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A non elliptic spectral problem related to the analysis of superconducting micro-strip lines. (English) Zbl 1070.35503
Summary: 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 $$-\infty$$. 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 $$-\infty$$.

##### MSC:
 35P20 Asymptotic distributions of eigenvalues in context of PDEs 47B25 Linear symmetric and selfadjoint operators (unbounded) 82D55 Statistical mechanical studies of superconductors 78A50 Antennas, waveguides in optics and electromagnetic theory
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