On the singular spectrum for adiabatic quasiperiodic Schrödinger operators. (English) Zbl 1201.81053

Summary: We study spectral properties of a family of quasiperiodic Schrödinger operators on the real line in the adiabatic limit. We assume that the adiabatic iso-energetic curve has a real branch that is extended along the momentum direction. In the energy intervals where this happens, we obtain an asymptotic formula for the Lyapunov exponent and show that the spectrum is purely singular. This result was conjectured and proved in a particular case by Fedotov and Klopp (2005).


81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis
34L40 Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.)
34D08 Characteristic and Lyapunov exponents of ordinary differential equations
Full Text: DOI arXiv EuDML


[1] L. Pastur and A. Figotin, Spectra of Random and Almost-Periodic Operators, vol. 297 of Grundlehren der Mathematischen Wissenschaften, Springer, Berlin, Germany, 1992. · Zbl 0752.47002
[2] J. Avron and B. Simon, “Almost periodic Schrödinger operators. II. The integrated density of states,” Duke Mathematical Journal, vol. 50, no. 1, pp. 369-391, 1983. · Zbl 0544.35030
[3] J. M. Ziman, Principles of the Theory of Solids, Cambridge University Press, London, UK, 2nd edition, 1972. · Zbl 0121.44801
[4] V. S. Buslaev and A. A. Fedotov, “Bloch solutions for difference equations,” Algebra i Analiz, vol. 7, no. 4, pp. 74-122, 1995. · Zbl 0859.39001
[5] A. Fedotov and F. Klopp, “On the singular spectrum for adiabatic quashuiti-periodic Schrödinger operators in the real line,” Annales Henri Poincaré, vol. 5, pp. 1-5, 2005. · Zbl 1101.34069
[6] R. Eastham, The Spectral Theory of Periodic Differential Operators, Scottish Academic Press, Edinburgh, UK, 1973. · Zbl 0287.34016
[7] M. Reed and B. Simon, Methods of Modern Mathemathical Vol. IV: Analysis of Operators, Academic Press, New York, NY, USA, 1978. · Zbl 0401.47001
[8] A. Fedotov and F. Klopp, “Anderson transitions for a family of almost periodic Schrödinger equations in the adiabatic case,” Communications in Mathematical Physics, vol. 227, no. 1, pp. 1-92, 2002. · Zbl 1004.81008
[9] H. L. Cycon, R. G. Froese, W. Kirsch, and B. Simon, Schrödinger Operators with Application to Quantum Mechanics and Global Geometry, Texts and Monographs in Physics, Springer, Berlin, Germany, 1987. · Zbl 0619.47005
[10] A. Fedotov and F. Klopp, “On the absolutely continuous spectrum of one-dimensional quasi-periodic Schrödinger operators in the adiabatic limit,” Transactions of the American Mathematical Society, vol. 357, no. 11, pp. 4481-4516, 2005. · Zbl 1101.34069
[11] H. P. McKean and P. van Moerbeke, “The spectrum of Hill’s equation,” Inventiones Mathematicae, vol. 30, no. 3, pp. 217-274, 1975. · Zbl 0319.34024
[12] E. C. Titschmarch, Eigenfunction Expansions Associated with Second-Order Differential Equations Part II, Clarendon Press, Oxford, UK, 1958, New York, NY, USA, 1978. · Zbl 0097.27601
[13] A. Fedotov and F. Klopp, “A complex WKB method for adiabatic problems,” Asymptotic Analysis, vol. 27, no. 3-4, pp. 219-264, 2001. · Zbl 1001.34082
[14] M. Marx, “On the eigenvalues for slowly varying perturbations of a periodic Schrödinger operator,” Asymptotic Analysis, vol. 48, no. 4, pp. 295-357, 2006. · Zbl 1124.34063
[15] M. V. Fedoryuk, Asymptotic Analysis, Springer, Berlin, Germany, 1993. · Zbl 0782.34001
[16] A. Fedotov and F. Klopp, “Geometric tools of the adiabatic complex WKB method,” Asymptotic Analysis, vol. 39, no. 3-4, pp. 309-357, 2004. · Zbl 1070.34124
[17] F. Klopp and M. Marx, “Resonances for slowly varying perturbations of one-dimensional periodic Schrödinger operators,” in Séminaire sur les Equations aux Dérivées Partielles de l’ Ecole Polytechnique (2005-2006), vol. 4, p. 18, Ecole Polytechnique, Palaiseau, France, 2006. · Zbl 1118.34082
[18] M. Marx, Etude de perturbations adiabatiques de l’équation de Schrödinger périodique, Ph.D. thesis, Université Paris XIII, Paris, France, 2004.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.