Complex oscillations of non-definite Sturm-Liouville problems, II

We correct and update a result of R. G. D. Richardson [Amer. J. Math. 40(1918), 283–316] dealing with the separation of zeros of the real and imaginary parts of non-real eigenfunctions of non-definite Sturm–Liouville eigenvalue problems. We then extend it to the case where the weight function is allowed to be identically zero on a subinterval that excludes the end-points and study the behavior of the zeros of the real and imaginary parts when the end-points are included. Examples are given illustrating the sharpness of the results along with open questions.