Scattering solutions in a network of thin fibers: small diameter asymptotics

TL;DR

This paper derives small diameter asymptotics for scattering solutions in a network of thin fibers, linking them to quantum graph problems and calculating vertex gluing conditions based on scattering data.

Contribution

It introduces a novel asymptotic analysis connecting thin fiber networks to quantum graph models with explicit vertex conditions.

Findings

Asymptotics expressed via solutions on the limiting quantum graph

Explicit calculation of Lagrangian gluing conditions at vertices

Gluing conditions depend on scattering data for each junction

Abstract

Small diameter asymptotics is obtained for scattering solutions in a network of thin fibers. The asymptotics is expressed in terms of solutions of related problems on the limiting quantum graph. We calculate the Lagrangian gluing conditions at vertices for the problems on the limiting graph. If the frequency of the incident wave is above the bottom of the absolutely continuous spectrum, the gluing conditions are formulated in terms of the scattering data for each individual junction of the network.