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An inverse problem for the matrix Schrodinger operator on the half-line with a general boundary condition

Xiao-Chuan Xu, Yi-Jun Pan

Abstract

In this work, we study the inverse spectral problem, using the Weyl matrix as the input data, for the matrix Schrodinger operator on the half-line with the boundary condition being the form of the most general self-adjoint. We prove the uniqueness theorem, and derive the main equation and prove its solvability, which yields a theoretical reconstruction algorithm of the inverse problem.

An inverse problem for the matrix Schrodinger operator on the half-line with a general boundary condition

Abstract

In this work, we study the inverse spectral problem, using the Weyl matrix as the input data, for the matrix Schrodinger operator on the half-line with the boundary condition being the form of the most general self-adjoint. We prove the uniqueness theorem, and derive the main equation and prove its solvability, which yields a theoretical reconstruction algorithm of the inverse problem.

Paper Structure

This paper contains 4 sections, 7 theorems, 105 equations, 2 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. The Weyl matrix
  3. The uniqueness theorem
  4. Solution of the inverse problem

Key Result

Lemma 1

The Jost matrix $J(\rho)$ defined in yj26 is analytic for ${\rm Im}\rho >0$, and continuous for $\rho \in \Omega$, and satisfies the asymptotics where

Figures (2)

  • Figure 4.1: The contour $\gamma$
  • Figure 4.2: The contour $\gamma_{R}^{0}$

Theorems & Definitions (14)

  • Lemma 1
  • proof
  • Proposition 1
  • proof
  • Lemma 2
  • proof
  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • ...and 4 more