Hokkaido Mathematical Journal

YOSHITOMI Kazushi,

Spectral gaps of the one-dimensional Schr\"odinger operators with periodic point interactions.

Hokkaido Mathematical Journal, 35 (2006) pp.365-378

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Abstract

We study the spectral gaps of the Schr{\"o}dinger operators $$H_{1}=-\frac{d^{2}}{dx^{2}}+\sum^{\infty}_{l=-\infty}( \beta_{1}\delta^{\prime}(x-\kappa-2\pi l)+\beta_{2}\delta^{\prime}(x-2\pi l))\quad {\rm in}\quad L^{2}({\mathbb R}),$$ $$H_{2}=-\frac{d^{2}}{dx^{2}}+\sum^{\infty}_{l=-\infty}( \beta_{1}\delta(x-\kappa-2\pi l)+\beta_{2}\delta(x-2\pi l))\quad {\rm in}\quad L^{2}({\mathbb R}),$$ where $\kappa\in (0,2\pi)$ and $\beta_{1},\beta_{2}\in{\mathbb R}\backslash\{0\}$ are parameters. Given $j\in{\mathbb N}$, we determine whether the $j$th gap of $H_{k}$ is absent or not for $k=1,2$.

MSC(Primary)34B37
MSC(Secondary)34D08, 34B30, 34L40
Uncontrolled KeywordsSchr\"odinger operators, periodic point interactions, spectral gaps