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The main objective of this dissertation is to analyse thoroughly theconstruction of self-adjoint extensions of the Laplace-Beltrami operatordefined on a compact Riemannian manifold with boundary and the role thatquadratic forms play to describe them. Moreover, we want to emphasise the rolethat quadratic forms can play in the description of quantum systems. Acharacterisation of the self-adjoint extensions of the Laplace-Beltramioperator in terms of unitary operators acting on the Hilbert space at theboundary is given. Using this description we are able to characterise a wideclass of self-adjoint extensions that go beyond the usual ones, i.e. Dirichlet,Neumann, Robin,.. and that are semi-bounded below. A numerical scheme tocompute the eigenvalues and eigenvectors in any dimension is proposed and itsconvergence is proved. The role of invariance under the action of symmetrygroups is analysed in the general context of the theory of self-adjointextensions of symmetric operators and in the context of closed quadratic forms.The self-adjoint extensions possessing the same invariance than the symmetricoperator that they extend are characterised in the most abstract setting. Thecase of the Laplace-Beltrami operator is analysed also in this case. Finally, away to generalise Kato's representation theorem for not semi-bounded, closedquadratic forms is proposed.

On the Theory of Self-Adjoint Extensions of the Laplace-Beltrami Operator, Quadratic Forms and Symmetry