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On the Theory of Self-Adjoint Extensions of the Laplace-Beltrami Operator, Quadratic Forms and Symmetry
Mathematical Physics · Submitted 9 Aug 2013
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim 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.
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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.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | 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. | contribution 0.96 | 0.66 | 0.16 | asserted 0.91 |
| s1 | Moreover, we want to emphasise the role that quadratic forms can play in the description of quantum systems. | interpretation 0.34 | 0.22 | 0.05 | asserted 0.39 |
| s2 | 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. | result 0.53 | 0.89 | 0.16 | asserted 1.00 |
| s3 | 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. | result 0.75 | 0.91 | 0.18 | asserted 1.00 |
| s4 | A numerical scheme to compute the eigenvalues and eigenvectors in any dimension is proposed and its convergence is proved. | method 0.65 | 0.91 | 0.07 | asserted 0.99 |
| s5 | 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. | contribution 0.99 | 0.77 | 0.05 | asserted 0.99 |
| s6 | The self-adjoint extensions possessing the same invariance than the symmetric operator that they extend are characterised in the most abstract setting. | result 0.64 | 0.85 | 0.10 | asserted 1.00 |
| s7 | The case of the Laplace-Beltrami operator is analysed also in this case. | contribution 0.95 | 0.82 | 0.04 | asserted 0.93 |
| s8 | Finally, a way to generalise Kato's representation theorem for not semi-bounded, closed quadratic forms is proposed. | contribution 0.98 | 0.87 | 0.11 | asserted 0.76 |
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Original 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.
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Such problem has been studied only recently in a series of papers [ref] .
Mode solutions for a Klein-Gordon field in anti-de Sitter spacetime with dynamical boundary conditions of Wentzell type · Interplay between the boundary conditions and the boundary isometry group · 0.84
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