Nagoya Mathematical Journal

Weak Bloch property for discrete magnetic Schrödinger operators

Yusuke Higuchi and Tomoyuki Shirai

Full-text: Open access


For a magnetic Schrödinger operator on a graph, which is a generalization of classical Harper operator, we study some spectral properties: the Bloch property and the behaviour of the bottom of the spectrum with respect to magnetic fields. We also show some examples which have interesting properties.

Article information

Nagoya Math. J. Volume 161 (2001), 127-154.

First available in Project Euclid: 27 April 2005

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 58J99: None of the above, but in this section
Secondary: 05C50: Graphs and linear algebra (matrices, eigenvalues, etc.) 39A70: Difference operators [See also 47B39] 82B41: Random walks, random surfaces, lattice animals, etc. [See also 60G50, 82C41]


Higuchi, Yusuke; Shirai, Tomoyuki. Weak Bloch property for discrete magnetic Schrödinger operators. Nagoya Math. J. 161 (2001), 127--154.

Export citation


  • R. Brooks, Combinatorial problems in spectral geometry , in the Proc. of Taniguchi Symp. “Curvature and topology of Riemannian manifolds” 1985, Lecture Note in Math., 1201 (1986), 14–32.
  • M. D. Choi, G. Elliott and N. Yui, Gauss polynomials and the rotation algebra , Invent. Math., 99 (1990), 225–246.
  • J. Dodziuk and L. Karp, Spectral and function theory for combinatorial Laplacians , Contemp. Math., 73 (1988), 25–40.
  • J. Dodziuk and W. S. Kendall, Combinatorial Laplacians and isoperimetric inequality , in “From local times to global geometry, control and physics” (K. D. Elworthy, ed.), Pitman Research Notes in Mathematics Series, 150 (1986), 68–74.
  • P. G. Harper, Single band motion of conduction electrons in a uniform magnetic field , Proc. Phys. Soc. London, A68 (1955), 874–878.
  • Yu. Higuchi, Boundary area growth and the spectrum of discrete Laplacian , preprint.
  • Yu. Higuchi and T. Shirai, The spectrum of magnetic Schrödinger operators on a graph with periodic structure , J. Funct. Anal., 169 (1999), 456–480.
  • ––––, A remark on the spectrum of magnetic Laplacian on a graph , Yokohama Math. J., 47 Special Issue (1999), 129–141.
  • D. R. Hofstadter, Energy levels and wave functions of Bloch electrons in rational and irrational magnetic fields , Phys. Rev., B14 (1976), 2239–2249.
  • T. Kobayashi, K. Ono and T. Sunada, Periodic Schrödinger operators on a manifold , Forum Math., 1 (1989), 69–79.
  • E. Lieb and M. Loss, Fluxes, Laplacians and Kasteleyn's theorem , Duke Math. J., 71 (1993), 337–363.
  • J. Milnor, A note on curvature and fundamental group , J. Differ. Geom., 2 (1968), 1–7.
  • M. C. Reed and B. Simon, Methods of Modern Mathematical Physics Vol. IV (1978, Academic Press, New York).
  • T. Sunada, A discrete analogue of periodic magnetic Schrödinger operators , in “Geometry of the spectrum” (P. Parry, R. Brooks, C. Gordon, eds.), Contemporary Mathematics, 173 (1994), 283–299.