Why does the attractive Dirac Delta distribution (function) potential $V = \alpha\delta$(x) (for negative $\alpha$) yield both bound AND scattered states? Is this due to the definition of the Dirac delta distribution? Or is it for a different reason?
Since the potential has a lowest value of -Infinity, the particle can be found in negative and positive energy. Now, for a bound state to occur, its corresponding energy of the eigenstate must be lower than both the value of the potential at x--> +Infinity and x--> -Infinity.
So, the negative energy eigenstates have corresponding energies that are lower than the value of the potential at minus and plus infinity, so the negative energy eigenstates are also bound eigenstates.
On the other side, the positive energy eigenstates are scattered states because their corresponding energies are higher than the value of the potential at plus and minus infinity.
So, we get both bound and scattered states.