ($E=2 \langle V \rangle$), consistent with the maintenance of $ E = \langle T \rangle + \langle V \rangle$.

In contrast to $L$, if $H$ is shifted enough, implosion is bypassed, and the system directly ionizes.