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 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:^[2]

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

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

References

1. Levinson_theorem
2. One_Dimension

• Levinson's Theorem
• M. Wellner, Levinson's Theorem (an Elementary Derivation, Atomic Energy Research Establishment, Harwell, England. March 1964.
• Shi-Hai Dong, Zhong-Qi Ma, #One_Dimension Levinson's Theorem for the Schrodinger Equation in One Dimension, International Journal of Theoretical Physics, Vol 39, No 2, 2000.