The topology of the monodromy map of a second order ODE

We consider the following question: given , 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 L2([0,2π]) to , the lift to the universal cover of of the fundamental matrix map , Let be the real infinite-dimensional separable Hilbert space: we present an explicit diffeomorphism such that the composition μ○Ψ is the projection on the first coordinate, where G0 is an explicitly given open subset of G diffeomorphic to . 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 be the set of potentials q for which the equation −u″+qu=0 admits a nonzero periodic solution: is diffeomorphic to the disjoint union of a hyperplane and Cartesian products of the usual cone in with .