%0 Journal Article
%T Levinson's theorem for the Schr£żdinger equation in one dimension
%A Shi-Hai Dong
%A Zhong-Qi Ma
%J Physics
%D 1999
%I arXiv
%R 10.1007/s100530070079
%X 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.
%U http://arxiv.org/abs/quant-ph/9903016v1