Instability of Pre-Existing Resonances Under a Small Constant Electric Field

Abstract

Two simple model operators are considered which have pre-existing resonances. A potential corresponding to a small electric field, f, is then introduced and the resonances of the resulting operator are considered as f → 0. It is shown that these resonances are not continuous in this limit. It is conjectured that a similar behavior will appear in more complicated models of atoms and molecules. Numerical results are presented.

References

  1. 1

    Avron J.E., Herbst I.W.: Spectral and scattering theory for Schrödinger operators related to the Stark effect. Commun. Math. Phys. 52, 239–254 (1977)

    MATH  MathSciNet  Article  ADS  Google Scholar 

  2. 2

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

    MATH  MathSciNet  Article  ADS  Google Scholar 

  3. 3

    Arnold V.I.: Mathematical Methods of Classical Mechanics. Graduate Texts in Mathematics 60. Springer, New York (1978)

    Book  Google Scholar 

  4. 4

    Bak J., Newman D.J.: Complex Analysis. 3rd edition, Undergraduate Texts in Mathematics. Springer, New York (2010)

    Google Scholar 

  5. 5

    Cerjan C., Reinhardt W.P., Avron J.E.: Spectra of atomic Hamiltonians in DC fields: Use of the numerical range to investigate the effect of a dilatation transformation. J. Phys. B: At. Mol. Phys. 11, L201–L205 (1978)

    Article  ADS  Google Scholar 

  6. 6

    Cycon H.L., Froese R.G., Kirsch W., Simon B.: Schrödinger operators with application to quantum mechanics and global geometry, Texts and Monographs in Physics Springer Study Edition. Springer, Berlin (1987)

    Google Scholar 

  7. 7

    Erdélyi A.: Asymptotic Expansions. Dover Publications, Inc., New York (1956)

    MATH  Google Scholar 

  8. 8

    Graffi S., Grecchi V.: Resonances in Stark effect and perturbation theory. Commun. Math. Phys. 62(1), 83–96 (1978)

    MathSciNet  Article  ADS  Google Scholar 

  9. 9

    Graffi S., Grecchi V.: Resonances in Stark effect of atomic systems. Commun. Math. Phys. 79(1), 91–109 (1981)

    MATH  MathSciNet  Article  ADS  Google Scholar 

  10. 10

    Harrell E., Simon B.: The mathematical theory of resonances whose widths are exponentially small. Duke Math. J. 47(4), 845–902 (1980)

    MATH  MathSciNet  Article  Google Scholar 

  11. 11

    Herbst I.: Dilation analyticity in constant electric field, I. The two body problem. Commun. Math. Phys. 64, 279–298 (1979)

    MATH  MathSciNet  Article  ADS  Google Scholar 

  12. 12

    Herbst I., Møller J.S., Skibsted E.: Spectral analysis of N-body Stark Hamiltonians. Commun. Math. Phys. 174, 261–294 (1995)

    MATH  Article  ADS  Google Scholar 

  13. 13

    Herbst I., Simon B.: Dilation analyticity in constant electric field, II. N-body problem, Borel summability. Commun. Math. Phys. 80, 181–216 (1981)

    MATH  MathSciNet  Article  ADS  Google Scholar 

  14. 14

    Herbst I., Simon B.: Stark effect revisited. Phys. Rev. Lett. 41, 67–69 (1978)

    MathSciNet  Article  ADS  Google Scholar 

  15. 15

    Hörmander L.: The Analysis of Linear Partial Differential Operators I. Springer Study Edition. Springer, Berlin (1990)

    Book  Google Scholar 

  16. 16

    Howland J.S.: Stationary scattering theory for time-dependent Hamiltonians. Math. Ann. 207, 315–335 (1974)

    MATH  MathSciNet  Article  Google Scholar 

  17. 17

    Nelson E.: Analytic vectors. Ann. Math. 70(3), 572–615 (1959)

    MATH  Article  Google Scholar 

  18. 18

    Olver F.W.J.: Asymptotics and special functions. Academic Press, New York (1974)

    Google Scholar 

  19. 19

    Reinhardt, W. P.: Complex scaling in atomic physics: a staging ground for experimental mathematics and for extracting physics from otherwise impossible computations. Spectral theory and mathematical physics: a Festschrift in honor of Barry Simon’s 60th birthday. Proc. Sympos. Pure Math., 76, Part 1, Am. Math. Soc., Providence, RI, pp. 357–381, (2007)

  20. 20

    Reed M., Simon B.: Methods of Modern Mathematical Physics II: Fourier Analysis, Self-Adjointness. Academic Press, New York (1975)

    MATH  Google Scholar 

  21. 21

    Rudin W.: Real and Complex Analysis. McGraw-Hill International Editions, Singapore (1987)

    MATH  Google Scholar 

  22. 22

    Simon B.: Resonances and complex scaling: a rigorous overview. Int. J. Quant. Chem. 14, 529–542 (1978)

    Article  ADS  Google Scholar 

  23. 23

    Weidmann J.: Lineare Operatoren in Hilberträumen. B. G. Teubner, Stuttgart (1976)

    MATH  Google Scholar 

  24. 24

    Yajima K.: Scattering theory for Schrödinger equations with potentials periodic in time. J. Math. Soc. Japan 29(4), 729–743 (1977)

    MATH  MathSciNet  Article  Google Scholar 

  25. 25

    Yajima K.: Resonances for the AC-Stark effect. Commun. Math. Phys. 87, 331–352 (1982)

    MATH  MathSciNet  Article  ADS  Google Scholar 

  26. 26

    Yamabe T., Tachibana A., Silverstone H.J.: Theory of the ionization of the hydrogen atom by an external electrostatic field. Phys. Rev. A 16, 877–890 (1977)

    Article  ADS  Google Scholar 

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Correspondence to Juliane Rama.

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J.R. was supported by the Deutsche Forschungsgemeinschaft (DFG), research grants RA 2020/1-1 and RA 2020/1-2 (and also grant RA 2020/3-1).

Communicated by Jan Dereziński.

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Herbst, I., Rama, J. Instability of Pre-Existing Resonances Under a Small Constant Electric Field. Ann. Henri Poincaré 16, 2783–2835 (2015). https://doi.org/10.1007/s00023-014-0389-2

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Keywords

  • Entire Function
  • Steep Descent
  • Half Complex Plane
  • Embed Eigenvalue
  • Entire Extension