Singular Schrödinger operators with prescribed spectral properties

Abstract

The paper deals with singular Schrödinger operators of the form \begin{gather*} -{\mathrm{d}^2\over \mathrm{d} x^2 } + \sum_{k\in\mathbb{Z} }γ_k δ(\cdot-z_k),\quad γ_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.

39 pages, 3 figures