|
Log-dimensional spectral properties of one-dimensional quasicrystals
Author(s):
David
Damanik;
Michael
Landrigan
Journal:
Proc. Amer. Math. Soc.
131
(2003),
2209-2216.
MSC (2000):
Primary 81Q10, 47B80
Posted:
November 6, 2002
Retrieve article in:
PDF
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.
References:
- 1.
- J. Bellissard, B. Iochum, E. Scoppola, and D. Testard, Spectral properties of one-dimensional quasi-crystals, Commun. Math. Phys. 125 (1989), 527-543. MR 90m:82043
- 2.
- J. M. Combes, Connections between quantum dynamics and spectral properties of time-evolution operators, in Differential Equations with Applications to Mathematical Physics, Eds. W. F. Ames, E. M. Harrel II, and J. V. Herod, Academic Press, Boston (1993), 59-68. MR 94g:81205
- 3.
- D. Damanik,
 -continuity properties of one-dimensional quasicrystals, Commun. Math. Phys. 192 (1998), 169-182. MR 99a:81031 - 4.
- D. Damanik, Gordon-type arguments in the spectral theory of one-dimensional quasicrystals, in Directions in Mathematical Quasicrystals, M. Baake, R. V. Moody, eds., CRM Monograph Series 13, AMS, Providence, RI (2000), 277-305. MR 2002c:81048
- 5.
- D. Damanik, R. Killip, and D. Lenz, Uniform spectral properties of one-dimensional quasicrystals, III.
 -continuity, Commun. Math. Phys. 212 (2000), 191-204. MR 2001g:81061 - 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), 30-56. MR 89a:34033
- 7.
- I. Guarneri, Spectral properties of quantum diffusion on discrete lattices, Europhys. Lett. 10 (1989), 95-100.
- 8.
- S. Jitomirskaya and Y. Last, Power-law subordinacy and singular spectra. I. Half-line operators, Acta Math. 183 (1999), 171-189. MR 2001a:47033
- 9.
- S. Jitomirskaya and Y. Last, Power-law subordinacy and singular spectra. II. Line operators, Commun. Math. Phys. 211 (2000), 643-658. MR 2001i:81079
- 10.
- A. Ya. Khinchin, Continued Fractions, Dover Publications, Mineola (1997). MR 98c:11008
- 11.
- M. Landrigan, Log-dimensional properties of spectral measures, Ph.D. thesis, UC Irvine (2001).
- 12.
- Y. Last, Quantum dynamics and decompositions of singular continuous spectra, J. Funct. Anal. 142 (1996), 406-445. MR 97k:81044
Similar Articles:
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:
10.1090/S0002-9939-02-06747-3
PII:
S 0002-9939(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
Posted:
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
Copyright of article:
Copyright
2002,
American Mathematical Society
|
Comments: webmaster@ams.org