  The main objective of this dissertation is to analyse thoroughly the
construction of self-adjoint extensions of the Laplace-Beltrami operator
defined on a compact Riemannian manifold with boundary and the role that
quadratic forms play to describe them. Moreover, we want to emphasise the role
that quadratic forms can play in the description of quantum systems. A
characterisation of the self-adjoint extensions of the Laplace-Beltrami
operator in terms of unitary operators acting on the Hilbert space at the
boundary is given. Using this description we are able to characterise a wide
class of self-adjoint extensions that go beyond the usual ones, i.e. Dirichlet,
Neumann, Robin,.. and that are semi-bounded below. A numerical scheme to
compute the eigenvalues and eigenvectors in any dimension is proposed and its
convergence is proved. The role of invariance under the action of symmetry
groups is analysed in the general context of the theory of self-adjoint
extensions of symmetric operators and in the context of closed quadratic forms.
The self-adjoint extensions possessing the same invariance than the symmetric
operator that they extend are characterised in the most abstract setting. The
case of the Laplace-Beltrami operator is analysed also in this case. Finally, a
way to generalise Kato's representation theorem for not semi-bounded, closed
quadratic forms is proposed.
