Hostname: page-component-cd9895bd7-mkpzs Total loading time: 0 Render date: 2024-12-27T11:38:13.454Z Has data issue: false hasContentIssue false
  • English
  • Français

A non elliptic spectral problem related to the analysisof superconducting micro-strip lines

Published online by Cambridge University Press:  15 August 2002

Anne-Sophie Bonnet-Bendhia  and
Karim Ramdani
Anne-Sophie Bonnet-Bendhia
Affiliation:
Laboratoire SMP, ENSTA, URA 853 du CNRS, 32 Boulevard Victor, 75739 Paris Cedex 15, France. [email protected].
Karim Ramdani
Affiliation:
Projet CORIDA, INRIA Lorraine, Campus Scientifique, BP 239, 54506 Vandœuvre-les-Nancy, France. [email protected].
Get access

Abstract

This paper is devoted to the spectral analysis of a non elliptic operator A , deriving from the study of superconducting micro-strip lines.Once a sufficient condition for the self-adjointness of operator A has been derived, we determine its continuous spectrum. Then, we show that A is unbounded from below and that it has a sequence of negative eigenvalues tending to -∞. Using the Min-Max principle, a characterization ofits positive eigenvalues is given. Thanks to this characterization, some conditions on the geometrical (large width) and physical (large dielectric permittivityin modulus) properties of the strip that ensure the existence of positive eigenvalues are derived. Finally, we analyze the asymptotic behavior of the eigenvaluesof A as the dielectric permittivity of the strip goes to -∞.

Keywords

Superconducting transmission lineswave-guidesself-adjointnessspectral analysisnon elliptic operators.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 36 , Issue 3 , May 2002 , pp. 461 - 487
Copyright
© EDP Sciences, SMAI, 2002

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Bamberger, A. and Bonnet, A.-S., Mathematical Analysis of the Guided Modes of an Optical Fiber. SIAM J. Math. Anal. 21 (1990) 1487-1510. CrossRef
Bonnet-Bendhia, A.-S., Caloz, G. and Mahé, F., Guided Modes of Integrated Optical Guides. A Mathematical Study. IMA J. Appl. Math. 60 (1998) 225-261. CrossRef
Bonnet-Bendhia, A.-S., Dauge, M. and Ramdani, K., Analyse Spectrale et Singularités d'un Problème de Transmission non Coercif. C. R. Acad. Sci. Paris Sér. I 328 (1999) 717-720. CrossRef
Bonnet-Bendhia, A.-S., Duterte, J. and Joly, P., Mathematical Analysis of Elastic Surface Waves in Topographic Waveguides. Math. Models Methods Appl. Sci. 9 (1999) 755-798.
Bonnet-Bendhia, A.-S. and Ramdani, K., Mathematical Analysis of Conducting and Superconducting Transmission Lines. SIAM J. Appl. Math. 60 (2000) 2087-2113.
J.-M. Cognet, Étude des modes guidés dans une ligne supraconductrice : le cas monodimensionnel. Rapport Interne n°295, ENSTA, Paris (1997).
R.E. Collins, Foundations for microwave engineering. Mc Graw-Hill Inc. (1992).
Joly, P. and Poirier, C., Mathematical Analysis of Electromagnetic Open Wave-guides. RAIRO Modèl. Math. Anal. Numér. 29 (1995) 505-575. CrossRef
Ma, J.G. and Wolff, I., Modeling the Microwave Properties of Superconductors. IEEE Trans. Microwave Theory Tech. 43 (1995) 1053-1059.
D. Marcuse, Theory of Dielectric Optical Waveguide. Academic Press, New-York (1974).
Mei, K.K and Liang, G., Electromagnetics of Superconductors. IEEE Trans. Microwave Theory Tech. 44 (1991) 1545-1552. CrossRef
A.D. Olver, Microwave and Optical Transmission. J. Wiley & Sons Ed. (1992).
K. Ramdani, Lignes Supraconductrices : Analyse Mathématique et Numérique. Ph.D. thesis, University of Paris VI, France (1999).
M. Reed and B. Simon, Methods of Modern Physics, Analysis of Operators. Academic Press (1980).