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Direct and inverse spectral continuity for Dirac operators

Roman Bessonov, Pavel Gubkin

TL;DR

The paper studies stability of direct and inverse spectral problems for half-line Dirac operators with $L^2$-potentials, establishing a concrete two-sided uniform estimate between potential differences and their Schur-function differences. Central to the method is reducing general problems to exactly solvable Kronig–Penney-type lattice models through Schur's algorithm, combined with careful lattice approximations and Wiener–Haagerup-type norm controls to pass to the $L^2$ setting. The work connects Dirac spectral data to canonical Hamiltonian systems and Szegő-type orthogonal-polynomial theory, yielding explicit continuity bounds and a rigorous framework for the spectral map as a homeomorphism. It further analyzes the Sylvester–Winebrenner metric to show that the direct and inverse spectral maps are not uniformly continuous on bounded sets, highlighting fundamental limits of stability despite the overall homeomorphic correspondence.

Abstract

The half-line Dirac operators with $L^2$-potentials can be characterized by their spectral data. It is known that the spectral correspondence is a homeomorphism: close potentials give rise to close spectral data and vice versa. We prove the first explicit two-sided uniform estimate related to this continuity in the general $L^2$-case. The proof is based on an exact solution of the inverse spectral problem for Dirac operators with $δ$-interactions on a half-lattice in terms of the Schur's algorithm for analytic functions.

Direct and inverse spectral continuity for Dirac operators

TL;DR

The paper studies stability of direct and inverse spectral problems for half-line Dirac operators with -potentials, establishing a concrete two-sided uniform estimate between potential differences and their Schur-function differences. Central to the method is reducing general problems to exactly solvable Kronig–Penney-type lattice models through Schur's algorithm, combined with careful lattice approximations and Wiener–Haagerup-type norm controls to pass to the setting. The work connects Dirac spectral data to canonical Hamiltonian systems and Szegő-type orthogonal-polynomial theory, yielding explicit continuity bounds and a rigorous framework for the spectral map as a homeomorphism. It further analyzes the Sylvester–Winebrenner metric to show that the direct and inverse spectral maps are not uniformly continuous on bounded sets, highlighting fundamental limits of stability despite the overall homeomorphic correspondence.

Abstract

The half-line Dirac operators with -potentials can be characterized by their spectral data. It is known that the spectral correspondence is a homeomorphism: close potentials give rise to close spectral data and vice versa. We prove the first explicit two-sided uniform estimate related to this continuity in the general -case. The proof is based on an exact solution of the inverse spectral problem for Dirac operators with -interactions on a half-lattice in terms of the Schur's algorithm for analytic functions.
Paper Structure (24 sections, 48 theorems, 261 equations)

This paper contains 24 sections, 48 theorems, 261 equations.

Table of Contents

  1. Introduction
  2. Dirac operators
  3. Sylvester-Winebrenner theorem
  4. The main result
  5. Historical remarks
  6. Schur's algorithm
  7. Kronig-Penney model
  8. Plan of the paper
  9. Dirac operators with periodic Weyl functions
  10. Canonical Hamiltonian systems
  11. Dirac operators with measures
  12. Auxiliary lemmas
  13. Proof of Theorem \ref{['t2']}
  14. Proof of Corollaries \ref{['c3']}-\ref{['c5']}
  15. Spectral continuity estimates for potentials in L2(R+)
  16. ...and 9 more sections

Key Result

Theorem 1.1

The correspondence $\mathcal{F}: q \mapsto f_{q}$ is a homeo-morphism from $L^2(\mathbb R_+)$ onto $S_{2}(\mathbb C_+)$. Moreover, we have

Theorems & Definitions (49)

  • Theorem 1.1: Sylvester-Winebrenner theorem
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Corollary 1.5: Rakhmanov
  • Corollary 1.6: Baxter
  • Corollary 1.7: Szegő-Golinskii-Ibragimov
  • Theorem 2.1: Krein -- de Branges theorem
  • Lemma 2.2
  • Lemma 2.3
  • ...and 39 more