Levinson's theorem and a bound state of a positron in electron gas

Positrons embedded in metals annihilate predominantly with conduction electrons. The annihilation rate reduces with decreasing density for the whole range of metallic densities. Controversial suggestions have been made on the annihilation rates at very low electron gas densities. We calculate the effective screened interaction between the positron impurity and electrons within a simplified jellium model where the electrons are treated as bosonic particles and show that the electrons attracted to the positron can form a bound cluster. The resonance states of the effective interaction are analysed by calculating the scattering phase-shifts. We find that the maximum of the phase-shift at zero energy approaches to π when r_s ≈ 14. According to the Levinson's theorem this indicates a bound state.