In non-relativistic quantum mechanics, it relates the number of bound states in channels with a definite orbital momentum to the difference in phase of a scattered wave at infinite and zero momenta.
[1] The theorem applies to a wide range of potentials that increase limitedly at zero distance and decrease sufficiently fast as the distance grows.
-wave phase shift of a scattered wave at infinite momentum,
-wave scattering, if a bound state with zero energy exists.
The following conditions are sufficient to guarantee the theorem:[2] Generalizations of Levinson's theorem include tensor forces, nonlocal potentials, and relativistic effects.
In relativistic scattering theory, essential information about the system is contained in the Jost function, whose analytical properties are well defined and can be used to prove and generalize Levinson's theorem.
The presence of Castillejo, Dalitz and Dyson (CDD) poles [3] and Jaffe and Low primitives [4] which correspond to zeros of the Jost function at the unitary cut modifies the theorem.
In general case, the phase difference at infinite and zero particle momenta is determined by the number of bound states,
: [5] The bound states and primitives give a negative contribution to the phase asymptotics, while the CDD poles give a positive contribution.
In the context of potential scattering, a decrease (increase) in the scattering phase shift due to greater particle momentum is interpreted as the action of a repulsive (attractive) potential.
, are essential to guarantee the generalized theorem: