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
-wave phase shift of a scattered wave at zero energy,
, and infinite energy,
, for a spherically symmetric potential
is related to the number of bound states
by:
:
where
or
. The case
is exceptional and it can only happen in
-wave scattering. The following conditions are sufficient to guarantee the theorem:
[A. Galindo and P. Pascual, Quantum Mechanics II (Springer-Verlag, Berlin, Germany, 1990).]
:
continuous in
except for a finite number of finite discontinuities
:
:
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