Quantum ergodicity for shrinking balls in arithmetic hyperbolic manifolds

TL;DR

This paper investigates quantum ergodicity on shrinking balls in arithmetic hyperbolic manifolds, revealing failures and bounds of equidistribution at various scales, with results depending on manifold dimension and number-theoretic hypotheses.

Contribution

It provides new bounds and conditional results on quantum ergodicity failure and equidistribution for shrinking balls in arithmetic hyperbolic manifolds across different dimensions.

Findings

Failure of quantum unique ergodicity near the Planck-scale for Eisenstein series on the modular surface.

Conditional and unconditional bounds for equidistribution on shrinking balls in 3-manifolds.

Extension of results to higher dimensions, showing failure of quantum ergodicity at specific shrinking scales.

Abstract

We study a refinement of the quantum unique ergodicity conjecture for shrinking balls on arithmetic hyperbolic manifolds, with a focus on dimensions $ 2 $ and $ 3 $. For the Eisenstein series for the modular surface $\mathrm{PSL}_2( {\mathbb Z}) \backslash \mathbb{H}^2$ we prove failure of quantum unique ergodicity close to the Planck-scale and an improved bound for its quantum variance. For arithmetic $ 3 $-manifolds we show that quantum unique ergodicity of Hecke-Maa{\ss} forms fails on shrinking balls centered on an arithmetic point and radius $ R \asymp t_j^{-\delta} $ with $ \delta > 3/4 $. For $ \mathrm{PSL}_2(\mathcal{O}_K) \setminus \mathbb{H}^3 $ with $ \mathcal{O}_K $ being the ring of integers of an imaginary quadratic number field of class number one, we prove, conditionally on the generalized Lindel\"of hypothesis, that equidistribution holds for Hecke-Maa{ss} forms if $ \delta < 2/5 $. Furthermore, we prove that equidistribution holds unconditionally for the Eisenstein series if $ \delta < (1-2\theta)/(34+4\theta) $ where $ \theta $ is the exponent towards the Ramanujan-Petersson conjecture. For $ \mathrm{PSL}_2(\mathbb{Z}[i]) $ we improve the last exponent to $ \delta < (1-2\theta)/(27+2\theta) $. Studying mean Lindel\"of estimates for $ L $-functions of Hecke-Maa{\ss} forms we improve the last exponent on average to $ \delta < 2/5$. Finally, we study massive irregularities for Laplace eigenfunctions on $ n $-dimensional compact arithmetic hyperbolic manifolds for $ n \geq 4 $. We observe that quantum unique ergodicity fails on shrinking balls of radii $ R \asymp t^{-\delta_n+\epsilon} $ away from the Planck-scale, with $ \delta_n = 5/(n+1) $ for $ n \geq 5 $.