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Spectral theory for self-adjoint Dirac operators with periodic potentials and inverse scattering transform for the defocusing nonlinear Schroedinger equation with periodic boundary conditions

Gino Biondini, Zechuan Zhang

TL;DR

This work develops a complete Riemann–Hilbert problem framework for the direct and inverse spectral theory of a periodic self-adjoint Zakharov–Shabat Dirac operator and uses it to solve the defocusing NLS IVP with periodic boundary conditions. It shows that a single Dirichlet-spectrum set suffices to reconstruct the potential and the associated NLS solution, extends the IST to infinite-genus potentials, and provides time-evolution and periodicity criteria within the same RHP formalism. The paper connects the spectral data to a time-dependent RHP for the IVP, derives scalar periodicity conditions, and links the approach to finite-gap (Baker–Akhiezer) theory while offering explicit genus-zero examples. Overall, it unifies periodic inverse spectral theory with IST for NLS, enabling rigorous reconstruction, periodicity analysis, and potential extensions to broader integrable systems and semiclassical regimes.

Abstract

The inverse spectral theory for a self-adjoint one-dimensional Dirac operator associated periodic potentials is formulated via a Riemann-Hilbert problem approach. The resulting formalism is also used to solve the initial value problem for the nonlinear Schrodinger (NLS) equation. A uniqueness theorem for the solutions of the Riemann-Hilbert problem is established, which provides a new method for obtaining the potential from the spectral data. Two additional, scalar Riemann-Hilbert problems are also formulated that provide conditions for the periodicity in space and time of the solution generated by arbitrary sets of spectral data. The formalism applies for both finite-genus and infinite-genus potentials. The formalism also shows that only a single set of Dirichlet eigenvalues is needed in order to uniquely reconstruct the potential of the Dirac operator and the corresponding solution of the defocusing NLS equation, in contrast with the representation of the solution of the NLS equation via the finite-genus formalism, in which two different sets of Dirichlet eigenvalues are used.

Spectral theory for self-adjoint Dirac operators with periodic potentials and inverse scattering transform for the defocusing nonlinear Schroedinger equation with periodic boundary conditions

TL;DR

This work develops a complete Riemann–Hilbert problem framework for the direct and inverse spectral theory of a periodic self-adjoint Zakharov–Shabat Dirac operator and uses it to solve the defocusing NLS IVP with periodic boundary conditions. It shows that a single Dirichlet-spectrum set suffices to reconstruct the potential and the associated NLS solution, extends the IST to infinite-genus potentials, and provides time-evolution and periodicity criteria within the same RHP formalism. The paper connects the spectral data to a time-dependent RHP for the IVP, derives scalar periodicity conditions, and links the approach to finite-gap (Baker–Akhiezer) theory while offering explicit genus-zero examples. Overall, it unifies periodic inverse spectral theory with IST for NLS, enabling rigorous reconstruction, periodicity analysis, and potential extensions to broader integrable systems and semiclassical regimes.

Abstract

The inverse spectral theory for a self-adjoint one-dimensional Dirac operator associated periodic potentials is formulated via a Riemann-Hilbert problem approach. The resulting formalism is also used to solve the initial value problem for the nonlinear Schrodinger (NLS) equation. A uniqueness theorem for the solutions of the Riemann-Hilbert problem is established, which provides a new method for obtaining the potential from the spectral data. Two additional, scalar Riemann-Hilbert problems are also formulated that provide conditions for the periodicity in space and time of the solution generated by arbitrary sets of spectral data. The formalism applies for both finite-genus and infinite-genus potentials. The formalism also shows that only a single set of Dirichlet eigenvalues is needed in order to uniquely reconstruct the potential of the Dirac operator and the corresponding solution of the defocusing NLS equation, in contrast with the representation of the solution of the NLS equation via the finite-genus formalism, in which two different sets of Dirichlet eigenvalues are used.
Paper Structure (16 sections, 41 theorems, 266 equations, 2 figures)

This paper contains 16 sections, 41 theorems, 266 equations, 2 figures.

Table of Contents

  1. Introduction and outline
  2. Direct spectral theory for the periodic self-adjoint Zakharov-Shabat problem
  3. Modified Bloch-Floquet solutions
  4. Inverse spectral theory for the periodic self-adjoint Zakharov-Shabat problem
  5. Initial value problem for the defocusing NLS equation with periodic BC
  6. Periodicity conditions in space and time
  7. Baker-Akhiezer functions and finite-gap potentials
  8. Concluding remarks
  9. Notation and standard definitions
  10. Basic properties of the spectrum
  11. Asymptotics of the modified fundamental matrix as $z\to\infty$
  12. Asymptotics of the modified Bloch-Floquet solutions as $z\to\infty$
  13. Order of growth of eigenfunctions and monodromy matrix at infinity
  14. Phragmen-Lindelöf theorem
  15. Alternative sets of Dirichlet eigenvalues
  16. ...and 1 more sections

Key Result

Proposition 2.2

The Dirichlet spectrum $\{\mu_j\}$ with base point $x_o=0$ coincides with the set

Figures (2)

  • Figure 1: Schematic representation of the behavior of the Floquet discriminant $\Delta(z)$ (vertical axis) as a function of $z$ (horizontal axis), together with the periodic and antiperiodic spectrum, the spectral bands (in red) and gaps and the Dirichlet eigenvalues (blue dots). See text for further details.
  • Figure 2: The branch cuts along the real $z$-axis for the functions $\sqrt{\Delta^2-1}$ (in red) and for $\sqrt[4]{\Delta^2-1}$ (in blue), as well as the segments (thick black) where $\sqrt{\Delta^2-1}$ and $\sqrt[4]{\Delta^2-1}$ are both positive. Top: $g=0$. Middle: $g=1$. Bottom: $g=2$.

Theorems & Definitions (88)

  • Remark 2.1
  • Proposition 2.2
  • proof
  • Theorem 2.3
  • Definition 2.4
  • Lemma 2.5
  • proof
  • Definition 2.6
  • Remark 2.7
  • Definition 2.8
  • ...and 78 more