  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.
