Improvements of Classical Eigenvalue Inequalities I: Quantitative Hersch-deficit and the Joint Spectral Range on the Two-Sphere
Year: 2026
Type: preprint
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 $(λ_1,λ_2)$.
Source: arXiv (Cornell University)
Authors: Wang, Zuoqin, Zeng, Qingwei
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7214755907