On the Spectra of One-Dimensional Schrodinger Operators With Singular Potentials

The paper is devoted to the spectral properties of one-dimensional Schrodinger operators S(q)u (x) = (-d(2)/dx(2) + q (x)) u (x), x is an element of R, (1) with potentials q = q(0) + q(s), where q(0) is an element of L-infinity (R) is a regular potential, and q(s) is an element of D' (R) is a singular potential with support on a discrete infinite set Y subset of R. We consider the extension H of formal operator (1) to an unbounded operator in L-2 (R) defined by the Schrodinger operator S-q0 with regular potential q(0) and interaction conditions at the points of the set Y. We study the closedness and self-adjointness of H. If the set Y similar or equal to Z has a periodic structure we give the description of the essential spectrum of operator H in terms of limit operators. For periodic potentials q(0) we consider the Floquet theory of H, and apply the spectral parameter power series method for determining the band-gap structure of the spectrum. We also consider the case when the regular periodic part of the potential is perturbed by a slowly oscillating at infinity term. We show that this perturbation changes the structure of the spectra of periodic operators significantly. This works presents several numerical examples to demonstrate the effectiveness of our approach.