Second order linear differential equations with a basis of solutions having only real zeros

TL;DR

This paper characterizes the order and growth of transcendental entire functions that produce second order linear differential equations with solutions having only real zeros, revealing a deep connection between zero distribution and function order.

Contribution

It establishes that such differential equations require the coefficient to have an odd or half-odd integer order and exhibit regular growth, extending to a geometric classification of related symmetric mappings.

Findings

Order of A must be an odd or half-odd integer.

A has completely regular growth in Levin and Pfluger's sense.

Classifies symmetric local homeomorphisms with real zeros and poles.

Abstract

Let $A$ be a transcendental entire function of finite order. We show that if the differential equation $w''+Aw=0$ has two linearly independent solutions with only real zeros, then the order of $A$ must be an odd integer or one half of an odd integer. Moreover, $A$ has completely regular growth in the sense of Levin and Pfluger. These results follow from a more general geometric theorem, which classifies symmetric local homeomorphisms from the plane to the sphere for which all zeros and poles lie on the real axis, and which have only finitely many singularities over finite non-zero values.