On the Spectra of One-Dimensional Schrödinger Operators With Singular Potentials - Rabinovich, Vladimir S.; Barrera-Figueroa, Víctor; Olivera Ramírez, Leticia (Frontiers in Physics, 2019)
The paper is devoted to the spectral properties of one-dimensional Schrödinger operators (1) Squ(x)=(-d2dx2+q(x))u(x), x∈ℝ, with potentials q = q0+qs, where q0∈L∞(ℝ) is a regular potential, and qs∈D′(ℝ) is a singular potential with support on a discrete infinite set Y⊂ℝ. We consider the extension H of formal operator (1) to an unbounded operator in L2(ℝ) defined by the Schrödinger operator Sq0 with regular potential q0 and interaction conditions at the points of the set Y. We study the closedness and self-adjointness of H. If the set Y≃ℤ has a periodic structure we give the description of the essential spectrum of operator H in terms of limit operators. For periodic potentials q0 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.