In this paper, we use renormalization methods to study self-similarity in the fluctuations of the Harper equation in the strong-coupling limit for quadratic irrationals of the form for Using the decimation method, we obtain a second-order functional recurrence which we prove rigorously has an entire fixed point. This fixed point governs the scaling of the fluctuations in the strong-coupling limit.
REFERENCES
1.
Aubry
, S.
and André
, G.
, “Analyticity breaking and Anderson localization in incommensurate lattices
,” Ann. Isr. Phys. Soc.
3
, 133
–164
(1980
).2.
Ketoja
, J. A.
and Satija
, I. I.
, “Renormalization approach to quasiperiodic quantum spin chains
,” Physica A
219
, 212
–233
(1995
).3.
Ketoja
, J. A.
and Satija
, I. I.
, “Self-similarity and localization
,” Phys. Rev. Lett.
75
, 2762
–2765
(1995
).4.
Mestel
, B. D.
, Osbaldestin
, A. H.
, and Winn
, B.
, “Golden mean renormalization for the Harper equation: The strong coupling fixed point
,” J. Math. Phys.
41
, 8304
–8330
(2000
).5.
Mestel
, B. D.
and Osbaldestin
, A. H.
, “Periodic orbits of renormalization for the correlations of strange nonchaotic attractors
,” Math. Phys. Electron. J.
6
, 27
(2000
).6.
Mestel
, B. D.
and Osbaldestin
, A. H.
, “Renormalization analysis of correlation properties in a quasiperiodically forced two-level system
,” J. Math. Phys.
43
, 3458
–3483
(2002
).7.
Mestel, B. D. and Osbaldestin, A. H., “Golden mean renormalization for a generalized Harper equation: The Ketoja-Satija orchid” (unpublished).
8.
Olds, C. D., Continued Fractions (Random House, New York, 1963).
9.
Simon
, B.
, “Schrödinger operators in the twentieth century
,” J. Math. Phys.
41
, 3523
–3555
(2000
).10.
Sokoloff
, J. B.
, “Unusual band-structure, wave-functions and electrical conductance in crystals with incommensurate periodic potentials
,” Phys. Rep.
126
, 189
–244
(1985
).
This content is only available via PDF.
© 2003 American Institute of Physics.
2003
American Institute of Physics
You do not currently have access to this content.
