Levinson theorem with the nonlocal Aharonov-Bohm effect

Abstract

Levinson theorem for a charged particle moving in an arbitrary short-range potential and the field of the Aharonov-Bohm magnetic flux is established. The theorem constructs the relation ${\ensuremath{\delta}}_{\ensuremath{\alpha}}{(0)=n}_{\ensuremath{\alpha}}\ensuremath{\pi}$ between the phase shift ${\ensuremath{\delta}}_{\ensuremath{\alpha}}(k)$ of scattering state at zero momentum and the total number ${n}_{\ensuremath{\alpha}}$ of bound states for the \ensuremath{\alpha}th angular-momentum channel, where $\ensuremath{\alpha}=|m+{\ensuremath{\mu}}_{0}|$ is a real number $(m=\mathrm{integer},$ and ${\ensuremath{\mu}}_{0}=\ensuremath{-}\ensuremath{\Phi}/{\ensuremath{\Phi}}_{0}$ with $\ensuremath{\Phi}$ being the magnetic flux and ${\ensuremath{\Phi}}_{0}=hc/e$ the fundamental flux quantum). The relation means that the phase shift at the threshold of zero momentum can serve as a counter for the bound states in the general angular-momentum channel.