Low-energy scattering and Levinson's theorem for a one-dimensional Dirac equation

Abstract

For a Dirac system on the line, the scattering matrix is defined in terms of Lippmann-Schwinger type solutions, which are also used to express an eigenfunction expansion. The S-matrix is shown to be continuous for potential functions whose first moments exist. Special attention is focused on the continuity of the S-matrix at the endpoints of the spectral gap (-c,c). Levinson's theorem follows as a corollary to this analysis. The author also obtains the leading-order behaviour for some specific power-law potential functions.