Levinson's theorem

Levinson's theorem is an important theorem in non-relativistic quantum scattering theory. It relates the number of bound states of a potential to the difference in phase of a scattered wave at zero and infinite energies. It was published by Norman Levinson in 1949.

Statement of theorem
The difference in the $$\ell$$-wave phase shift of a scattered wave at zero energy, $$\varphi_\ell(0)$$, and infinite energy, $$\varphi_\ell(\infty)$$, for a spherically symmetric potential $$V(r)$$ is related to the number of bound states $$n_\ell$$ by:


 * $$ \varphi_\ell(0) - \varphi_\ell(\infty) = ( n_\ell + \frac{1}{2}N )\pi \ $$

where $$N = 0$$ or $$1$$. The case $$N = 1$$ is exceptional and it can only happen in $$s$$-wave scattering. The following conditions are sufficient to guarantee the theorem:
 * $$ V(r) $$ continuous in $$(0,\infty)$$ except for a finite number of finite discontinuities
 * $$ V(r) = O(r^{ -3/2 + \varepsilon}) ~\text{ as } ~r\rightarrow 0 \varepsilon>0 $$
 * $$ V(r) = O(r^{ -3 - \varepsilon}) ~\text{ as } ~r \rightarrow \infty  \varepsilon>0 $$