Simple Lyapunov spectrum for linear homogeneous differential equations with $$L^p$$ parameters

Abstract

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 \(\Phi :\mathbb {R}\times M\rightarrow {{\,\textrm{GL}\,}}(2,\mathbb {R})\) induced by second order linear homogeneous differential equations \(\ddot{x}+\alpha (\varphi ^t(\omega ))\dot{x}+\beta (\varphi ^t(\omega ))x=0\) are almost everywhere distinct. The coefficients \(\alpha ,\beta \) evolve along the \(\varphi ^t\)-orbit for \(\omega \in M\) and \(\varphi ^t: M\rightarrow M\) is an ergodic flow defined on a probability space. We also obtain the corresponding version for the frictionless equation \(\ddot{x}+\beta (\varphi ^t(\omega ))x=0\) and for a Schrödinger equation \(\ddot{x}+(E-Q(\varphi ^t(\omega )))x=0\), inducing a cocycle \(\Phi :\mathbb {R}\times M\rightarrow {{\,\textrm{SL}\,}}(2,\mathbb {R})\).