Tokyo Journal of Mathematics

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

Tomohiro ICHIMURA

Full-text: Open access

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

Permanent link to this document
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


Export citation

References

  • 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.