A note on the zeros of solutions [wacute]+P(z)w= 0 where P is a polynomial

Abstract

For the class of equations described in the title, there is a classical result which states that there are (deg P) + 2 critical rays, arg z = θ_j, such that for any ε > 0, all but finitely many zeros of any solution f(z) ≢ 0 must lie in the union of the sectors |arg z - θ_j| < ε. We prove that any infinite set of zeros in such a sector (for sufficiently small ε) must actually approach a definite ray, which is a translate ofthe critical ray (and which can be explicitely calculated). In addition, we estimate the rate at which the zeros approach the ray, thus obtaining new information on the exact location of the zeros. The class of equations treated here contains equations which arise in applications in other areas, such as Airy's equation and Titchmarsh's equation

Keywords:

• Complex oscillation theory
• AMS (MOS) 34A20
• 34C10

¹This research was supported in part by the National Science Foundation (DMS 84-20561)

¹This research was supported in part by the National Science Foundation (DMS 84-20561)

Notes

¹This research was supported in part by the National Science Foundation (DMS 84-20561)