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

Anne-Sophie Bonnet-Bendhia, Karim Ramdani (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

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...

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

Anne-Sophie Bonnet-Bendhia, Karim Ramdani (2002)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

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...

Accurate Spectral Asymptotics for periodic operators

Victor Ivrii (1999)

Journées équations aux dérivées partielles

Asymptotics with sharp remainder estimates are recovered for number 𝐍 ( τ ) of eigenvalues of operator A ( x , D ) - t W ( x , x ) crossing level E as t runs from 0 to τ , τ . Here A is periodic matrix operator, matrix W is positive, periodic with respect to first copy of x and decaying as second copy of x goes to infinity, E either belongs to a spectral gap of A or is one its ends. These problems are first treated in papers of M. Sh. Birman, M. Sh. Birman-A. Laptev and M. Sh. Birman-T. Suslina.

Almost sure Weyl asymptotics for non-self-adjoint elliptic operators on compact manifolds

William Bordeaux Montrieux, Johannes Sjöstrand (2010)

Annales de la faculté des sciences de Toulouse Mathématiques

In this paper, we consider elliptic differential operators on compact manifolds with a random perturbation in the 0th order term and show under fairly weak additional assumptions that the large eigenvalues almost surely distribute according to the Weyl law, well-known in the self-adjoint case.

Asymptotic distribution of eigenfrequencies for damped wave equations

Johannes Sjöstrand (2000)

Journées équations aux dérivées partielles

Il est bien connu que les fréquences propres associées à un d'Alembertien amorti sont confinées dans une bande parallèle à l'axe réel. Nous rappelons l'asymptotique de Weyl pour la distribution des parties réelles des fréquences propres, nous montrons que «presque toutes» les fréquences propres appartiennent à une bande déterminée par la limite de Birkhoff du coefficient d'amortissement. Nous montrons aussi que certaines moyennes des parties imaginaires convergent vers la moyenne du coefficient...

Asymptotic distribution of negative eigenvalues for two dimensional Pauli operators with nonconstant magnetic fields

Akira Iwatsuka, Hideo Tamura (1998)

Annales de l'institut Fourier

This article studies the asymptotic behavior of the number N ( λ ) of the negative eigenvalues < - λ as λ + 0 of the two dimensional Pauli operators with electric potential V ( x ) decaying at and with nonconstant magnetic field b ( x ) , which is assumed to be bounded or to decay at . In particular, it is shown that N ( λ ) = ( 1 / 2 π ) V ( x ) > λ b ( x ) d x ( 1 + o ( 1 ) ) , when V ( x ) decays faster than b ( x ) under some additional conditions.

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