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.[1]

Statement of theorem

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

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

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

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

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

References

1. Levinson's Theorem
2. A. Galindo and P. Pascual, Quantum Mechanics II (Springer-Verlag, Berlin, Germany, 1990).

External links

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