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Integral formulation of Klein-Gordon singular waveguides

Guillaume Bal, Jeremy Hoskins, Solomon Quinn, Manas Rachh

TL;DR

This work develops a robust integral-equation framework for surface waves localized along curved interfaces separating insulating regions in the 2D Klein-Gordon setting. By introducing an analytic preconditioner built from a one-dimensional interface Green’s function, it transforms the ill-posed continuous-spectrum problem into a well-posed, invertible system on exponentially weighted spaces, enabling fast high-order numerics. The authors prove bounded invertibility for relevant parameter regimes, establish regularity and radiation properties, and extend the construction to a two-mass scenario. Numerically, the method combines boundary-integral discretization with fast multipole and sweeping techniques, achieving linear-time performance in the number of interface discretization points and demonstrating accurate, efficient simulations on curved interfaces and scattering configurations with clear physical interpretation. The framework is general and is expected to extend to Dirac-type models, 3D settings, and multiple interfaces, offering a practical tool for studying topological edge states and related waveguide phenomena.

Abstract

We consider the analysis of singular waveguides separating insulating phases in two-space dimensions. The insulating domains are modeled by a massive Schrödinger equation and the singular waveguide by appropriate jump conditions along the one-dimensional interface separating the insulators. We present an integral formulation of the problem and analyze its mathematical properties. We also implement a fast multipole and sweeping-accelerated iterative algorithm for solving the integral equations, and demonstrate numerically the fast convergence of this method. Several numerical examples of solutions and scattering effects illustrate our theory.

Integral formulation of Klein-Gordon singular waveguides

TL;DR

This work develops a robust integral-equation framework for surface waves localized along curved interfaces separating insulating regions in the 2D Klein-Gordon setting. By introducing an analytic preconditioner built from a one-dimensional interface Green’s function, it transforms the ill-posed continuous-spectrum problem into a well-posed, invertible system on exponentially weighted spaces, enabling fast high-order numerics. The authors prove bounded invertibility for relevant parameter regimes, establish regularity and radiation properties, and extend the construction to a two-mass scenario. Numerically, the method combines boundary-integral discretization with fast multipole and sweeping techniques, achieving linear-time performance in the number of interface discretization points and demonstrating accurate, efficient simulations on curved interfaces and scattering configurations with clear physical interpretation. The framework is general and is expected to extend to Dirac-type models, 3D settings, and multiple interfaces, offering a practical tool for studying topological edge states and related waveguide phenomena.

Abstract

We consider the analysis of singular waveguides separating insulating phases in two-space dimensions. The insulating domains are modeled by a massive Schrödinger equation and the singular waveguide by appropriate jump conditions along the one-dimensional interface separating the insulators. We present an integral formulation of the problem and analyze its mathematical properties. We also implement a fast multipole and sweeping-accelerated iterative algorithm for solving the integral equations, and demonstrate numerically the fast convergence of this method. Several numerical examples of solutions and scattering effects illustrate our theory.
Paper Structure (24 sections, 11 theorems, 138 equations, 12 figures)

This paper contains 24 sections, 11 theorems, 138 equations, 12 figures.

Table of Contents

  1. Introduction
  2. Contributions
  3. Relation to other work
  4. Paper outline
  5. Mathematical preliminaries
  6. Detailed formulation of the problem
  7. Relation to Dirac equations and topological insulators
  8. Boundary integral operators and their properties
  9. Analytical results
  10. The boundary integral equations
  11. Intuition for the integral formulation
  12. Extension to two masses
  13. Proofs of main analytical results
  14. Proof of Theorem \ref{['thm:invL']}
  15. Proofs of Theorems \ref{['thm:reg0']} and \ref{['thm:outgoing']}
  16. ...and 9 more sections

Key Result

Theorem 3.3

Fix $m_0 > 0$ and $E_0 \in (-m_0, m_0) \setminus \{0\}$. Define $\omega_0 := \sqrt{m_0^2 - E_0^2}$ and set $m=\lambda m_0$ and $E = \lambda E_0$ for $\lambda \in \mathbb{R}$. Define the function $\alpha_*$ by where we recall the definitions of $\beta$ and $c$ in eq:beta and eq:c. Then for any $0 < \alpha < \alpha_* (\omega_0)$, the integral equation eq:bif admits a unique solution $\rho \in L^2_\

Figures (12)

  • Figure 2.1: Geometry
  • Figure 5.2: Schematic of the discretization approach used by chunkie. In the inlay, bounds between 'chunks' are shown with vertical lines, and discretization nodes are denoted by red triangles. For clarity, the 'panel' shown is $8^{\rm th}$ rather than $16^{\rm th}$.
  • Figure 5.3: The factorized linear system, after discretization.
  • Figure 6.4: The interfaces $\Gamma_0$ (top left), $\Gamma_1$ (top right), $\Gamma_2$ (bottom left) and $\Gamma_3$ (bottom right), with respective sources at $(0,2.5)$, $(0,1)$, $(0,-7)$ and $(0,3)$ as indicated by the red dot. Outside the plotted region, the interfaces extend linearly to infinity.
  • Figure 6.5: Densities $\mu$ and $\rho$ (top left panel), relative error of the computed solution at the indicated points as a function of $n_c$ (top right panel), and Green's function $u$ (bottom two rows) corresponding to the flat interface $\Gamma_0$ with $m=2$ and $E=1$. The top left panel zooms in on the region $[10,10]\times \{0\} \subset \Gamma_0$, with $t=60$ corresponding to the point $(0,0) \in \Gamma_0$.
  • ...and 7 more figures

Theorems & Definitions (25)

  • Remark 2.1
  • Remark 2.2
  • Remark 3.1
  • Remark 3.2
  • Theorem 3.3
  • Remark 3.4
  • Theorem 3.5
  • Theorem 3.6
  • Theorem 3.7
  • Theorem 3.8
  • ...and 15 more