ONE-DIMENSIONAL SCHR "ODINGER OPERATORS WITH SINGULAR PERIODIC POTENTIALS

We study the one-dimensional Schrodinger operators S(q) u := -u" + q( x) u, u is an element of Dom(S( q)), with 1-periodic real- valued singular potentials q( x) is an element of H-1 (per) (R, R) on the Hilbert space L-2 (R). We show equivalence of five basic definitions of the operators S(q) and prove that they are self- adjoint. A new proof of continuity of the spectrum of the operators S(q) is found. Endpoints of spectrum gaps are precisely described.