Complex WKB method for adiabatic perturbations of a periodic Schrödinger operator

A. A. Fedotov

Zapiski Nauchnykh Seminarov POMI, Mathematical problems in the theory of wave propagation. Part 39, Tome 379 (2010), pp. 142-178

Résumé

The review is devoted to an analog of the complex WKB method developed to study solutions of an one dimensional periodic Schrödinger equation with an adiabatic perturbation. We illustrate how to use the method by examples. Bibl. 30 titles.

Bibliographie

[1] J. Avron, B. Simon, “Almost periodic Hill's equation and the rings of Saturn”, Phys. Rev. Lett., 46:17 (1981), 1166–1168 | DOI | MR

[2] V. S. Buslaev, “Adiabaticheskoe vozmuschenie periodicheskogo potentsiala”, Teor. mat. fiz., 58:2 (1984), 233–243 | MR | Zbl

[3] V. S. Buslaev, L. A. Dmitrieva, “Adiabaticheskoe vozmuschenie periodicheskogo potentsiala. II”, Teor. mat. fiz., 73:3 (1987), 430–442 | MR | Zbl

[4] V. S. Buslaev, “Kvaziklassicheskoe priblizhenie dlya uravnenii s periodicheskimi koeffitsientami”, Uspekhi mat. nauk, 42:6 (1987), 77–98 | MR | Zbl

[5] V. S. Buslaev, L. A. Dmitrieva, “Blokhovskii elektron vo vneshnem pole”, Algebra i analiz, 1:2 (1989), 1–29 | MR | Zbl

[6] V. S. Buslaev, “Spectral properties of the operators $H\psi=-\psi_{xx}+p(x)\psi+v(\varepsilon x)\psi$, $p$ is periodic”, Oper. Theory Adv. Appl., 46 (1990), 85–107 | MR | Zbl

[7] V. Buslaev, “On spectral properties of adiabatically perturbed Schrödinger operators with periodic potentials”, Séminaires Équations aux Dérivées Partielles (1990–1991), École Polytechnique, Palaiseau, 1991, Exp. No. XXIII, 15 pp. | MR

[8] V. Buslaev, A. Grigis, “Imaginary parts of Stark-Wannier resonances”, J. Math. Phys., 39:5 (1998), 2520–2550 | DOI | MR | Zbl

[9] V. S. Buslaev, M. V. Buslaeva, A. Grizhis, “Asimptotiki koeffitsienta otrazheniya”, Algebra i analiz, 16:3 (2004), 1–23 | MR | Zbl

[10] V. Buslaev, A. Fedotov, The complex WKB method for Harper's equation, Preprint, Mittag-Leffler Institute, Stocholm, 1993

[11] V. S. Buslaev, A. A. Fedotov, “Kompleksnyi metod VKB dlya uravneniya Kharpera”, Algebra i analiz, 6:3 (1994), 59–83 | MR | Zbl

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

[13] M. Fedoryuk, Asymptotic analysis, 1-ed., Springer Verlag, Berlin, 1993 | MR | Zbl

[14] A. Fedotov, F. Klopp, “A complex WKB method for adiabatic problems”, Asymptotic analysis, 27 (2001), 219–264 | MR | Zbl

[15] A. Fedotov, F. Klopp, “Anderson transitions for a family of almost periodic Schrödinger equations in the adiabatic case”, Commun. Math. Phys., 227 (2002), 1–92 | DOI | MR | Zbl

[16] A. Fedotov, F. Klopp, “On the singular spectrum of quasi-periodic Schrödinger operator in adiabatic limit”, Annales Henri Poincaré, 5 (2004), 929–978 | DOI | MR | Zbl

[17] A. Fedotov, F. Klopp, “Geometric tools of the adiabatic complex WKB method”, Asymptotic Analysis, 39:3–4 (2004), 309–357 | MR | Zbl

[18] A. Fedotov, F. Klopp, “On the absolutely continuous spectrum of one dimensional quasi-periodic Schrödinger operator in adiabatic limit”, Transactions Amer. Math. Soc., 357 (2005), 4481–4516 | DOI | MR | Zbl

[19] A. Fedotov, F. Klopp, “Strong resonant tunneling, level repulsion and spectral type for one-dimensional adiabatic quasi-periodic Schrödinger operators”, Annales Scientifiques de l'Ecole Normale Supérieure, 4e série, 38:6 (2005), 889–950 | DOI | MR | Zbl

[20] A. Fedotov, F. Klopp, “Weakly resonant tunneling interactions for adiabatic quasi-periodic Schrödinger operators”, Mémoires de la S.M.F., 104 (2006), 1–108 | MR

[21] F. Klopp, M. Marx, “The width of resonances for slowly varying perturbations of one-dimensional periodic Schrödinger operators”, Séminaires Équations aux Dérivées Partielles (2005–2006), École Polytechnique, Palaiseau, 2006, Exp. No. IV, 18 pp. | MR

[22] V. Marchenko, I. Ostrovskii, “Kharakteristika spektra operatora Khilla”, Mat. sbornik, 97(139):4 (1975), 540–606 | MR | Zbl

[23] M. Marx, Étude de perturbations adiabatiques de l'équation de Schrödinger pèriodique, PhD Thesis, Université Paris 13, Villetaneuse, 2004

[24] M. Marx, “On the eigenvalues for slowly varying perturbations of a periodic Schrödinger operator”, J. Asymptotic Analysis, 48:4 (2006), 295–357 | MR | Zbl

[25] M. Marx, H. Najar, “On the singular spectrum for adiabatic quasi-periodic Schrödinger operators”, Adv. Math. Phys., 2010, Art. ID 145436, 30 pp. | MR

[26] A. Metelkina, “Lyapunov exponent and integrated density of states for the slowly oscillating perturbations of the periodic Schrödinger operators”, International Conference in Spectral Theor, Program and abstracts of the conference, Euler International Mathematical Institute, St. Petersburg, 2010, 55–56

[27] H. McKean, P. van Moerbeke, “The spectrum of Hill's equation”, Inventiones Mathematicae, 30 (1975), 217–274 | DOI | MR | Zbl

[28] H. P. McKean, E. Trubowitz, “Hill's surfaces and their theta functions”, Bull. Amer. Math. Soc., 84:6 (1978), 1042–1085 | DOI | MR | Zbl

[29] Y. Sibuya, Global theory of second order linear ordinary differential equations with a polynomial coefficient, North-Holland, Amsterdam, 1975 | Zbl

[30] E. C. Titschmarch, Eigenfunction expansions associated with second-order differential equations, Part II, Clarendon Press, Oxford, 1958