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

Abstract

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.

18 pages

References in corpus (6)

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• On the structure of the wave operators in one dimensional potential scattering
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Cited by in corpus (14)

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• Two-dimensional Dirac operators with singular interactions supported on closed curves
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• Dirac operators with Lorentz scalar shell interactions
• One-dimensional scattering of fermions on $\delta$-impurities
• Two-dimensional Dirac operators with general $\delta$-shell interactions supported on a straight line
• Approximation of one-dimensional relativistic point interactions by regular potentials revised
• An index theorem on asymptotically static spacetimes with compact Cauchy surface
• Levinson's theorem: an index theorem in scattering theory
• Levinson's theorem as an index pairing
• Limiting absorption principle and scattering matrix for Dirac operators with $\delta$-shell interactions
• Effective quantum electrodynamics: One-dimensional model of the relativistic hydrogen-like atom
• Zero-range potentials for Dirac particles: Bound-state problems