Groupe de Travail Guide d’ondes, milieux stratifiés et problèmes inverses (GOMS)
Date(s) : 23/03/2016 iCal
15h30 - 16h30
{\bf Periodic differential operators with predefined spectral gaps}\\
It is well-known that the spectrum of self-adjoint periodic differential operators has a
band structure, i.e. it is a locally finite union of compact intervals called \textit{bands}. In general the bands may overlap. The bounded open interval $(a,b)\subset\mathbb{R}$ is called a \textit{gap} in the spectrum of the operator $\mathcal{H}$ if $(a,b)\cap\mathcal{H}=\emptyset$ 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 $\sigma(-\Delta_{\mathbb{R}^n})=[0,\infty)$. Therefore the natural problem is a
construction 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 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 \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 spectral gaps are close (in some natural sense) to predefined intervals.
\begin{thebibliography}{99}
\bibitem{HP} R. Hempel, O. Post, Spectral Gaps for Periodic Elliptic Operators
with High Contrast: an Overview, Progress in Analysis, Proceedings
of the 3rd International ISAAC Congress Berlin 2001, Vol. 1,
577-587, 2003; arXiv:math-ph/0207020.
\bibitem{1}
A. Khrabustovskyi, 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 operators with asymptotically preassigned spectrum, {Asymptotic Analysis, 82(1-2) (2013), 1-37.}
\bibitem{3}
A. Khrabustovskyi,
Opening up and control of spectral gaps of the Laplacian in periodic domains,
{Journal of Mathematical Physics, 55(12) (2014), 121502.}
\bibitem{4}
D. Barseghyan, A. Khrabustovskyi,
{Gaps in the spectrum of a periodic quantum graph with periodically distributed $\delta’$-type interactions}, Journal of Physics A: Mathematical and Theoretical, 48(25) (2015), 255201.
\end{thebibliography}
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