For ergodic 1D Jacobi operators we prove that the random singular components of any spectral measure are almost surely (a.s.) mutually disjoint as long as one restricts to the set of positive Lyapunov exponent. In the context of extended Harper's equation this yields the first rigorous proof of the Thouless' formula for the Lyapunov exponent in the dual regions.

1.
Carmona
,
R.
and
Kotani
,
S.
, “
Inverse spectral theory for random Jacobi matrices
,”
J. Stat. Phys.
46
,
1091
1114
(
1987
).
2.
Chulaevsky
,
V.
and
Delyon
,
F.
, “
Purely absolutely continuous spectrum for almost Mathieu operators
,”
J. Stat. Phys.
55
,
1279
1284
(
1989
).
3.
Cycon
,
H. L.
,
Froese
,
R. G.
,
Kirsch
,
W.
, and
Simon
,
B.
,
Schrödinger Operators with Application to Quantum Mechanics and Global Geometry
(
Springer
,
Heidelberg
,
1987
).
4.
Deift
,
P.
and
Simon
,
B.
, “
Almost periodic Schrdinger operators, III. The absolutely continuous spectrum in one dimension
,”
Commun. Math. Phys.
90
,
389
411
(
1983
).
5.
Gesztesy
,
F.
and
Simon
,
B.
, “
m-functions and inverse spectral analysis for finite and semi-infinite Jacobi matrices
,”
J. Anal. Math.
73
,
267
297
(
1997
).
6.
Gordon
,
A. Y.
,
Jitomirskaya
,
S.
,
Last
,
Y.
, and
Simon
,
B.
, “
Duality and singular continuous spectrum in the almost Mathieu equation
,”
Acta Math.
178
,
169
183
(
1997
).
7.
Han
,
J. H.
,
Thouless
,
D. J.
,
Hiramoto
,
H.
, and
Kohmoto
,
M.
, “
Critical and bicritical properties of Harper's equation with next-nearest neighbor coupling
,”
Phys. Rev. B
50
,
11365
11380
(
1994
).
8.
Ishii
,
K.
, “
Localization of eigenstates and transport phenomena in one-dimensional disordered systems
,”
Suppl. Prog. Theor. Phys.
53
,
77
138
(
1973
).
9.
Jitomirskaya
,
S.
,
Koslover
,
D. A.
, and
Schulteis
,
M. S.
, “
Localization for a Family of One-dimensional Quasi-periodic Operators of Magnetic Origin
,”
Ann. Henri Poincarè
6
,
103
124
(
2005
).
10.
Jitomirskaya
,
S.
,
Koslover
,
D. A.
, and
Schulteis
,
M. S.
, “
Continuity of the Lyapunov Exponent for analytic quasi-periodic cocycles
,”
Ergod. Theory Dyn. Syst.
29
,
1881
1905
(
2009
).
11.
Jitomirskaya
,
S.
and
Marx
,
C. A.
, “
Continuity of the Lyapunov Exponent for analytic quasi-periodic cocycles with singularities
,”
J. Fixed Point Theory and Appl. (JFPTA)
, Festschrift in honor of the 80th birthday of Richard Palais, (to appear).
12.
Mandelshtam
,
V. A.
and
Zhitomirskaya
,
S. Ya.
, “
1D-Quaisperiodic Operators. Latent Symmetries
,”
Commun. Math. Phys.
139
,
589
604
(
1991
).
13.
Oseledets
,
V.
, “
Oseledets theorem
,”
Scholarpedia J.
3
(
1
),
1846
(
2008
).
14.
Pastur
,
L.
, “
Spectral properties of disordered systems in one-body approximation
,”
Commun. Math. Phys.
75
,
179
196
(
1980
).
15.
Simon
,
B.
, “
Kotani theory for one dimensional stochastic Jacobi matrices
,”
Commun. Math. Phys.
89
,
227
234
(
1983
).
16.
Simon
,
B.
,
Trace Ideals and their Applications
, 2nd ed. (
American Mathematical Society
,
Providence, RI
,
2005
).
17.
Teschl
,
G.
,
Jacobi Operators and Completely Integrable Nonlinear Lattices
,
Mathematical Surveys and Monographs
Vol
72
(
American Mathematical Society
,
Providence, RI
,
2000
).
18.
Thouless
,
D. J.
, “
Bandwidths for a quasiperiodic tight-binding model
,”
Phys. Rev. B
28
,
4272
4276
(
1983
).
19.
It is a well-known fact that given a sequence (An) of bounded self adjoint operators approximating a bounded operator A in norm topology, the spectra satisfy spec(An) → spec(A) in the Hausdorff metric.
You do not currently have access to this content.