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 phase 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 ${\bar {n}}_{l}$ by:

$\phi _{l}(0)-\phi _{l}(\infty )=(n+{\frac {1}{2}}N)\pi \$

where $N=1$ for $s$-wave scattering, $N=2$ for $l\geq 1$ and $N=0$ otherwise. Furthermore, the potential must satisfy the following asymptotic conditions:

$r^{2}|V(r)|\rightarrow 0{\text{ at }}r\rightarrow 0$

$r^{3}|V(r)|\rightarrow 0{\text{ at }}r\rightarrow \infty$

External links

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