  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.
