Details
| Original language | English |
|---|---|
| Pages (from-to) | 361-408 |
| Number of pages | 48 |
| Journal | Analysis and PDE |
| Volume | 18 |
| Issue number | 2 |
| Publication status | Published - 5 Feb 2025 |
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 R3. 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 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 clarify the function spaces on which the determinants need to be taken.
Keywords
- Casimir energy, layer potential, Maxwell equations, trace formula
ASJC Scopus subject areas
- Mathematics(all)
- Analysis
- Mathematics(all)
- Numerical Analysis
- Mathematics(all)
- Applied Mathematics
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In: Analysis and PDE, Vol. 18, No. 2, 05.02.2025, p. 361-408.
Research output: Contribution to journal › Article › Research › peer review
}
TY - JOUR
T1 - The relative trace formula in electromagnetic scattering and boundary layer operators
AU - Strohmaier, Alexander
AU - Waters, Alden
N1 - Publisher Copyright: © 2025 The Authors, under license to MSP (Mathematical Sciences Publishers). Distributed under the Creative Commons Attribution License 4.0 (CC BY).
PY - 2025/2/5
Y1 - 2025/2/5
N2 - 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 R3. 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 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 clarify the function spaces on which the determinants need to be taken.
AB - 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 R3. 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 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 clarify the function spaces on which the determinants need to be taken.
KW - Casimir energy
KW - layer potential
KW - Maxwell equations
KW - trace formula
UR - http://www.scopus.com/inward/record.url?scp=85217515825&partnerID=8YFLogxK
U2 - 10.48550/arXiv.2111.15331
DO - 10.48550/arXiv.2111.15331
M3 - Article
AN - SCOPUS:85217515825
VL - 18
SP - 361
EP - 408
JO - Analysis and PDE
JF - Analysis and PDE
SN - 2157-5045
IS - 2
ER -