The charged particle Levinson theorem and the quantum defect

Abstract

Levinson's theorem is considered for the single-channel scattering of charged particles by a short-range central potential U( r) with finite first and second moment integrals. For a repulsive Coulomb field, the Levinson relation for the phase shift η l(k) for l ≧ 1 is η l (0) − η l (∞) = n l π, where η l (∞) = 0 and n l is the number of composite bound states with energies E n ≦ 0. For l = 0, n 0 counts only the composite bound states with energy E n < 0, but if a resonant state at E = 0 is present, the relation is modified to η 0(0) − η 0(∞) = (n 0 + 1 2 )π . An attractive Coulomb field has an infinite number of bound state energies E n = − h ̷ 2β 2/8un 2 . Switching on the short range potential field U( r) displaces these levels to E′ n = − h ̷ 2β 2/8u n ́ 2 , where n′ = n − μ l ( n) and μ l ( n) is called the quantum defect. In terms of μ l ( k 2), the Levinson theorem for l ≧ 0 becomes η l(0) − η l(∞) = μ l(0)π, with η l(∞) = 0 and η l(0) in general an irrational number. The quantum defect μ: l (0) is found most simply via solution of the zero-energy wave equation. Alternatively, a first order, non-linear differential equation for a phase function η l ( k, r) gives the zero-energy phase shift η l (0, ∞) = η l (0) via numerical integration. This determines n l for repulsive and μ l (0) for attractive Coulomb fields.