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 ${\eta }_{+}(0)[{\eta }_{-}(0)]$ 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.

Revtex 11 pages and submitted to Phys. Rev. A

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