In the classical Riemannian context, the Laplacian is an elliptic differential operator, ensuring that all eigenfunctions are real analytic. Furthermore, if the manifold is complete, the Laplacian is essentially self-adjoint, enabling the unique expansion of L2 functions into a direct integral of real analytic eigenfunctions of the Laplacian. In contrast, on pseudo-Riemannian manifolds where the Laplacian does not behave as an elliptic differential operator, eigenfunctions are not necessarily real analytic. Moreover, in the absence of a general theory, it is unclear whether the Laplacian is essentially self-adjoint, and consequently, the question whether there exists a spectral decomposition theory of the Laplacian remains open in general. In this chapter, we establish that the pseudo-Riemannian Laplacian is essentially self-adjoint for any standard locally homogeneous space XΓ = Γ\G/H defined by a proper spherical action. The main strategy in the proof is to use a decomposition theorem for the Laplacian introduced in the preceding chapter. This theorem allows for a more detailed decomposition of eigenfunctions of the Laplacian in relation to another commuting elliptic differential operator.