Levinson theorem for Aharonov-Bohm scattering in two dimensions

In 1949 Levinson 1 established one of the most beautiful results of scattering theory: the Levinson theorem sets up a relation between the number of bound states, Nl , in a given lth partial wave and the phase shift l k : namely, l 0 − l = Nl . Ten years later, in 1959, Aharonov and Bohm 2 discovered the global properties of the magnetic flux. Nowadays the Aharonov-Bohm AB effect is often involved to understand different quantum-mechanical phenomena 3 . The aim of this paper is to generalize the Levinson theorem to systems which exhibit AB effects. We denote such systems as AB systems. Recently, the analog of the Levinson theorem was established by Lin 4 for the simplest AB system with constant magnetic flux . Here we establish a more general relation, valid for a magnetic field with a vector potential of the form