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

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

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

External links

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