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arXiv · 2609.32443

Improvements of Classical Eigenvalue Inequalities I: Quantitative Hersch-deficit and the Joint Spectral Range on the Two-Sphere

Zuoqin Wang·Qingwei Zeng
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Abstract

For area-$4π$ metrics on the two-sphere, we obtain quantitative refinements of classical eigenvalue inequalities and study the joint behavior of the first two Laplace eigenvalues. We prove a sharp quadratic improvement of Hersch's reciprocal-sum inequality in terms of the first eigenvalue, together with a simultaneous quadratic estimate for the first three eigenvalues and a geometric stability result for admissible measures. We also establish a quantitative refinement of Nadirashvili's second-eigenvalue inequality with sharp exponential scale. Finally, using explicit spectral families due to Petrides, a degree argument, and a refined finite-dimensional matrix method, we obtain nontrivial inner and outer bounds for the joint range of $(λ_1,λ_2)$.

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Zuoqin Wang, Qingwei Zeng. 2026-09-26. Improvements of Classical Eigenvalue Inequalities I: Quantitative Hersch-deficit and the Joint Spectral Range on the Two-Sphere. https://arxiv.org/abs/2609.32443

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