Quantum Limits of Eisenstein Series in H^3

TL;DR

This paper investigates the quantum limits of Eisenstein series in hyperbolic 3-space over imaginary quadratic fields, revealing conditions for equidistribution and convergence to continuous measures under the Generalized Riemann Hypothesis.

Contribution

It generalizes previous results from hyperbolic surfaces to hyperbolic 3-space and analyzes the conditions for quantum ergodicity of Eisenstein series off the critical line.

Findings

Measures become equidistributed only if _t ightarrow 1 as t ightarrow \u221e

Measures defined via scattering states converge to a specific continuous measure under GRH

Extends quantum limit results from _t ightarrow 1 in _t ightarrow 1 in _t ightarrow 1 in _t ightarrow 1 in _t ightarrow 1 in _t ightarrow 1 in the setting of _t ightarrow 1 in hyperbolic 3-space.

Abstract

We study the quantum limits of Eisenstein series off the critical line for $\mathrm{PSL}_{2}(\mathcal{O}_{K})\backslash\mathbb{H}^{3}$, where $K$ is an imaginary quadratic field of class number one. This generalises the results of Petridis, Raulf and Risager on $\mathrm{PSL}_{2}(\mathbb{Z})\backslash\mathbb{H}^{2}$. We observe that the measures $\lvert E(p,\sigma_{t}+it)\rvert^{2}d\mu(p)$ become equidistributed only if $\sigma_{t}\rightarrow 1$ as $t\rightarrow\infty$. We use these computations to study measures defined in terms of the scattering states, which are shown to converge to the absolutely continuous measure $E(p,3)d\mu(p)$ under the GRH.