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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 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: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