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

$\phi _{l}(0)-\phi _{l}(\infty )=(n_{l}+{\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

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

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

External links

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