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, Volume 38 (2005) no. 6, p. 889-950
@article{ASENS_2005_4_38_6_889_0,
     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},
     publisher = {Elsevier},
     volume = {Ser. 4, 38},
     number = {6},
     year = {2005},
     pages = {889-950},
     doi = {10.1016/j.ansens.2005.10.002},
     zbl = {05078681},
     mrnumber = {2216834},
     language = {en},
     url = {http://www.numdam.org/item/ASENS_2005_4_38_6_889_0}
}
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, Volume 38 (2005) no. 6, pp. 889-950. doi : 10.1016/j.ansens.2005.10.002. http://www.numdam.org/item/ASENS_2005_4_38_6_889_0/

[1] Avron J., Simon B., Almost periodic Schrödinger operators, II. The integrated density of states, Duke Math. J. 50 (1983) 369-391. | MR 700145 | Zbl 0544.35030

[2] Bellissard J., Lima R., Testard D., Metal-insulator transition for the Almost Mathieu model, Comm. Math. Phys. 88 (1983) 207-234. | MR 696805 | Zbl 0542.35059

[3] Buslaev V., Fedotov A., On the difference equations with periodic coefficients, Adv. Theor. Math. Phys. 5 (6) (2001) 1105-1168. | MR 1926666 | Zbl 1012.39008

[4] Buslaev V.S., Fedotov A.A., Bloch solutions for difference equations, Algebra Anal. 7 (4) (1995) 74-122. | MR 1356532 | Zbl 0847.39002

[5] Dinaburg E.I., Sinaĭ J.G., The one-dimensional Schrödinger equation with quasiperiodic potential, Funkcional. Anal. Priložen. 9 (4) (1975) 8-21. | MR 470318 | Zbl 0333.34014

[6] Eastham M., The Spectral Theory of Periodic Differential Operators, Scottish Academic Press, Edinburgh, 1973. | Zbl 0287.34016

[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 2156718 | Zbl 1101.34069

[8] Fedotov A., Klopp F., Geometric tools of the adiabatic complex WKB method, Asymptot. Anal. 39 (3-4) (2004) 309-357. | MR 2097997 | Zbl 1070.34124

[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 2091984 | Zbl 1059.81057

[10] Fedotov A., Klopp F., A complex WKB method for adiabatic problems, Asymptot. Anal. 27 (3-4) (2001) 219-264. | MR 1858917 | Zbl 1001.34082

[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 1903839 | Zbl 1004.81008

[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 1775625 | Zbl 0996.81124

[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 915965 | Zbl 0666.34023

[15] Harrell E.M., Double wells, Comm. Math. Phys. 75 (3) (1980) 239-261. | MR 581948 | Zbl 0445.35036

[16] Helffer B., Sjöstrand J., Multiple wells in the semi-classical limit I, Comm. Partial Differential Equations 9 (1984) 337-408. | MR 740094 | Zbl 0546.35053

[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 727713 | Zbl 0554.58034

[18] Its A.R., Matveev V.B., Hill operators with a finite number of lacunae, Funkcional. Anal. Priložen. 9 (1) (1975) 69-70. | MR 390355 | Zbl 0318.34038

[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 1666767 | Zbl 0931.34066

[20] Marchenko V., Ostrovskii I., A characterization of the spectrum of Hill's equation, Math. USSR Sb. 26 (1975) 493-554. | Zbl 0343.34016

[21] Mckean H., Van Moerbeke P., The spectrum of Hill's equation, Invent. Math. 30 (1975) 217-274. | MR 397076 | Zbl 0319.34024

[22] Mckean H.P., Trubowitz E., Hill's surfaces and their theta functions, Bull. Amer. Math. Soc. 84 (6) (1978) 1042-1085. | MR 508448 | Zbl 0428.34026

[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 779467 | Zbl 0598.35002

[24] Pastur L., Figotin A., Spectra of Random and Almost-Periodic Operators, Grundlehren der Mathematischen Wissenschaften, vol. 297, Springer, Berlin, 1992. | MR 1223779 | Zbl 0752.47002

[25] Shabat B.V., Vvedenie v kompleksnyi analiz. Chast I, Nauka, Moscow, 1985, Funktsii odnogo peremennogo (Functions of a single variable). | MR 819482

[26] Simon B., Instantons, double wells and large deviations, Bull. Amer. Math. Soc. (N.S.) 8 (2) (1983) 323-326. | MR 684899 | Zbl 0529.35059

[27] Sorets E., Spencer T., Positive Lyapunov exponents for Schrödinger operators with quasi-periodic potentials, Comm. Math. Phys. 142 (3) (1991) 543-566. | MR 1138050 | Zbl 0745.34046

[28] Titschmarch E.C., Eigenfunction Expansions Associated with Second-Order Differential Equations. Part II, Clarendon Press, Oxford, 1958. | Zbl 0097.27601