In all the possible potentials I have encountered so far, it seems that the bound states (i.e. E < [V(-infinity) and V(infinity)]) always results in a discrete spectrum of energies, whereas the scattering states (E > [V(-infinity) and V(infinity)]) always results in a continuous spectrum of energies. I can't seem to find a logical explanation for this. If we use the anove defintion of bound and scattering states: The potential at plus/minus infinity of the harmonic oscillator is infinite, but so is the energy (for infinite n). But the harmonic oscillator has a bound spectrum. I can't quite see this. I've taken the above from Griffith's, and sadly he never mentions whether "bound states = discrete spectrum" or not.