  Given a compact manifold $M$ with boundary $\partial M$, in this paper we
introduce a global symbolic calculus of pseudo-differential operators
associated to $(M,\partial M)$. The symbols of operators with boundary
conditions on $\partial M$ are defined in terms of the biorthogonal expansions
in eigenfunctions of a fixed operator $L$ with the same boundary conditions on
$\partial M$. The boundary $\partial M$ is allowed to have (arbitrary)
singularities. As an application, several criteria for the membership in
Schatten classes on $L^2(M)$ and $r$-nuclearity on $L^p(M)$ are obtained. We
also describe a new addition to the Grothendieck-Lidskii formula in this
setting. Examples and applications are given to operators on $M=[0,1]^n$ with
non-periodic boundary conditions, and of operators with non-local boundary
conditions.
