On a Sturm Liouville periodic boundary values problem

Consider the operator, acting in the Hilbert space, where v is a constant, with the periodic boundary values conditions, related to the maximum and minimum of the even Hermite Orthogonal polynomial(s) (OP). The problem is of interest in the field of optics when spiral geometries are involved. Specifically, the solutions represent the azimuthal component of the propagating modes into spiral optical fibres, previously studied. We establish two-sided asymptotic bounds for the relative extrema of the even Hermite OP and relate them to the values of ν = ν_k,j. The ratio between the upper and the lower bounds does not exceed, √3 near to the least and the largest maxima, and is very close to one in the bulk of. We also apply the same method to prove the inequality, for the Bessel function, and show that it is sharp outside monotonicity region.