Levinson's Theorem for the Klein-Gordon Equation in Two Dimensions

Abstract

The two-dimensional Levinson theorem for the Klein-Gordon equation with a cylindrically symmetric potential $V(r)$ is established. It is shown that $N_{m}π=π(n_{m}^{+}-n_{m}^{-})= [δ_{m}(M)+β_{1}]-[δ_{m}(-M)+β_{2}]$, where $N_{m}$ denotes the difference between the number of bound states of the particle $n_{m}^{+}$ and the ones of antiparticle $n_{m}^{-}$ with a fixed angular momentum $m$, and the $δ_{m}$ is named phase shifts. The constants $β_{1}$ and $β_{2}$ are introduced to symbol the critical cases where the half bound states occur at $E=\pm M$.