Adiabatic almost-periodic Schrödinger operators
Zapiski Nauchnykh Seminarov POMI, Mathematical problems in the theory of wave propagation. Part 39, Tome 379 (2010), pp. 103-141 Cet article a éte moissonné depuis la source Math-Net.Ru

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In this review we describe spectral results for an one-dimensional almost-periodic Schrödinger operator with two periods, one of the periods being much larger, than the second one. Bibl. 35 titles.
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A. A. Fedotov. Adiabatic almost-periodic Schrödinger operators. Zapiski Nauchnykh Seminarov POMI, Mathematical problems in the theory of wave propagation. Part 39, Tome 379 (2010), pp. 103-141. http://geodesic.mathdoc.fr/item/ZNSL_2010_379_a5/

[1] V. Buslaev, A. Fedotov, “The monodromization and {H}arper equation”, Séminaires Équations aux Dérivées Partielles (1993–1994), École Polytechnique, Palaiseau, 1994, Exp. No. XXI, 23 pp. | MR

[2] V. S. Buslaev, A. A. Fedotov, “Blokhovskie resheniya dlya raznostnykh uravnenii”, Algebra i analiz, 7:4 (1994), 74–122 | MR | Zbl

[3] V. S. Buslaev, A. A. Fedotov, “Uravnenie Kharpera: monodromizatsiya bez kvaziklassiki”, Algebra i analiz, 8:2 (1996), 65–97 | MR | Zbl

[4] V. Buslaev, A. Fedotov, “On the difference equations with periodic coefficients”, Advances in Theoretical and Mathematical Physics, 5:6 (2001), 1105–1168 | MR | Zbl

[5] H. L. Cycon, R. G. Froese, W. Kirsch, B. Simon, Schrödinger Operators, Springer Verlag, Berlin, 1987 | MR | Zbl

[6] E. Sorets, T. Spencer, “Positive Lyapunov exponents for Schrödinger operators with quasi-periodic potentials”, Comm. Math. Phys., 142:3 (1991), 543–566 | DOI | MR | Zbl

[7] E. I. Dinaburg, Ja. G. Sinaĭ, “The one-dimensional Schrödinger equation with quasiperiodic potential”, Funkcional. Anal. i Priložen., 9:4 (1975), 8–21 | MR | Zbl

[8] M. Eastham, The spectral theory of periodic differential operators, Scottish Academic Press, Edinburgh, 1973 | Zbl

[9] L. H. Eliasson, “Floquet solutions for the 1-dimensional quasi-periodic Schrödinger equation”, Communications in Mathematical Physics, 146 (1992), 447–482 | DOI | MR | Zbl

[10] A. Fedotov, F. Klopp, “Coexistence of different spectral types for almost periodic Schrodinger equations in dimension one”, Mathematical Results in Quantum Mechanics, Proceedings of the QMath7 conference held 22–26 June 1998 in Prague, Operator Theory: Advances and Applications, 108, eds. J. Dittrich, P. Exner, M. Tater, Birkhäuser Verlag, Basel, Switzerland, 1999, 243–252 | MR

[11] A. Fedotov, F. Klopp, “Transitions d'Anderson pour des opérateurs de Schrödinger quasi-périodiques en dimension 1”, Séminaires Équations aux Dérivées Partielles (1998–1999), École Polytechnique, Palaiseau, 1999, Exp. No. IV, 12 pp. | MR

[12] A. Fedotov, F. Klopp, “A complex WKB method for adiabatic problems”, Asymptotic analysis, 27 (2001), 219–264 | MR | Zbl

[13] A. Fedotov, F. Klopp, “Anderson transitions for a family of almost periodic Schrödinger equations in the adiabatic case”, Communications in Mathematical Physics, 227 (2002), 1–92 | DOI | MR | Zbl

[14] A. Fedotov, F. Klopp, “The spectral teory of the adiabatic quasi-periodic operators on the real line”, Markov Processes and Related Fields, 9:4 (2004), 579–615 | MR

[15] A. Fedotov, F. Klopp, “On the singular spectrum of quasi-periodic Schrödinger operator in adiabatic limit”, Annales Henri Poincaré, 5 (2004), 929–978 | DOI | MR | Zbl

[16] A. Fedotov, F. Klopp, “Geometric tools of the adiabatic complex WKB method”, Asymptotic analysis, 39:3–4 (2004), 309–357 | MR | Zbl

[17] A. Fedotov, F. Klopp, “Opérateurs de Schrödinger quasi-périodiques adiabatiques: interactions entre les bandes spectrales d'un opérateur périodique”, Séminaires Équations aux Dérivées Partielles (2003–2004), Ecole Polytechnique, Palaiseau, 2004, Exp. No. VII, 23 pp. | MR

[18] A. Fedotov, F. Klopp, “On the absolutely continuous spectrum of one dimensional quasi-periodic Schrödinger operator in adiabatic limit”, Transactions of AMS, 357 (2005), 4481–4516 | DOI | MR | Zbl

[19] A. Fedotov, F. Klopp, “Strong resonant tunneling, level repulsion and spectral type for one-dimensional adiabatic quasi-periodic Schrödinger operators”, Annales Scientifiques de l'Ecole Normale Supérieure, 4e série, 38:6 (2005), 889–950 | DOI | MR | Zbl

[20] A. Fedotov, F. Klopp, “Level repulsion and spectral type for one-dimensional adiabatic quasi-periodic Schrödinger operators”, Mathematical Physics of Quantum Mechanics, Selected and Refereed Lectures from QMath9, Lect. Notes in Physics, 690, Springer Verlag, Berlin, 2006, 383–402 | DOI | MR | Zbl

[21] A. Fedotov, F. Klopp, “Weakly resonant tunneling interactions for adiabatic quasi-periodic Schrödinger operators”, Mémoires de la S.M.F., 104 (2006), 1–108 | MR

[22] J. Fröhlich, T. Spencer, P. Wittwer, “Localization for a class of one dimensional quasi-periodic Schrödinger operators”, Communications in Mathematical Physics, 132 (1990), 5–25 | DOI | MR | Zbl

[23] Michael-R. Herman, “Une méthode pour minorer les exposants de Lyapounov et quelques exemples montrant le caractère local d'un théorème d'Arnol'd et de Moser sur le tore de dimension 2”, Comment. Math. Helv., 58:3 (1983), 453–502 | MR | Zbl

[24] H. Hiramoto, M. Kohmoto, “Electronic spectral and wavefunction properties of one-dimensional quasi-periodic systems: a scaling approach”, International Journal of Modern Physics B, 164:3–4 (1992), 281–320 | DOI | MR

[25] A. R. Its, V. B. Matveev, “Hill operators with a finite number of lacunae”, Funkcional. Anal. i Priložen., 9:1 (1975), 69–70 | DOI | MR | Zbl

[26] T. Janssen, “Aperiodic Schrödinger operators”, The Mathematics of Long-Range Aperiodic Order, ed. R. Moody, Kluwer, 1997, 269–306 | DOI | MR | Zbl

[27] S. Jitomirskaya, “Almost everything about the almost Mathieu operator. II”, XIth International Congress of Mathematical Physics (Paris, 1994), Internat. Press, Cambridge, 1995, 373–382 | MR | Zbl

[28] S. Ya. Jitomirskaya, “Metal-insulator transition for the almost Mathieu operator”, Ann. of Math. (2), 150:3 (1999), 1159–1175 | DOI | MR | Zbl

[29] P. Kargaev, E. Korotyaev, “Effective masses and conformal mappings”, Communications in Mathematical Physics, 169 (1995), 597–625 | DOI | MR | Zbl

[30] Y. Last, “Almost everything about the almost Mathieu operator. I”, XIth International Congress of Mathematical Physics (Paris, 1994), Internat. Press, Cambridge, 1995, 366–372 | MR | Zbl

[31] V. Marchenko, I. Ostrovskii, “A characterization of the spectrum of Hill's equation”, Math. USSR Sbornik, 26:4 (1975), 493–554 | DOI | MR | Zbl

[32] M. Marx, H. Najar, “On the singular spectrum for adiabatic quasi-periodic Schrödinger operators”, Adv. Math. Phys., 2010, Art. ID 145436, 30 pp. | MR

[33] H. McKean, P. van Moerbeke, “The spectrum of Hill's equation”, Inventiones Mathematicae, 30 (1975), 217–274 | DOI | MR | Zbl

[34] L. Pastur, A. Figotin, Spectra of Random and Almost-Periodic Operators, Springer Verlag, Berlin, 1992 | MR

[35] Joachim Puig, Reducibility of quasi-periodic skew products and the spectrum of Schrödinger operators, PhD Thesis, Univ. of Barcelona, Barcelona, Spain, 2004 http://www.maia.ub.es/dsg/2004/puig0402.pdf