# Boundary Control method and De Branges spaces. Schr\"odinger equation, Dirac system and Discrete Schr\"odinger operator

## TL;DR

This paper explores the application of the Boundary Control method to inverse problems for Schr"odinger, Dirac, and discrete Schr"odinger operators, establishing links with De Branges spaces and their dynamical interpretations.

## Contribution

It constructs De Branges spaces for these operators and provides a dynamical interpretation of their elements, connecting inverse problems with functional analysis.

## Key findings

- Constructed De Branges spaces for Schr"odinger, Dirac, and discrete Schr"odinger operators.
- Established a natural dynamical interpretation of De Branges space components.
- Linked the Boundary Control method with De Branges spaces in inverse spectral problems.

## Abstract

In the framework of the application of the Boundary Control method to solving the inverse dynamical problems for the one-dimensional Schr\"odinger and Dirac operators on the half-line and semi-infinite discrete Schr\"odinger operator, we establish the connections with the method of De Branges: for each of the system we construct the De Branges space and give a natural dynamical interpretation of all its ingredients: the set of function the De Brange space consists of, the scalar product, the reproducing kernel.