Log-dimensional spectral properties of one-dimensional quasicrystals

Authors:
David Damanik and Michael Landrigan

Journal:
Proc. Amer. Math. Soc. **131** (2003), 2209-2216

MSC (2000):
Primary 81Q10, 47B80

Published electronically:
November 6, 2002

MathSciNet review:
1963769

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We consider discrete one-dimensional Schrödinger operators on the whole line and establish a criterion for continuity of spectral measures with respect to log-Hausdorff measures. We apply this result to operators with Sturmian potentials and thereby prove logarithmic quantum dynamical lower bounds for all coupling constants and almost all rotation numbers, uniformly in the phase.

**1.**J. Bellissard, B. Iochum, E. Scoppola, and D. Testard,*Spectral properties of one-dimensional quasi-crystals*, Comm. Math. Phys.**125**(1989), no. 3, 527–543. MR**1022526****2.**Jean-Michel Combes,*Connections between quantum dynamics and spectral properties of time-evolution operators*, Differential equations with applications to mathematical physics, Math. Sci. Engrg., vol. 192, Academic Press, Boston, MA, 1993, pp. 59–68. MR**1207148**, 10.1016/S0076-5392(08)62372-3**3.**David Damanik,*𝛼-continuity properties of one-dimensional quasicrystals*, Comm. Math. Phys.**192**(1998), no. 1, 169–182. MR**1612089**, 10.1007/s002200050295**4.**David Damanik,*Gordon-type arguments in the spectral theory of one-dimensional quasicrystals*, Directions in mathematical quasicrystals, CRM Monogr. Ser., vol. 13, Amer. Math. Soc., Providence, RI, 2000, pp. 277–305. MR**1798997****5.**David Damanik, Rowan Killip, and Daniel Lenz,*Uniform spectral properties of one-dimensional quasicrystals. III. 𝛼-continuity*, Comm. Math. Phys.**212**(2000), no. 1, 191–204. MR**1764367**, 10.1007/s002200000203**6.**D. J. Gilbert and D. B. Pearson,*On subordinacy and analysis of the spectrum of one-dimensional Schrödinger operators*, J. Math. Anal. Appl.**128**(1987), no. 1, 30–56. MR**915965**, 10.1016/0022-247X(87)90212-5**7.**I. Guarneri, Spectral properties of quantum diffusion on discrete lattices,*Europhys. Lett.***10**(1989), 95-100.**8.**Svetlana Jitomirskaya and Yoram Last,*Power-law subordinacy and singular spectra. I. Half-line operators*, Acta Math.**183**(1999), no. 2, 171–189. MR**1738043**, 10.1007/BF02392827**9.**Svetlana Ya. Jitomirskaya and Yoram Last,*Power law subordinacy and singular spectra. II. Line operators*, Comm. Math. Phys.**211**(2000), no. 3, 643–658. MR**1773812**, 10.1007/s002200050830**10.**A. Ya. Khinchin,*Continued fractions*, Translated from the third (1961) Russian edition, Dover Publications, Inc., Mineola, NY, 1997. With a preface by B. V. Gnedenko; Reprint of the 1964 translation. MR**1451873****11.**M. Landrigan, Log-dimensional properties of spectral measures, Ph.D. thesis, UC Irvine (2001).**12.**Yoram Last,*Quantum dynamics and decompositions of singular continuous spectra*, J. Funct. Anal.**142**(1996), no. 2, 406–445. MR**1423040**, 10.1006/jfan.1996.0155

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (2000):
81Q10,
47B80

Retrieve articles in all journals with MSC (2000): 81Q10, 47B80

Additional Information

**David Damanik**

Affiliation:
Department of Mathematics 253–37, California Institute of Technology, Pasadena, California 91125

Email:
damanik@its.caltech.edu

**Michael Landrigan**

Affiliation:
Department of Mathematics, Idaho State University, Pocatello, Idaho 83209

Email:
landmich@isu.edu

DOI:
https://doi.org/10.1090/S0002-9939-02-06747-3

Keywords:
Schr\"odinger operators,
Hausdorff dimensional spectral properties,
Sturmian potentials

Received by editor(s):
October 5, 2001

Received by editor(s) in revised form:
February 23, 2002

Published electronically:
November 6, 2002

Additional Notes:
The first author was supported in part by the National Science Foundation through Grant DMS–0010101

The second author was supported in part by the National Science Foundation through Grant DMS-0070755

Communicated by:
Joseph A. Ball

Article copyright:
© Copyright 2002
American Mathematical Society