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Analytic expansion of Lyapunov exponents associated to the Schrödinger operator

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Part of the Lecture Notes in Mathematics book series (LNM,volume 1109)

Abstract

Let λ(σ) be the Lyapunov exponent associated to Hy = Ey, H = −d2/dx2 + ση the one-dimensional Schrödinger operator with random potential ση. For positive eigenvalues E (or non-overdamped oscillator \(\ddot z\)+2βż+(c-ση)z=0, z=y exp(βt), E = c-β2>0) we expand λ(σ) analytically in terms of σ and compute the coefficients using regular perturbation theory.

Keywords

  • Weak Solution
  • Lyapunov Exponent
  • Regular Semigroup
  • Continuous Semigroup
  • Analytic Semigroup

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This paper is part of a research project supported by Stiftung Volkswagenwerk

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References

  1. Arnold, L.; Crauel, H. and Wihstutz, V. Stabilization of linear systems by noise, SIAM J. Control and Optimization 21 (1983), 451–461

    CrossRef  MathSciNet  MATH  Google Scholar 

  2. Bensoussan, A. and Lions, J.L. Applications of variational inequalities in stochastic control, North-Holland, 1982

    Google Scholar 

  3. Curtain, R.F. and Pritchard, A.J. Infinite dimensional linear system theory, Lecture Notes in Control and Information Sciences, vol. 8, Springer, 1978

    Google Scholar 

  4. Davies, E.B. One parameter semigroups, Academic Press, 1980

    Google Scholar 

  5. Kato, T. Pertubation theory for linear operators, Springer (Grundlehren vol. 132), 2nd ed., 1980

    Google Scholar 

  6. Kliemann, W. and Arnold, L. Lyapunov exponents of linear stochastic systems. Report No. 93, Forschungsschwerpunkt Dynamische Systeme, Universität Bremen, 1983

    Google Scholar 

  7. Lions, J.L. and Magenes, E. Non-homogenous boundary value problems and applications (vol. I), Springer (Grundlehren, vol. 181), 1972

    Google Scholar 

  8. Rellich, F. Störungstheorie der Spektralzerlegung I Math. Ann. 113 (1937), 600–619

    CrossRef  MathSciNet  MATH  Google Scholar 

  9. Tanabe, H. Equations of evolution, Pitman 1979

    Google Scholar 

  10. Wihstutz, V. Quantitative results on Lyapunov exponents, Report No. 99, Forschungsschwerpunkt Dynamische Systeme, Universität Bremen, 1983

    Google Scholar 

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© 1985 Springer-Verlag

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Wihstutz, V. (1985). Analytic expansion of Lyapunov exponents associated to the Schrödinger operator. In: Albeverio, S., Combe, P., Sirugue-Collin, M. (eds) Stochastic Aspects of Classical and Quantum Systems. Lecture Notes in Mathematics, vol 1109. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0101544

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  • DOI: https://doi.org/10.1007/BFb0101544

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-13914-0

  • Online ISBN: 978-3-540-39138-8

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