I'm kind of self studying from Griffiths's QM book, and I'd like to clarify a few things I find confusing. As I understand it, for any potential V, there can exist bound states or scattering states. In the case of the bound states, the solutions to the time-independent schrodinger equation are normalizable, whereas with scattering states they are not. Is this true for all potentials? Why is that true? Anyway, in that case, you can immediately reject the solutions which are not normalizable for the bounded states, since they do not represent physical states. But for the scattering states, there is no reason to throw out the non normalizable solutions.