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Turning points for adiabatically perturbed periodic equations

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Abstract

We make a study of uniform asymptotic solutions of some general adiabatic differential equations on the intervals containing a single turning point or a pair of turning points. We reduce this to the study of models which can be explicitly solved. We apply these results to the case of an adiabatically perturbed differential equation with periodic coefficients.

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References

  1. M. Abramowitz and I. A. Stegun,Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Dover Publications, Inc., New York, 1992.

    Google Scholar 

  2. V. I. Arnold,Geometrical Methods in the Theory of Ordinary Differential Equations, Springer-Verlag, New York-Berlin, 1988.

    Google Scholar 

  3. J. E. Avron, R. Seiler and L. G. Jaffe,Adiabatic theorems and applications to the quantum Hall effect, Comm. Math. Phys.110 (1987), 33–49.

    Article  MATH  MathSciNet  Google Scholar 

  4. V. S. Buldyrev and S. Yu. Slavjanov,Uniform asymptotic expansions for solutions of an equation of Schrödinger type with two transition points. I (in Russian), Vestnik Leningrad. Univ.23 (1968), no. 22, 70–84.

    MathSciNet  Google Scholar 

  5. V. S. Buldyrev and S. Yu. Slavjanov,Regularization of the phase integrals near the barrier top (in Russian), Problemi Matemat. Fiziki, Leningrad Univ. No. 10 (M. Sh. Birman, ed.) (1982), 50–70.

  6. V. S. Buslaev,Adiabatic perturbation of a periodic potential (in Russian), Teoret. Mat. Fiz.58 (1984), no. 2, 233–243.

    MATH  MathSciNet  Google Scholar 

  7. V. S. Buslaev,Quasiclassical approximation for equations with periodic coefficients (in Russian) Uspekhi Mat. Nauk42 (1987), no. 6 (258), 77–98.

    MathSciNet  Google Scholar 

  8. V. S. Buslaev,On spectral properties of adiabatically perturbed Schrödinger operators with periodic potential, Séminaire EDP, Ecole Polytechnique, 1990–91, no. 23.

  9. V. S. Buslaev and L. A. Dmitrieva,Adiabatic perturbation of a periodic potential. II (in Russian), Teoret. Mat. Fiz.73 (1987), no. 3, 430–442.

    MATH  MathSciNet  Google Scholar 

  10. V. S. Buslaev and L. A. Dmitrieva,A Bloch electron in an external field, Algebra i Analiz.1, No. 2 (1989), 1–29; translated in Leningrad Math. J.1 (1990), 287–320.

    MATH  MathSciNet  Google Scholar 

  11. V. S. Buslaev and A. Grigis,Imaginary parts of Stark-Wannier resonances, J. Math. Phys.39, No. 5 (1998), 2520–2550.

    Article  MATH  MathSciNet  Google Scholar 

  12. T. M. Cherry,Uniform asymptotic formulae for functions with transition points, Trans. Amer. Math. Soc.68 (1950), 224–257.

    Article  MATH  MathSciNet  Google Scholar 

  13. Y. Colin de Verdière, M. Lombardi and J. Pollet,The microlocal Landau-Zener formula, Ann. Inst. H. Poincaré Phys. Théor.71 (1999), 95–127.

    MATH  Google Scholar 

  14. Yu. Daleckii and M. G. Krein,Stability of solutions of differential equations in Banach spaces, Amer. Math. Soc. Transl. Math. Monographs43 (1974).

  15. M. V. Fedoryuk,Asymptotic Analysis. Linear Ordinary Differential Equations, Springer-Verlag, Berlin, 1993.

    MATH  Google Scholar 

  16. A. Grigis,Points tournants et résonances de Stark-Wannier, Séminaire EDP, Ecole Polytechnique, 1997–98, no. 11.

  17. J.-C. Guillot, J. Ralston and E. Trubowitz,Semiclassical methods in solide State Physics, Comm. Math. Phys.116 (1988), 401–405.

    Article  MATH  MathSciNet  Google Scholar 

  18. G. A. Hagedron,Proof of the Landau-Zener formula in an adiabatic limit with small eigenvalue gaps, Comm. Math. Phys.136 (1991), 433–449.

    Article  MathSciNet  Google Scholar 

  19. G. A. Hagedorn,Molecular propagation through electron energy level crossings, Mem. Amer. Math. Soc.111 (1994), no. 536. 0

    Google Scholar 

  20. B. Helffer,Formes normales pour des opérateurs pseudodifferentiels semiclassiques en dimension 1, Séminaire EDP, Ecole Polytechnique, 1988–89, no. 2.

  21. B. Helffer and J. Sjöstrand,Semiclassical analysis for Harper's equation III. Cantor structure of the spectrum, Mém. Soc. Math. France, No. 39, Suppl. au Bull. Soc. Math. France117, No. 4 (1989), 1–124.

    Google Scholar 

  22. A. Joye,Proof of the Landau-Zener formula, Asymptotic Anal.9 (1994), 209–258.

    MATH  MathSciNet  Google Scholar 

  23. T. Kato,Perturbation Theory, Springer-Verlag, New York, 1966.

    MATH  Google Scholar 

  24. L. Landau,Collected Papers of L. Landau, Pergamon Press, Oxford, 1965.

    Google Scholar 

  25. V. A. Marchenko and I. V. Ostrovskii,Characteristics of the spectrum of the Hill operator, Mat. Sb.97, No. 4 (1975), 540–606.

    MathSciNet  Google Scholar 

  26. A. Martinez and G. Nenciu,On adiabatic reduction theory, Oper. Theory Adv. Appl.78 (1995), 243–252.

    MathSciNet  Google Scholar 

  27. V. P. Maslov and M. V. Fedoriuk,Semiclassical Approximation in Quantum Mechanics, D. Reidel Publishing Co., Dordrecht-Boston, Mass., 1981.

    Google Scholar 

  28. E. C. Titchmarsh,Eigenfunction Expansions Associated with Second-order Differential Equations, Clarendon Press, Oxford, 1962.

    MATH  Google Scholar 

  29. W. Wasow,Linear Turning Point Theory, Springer-Verlag, Berlin-Heidelberg-New York, 1985.

    MATH  Google Scholar 

  30. I. N. Yakushina,Uniform asymptotic expansions for the solutions of second-order differential equations with two turning points and a spectral parameter (in Russian), Differentsial'nye Uravneniya23 (1987), no. 6, 1014–1020.

    MathSciNet  Google Scholar 

  31. C. Zener,Non-adiabatic crossing of energy levels, Proc. Roy. Soc. London137 (1932), 696–702.

    Article  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematical Physics, NIIF, Saint-Petersburg State University, 198904, Saint-Petersberg, Russia

    Vladimir Buslaev

  2. Département de Mathématiques CNRS-UMR 7539, Université Paris 13, 99 Avenue Jean-Baptiste Clément, 93430, Villetaneuse, France

    Vladimir Buslaev & Alain Grigis

Authors
  1. Vladimir Buslaev
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  2. Alain Grigis
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Buslaev, V., Grigis, A. Turning points for adiabatically perturbed periodic equations. J. Anal. Math. 84, 67–143 (2001). https://doi.org/10.1007/BF02788107

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  • DOI: https://doi.org/10.1007/BF02788107

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