  We study the spectrum of the Laplace operator of a complete minimal properly
immersed hypersurface $M$ in $\R^{n+1}$. (1) Under a volume growth condition on
extrinsic balls and a condition on the unit normal at infinity, we prove that
$M$ has only essential spectrum consisting of the half line $[0, +\infty)$.
This is the case when $\lim_{\tilde{r}\to +\infty}\tilde{r}\kappa_i=0$, where
$\tilde{r}$ is the extrinsic distance to a point of $M$ and $\kappa_i$ are the
principal curvatures. (2) If the $\kappa_i$ satisfy the decay conditions
$|\kappa_i|\leq 1/\tilde{r}$, and strict inequality is achieved at some point
$y\in M$, then there are no eigenvalues. We apply these results to minimal
graphic and multigraphic hypersurfaces.
