Levinson's theorem for the Schrödinger equation in two dimensions

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 $\pi$.

Latex 11 pages, no figure and accepted by P.R.A (in August); Email: [email protected], [email protected]

References in corpus (2)

• Variable-phase method and Levinson's theorem in two dimensions: Application to a screened Coulomb potential
• Levinson theorem in two dimensions

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• Asymptotic phase shifts and Levinson theorem for 2D potentials with inverse square singularities
• Levinson theorem for Aharonov-Bohm scattering in two dimensions