Given a self-adjoint bounded operator, its spectrum is a compact subset of the real numbers. The space of compact subsets of the real numbers is naturally equipped with the Hausdorff metric. Let $T$ be a topological (metric) space and $(A_t)$ be a family of self-adjoint, bounded operators. In the talk, we study the (Hölder-)continuity of the map assigning to each $t\in T$ the spectrum of the operator $A_t$.
As application, we study Schroedinger operators over dynamical systems. Using the previous characterization, we show that the spectra converges if and only if the underlying dynamical systems converge in a suitable topology. This has a wide range of applications for numerical as well as analytic questions. A particular focus is put on Schrödinger operators arising by quasi crystals.
Wed, 01/11/2017 - 12:00 to 13:00