A note on Mellin transform, Eisenstein Series and distribution $d\epsilon_{it}$ on $PSL(2,\mathbb{Z}[i]) \backslash PSL(2,\mathbb{C})$

TL;DR

This paper develops a Mellin transform formula for functions on a quotient of PSL(2,C), defines a micro-local lift, computes pairings with cuspidal and Eisenstein series, and discusses implications for quantum ergodicity.

Contribution

It introduces a new Mellin transform formula, defines a micro-local lift on PSL(2,C), and conjectures a positive distribution satisfying asymptotic estimates related to quantum ergodicity.

Findings

Derived a Mellin transform formula for smooth functions on the quotient space.

Computed pairings with cuspidal forms and Eisenstein series.

Established asymptotic estimates as the spectral parameter tends to infinity.

Abstract

Let $f$ a smooth function with compact support defined on $PSL(2,\mathbb{Z}[i]) \backslash PSL(2,\mathbb{C})$, we prove a formula for the Mellin transform of $f$, then we can define the micro-local lift $d\epsilon_{it}$ to $SL(2,\mathbb{C})$. We calculate $(f,d\epsilon_{it})$ for $f$ a cuspidal form and for $f$ an incomplete Eisenstein series. We also establish asymptotic estimates when $t$ tends to $\infty$. We conjecture that a new positive distribution $d\epsilon_{it}^F$, constructed with the Friedrichs' symmetrization technique, satisfies the same asymptotic estimates that $d\epsilon_{it}$. This would imply the quantum ergodicity for Eisenstein series on $PSL(2,\mathbb{Z}[i]) \backslash PSL(2,\mathbb{C})$