Levinson's theorem for the nonlocal interaction in one dimension

The Levinson theorem for the one-dimensional Schrodinger equation with both local and the nonlocal symmetric potentials is established by the Sturm-Liouville theorem. The critical case where the Schrodinger equation has a finite zero-energy solution is also analyzed. It is shown that the number n(+) (n(-)) of bound states with even (odd) parity is related to the phase shift eta(+)(0)[eta-(0)] of the scattering states with the same parity at zero momentum as eta+(0) = {(n+ - 1/2)pi noncritical case n(+)pi critical case and eta-(0) = {n-pi noncritical case (n(-) + 1.2)pi critical case The problems on the positive-energy bound states and the physically redundant state related to the nonlocal interaction are also discussed.