Inst High Energy Phys, Beijing 100039, Peoples R China
; Chinese Ctr Adv Sci & Technol, World Lab, Beijing 100080, Peoples R China
Levinson's theorem for the one-dimensional Schrodinger 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 Schrodinger 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 and eta(-)(0) = n(-)(0)for the noncritical case, and eta(+)(0) = n(+)pi and eta(-)(0) - pi/2 = n-pi for the critical case.