One-dimensional Dirac operators with zero-range interactions: Spectral, scattering, and topological results

The spectral and scattering theory for 1-dimensional Dirac operators with mass $m$ and with zero-range interactions are fully investigated. Explicit expressions for the wave operators and for the scattering operator are provided. These new formulae take place in a representation which links, in a suitable way, the energies $-\infty$ and $+\infty$, and which emphasizes the role of $\pm m$. Finally, a topological version of Levinson's theorem is deduced, with the threshold effects at $\pm m$ automatically taken into account.