Addendum to "a note on the zeros of solutions of w″+P(z)w=0 where P is a polynomial

Abstract

In the original paper [1], it was shown that the zeros of solutions of w″ + P(z)w = 0, where P(z) is a polynomial of degree n ≥ 1, must approach certain rays. This was proved by first obtaining asymptotic formulas for a fundamental set of solutions in sectors, and then using them to derive estimates on the rate at which the nearby zeros approach the ray. The estimates derived in [1] for the rate of approach were rough estimates which were sufficient to prove the main result but simple enough to avoid unnecessary complications in the proof. The present note is intended to give the best estimate which can be derived from the asymptotic formulas for the rate of approach of the zeros. The main reason for deriving these estimates is that they show that for many equations (e.g., the Titchmarsh equation) the rate of approach is actually much faster than that indicated by the rough estimate in [1]. In fact, we show that the estimate dramatically improves whenever P(z) has the property that the translate P(z − c) which eliminates the term of degree n − 1 also eliminates the term of degree n − 2.

AMS(MOS)::

• 34A20
• 34C10

Notes

This Research was supported in part by the national science foundation (DMS 84-20561).