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Inverse nodal problems for Dirac differential operators with jump condition

Baki Keskin

TL;DR

This work addresses the inverse nodal problem for a one-dimensional Dirac system with an internal discontinuity and jump conditions. It develops new uniform-asymptotics for the eigenfunctions, defines a characteristic function $\Delta(\mu)$, and derives precise eigenvalue and nodal expansions in terms of the potential and boundary data. The main theoretical contribution is a uniqueness result: a dense nodal subset of zeros of the first eigenfunction component uniquely determines the coefficient function $\Omega(x)$ and the boundary parameter $\theta$, accompanied by an explicit reconstruction algorithm to recover $V(x)$, $m$, and $\theta$ from nodal data. A worked example demonstrates the reconstruction, highlighting the practical applicability of the method to discontinuous Dirac operators. Overall, the paper advances inverse nodal theory for discontinuous Dirac systems and provides a concrete, data-driven reconstruction procedure.

Abstract

This paper deals with an inverse nodal problem for the Dirac differential operator with the discontinuity conditions inside the interval. We obtain a new approach for asymptotic expressions of the solutions and prove that the coefficients of the Dirac system can be determined uniquely by a dense subset of the nodal points (zeros of the first component of the eigenfunction). We also provide an algorithm for constructing the solution of this inverse nodal problem.

Inverse nodal problems for Dirac differential operators with jump condition

TL;DR

This work addresses the inverse nodal problem for a one-dimensional Dirac system with an internal discontinuity and jump conditions. It develops new uniform-asymptotics for the eigenfunctions, defines a characteristic function , and derives precise eigenvalue and nodal expansions in terms of the potential and boundary data. The main theoretical contribution is a uniqueness result: a dense nodal subset of zeros of the first eigenfunction component uniquely determines the coefficient function and the boundary parameter , accompanied by an explicit reconstruction algorithm to recover , , and from nodal data. A worked example demonstrates the reconstruction, highlighting the practical applicability of the method to discontinuous Dirac operators. Overall, the paper advances inverse nodal theory for discontinuous Dirac systems and provides a concrete, data-driven reconstruction procedure.

Abstract

This paper deals with an inverse nodal problem for the Dirac differential operator with the discontinuity conditions inside the interval. We obtain a new approach for asymptotic expressions of the solutions and prove that the coefficients of the Dirac system can be determined uniquely by a dense subset of the nodal points (zeros of the first component of the eigenfunction). We also provide an algorithm for constructing the solution of this inverse nodal problem.

Paper Structure

This paper contains 2 sections, 4 theorems, 43 equations.

Table of Contents

  1. Introduction
  2. Main Results

Key Result

Theorem 1

For $\left\vert \mu \right\vert \rightarrow \infty ,$ uniformly in $x,$the functions $\psi _{1}(x,\mu )$ and $\psi _{2}(x,\mu )$ have the following representations : for $x<\dfrac{\pi }{2}$ and for $x>\dfrac{\pi }{2}$ where, $\rho (x)=\dfrac{1}{2}\int\limits_{0}^{x}(p(t)+q(t))dt$ and $\tau =\func{Im}\mu .$

Theorems & Definitions (7)

  • Theorem 1
  • proof
  • Lemma 1
  • proof
  • Theorem 2
  • Theorem 3
  • Example 1