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Inverse Spectral Analysis of Singular Radial AKNS Operators

Damien Gobin, Benoît Grébert, Bernard Helffer, François Nicoleau

Abstract

We study an inverse spectral problem for singular AKNS operators based on spectral data associated with two distinct values of the effective angular momentum parameter $κ\,$. Our main focus is the local inverse problem near the zero potential. For the pairs $(κ_1,κ_2)=(0,1)$, $(1,2)$ and $(0,3)\,$, we establish local uniqueness. For $(0,2)\,$, we prove that the Fréchet differential of the spectral map at the origin is injective, while the question whether its range is closed remains open.

Inverse Spectral Analysis of Singular Radial AKNS Operators

Abstract

We study an inverse spectral problem for singular AKNS operators based on spectral data associated with two distinct values of the effective angular momentum parameter . Our main focus is the local inverse problem near the zero potential. For the pairs , and , we establish local uniqueness. For , we prove that the Fréchet differential of the spectral map at the origin is injective, while the question whether its range is closed remains open.
Paper Structure (32 sections, 17 theorems, 340 equations, 1 figure)

This paper contains 32 sections, 17 theorems, 340 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Eigenvalue analysis in the unperturbed case $V=0$
  3. The case $\lambda=0$
  4. The case $\lambda \not=0$
  5. Symmetries
  6. Summary and notation
  7. The case $\kappa=0$
  8. Spectral map and the linearized problem at $V=0$.
  9. Differential of $\lambda_{\kappa,n}(p,q)$ and the spectral map
  10. The linearized problem at $V=0$
  11. Decoupling of the differential via a continuous isomorphism
  12. Notation.
  13. Reformulation of the injectivity problem
  14. Kneser--Sommerfeld--Type Expansions
  15. Application of the Kneser--Sommerfeld representation
  16. ...and 17 more sections

Key Result

Theorem 1.1

Let $(\kappa_1,\kappa_2)=(0,1)$, $(1,2)$ or $(0,3)\,$. Then the knowledge of the spectra associated with the effective angular momenta $\kappa_1$ and $\kappa_2$ uniquely determines the potential $V=(p,q)\in L^2(0,1)\times L^2(0,1)$ in a neighborhood of the zero potential $V_0=0\,$.

Figures (1)

  • Figure 1: Numerical solutions $w(x)=v_2'(x)$ (left) and $v_2(x)$ (right), computed with Mathematica.

Theorems & Definitions (28)

  • Theorem 1.1: Local uniqueness for the pairs $(0,1)$, $(1,2)$ and $(0,3)$
  • Theorem 1.2: Behavior of the differential of the spectral map
  • Remark 2.1: Normalization constants
  • Remark 2.2
  • Proposition 4.1
  • proof
  • Corollary 4.2
  • proof
  • Proposition 5.1
  • Theorem 5.2
  • ...and 18 more