  Levinson's theorem for the one-dimensional Schr\"{o}dinger equation with a
symmetric potential, which decays at infinity faster than $x^{-2}$, is
established by the Sturm-Liouville theorem. The critical case, where the
Schr\"{o}dinger equation has a finite zero-energy solution, is also analyzed.
It is demonstrated that the number of bound states with even (odd) parity
$n_{+}$ ($n_{-}$) is related to the phase shift $\eta_{+}(0)[\eta_{-}(0)]$ of
the scattering states with the same parity at zero momentum as
$\eta_{+}(0)+\pi/2=n_{+}\pi, \eta_{-}(0)=n_{-}\pi$, for the non-critical case,
$\eta_{+}(0)=n_{+}\pi, \eta_{-}(0)-\pi/2=n_{-}\pi$, for the critical case.
