  Let $G=\operatorname{SO}^\circ(n,1) \times \operatorname{SO}^\circ(n,1)$ and
$X={\mathbb H}^{n}\times {\mathbb H}^{n}$ for $n\ge 2$. For a pair $(\pi_1,
\pi_2)$ of non-elementary convex cocompact representations of a finitely
generated group $\Sigma$ into $\operatorname{SO}^\circ(n,1)$, let
$\Gamma=(\pi_1\times \pi_2)(\Sigma)$. Denoting the bottom of the $L^2$-spectrum
of the negative Laplacian on $\Gamma\backslash X$ by $\lambda_0$, we show:
  (1) $L^2(\Gamma\backslash G)$ is tempered and $\lambda_0=\frac{1}{2}(n-1)^2$;
  (2) There exists no positive Laplace eigenfunction in $L^2(\Gamma \backslash
X)$.
  In fact, analogues of (1)-(2) hold for any Anosov subgroup $\Gamma$ in the
product of at least two simple algebraic groups of rank one as well as for
Hitchin subgroups $\Gamma<\operatorname{PSL}_d(\mathbb R)$, $d\ge 3$. Moreover,
if $G$ is a semisimple real algebraic group of rank at least $2$, then (2)
holds for any Anosov subgroup $\Gamma$ of $G$.
