{"paper":{"title":"On the Theory of Self-Adjoint Extensions of the Laplace-Beltrami Operator, Quadratic Forms and Symmetry","arxiv_id":"1308.2158","primary_cat":"math-ph","submitted_at":"2013-08-09T15:25:53+00:00","version_compiled":"v2"},"notice":"Every quoted sentence is the authors' text from the arXiv listing abstract. Labels and probabilities are the outputs of classifier-1.13-20260917 on the question bank abstract-p1-20260923, exactly as returned. 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systems.","hedge":{"p":0.39,"label":"asserted"},"own_p":0.22,"main_claim_p":0.05,"quantitative":false,"quantitative_p":0.03,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.0,"background":0.16,"limitation":0.0,"future_work":0.16,"contribution":0.34,"interpretation":0.34}},{"i":2,"p":0.53,"id":"s2","own":true,"role":"result","text":"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.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.89,"main_claim_p":0.16,"quantitative":false,"quantitative_p":0.03,"role_probabilities":{"gap":0.0,"method":0.03,"result":0.53,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.44,"interpretation":0.0}},{"i":3,"p":0.75,"id":"s3","own":true,"role":"result","text":"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.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.91,"main_claim_p":0.18,"quantitative":true,"quantitative_p":0.52,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.75,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.25,"interpretation":0.0}},{"i":4,"p":0.65,"id":"s4","own":true,"role":"method","text":"A numerical scheme to compute the eigenvalues and eigenvectors in any dimension is proposed and its convergence is proved.","hedge":{"p":0.99,"label":"asserted"},"own_p":0.91,"main_claim_p":0.07,"quantitative":false,"quantitative_p":0.1,"role_probabilities":{"gap":0.0,"method":0.65,"result":0.0,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.35,"interpretation":0.0}},{"i":5,"p":0.99,"id":"s5","own":true,"role":"contribution","text":"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.","hedge":{"p":0.99,"label":"asserted"},"own_p":0.77,"main_claim_p":0.05,"quantitative":false,"quantitative_p":0.03,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.0,"background":0.01,"limitation":0.0,"future_work":0.0,"contribution":0.99,"interpretation":0.0}},{"i":6,"p":0.64,"id":"s6","own":true,"role":"result","text":"The self-adjoint extensions possessing the same invariance than the symmetric operator that they extend are characterised in the most abstract setting.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.85,"main_claim_p":0.1,"quantitative":false,"quantitative_p":0.04,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.64,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.36,"interpretation":0.0}},{"i":7,"p":0.95,"id":"s7","own":true,"role":"contribution","text":"The case of the Laplace-Beltrami operator is analysed also in this case.","hedge":{"p":0.93,"label":"asserted"},"own_p":0.82,"main_claim_p":0.04,"quantitative":false,"quantitative_p":0.03,"role_probabilities":{"gap":0.0,"method":0.01,"result":0.02,"background":0.02,"limitation":0.0,"future_work":0.0,"contribution":0.95,"interpretation":0.0}},{"i":8,"p":0.98,"id":"s8","own":true,"role":"contribution","text":"Finally, a way to generalise Kato's representation theorem for not semi-bounded, closed quadratic forms is proposed.","hedge":{"p":0.76,"label":"asserted"},"own_p":0.87,"main_claim_p":0.11,"quantitative":false,"quantitative_p":0.05,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.02,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.98,"interpretation":0.0}}]},"compiled":{"main_claim":{"p":0.68,"text":"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.","location":"abstract 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