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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 (λ1,λ2).

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Cite

BibTeXRIS
BibTeX
@misc{wang2026improvements,
  title = {{Improvements of Classical Eigenvalue Inequalities I: Quantitative Hersch-deficit and the Joint Spectral Range on the Two-Sphere}},
  author = {Zuoqin Wang and Qingwei Zeng},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.32443},
  eprint = {2609.32443},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.32443},
}
RIS
TY  - UNPB
TI  - Improvements of Classical Eigenvalue Inequalities I: Quantitative Hersch-deficit and the Joint Spectral Range on the Two-Sphere
AU  - Zuoqin Wang
AU  - Qingwei Zeng
PY  - 2026
DA  - 2026/09/26
PB  - arXiv
DO  - 10.48550/arxiv.2609.32443
UR  - https://arxiv.org/abs/2609.32443
AB  - 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).
ER  -