  The two-dimensional Levinson theorem for the Klein-Gordon equation with a
cylindrically symmetric potential $V(r)$ is established. It is shown that
$N_{m}\pi=\pi (n_{m}^{+}-n_{m}^{-})=
[\delta_{m}(M)+\beta_{1}]-[\delta_{m}(-M)+\beta_{2}]$, 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 $\delta_{m}$ is named phase shifts. The constants $\beta_{1}$ and
$\beta_{2}$ are introduced to symbol the critical cases where the half bound
states occur at $E=\pm M$.
