Macroscopic limits of chaotic eigenfunctions

TL;DR

This work analyzes macroscopic behavior of high-energy Laplacian eigenfunctions on compact manifolds through semiclassical measures on the cosphere bundle $S^*M$, tying quantum statistics to the classical geodesic flow. It develops the semiclassical quantization framework with $\mathrm{Op}_h(a)$ and demonstrates that semiclassical measures are invariant under geodesic flow and push forward to weak limits on $M$, enabling rigorous statements like Quantum Ergodicity under ergodicity of the flow. The article surveys entropy bounds (Anantharaman–Nonnenmacher) showing positive $h_{KS}(\mu)$ and a universal lower bound in the hyperbolic setting, as well as full support results (via fractal uncertainty principles) for Anosov flows; it also discusses the Quantum Unique Ergodicity conjecture and known counterexamples in quantum cat maps. Finally, it contrasts these results with the cat-map toy model, where QE can fail, yet partial results (like Hecke-operator–assisted QUE) and full-support phenomena provide nuanced insights into macroscopic eigenfunction distribution and quantum chaos.

Abstract

We give an overview of the interplay between the behavior of high energy eigenfunctions of the Laplacian on a compact Riemannian manifold and the dynamical properties of the geodesic flow on that manifold. This includes the Quantum Ergodicity theorem, the Quantum Unique Ergodicity conjecture, entropy bounds, and uniform lower bounds on mass of eigenfunctions. The above results belong to the domain of quantum chaos and use microlocal analysis, which is a theory behind the classical/quantum, or particle/wave, correspondence in physics. We also discuss the toy model of quantum cat maps and the challenges it poses for Quantum Unique Ergodicity.

Figures (5)

• Figure 1: Top: typical eigenfunctions (with Dirichlet boundary conditions) for two planar domains. The picture on the left (courtesy of Alex Barnett, see Barnett-Billiard and Barnett-Hassell for a description of the method used and for a numerical investigation of Quantum Ergodicity, showing empirically $\mathcal{O}(\lambda^{-1/2})$ convergence to equidistribution) shows equidistribution, i.e. convergence to the volume measure in the sense of Definition \ref{['d:weak-limit-1']}. The picture on the right (where the domain is a disk) shows lack of equidistribution, with the limiting measure supported in an annulus. This difference in quantum behavior is related to the different behavior of the billiard-ball flows on the two domains (which replace geodesic flows in this setting). Bottom: two typical billiard-ball trajectories on the domains in question. On the left we see ergodicity (equidistribution of the trajectory for long time) and on the right we see completely integrable behavior.
• Figure 2: Two Dirichlet eigenfunctions for a Bunimovich stadium, courtesy of Alex Barnett (see the caption to Figure \ref{['f:billiards']}): the right one shows equidistribution but the left one does not. Quantum Ergodicity implies that most eigenfunctions look from afar like the one on the right.
• Figure 3: Two Laplacian eigenfunctions on a hyperbolic surface, courtesy of Alex Strohmaier (see Strohmaier--Uski Strohmaier-Uski). Here we view the surface as a quotient of the hyperbolic plane by a group of isometries, or equivalently as the result of gluing together appropriate sides of the pictured fundamental domain. On a microscopic level the two eigenfunctions look different, but the macroscopic features are the same -- both show equidistribution.
• Figure 4: Phase space concentration for two eigenfunctions of the quantum cat map with $A$ given by \ref{['e:basic-cat']} and $N=1292$. More specifically, we plot the absolute value of a smoothened out Wigner transform of the eigenfunction on the logarithmic scale (see e.g. highcat). On the left is a typical eigenfunction, showing equidistribution. On the right is a particular eigenfunction of the type constructed in Faure-Nonnenmacher-dB, corresponding to a measure of the type \ref{['e:cat-que-fails']} featuring the closed trajectory $\{({1\over 3},0),({2\over 3},{1\over 3}),({2\over 3},0),({1\over 3},{2\over 3})\}$. The existence of such an eigenfunction relies on the careful choice of $N$: $A^{18}$ is the identity matrix modulo $2N$.
• Figure 5: A set $\mathcal{U}\subset\mathbb T^2$ (center picture, in white) and the corresponding sets $\Omega_+(N),\Omega_-(N)$ (left/right picture). The set $\Omega_+(N)$ is 'smooth' in the unstable direction of the matrix $A$ and porous in the stable direction, with the porosity constant depending only on $\mathcal{U}$. Same is true for $\Omega_-(N)$ but switching the roles of the stable/unstable directions. The fractal uncertainty principle of Theorem \ref{['t:fup']} can be used to show that no function can be localized on both $\Omega_+(N)$ and $\Omega_-(N)$.

Theorems & Definitions (16)

• Definition 1.1
• Definition 2.1
• Proposition 2.2
• Theorem 1
• Definition 4.1
• Conjecture 4.2
• Theorem 2
• Theorem 3
• Conjecture 4.3
• Conjecture 4.4