Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2609.25027

Fixed Convex-Lens Spectral Constants: Möbius Reduction, Sharp Model Theorems, and Angle-Dependent Bounds

Zijian Zeng
View original source ↗Read source document ↗Export metadata ↓

Abstract

For the intersection of two disks meeting at angle $2α$, let $C(α)$ be the least constant in the associated spectral-set inequality. We give a self-contained M"obius reduction to the corresponding numerical-range problem on a sector and determine the sharp constant for affine square-zero operators $B=λI+N$, $N^2=0$: $C_{\mathrm{sq0}}(α)=π\sinα/(2α)$. A $2\times2$ matrix attains equality and yields an explicit lens lower-bound certificate. At the right angle, we prove the conjectural $\sqrt2$ bound in arbitrary dimension for the full palindromic quadratic family, and give exact rational certificates for several larger parameter families, including a complex post-automorphism disk, the complete imaginary diameter, boundary-reaching phase arcs, and symmetric and asymmetric three-node admissible-kernel problems. We also obtain a strict central bound $|w^2|\leqκ_0<\sqrt2$, a two-small-zero extension, an exact two-moment criterion for the remaining boundary layer, and a verified angle-dependent envelope. Every computer-assisted assertion has an exact rational verifier. These results are dimension-free but do not determine the unrestricted fixed-lens constant.

Explore related subjects

math.SP

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Zijian Zeng. 2026-08-18. Fixed Convex-Lens Spectral Constants: Möbius Reduction, Sharp Model Theorems, and Angle-Dependent Bounds. https://arxiv.org/abs/2609.25027

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

The exterior Steklov problem for Euclidean domains

We investigate the Steklov eigenvalue problem in an exterior Euclidean domain. First, we present several formulations of this problem and establish the equivalences between them. Next, we examine various properties of the exterior Steklov eigenvalues and eigenfunctions. One of our main findings is an Escobar-type lower bound for the first exterior Steklov eigenvalue on convex domains in dimensions three and higher. This bound is expressed in terms of the principal curvatures of the boundary and is sharp, with equality attained for a ball. Moreover, it implies the existence of a sequence of convex domains with fixed volume and the first exterior Steklov eigenvalues tending to infinity. This contrasts with the interior case, as well as with the two-dimensional exterior case, for which we show that an analogue of the Weinstock isoperimetric inequality holds.

math.SP↗
arXiv2026

Lieb-Thirring bounds for Melik-Adamyan canonical Hamiltonians

We study a class of positive matrix Hamiltonians arising from the canonical differential expressions of Melik--Adamyan and appearing in the appendix of Alpay--Gohberg. Let $J$ and $B$ be self-adjoint involutions on $\mathbb C^{2n}$ satisfying $JB=-BJ$, and let $H>0$ satisfy $HJH=J$. For $m>0$ we consider $$ \mathcal A_{m,H}=H^{-1}\left(-iJ\frac{d}{dt}+mB\right) $$ in the weighted space $L^2_H$. A locally absolutely continuous $J$-unitary gauge $Θ$ representing $H$ reduces this expression to the free massive Dirac operator plus the Hermitian coefficient $$ P_{m,Θ}=-iΘ^*JΘ'+m(Θ^*BΘ-B). $$ Whenever this coefficient belongs to $L^2$, the corresponding self-adjoint realization, including its operator domain, is independent of the chosen representing gauge. Minimizing $\int\mathrm{Tr}|P_{m,Θ}|^2$ over the gauge fibre defines an intrinsic energy. A two-sided Birman--Schwinger decoupling, combined with a truncated pseudo-relativistic estimate proved here, gives a $3/2$-moment bound for all eigenvalues in the gap $(-m,m)$ in terms of this energy. The Dirac estimate applies to arbitrary Hermitian matrix coefficients in $L^2$ and requires no sign condition. On the half-line we treat every self-adjoint Lagrangian boundary condition. Two reflection-compatible conditions require no endpoint correction, while an arbitrary condition contributes at most $2nm^{3/2}$. At zero mass, the optimal-gauge energy is computed explicitly in terms of $H^{-1/2}H'H^{-1/2}$. For a scalar hyperbolic-rotation family the massive gauge minimization reduces exactly to a one-dimensional phase functional. We prove existence of a minimizer in the principal phase sector and give an explicit trial phase that strictly and quantitatively improves the positive lift whenever the corresponding first variation is nonzero.

math.SP↗
arXiv2026

Neumann eigenvalues of isosceles triangles: monotonicity and asymptotics

We prove Laugesen and Siudeja's conjecture on Neumann eigenvalues of isosceles triangles. After multiplication by the squared diameter, the first positive symmetric eigenvalue increases strictly with the apex angle $θ$, while the first antisymmetric eigenvalue decreases strictly up to the equilateral triangle and increases strictly thereafter. \rev{The proof combines a slice-average estimate of directional energies with the explicit equilateral eigenfunction.} A quadratic correction across slices also gives a two-term expansion for every fixed positive spectral index $k$ as $θ\downarrow0$, with relative correction $θ^2/6$ and remainder $O_k(θ^4)$.

math.SP↗