Upcoming Talks

KOLLOQUIUMSVORTRAG im Vorfeld eines Habilitationsantrags

Title: Spectral and asymptotic properties of periodic media
Speaker: Andrii KHRABUSTOVSKYI (Institut für Angewandte Mathematik)
Date: Donnerstag, 15.3.2018, 14:00 Uhr
Room: TU Graz, Seminarraum AE02, Steyrergasse 30, EG
Abstract:

It is well-known that the spectrum of self-adjoint periodic differential operators has the form of a locally finite union of compact intervals called bands. In general the bands may overlap. A bounded open interval (a,b) is called a gap in the spectrum σ(H) of the operator H if (a,b)∩σ(H)=Ø with a,b belonging to σ(H).

The presence of gaps in the spectrum is not guaranteed: for example, the spectrum of the Laplacian in ℝn has no gaps. 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.

Another interesting question arising in this area is how to control the location of spectral gaps via a suitable choice of coefficients of the underlying operators or/and via a suitable choice of geometric parameters of the medium. In the talk we give an overview of the results, 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.