Ashbaugh–Benguria eigenvalue-sum conjecture for Euclidean domains

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Let Ω\Omega be a bounded domain in Euclidean space Rn\mathbb{R}^{n}, and let λi\lambda_i be the ii-th Dirichlet eigenvalue of the Laplacian on Ω\Omega. Let Bn\mathbb{B}^{n} be a ball with the same volume as Ω\Omega, so that vol⁡(Ω)=vol⁡(Bn)\operatorname{vol}(\Omega)=\operatorname{vol}(\mathbb{B}^{n}). Ashbaugh–Benguria conjecture.

λ2+λ3+⋯+λn+1≤nλ2(Bn)λ1(Bn)λ1.\lambda_{2}+\lambda_{3}+\cdots+\lambda_{n+1}\leq n\frac{\lambda_{2}(\mathbb{B}^{n})}{\lambda_{1}(\mathbb{B}^{n})}\lambda_{1}.

This conjecture concerns sharp universal bounds for sums of Dirichlet eigenvalues; the supplied source states that it has been solved.

References

Primary source

Lingzhong Zeng, “Spectrum of the Dirac operator on Compact Riemannian Manifolds”, arXiv:2402.14247 (2024).

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