# 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.
Original language English 3403-3420 18 Proceedings A: Mathematical, Physical and Engineering Sciences 460 2052 https://doi.org/10.1098/rspa.2003.1272 Published - Dec 2004

### Fingerprint

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

TY - JOUR

T1 - Spectral concentrations and resonances of a second order block operator matrix and an associated λ-rational Sturm-Liouville problem

AU - Brown, B.Malcolm

AU - Langer, M.

AU - Marletta, Marco

PY - 2004/12

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N2 - 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.

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.

KW - resonance

KW - spectral concentration

KW - embedded eigenvalue

KW - block operator matrix

KW - λ-rational eigenvalue problem

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