Optimal weight threshold for Waveholtz convergence (s > 1)

Establish that the convergence estimates in Theorem 3.1 for the Waveholtz iteration applied to the constant-coefficient Helmholtz equation on R^d hold under the weaker assumption s > 1 (instead of s > 3/2), thereby determining the optimal lower bound on the Agmon-type weight exponent s for which the results remain valid.

Background

The paper proves convergence of the real parts of the Waveholtz iterates to the outgoing solution of the constant-coefficient Helmholtz equation on Rd under the condition s > 3/2 in weighted Sobolev spaces. The frequency-explicit analysis implies an iteration count scaling n ~ ω{2s−1}, which with s > 3/2 yields at best n ~ ω{2+}.

Based on numerical observations suggesting n ~ ω, the authors conjecture that the theoretical requirement on the weight exponent s can be lowered to s > 1, which would imply an almost linear iteration scaling with frequency. The open problem asks to rigorously prove that the theorem’s convergence bounds hold for all s > 1.

References

We therefore conjecture that the optimal limit for s in Theorem~\ref{thm:convergence} is actually $s>1$, as this would give the scaling $n\sim \omega{1+}$.

— Convergence of the Waveholtz Iteration on $\mathbb{R}^d$ (2510.15606 - Runborg et al., 17 Oct 2025) in Remark after Theorem 3.1, Section 3 (Convergence of Waveholtz)