Spectral theory of automorphic forms and analysis of invariant operators on $SL_3({\cal{Z}}$ with applications

TL;DR

This paper investigates the spectral properties of automorphic forms on $SL_3({f Z})$ using analytic methods, providing bounds on eigenvalues and error estimates, with implications for high energy physics.

Contribution

It introduces new bounds on the first non-trivial eigenvalue of the Laplacian for $SL_3({f Z})$ and extends Selberg's eigenvalue conjecture to higher rank groups.

Findings

Proved $ ext{lambda}_1 > 2.96088$ for $SL_3({f Z})$.

Derived error terms for Weyl's law in this setting.

Established an analogue of Selberg's eigenvalue conjecture for $SL_3({f Z})$.

Abstract

We study a variety of problems in the spectral theory of automorphic forms using entirely analytic techniques such as Selberg trace formula, asymptotics of Whittaker functions and behavior of heat kernels. Error terms for Weyl's law and an analog of Selberg's eigenvalue conjecture for $SL_3({\bf Z})$ is given. We prove the following: Let $\cal H$ be the homogeneous space associated to the group $PGL_3(\bf R)$. Let $X = \Gamma{\backslash SL_3({\bf Z}})$ and consider the first non-trivial eigenvalue $\lambda_1$ of the Laplacian on $L^2(X)$. Using geometric considerations, we prove the inequality $\lambda_1 > 3pi^2/10> 2.96088.$ Since the continuous spectrum is represented by the band $[1,\infty)$, our bound on $\lambda_{1}$ can be viewed as an analogue of Selberg's eigenvalue conjecture for quotients of the hyperbolic half space. Brief comment on relevance of automorphic forms to applications in high energy physics is given.