Simple zeros and parity interlacing for the negative Connes–Moscovici prolate spectrum

Abstract

Ramis, Richard-Jung and Thomann (C. R. Math. 363 (2025), 1065–1081) introduced parity-separated spectral determinants for the negative (non-classical) part of the Connes–Moscovici prolate spectrum: entire functions D_even(μ) = y⁺{τ,μ}(0) and D_odd(μ) = (y⁺)'{τ,μ}(0) of order ≤ 1/2 whose zeros are, respectively, the negative even and negative odd CM eigenvalues; they conjectured that these zeros are simple. We prove this simplicity assertion for every fixed τ > 0 and prove in addition that the zeros of D_even and D_odd strictly interlace. The proof uses the imaginary-axis Sturm–Liouville reduction and identifies D_odd/D_even, up to a nonzero constant, with the Weyl m-function of the associated half-line problem. Its Herglotz property yields simplicity and strict interlacing. Consequently, the negative CM spectrum ordered by increasing |μ| alternates strictly in parity, beginning with the odd sector. No claim is made concerning the Weil quadratic form of the Connes–Consani–Moscovici program or the Riemann Hypothesis. Version 4 clarifies the regular-endpoint argument at t = 0. It defines AC_loc and H¹ = W¹,² explicitly, states the Neumann and Dirichlet form domains, and proves from the first representation theorem and integration by parts that every element of the Neumann operator domain, in particular its fundamental state, satisfies u′(0) = 0. The proof of ν₀ᴺ < ν₀ᴰ is rewritten to invoke this lemma explicitly. An acknowledgment to Professor Jean-Pierre Ramis has also been added. No theorem statement or conclusion is changed.