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 ✩ 635 Boundary conditions the literature and various examples and applications are discussed.

The dynamics of the magnetic field in a superconducting phase is described by an effective massive bosonic field theory. If the superconductor is confined in a domain M with boundary ∂M , the boundary conditions of the electromagnetic fields are predetermined by physics. They are time-reversal and also parity invariant for adapted geometry. They lead to edge excitations while in comparison, the bulk energies have a large gap. A similar phenomenon occurs for topological insulators where appropriate boundary conditions for the Dirac Hamiltonian also lead to similar edge states and an "incompressible bulk". They give spin-momentum locking as well. In addition time-reversal and parity invariance emerge for adapted geometry. Similar edge states appear in QCD bag models with chiral boundary conditions.