Levinson's theorem for the Schrödinger equation in one dimension

Abstract

Levinson's theorem for the one-dimensional Schrödinger 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 Schrödinger 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 $η_{+}(0)[η_{-}(0)]$ of the scattering states with the same parity at zero momentum as $η_{+}(0)+π/2=n_{+}π, η_{-}(0)=n_{-}π$, for the non-critical case, $η_{+}(0)=n_{+}π, η_{-}(0)-π/2=n_{-}π$, for the critical case.