Levinson's theorem for the Schrödinger equation in one dimension

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 and η_-(0) = n_-π for the noncritical case, and η₊(0) = n₊π and η_-(0) - π/2 = n_-π for the critical case.