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.