Spectral geometry of intermediate quantum systems – SpInQS

The key objective of this research proposal is to develop and apply methods from number theory to study the geometry of eigenfunctions and the spectral properties of quantum sys- tems in a state of transition. The project will be of a duration of 4 years and consists of two main tasks: to prove delocalization for various types of random Schrödinger operators; and to investigate the structure of eigenfunctions of pseudo-integrable billiards.
The first task is to study the important problem of the Anderson transition for Schrödinger operators with random potentials which are popular models to study disordered quantum systems.

The spectral theory of such random Schrödinger operators is very active and a thriving area of research in Analysis. The main objective of this project is to apply completely new methods (lattice point problems in analytic number theory) to the setting of random Schrödinger operators. The key objective is to obtain a proof of the Anderson transition for Schrödinger operators with random delta potentials in dimension d=3 for a wide class of stochastic processes.

Further objectives are a generalization of these results to the general problem for smooth compactly supported potentials in a suitable energy range, to understand the critical case d=2 and to construct delocalized eigenfunctions.

The second task is concerned with the geometry of eigenfunctions in pseudo-integrable billiards. These are quantum billiards which are classically close to integrable yet show chaotic features at the quantum level, such as quantum ergodic eigenfunctions and level repulsion on the scale of the mean spacing. One example of such systems are rational polygonal billiards. Another example is Seba's billiard.

The first milestone of this project is to arrive at an answer as to the validity of Berry's random wave model for arithmetic Seba billiards (e. g. a square torus with a delta potential). This part of the project is designated as a research project for a postdoctoral researcher to be hired within the framework of this project for a duration of 24 months. A second objective is to obtain a proof of the existence of superscars – localized eigenfunctions at high energy – in rational polygonal quantum billiards by making a connection with the rich dynamics of flat surfaces with conic singularities which is currently a topic of much interest in the theory of dynamical systems.

Within the framework of this project I will also organise a summer school to be held at CIRM, Luminy, about problems in the field of quantum chaos which lie at the intersection of cutting edge research in spectral theory, number theory and dynamical systems. In particular, I plan to invite several of the leading mathematicians in the field from France and abroad. The goal will be to support the participation of young PhD students and postdocs from all over France in order to introduce them to the field, to disseminate the results of the most recent work and to encourage the next generation of mathematicians to begin their own work on current problems in quantum chaos.