Strong resonant tunneling, level repulsion and spectral type for one-dimensional adiabatic quasi-periodic Schrödinger operators

Fedotov, Alexander; Klopp, Frédéric

doi:10.1016/j.ansens.2005.10.002

Fedotov, Alexander ; Klopp, Frédéric

Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 38 (2005) no. 6, pp. 889-950.

MR Zbl | 1 citation dans Numdam

DOI : 10.1016/j.ansens.2005.10.002

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     author = {Fedotov, Alexander and Klopp, Fr\'ed\'eric},
     title = {Strong resonant tunneling, level repulsion and spectral type for one-dimensional adiabatic quasi-periodic {Schr\"odinger} operators},
     journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure},
     pages = {889--950},
     publisher = {Elsevier},
     volume = {Ser. 4, 38},
     number = {6},
     year = {2005},
     doi = {10.1016/j.ansens.2005.10.002},
     mrnumber = {2216834},
     zbl = {05078681},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1016/j.ansens.2005.10.002/}
}

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Fedotov, Alexander; Klopp, Frédéric. Strong resonant tunneling, level repulsion and spectral type for one-dimensional adiabatic quasi-periodic Schrödinger operators. Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 38 (2005) no. 6, pp. 889-950. doi : 10.1016/j.ansens.2005.10.002. http://archive.numdam.org/articles/10.1016/j.ansens.2005.10.002/

Bibliographie
Cité par

[1] Avron J., Simon B., Almost periodic Schrödinger operators, II. The integrated density of states, Duke Math. J. 50 (1983) 369-391. | MR | Zbl

[2] Bellissard J., Lima R., Testard D., Metal-insulator transition for the Almost Mathieu model, Comm. Math. Phys. 88 (1983) 207-234. | MR | Zbl

[3] Buslaev V., Fedotov A., On the difference equations with periodic coefficients, Adv. Theor. Math. Phys. 5 (6) (2001) 1105-1168. | MR | Zbl

[4] Buslaev V.S., Fedotov A.A., Bloch solutions for difference equations, Algebra Anal. 7 (4) (1995) 74-122. | MR | Zbl

[5] Dinaburg E.I., Sinaĭ J.G., The one-dimensional Schrödinger equation with quasiperiodic potential, Funkcional. Anal. Priložen. 9 (4) (1975) 8-21. | MR | Zbl

[6] Eastham M., The Spectral Theory of Periodic Differential Operators, Scottish Academic Press, Edinburgh, 1973. | Zbl

[7] Fedotov A., Klopp F., On the absolutely continuous spectrum of one-dimensional quasi-periodic Schrödinger operators in the adiabatic limit, Trans. Amer. Math. Soc. 357 (2005) 4481-4516. | MR | Zbl

[8] Fedotov A., Klopp F., Geometric tools of the adiabatic complex WKB method, Asymptot. Anal. 39 (3-4) (2004) 309-357. | MR | Zbl

[9] Fedotov A., Klopp F., On the singular spectrum of one-dimensional quasi-periodic Schrödinger operators in the adiabatic limit, Ann. H. Poincaré 5 (2004) 929-978. | MR | Zbl

[10] Fedotov A., Klopp F., A complex WKB method for adiabatic problems, Asymptot. Anal. 27 (3-4) (2001) 219-264. | MR | Zbl

[11] Fedotov A., Klopp F., Anderson transitions for a family of almost periodic Schrödinger equations in the adiabatic case, Comm. Math. Phys. 227 (1) (2002) 1-92. | MR | Zbl

[12] Fedotov A., Klopp F., Weakly resonant tunneling interactions for adiabatic quasi-periodic Schrödinger operators, Mémoires SMF, in press.

[13] Firsova N.E., On the global quasimomentum in solid state physics, in: Mathematical Methods in Physics, Londrina, 1999, World Scientific, River Edge, NJ, 2000, pp. 98-141. | MR | Zbl

[14] Gilbert D., Pearson D., On subordinacy and analysis of the spectrum of one-dimensional Schrödinger operators, J. Math. Anal. Appl. 128 (1987) 30-56. | MR | Zbl

[15] Harrell E.M., Double wells, Comm. Math. Phys. 75 (3) (1980) 239-261. | MR | Zbl

[16] Helffer B., Sjöstrand J., Multiple wells in the semi-classical limit I, Comm. Partial Differential Equations 9 (1984) 337-408. | MR | Zbl

[17] Herman M., 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

[18] Its A.R., Matveev V.B., Hill operators with a finite number of lacunae, Funkcional. Anal. Priložen. 9 (1) (1975) 69-70. | MR | Zbl

[19] Last Y., Simon B., Eigenfunctions, transfer matrices, and absolutely continuous spectrum of one-dimensional Schrödinger operators, Invent. Math. 135 (2) (1999) 329-367. | MR | Zbl

[20] Marchenko V., Ostrovskii I., A characterization of the spectrum of Hill's equation, Math. USSR Sb. 26 (1975) 493-554. | Zbl

[21] Mckean H., Van Moerbeke P., The spectrum of Hill's equation, Invent. Math. 30 (1975) 217-274. | MR | Zbl

[22] Mckean H.P., Trubowitz E., Hill's surfaces and their theta functions, Bull. Amer. Math. Soc. 84 (6) (1978) 1042-1085. | MR | Zbl

[23] Novikov S., Manakov S.V., Pitaevskiĭ L.P., Zakharov V.E., Theory of Solitons, Contemporary Soviet Mathematics, Consultants Bureau (Plenum), New York, 1984, The inverse scattering method. Translated from the Russian. | MR | Zbl

[24] Pastur L., Figotin A., Spectra of Random and Almost-Periodic Operators, Grundlehren der Mathematischen Wissenschaften, vol. 297, Springer, Berlin, 1992. | MR | Zbl

[25] Shabat B.V., Vvedenie v kompleksnyi analiz. Chast I, Nauka, Moscow, 1985, Funktsii odnogo peremennogo (Functions of a single variable). | MR

[26] Simon B., Instantons, double wells and large deviations, Bull. Amer. Math. Soc. (N.S.) 8 (2) (1983) 323-326. | MR | Zbl

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

[28] Titschmarch E.C., Eigenfunction Expansions Associated with Second-Order Differential Equations. Part II, Clarendon Press, Oxford, 1958. | Zbl

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