Galoisian approach to complex oscillation theory of some Hill equations

TL;DR

This paper uses differential Galois theory to classify and explicitly construct non-oscillatory solutions of certain Hill equations, revealing their Liouvillian nature and orthogonality properties.

Contribution

It establishes a full correspondence between non-oscillatory and Liouvillian solutions for specific Hill equations and derives explicit solutions with novel orthogonality features.

Findings

All non-oscillatory solutions are Liouvillian.

Explicit closed-form solutions are obtained.

Liouvillian solutions exhibit novel orthogonality properties.

Abstract

We apply Kovacic's algorithm from differential Galois theory to show that all complex non-oscillatory solutions (finite exponential of convergence of zeros) of certain Hill equations considered by Bank and Laine using Nevanlinna theory must be Liouvillian solutions. That is, solutions are obtainable by suitable differential field extensions construction. In particular, we have established a full correspondence between solutions of non-oscillatory type and Liouvillian solutions for a particular Hill equation. Explicit closed-form solutions are obtained via both methods for this Hill equation whose potential has four exponential functions in the Bank-Laine theory. The differential equation is a periodic form of biconfluent Heun equation. We further show that these Liouvillian solutions exhibit novel single and double orthogonality and a Fredholm integral equation over suitable integration regions in $\mathbf{C}$ that mimic single/double orthogonality for the corresponding Liouvillian solutions of the Lam\'e and Whittaker-Hill equations, discovered by Whittaker and Ince almost a century ago.