Suvarna Garge (Editor)

Levinson's Theorem

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Levinson's Theorem is an important theorem in non-relativistic quantum scattering theory. It relates the number of bound states of a potential to the difference in phase of a scattered wave at zero and infinite energies. It was published by Norman Levinson in 1949.

Statement of theorem

The difference in phase of a scattered wave at zero energy, ϕ l ( 0 ) , and infinite energy, ϕ l ( ∞ ) , for a spherically symmetric potential V ( r ) is related to the number of bound states n ¯ l by:

ϕ l ( 0 ) − ϕ l ( ∞ ) = ( n + 1 2 N ) π  

where N = 1 for s -wave scattering, N = 2 for l ≥ 1 and N = 0 otherwise. Furthermore, the potential must satisfy the following asymptotic conditions:

r 2 | V ( r ) | → 0  at  r → 0 r 3 | V ( r ) | → 0  at  r → ∞

References

Levinson's Theorem Wikipedia
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