Consider the delta well. You have a bound state and scattering states.
Can I get a normalizable solution to the schrodinger equation that is some sort of a combination of the bound state and continuous states (maybe of the form bound state wave function + fourier integral for scattered states)?
Since even the bound state can be written as a linear combo of the scattered states, does this mean I can't interpret the fourier inverse transform as a probability density? Because if I could, then that would imply the bound state can give a range of positive energies, which makes no sense.
In general, can I get normalizable solutions to the schrodinger equation that are combos of bound states and scattered states? What is the probability interpretation then?
Note, the delta well has a half discrete/ half continous spectrum, which is where the issue of interpretation arises. (for discrete, I have orthonormality that explains interpretation and for totally continuous, I have dirac orthonormality).
The Hydrogen isotope has bound states with energies that cluster around a point, beyond which there are only unbound states. Any complete expansion requires both the bound eigenstates as well as the unbound. The bound states are summed with an ordinary sum, and the unbound states are summed with an integral and a denstity function. Combining these two allows you to represent any state.