Levinson's theorem

The difference in the $\backslash ell$-wave phase shift of a scattered wave at infinite momentum, $\backslash varphi(+\backslash infty)$, and zero momentum, $\backslash varphi(0)$, for a spherically symmetric potential $V(r)$ is related to the number of bound states $n\_b$ by:

: $\backslash varphi(+\backslash infty)\; -\; \backslash varphi(0)\; =\; -\; \backslash pi\; \backslash left(\; \backslash frac\{1\}\{2\}n\_\{0\}\; +\; n\_\{b\}\; \backslash right)$,

where $n\_\{0\}\; =\; 0$ or $1$.

The scenario $n\_\{0\}\; =\; 1$ is uncommon and can only occur in $s$-wave scattering, if a bound state with zero energy exists. 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 $(0,+\backslash infty)$ except for a finite number of finite discontinuities,

:$V(r)\; =\; O(r^\{\; -3/2\; +\; \backslash varepsilon\})\; ~\backslash text\{\; as\; \}\; ~r\backslash rightarrow\; 0,\; ~~\backslash varepsilon>0,$

:$V(r)\; =\; O(r^\{\; -3\; -\; \backslash varepsilon\})\; ~\backslash text\{\; as\; \}\; ~r\; \backslash rightarrow\; \backslash infty,\; ~~\backslash varepsilon>0.$

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

{{cite journal | first1 = L. | last1 = Castillejo| first2 = R. H. | last2 = Dalitz| first3 = F. J. | last3 = Dyson| year = 1956 |

title= Low's Scattering Equation for the Charged and Neutral Scalar Theories | journal = Phys. Rev. | volume = 101 | pages = 453–458 | issue = 1

| doi = 10.1103/physrev.101.453| bibcode = 1956PhRv..101..453C}}

and Jaffe and Low primitives

{{cite journal |last1=Jaffe |first1=R. L. |author1-link=Robert Jaffe (physicist) |author2-link=Francis E. Low |last2=Low |first2=F. E. |year=1979 |title=Connection between quark-model eigenstates and low-energy scattering |journal=Phys. Rev. D |volume=19 |issue=7| page=2105 |doi=10.1103/PhysRevD.19.2105 |bibcode=1979PhRvD..19.2105J }}

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, $n\_\{b\}$,

the number of primitives, $n\_\{p\}$, and the number of CDD poles, $n\_\{\backslash text\{CDD\}\}$:

{{cite journal |last1=Krivoruchenko |first1=M. I. |last2=Tyrin |first2=K. S. |year=2021 |title= Generalization of the Levinson Theorem on the Asymptotic Value of the Scattering-Amplitude Phase Shift |journal=Physics of Atomic Nuclei |volume=84 |issue=1| pages=29–33 |doi=10.1134/S1063778821010130|arxiv=2006.09366 |bibcode=2021PAN....84...29K }}

: $\backslash varphi(+\backslash infty)\; -\; \backslash varphi(0)\; =\; -\; \backslash pi\; \backslash left(\; \backslash frac\{1\}\{2\}n\_\{0\}\; +\; n\_\{b\}\; +\; n\_\{p\}\; -\; n\_\{\backslash text\{CDD\}\}\backslash right)$.

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. The following universal properties of the Jost function, $\backslash omega\_J\; (s)$, are essential to guarantee the generalized theorem:

: $\backslash omega\_J(s)$ an analytic function of the square of energy, $s$, in the center-of-mass frame of the scattered particles with a cut from threshold to infinity, $n\_b$ simple zeros below the threshold, $n\_p$ simple zeros above the threshold, and $n\_\{\backslash text\{CDD\}\}$ simple poles on the real axis. The zeros correspond to bound states and primitives in a fixed channel with total angular momentum $J$.

{{Reflist}}

• Larry Spruch, "Levinson's Theorem", http://physics.nyu.edu/LarrySpruch/LevinsonsTheorem.PDF#Levinson_theorem.
• M. Wellner, [http://scitation.aip.org/content/aapt/journal/ajp/32/10/10.1119/1.1969857 "Levinson's Theorem (an Elementary Derivation)]," Atomic Energy Research Establishment, Harwell, England. March 1964.

Category:Theorems in quantum mechanics

de:Compton-Effekt#Compton-Wellenlänge