Uniform quantitative estimates for the remainder in Weyl’s law

Determine whether uniform and quantitative estimates exist for the remainder term R_Ω(λ) = N^D_Ω(λ) − (|Ω| ω(n)/(2π)^n) λ^{n/2} in Weyl’s law for bounded open domains Ω ⊂ ℝ^n (n ≥ 2), by deriving explicit bounds valid for large λ with clearly quantified thresholds.

Background

The authors highlight that classical asymptotic results for the spectral counting function lack explicit quantitative thresholds on λ and do not provide uniform remainder bounds across Lipschitz domains. They pose this question to motivate their later theorems giving the first quantitative estimate for the remainder since Weyl’s original work.

Subsequently, the paper offers a solution with explicit constants and dependencies on geometric features, but the question is stated explicitly in the introductory section as a motivating open challenge.