Harmonic analysis techniques in spectral theory

The circle of problems around the "Fourier restriction conjecture" has been a driving force behind the development of modern harmonic analysis.
Recent years have seen exciting advances that have lead to spectacular applications in PDE, combinatorics, geometric analysis and number theory. However, most applications in spectral theory are still based on the classical Stein-Tomas theorem. It is currently not understood if and how the state-of-the-art advances in harmonic analysis (such as decoupling, polynomial partitioning, multilinear methods and the Bourgain--Guth induction on scales) can be applied to spectral theory. The research programme aims to address this issue by developing a new approach that combines modern harmonic analysis techniques with classical functional and complex analysis.

These techniques are expected to find applications in a variety of spectral problems involving differential operators. The focus is on Schrödinger operators with complex potentials, a subject of considerable contemparary interest. The techniques to deal with these operators are limited, mainly due to a lack of the powerful spectral theorem and variational methods. Harmonic analysis tools have turned out to be extremely fruitful in this regard and have been instrumental in solving major open problems in the field. In a broad sense, one would like to understand how much of the classical theory of Schrödinger operators with real-valued potentials survives for complex-valued potentials; the answer in many cases is "not very much". Complex potentials appear naturally in many different problems, for instance, local solvability of PDE, the Kramers-Fokker-Planck equation, the damped wave equation, the study of resonances and in the stability analysis of nonlinear PDE. Their spectral properties can be very wild and unintuitive, and they are still far from being completely understood.