Revisiting second-order linear differential equations over Hardy fields

TL;DR

This paper investigates second-order linear differential equations with coefficients in Hardy fields, proving a conjecture about the regularity of oscillating solutions and analyzing their zeros and asymptotic behavior.

Contribution

It proves Boshernitzan's conjecture that oscillating solutions have amplitude and phase functions in a larger Hardy field, revealing their regular oscillatory nature.

Findings

Oscillating solutions are characterized by amplitude and phase functions in a bigger Hardy field.

Sharp conditions for the uniqueness of amplitude and phase germs are established.

The asymptotic behavior and zero distribution of solutions are analyzed.

Abstract

We review second-order homogeneous linear differential equations with coefficient functions whose germs lie in a Hardy field (and hence are strongly non-oscillating). We prove a conjecture of Boshernitzan (1982): the oscillating solutions to such an equation are given by amplitude and phase functions with germs in a bigger Hardy field, and hence oscillate in a very regular way. We give sharp conditions for the uniqueness of such germs, study their asymptotic behavior, and use this to obtain information about the zeros and critical points of oscillating solutions.