Singular integral operators on non-compact manifolds and analysis on polyhedral domains

We review the definition of a Lie manifold $(M,\text{VV})$ and the construction of the algebra $\mathrm{\Psi }\text{sp}\mathrm{\infty }\text{sb}\text{VV}(M)$ of pseudodifferential operators on a Lie manifold $(M,\text{VV})$. We give some concrete Fredholmness conditions for pseudodifferential operators in $\mathrm{\Psi }\text{sp}\mathrm{\infty }\text{sb}\text{VV}(M)$ for a large class of Lie manifolds $(M,\text{VV})$. These Fredholmness conditions have applications to boundary value problems on polyhedral domains and to non-linear PDEs on non-compact manifolds. As an application, we determine the spectrum of the Dirac operator on a manifold with multi-cylindrical ends.