When do the spectra of self-adjoint operators converge?

24.04.2018, 13:00  –  Haus 9, Raum 2.22

Siegfried Beckus (Technion Haifa, Israel)

The central object of the talk are operator families indexed over a topological space where each single operator is self-adjoint and bounded. Thus, the spectrum of each single operator is a compact subset of the real numbers. Consequently, the distance of two of their spectra is naturally measured with respect to the Hausdorff metric on the compact subsets of the real numbers. During the talk, a characterization of the continuous variation of the spectra in the index is provided and proven. This result has various applications in order to develop an approximation theory for the corresponding operators. In view of this, we will focus in the second part of the talk on Schrödinger operators associated with dynamical systems or graphs that are considered in solid state physics. Based on the previously discussed characterization, one can show that the spectra of such Schrödinger operators converge if the underlying dynamical systems converge in a suitable topology. The talk is based on joint works with Jean Bellissard and Giuseppe de Nittis.

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