# The inverse approach to Dirac-type systems based on the $A$-function concept

## TL;DR

This paper introduces a novel inverse method for Dirac-type systems using the A-function, enabling unique recovery of potential coefficients and extending to matrix-valued Schrödinger operators.

## Contribution

It develops an A-equation framework for Dirac-type systems, proving its unique solvability and applicability to matrix Schrödinger operators, inspired by Simon's inverse approach.

## Key findings

- Derived the A-equation for Dirac systems
- Proved unique solvability under positivity conditions
- Extended method to matrix Schrödinger operators

## Abstract

The principal objective in this paper is a new inverse approach to general Dirac-type systems modeled after B. Simon's 1999 inverse approach to half-line Schr\"odinger operators. In particular, we derive the so-called A-equation associated to Dirac-type systems and, given a fundamental positivity condition, we prove that this integro-differential equation for A is uniquely solvable. We show how to recover the matrix-valued potential coefficient from A. This approach is also applicable to matrix-valued Schr\"odinger operators on a half-line.