On Unitary Monodromy of Second-Order Ordinary Differential Equations

Given a second-order, holomorphic, linear differential equation $Lf=0$ on a punctured Riemann surface, we say that its monodromy group $G\subset\operatorname{GL}(2,\mathbb{C})$ is `unitary' if it preserves a non-degenerate Hermitian form $H$ on $\mathbb{C}^2$ under the action $g\circ H=g^\dagger H g$. In the present work, we give two sets of necessary and sufficient conditions for a monodromy group $G\subset\operatorname{GL}(2,\mathbb{C})$ to be unitary. 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 the representation is reducible, we show that $G$ is unitary if and only if it is contained in one of two model subgroups of $\operatorname{GL}(2,\mathbb{C})$. Second, we show that unitarity 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. Our results directly extend the recent work of Adachi (2022, 2024), which treated Fuchsian equations with irreducible monodromy representation on the punctured Riemann sphere. We leverage these results to provide evidence that the spectrum of any Darboux operator should belong to a perturbed, squared lattice in the plane, extending a conjecture of Frits Beukers (2007). Our work makes progress towards characterizing the spectra of second-order operators on Riemann surfaces, and in particular, towards answering the \emph{accessory parameter problem} for Darboux equations.