The topology of the monodromy map of the second order ODE

Abstract

We consider the following question: given $A \in SL(2,R)$, which potentials $q$ for the second order Sturm-Liouville problem have $A$ as its Floquet multiplier? More precisely, define the monodromy map $μ$ taking a potential $q \in L^2([0,2π])$ to $μ(q) = \tildeΦ(2π)$, the lift to the universal cover $G = \widetilde{SL(2,R)}$ of $SL(2,R)$ of the fundamental matrix map $Φ: [0,2π] \to SL(2,R)$, \[ Φ(0) = I, \quad Φ'(t) = \begin{pmatrix} 0 & 1 q(t) & 0 \end{pmatrix} Φ(t). \] Let $H$ be the real infinite dimensional separable Hilbert space: we present an explicit diffeomorphism $Ψ: G_0 \times H \to H^0([0,2π])$ such that the composition $μ\circ Ψ$ is the projection on the first coordinate. The key ingredient is the correspondence between potentials $q$ and the image in the plane of the first row of $Φ$, parametrized by polar coordinates, which we call the Kepler transform. As an application among others, let $C_1 \subset L^2([0,2π])$ be the set of potentials $q$ for which the equation $-u'' + qu = 0$ admits a nonzero periodic solution: $C_1$ is diffeomorphic to the disjoint union of a hyperplane and cartesian products of the usual cone in $R^3$ with $H$.