2015/08/28 by Neal Coleman, Coleman, Neal
Computer Science · Materials Science · Mathematics · #35P15 #Advanced Mathematical Modeling in Engineering #FOS: Mathematics #Mathematical Approximation and Integration #Quasicrystal Structures and Properties #Spectral Theory (math.SP) #math.SP #msc:35P15
paper · pdf · doi:10.48550/arxiv.1508.07346
6 pages + references. Incorporated new reference info and rephrased proof to use eigenvalue counting function
openalex publication_date 2015/08/28 · arxiv created 2015/12/25 · arxiv updated 2015/12/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We prove a lower bound on the eigenvalues λk, k∈ℕ, of the Dirichlet Laplacian of a bounded domain Ω⊂ℝn of volume V: λk ≥ Cn( δ(k)/(V))2/n where δ is a constant that measures how efficiently Ω can be packed into ℝn and Cn is the constant found in Weyl's law. This generalizes a result of Urakawa in 1984. If δ2/n > n/(n+2), this bound is stronger than the eigenvalue bound proven by Li and Yau in 1983. For example, in the case of convex planar domains, we have for all k∈ℕ, λk ≥ (2√(3)πk)/(V).