The width of resonances for slowly varying perturbations of one-dimensional periodic Schrödinger operators
Séminaire Équations aux dérivées partielles (Polytechnique) (2005-2006), Talk no. 4, 16 p.
Classification:  34E05,  34E20,  34L05,  34L40
Keywords: 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)},
     publisher = {Centre de math\'ematiques Laurent Schwartz, \'Ecole polytechnique},
     year = {2005-2006},
     note = {talk:4},
     mrnumber = {2276070},
     language = {en},
     url = {http://www.numdam.org/item/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) (2005-2006), Talk no. 4, 16 p. http://www.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 1611735 | Zbl 1001.34075

[2] V. Buslaev and A. Grigis. Turning points for adiabatically perturbed periodic equations. J. Anal. Math., 84:67–143, 2001. | MR 1849199 | Zbl 0987.35013

[3] M. Dimassi. Resonances for slowly varying perturbations of a periodic Schrödinger operator. Canad. J. Math., 54(5):998–1037, 2002. | MR 1924711 | Zbl 1025.81016

[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 1962356 | Zbl 1090.35065

[5] M. Eastham. The spectral theory of periodic differential operators. Scottish Academic Press, Edinburgh, 1973. | Zbl 0287.34016

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

[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 1903839 | Zbl 1004.81008

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

[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 2091984 | Zbl 1059.81057

[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 2156718 | Zbl 1101.34069

[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 1775625 | Zbl 0996.81124

[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 1635811 | Zbl 0916.34071

[13] S. Fujiié and T. Ramond. Breit-Wigner formula at barrier tops. J. Math. Phys., 44(5):1971–1983, 2003. | MR 1972758 | Zbl 1062.81057

[14] B. Helffer and J. Sjöstrand. Résonances en limite semi-classique. Mém. Soc. Math. France (N.S.), (24-25):iv+228, 1986. | Numdam | MR 871788 | Zbl 0631.35075

[15] P. Hislop and I. Sigal. Semi-classical theory of shape resonances in quantum mechanics. Memoirs of the American Mathematical Society, 78, 1989. | Zbl 0704.35115

[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 0343.34016

[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 2256576 | Zbl 05144764

[21] H. P. McKean and E. Trubowitz. Hill’s surfaces and their theta functions. Bull. Amer. Math. Soc., 84(6):1042–1085, 1978. | Zbl 0428.34026

[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 0097.27601

[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 1291649 | Zbl 0823.35139

[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 1957536 | Zbl 01789890

[26] M. Zworski. Resonances in physics and geometry. Notices Amer. Math. Soc., 46(3):319–328, 1999. | MR 1668841 | Zbl 1177.58021