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Spectral analysis of Dirac operators under integral conditions on the potential

Hughes, Daniel Gordon John 2012. Spectral analysis of Dirac operators under integral conditions on the potential. PhD Thesis, Cardiff University.
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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 $(-\infty,-1]\cup[1,\infty)$. 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 establishing the result for spherically symmetric Dirac operators in higher dimensions, too. Finally, with regard to this problem, we show that a sparse perturbation of a square integrable potential does not cause the absolutely continuous spectrum to become larger in the one-dimensional case. The final problem considered is regarding bound states, where we show that if the electric potential obeys the asymptotic bound $C:=\limsup_{x\rightarrow\infty} x|q(x)|<\infty$ then the eigenvalues outside of the spectral gap $[-m,m]$ must obey $\sum_{n} (\lambda_n^2-1)<\frac{C^2}{2}$, where $m$ is the constant mass.

Item Type: Thesis (PhD)
Status: Unpublished
Schools: Mathematics
Subjects: Q Science > QA Mathematics
Funders: EPSRC
Date of First Compliant Deposit: 30 March 2016
Last Modified: 19 Mar 2016 23:13
URI: https://orca.cardiff.ac.uk/id/eprint/43141

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