Riccati map on L20 (T) and its applications

This paper is concerned with the spectral properties of the Schrödinger operator Lq def = − d2 dx2 +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 L20 (T) and the Sobolev space H−1 0 (T).  2004 Elsevier Inc. All rights reserved.