# Functional Analysis, Linear Structures and Applications Seminar

### Index formula on compactified domains and perturbed Dirac operators

We consider the Fredholm index problem on open domains in $\mathbb{R}^n$ with a compactification to a domain with boundary, for a class of pseudodifferential operators suitably generated by an algebra of vector fields tangent to the boundary. This class falls in the framework of pseudodifferential calculi on so-called Lie manifolds, defined by Ammann, Lauter, Nistor, building on work by Melrose. Examples are, e.g., domains with cylindrical ends, or asymptotically Euclidean.

We present an index formula in case the generating vector fields vanish at infinity, that is, at the boundary. There exists a commutative full symbol, defining an invertible function on a $2n-1$-sphere and the index depends only on a topological invariant associated to this function, e.g., if $n=1$, we have the winding number.

As an application, we study the index of Dirac operators coupled with an unbounded potential on an even dimensional domain. We reduce to the computation of the index of a perturbed Dirac operator on a commutative domain, now with a bounded potential, and we obtain a Callias-type index formula in these two cases.

Current organizers: Helena Mascarenhas, Ângela Mestre.