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 journal › Article

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

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

Brown, B.Malcolm ; Langer, M. ; Marletta, Marco. / Spectral concentrations and resonances of a second order block operator matrix and an associated λ-rational Sturm-Liouville problem. In: Proceedings A: Mathematical, Physical and Engineering Sciences. 2004 ; Vol. 460, No. 2052. pp. 3403-3420.
@article{c0a9637f52324c17a516a14284612657,
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.",
keywords = "resonance, spectral concentration, embedded eigenvalue, block operator matrix, λ-rational eigenvalue problem",
author = "B.Malcolm Brown and M. Langer and Marco Marletta",
year = "2004",
month = "12",
doi = "10.1098/rspa.2003.1272",
language = "English",
volume = "460",
pages = "3403--3420",
journal = "Proceedings A: Mathematical, Physical and Engineering Sciences",
issn = "1364-5021",
publisher = "Royal Society of London",
number = "2052",

}

Brown, BM, Langer, M & Marletta, M 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, vol. 460, no. 2052, pp. 3403-3420. https://doi.org/10.1098/rspa.2003.1272

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 journal › Article

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

Y1 - 2004/12

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

U2 - 10.1098/rspa.2003.1272

DO - 10.1098/rspa.2003.1272

M3 - Article

VL - 460

SP - 3403

EP - 3420

JO - Proceedings A: Mathematical, Physical and Engineering Sciences

T2 - Proceedings A: Mathematical, Physical and Engineering Sciences

JF - Proceedings A: Mathematical, Physical and Engineering Sciences

SN - 1364-5021

IS - 2052

ER -

Brown BM, Langer M, Marletta M. 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. 2004 Dec;460(2052):3403-3420. https://doi.org/10.1098/rspa.2003.1272

Access to Document