How to understand the energy derivative of the logarithmic derivative of a wavefunction when deriving norm-conserving pseudopotentials

Question

The 4th desirable property of a norm-conserving pseudopotential given by Hamann et al is for the 'logarithmic derivatives of the real and pseudo wave function and their first energy derivatives agree for $r>r_c$'. However I do not understand how the energy derivative of the log derivative $$ \frac{d}{d\epsilon}r\frac{d}{dr}\ln\psi(\epsilon;r) $$ can be defined. In the all-electron case, my understanding is that the valence states are discretised (traditionally labelled with $(n,l,m)$). So what are the states $\psi(\epsilon;r)$ with continuous $\epsilon$ that are presumably required to allow us to define the energy derivative of the log derivative above?

EDIT

Link to follow-up question

Answer 1

Related prior question about norm-conserving potentials. Hopefully someone more informed on pseudopotenials can give a more descriptive answer. However if I'm understanding the question correctly, I think the issue is what you are considering discretized.

While the states for an atom/molecule are discretized, the energy of any given state can still vary continuously as you perturb the atom or molecule. Taking the example of an atom, while it has discrete set of orbitals, the energies of these individual orbitals can continuously vary if an electric field was applied to the atom or you embedded it within some larger molecule or structure.