Number Theory Related to Quantum Chaos and Dynamics

The goal of this project is to study number theoretical questions related to mathematical physics and dynamical systems, and potentially prove/disprove conjectures by the physics community. Quantum Chaos asks how chaos in classical systems manifests itself in terms of quantum mechanics.

E.g., can chaos be detected in the statistical behavior of the gaps between eigenvalues (which can be interpreted as energy levels) of quantized Hamiltonians?

Pursuing this question for some arithmetic systems leads to intriguing questions regarding "deterministic randomness" - even though the sequence of eigenvalues is determined by simple rules, in many ways it behaves as a truly random sequence. Another question is properties of eigenfunctions of quantized Hamiltonians.

If the classical dynamics are ergodic, then most eigenfunctions should be equidistributed in a certain sense.

Other fingerprints of chaos is the distribution of zero sets (nodal lines) of eigenfunctions, to be compared with predictions given by Berry´s Random Wave model (RWM) for chaotic eigenfunctions.

An interesting class of models is given by Laplace eigenfunctions on tori when perturbed by delta potentials; the aritmetic nature of the unperturbed Laplace multiplicities gives rise to some subtle and interesting number theoretical questions.

The project will be carried out by the PI and a PhD student.Rigorous proof of level repulsion, or validy of Berry´s RWM for deterministic eigenfunctions, would be major breakthroughs.