ANDRII KHRABUSTOVSKYI – Groupe de Travail Guide d’ondes, milieux stratifiés et problèmes inverses (GOMS)

Carte non disponible

Date/heure
Date(s) - 23/03/2016
15 h 30 min - 16 h 30 min

Catégories Pas de Catégories


\bf Periodic differential operators with predefined spectral gaps\ It is well-known that the spectrum of self-adjoint periodic differential operators has aband structure, i.e. it is a locally finite union of compact intervals called \textitbands. In general the bands may overlap. The bounded open interval is called a \textitgap in the spectrum of the operator if and . The presence of gaps in the spectrum is not guaranteed : for example, the spectrum of the Laplacian in has no gaps, namely . Therefore the natural problem is aconstruction of periodic operators with non-void spectral gaps. The importance of this problem is caused by various applications, for example in physics of photonic crystals. We refer to the overview \citeHP, where a lot of examples are discussed in detail. Another important question arising here is how to control the location of the gaps via a suitable choice of the coefficients of the operators or/and via a suitable choice of the geometry of the medium. In the talk we give an overview of the results obtained in \cite1,2,3,4, where this problem is studied for various classes of periodic differential operators. In a nutshell, our goal is to construct an operator (from some given class of periodic operators) such that its spectral gaps are close (in some natural sense) to predefined intervals. \beginthebibliography99 \bibitemHP R. Hempel, O. Post, Spectral Gaps for Periodic Elliptic Operatorswith High Contrast : an Overview, Progress in Analysis, Proceedingsof the 3rd International ISAAC Congress Berlin 2001, Vol. 1,577-587, 2003 ; arXiv:math-ph/0207020. \bibitem1A. Khrabustovskyi, Periodic Riemannian manifold with preassigned gaps in spectrum of Laplace-Beltrami operator, Journal of Differential Equations, 252(3) (2012), 2339—2369. \bibitem2A. Khrabustovskyi, Periodic elliptic operators with asymptotically preassigned spectrum, Asymptotic Analysis, 82(1-2) (2013), 1-37. \bibitem3A. Khrabustovskyi,Opening up and control of spectral gaps of the Laplacian in periodic domains,Journal of Mathematical Physics, 55(12) (2014), 121502. \bibitem4D. Barseghyan, A. Khrabustovskyi,Gaps in the spectrum of a periodic quantum graph with periodically distributed -type interactions, Journal of Physics A : Mathematical and Theoretical, 48(25) (2015), 255201. \endthebibliography[

ANDRII KHRABUSTOVSKYI – Groupe de Travail Guide d’ondes, milieux stratifiés et problèmes inverses (GOMS)

Carte non disponible

Date/heure
Date(s) - 23/03/2016
15 h 30 min - 16 h 30 min

Catégories Pas de Catégories


\bf Periodic differential operators with predefined spectral gaps\ It is well-known that the spectrum of self-adjoint periodic differential operators has aband structure, i.e. it is a locally finite union of compact intervals called \textitbands. In general the bands may overlap. The bounded open interval is called a \textitgap in the spectrum of the operator if and . The presence of gaps in the spectrum is not guaranteed : for example, the spectrum of the Laplacian in has no gaps, namely . Therefore the natural problem is aconstruction of periodic operators with non-void spectral gaps. The importance of this problem is caused by various applications, for example in physics of photonic crystals. We refer to the overview \citeHP, where a lot of examples are discussed in detail. Another important question arising here is how to control the location of the gaps via a suitable choice of the coefficients of the operators or/and via a suitable choice of the geometry of the medium. In the talk we give an overview of the results obtained in \cite1,2,3,4, where this problem is studied for various classes of periodic differential operators. In a nutshell, our goal is to construct an operator (from some given class of periodic operators) such that its spectral gaps are close (in some natural sense) to predefined intervals. \beginthebibliography99 \bibitemHP R. Hempel, O. Post, Spectral Gaps for Periodic Elliptic Operatorswith High Contrast : an Overview, Progress in Analysis, Proceedingsof the 3rd International ISAAC Congress Berlin 2001, Vol. 1,577-587, 2003 ; arXiv:math-ph/0207020. \bibitem1A. Khrabustovskyi, Periodic Riemannian manifold with preassigned gaps in spectrum of Laplace-Beltrami operator, Journal of Differential Equations, 252(3) (2012), 2339—2369. \bibitem2A. Khrabustovskyi, Periodic elliptic operators with asymptotically preassigned spectrum, Asymptotic Analysis, 82(1-2) (2013), 1-37. \bibitem3A. Khrabustovskyi,Opening up and control of spectral gaps of the Laplacian in periodic domains,Journal of Mathematical Physics, 55(12) (2014), 121502. \bibitem4D. Barseghyan, A. Khrabustovskyi,Gaps in the spectrum of a periodic quantum graph with periodically distributed -type interactions, Journal of Physics A : Mathematical and Theoretical, 48(25) (2015), 255201. \endthebibliography[