  Let $X=G/H$ be a reductive homogeneous space with $H$ noncompact, endowed
with a $G$-invariant pseudo-Riemannian structure. Let $L$ be a reductive
subgroup of $G$ acting properly on $X$ and $\Gamma$ a torsion-free discrete
subgroup of $L$. Under the assumption that the complexification $X_{\mathbb C}$
is $L_{\mathbb C}$-spherical, we prove an explicit correspondence between
spectral analysis on the standard locally homogeneous space
$X_{\Gamma}=\Gamma\backslash X$ and on $\Gamma\backslash L$ via branching laws
for the restriction to $L$ of irreducible representations of $G$. In
particular, we prove that the pseudo-Riemannian Laplacian on $X_{\Gamma}$ is
essentially self-adjoint, and that it admits an infinite point spectrum when
$X_{\Gamma}$ is compact or $\Gamma\subset L$ is arithmetic. The proof builds on
structural results for invariant differential operators on spherical
homogeneous spaces with overgroups.
