Simple Lyapunov spectrum for linear homogeneous differential equations with Lp parameters

In the present paper we prove that densely, with respect to an $L^p$-like topology, the Lyapunov exponents associated to linear continuous-time cocycles $Φ:\mathbb{R}\times M\to \text{GL}(2,\mathbb{R})$ induced by second order linear homogeneous differential equations $\ddot x+α(φ^t(ω))\dot x+β(φ^t(ω))x=0$ are almost everywhere distinct. The coefficients $α,β$ evolve along the $φ^t$-orbit for $ω\in M$ and $φ^t: M\to M$ is an ergodic flow defined on a probability space. We also obtain the corresponding version for the frictionless equation $\ddot x+β(φ^t(ω))x=0$ and for a Schrödinger equation $\ddot x+(E-Q(φ^t(ω)))x=0$, inducing a cocycle $Φ:\mathbb{R}\times M\to \text{SL}(2,\mathbb{R})$.

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