Limit-periodic Schrödinger operators with Lipschitz continuous IDS

Damanik, David; Fillman, Jake

doi:10.1090/proc/14354

Limit-periodic Schrödinger operators with Lipschitz continuous IDS
HTML articles powered by AMS MathViewer

by David Damanik and Jake Fillman

Proc. Amer. Math. Soc. 147 (2019), 1531-1539

DOI: https://doi.org/10.1090/proc/14354

Published electronically: December 12, 2018

PDF

Abstract:

We show that there exist limit-periodic Schrödinger operators such that the associated integrated density of states is Lipschitz continuous. These operators arise in the inverse spectral theoretic KAM approach of Pöschel.

References

Michael Aizenman and Simone Warzel, Random operators, Graduate Studies in Mathematics, vol. 168, American Mathematical Society, Providence, RI, 2015. Disorder effects on quantum spectra and dynamics. MR 3364516, DOI 10.1090/gsm/168
Artur Avila and Svetlana Jitomirskaya, Almost localization and almost reducibility, J. Eur. Math. Soc. (JEMS) 12 (2010), no. 1, 93–131. MR 2578605, DOI 10.4171/JEMS/191
J. Bourgain, Hölder regularity of integrated density of states for the almost Mathieu operator in a perturbative regime, Lett. Math. Phys. 51 (2000), no. 2, 83–118. MR 1774640, DOI 10.1023/A:1007641323456
Lennart Carleson, On $H^{\infty }$ in multiply connected domains, Conference on harmonic analysis in honor of Antoni Zygmund, Vol. I, II (Chicago, Ill., 1981) Wadsworth Math. Ser., Wadsworth, Belmont, CA, 1983, pp. 349–372. MR 730079
René Carmona, Abel Klein, and Fabio Martinelli, Anderson localization for Bernoulli and other singular potentials, Comm. Math. Phys. 108 (1987), no. 1, 41–66. MR 872140
René Carmona and Jean Lacroix, Spectral theory of random Schrödinger operators, Probability and its Applications, Birkhäuser Boston, Inc., Boston, MA, 1990. MR 1102675, DOI 10.1007/978-1-4612-4488-2
H. L. Cycon, R. G. Froese, W. Kirsch, and B. Simon, Schrödinger operators with application to quantum mechanics and global geometry, Springer Study Edition, Texts and Monographs in Physics, Springer-Verlag, Berlin, 1987. MR 883643
D. Damanik and J. Fillman, Spectral theory of discrete one-dimensional ergodic Schrödinger operators, monograph in preparation.
David Damanik and Zheng Gan, Limit-periodic Schrödinger operators with uniformly localized eigenfunctions, J. Anal. Math. 115 (2011), 33–49. MR 2855032, DOI 10.1007/s11854-011-0022-y
David Damanik and Zheng Gan, Limit-periodic Schrödinger operators on $\Bbb {Z}^d$: uniform localization, J. Funct. Anal. 265 (2013), no. 3, 435–448. MR 3056711, DOI 10.1016/j.jfa.2013.05.020
R. del Rio, S. Jitomirskaya, Y. Last, and B. Simon, Operators with singular continuous spectrum. IV. Hausdorff dimensions, rank one perturbations, and localization, J. Anal. Math. 69 (1996), 153–200. MR 1428099, DOI 10.1007/BF02787106
Michael Goldstein and Wilhelm Schlag, Hölder continuity of the integrated density of states for quasi-periodic Schrödinger equations and averages of shifts of subharmonic functions, Ann. of Math. (2) 154 (2001), no. 1, 155–203. MR 1847592, DOI 10.2307/3062114
Sana Hadj Amor, Hölder continuity of the rotation number for quasi-periodic co-cycles in $\textrm {SL}(2,\Bbb R)$, Comm. Math. Phys. 287 (2009), no. 2, 565–588. MR 2481750, DOI 10.1007/s00220-008-0688-x
Svetlana Jitomirskaya and Wencai Liu, Arithmetic spectral transitions for the Maryland model, Comm. Pure Appl. Math. 70 (2017), no. 6, 1025–1051. MR 3639318, DOI 10.1002/cpa.21688
Werner Kirsch, An invitation to random Schrödinger operators, Random Schrödinger operators, Panor. Synthèses, vol. 25, Soc. Math. France, Paris, 2008, pp. 1–119 (English, with English and French summaries). With an appendix by Frédéric Klopp. MR 2509110
Jürgen Pöschel, Examples of discrete Schrödinger operators with pure point spectrum, Comm. Math. Phys. 88 (1983), no. 4, 447–463. MR 702563
Joaquim Puig, A nonperturbative Eliasson’s reducibility theorem, Nonlinearity 19 (2006), no. 2, 355–376. MR 2199393, DOI 10.1088/0951-7715/19/2/007
Barry Simon, Almost periodic Schrödinger operators. IV. The Maryland model, Ann. Physics 159 (1985), no. 1, 157–183. MR 776654, DOI 10.1016/0003-4916(85)90196-4
Mikhail Sodin and Peter Yuditskii, Almost periodic Jacobi matrices with homogeneous spectrum, infinite-dimensional Jacobi inversion, and Hardy spaces of character-automorphic functions, J. Geom. Anal. 7 (1997), no. 3, 387–435. MR 1674798, DOI 10.1007/BF02921627
Franz Wegner, Bounds on the density of states in disordered systems, Z. Phys. B 44 (1981), no. 1-2, 9–15. MR 639135, DOI 10.1007/BF01292646
Jiangong You and Shiwen Zhang, Hölder continuity of the Lyapunov exponent for analytic quasiperiodic Schrödinger cocycle with weak Liouville frequency, Ergodic Theory Dynam. Systems 34 (2014), no. 4, 1395–1408. MR 3227161, DOI 10.1017/etds.2013.4