Scattering at low energies on manifolds with cylindrical ends and stable systoles

TL;DR

This paper explores the relationship between scattering theory for p-forms on manifolds with cylindrical ends and cohomology, revealing how scattering lengths relate to geometric data like systoles.

Contribution

It provides a cohomological interpretation of the scattering matrix and scattering length, connecting spectral data with geometric invariants of the manifold.

Findings

Scattering matrix at zero energy corresponds to boundary cohomology.

Scattering length relates to the norm of cohomology classes and connecting homomorphisms.

Scattering lengths can be estimated using volumes of homological systoles.

Abstract

Scattering theory for p-forms on manifolds with cylindrical ends has a direct interpretation in terms of cohomology. Using the Hodge isomorphism,the scattering matrix at low energy may be regarded as operator on the cohomology of the boundary. Its value at zero describes the image of the absolute cohomology in the cohomology of the boundary. We show that the so-called scattering length, the Eisenbud-Wigner time delay at zero energy, has a cohomological interpretation as well. Namely, it relates the norm of a cohomology class on the boundary to the norm of its image under the connecting homomorphism in the long exact sequence in cohomology. An interesting consequence of this is that one can estimate the scattering lengths in terms of geometric data like the volumes of certain homological systoles.