Edge spectral invariants and second-order Steklov asymptotics for the regular tetrahedron
Abstract
We give a two-term Weyl asymptotic for the Steklov spectrum of a solid regular tetrahedron. Its linear coefficient is the total edge length times an explicit, absolutely convergent dihedral invariant. A trace-class difference of massive Neumann-to-Dirichlet operators computes the full invariant, including the continuous spectrum. The main analytic step is a fixed-time cumulative trace comparison between the compact tetrahedron and its exact half-space or dihedral models. Localized Hilbert-Schmidt defects have vanishing high-energy tails; retaining the time factor in Duhamel's formula makes this sufficient for an o(L) cumulative error. A separate trace-class face comparison fixes the model's infinite exterior primitive and permits one observation time independent of the inner collar width. Compact-band Mourre estimates and a harmonic comparison establish immediate reflection/transmission support for incident face waves. Triangular unfolding gives null periodic and dead-end sets for the maximal branching relation. Actual edge-window bounds, adapted conical annuli and an ordered-limit Fourier-Tauberian argument then prove the sharp remainder. The proof treats guided and continuum channels together in the edge trace and requires neither a guided metric graph nor a lower-departure estimate uniform at zero tangential momentum.
Related
- Discreteness of the Steklov Spectrum for Exterior Non-Compact Free-Boundary Minimal Surfaces with Regular Ends of Finite Total Curvature
- An exact hypergeometric Weierstrass representation for Smyth's free-boundary minimal disk in the regular tetrahedron
- Computation of leading coefficients in asymptotics of relative quantum invariants
- Equivariant Steklov homogenisation and stability of spectral representations
- Spectral Convergence of the Discrete Steklov Problem
BibTeX
@misc{park2026edge,
title = {{Edge spectral invariants and second-order Steklov asymptotics for the regular tetrahedron}},
author = {Josiah Park},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.02733},
eprint = {2611.02733},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.02733},
note = {CC BY-SA 4.0},
}
RIS
TY - UNPB
TI - Edge spectral invariants and second-order Steklov asymptotics for the regular tetrahedron
AU - Josiah Park
PY - 2026
DA - 2026/11/02
PB - arXiv
DO - 10.48550/arxiv.2611.02733
UR - https://arxiv.org/abs/2611.02733
AB - We give a two-term Weyl asymptotic for the Steklov spectrum of a solid regular tetrahedron. Its linear coefficient is the total edge length times an explicit, absolutely convergent dihedral invariant. A trace-class difference of massive Neumann-to-Dirichlet operators computes the full invariant, including the continuous spectrum. The main analytic step is a fixed-time cumulative trace comparison between the compact tetrahedron and its exact half-space or dihedral models. Localized Hilbert-Schmidt defects have vanishing high-energy tails; retaining the time factor in Duhamel's formula makes this sufficient for an o(L) cumulative error. A separate trace-class face comparison fixes the model's infinite exterior primitive and permits one observation time independent of the inner collar width. Compact-band Mourre estimates and a harmonic comparison establish immediate reflection/transmission support for incident face waves. Triangular unfolding gives null periodic and dead-end sets for the maximal branching relation. Actual edge-window bounds, adapted conical annuli and an ordered-limit Fourier-Tauberian argument then prove the sharp remainder. The proof treats guided and continuum channels together in the edge trace and requires neither a guided metric graph nor a lower-departure estimate uniform at zero tangential momentum.
ER -