Suppose a system is under the influence of a potential which vanishes at $\pm \infty$. Now we know that if the energy of the system is negative ($E<0$) then the system is in a bound state and the bound state energy spectrum is discrete. For $E>0$, the system is in scattering states (with continuous energy spectrum). My question is:
What happens at $E=0$? Can we consider it a discrete bound state or should we consider it as a scattering state and why?