Mixed quantization and partial hyperbolicity

TL;DR

This work develops a stable quantum ergodicity theory for spin Pauli–Schrödinger operators by coupling mixed quantization (Berezin–Toeplitz fibers with Weyl base quantization) to ergodic partially hyperbolic dynamics. It proves that under ergodic energy flows and small perturbations, a density-one subsequence of eigenfunctions equidistributes on energy shells; the analytical backbone combines a local Weyl law, Egorov-type propagation, and a QE concentration argument. Dynamically, the authors establish leavewise ergodicity for perturbed horizontal flows by exploiting a dense SU(n+1) fiber action and a symplectic quotient, then apply stable ergodicity results to deduce QE stability. The framework connects semiclassical, geometric, and dynamical methods, with explicit examples on CP^n and hyperbolic surfaces and broad moment-map applications, illustrating deep links between quantum ergodicity, group actions, and foliations.

Abstract

We establish stable quantum ergodicity for spin Hamiltonians, also known as Pauli-Schrödinger operators. Our approach combines new analytic techniques of mixed quantization, inspired by local index theory, with stable ergodicity results for partially hyperbolic systems.