Tokyo Journal of Mathematics

Asymptotic Estimates for the Spectral Gaps of the Schrödinger Operators with Periodic $\delta^{\prime}$-Interactions

Tomohiro ICHIMURA

Abstract

In this note we investigate the spectral gaps of the Schrödinger operator $$H=-\frac{d^2}{dx^2}+\sum_{l=-\infty}^{\infty}\big(\beta_1\delta^{\prime}(x-2\pi l)+\beta_2\delta^{\prime}(x-\kappa-2\pi l)\big) \quad \textrm{in} \quad L^2(\mathbf{R})\,,$$ where $\beta_1$, $\beta_2 \in \mathbf{R}\setminus\{0\}$ and $\kappa/\pi \in \mathbf{Q}$. By $G_{j}$ we denote the $j$-th gap of the spectrum of $H$. We provide the asymptotic expansion of the length of $G_{j}$ as $j\rightarrow\infty$.

Article information

Source
Tokyo J. Math., Volume 30, Number 1 (2007), 121-138.

Dates
First available in Project Euclid: 20 July 2007

https://projecteuclid.org/euclid.tjm/1184963651

Digital Object Identifier
doi:10.3836/tjm/1184963651

Mathematical Reviews number (MathSciNet)
MR2328059

Zentralblatt MATH identifier
1132.34063

Citation

ICHIMURA, Tomohiro. Asymptotic Estimates for the Spectral Gaps of the Schrödinger Operators with Periodic $\delta^{\prime}$-Interactions. Tokyo J. Math. 30 (2007), no. 1, 121--138. doi:10.3836/tjm/1184963651. https://projecteuclid.org/euclid.tjm/1184963651

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