  In this paper we study the spectrum of the random geometric graph $G(n,r)$,
in a regime where the graph is dense and highly connected. In the \erdren
$G(n,p)$ random graph it is well known that upon connectivity the spectrum of
the normalized graph Laplacian is concentrated around $1$. We show that such
concentration does not occur in the $G(n,r)$ case, even when the graph is dense
and almost a complete graph. In particular, we show that the limiting spectral
gap is strictly smaller than $1$. In the special case where the vertices are
distributed uniformly in the unit cube and $r=1$, we show that for every $0\le
k \le d$ there are at least $\binom{d}{k}$ eigenvalues near $1-2^{-k}$, and the
limiting spectral gap is exactly $1/2$. We also show that the corresponding
eigenfunctions in this case are tightly related to the geometric configuration
of the points.
