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Half-delocalization of eigenfunctions for the Laplacian on an Anosov manifold. (English) Zbl 1145.81033
In the present paper the authors study high energy eigenfunctions $$\psi$$, of the Laplacian on a compact Riemannian manifold $$M$$ with Anosov geodesic flow. The Quantum Unique Ergodicity conjecture asserts that for such eigenfunctions the corresponding probability density $$| \psi| ^2dx$$ (or rather the corresponding Wigner distribution $$W_\psi$$ on the cotangent bundle $$T^*M$$) should (weakly) approach the Riemannian volume, when the corresponding eigenvalue tends to infinity. It has been shown by Schnirelman, Zelditch and Colin de Verdière that this is true for almost all eigenfunctions. However, the question whether there could be exceptional sequences of eigenfunctions with different semiclassical limits remains open in general, and has been shown only for special arithmetic surfaces.
The authors study the limiting measures of these eigenfunctions, which are shown to be probability measures on the unit cotangent bundle $$S^*M$$, and the Kolmogorov-Sinai entropy of these measures. An equivalent condition to the quantum unique ergodicity conjecture in terms of entropy would be that any limiting measure $$\mu$$ must have the maximal possible entropy, that is $h_{KS}(\mu)=\left| \int_{S^*M}\log J^u(x)\,d\mu(x)\right| ,$ where $$J^u(x)$$ is the unstable Jacobian of the flow at the point $$x\in S^*M$$. The main result in this paper is a lower bound for the entropy given by
$h_{KS}(\mu)\geq \frac{3}{2}\left| \int_{S^*M}\log J^u(x)\,d\mu(x)\right| -(d-1)\lambda_{\max}$ where $$d=\dim{M}$$ and $$\lambda_{\max}$$ is the maximal expansion rate of the geodesic flow. In particular for constant sectional curvature $$-1$$ this means that $$h_{KS}(\mu)\geq \frac{d-1}{2}$$ is bounded by half of the maximal entropy.
The authors remark that it should be possible to extend their methods to obtain the bound of half the maximal entropy
$h_{KS}(\mu)\geq \frac{1}{2}\left| \int_{S^*M}\log J^u(x)\,d\mu(x)\right| ,$ also for variable curvature. This would be the optimal bound one could hope to obtain in general without assuming more precise information. Indeed, there are Anosov systems, such as the quantum cat map, where quantum unique ergodicity fails and the bound of half the maximal entropy is actually sharp.

MSC:
 81Q50 Quantum chaos 35Q40 PDEs in connection with quantum mechanics 35P20 Asymptotic distributions of eigenvalues in context of PDEs 37D40 Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.) 58J30 Spectral flows 28D20 Entropy and other invariants 81S30 Phase-space methods including Wigner distributions, etc. applied to problems in quantum mechanics
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