Semiclassical behaviour of quantum eigenstates

Abstract

Given a quantum Hamiltonian, we explain how the dynamical properties of the underlying classical system affect the behaviour of quantum eigenstates in the semi-classical limit. We study this problem via the notion of semiclassical measures. We mostly focus on two opposite dynamical paradigms: completely integrable systems and chaotic ones. We recall standard tools from microlocal analysis and from dynamical systems. We show how to use them in order to illustrate the classical-quantum correspondance and to compare properties of completely integrable and chaotic systems.

arXiv commentLecture notes for the Spring School ''From quantum to classical'' (CIRM, April 2019)

References in corpus (13)

• Scarred eigenstates for quantum cat maps of minimal periods
• Spectral gaps without the pressure condition
• High-frequency propagation for the Schroedinger equation on the torus
• Entropy of semiclassical measures of the Walsh-quantized baker's map
• Entropy of semiclassical measures in dimension 2
• Spectral asymptotics for sub-Riemannian Laplacians. I: quantum ergodicity and quantum limits in the 3D contact case
• L^p norms of eigenfunctions in the completely integrable case
• Small scale quantum ergodicity in negatively curved manifolds
• An introduction to fractal uncertainty principle
• Quantum ergodicity and Benjamini-Schramm convergence of hyperbolic surfaces
• Arithmetic and equidistribution of measures on the sphere
• Eigenmodes of the damped wave equation and small hyperbolic subsets
• Eigenfunctions for partially rectangular billiards