These notes are based in the lecture series on the subject given by the second author during the Summer School Geometric, Algebraic and Topological Methods for Quantum Field Theory, in Villa de Leyva (Colombia), July 2017. The goal of the lectures was to give a first overview of the algebra of global pseudo-differential operators associated to the boundary value problem on a domain \(\Omega \) , associated to the spectral data furnished by a differential operator \(\mathfrak {L}\) with fixed boundary conditions, giving rise to biorthogonal systems of eigenfunctions in \(L^2(\Omega )\) .