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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Eigenvalue asymptotics of perturbed periodic Dirac systems in the slow-decay limit


Author: Karl Michael Schmidt
Journal: Proc. Amer. Math. Soc. 131 (2003), 1205-1214
MSC (2000): Primary 34L20, 34L40, 47E05, 81Q10, 81Q15
Published electronically: July 26, 2002
MathSciNet review: 1948112
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Abstract: A perturbation decaying to $0$ at $\infty $ and not too irregular at $0$introduces at most a discrete set of eigenvalues into the spectral gaps of a one-dimensional Dirac operator on the half-line. We show that the number of these eigenvalues in a compact subset of a gap in the essential spectrum is given by a quasi-semiclassical asymptotic formula in the slow-decay limit, which for power-decaying perturbations is equivalent to the large-coupling limit. This asymptotic behaviour elucidates the origin of the dense point spectrum observed in spherically symmetric, radially periodic three-dimensional Dirac operators.


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Additional Information

Karl Michael Schmidt
Affiliation: School of Mathematics, Cardiff University, 23 Senghennydd Rd., Cardiff CF24 4YH, United Kingdom
Email: SchmidtKM@Cardiff.ac.uk

DOI: http://dx.doi.org/10.1090/S0002-9939-02-06679-0
PII: S 0002-9939(02)06679-0
Received by editor(s): August 3, 2001
Received by editor(s) in revised form: November 16, 2001
Published electronically: July 26, 2002
Communicated by: Joseph A. Ball
Article copyright: © Copyright 2002 American Mathematical Society