Abstract
Levinson's theorem for the one-dimensional Schrödinger equation with asymmetric potential which decays at infinity faster thanx −2 is established by theSturm-Liouville theorem. The critical case where the Schrödinger equation hasa finite zero-energy solution is also analyzed. It is demonstrated that the numberof bound states with even (odd) parityn +(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 − πfor the noncritical case, and η+ (0) =n + π andη− (0) − π/2 =n − π for the critical case.
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Dong, SH., Ma, ZQ. Levinson's Theorem for the Schrödinger Equation in One Dimension. International Journal of Theoretical Physics 39, 469–481 (2000). https://doi.org/10.1023/A:1003604830131
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DOI: https://doi.org/10.1023/A:1003604830131