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

TL;DR

This paper develops a framework for constructing semi-bounded self-adjoint extensions of the Laplace-Beltrami operator on manifolds with boundary using boundary unitary operators, expanding the understanding of boundary conditions in geometric analysis.

Contribution

It introduces a new class of quadratic forms parametrized by boundary unitaries, providing a systematic way to characterize self-adjoint extensions of the Laplace-Beltrami operator.

Findings

Constructed semi-bounded quadratic forms based on boundary unitaries.

Established correspondence between quadratic forms and self-adjoint extensions.

Compared new extensions with existing results and discussed applications.

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 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.