Abstract
Anomalies are known to appear in the perturbation theory for the one-dimensional Anderson model. A systematic approach to anomalies at critical points of products of random matrices is developed, classifying and analysing their possible types. The associated invariant measure is calculated formally. For an anomaly of so-called second degree, it is given by the ground-state of a certain Fokker-Planck equation on the unit circle. The Lyapunov exponent is calculated to lowest order in perturbation theory with rigorous control of the error terms.
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References
P. Bougerol, J. Lacroix, Products of Random Matrices with Applications to Schrödinger Operators, (Birkhäuser, Boston, 1985).
A. Bovier, A. Klein, Weak disorder expansion of the invariant measure for the one-dimensional Anderson model, J. Stat. Phys. 51, 501–517 (1988).
M. Campanino, A. Klein, Anomalies in the one-dimensional Anderson model at weak disorder, Commun. Math. Phys. 130, 441–456 (1990).
B. Derrida, E.J. Gardner, Lyapunov exponent of the one dimensional Anderson model: weak disorder expansion, J. Physique 45, 1283 (1984).
S. Jitomirskaya, H. Schulz-Baldes, G. Stolz, Delocalization in random polymer chains, Commun. Math. Phys. 233, 27–48 (2003).
M. Kappus, F. Wegner, Anomaly in the band centre of the one-dimensional Anderson model, Z. Phys. B 45, 15–21 (1981).
H. Risken, The Fokker-Planck equation, Second Edition, (Springer, Berlin, 1988).
R. Schrader, H. Schulz-Baldes, A. Sedrakyan, Perturbative test of single parameter scaling for 1D random media, Ann. H. Poincaré 5, 1159–1180 (2004).
C. Shubin, R. Vakilian, T. Wolff, Some harmonic analysis questions suggested by Anderson-Bernoulli models, Geom. Funct. Anal. 8, 932–964 (1998).
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© 2007 Birkhäuser Verlag Basel/Switzerland
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Schulz-Baldes, H. (2007). Lyapunov Exponents at Anomalies of SL(2, ℝ)-actions. In: Janas, J., Kurasov, P., Laptev, A., Naboko, S., Stolz, G. (eds) Operator Theory, Analysis and Mathematical Physics. Operator Theory: Advances and Applications, vol 174. Birkhäuser Basel. https://doi.org/10.1007/978-3-7643-8135-6_10
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DOI: https://doi.org/10.1007/978-3-7643-8135-6_10
Publisher Name: Birkhäuser Basel
Print ISBN: 978-3-7643-8134-9
Online ISBN: 978-3-7643-8135-6
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