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We study one-dimensional Schr\"odinger operators with complex measures aspotentials and present an improved criterion for absence of eigenvalues whichinvolves a weak local periodicity condition. The criterion leads to sharpquantitative bounds on the eigenvalues. We apply our result to quasiperiodicmeasures as potentials.

A weak Gordon type condition for absence of eigenvalues of one-dimensional Schr\"odinger operators