Levinson's theorem for the Schrödinger equation in two dimensions

Details Levinson's theorem for the Schrödinger equation in two dimensions Levinson's theorem for the Schrödinger equation in two dimensions

1998-06-02

arxiv.org/abs/quant-ph/9806004v1

Physics

Quantum Physics

Latex 11 pages, no figure and accepted by P.R.A (in August); Email: dongsh@bepc4.ihep.ac.cn, mazq@bepc4.ihep.ac.cn

Scientific paper

10.1103/PhysRevA.58.2790

Levinson's theorem for the Schr\"{o}dinger equation with a cylindrically symmetric potential in two dimensions is re-established by the Sturm-Liouville theorem. The critical case, where the Schr\"{o}dinger 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 $\pi$.

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