Singular Schr\"odinger operators with prescribed spectral properties

TL;DR

This paper demonstrates how to construct singular Schr"odinger operators with delta interactions that have prescribed spectral properties, including specific essential and discrete spectra, by appropriately choosing interaction strengths and support points.

Contribution

It provides a method to design singular Schr"odinger operators with tailored spectral characteristics through parameter selection.

Findings

Essential spectrum can be prescribed independently.

Discrete spectrum can be controlled within bounds.

Construction method for operators with desired spectral sets.

Abstract

The paper deals with singular Schr\"odinger operators of the form \begin{gather*} -{\mathrm{d}^2\over \mathrm{d} x^2 } + \sum_{k\in\mathbb{Z} }\gamma_k \delta(\cdot-z_k),\quad \gamma_k\in\mathbb{R}, \end{gather*} in $\mathsf{L}^2(\ell_-,\ell_+)$, where $(\ell_-,\ell_+)$ is a bounded interval, and $ \delta(\cdot-z_k)$ is the Dirac delta-function supported at $z_k\in (\ell_-,\ell_+)$. It will be shown that the interaction strengths $\gamma_k$ and the points $z_k$ can be chosen in such a way that the essential spectrum and a bounded part of the discrete spectrum of this self-adjoint operator coincide with prescribed sets on a real line.