Quantum mixing for eigenfunctions of rational polygons in configuration space

Technical Summary: Quantum Mixing for Eigenfunctions of Rational Polygons in Configuration Space

Problem Statement
The paper addresses the relationship between classical dynamical properties (ergodicity and weak mixing) and their quantum counterparts (quantum ergodicity and quantum mixing) for specific geometric systems. While the Shnirelman–Zelditch–Colin de Verdière theorem establishes that ergodic geodesic flows imply quantum ergodicity for general compact Riemannian manifolds, integrable systems (such as the 2-torus and 2-sphere) generally fail to satisfy these properties.

The authors focus on rational polygons (polygons where all vertex angles are rational multiples of $\pi$). Although these systems are not quantum ergodic in the general sense (due to the existence of invariant tori in phase space), Marklof and Rudnick previously established that for rational polygons, almost all eigenfunctions equidistribute in configuration space when restricted to position-dependent observables (multiplication operators). The central problem of this note is to extend these results to off-diagonal elements (matrix elements $\langle a\psi_j, \psi_k \rangle$ with $j \neq k$) to establish quantum mixing properties in configuration space. Specifically, the authors aim to prove that for a subset of rational polygons, the off-diagonal matrix elements vanish in the high-energy limit, analogous to the weak mixing property of the classical directional billiard flow.

Methodology
The paper employs a combination of spectral theory, microlocal analysis, and dynamical systems theory:

1. Reduction to Specific Bases: The authors utilize a lemma (Lemma 2.1) demonstrating that if the desired properties hold for one orthonormal eigenbasis, they hold for any other. This allows them to work with specific bases convenient for the geometry (e.g., Fourier bases for tori).
2. Microlocal Analysis and Propagation of Singularities: For rational polygons, the authors use pseudodifferential operators of order zero ($\Psi^0_{phg}$) with Schwartz kernels compactly supported in the interior of the polygon. They rely on the propagation of singularities along bicharacteristics (Lemma 2.6) to relate the time-evolution of operators to the classical billiard flow $\Phi_t$.
3. Dynamical Inputs:
   • For general rational polygons, the proof relies on the unique ergodicity of the directional flow for almost all directions (Kerckhoff, Masur, and Smillie).
   • For the weak mixing results, the authors restrict the domain to polygons not belonging to a specific set $\mathcal{E}$ (defined as almost integrable polygons and polygons with commensurable side lengths in specific orientations). For these polygons, Arana-Herrera, Chaika, and Forni established that the directional flow is weakly mixing for almost all directions.
4. Counting Arguments for Tori: For the 2-torus case, where the classical flow is not weakly mixing, the authors bypass microlocal analysis. Instead, they perform explicit counting of lattice points in the dual lattice $\Gamma^*_l$ to bound the number of eigenvalue pairs satisfying specific spectral gap conditions (Lemma 4.1).

Key Contributions and Results

The paper establishes three main properties for the matrix elements of observables $a$ (restricted to configuration space) as the eigenvalue parameter $\lambda \to \infty$:

1. Property (1) - Quantum Ergodicity in Configuration Space:

   • Result: For any rational polygon $D$, the diagonal elements equidistribute.
   • Statement: $\lim_{\lambda \to \infty} \frac{1}{N(\lambda)} \sum_{\lambda_n \le \lambda} |\langle a\psi_n, \psi_n \rangle - \frac{1}{\text{area}(D)}\int_D a|^2 = 0$.
   • Note: This reproduces the result of Marklof and Rudnick [11].

2. Property (2) - Quantum Mixing (Diagonal Gap):

   • Result: For any rational polygon $D$, off-diagonal elements with small spectral gaps vanish.
   • Statement: For every $\epsilon > 0$, there exists $\delta(\epsilon)$ such that $\limsup_{\lambda \to \infty} \frac{1}{N(\lambda)} \sum_{j \neq k, |\lambda_j - \lambda_k| < \delta} |\langle a\psi_j, \psi_k \rangle|^2 < \epsilon$.
   • Significance: This extends the equidistribution result to off-diagonal terms for small energy differences.

3. Property (3) - Quantum Weak Mixing (Arbitrary Spectral Shift):

   • Result: For rational polygons $D \notin \mathcal{E}$ (excluding almost integrable polygons and specific commensurable cases), off-diagonal elements vanish for any fixed spectral shift $\tau$.
   • Statement: For every $\epsilon > 0$ and $\tau \in \mathbb{R}$, there exists $\delta(\epsilon)$ such that $\limsup_{\lambda \to \infty} \frac{1}{N(\lambda)} \sum_{j \neq k, |\lambda_j - \lambda_k - \tau| < \delta} |\langle a\psi_j, \psi_k \rangle|^2 < \epsilon$.
   • Significance: This establishes a direct link between the classical weak mixing of the directional billiard flow and the quantum mixing of eigenfunctions in configuration space for this specific class of polygons.

4. Extension to 2-Tori (Theorem 1.5):

   • Result: Despite the 2-torus being an integrable system with a non-weakly-mixing directional flow, Properties (1), (2), and (3) still hold for the 2-torus when restricted to configuration space observables.
   • Method: The proof uses explicit lattice point counting rather than dynamical mixing arguments.
   • Significance: This contrasts with results on "large scale" tori where such properties fail, highlighting that the restriction to configuration space observables (multiplication operators) is crucial for these mixing phenomena to appear even in integrable systems.

Significance and Claims
The authors claim that this work extends the understanding of quantum chaos in integrable and near-integrable systems. Specifically:

• It demonstrates that quantum mixing in configuration space can occur even when the full phase-space dynamics are not mixing (as seen in the torus case).
• It clarifies the role of the observable class: while general pseudodifferential operators may not exhibit mixing for integrable systems, restricting to position-dependent observables (functions on configuration space) allows for equidistribution and mixing properties to emerge.
• The paper provides a rigorous proof that the weak mixing of the directional billiard flow for almost all directions in non-integrable rational polygons implies the corresponding quantum weak mixing property for off-diagonal matrix elements.
• The authors note that the proofs for Theorems 1.1 and 1.3 can be generalized to pseudodifferential operators of order zero, provided the underlying billiard flow is ergodic or weakly mixing in phase space.

The paper does not propose new experimental applications or future directions beyond the mathematical scope of establishing these spectral properties for rational polygons and tori.