  We consider the spectrum of the Laplace operator acting on $\mathcal{L}^p$
over a conformally compact manifold for $1 \leq p \leq \infty$. We prove that
for $p \neq 2$ this spectrum always contains an open region of the complex
plane. We further show that the spectrum is contained within a certain
parabolic region of the complex plane. These regions depend on the value of
$p$, the dimension of the manifold, and the values of the sectional curvatures
approaching the boundary.
