Spectral gaps of Schr\"odinger operators with periodic singular potentials

By using quasi--derivatives we develop a Fourier method for studying the spectral gaps of one dimensional Schr\"odinger operators with periodic singular potentials $v.$ Our results reveal a close relationship between smoothness of potentials and spectral gap asymptotics under a priori assumption $v \in H^{-1}_{loc} (\mathbb{R}).$ They extend and strengthen similar results proved in the classical case $v \in L^2_{loc}(\mathbb{R}).$