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Levinson's theorem for the Schrödinger equation in two dimensions

Shi-Hai dong·Xi-Wen Hou·Zhong-Qi Ma
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Abstract

Levinson's theorem for the Schrödinger equation with a cylindrically symmetric potential in two dimensions is re-established by the Sturm-Liouville theorem. The critical case, where the Schrödinger equation has a finite zero-energy solution, is analyzed in detail. It is shown that, in comparison with Levinson's theorem in non-critical case, the half bound state for $P$ wave, in which the wave function for the zero-energy solution does not decay fast enough at infinity to be square integrable, will cause the phase shift of $P$ wave at zero energy to increase an additional $π$.

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Shi-Hai dong, Xi-Wen Hou, Zhong-Qi Ma. 1998-06-02. Levinson's theorem for the Schrödinger equation in two dimensions. https://doi.org/10.1103/physreva.58.2790

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