  We use algebras of pseudodifferential operators on groupoids to study
geometric operators on non-compact manifolds and singular spaces. The first
step is to establish that the geometric operators are in our algebras. This
then leads to criteria for Fredholmness for geometric operators on suitable
non-compact manifolds, as well as to an inductive procedure to study their
essential spectrum. As an application, we answer a question of Melrose on the
essential spectrum of the Laplace operator on manifolds with multi-cylindrical
ends.

Spectral Theory