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

TL;DR

This paper thoroughly analyzes the construction and characterization of self-adjoint extensions of the Laplace-Beltrami operator on manifolds with boundary, emphasizing quadratic forms, symmetry, and numerical eigenvalue computation.

Contribution

It introduces a boundary unitary operator framework for self-adjoint extensions, extends the class beyond classical types, and proposes a numerical scheme with convergence proof.

Findings

Characterization of self-adjoint extensions via boundary unitaries.

Extension of the class of semi-bounded self-adjoint extensions.

A convergent numerical scheme for eigenvalues and eigenvectors.

Abstract

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.