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THE ESSENTIAL SPECTRUM OF A PERTURBED OPERATOR ARISING IN TWO-DIMENSIONAL MAGNETOHYDRODYNAMICS

Part of: Close-to-elliptic equations and systems

Published online by Cambridge University Press:  16 April 2010

M. FAIERMAN  and
R. MENNICKEN
M. FAIERMAN*
Affiliation:
School of Mathematics and Statistics, University of New South Wales, UNSW Sydney NSW 2052, Australia (email: [email protected])
R. MENNICKEN
Affiliation:
Department of Mathematics, University of Regensburg, D-93040 Regensburg, Germany (email: [email protected])
*
✉For correspondence; e-mail: [email protected]
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Abstract

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Descloux and Geymonat considered a model problem in two-dimensional magnetohydrodynamics and conjectured that the essential spectrum has an explicitly given band structure. This conjecture was recently proved by Faierman, Mennicken, and Möller by reducing the problem to that for a 2×2 block operator matrix. In a subsequent paper Faierman and Mennicken investigated the essential spectrum for the problem arising from a particular type of perturbation of precisely one of the operator entries in the matrix representation cited above of the original problem considered by Descloux and Geymonat. In this paper we extend the results of that work by investigating the essential spectrum for the problem arising from particular types of perturbations of all but one of the aforementioned operators. It remains an open question whether one can perturb the exceptional operator in such a way as to leave the essential spectrum unchanged.

Keywords

essential spectrumperturbed operatormagnetohydrodynamics

MSC classification

Secondary: 35H99: None of the above, but in this section
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 88 , Issue 2 , April 2010 , pp. 169 - 182
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2010

References

[1]Descloux, J. and Geymonat, G., ‘On the essential spectrum of an operator relative to the stability of a plasma in toroidal geometry’, École Polytechnique Fédérale de Lausanne (1979).Google Scholar
[2]Descloux, J. and Geymonat, G., ‘Sur le spectre essentiel d’un opérateur relatif à la stabilité d’un plasma en géométrie toroïdale’, C. R. Acad. Sci. Paris, Sér. A 290 (1980), 795–797.Google Scholar
[3]Faierman, M. and Mennicken, R., ‘The essential spectrum of an operator arising in two-dimensional magnetohydrodynamics’, Math. Nachr. 282 (2009), 422–429.CrossRefGoogle Scholar
[4]Faierman, M., Mennicken, R. and Möller, M., ‘The essential spectrum of a model problem in two-dimensional magnetohydrodynamics: a proof of a conjecture of J. Descloux and G. Geymonat’, Math. Nachr. 269–270 (2004), 129–149.CrossRefGoogle Scholar
[5]Gohberg, I. and Kreĭn, M. G., ‘Systems of integral equations on a half line with kernels depending on the difference of arguments’, Amer. Math. Soc. Transl. (2) 14 (1960), 217–287.Google Scholar
[6]Kato, T., Perturbation Theory for Linear Operators (Springer, New York, 1976).Google Scholar