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 $\mu$ taking a potential $q\in L^2([0,2\pi ])$ to $\mu (q)=\tilde{\Phi }(2\pi )$, the lift to the universal cover $G=\widetilde{SL(2,R)}$ of $SL(2,R)$ of the fundamental matrix map $\Phi :[0,2\pi ]\to SL(2,R)$, \[ Φ(0) = I, \quad Φ'(t) = \begin{pmatrix} 0 & 1 q(t) & 0 \end{pmatrix} Φ(t). \] Let $H$ be the real infinite dimensional separable Hilbert space: we present an explicit diffeomorphism $\Psi :G_0\times H\to H^0([0,2\pi ])$ such that the composition $\mu \circ \Psi$ 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 $\Phi$, parametrized by polar coordinates, which we call the Kepler transform. As an application among others, let $C_1\subset L^2([0,2\pi ])$ be the set of potentials $q$ for which the equation $-u^{\prime \prime }+qu=0$ 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 $H$.

19 pages, 3 figures