2003/04/15 by Sultan Catto, Catto, Sultan, Jonathan Huntley +5
Physics and Astronomy · #FOS: Physical sciences #High Energy Physics - Theory (hep-th) #hep-th
paper · pdf · doi:10.48550/arxiv.hep-th/0304130
6 pages, revtex
arxiv created 2003/04/15 · arxiv updated 2009/11/30
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 SL3(\bf Z) is given. We prove the following: Let \cal H be the homogeneous space associated to the group PGL3(\bf R). Let X = Γ\backslash SL3(\bf Z) and consider the first non-trivial eigenvalue λ1 of the Laplacian on L2(X). Using geometric considerations, we prove the inequality λ1 > 3pi2/10> 2.96088. Since the continuous spectrum is represented by the band [1,∞), our bound on λ1 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.