Advertisement

Quantum Tunnelling for Bloch Electrons in Small Electric Fields

  • J. M. Combes
  • P. Hislop
Chapter
  • 150 Downloads
Part of the Mathematical Physics Studies book series (MPST, volume 12)

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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [AC]
    Aguilar J., Combes J.M.: A class of analytic perturbations for one body Schrödinger Operators. Corn. Math. Phys. 22, 269–279 (1971)MathSciNetADSzbMATHCrossRefGoogle Scholar
  2. [Ag]
    Agmon S. Lectures on exponential decay of second order elliptic Equation. Princeton Math. Notes. 29 (1982)Google Scholar
  3. [AF]
    Agler J., Froese R.: Existence of Stark ladder resonances. Corn. Math. Phys. 100, 161–171 (1985)MathSciNetADSzbMATHCrossRefGoogle Scholar
  4. [Av]
    Avron J.: The lifetime of Wannier Ladder States. Ann. Phys. 143, 33–53 (1982)ADSCrossRefGoogle Scholar
  5. [B]
    Bentosela F.: This volume.Google Scholar
  6. [BVD]
    Bleuse J., Bastard G., Voisin P. Phys. Rev. Letters 60, 220 (1988)ADSCrossRefGoogle Scholar
  7. [BCD1]
    Briet Ph., Combes J.M., Duclos P.: On the location of resonances for Schrödinger Operators.Google Scholar
  8. I.
    Resonance free domains. J. Math. Anal. Appl. 126, 90–99 (1987)MathSciNetCrossRefGoogle Scholar
  9. II.
    Barrier top resonances. Com. in P.D.E. 12, 201–222 (1987)CrossRefGoogle Scholar
  10. III.
    Shape resonances. To appearGoogle Scholar
  11. [BCD2]
    Briet Ph., Combes J.M., Duclos P.: Spectral Stability under tunnelling. Com. Math. Phys. 126, 133–156 (1989)MathSciNetADSzbMATHCrossRefGoogle Scholar
  12. [BD]
    Buslaev V.S.: Dimitrieva L.A. This volumeGoogle Scholar
  13. [CDKS]
    Combes J.M., Duclos P., Klein M., Seiler R.: The shape resonance. Com. Math. Phys. 110, 215–236 (1987)MathSciNetADSzbMATHCrossRefGoogle Scholar
  14. [CH]
    Combes J.M., Hislop P. Stark ladder resonances for small electric fields. Preprint, Univ. Kentucky (1989)Google Scholar
  15. [DeBHi]
    De Bièvre S., Hislop P.:Spectral resonances for the Laplace-Beltrami operator. Ann. Inst. H. Poincaré 48, 105–145 (1988)zbMATHGoogle Scholar
  16. [HH]
    Herbst I.W., Howland J.S.: The Stark ladder and other one dimensional external field problems. Com. Math. Phys. 80, 23–40 (1981)MathSciNetADSzbMATHCrossRefGoogle Scholar
  17. [HS1]
    Hellfer B. Sjostrand J.: Resonances en limite semi-clanique. Supplem Bulletin SMF 114 (1986)Google Scholar
  18. [HS2]
    Hellfer B., Sjostrand J. Multiple wells in the semi-classical limit I. Corn. in P.D.E. 9, 337–408 (1984)CrossRefGoogle Scholar
  19. [HiSi]
    Hislop P., Sigal I.M.: Semi-classical theory of shape resonances in quantum mechanics, Memoirs. Am. Math. Soc. 399 (1989)Google Scholar
  20. [Hu]
    Hunziker W.: Distortion analyticity and molecular resonance curves. Ann. Inst. M. Poincaré, 45, 339–358 (1986)MathSciNetzbMATHGoogle Scholar
  21. [J]
    Jensen A. Asymptotic completeness for a new class of Stark effect Hamiltonians. 107, 21–28 (1986)zbMATHGoogle Scholar
  22. [NA]
    Nakamura S.: A note on the absence of resonances for Schrödinger Operators. Lett. Math. Phys. 16, 217–223 (1988)MathSciNetADSzbMATHCrossRefGoogle Scholar
  23. [N]
    Nenciu G. This volumeGoogle Scholar
  24. [JMS]
    Jona-Lasinio, Martinelli F., Scoppola E. New approach to the semi-classical limit of quantum mechanics I. Corn. Math. Phys. 80, 223 (1981)MathSciNetADSzbMATHCrossRefGoogle Scholar
  25. [Kl]
    Klein M.: On the absence of resonances for Schrödinger Operators in the semiclassical limit. Corn. Math. Phys. 106, 485–494 (1985)ADSCrossRefGoogle Scholar
  26. [RS]
    Reed M., Simon B.: Methods of Modern Mathematical Physics IV: Analysis of Operators (Academic, N.Y, 1978 )zbMATHGoogle Scholar
  27. [Si1]
    Sigal I.M.: Sharp exponential bounds on resonance states and width of resonances. Adv. Appl. Math. 9, 127–166 (1988)MathSciNetzbMATHCrossRefGoogle Scholar
  28. [Si2]
    Sigal I.M.: Geometric Theory of Stark resonances in multielectron systems. Corn. Math. Phys. 119, 287–314 (1988)MathSciNetADSzbMATHCrossRefGoogle Scholar
  29. [Si3]
    Sigal I.M.: Geometric methods in the quantum many body problems. Corn. Math. Phys. 85, 309–324 (1982)MathSciNetADSzbMATHCrossRefGoogle Scholar
  30. [Sim]
    Simon B.: Semi-classical analysis of low lying eigenvalues I. Ann. Inst. H. Poincaré 38, 295–307 (1983)zbMATHGoogle Scholar
  31. [Z]
    Zak J.: Stark ladders in solids ? Phys. Rev. Lett. 20, 1477–1481 (1968)ADSCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 1991

Authors and Affiliations

  • J. M. Combes
    • 1
  • P. Hislop
    • 2
  1. 1.PHYMAT, Departement de MathématiquesUniversité de Toulon et du VarLa GardeFrance
  2. 2.Mathematics DepartmentUniversity of KentuckyLexingtonUSA

Personalised recommendations