How to prove the bound and scattering states theorem?

Question

In Griffiths it is mentioned that if the energy eigenvalue is less than the value of the potential at + and - infinity, then we have bound states. If however the energy is bigger than the potential at + and - infinity, we get scattered states.

I can understand how to prove this in the case that the potential goes to zero at +- infinity - but how can I prove this in general, for any potential?

Answer 1

Assume we have a potential $V(x)$ where $$\lim_{x\to\pm\infty}V(x)=V_{\pm},$$ then in the limit of large $|x|$ the Schrodinger equation approaches $$\psi''=-\frac{2m}{\hbar^2}(E-V_{\pm})\psi.$$ If $E<V_{\pm}$ on either end of the real axis then $\psi$ will decay exponentially on that side. If this is true on both ends then we have a bound state, otherwise we have a scattering state.