Quantitative unique continuation and applications

Abstract

This thesis treats quantitative unique continuation principles for functions in spectal subspaces of Schrödinger operators. The first main theorem improves and generalizes several existing results in this field. It treats all finite energy spectral subspaces and Schrödinger operators on bounded as well as on unbounded unbounded domains. The appearing constant is scale-free and uniform over a large class of geometric configuations. Furthermore, the issue of the optimality of its dependece on the energy and on properties of the Schrödinger operators is discussed.

The first application are lower bounds on the movement of spectra - in particular of the essential spectrum - of Schröodinger operators under particular non-negative perturbations. For that purpose, abstract results on perturbations of spectra of self-adjoint operators are developed which might be interesting in a broader context.

The second application is about random Schrödinger operators. A Wegner estimate, an important step in proving Anderson localization, is established for new classes of such operators. A particular example is the random breather model where the random potential consists of characteristic functions of balls with random radii. Furthermore, Wegner estimates for so-called crooked magnetic alloy-type operators with bounded magnetic potential and for the Landau-breather model are proved.

The last application concerns control theory for equations of heat-type with interior control. First, in an abstract framework, null-controllability of some Cauchy problems with explicit estimates on the control cost at all times is proved. The resulting estimate on the control cost is - to our knowledge - the best one with respect to the existing literature. Then, combining this with quantitative unique continuation principles, we obtain explicit estimates on the control cost of heat-type equations on bounded and unbounded domains at all times. This result in this quantitative form is new even for the classic heat equation and enables to study asymptotics of the control cost in the homogenization and the complementary regime.