On eigenvalue bounds for a general class of Sturm-Liouville operators

We consider Sturm-Liouville operators with measure-valued weight and potential, and positive, bounded diffusion coefficient which is bounded away from zero. By means of a local periodicity condition, which can be seen as a quantitative Gordon condition, we prove a bound on eigenvalues for the corresponding operator in $L_p$, for $1\le p<\mathrm{\infty }$. We also explain the sharpness of our quantitative bound, and provide an example for quasiperiodic operators.