Abstract
We consider an effective HamiltonianH representing the motion of a single-band-two-dimensional electron in a uniform magnetic field. ThenH belongs to the rotation algebra, namely the algebra of continuous functions over a non-commutative 2-torus. We define a non-commutative analog of smooth functions by mean of elements of classC l,n, wherel andn characterize respectively the degree of differentiability with respect to the magnetic field and the torus variables. We show that ifH is of classC 1,3+ε, the gap boundaries of the spectrum ofH are Lipshitz continuous functions of the magnetic field at each point for which the gap is open.
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References
Avron, J.E., Simon, B.: Stability of gaps for periodic potentials under variation of a magnetic field. J. Phys. A.: Math. Gen.18, 2199–2205 (1985)
Avron, J., van Mouche, P.H.M., Simon, B.: On the measure of the spectrum for the almos Mathieu operator. Commun. Math. Phys.132, 103–118, (1990). Erratum in Commun. Math. Phys.139, 215 (1991)
Barelli, A.: Approche algébrique de la limite semi-classique: electrons bidimensionnels en champ magnétique et localisation dynamique. Thèse. Univ. Paul Sabatier, Toulouse, Sept. 1992
Barelli, A., Bellissard, J. Fleckinger, R.: 2D Bloch electrons in a uniform magnetic field. In preparation 1993
Bellissard, J.:C *-algebras in solid state Physics. In: “Operator algebras and applications”, Vol. II. Evans, D.E., Takesaki, (eds.) Cambridge: University Press, 1988
Bellissard, J., Iochum, B., Testard, D.: Continuity properties of the electronic spectrum of 1D quasicrystals. Commun. Math. Phys.141, 353–380 (1991)
Bourbaki, N.: Topologie générale. Chap. IX, §4, n0 1. Paris: Hermann 1948
Choi, M.D., Elliott, G.A., Yui, N.: Gauss polynomials and the rotation algebra. Invent. Math.99, 225–246 (1990)
Connes, A.: A survey of foliation algebras. In “Operator algebras and applications.” Proc. Symposia Pure Math., Vol. I,38, 521–628 (1982)
Dixmier, J.: LesC *-algèbres et leurs représentations. Paris: Gauthiers-Villars 1969
Elliott, G.: Gaps in the spectrum of an almost periodic Schrödinger operator. C.R. Math. Rep. Acad. Sci. Canada,4, 255–259 (1982)
Harper, P.G.: Single band motion of conduction electrons in a uniform magnetic field. Proc. Phys. Soc. Lon.A68, 874–892 (1955)
Helffer, B., Sjöstrand, J.: Analyse semi-classique pour l'équation de Harper. II. Bull. Soc. Math. France117, Fasc. 4, Mémoire 40 (1990)
Helffer, B., Sjöstrand, B.: Equation de Schrödinger avec champ magnétique et équation de Harper. Springer Lecture Notes in Physics345 1989, pp. 118–197
Hofstadter, D.G.: Energy levels and wave functions of Bloch electrons in rational or irrational magnetic field. Phys. Rev.B14, 2239–2249 (1976)
Nenciu, G.: Stability of energy gaps under variation of the magnetic field. Lett. Math. Phys.11, 127–132, (1986)
Pedersen, G.:C *-algebras and their automorphism groups. New York: Academic Press 1979
Peierls, R.: Zur Theorie des Diamagnetismus von Leitungselektronen. Z. Phys.80, 763–791, 1933
Rammal, R.: In: Physics and fabrication of microstructures. Kelly, M., Weisbuch, C., eds., Berlin-Heidelberg-New York: Springer 1986, p. 303
Rammal, R., Bellissard, J.: An algebraic semiclassical approach to Bloch electrons in a magnetic field. J. Phys. France,51, 1803–1830, (1990)
Reed, M., Simon, B.: Methods of modern mathematical physics, Vol. 1, Chap. VIII. New York: Academic Press 1980
Rieffel, M.A.:C *-algebras associated with the irrational rotation. Pac. J. Math.95, 415–419 (1981)
Tomiyama, J.: Topological representations ofC *-algebras. Tokohu Math. J.14, 187–204 (1962)
Wilkinson, M.: An example of phase holonomy in WKB theory, J. Phys. A.: Math. Gen.17, 3459–3476 (1984)
Zak, J.: Magnetic translation group. Phys. Rev.A134, 1602–1607 (1964); Magnetic translation Group. II. Irreducible representations. Phys. RevA134, 1607–1611 (1964)
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Communicated by H. Araki
URA 505, CNRS
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Bellissard, J. Lipshitz continuity of gap boundaries for Hofstadter-like spectra. Commun.Math. Phys. 160, 599–613 (1994). https://doi.org/10.1007/BF02173432
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DOI: https://doi.org/10.1007/BF02173432