Levinson's theorem is an important

theorem In mathematics, a theorem is a statement that has been proved, or can be proved. The ''proof'' of a theorem is a logical argument that uses the inference rules of a deductive system to establish that the theorem is a logical consequence of ...

in non-relativistic quantum scattering theory. It relates the number of

bound state Bound or bounds may refer to: Mathematics * Bound variable * Upper and lower bounds, observed limits of mathematical functions Physics * Bound state, a particle that has a tendency to remain localized in one or more regions of space Geography * ...

s of a

potential Potential generally refers to a currently unrealized ability. The term is used in a wide variety of fields, from physics to the social sciences to indicate things that are in a state where they are able to change in ways ranging from the simple r ...

to the difference in phase of a scattered wave at zero and infinite energies. It was published by

Norman Levinson Norman Levinson (August 11, 1912 in Lynn, Massachusetts – October 10, 1975 in Boston) was an American mathematician. Some of his major contributions were in the study of Fourier transforms, complex analysis, non-linear differential equations, ...

in 1949.

Statement of theorem

The difference in the

\ell

-wave phase shift of a scattered wave at zero energy,

\varphi_\ell(0)

, and infinite energy,

\varphi_\ell(\infty)

, for a spherically symmetric potential

V(r)

is related to the number of bound states

n_\ell

by: :

\varphi_\ell(0) - \varphi_\ell(\infty) = ( n_\ell + \fracN )\pi \

where

N = 0

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:A. Galindo and P. Pascual, Quantum Mechanics II (Springer-Verlag, Berlin, Germany, 1990). :

V(r)

continuous in

(0,\infty)

except for a finite number of finite discontinuities :

V(r) = O(r^) ~\text ~r\rightarrow 0  ~~\varepsilon>0

V(r) = O(r^) ~\text ~r \rightarrow \infty   ~~\varepsilon>0

References

{{Reflist

External links

*M. Wellner
"Levinson's Theorem (an Elementary Derivation
" Atomic Energy Research Establishment, Harwell, England. March 1964. Theorems in quantum mechanics de:Compton-Effekt#Compton-Wellenlänge