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M-function behaviour for a periodic Dirac system

Published online by Cambridge University Press:  14 November 2011

Dominic P. Clemence
Dominic P. Clemence
Affiliation:
Department of Mathematics, University of Zimbabwe, P.O. Box MP 167, Mount Pleasant, Harare, Zimbabwe
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Abstract

For a 2 × 2 periodic system with a perturbation P whose first moment is finite, Jy′ = [λI + R(x) + P(x)]y, we study the behaviour of the Titchmarsh–Weyl m(λ)-coefficient at the spectral gap endpoints. Assuming gap nondegeneracy, our main result is that as λ → λ0, (λ − λ0)½(m(λ) → c ≠ 0 if and only if λ0 is a φ-half-bound state, which follows from an analysis of Jost-type functions.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 124 , Issue 1 , 1994 , pp. 149 - 159
Copyright
Copyright © Royal Society of Edinburgh 1994

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References

1Harris, B. J.. On the spectra and stability of periodic differential equations. Proc. London Math. Soc. (3) 41 (1980), 161–192.CrossRefGoogle Scholar
2Hinton, D. B., Klaus, M. and Shaw, J. K.. On the Titchmarsh-Weyl function for the half-line perturbed periodic Hill's equation. Quart. J. Math. Oxford. 41 (1990), 189–224.CrossRefGoogle Scholar
3Hinton, D. B. and Shaw, J. K.. Absolutely continuous spectra of perturbed periodic Hamiltonian systems. Rocky Mountain J. Math. (4) 17 (1987), 727–748.CrossRefGoogle Scholar