2024/12/10 by Darrow, David, Chen, Eric, Zitzewitz, Alex
#34M03 #34M35 #34M45 #Classical Analysis and ODEs (math.CA) #Complex Variables (math.CV) #FOS: Mathematics
paper · doi:10.48550/arxiv.2412.07932
Given a second-order, holomorphic, linear differential equation Lf=0 on a Riemann surface, we say that its monodromy group G\subsetGL(2,ℂ) is unitary if it preserves a non-degenerate (though not necessarily positive) Hermitian form H on ℂ2 under the action g∘ H\doteq g^† 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 ℂ2 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 GL(2,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'\subsetSL(2,ℂ): that dim(A)=1 if G⊂ S1 is scalar, dim(A)=2 if G is abelian, dim(A)=3 if G is non-abelian but its action on ℂ2 is reducible, and dim(A)=4 otherwise. We leverage these results to extend a conjecture of Frits Beukers on the spectrum of Lamé 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.