Levinson theorem for Dirac particles in one dimension

Abstract

The scattering of Dirac particles by symmetric potentials in one dimension is studied. A Levinson theorem is established. By this theorem, the number of bound states with even (odd) parity, $n_{+}$ ($n_{-}$), is related to the phase shifts ${\eta }_{+}(\pm E_k)$ [${\eta }_{-}(\pm E_k)$] of scattering states with the same parity at zero momentum as follows: $${\eta }_{\pm }(\mu )+{\eta }_{\pm }(-\mu )\pm \frac{\pi }{2}[{\sin }^2{\eta }_{\pm }(\mu )-{\sin }^2{\eta }_{\pm }(-\mu )]=n_{\pm }\pi .$$ The theorem is verified by several simple examples.

REVTeX, 17 pages, no figure

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