For cylindrical strips S^1×[0,h], Pólya's conjecture holds exactly for h in (0,≈3.048] ∪ [≈3.238,≈4.046] ∪ [≈4.082,π^2/2], failing only for the 8th and 13th eigenvalues in two narrow gaps.
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P\'{o}lya's conjecture on $\mathbb{S}^1 \times \R$
For cylindrical strips S^1×[0,h], Pólya's conjecture holds exactly for h in (0,≈3.048] ∪ [≈3.238,≈4.046] ∪ [≈4.082,π^2/2], failing only for the 8th and 13th eigenvalues in two narrow gaps.