[Effet tunnel faiblement résonant pour des opérateurs de Schrödinger quasi-périodiques adiabatiques]
Cet article est consacré à l’étude du spectre d’une famille d’opérateurs quasi-périodiques obtenus comme perturbations adiabatiques d’un opérateur périodique fixé. Nous montrons que, dans certaines régions d’énergies, la perturbation entraîne des phénomènes de résonance similaires à ceux observés dans le cas de deux puits. Ces effets s’observent autant sur la géométrie du spectre que sur sa nature. En particulier, on peut observer un entrelacement de types spectraux i.e. une alternance entre du spectre singulier et du spectre absolument continu. Un autre phénomène observé est l’apparition d’îlots de spectre absolument continu dans du spectre singulier dus aux résonances.
In this paper, we study spectral properties of the one dimensional periodic Schrödinger operator with an adiabatic quasi-periodic perturbation. We show that in certain energy regions the perturbation leads to resonance effects related to the ones observed in the problem of two resonating quantum wells. These effects affect both the geometry and the nature of the spectrum. In particular, they can lead to the intertwining of sequences of intervals containing absolutely continuous spectrum and intervals containing singular spectrum. Moreover, in regions where all of the spectrum is expected to be singular, these effects typically give rise to exponentially small “islands” of absolutely continuous spectrum.
Keywords: Quasi-periodic Schrödinger equation, two resonating wells, pure point spectrum, absolutely continuous spectrum, complex WKB method, monodromy matrix
Mot clés : Équations de Schrödinger quasi-périodique, double puis résonnant, spectre purement ponctuel, spectre absolument continu, méthode BKW complexe, matrice de monodromie
@book{MSMF_2006_2_104__1_0,
author = {Fedotov, Alexander and Klopp, Fr\'ed\'eric},
title = {Weakly resonant tunneling interactions for adiabatic quasi-periodic {Schr\"odinger} operators},
series = {M\'emoires de la Soci\'et\'e Math\'ematique de France},
publisher = {Soci\'et\'e math\'ematique de France},
number = {104},
year = {2006},
doi = {10.24033/msmf.416},
mrnumber = {2247399},
zbl = {1129.34001},
language = {en},
url = {http://www.numdam.org/item/MSMF_2006_2_104__1_0/}
}
TY - BOOK AU - Fedotov, Alexander AU - Klopp, Frédéric TI - Weakly resonant tunneling interactions for adiabatic quasi-periodic Schrödinger operators T3 - Mémoires de la Société Mathématique de France PY - 2006 IS - 104 PB - Société mathématique de France UR - http://www.numdam.org/item/MSMF_2006_2_104__1_0/ DO - 10.24033/msmf.416 LA - en ID - MSMF_2006_2_104__1_0 ER -
%0 Book %A Fedotov, Alexander %A Klopp, Frédéric %T Weakly resonant tunneling interactions for adiabatic quasi-periodic Schrödinger operators %S Mémoires de la Société Mathématique de France %D 2006 %N 104 %I Société mathématique de France %U http://www.numdam.org/item/MSMF_2006_2_104__1_0/ %R 10.24033/msmf.416 %G en %F MSMF_2006_2_104__1_0
Fedotov, Alexander; Klopp, Frédéric. Weakly resonant tunneling interactions for adiabatic quasi-periodic Schrödinger operators. Mémoires de la Société Mathématique de France, Série 2, no. 104 (2006), 119 p. doi : 10.24033/msmf.416. http://numdam.org/item/MSMF_2006_2_104__1_0/
[1] & – “Almost periodic Schrödinger operators II, the integrated density of states”, Duke Math. J. 50 (1983), p. 369–391. | MR | Zbl
[2] , & – “Metal-insulator transition for the Almost Mathieu model”, Comm. Math. Phys. 88 (1983), p. 207–234. | MR | Zbl
[3] – “Adiabatic perturbation of a periodic potential”, Teoret. Mat. Fiz. 58 (1984), p. 223–243, in Russian. | MR | Zbl
[4] – “Quasiclassical approximation for equations with periodic coefficients”, Uspekhi Mat. Nauk 42 (1987), p. 77–98, in Russian. | MR
[5] and – “Bloch solutions for difference equations”, Algebra i Analiz 7 (1995), 4, p. 74–122. | MR | Zbl
[6] —, “On the difference equations with periodic coefficients”, Adv. Theor. Math. Phys. 5 (2001), 6, p. 1105–1168. | MR | Zbl
[7] and – “The one-dimensional Schrödinger equation with quasiperiodic potential”, Funktsional. Anal. i Prilozhen. 9 (1975), 4, p. 8–21. | MR
[8] – The spectral theory of periodic differential operators, Scottish Academic Press, 1973. | MR
[9] & – “Strong resonant tunneling, level repulsion and spectral type for one-dimensional adiabatic quasi-periodic Schrödinger operator”, To appear in Ann. Sci. École Norm. Sup. | MR | EuDML | Zbl | Numdam
[10] —, “A complex WKB method for adiabatic problems”, Asymptot. Anal. 27 (2001), 3-4, p. 219–264. | MR | Zbl
[11] —, “Anderson transitions for a family of almost periodic Schrödinger equations in the adiabatic case”, Comm. Math. Phys. 227 (2002), 1, p. 1–92. | MR | Zbl
[12] —, “Geometric tools of the adiabatic complex WKB method”, Asymptot. Anal. 39 (2004), p. 309–357. | MR | Zbl
[13] —, “On the singular spectrum of one dimensional quasi-periodic Schrödinger operators in the adiabatic limit”, Ann. Henri Poincaré 5 (2004), p. 929–978. | MR | Zbl
[14] —, “On the absolutely continuous spectrum of one dimensional quasi-periodic Schrödinger operators in the adiabatic limit”, Trans. Amer. Math. Soc. 357 (2005), p. 4481–4516. | MR | Zbl
[15] – “On the global quasimomentum in solid state physics”, in Mathematical methods in physics, Londrina/World Sci. Publishing, 1999/2000, p. 98–141. | MR | Zbl
[16] – “Double wells”, Comm. Math. Phys. 75 (1980), 3, p. 239–261. | MR | Zbl
[17] & – “Multiple wells in the semi-classical limit I”, Communications in Partial Differential Equations 9 (1984), p. 337–408. | MR | Zbl
[18] – “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 ”, Comment. Math. Helv. 58 (1983), 3, p. 453–502. | EuDML | Zbl
[19] and – “Hill operators with a finite number of lacunae”, Funkcional. Anal. i Priložen. 9 (1975), 1, p. 69–70. | MR
[20] & – “Eigenfunctions, transfer matrices, and absolutely continuous spectrum of one-dimensional Schrödinger operators”, Invent. Math. 135 (1999), 2, p. 329–367. | MR | Zbl
[21] & – “A characterization of the spectrum of Hill’s equation”, Math. USSR Sbornik 26 (1975), p. 493–554. | Zbl
[22] and – “Hill’s surfaces and their theta functions”, Bull. Amer. Math. Soc. 84 (1978), 6, p. 1042–1085. | MR | Zbl
[23] and – “The spectrum of Hill’s equation”, Inventiones Mathematicae 30 (1975), p. 217–274. | MR | EuDML | Zbl
[24] & – Spectra of random and almost-periodic operators, Springer Verlag, 1992. | MR | Zbl
[25] —, “Spectra of random and almost-periodic operators”, in Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 297, Springer-Verlag, 1992. | Zbl
[26] – “Instantons, double wells and large deviations”, Bull. Amer. Math. Soc. 8 (1983), 2, p. 323–326. | MR | Zbl
[27] & – “Positive Lyapunov exponents for Schrödinger operators with quasi-periodic potentials”, Comm. Math. Phys. 142 (1991), 3, p. 543–566. | MR | Zbl
[28] – Eigenfunction expansions associated with second-order differential equations. part ii, Clarendon Press, 1958.
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