Non-frame obstructions on every density hyperbola p/(p+1)

Establish that for every positive integer p, there exists a lattice point (α_p, β_p) with α_pβ_p = p/(p+1) such that the Gabor system generated by g(x) = (x^2 − 1)/[(x^2 + 1)(x^2 + 4)(x^2 + 9)] is not a frame for L²(ℝ).

Background

The paper studies the frame set of the even rational window g(x) = (x² − 1)/[(x² + 1)(x² + 4)(x² + 9)]. Unlike the previously studied odd rational window, parity does not automatically force non-frame behavior on the density curves αβ = p/(p+1).

The authors prove the conjecture analytically for p = 1, 2, and 3 by constructing explicit obstruction points on the hyperbolas αβ = 1/3, 1/2, and 3/4. For general p ≥ 2, they reduce rank loss of the Zibulski–Zeevi matrix to the vanishing of a determinant Δ_p(β), and report numerical evidence of a positive zero for every 2 ≤ p ≤ 200. A proof for all positive integers p remains unresolved.