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Self-adjoint extensions of the Laplace-Beltrami operator and unitaries at the boundary

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URI: https://hdl.handle.net/10016/23360
ISSN: 0022-1236
DOI: 10.1016/j.jfa.2014.10.013
UXXI: AR/0000016241
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2015-02-01
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Elsevier
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Abstract
We construct in this article a class of closed semi-bounded quadratic forms on the space of square integrable functions over a smooth Riemannian manifold with smooth compact boundary. Each of these quadratic forms specifies a semibounded self-adjoint extension of the Laplace-Beltrami operator. These quadratic forms are based on the Lagrange boundary form on the manifold and a family of domains parametrized by a suitable class of unitary operators on the boundary that will be called admissible. The corresponding quadratic forms are semi-bounded below and closable. Finally, the representing operators correspond to semi-bounded self-adjoint extensions of the Laplace-Beltrami operator. This family of extensions is compared with results existing in the literature and various examples and applications are discussed.
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Self-adjoint extensions, Laplace-Beltrami operator, Quadratic forms, Boundary conditions
Bibliographic citation
Journal of Functional Analysis, 2015, v. 268. Issue 3, pp. 634-670
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