Physics: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, ${\phi }_{\ell }(0)$, and infinite energy, ${\phi }_{\ell }(\mathrm{\infty })$, for a spherically symmetric potential $V(r)$ is related to the number of bound states $n_{\ell }$ by:

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

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 $(0,\mathrm{\infty })$ except for a finite number of finite discontinuities

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

$V(r)=O(r^{-3-\epsilon })\text{ as }r\to \mathrm{\infty }\epsilon >0$

External links

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

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