On Leighton's comparison theorem (Q2400594)

In this paper, the celebrated Sturm comparison theorem for Sturm-Liouville equations is extended to equations of the form \[ -(p(u'+su))'+rp(u'+su)+qu=0. \] A criterion of Leighton's type is obtained which involve some flexible choices of two functions \(F\) and \(G\). By picking special \(F\) and \(G\), the authors derive a generalized Sturm-Picone theorem and the corresponding separation theorem which cover most existing results in the literature. These results are then illustrated to special cases such as Schrödinger equations with distributional potentials and Jacobi difference equations.