  We give the first numerical calculation of the spectrum of the Laplacian
acting on bundle-valued forms on a Calabi-Yau three-fold. Specifically, we show
how to compute the approximate eigenvalues and eigenmodes of the Dolbeault
Laplacian acting on bundle-valued $(p,q)$-forms on K\"ahler manifolds. We
restrict our attention to line bundles over complex projective space and
Calabi-Yau hypersurfaces therein. We give three examples. For two of these,
$\mathbb{P}^3$ and a Calabi-Yau one-fold (a torus), we compare our numerics
with exact results available in the literature and find complete agreement. For
the third example, the Fermat quintic three-fold, there are no known analytic
results, so our numerical calculations are the first of their kind. The
resulting spectra pass a number of non-trivial checks that arise from Serre
duality and the Hodge decomposition. The outputs of our algorithm include all
the ingredients one needs to compute physical Yukawa couplings in string
compactifications.
