Regarding normalization of the eigen basis vectors

[ashokanand_n]

For a continuous eigen-basis the basis vectors are not normalizable to unity length. They can be normalized only upto a delta function. At the same time for discrete eigen basis the basis vectors are normalizable to unity length.

What about the systems with both discrete as well as continuous spectrum (For eg., a finite potential well) ?

[Avodyne]

In this case the spectrum has both a continuous part (and the corresponding eigenstates can only be normalized to a delta-function) and a discrete part (and the corresponding eigenstates can only be normalized to one).