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Quantum Tunnelling for Bloch Electrons in Small Electric Fields

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Recent Developments in Quantum Mechanics

Part of the book series: Mathematical Physics Studies ((MPST,volume 12))

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Abstract

We consider the quantum Hamiltonian describing a one dimensional particle in a periodic potential plus constant electric field and show that it exhibits an infinite ladder of resonances in the semi-classical regime. The lifetime of these resonances is related to tunnelling phenomena. This provides in this particular case a positive answer to the question of the existence of Stark Ladders.

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References

  1. Aguilar J., Combes J.M.: A class of analytic perturbations for one body Schrödinger Operators. Corn. Math. Phys. 22, 269–279 (1971)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  2. Agmon S. Lectures on exponential decay of second order elliptic Equation. Princeton Math. Notes. 29 (1982)

    Google Scholar 

  3. Agler J., Froese R.: Existence of Stark ladder resonances. Corn. Math. Phys. 100, 161–171 (1985)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  4. Avron J.: The lifetime of Wannier Ladder States. Ann. Phys. 143, 33–53 (1982)

    Article  ADS  Google Scholar 

  5. Bentosela F.: This volume.

    Google Scholar 

  6. Bleuse J., Bastard G., Voisin P. Phys. Rev. Letters 60, 220 (1988)

    Article  ADS  Google Scholar 

  7. Briet Ph., Combes J.M., Duclos P.: On the location of resonances for Schrödinger Operators.

    Google Scholar 

  8. Resonance free domains. J. Math. Anal. Appl. 126, 90–99 (1987)

    Article  MathSciNet  Google Scholar 

  9. Barrier top resonances. Com. in P.D.E. 12, 201–222 (1987)

    Article  Google Scholar 

  10. Shape resonances. To appear

    Google Scholar 

  11. Briet Ph., Combes J.M., Duclos P.: Spectral Stability under tunnelling. Com. Math. Phys. 126, 133–156 (1989)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  12. Buslaev V.S.: Dimitrieva L.A. This volume

    Google Scholar 

  13. Combes J.M., Duclos P., Klein M., Seiler R.: The shape resonance. Com. Math. Phys. 110, 215–236 (1987)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  14. Combes J.M., Hislop P. Stark ladder resonances for small electric fields. Preprint, Univ. Kentucky (1989)

    Google Scholar 

  15. De Bièvre S., Hislop P.:Spectral resonances for the Laplace-Beltrami operator. Ann. Inst. H. Poincaré 48, 105–145 (1988)

    MATH  Google Scholar 

  16. Herbst I.W., Howland J.S.: The Stark ladder and other one dimensional external field problems. Com. Math. Phys. 80, 23–40 (1981)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  17. Hellfer B. Sjostrand J.: Resonances en limite semi-clanique. Supplem Bulletin SMF 114 (1986)

    Google Scholar 

  18. Hellfer B., Sjostrand J. Multiple wells in the semi-classical limit I. Corn. in P.D.E. 9, 337–408 (1984)

    Article  Google Scholar 

  19. Hislop P., Sigal I.M.: Semi-classical theory of shape resonances in quantum mechanics, Memoirs. Am. Math. Soc. 399 (1989)

    Google Scholar 

  20. Hunziker W.: Distortion analyticity and molecular resonance curves. Ann. Inst. M. Poincaré, 45, 339–358 (1986)

    MathSciNet  MATH  Google Scholar 

  21. Jensen A. Asymptotic completeness for a new class of Stark effect Hamiltonians. 107, 21–28 (1986)

    MATH  Google Scholar 

  22. Nakamura S.: A note on the absence of resonances for Schrödinger Operators. Lett. Math. Phys. 16, 217–223 (1988)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  23. Nenciu G. This volume

    Google Scholar 

  24. Jona-Lasinio, Martinelli F., Scoppola E. New approach to the semi-classical limit of quantum mechanics I. Corn. Math. Phys. 80, 223 (1981)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  25. Klein M.: On the absence of resonances for Schrödinger Operators in the semiclassical limit. Corn. Math. Phys. 106, 485–494 (1985)

    Article  ADS  Google Scholar 

  26. Reed M., Simon B.: Methods of Modern Mathematical Physics IV: Analysis of Operators (Academic, N.Y, 1978 )

    MATH  Google Scholar 

  27. Sigal I.M.: Sharp exponential bounds on resonance states and width of resonances. Adv. Appl. Math. 9, 127–166 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  28. Sigal I.M.: Geometric Theory of Stark resonances in multielectron systems. Corn. Math. Phys. 119, 287–314 (1988)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  29. Sigal I.M.: Geometric methods in the quantum many body problems. Corn. Math. Phys. 85, 309–324 (1982)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  30. Simon B.: Semi-classical analysis of low lying eigenvalues I. Ann. Inst. H. Poincaré 38, 295–307 (1983)

    MATH  Google Scholar 

  31. Zak J.: Stark ladders in solids ? Phys. Rev. Lett. 20, 1477–1481 (1968)

    Article  ADS  Google Scholar 

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Author information

Author notes

  1. P. Hislop

    Present address: Mathematics Department, University of Kentucky, Lexington, KY, 40506, USA

Authors and Affiliations

  1. PHYMAT, Departement de Mathématiques, Université de Toulon et du Var, La Garde, France

    J. M. Combes

Authors
  1. J. M. Combes
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  2. P. Hislop
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Editor information

Editors and Affiliations

  1. University Paris VII, Paris, France

    Anne Boutet de Monvel

  2. Bucharest University, Bucharest, Romania

    Petre Dita

  3. Theoretical Physics Department, Institute of Atomic Physics, Bucharest, Romania

    Gheorghe Nenciu

  4. Bucharest University, Bucharest, Romania

    Radu Purice

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© 1991 Springer Science+Business Media Dordrecht

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Combes, J.M., Hislop, P. (1991). Quantum Tunnelling for Bloch Electrons in Small Electric Fields. In: Boutet de Monvel, A., Dita, P., Nenciu, G., Purice, R. (eds) Recent Developments in Quantum Mechanics. Mathematical Physics Studies, vol 12. Springer, Dordrecht. https://doi.org/10.1007/978-94-011-3282-4_6

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  • DOI: https://doi.org/10.1007/978-94-011-3282-4_6

  • Publisher Name: Springer, Dordrecht

  • Print ISBN: 978-94-010-5449-2

  • Online ISBN: 978-94-011-3282-4

  • eBook Packages: Springer Book Archive

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