Quantum ergodicity for contact metric structures

2025/07/24 by Lino Benedetto, Benedetto, Lino
Mathematics · Physics and Astronomy · #Advanced Operator Algebra Research #Analysis of PDEs (math.AP) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Quantum chaos and dynamical systems #Spectral Theory (math.SP) #Spectral Theory in Mathematical Physics

paper · pdf · doi:10.48550/arxiv.2507.18216

openalex publication_date 2025/07/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01

Abstract

This paper is dedicated to the proof of a Quantum Ergodicity (QE) theorem for the eigenfunctions of subLaplacians on contact metric manifolds, under the assumption that the Reeb flow is ergodic. To do so, we rely on a semiclassical pseudodifferential calculus developed for general filtered manifolds that we specialize to the setting of contact manifolds. Our strategy is then reminiscent of an implementation of the Born-Oppenheimer approximation as we rely on the construction of microlocal projectors in our calculus which commute with the subLaplacian, called Landau projectors. The subLaplacian is then shown to act effectively on the range of each Landau projector as the Reeb vector field does. The remainder of the proof follows the classical path towards QE, once microlocal Weyl laws have been established.

Citations

• The superadiabatic projectors method applied to the spectral theory of magnetic operators
• A microlocal calculus on filtered manifolds
• Semiclassical functional calculus on nilpotent Lie groups and their compact nilmanifolds
• Asymptotic regularity of sub-Riemannian eigenfunctions in dimension 3: the periodic case
• Quantum limits of perturbed sub-Riemannian contact Laplacians in dimension 3
• Spectral asymptotics for sub-Riemannian Laplacians
• Local dispersive and Strichartz estimates for the Schrödinger operator on the Heisenberg group
• Observability and Controllability for the Schrödinger Equation on Quotients of Groups of Heisenberg Type
• Quantum limits of sub-Laplacians via joint spectral calculus
• Quantum Evolution And Sub-laplacian Operators On Groups Of Heisenberg\n Type
• Semi-classical analysis on H-type groups
• Semiclassical functional calculus for h-dependent functions
• Geometry and Spectrum in 2D Magnetic Wells

Related

• Quantum ergodicity and quantum limits for sub-Riemannian Laplacians
• Remarks on quantum ergodicity
• Quantum Ergodicity and the Analysis of Semiclassical Pseudodifferential Operators
• Quantum limits of perturbed sub-Riemannian contact Laplacians in dimension 3
• Riemann moduli spaces are quantum ergodic