Number Theory Related to Quantum Chaos and Dynamics

The goal is to study number theoretical questions related to mathematical physics and dynamical systems, and prove or disprove conjectures by the physics community.

Quantum Chaos is concerned with 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 ("energy levels") of quantized Hamiltonians?

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

For ergodic classical dynamics, 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 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 subtle and interesting number theoretical questions.

The research consists of several subprojects that will be carried out by the PI and a student or postdoc.Rigorous proof of level repulsion, or validy of Berry´s random wave model for deterministic eigenfunctions, would be major breakthroughs.