Spectral gaps of the one-dimensional Schrödinger operators with periodic point interactions
The spectral gaps of the Schrödinger operators \[ H_{1}=-\frac{d^{2}}{dx^{2}}+\sum^{\infty}_{l=-\infty}( \beta_{1}\delta^{\prime}(x-\kappa-2\pi l)+\beta_{2}\delta^{\prime}(x-2\pi l))\quad \text{ in}\quad L^{2}({\mathbb R}), \] \[ H_{2}=-\frac{d^{2}}{dx^{2}}+\sum^{\infty}_{l=-\infty}( \beta_{1}\delta(x-\kappa-2\pi l)+\beta_{2}\delta(x-2\pi l))\quad \text{ in}\quad L^{2}({\mathbb R}), \] are studied, where \(\kappa\in (0,2\pi)\) and \(\beta_{1},\beta_{2}\in{\mathbb R}\backslash\{0\}\) are parameters. Given \(j\in{\mathbb N}\), it is determined that the \(j\)th gap of \(H_{k}\) is absent or not for \(k=1,2\).
scientific article; zbMATH DE number 5076741