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

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

{\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.


\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.

A. Khrabustovskyi, Periodic Riemannian manifold with preassigned gaps in spectrum of Laplace-Beltrami operator, {Journal of Differential Equations, 252(3) (2012), 2339–2369.}

A. Khrabustovskyi, Periodic elliptic operators with asymptotically preassigned spectrum, {Asymptotic Analysis, 82(1-2) (2013), 1-37.}

A. Khrabustovskyi,
Opening up and control of spectral gaps of the Laplacian in periodic domains,
{Journal of Mathematical Physics, 55(12) (2014), 121502.}

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.



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

Date(s) : 23/03/2016
14 h 00 min - 15 h 00 min

« Uniform resolvent convergence for a strip with fast oscillating boundary »
In a planar infinite strip with a fast oscillating boundary we consider an elliptic operator assuming that both the period and the amplitude of the oscillations are small. On the oscillating boundary we impose Dirichlet, Neumann or Robin boundary condition. In all cases we describe the homogenized operator, establish the uniform resolvent convergence of the perturbed resolvent to the homogenized
one, and prove the estimates for the rate of convergence. These results are obtained as the order of the amplitude of the oscillations is less, equal or greater than that of the period. It is shown that under the homogenization the type of the boundary condition can change


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

Date(s) : 03/12/2015 - 04/12/2015
13 h 30 min - 18 h 00 min

Le troisième workshop « Problèmes inverses et domaines associés » aura lieu du jeudi 3 Décembre au Vendredi 4 Décembre 2015 dans les locaux de la Frumam.
Le programme sera disponible en ligne très rapidement.


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

Date(s) : 04/06/2014
16 h 00 min - 17 h 00 min

Titre: Eigenvalue statistics for some one-dimensional random Schrodinger operators

resume: One-dimensional random Schrodinger operators on the lattice exhibit various local eigenvalue statistics depending on the rate of decay of the randomness. For example, if the disorder is scaled with the length of the interval on which the Hamiltonian is restricted, then in the limit of infinite length, the eigenvalue statistics varies from Poisson to clock as the power of the scaling varies from zero to one. The critical case of one-half was treated by Kritchevski, Valko, and Virag. The eigenvalue statistics for the other values of the scaling will be presented and are joint work with F. Klopp.


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