Abstract
The localisation properties of the solutions of almost periodic Schrödinger equations are related to the theory of dynamical systems. Here we shall proceed in a reverse way. We shall first discuss some algebraic properties of the iterations of polynomials and show that these properties are related to almost periodic schrödinger operators. The exactly solvable model introduced displays interesting features: almost periodicity, singular spectrum, chaotic states, exact renormalization group property. The relevance of this model with physical systems is also discussed.
Keywords
- Orthogonal Polynomial
- Bernoulli Shift
- Singular Continuous Spectrum
- Renormalization Group Equa
- Liouville Number
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
This is a preview of subscription content, access via your institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
D. Bessis, M.L. Mehta, P. Moussa, C.R. Acad. Sci. Paris 293, Ser. 1, 705–708 (1981)
D. Bessis, M.L. Mehta, P. Moussa, Letters. Math. Phys. 6, 123–140 (1982)
M.F. Barnsley, J.S. Geronimo, A.N. Harrington, Commun. Math. Phys. 88, 479–501 (1983)
M.F. Barnsley, J.S. Geronimo, A.N. Harrington, Bull. Amer. Math. Soc. 7, 381–384 (1982)
D. Bessis, P. Moussa, Commun. Math. Phys. 88, 503–529 (1983)
G. Julia, J. Maths. Ser. 7 (Paris), 47–245 (1918)
P. Fatou, Bull. Soc. Math. France, 47, 161–271 (1919); 48 33–94 (1920); 48 208–314 (1920)
A. Douady, Systemes Dynamiques Holomorphes, Séminaire Bourbaki no599, November 1982
H. Brolin, Ark. Mat. 6, 103–144 (1965)
N.I. Akhiezer, The Classical Moment Problem, Oliver and Boyd, Edmburgh 1965, see p. 85–86
G. Szegö, Orthogonal polynomials, Amer. Math. Soc. Colloquim publications, 23 (1939)
M.F. Barnsley, J.S. Geronimo, A.N. Harrington, to appear in Ergodic Theory and Dynamical Systems.
R.L. Adler, T.J. Rivlin, Proc. Am. Math. Soc., 15, 794–796 (1964)
T.S. Pitcher, J.R. Kinney, Ark. Mat., 8, 25–32 (1968)
J. Bellissard, D. Bessis, P. Moussa, Phys. Rev. Lett. 49, 701–704 (1982)
G.A. Baker, D. Bessis, P. Moussa, to appear in proceedings of the VIIth Conference on Mathematics and Physics, Boulder, Colorado, August (1983)
H. Bohr, Almost periodic functions, Chelsea New York (1951)
D.J. Thouless, J. Phys. C, 5, 77–81 (1972)
D.J. Thouless, Q. Niu, J. Phys. A, 16, 1911–1919 (1983)
M. Kohmoto, Y. Oono, Cantor Spectrum for an almost periodic Schrödinger Equation and a Dynamical Map, Illinois University at Urbana Preprint (1983)
P.G. Harper, Proc. Phys. Soc. London, Sect A, 68, 874–878 (1955)
S. Aubry, G. André, Ann. Israel Phys. Soc., 3, 133–164 (1979)
E. Dinaburg, Ya.G. Sinai, Funct. Anal. Appl. 9, 279–289 (1976)
J. Bellissard, R. Lima, D. Testard, Commun. Math. Phys., 88, 207–234 (1983)
B. Simon, Advances in Appl. Maths., 3, 463–490 (1982)
J. Avron, B. Simon, Bull. Amer. Math. Soc., 6, 81–85 (1982)
E. Domany, B. Alexander, D. Bensimon, L.P. Kadanoff, Phys. Rev. B 28, 3110–3123 (1983)
R. Rammal, Phys. Rev. B28, 4871–1873 (1983)
R. Rammal, G. Toulouse, J. Phys. Lettres, Paris, 44, L13–L22 (1983)
B. Derrida, J.P. Eckmann, A. Erzan, J. Phys. A, 16, 893–906 (1983)
S. Alexander, R. Orbach, J. Physique Lettres, Paris, 43, L625–L631 (1982)
R. Rammal, T. Lubensky, G. Toulouse, J. Phys. Lettres, Paris, 44, L65–L71 (1983)
D. Bessis, J.S. Geronimo, P. Moussa, Complex dimensionality on fractal structures Saclay SPhT, Preprint (1983), to appear in J. Phys. Lettres Paris
D. Bessis, J.S. Geronimo, P. Moussa, Mellin transform associated with Julia sets and physical applications,Saclay SPhT Preprint (1983) to appear in J. Stat. Phys.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1985 Springer-Verlag
About this paper
Cite this paper
Moussa, P., Bessis, D. (1985). A solvable almost periodic Schrödinger operator. In: Albeverio, S., Combe, P., Sirugue-Collin, M. (eds) Stochastic Aspects of Classical and Quantum Systems. Lecture Notes in Mathematics, vol 1109. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0101541
Download citation
DOI: https://doi.org/10.1007/BFb0101541
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-13914-0
Online ISBN: 978-3-540-39138-8
eBook Packages: Springer Book Archive
