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Absolutely continuous spectrum of Dirac operators with square-integrable potentials

Published online by Cambridge University Press:  16 May 2014

Daniel Hughes  and
Karl Michael Schmidt
Daniel Hughes
Affiliation:
School of Mathematics, Cardiff University, Senghennydd Road, Cardiff CF24 4AG, UK, ([email protected])
Karl Michael Schmidt
Affiliation:
School of Mathematics, Cardiff University, Senghennydd Road, Cardiff CF24 4AG, UK, ([email protected])
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Abstract

We show that the absolutely continuous part of the spectral function of the one-dimensional Dirac operator on a half-line with a constant mass term and a real, square-integrable potential is strictly increasing throughout the essential spectrum (−∞, −1] ∪ [1, ∞). The proof is based on estimates for the transmission coefficient for the full-line scattering problem with a truncated potential and a subsequent limiting procedure for the spectral function. Furthermore, we show that the absolutely continuous spectrum persists when an angular momentum term is added, thus also establishing the result for spherically symmetric Dirac operators in higher dimensions.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 144 , Issue 3 , June 2014 , pp. 533 - 555
Copyright
Copyright © Royal Society of Edinburgh 2014 

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