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Spectral Concentrations and Resonances of a Second-Order Block Operator Matrix and an Associated λ-Rational Sturm-Liouville Problem

B. Malcolm Brown, Matthias Langer and Marco Marletta
Proceedings: Mathematical, Physical and Engineering Sciences
Vol. 460, No. 2052 (Dec. 8, 2004), pp. 3403-3420
Published by: Royal Society
Stable URL: http://www.jstor.org/stable/4143246
Page Count: 18
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Since scans are not currently available to screen readers, please contact JSTOR User Support for access. We'll provide a PDF copy for your screen reader.
Spectral Concentrations and Resonances of a Second-Order Block Operator Matrix and an Associated λ-Rational Sturm-Liouville Problem
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Abstract

This paper studies the resonances and points of spectral concentration of the block operator matrix $\pmatrix{-{d^2\over dx^2}+ q&\sqrt{tw} \cr\sqrt{tw}&u}$ in the space $L^2(0,1) \bigoplus L^2(0,1)$. 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.

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