Conjectured link between limiting Turán determinants and the spectral measure

Determine an explicit relation, analogous to the Jacobi matrix case, between the limiting function h(t, η; λ) defined by lim_{n→∞} p_n(t) |D_n(t, η; λ)| and the spectral measure μ_α associated with the Sturm–Liouville operator H_α for ω-periodically modulated parameters, with the aim of directly establishing continuity and positivity of the density μ_α′.

Background

The paper introduces generalized Turán determinants for Sturm–Liouville operators with periodically modulated parameters and proves the existence of a positive continuous limit h(t, η; λ) = lim_{n→∞} p_n(t) |D_n(t, η; λ)| (Theorem C).

In the classical periodic Schrödinger/Jacobi settings, limits of Turán-type determinants are tightly connected to the spectral measure’s density. Motivated by an explicit identity in the free case (Example 1), the authors conjecture an analogous relation in the present continuous setting. Proving such a link would yield a direct route to continuity and positivity of μ_α′ beyond the more involved approach via Christoffel–Darboux kernels and density of states used in the paper.