Spectral concentrations and resonances of a second order block operator matrix and an associated λ-rational Sturm-Liouville problem

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Brown, B.Malcolm and Langer, M. and Marletta, Marco (2004) Spectral concentrations and resonances of a second order block operator matrix and an associated λ-rational Sturm-Liouville problem. Proceedings A: Mathematical, Physical and Engineering Sciences, 460 (2052). pp. 3403-3420. ISSN 1471-2962 (https://doi.org/10.1098/rspa.2003.1272)

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Abstract

This paper studies the resonances and points of spectral concentration of the block operator matrix $$\egin{pmatrix} -\frac{d^2}{d x^2}+q & \sqrt{tw} \\ \sqrt{tw} & u \end{pmatrix} $$ in the space $L^2(0,1) \oplus L^2(0,1)$. In particular we study the dynamics of the resonance/eigenvalue λ(t), showing that an embedded eigenvalue can evolve into a resonance and that eigenvalues which are absorbed by the essential spectrum give rise to resonance points. A connection is also established between resonances and points of spectral concentration. Finally, some numerical examples are given which show that each of the above theoretical possibilities can be realized.

ORCID iDs

Brown, B.Malcolm, Langer, M. ORCID logoORCID: https://orcid.org/0000-0001-8813-7914 and Marletta, Marco;
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