A generalization of Gordon's theorem and applications to quasiperiodic Schr\"odinger operators

Abstract

We prove a criterion for absence of eigenvalues for one-dimensional Schr\"odinger operators. This criterion can be regarded as an $L^1$-version of Gordon's theorem and it has a broader range of application. Absence of eigenvalues is then established for quasiperiodic potentials generated by Liouville frequencies and various types of functions such as step functions, H\"older continuous functions and functions with power-type singularities. The proof is based on Gronwall-type a priori estimates for solutions of Schr\"odinger equations.