Essential spectrum of a system of singular differential operators and the asymptotic Hain–Lüst operator
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- by Reinhard Mennicken, Serguei Naboko and Christiane Tretter PDF
- Proc. Amer. Math. Soc. 130 (2002), 1699-1710 Request permission
Abstract:
We consider a matrix differential operator with singular entries which arises in magnetohydrodynamics. By means of the asymptotic Hain-Lüst operator and some pseudo-differential operator techniques, we determine the essential spectrum of this operator. Whereas in the regular case, the essential spectrum consists of two intervals, it turns out that in the singular case two additional intervals due to the singularity may arise. In addition, we establish criteria for the essential spectrum to lie in the left half-plane.References
- Jean Descloux and Giuseppe Geymonat, Sur le spectre essentiel d’un opérateur relatif à la stabilité d’un plasma en géométrie toroïdale, C. R. Acad. Sci. Paris Sér. A-B 290 (1980), no. 17, A795–A797 (French, with English summary). MR 580568
- Melvin Faierman, Reinhard Mennicken, and Manfred Möller, The essential spectrum of a system of singular ordinary differential operators of mixed order. I. The general problem and an almost regular case, Math. Nachr. 208 (1999), 101–115. MR 1719791, DOI 10.1002/mana.3212080105
- Melvin Faierman, Reinhard Mennicken, and Manfred Möller, The essential spectrum of a system of singular ordinary differential operators of mixed order. II. The generalization of Kako’s problem, Math. Nachr. 209 (2000), 55–81. MR 1734359, DOI 10.1002/(SICI)1522-2616(200001)209:1<55::AID-MANA55>3.3.CO;2-N
- Volker Hardt, Reinhard Mennicken, and Serguei Naboko, Systems of singular differential operators of mixed order and applications to $1$-dimensional MHD problems, Math. Nachr. 205 (1999), 19–68. MR 1709162, DOI 10.1002/mana.3212050103
- Volker Hardt and Ekkehard Wagenführer, Spectral properties of a multiplication operator, Math. Nachr. 178 (1996), 135–156. MR 1380708, DOI 10.1002/mana.19961780108
- T. Kako, On the absolutely continuous spectrum of MHD plasma confined in the flat torus, Math. Methods Appl. Sci. 7 (1985), no. 4, 432–442. MR 827203, DOI 10.1002/mma.1670070130
- Takashi Kako, Essential spectrum of linearized operator for MHD plasma in cylindrical region, Z. Angew. Math. Phys. 38 (1987), no. 3, 433–449 (English, with French summary). MR 894249, DOI 10.1007/BF00944961
- Tosio Kato, Perturbation theory for linear operators, Classics in Mathematics, Springer-Verlag, Berlin, 1995. Reprint of the 1980 edition. MR 1335452, DOI 10.1007/978-3-642-66282-9
- Kurasov, P., Naboko, S.N., On the spectrum of a class of singular matrix-differential operators. Quasiregularity conditions and essential self-adjointness, to be submitted to Trans. Amer. Math. Soc.
- Yoshida, K., Functional Analysis, Springer-Verlag, Berlin, Heidelberg, New York, 1968.
Additional Information
- Reinhard Mennicken
- Affiliation: NWF I – Mathematik, University of Regensburg, D-93040 Regensburg, Germany
- Email: reinhard.mennicken@mathematik.uni-regensburg.de
- Serguei Naboko
- Affiliation: Department of Mathematical Physics, Institute for Physics, St. Petersburg University, ul. Ulianovskaja 1, 198904 St. Petergoff, St. Petersburg, Russia
- Email: naboko@snoopy.phys.spbu.ru
- Christiane Tretter
- Affiliation: Department of Mathematics and Computer Science, University of Leicester, University Road, Leicester LE1 7RH, United Kingdom
- Email: c.tretter@mcs.le.ac.uk
- Received by editor(s): May 3, 2000
- Received by editor(s) in revised form: December 5, 2000
- Published electronically: December 20, 2001
- Additional Notes: The authors acknowledge the support of the Regensburger Universitätsstiftung Hans Vielberth.
- Communicated by: Joseph A. Ball
- © Copyright 2001 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 130 (2002), 1699-1710
- MSC (1991): Primary 47A10, 47B25, 76W05
- DOI: https://doi.org/10.1090/S0002-9939-01-06239-6
- MathSciNet review: 1887017