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 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.
Original language | English |
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Pages (from-to) | 469-481 |
Number of pages | 13 |
Journal | International Journal of Theoretical Physics |
Volume | 39 |
Issue number | 2 |
DOIs | |
State | Published - Feb 2000 |
Externally published | Yes |