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.^[1]

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

${\displaystyle \varphi _{\ell }(0)-\varphi _{\ell }(\infty )=(n_{\ell }+{\frac {1}{2}}N)\pi \ }$

where ${\displaystyle N=0}$ or ${\displaystyle 1}$. The case ${\displaystyle N=1}$ is exceptional and it can only happen in ${\displaystyle s}$-wave scattering. The following conditions are sufficient to guarantee the theorem:^[2]

${\displaystyle V(r)}$ continuous in ${\displaystyle (0,\infty )}$ except for a finite number of finite discontinuities

${\displaystyle V(r)=O(r^{-3/2+\varepsilon })~{\text{ as }}~r\rightarrow 0~~\varepsilon >0}$

${\displaystyle V(r)=O(r^{-3-\varepsilon })~{\text{ as }}~r\rightarrow \infty ~~\varepsilon >0}$

• M. Wellner, "Levinson's Theorem (an Elementary Derivation," Atomic Energy Research Establishment, Harwell, England. March 1964.