AUTHOR=Rabinovich Vladimir S. , Barrera-Figueroa Víctor , Olivera Ramírez Leticia 

TITLE=On the Spectra of One-Dimensional Schrödinger Operators With Singular Potentials

JOURNAL=Frontiers in Physics

VOLUME=Volume 7 - 2019

YEAR=2019

URL=https://www.frontiersin.org/journals/physics/articles/10.3389/fphy.2019.00057

DOI=10.3389/fphy.2019.00057

ISSN=2296-424X

ABSTRACT=The paper is devoted to the spectral properties of one-dimensional
Schr\"{o}dinger operators
\begin{equation}
S_{q}u\left(x\right)=\left(-\frac{d^{2}}{dx^{2}}+q\left(x\right)\right)u\left(x\right),\quad x\in\mathbb{R},\label{eq1}
\end{equation}
with potentials $q=q_{0}+q_{s}$, where $q_{0}\in L^{\infty}\left(\mathbb{R}\right)$
is a regular potential, and $q_{s}\in\mathcal{D}^{\prime}\left(\mathbb{R}\right)$
is a singular potential with support on a discrete infinite set $\mathcal{Y}\subset\mathbb{R}$.
We consider the extension $\mathcal{H}$ of formal operator (\ref{eq1})
to an unbounded operator in $L^{2}\left(\mathbb{R}\right)$ defined
by the Schr\"{o}dinger operator $S_{q_{0}}$ with regular potential
$q_{0}$ and interaction conditions at the points of the set $\mathcal{Y}$.
We study the closedness and self-adjointness of $\mathcal{H}$. If
the set $\mathcal{Y}\simeq\mathbb{Z}$ has a periodic structure we
give the description of the essential spectrum of operator $\mathcal{H}$
in terms of limit operators. For periodic potentials $q_{0}$ we consider
the Floquet theory of $\mathcal{H}$, and apply the spectral parameter
power series method for definition of 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.