Quantum mixing for eigenfunctions of rational polygons in configuration space

Kai Hippi; Søren Mikkelsen

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arXiv · 2608.11728

Quantum mixing for eigenfunctions of rational polygons in configuration space

Kai Hippi·Søren Mikkelsen

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Abstract

The Shnirelman-Zelditch-Colin de Verdi\`ere theorem and its weak mixing extension relate quantum ergodicity and quantum mixing with ergodicity and weak mixing of the geodesic flow. Integrable systems, such as the flat torus, do not satisfy either in general. Restricting to position-dependent observables, Marklof and Rudnick established equidistribution for almost all eigenfunctions of rational polygons. In this note, we extend this result to off-diagonal elements for a subset of rational polygons not including the torus using weak mixing of the directional billiard flow for almost all directions. Additionally, we provide a different proof that establishes the same result for $2$-tori.

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BibTeX RIS

Kai Hippi, Søren Mikkelsen. 2026-08-12. Quantum mixing for eigenfunctions of rational polygons in configuration space. https://arxiv.org/abs/2608.11728

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