A degree counting formula for Fuchsian ODEs with unitarizable monodromy

Abstract

In this paper, we study the problem of whether the monodromy matrices of second-order Fuchsian ordinary differential equations (ODEs) are unitary. Let k be the number of non-integer differences of the local exponents at the singular points of the ODEs. By employing the Leray-Schauder degree formulas for the corresponding curvature equations, we show that under certain assumptions, the degree does not vanish when $k=3,4,5$, which implies that the corresponding monodromy matrices are unitary. Among others, we show the form of the degree counting formulas. To the best of our knowledge, this is the first work that assigns the Leray-Schauder degree to Fuchsian ODEs from the perspective of the corresponding curvature equations.