# Functional model for extensions of symmetric operators and applications to scattering theory

## TL;DR

This paper develops a new representation for the scattering matrix of symmetric operator extensions and applies it to solve inverse scattering problems on quantum graphs with delta-type conditions.

## Contribution

It introduces an explicit formula for the scattering matrix of almost solvable symmetric operator extensions and uses it to recover coupling constants in inverse quantum graph problems.

## Key findings

- Derived a new representation for the scattering matrix
- Explicitly recovered coupling constants in quantum graphs
- Enhanced understanding of scattering theory for operator extensions

## Abstract

On the basis of the explicit formulae for the action of the unitary group of exponentials corresponding to almost solvable extensions of a given closed symmetric operator with equal deficiency indices, we derive a new representation for the scattering matrix for pairs of such extensions. We use this representation to explicitly recover the coupling constants in the inverse scattering problem for a finite non-compact quantum graph with $\delta$-type vertex conditions.