The Friedrichs extension is a concept from functional analysis and operator theory, particularly related to self-adjoint operators in the context of quantum mechanics and partial differential equations. It provides a way to extend an unbounded symmetric operator to a self-adjoint operator, which is crucial because self-adjoint operators have well-defined spectral properties and their associated physical observables are mathematically rigorous.
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A densely defined semibounded symmetric operator on a Hilbert space has a canonical semibounded extension that is a self-adjoint operator obtained by closing its quadratic form. After shifting the lower bound to zero, complete the initial form domain in the norm ; the associated closed form determines the Friedrichs extension. For the positive Laplace-Beltrami operator on a polygonal surface, this specifies the finite-energy realization at corners. Different choices of the initial form domain impose Dirichlet boundary conditions, Neumann boundary conditions, or mixed boundary conditions; the extension does not choose the boundary condition independently of that domain.