Quantum Limits of Eisenstein Series in H^3

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 $|E(p,{\sigma }_t+it){|}^{2}d\mu (p)$ become equidistributed only if ${\sigma }_t\to 1$ as $t\to \infty$. 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.

12 pages

References in corpus (1)

• The quantum unique ergodicity conjecture for thin sets