The topology of the monodromy map of the second order ODE

Details The topology of the monodromy map of the second order ODE The topology of the monodromy map of the second order ODE

2005-07-06

arxiv.org/abs/math/0507120v1

J. Differential Equations 227, 581-597 (2006).

Mathematics

Classical Analysis and ODEs

19 pages, 3 figures

Scientific paper

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 $\Phi: [0,2\pi] \to SL(2,R)$, \[ \Phi(0) = I, \quad \Phi'(t) = \begin{pmatrix} 0 & 1 q(t) & 0 \end{pmatrix} \Phi(t). \] 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 $-u'' + qu = 0$ 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 $H$.

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