Riccati map on L02(T) and its applications

This paper is concerned with the spectral properties of the Schrödinger operator Lq=def−d2dx2+q with periodic potential q from the Sobolev space H−1(T). We obtain asymptotic formulas and a priori estimates for the periodic and Dirichlet eigenvalues which generalize known results for the case of potentials q∈L2(T). The key idea is to reduce the problem to a known one-the spectrum of the impedance operator-via a nonlinear analytic isomorphism between L02(T) and the Sobolev space H0−1(T).