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 delta(alpha)(0)=n(alpha)pi between the phase shift delta(alpha)(k) of scattering state at zero momentum and the total number n(alpha) of bound states for the alphath angular-momentum channel, where alpha=\m+mu(0)\ is a real number (m=integer, and mu(0)=-Phi/Phi(0) with Phi being the magnetic flux and 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.