Universal k^{1−1/n} index-gap for Euclidean domains

Establish the existence of dimension-dependent constants C_n in (0,1] such that for every bounded Euclidean domain Ω ⊂ R^n, the Laplacian Dirichlet and Neumann spectra satisfy λ_k(Ω) ≥ μ_{k + ⌊ C_n (n ω_{n-1} / (2 ω_n^{1-2/n})) k^{1−1/n} ⌋}(Ω) for all positive integers k, and prove that the exponent 1−1/n cannot be improved.

Background

The paper proves that for domains satisfying a non-periodicity condition and not being a ball, there exists an index shift p(k) of order k^{1−1/n} such that λk ≥ μ{k+p(k)} for all sufficiently large k. For general Lipschitz domains in dimensions n ≥ 4, a weaker bound of order k^{1−3/n} is shown to hold for all k. Motivated by these results and numerical evidence (especially for disks), the authors formulate a conjecture asserting a universal gap with index shift of order k^{1−1/n}, with a dimension-dependent prefactor C_n ≤ 1, valid for all bounded Euclidean domains and all k.

They further assert that the exponent 1−1/n is optimal (i.e., cannot be increased), in line with the behavior suggested by two-term Weyl asymptotics and the isoperimetric inequality. This conjecture would extend the main theorem beyond the non-periodicity and large-k regime, providing a unified statement across all bounded Euclidean domains.