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

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

2003-04-15

arxiv.org/abs/hep-th/0304130v1

Physics

High Energy Physics

High Energy Physics - Theory

6 pages, revtex

Scientific paper

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.

Affiliated with

Also associated with

No associations

Rating

Spectral theory of automorphic forms and analysis of invariant operators on $SL_3({\cal{Z}}$ with applications does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Spectral theory of automorphic forms and analysis of invariant operators on $SL_3({\cal{Z}}$ with applications, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Spectral theory of automorphic forms and analysis of invariant operators on $SL_3({\cal{Z}}$ with applications will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-478136