Harmonic analysis and random schrödinger operators

Abstract

This survey is based on a series of lectures given during the School on Random Schrödinger Operators and the International Conference on Spectral Theory and Mathematical Physics at the Pontificia Universidad Catolica de Chile, held in Santiago in November 2014. As the title suggests, the presented material has two foci: Harmonic analysis, more precisely, unique continuation properties of several natural function classes and Schrödinger operators, more precisely properties of their eigenvalues, eigenfunctions and solutions of associated differential equations. It mixes topics from (rather) pure to (rather) applied mathematics, as well as classical questions and results dating back a whole century to very recent and even unpublished ones. The selection of material covered is based on the selection made for the minicourse, and is certainly a personal choice corresponding to the research interests of the authors. Emphasis is laid not so much on proofs, but rather on concepts, questions, results, examples and applications. In several cases, however, we do supply proofs of special cases or sketches of proofs, and use them to illustrate the underlying concepts. As the minicourse Harmonic Analysis and Random Schrödinger Operators itself, we designed the text to be accessible to advanced graduate students who have already acquired some experience with partial differential equations. On the other hand, even experts in the field will find new results, mostly toward the end of the text. The line of thought starts with discussing unique continuation properties of holomorphic and harmonic functions. Already here we illustrate different notions of unique continuation. Hereafter, elliptic partial differential equations are introduced and unique continuation properties of their solutions are discussed. Then we shift our attention to domains and differential equations with an inherent multiscale structure. The question here is, whether appropriately collected local data of a function give good estimates to global properties of the function. In the framework of harmonic analysis the Whittaker–Nyquist– Kotelnikov–Shannon Sampling and the Logvinenko–Sereda Theorem are examples of such results. From here it is natural to pursue the question whether similar and related results can be expected for (classes of) solutions of differential equations. This leads us to quantitative unique countinuation bounds which are obtained by the use of Carleman estimates. In the context of random Schrödinger operators they have risen to some prominence recently since they facilitated the resolution of some long-standing problems in the field. We present several unique continuation theorems tailored for this applications. Finally, after several results on the spectral properties of random Schrödinger operators, an application to control of the heat equation is given.