On the number of roots of Sturm-Liouville random sums

We consider the number of roots of linear combinations of a system of $n$ orthogonal eigenfunctions of a Sturm-Liouville initial value problem with i.i.d. standard Gaussian coefficients. We prove that its distribution inherits the asymptotic behavior of the number of roots of Quall’s random trigonometric polynomials. This result can be thought as a robustness result for the central limit theorem for the number of roots of Quall’s random trigonometric polynomials in the sense that small uniform perturbations of sines and cosines do not change the limit distribution.