The topology of the monodromy map of a second order ODE

We consider the following question: given A∈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∈L2([0,2π]) to μ(q)=Φ˜(2π), the lift to the universal cover G=SL(2,R)˜ of SL(2,R) of the fundamental matrix map Φ:[0,2π]→SL(2,R), Φ(0)=I,Φ′(t)=(01q(t)0)Φ(t). Let H be the real infinite-dimensional separable Hilbert space: we present an explicit diffeomorphism Ψ:G0×H→H0([0,2π]) 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 R3. 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 C1⊂L2([0,2π]) be the set of potentials q for which the equation −u″+qu=0 admits a nonzero periodic solution: C1 is diffeomorphic to the disjoint union of a hyperplane and Cartesian products of the usual cone in R3 with H.