Scattering states are not subject to Normalised Wavefunction boundary conditions, so any value of energy gives a physically real scattering state.
Energy:Value: 5 units
Bound States must be such that they go to zero at infinities. With these boundary conditions, and depending on the potential, only few energy values are allowed.
We take the example of finite square well and try to estimate one bound state energy.
If E is a bound state energy, then the wavefunction will be unbounded for E + dE and E - dE. However this unbounded will change its direction.
To see this in action, try to set E = 1.1 and E = 1.0 in the simulation below. The change in direction of the divergence indicates that there is a bound state for a finite value of energy that lies between 1.1 and 1.0. The error in this simulation is of the order of sqrt(N) ~ 0.1. So we estimate the bound energy to be 1.05 +- 0.1 units. Approximating this potential with an infinite square well potential shows that the first bound state energy is given by pi/3 units = 1.047 units.
The higher energy states are increasingly less effective due to the infinite square well being a bad approximation for the finite well. The 2nd bound state energy predicted by this simulation is 4.05 +- 0.1 units, while the actual energy comes out to be around 4.386 units.
Energy:Value: 5 units