Uncertainty Principles in Quantum Chaos2025.7.7
地点：数学高等研究院报告厅
时间：16:00 - 17:00
报告人：Semyon Dyatlov（麻省理工学院）
摘要: In quantum mechanics, the pure states of a particle are the eigenfunctions of the associated quantum Hamiltonian. A commonly studied setting is that of compact Riemannian manifolds, where the pure quantum states are the eigenfunctions of the Laplace-Beltrami operator. The behavior of eigenfunctions in the high energy limit (i.e. with the eigenvalue going to infinity) is known to be connected to the dynamics of the geodesic flow on the manifold.
A fundamental problem in quantum chaos is to understand localization of eigenfunctions when the geodesic flow has chaotic behavior. This can be done using semiclassical measures, which are limiting objects on the cotangent bundle of the manifold capturing concentration of a high energy sequence of eigenfunctions in the position/frequency space. The Quantum Ergodicity theorem, going back to the 1970s--80s, states that most eigenfunctions equidistribute, that is, converge to the volume measure. The Quantum Unique Ergodicity (QUE) conjecture claims that the entire sequence of eigenfunctions equidistributes, that is the only semiclassical measure is the volume measure.
In this talk I will discuss two partial results towards the QUE conjecture. Each of these says that every semiclassical measure is delocalized: one is the statement that the measure has full support, and the other one is a lower bound on the entropy of the measure. I will highlight the role played in the proofs by uncertainty principles such as the Fractal Uncertainty Principle and the Entropic Uncertainty Principle.
Featuring joint works with Jean Bourgain, Long Jin, Stéphane Nonnenmacher, Jayadev Athreya, Nicholas Miller, and Alex Cohen.
