The course covers advanced topics in classical/quantum correspondence, focusing on results related to chaotic classical dynamics without classical counterparts. Students will learn about lower bounds on Laplacian eigenfunctions, observability for the Schrödinger equation, and spectral gaps on specific surfaces. The key tool used in the proofs is the fractal uncertainty principle. The teaching method includes theoretical explanations, illustrations, and applications to partial differential equations. This course is intended for individuals with a strong background in mathematics and an interest in quantum chaos and spectral theory.
Overview
Syllabus
Intro
Overview
Control of eigenfunctions
An illustration
Applications to PDE
Semiclassical measures II
Main tool fractal uncertainty principle (FUP)
Main tool: fractal uncertainty principle (FUP)
A bit about proof of Theorem 1
Illustration: Arnold cat map
Open quantum chaos and resonances
Spectral gap
Higher dimensional FUP?
Taught by
International Mathematical Union
