The coexistence problem for the discrete Mathieu operator. (English) Zbl 0669.34016

We solve the coexistence problem for the periodic discrete Mathieu operator in all parametric cases. The main tool in the proof is Bezout’s theorem for projective plane curves. As an additional result we obtain the gap opening and gap growth powers for this operator.


34A30 Linear ordinary differential equations and systems
47E05 General theory of ordinary differential operators
Full Text: DOI


[1] Aubry, S., Andre, G.: Analyticity breaking and Anderson localization in incommensurate lattices, Ann. Israel Phys. Soc.3, 133-164 (1980) · Zbl 0943.82510
[2] Bellissard, J.: Almost periodicity in solid state physics and C* algebras, Proceedings of the Harald Bohr Centenary conference on almost periodic functions, University of Copenhagen, to appear (1987) · Zbl 0678.42007
[3] Bellissard, J., Simon, B.: Cantor spectrum for the almost Mathieu potential, J. Funct. Anal.48, 408-419 (1982) · Zbl 0516.47018
[4] Butler, F., Brown, E.: Model calculations of magnetic band structure, Phys. Rev.166, 630-636 (1968)
[5] Chambers, W.: Linear-network model for magnetic breakdown in two dimensions. Phys. Rev.140, A135-A143 (1965)
[6] Claro, F., Wannier, G.: Closure of bands for Bloch electrons in a magnetic field, Phys. Stat. Sol. (b)88, K147-K151 (1978)
[7] Cycon, H., Froese, R., Kirsch, W., Simon, B.: Schrödinger operators with applications to quantum mechanics and global geometry. Texts and Monographs in Physics. Berlin, Heidelberg, New York: Springer (1987) · Zbl 0619.47005
[8] Duistermaat, J.: Fourier integral operators, Courant Institute of Mathematical Sciences. Lecture Notes (1973) · Zbl 0272.47028
[9] Elliott, G., Choi, M., Yui, N.: Gauss polynomials and the rotation algebra, Preprint · Zbl 0665.46051
[10] Flaschka, H.: Discrete and periodic illustratons of some aspects of the inverse method, Lecture Notes in Physics, vol38, pp. 441-466. Berlin, Heidelberg, New York: Springer 1974
[11] Fulton, W.: Algebraic curves, Benjamin Mathematics lecture note series (1969) · Zbl 0181.23901
[12] Helffer, B., Sjostrand, J.: Semi-classical analysis for Harper’s equation III. Cantor structure of the spectrum, Prepublications Univ. de Nantes
[13] Hochstadt, H.: On the theory of Hill’s matrices and related inverse problems, Lin. Alg. Appl.11, 41-52 (1975) · Zbl 0313.15008
[14] Magnus, W., Winkler, S.: Hill’s equation, Dover publications 1979
[15] van Moerbeke, P.: The spectrum of Jacobi-matrices, Invent. Math.37, 45-81 (1976) · Zbl 0361.15010
[16] v. Mouche, P.: Sur les régions interdites du spectre de l’opérateur périodique et discret de Mathieu, Thesis University of Utrecht (1988)
[17] Potts, R.: Mathieu’s difference equation, Schlesinger M., Weiss, G.: (eds.) The wonderful world of stochastics, pp. 111-125 Amsterdam: North-Holland 1985
[18] Simon, B.: Almost periodic Schrödinger operators, A review. Adv. Appl. Math.3, 463-490 (1982) · Zbl 0545.34023
[19] Sokoloff, J.: Unusual band structure, wave functions and electrical conductance in crystals with incommensurate periodic potentials, Phys. Rep.126, 183-244 (1985)
[20] Toda, M.: Theory of nonlinear lattices, Berlin, Heidelberg, New York: Springer 1981 · Zbl 0465.70014
[21] Wannier, G., Obermair, G., Ray, R.: Magneto electronic density of states for a model crystal. Phys. Stat. Sol (b)93, 337-342 (1979)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.