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Title: Spectral analysis of Dirac operators under integral conditions on the potential
Author: Hughes, Daniel Gordon John
ISNI:       0000 0004 2735 2495
Awarding Body: Cardiff University
Current Institution: Cardiff University
Date of Award: 2012
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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]U[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 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:=\lim sup_x→∞_ x|q(x)|<∞ then the eigenvalues outside of the spectral gap [-m,m] must obey Σ_n_(λ²_n-1)
Supervisor: Not available Sponsor: Not available
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
EThOS ID:  DOI: Not available
Keywords: QA Mathematics