Simple Lyapunov spectrum for linear homogeneous differential equations with \(L^p\) parameters

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