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The relative trace formula in electromagnetic scattering and boundary layer operators

Alexander Strohmaier, Alden Waters

TL;DR

This work extends trace-formula techniques to Maxwell scattering on Lipschitz obstacles by formulating and analyzing relative and absolute Laplacians on divergence-free vector fields. It introduces Maxwell boundary layer operators and a holomorphic determinant framework, defining $\Xi(\lambda)$ to capture spectral differences and linking traces to contour integrals of $\Xi$ and its derivative. The authors prove that the relevant D$_{\mathrm{rel},f}$ and D$_{\mathrm{abs},f}$ are trace-class for suitable $f$ and derive a relative zeta function as well as determinant-based expressions for Casimir-type energies. The results clarify function spaces and provide Maxwell analogues of Birman-Krein-type formulas, with explicit handling of non-smooth Lipschitz boundaries and topology via the boundary-layer machinery.

Abstract

This paper establishes trace-formulae for a class of operators defined in terms of the functional calculus for the Laplace operator on divergence-free vector fields with relative and absolute boundary conditions on Lipschitz domains in $\mathbb{R}^3$. Spectral and scattering theory of the absolute and relative Laplacian is equivalent to the spectral analysis and scattering theory for Maxwell equations. The trace-formulae allow for unbounded functions in the functional calculus that are not admissible in the Birman-Krein formula. In special cases the trace-formula reduces to a determinant formula for the Casimir energy that is being used in the physics literature for the computation of the Casimir energy for objects with metallic boundary conditions. Our theorems justify these formulae in the case of electromagnetic scattering on Lipschitz domains, give a rigorous meaning to them as the trace of certain trace-class operators, and clarifies the function spaces on which the determinants need to be taken.

The relative trace formula in electromagnetic scattering and boundary layer operators

TL;DR

This work extends trace-formula techniques to Maxwell scattering on Lipschitz obstacles by formulating and analyzing relative and absolute Laplacians on divergence-free vector fields. It introduces Maxwell boundary layer operators and a holomorphic determinant framework, defining to capture spectral differences and linking traces to contour integrals of and its derivative. The authors prove that the relevant D and D are trace-class for suitable and derive a relative zeta function as well as determinant-based expressions for Casimir-type energies. The results clarify function spaces and provide Maxwell analogues of Birman-Krein-type formulas, with explicit handling of non-smooth Lipschitz boundaries and topology via the boundary-layer machinery.

Abstract

This paper establishes trace-formulae for a class of operators defined in terms of the functional calculus for the Laplace operator on divergence-free vector fields with relative and absolute boundary conditions on Lipschitz domains in . Spectral and scattering theory of the absolute and relative Laplacian is equivalent to the spectral analysis and scattering theory for Maxwell equations. The trace-formulae allow for unbounded functions in the functional calculus that are not admissible in the Birman-Krein formula. In special cases the trace-formula reduces to a determinant formula for the Casimir energy that is being used in the physics literature for the computation of the Casimir energy for objects with metallic boundary conditions. Our theorems justify these formulae in the case of electromagnetic scattering on Lipschitz domains, give a rigorous meaning to them as the trace of certain trace-class operators, and clarifies the function spaces on which the determinants need to be taken.
Paper Structure (21 sections, 33 theorems, 258 equations, 2 figures)

This paper contains 21 sections, 33 theorems, 258 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Statement of main results
  3. Discussion
  4. Relation to the Birman-Krein formula
  5. Organisation of the paper
  6. Function spaces on Lipschitz domains
  7. Laplace operators on the interior domain
  8. The relative Laplacian
  9. The absolute Laplacian
  10. Relation to singular and de Rham cohomology groups
  11. Laplace operators on the exterior domain
  12. The exterior absolute Laplacian
  13. The combined relative operators and the Birman-Krein formula
  14. Maxwell boundary layer operators
  15. Estimates and low energy expansions for the Layer potential operators
  16. ...and 6 more sections

Key Result

Theorem 1.1

The operator $\mathcal{L}_\lambda \mathcal{L}^{-1}_{D,\lambda}$ is well-defined and a trace-class perturbation of the identity for any complex $\lambda$ with $\operatorname{Im}(\lambda)>0$. It therefore has a well-defined Fredholm determinant $\mathrm{det}(\mathcal{L}_\lambda \mathcal{L}^{-1}_{D,\la where the branch of the logarithm has been fixed by continuity, extends to a holomorphic function i

Figures (2)

  • Figure 1: A Lipschitz domain $\Omega$ consisting of four connected components $\Omega_1,\Omega_2,\Omega_3,\Omega_4$.
  • Figure 2: The sectors $\mathfrak{S}_\epsilon$, $\mathfrak{D}_\frac{\epsilon}{2}$ and the corresponding contours.

Theorems & Definitions (67)

  • Theorem 1.1
  • Definition 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Lemma 3.1
  • proof
  • Theorem 5.1: Theorem 1.5 in SWB
  • Lemma 6.1
  • Lemma 6.2
  • ...and 57 more