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

B.Malcolm Brown, M. Langer, Marco Marletta

Research output: Contribution to journalArticle

3 Citations (Scopus)

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.
LanguageEnglish
Pages3403-3420
Number of pages18
JournalProceedings A: Mathematical, Physical and Engineering Sciences
Volume460
Issue number2052
DOIs
Publication statusPublished - Dec 2004

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Operator Matrix
Sturm-Liouville Problem
Block Matrix
operators
P-matrix
eigenvalues
Eigenvalue
Essential Spectrum
L-space
Numerical Examples

Keywords

  • resonance
  • spectral concentration
  • embedded eigenvalue
  • block operator matrix
  • λ-rational eigenvalue problem

Cite this

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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.",
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Spectral concentrations and resonances of a second order block operator matrix and an associated λ-rational Sturm-Liouville problem. / Brown, B.Malcolm; Langer, M.; Marletta, Marco.

In: Proceedings A: Mathematical, Physical and Engineering Sciences, Vol. 460, No. 2052, 12.2004, p. 3403-3420.

Research output: Contribution to journalArticle

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AB - 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.

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