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UID:7858@i2m.univ-amu.fr
DTSTART;TZID=Europe/Paris:20160323T153000
DTEND;TZID=Europe/Paris:20160323T163000
DTSTAMP:20241120T205553Z
URL:https://www.i2m.univ-amu.fr/evenements/groupe-de-travail-guide-d-ondes
 -milieux-stratifies-et-problemes-inverses-goms-3/
SUMMARY: (...): Groupe de Travail Guide d'ondes\, milieux stratifiés et pr
 oblèmes inverses (GOMS)
DESCRIPTION:: {\\bf Periodic differential operators with predefined spectra
 l gaps}\\\\It is well-known that the spectrum of self-adjoint periodic dif
 ferential operators has aband structure\, i.e. it is a locally finite unio
 n of compact intervals called \\textit{bands}. In general the bands may ov
 erlap. The bounded open interval $(a\,b)\\subset\\mathbb{R}$ is called a \
 \textit{gap} in the spectrum of the operator $\\mathcal{H}$ if $(a\,b)\\ca
 p\\mathcal{H}=\\emptyset$ and $a\,b\\in\\sigma(\\mathcal{H})$.The presence
  of gaps in the spectrum is not guaranteed: for example\, the spectrum of 
 the Laplacian in $L^2(\\mathbb{R}^n)$ has no gaps\, namely $\\sigma(-\\Del
 ta_{\\mathbb{R}^n})=[0\,\\infty)$. Therefore the natural problem is aconst
 ruction 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 \\cite{HP}\, where  a lot 
 of examples are discussed in detail.Another important question arising her
 e 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
  \\cite{1\,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 s
 pectral gaps are close (in some natural sense) to predefined intervals. \\
 begin{thebibliography}{99}\\bibitem{HP} R. Hempel\, O. Post\, Spectral Gap
 s for Periodic Elliptic Operatorswith High Contrast: an Overview\, Progres
 s in Analysis\, Proceedingsof the 3rd International ISAAC Congress Berlin 
 2001\, Vol. 1\,577-587\, 2003\; arXiv:math-ph/0207020.\\bibitem{1}A. Khrab
 ustovskyi\, Periodic Riemannian manifold with preassigned gaps in spectrum
  of Laplace-Beltrami operator\, {Journal of Differential Equations\, 252(3
 ) (2012)\, 2339--2369.}\\bibitem{2}A. Khrabustovskyi\, Periodic elliptic o
 perators with asymptotically preassigned spectrum\, {Asymptotic Analysis\,
  82(1-2) (2013)\, 1-37.}\\bibitem{3}A. Khrabustovskyi\,Opening up and cont
 rol of spectral gaps of the Laplacian in periodic domains\,{Journal of Mat
 hematical Physics\, 55(12) (2014)\, 121502.}\\bibitem{4}D. Barseghyan\, A.
  Khrabustovskyi\,{Gaps in the spectrum of a periodic quantum graph with pe
 riodically distributed $\\delta'$-type interactions}\, Journal of Physics 
 A: Mathematical and Theoretical\, 48(25) (2015)\, 255201.\\end{thebibliogr
 aphy} 
CATEGORIES:Groupe de travail,Analyse Spectrale et Problèmes Inverses
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