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

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 semi-bounded 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.