The Relativistic Levinson Theorem in Two Dimensions

Abstract

In the light of the generalized Sturm-Liouville theorem, the Levinson theorem for the Dirac equation in two dimensions is established as a relation between the total number $n_j$ of the bound states and the sum of the phase shifts ${\eta }_j(\pm M)$ of the scattering states with the angular momentum $j$: $${\eta }_j(M)+{\eta }_j(-M)\tilde{}$$ $$=\{\begin{array}{ll}(n_j+1)\pi & \mathrm{when}\mathrm{a}\mathrm{half}\mathrm{bound}\mathrm{state}\mathrm{occurs}\mathrm{at}E=M\mathrm{and}j=3/2\mathrm{or}-1/2\\ (n_j+1)\pi & \mathrm{when}\mathrm{a}\mathrm{half}\mathrm{bound}\mathrm{state}\mathrm{occurs}\mathrm{at}E=-M\mathrm{and}j=1/2\mathrm{or}-3/2\\ n_j\pi & \mathrm{the}\mathrm{rest}\mathrm{cases}.\end{array}$$ \noindent The critical case, where the Dirac equation has a finite zero-momentum solution, is analyzed in detail. A zero-momentum solution is called a half bound state if its wave function is finite but does not decay fast enough at infinity to be square integrable.

Latex 14 pages, no figure, submitted to Phys.Rev.A; 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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