Tokyo Journal of Mathematics

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


Full-text: Open access


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

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

First available in Project Euclid: 20 July 2007

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


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.

Export citation


  • S. Albeverio, F. Gesztesy, R. Høegh-Krohn and H. Holden, Solvable models in quantum mechanics, Springer, Heidelberg, 1988.
  • S. Albeverio and P. Kurasov, Singular perturbations of differential operators, London Mathematical Society Lecture Note Series 271, Cambridge Univ. Press, 1999.
  • F. Gesztesy and H. Holden, A new class of solvable models in quantum mechanics describing point interactions on the line, J. Phys. A: Math. Gen. 20 (1987), 5157–5177.
  • F. Gesztesy, H. Holden and W. Kirsch, On energy gaps in a new type of analytically solvable model in quantum mechanics, J. Math. Anal. Appl. 134 (1988), 9–29.
  • G. Sh. Guseinov and I. Y. Karaca, Instability intervals of a Hill's equation with piecewise constant and alternating coefficient, Comput. Math. Appl. 47 (2004), 319–326.
  • R. Hughes, Generalized Kronig-Penney Hamiltonians, J. Math. Anal. Appl. 222 (1998), 151–166.
  • T. Kappeler and C. Möhr, Estimates for periodic and Dirichlet eigenvalues of the Schrödinger operator with singular potentials, J. Funct. Anal. 186 (2001), 62–91.
  • R. Kronig and W. Penney, Quantum mechanics in crystal lattices, Proc. Royal Soc. London 130 (1931), 499–513.
  • W. Magnus and S. Winkler, Hill's equations, Wiley, New York, 1966.
  • K. Yoshitomi, Spectral gaps of the one-dimensional Schrödinger operators with periodic point interactions, Hokkaido Math. J., to appear.