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Rectangle gap with plus-one improvement

Show that for every rectangle, the Laplacian Dirichlet and Neumann spectra satisfy λ_k ≥ μ_{k + ⌊ 4√k/π ⌋ + 1} for all positive integers k.

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Background

The authors prove that for any rectangle, λk ≥ μ{k + ⌊ 4√k/π ⌋} holds for all k, using the quadrilateral isoperimetric inequality. Observing that the constant 4/√π is optimal asymptotically (as seen from the square), they conjecture that a strengthened bound adding +1 to the index shift holds universally for rectangles.

They verify the conjecture in a special case (a ≥ 4b) via direct lattice counting, and provide further numerical support, but a general proof remains open.

References

The constant 4/√π appearing in the inequality in Theorem 3.2 as the coefficient of the term in √k is optimal for a sequence p of this type, as may be seen from the asymptotic behaviour of the eigenvalues of the square. We conjecture that it is possible to take p(k) = ⌊ 4/π √k ⌋ + 1, but have only been able to prove it in the case where a≥4b, where this is a direct consequence of (3.1).

— On the (growing) gap between Dirichlet and Neumann eigenvalues (2405.18079 - Freitas et al., 28 May 2024) in Section 3.1 (Rectangles), after Corollary 3.2