Mots-clés : resonances, complex WKB method
@article{SEDP_2005-2006____A4_0,
author = {Klopp, Fr\'ed\'eric and Marx, Magali},
title = {The width of resonances for slowly varying perturbations of one-dimensional periodic {Schr\"odinger} operators},
journal = {S\'eminaire \'Equations aux d\'eriv\'ees partielles (Polytechnique) dit aussi "S\'eminaire Goulaouic-Schwartz"},
note = {talk:4},
pages = {1--16},
publisher = {Centre de math\'ematiques Laurent Schwartz, \'Ecole polytechnique},
year = {2005-2006},
mrnumber = {2276070},
language = {en},
url = {http://archive.numdam.org/item/SEDP_2005-2006____A4_0/}
}
TY - JOUR AU - Klopp, Frédéric AU - Marx, Magali TI - The width of resonances for slowly varying perturbations of one-dimensional periodic Schrödinger operators JO - Séminaire Équations aux dérivées partielles (Polytechnique) dit aussi "Séminaire Goulaouic-Schwartz" N1 - talk:4 PY - 2005-2006 SP - 1 EP - 16 PB - Centre de mathématiques Laurent Schwartz, École polytechnique UR - http://archive.numdam.org/item/SEDP_2005-2006____A4_0/ LA - en ID - SEDP_2005-2006____A4_0 ER -
%0 Journal Article %A Klopp, Frédéric %A Marx, Magali %T The width of resonances for slowly varying perturbations of one-dimensional periodic Schrödinger operators %J Séminaire Équations aux dérivées partielles (Polytechnique) dit aussi "Séminaire Goulaouic-Schwartz" %Z talk:4 %D 2005-2006 %P 1-16 %I Centre de mathématiques Laurent Schwartz, École polytechnique %U http://archive.numdam.org/item/SEDP_2005-2006____A4_0/ %G en %F SEDP_2005-2006____A4_0
Klopp, Frédéric; Marx, Magali. The width of resonances for slowly varying perturbations of one-dimensional periodic Schrödinger operators. Séminaire Équations aux dérivées partielles (Polytechnique) dit aussi "Séminaire Goulaouic-Schwartz" (2005-2006), Exposé no. 4, 16 p. http://archive.numdam.org/item/SEDP_2005-2006____A4_0/
[1] V. Buslaev and A. Grigis. Imaginary parts of Stark-Wannier resonances. J. Math. Phys., 39(5):2520–2550, 1998. | MR | Zbl
[2] V. Buslaev and A. Grigis. Turning points for adiabatically perturbed periodic equations. J. Anal. Math., 84:67–143, 2001. | MR | Zbl
[3] M. Dimassi. Resonances for slowly varying perturbations of a periodic Schrödinger operator. Canad. J. Math., 54(5):998–1037, 2002. | MR | Zbl
[4] M. Dimassi and M. Zerzeri. A local trace formula for resonances of perturbed periodic Schrödinger operators. J. Funct. Anal., 198(1):142–159, 2003. | MR | Zbl
[5] M. Eastham. The spectral theory of periodic differential operators. Scottish Academic Press, Edinburgh, 1973. | Zbl
[6] A. Fedotov and F. Klopp. A complex WKB method for adiabatic problems. Asymptot. Anal., 27(3-4):219–264, 2001. | MR | Zbl
[7] A. Fedotov and F. Klopp. Anderson transitions for a family of almost periodic Schrödinger equations in the adiabatic case. Comm. Math. Phys., 227(1):1–92, 2002. | MR | Zbl
[8] A. Fedotov and F. Klopp. Geometric tools of the adiabatic complex WKB method. Asymptot. Anal., 39(3-4):309–357, 2004. | MR | Zbl
[9] A. Fedotov and F. Klopp. On the singular spectrum for adiabatic quasi-periodic Schrödinger operators on the real line. Ann. Henri Poincaré, 5(5):929–978, 2004. | MR | Zbl
[10] A. Fedotov and F. Klopp. On the absolutely continuous spectrum of one-dimensional quasi-periodic Schrödinger operators in the adiabatic limit. Trans. Amer. Math. Soc., 357(11):4481–4516 (electronic), 2005. | MR | Zbl
[11] N. E. Firsova. On the global quasimomentum in solid state physics. In Mathematical methods in physics (Londrina, 1999), pages 98–141. World Sci. Publishing, River Edge, NJ, 2000. | MR | Zbl
[12] S. Fujiié and T. Ramond. Matrice de scattering et résonances associées à une orbite hétérocline. Ann. Inst. H. Poincaré Phys. Théor., 69(1):31–82, 1998. | Numdam | MR | Zbl
[13] S. Fujiié and T. Ramond. Breit-Wigner formula at barrier tops. J. Math. Phys., 44(5):1971–1983, 2003. | MR | Zbl
[14] B. Helffer and J. Sjöstrand. Résonances en limite semi-classique. Mém. Soc. Math. France (N.S.), (24-25):iv+228, 1986. | Numdam | MR | Zbl
[15] P. Hislop and I. Sigal. Semi-classical theory of shape resonances in quantum mechanics. Memoirs of the American Mathematical Society, 78, 1989. | Zbl
[16] F. Klopp and M. Marx. Resonances for slowly varying perturbations of one-dimensional periodic Schrödinger operators. in progress.
[17] F. Klopp and M. Marx. Resonances for slowly varying perturbations of one-dimensional periodic Schrödinger operators II: oscillation of resonances. in progress.
[18] V. Marchenko and I. Ostrovskii. A characterization of the spectrum of Hill’s equation. Math. USSR Sbornik, 26:493–554, 1975. | Zbl
[19] M. Marx. Étude de perturbations adiabatiques de l’équation de Schrödinger périodique. PhD thesis, Université Paris 13, Villetaneuse, 2004.
[20] M. Marx. On the eigenvalues for slowly varying perturbations of a periodic Schrödinger operator. To appear in Asymptotic Analysis, 2006. | MR | Zbl
[21] H. P. McKean and E. Trubowitz. Hill’s surfaces and their theta functions. Bull. Amer. Math. Soc., 84(6):1042–1085, 1978. | Zbl
[22] J. Sjöstrand. Lectures on resonances, 2002. http://www.math.polytechnique.fr/~sjoestrand/CoursgbgWeb.pdf
[23] E.C. Titschmarch. Eigenfunction expansions associated with second-order differential equations. Part II. Clarendon Press, Oxford, 1958. | Zbl
[24] M. Zworski. Counting scattering poles. In Spectral and Scattering, volume 161 of Lecture Notes in Pure and Applied Mathematics, pages 301–331, New-York, 1994. Marcel Dekker. | MR | Zbl
[25] M. Zworski. Quantum resonances and partial differential equations. In Proceedings of the International Congress of Mathematicians, Vol. III (Beijing, 2002), pages 243–252, Beijing, 2002. Higher Ed. Press. | MR | Zbl
[26] M. Zworski. Resonances in physics and geometry. Notices Amer. Math. Soc., 46(3):319–328, 1999. | MR | Zbl
