Complex WKB method for adiabatic perturbations of a periodic Schrödinger operator. (English. Russian original) Zbl 1237.34148

J. Math. Sci., New York 173, No. 3, 320-339 (2011); translation from Zap. Nauchn. Semin. POMI 379, 142-178 (2010).
The paper considers the periodic Schrödinger operator \[ -\frac{d^{2}\psi }{dx^{2}}+(V(x)+W(\varepsilon x))\psi =E\psi ,\;x\in\mathbb{R}, \] where \(E\) is the spectral parameter, \(\varepsilon \) is a small parameter, \(V\) is a real valued \(1\)-periodic function from \(L_{loc}^{2}(\mathbb{R})\) and \(W\) is an analytic function in the strip of the form \(\delta =\left\{ \xi \in\mathbb{C}:y_{1}<\text{Im}\xi <y_{2}\right\}\). The author presents an analog of the complex WKB method developed for studying effects of adiabatic perturbations of the periodic Schrödinger operator.


34L40 Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.)
34M60 Singular perturbation problems for ordinary differential equations in the complex domain (complex WKB, turning points, steepest descent)
47E05 General theory of ordinary differential operators
Full Text: DOI


[1] J. Avron and B. Simon, ”Almost periodic Hill’s equation and the rings of Saturn,” Phys. Rev. Lett., 46, No. 17, 1166–1168 (1981).
[2] V. S. Buslaev, ”Adiabatic perturbation of a periodic potential,” Teor. Mat. Fiz., 58, No. 2, 233–243 (1984). · Zbl 0534.34064
[3] V. S. Buslaev and L. A. Dmitrieva, ”Adiabatic perturbation of a periodic potential. II,” Teor. Mat. Fiz., 73, No. 3, 439–442 (1987). · Zbl 0643.34068
[4] V. S. Buslaev, ”Quasiclassical approximation for equations with periodic coefficients,” Usp. Mat. Nauk, 42, No. 6, 77–98 (1987).
[5] V. S. Buslaev and L. A. Dmitrieva, ”Bloch electron in an external field,” Algebra Analiz, 1, No. 1, 1–29 (1989). · Zbl 0714.34128
[6] V. S. Buslaev, ”Spectral properties of the operators H{\(\psi\)} = {\(\psi\)} xx + p(x){\(\psi\)} + {\(\nu\)}(x){\(\psi\)}, p: is periodic,” Oper. Theory Adv. Appl., 46, 85–197 (1999).
[7] V. Buslaev, ”On spectral properties of adiabatically perturbed Schrödinger operators with periodic potentials,” in: Équations auz Dérinées Partielles, École Polytech., Palaiseau (1991), pp. 1–15. · Zbl 0739.35053
[8] V. Buslaev and A. Grigis, ”Imaginary parts of Stark-Wannier resonances,” J. Math. Phys, 39, No. 5, 2529–2559 (1998). · Zbl 1001.34075
[9] V. S. Buslaev, M. V. Buslaeva, and A. Grigis, ”Asymptotics of a reflection coefficient,” Algebra Analiz, 16, No. 3, 1–23 (2004). · Zbl 1084.34026
[10] V. Buslaev and A. Fedotov, ”The complex WKB method for Harper’s equation,” Preprint, Mittag-Leffler Institute, Stockholm (1993). · Zbl 0821.34062
[11] V. S. Buslaev and A. A. Fedotov. ”Complex WKB method for the Harper equation,” Algebra Analiz, 6, No. 3, 59–89 (1994). · Zbl 0839.34066
[12] M. Eastham, The Spectral Theory of Periodic Differential Operators, Scottish Academic Press, Edinburgh (1973). · Zbl 0287.34016
[13] M. Fedoryuk, Asymptotic Analysis, led., Springer Verlag, Berlin (1993). · Zbl 0782.34001
[14] A. Fedotov and F. Klopp, ”A complex WKB method for adiabatic problems,” Asympt. Analysis, 27, 219–264 (2001). · Zbl 1001.34082
[15] A. Fedotov and F. Klopp, ”Anderson transitions for a family of almost periodic Schrödinger equations in the adiabatic case.” Comm. Math. Phys., 227, 1–92 (2002). · Zbl 1004.81008
[16] A. Fedotoy and F. Klopp, ”On the singular spectrum of quasi-periodic Schrödinger operator in adiabatic limit,” Annales Henri Poincaré, 5, 929–978 (2994). · Zbl 1059.81057
[17] A. Fedotov and F. Klopp, ”Geometric tools of the adiabatic complex WKB method,” Asympt. Analysis, 39, No. 3–4, 309–357 (2004). · Zbl 1070.34124
[18] A. Fedotov and F. Klopp, ”On the absolutely continuous spectrum of one dimensional quasi-periodic Schrödinger operator in adiabatic limit,” Trans. Amer. Math. Soc., 357, 4481–4516 (2005). · Zbl 1101.34069
[19] A. Fedotov and F. Klopp, ”Strong resonant tunneling, level repulsion and spectral type for one-dimensional adiabatic quasi-periodic Schrödinger operators,” Annales Sei. l’Ecole Norm. Supér., 4e série, 38, No. 6, 889–959 (2005). · Zbl 1112.47038
[20] A. Fedotov and F. Klopp, ”Weakly resonant tunneling interactions for adiabatic quasi-periodic Schrödinger operators,” Mérnoires de la S.M.F., 194, 1–198 (2006). · Zbl 1129.34001
[21] F. Klopp and M. Marx, ”The width of resonances for slowly varying perturbations of one-dimensional periodic Schrödinger operators,” in: Séminaire Equations aux Dérivées Partielles, Ècole Polytech. (2006), pp. 1–16.
[22] V. Marchenko and I. Ostrovski, ”Characterization of the spectrum of the Hill operator,” Mat. Sb., 97, No. 4, 493–554 (1975). · Zbl 0343.34016
[23] M. Marx, ”Étude de perturbations adiabatiques de l’équation de Schrödinger pèriodique,” PhD Thesis, Univ. Paris 13, Villetaneuse (2004).
[24] M. Marx, ”On the eigenvalues for slowly varying perturbations of a periodic Schrödinger operator,” J. Asympt. Anal., 48, No. 4, 295–357 (2006). · Zbl 1124.34063
[25] M. Marx and H. Najar, ”On the singular spectrum for adiabatic quasi-periodic Schrödinger operators,” Adv. Math. Phys., accepted February 28 (2010). · Zbl 1201.81053
[26] A. Metelkina, ”Lyapunov exponent and integrated density of states for the slowly oscillating perturbations of the periodic Schrödinger operators,” in: International Conference in Spectral Theory, Program and abstracts of the conference, Euler Intern. Math. Inst., St. Petersburg (2010), pp. 55–56.
[27] H. McKean and P. van Moerbeke, ”The spectrum of Hill’s equation,” Invent. Math., 39, 217–274 (1975). · Zbl 0319.34024
[28] H. P. McKean and E. Trubowitz, ”Hill’s surfaces and their theta functions” Bull. Amer. Math., Soc., 84, No., 6, 1042–1085 (1978). · Zbl 0428.34026
[29] Y. Sibuya, Global Theory of Second Order Linear Ordinary Differential Equations with a Polynomial Coefficient, North-Holland, Amsterdam (1975). · Zbl 0322.34006
[30] E. C. Titschmarch, Eigenfunction Expansions Associated with Second-Order Differential Equations, Part II, Clarendon Press, Oxford (1958).
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