On Unitary Monodromy of Second-Order ODEs

TL;DR

This paper characterizes when the monodromy group of second-order linear differential equations on Riemann surfaces is unitary, providing necessary and sufficient conditions, explicit constructions, and applications to spectral problems of Heun operators.

Contribution

It offers new criteria for unitarity of monodromy groups, including trace conditions and algebraic dimension analysis, extending understanding of second-order ODEs on Riemann surfaces.

Findings

Derived trace conditions for irreducible monodromy groups.

Classified reducible monodromy groups via conjugation to model subgroups.

Connected monodromy unitarity to algebraic dimension of generated algebras.

Abstract

Given a second-order, holomorphic, linear differential equation $Lf=0$ on a Riemann surface, we say that its monodromy group $G\subset\operatorname{GL}(2,\mathbb{C})$ is \emph{unitary} if it preserves a non-degenerate (though not necessarily positive) Hermitian form $H$ on $\mathbb{C}^2$ under the action $g\circ H\doteq g^\dagger H g$. In the present work, we give two sets of necessary and sufficient conditions for a differential operator $L$ to have a unitary monodromy group, and we construct the form $H$ explicitly. First, in the case that the natural representation of $G$ on $\mathbb{C}^2$ is irreducible, we show that unitarity is equivalent to a set of easily-verified trace conditions on local monodromy matrices; in the case that it is reducible, we show that $G$ is unitary if and only if it is conjugate to a subgroup of one of two model subgroups of $\operatorname{GL}(2,\mathbb{C})$. Second, we show that unitarity of $G$ is equivalent to a criterion on the real dimension of the algebra $A$ generated by a rescaled group $G'\subset\operatorname{SL}(2,\mathbb{C})$: that $\dim(A)=1$ if $G\subset S^1$ is scalar, $\dim(A)=2$ if $G$ is abelian, $\dim(A)=3$ if $G$ is non-abelian but its action on $\mathbb{C}^2$ is reducible, and $\dim(A)=4$ otherwise. We leverage these results to extend a conjecture of Frits Beukers on the spectrum of Lam\'e operators -- namely, we give asymptotic and numerical evidence that his conjecture should apply similarly to a wider class of Heun operator. Our work makes progress towards characterizing the spectra of second-order operators on Riemann surfaces, and in particular, towards answering the accessory parameter problem for Heun equations.