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

Anne-Sophie Bonnet-Bendhia; Karim Ramdani

ESAIM: Mathematical Modelling and Numerical Analysis (2010)

  • Volume: 36, Issue: 3, page 461-487
  • ISSN: 0764-583X

Abstract

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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 of its positive eigenvalues is given. Thanks to this characterization, some conditions on the geometrical (large width) and physical (large dielectric permittivity in modulus) properties of the strip that ensure the existence of positive eigenvalues are derived. Finally, we analyze the asymptotic behavior of the eigenvalues of A as the dielectric permittivity of the strip goes to -∞.

How to cite

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Bonnet-Bendhia, Anne-Sophie, and Ramdani, Karim. "A non elliptic spectral problem related to the analysis of superconducting micro-strip lines." ESAIM: Mathematical Modelling and Numerical Analysis 36.3 (2010): 461-487. <http://eudml.org/doc/194112>.

@article{Bonnet2010,
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 of its positive eigenvalues is given. Thanks to this characterization, some conditions on the geometrical (large width) and physical (large dielectric permittivity in modulus) properties of the strip that ensure the existence of positive eigenvalues are derived. Finally, we analyze the asymptotic behavior of the eigenvalues of A as the dielectric permittivity of the strip goes to -∞. },
author = {Bonnet-Bendhia, Anne-Sophie, Ramdani, Karim},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {Superconducting transmission lines; wave-guides; self-adjointness; spectral analysis; non elliptic operators.; superconducting transmission lines; non elliptic operators},
language = {eng},
month = {3},
number = {3},
pages = {461-487},
publisher = {EDP Sciences},
title = {A non elliptic spectral problem related to the analysis of superconducting micro-strip lines},
url = {http://eudml.org/doc/194112},
volume = {36},
year = {2010},
}

TY - JOUR
AU - Bonnet-Bendhia, Anne-Sophie
AU - Ramdani, Karim
TI - A non elliptic spectral problem related to the analysis of superconducting micro-strip lines
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2010/3//
PB - EDP Sciences
VL - 36
IS - 3
SP - 461
EP - 487
AB - 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 of its positive eigenvalues is given. Thanks to this characterization, some conditions on the geometrical (large width) and physical (large dielectric permittivity in modulus) properties of the strip that ensure the existence of positive eigenvalues are derived. Finally, we analyze the asymptotic behavior of the eigenvalues of A as the dielectric permittivity of the strip goes to -∞.
LA - eng
KW - Superconducting transmission lines; wave-guides; self-adjointness; spectral analysis; non elliptic operators.; superconducting transmission lines; non elliptic operators
UR - http://eudml.org/doc/194112
ER -

References

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  1. A. Bamberger and A.-S. Bonnet, Mathematical Analysis of the Guided Modes of an Optical Fiber. SIAM J. Math. Anal.21 (1990) 1487-1510.  Zbl0729.35090
  2. A.-S. Bonnet-Bendhia, G. Caloz and F. Mahé, Guided Modes of Integrated Optical Guides. A Mathematical Study. IMA J. Appl. Math.60 (1998) 225-261.  Zbl0914.35130
  3. A.-S. Bonnet-Bendhia, M. Dauge and K. Ramdani, Analyse Spectrale et Singularités d'un Problème de Transmission non Coercif. C. R. Acad. Sci. Paris Sér. I328 (1999) 717-720.  Zbl0932.35153
  4. A.-S. Bonnet-Bendhia, J. Duterte and P. Joly, Mathematical Analysis of Elastic Surface Waves in Topographic Waveguides. Math. Models Methods Appl. Sci.9 (1999) 755-798.  Zbl0946.74034
  5. A.-S. Bonnet-Bendhia and K. Ramdani, Mathematical Analysis of Conducting and Superconducting Transmission Lines. SIAM J. Appl. Math.60 (2000) 2087-2113.  Zbl1136.78315
  6. J.-M. Cognet, Étude des modes guidés dans une ligne supraconductrice : le cas monodimensionnel. Rapport Interne n°295, ENSTA, Paris (1997).  
  7. R.E. Collins, Foundations for microwave engineering. Mc Graw-Hill Inc. (1992).  
  8. P. Joly and C. Poirier, Mathematical Analysis of Electromagnetic Open Wave-guides. RAIRO Modèl. Math. Anal. Numér.29 (1995) 505-575.  Zbl0834.35126
  9. J.G. Ma and I. Wolff, Modeling the Microwave Properties of Superconductors. IEEE Trans. Microwave Theory Tech.43 (1995) 1053-1059.  
  10. D. Marcuse, Theory of Dielectric Optical Waveguide. Academic Press, New-York (1974).  
  11. K.K Mei and G. Liang, Electromagnetics of Superconductors. IEEE Trans. Microwave Theory Tech.44 (1991) 1545-1552.  
  12. A.D. Olver, Microwave and Optical Transmission. J. Wiley & Sons Ed. (1992).  
  13. K. Ramdani, Lignes Supraconductrices : Analyse Mathématique et Numérique. Ph.D. thesis, University of Paris VI, France (1999).  
  14. M. Reed and B. Simon, Methods of Modern Physics, Analysis of Operators. Academic Press (1980).  Zbl0459.46001

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