From what I understand the core cutoff radius is the distance where the pseudo wavefunction and the full electron wavefunction start to resemble each other. In other words, the pseudo wavefunction and full electron wavefunction have the same energy and electron densities for distances larger than the core cutoff radius. The energy cutoff radius is the energy that determines the size of the plane wave basis set.
My question is how they are related. Most computational software control the energy cutoff radius, and the energy cutoff radius is lower for ultrasoft pseudopotentials. My question is why is this so? I understand conceptually that ultrasoft pseudopotentials are smoother. Hence the Fourier series is less complicated. Hence the basis set does not need to be large. However, I have trouble putting this vague (if correct) concept into concrete reasoning. What are relevant equations that directly relate the energy cutoff and core cutoff?