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Schrödinger operators with electric or magnetic fields

  • W. Hunziker
Schrödinger Operators Main Lectures
Part of the Lecture Notes in Physics book series (LNP, volume 116)

Abstract

The Zeeman- and Stark effect are first examples of quantum mechanical perturbation theory. Nevertheless it has taken half a century to develop an adequate mathematical description. Here we summarize the results of a systematic effort in recent years, notably by Avron, Herbst and Simon [2,35].

Keywords

Essential Spectrum Stark Effect Perturbation Series Ground State Binding Energy Borel Summability 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References for Magnetic Fields

  1. [1]
    Avron J.E.; Adams B.G.; Čížek J.; Clay M.; Glasser M.L.; Otto P.; Paldus J.; Vrscay E.: “The Bender-Wu formula, SO(4,2) dynamical group and the Zeeman effect in hydrogen”; subm. to Phys.Rev.Lett.Google Scholar
  2. [2]
    Avron J.E.; Herbst I.W.; Simon B.: Phys. Rev. Lett. 62A, 214–216 (1977)Google Scholar
  3. [3]
    Avron J.E.; Herbst I.W.; Simon B.: Phys. Rev. Lett. 39, 1068–1070 (1977)Google Scholar
  4. [4]
    Avron J.E.; Herbst I.W.; Simon B.: Duke Math. J. 45, 847–883 (1978)Google Scholar
  5. [5]
    Avron J.W.; Herbst I.W.; Simon B.: Ann. Phys. 114, 431–451 (1978)Google Scholar
  6. [6]
    Avron J.E.; Herbst I.W.; Simon B.: “The strongly bound states of hydrogen in intense magnetic fields”; Phys.Rev. A, to appearGoogle Scholar
  7. [7]
    Avron J.E.; Herbst I.W.; Simon B.: “Schrödinger Operators with Magnetic Fields. III. Atoms in magnetic fields”; to be submitted to Commun.math.Phys.Google Scholar
  8. [8]
    Avron J.E.; Seiler R.: Phys. Rev. Lett. 42, 931–933 (1979)Google Scholar
  9. [9]
    Avron J.E.; Simon B.: “A counterexample to the paramagnetic conjecture”; submitted to Phys. Lett.Google Scholar
  10. [10]
    Combes J.M.; Schrader R.; Seiler R.: Ann. Phys. 111, 1–18 (1978)Google Scholar
  11. [11]
    Hess H.; Schrader R.; Uhlenbrock D.A.: Duke Math. J. 44, 893–904 (1977)Google Scholar
  12. [12]
    Hogreve H.; Schrader R.; Seiler R.: Nucl. Phys. B 142, 525–534 (1978)Google Scholar
  13. [13]
    Kato T.: Israel J. Math. 13, 135–148 (1972)Google Scholar
  14. [14]
    Kato T.: Integral Eqn. and Operator Th. 1, 103–113 (1978)Google Scholar
  15. [15]
    Schechter, M.: J. Functional Analysis 20, 93–104 (1975)Google Scholar
  16. [16]
    Schrader R.; Seiler R.: Commun. math. Phys. 61, 169–175 (1978)Google Scholar
  17. [17]
    Simon B.: Phys. Rev. Lett. 36, 1083–1084 (1976)Google Scholar
  18. [18]
    Simon B.: Indiana Univ. Math. J. 26, 1067–1073 (1977)Google Scholar
  19. [19]
    Simon B.: J. Functional Analysis 32, 97–101 (1979)Google Scholar
  20. [20]
    Simon B.: J. Operator Theory 1, 37–47 (1979)Google Scholar

References for Electric Fields

  1. [21]
    Avron J.E.; Herbst I.W.: Commun. math. Phys. 52, 239–254 (1977)Google Scholar
  2. [22]
    Benassi L.; Grecchi V.: “Resonances in Stark Effect and Strongly Asymptotic Approximants”; to appear in J. Phys. BGoogle Scholar
  3. [23]
    Benassi L.; Grecchi V.; Harrell E.; Simon B.: Phys. Rev. Lett. 42, 704–707 (1979)Google Scholar
  4. [24]
    Brändas E.; Froelich P.: Phys. Rev. A 16, 2207–2210 (1977)Google Scholar
  5. [25]
    Conley C.C.; Rejto P.A.: “Spectral concentration II: General theory” in: Perturbation Theory and its Applications in Quantum Mechanics (C.H. Wilcox ed.) Wiley, New York 1966Google Scholar
  6. [26]
    Damburg R.J.; Kolosov V.V.: J. Phys. B 9, 3149–3157 (1976), and J. Phys. B 11, 1921 (1978)Google Scholar
  7. [27]
    Graffi S.; Grecchi V.: Lett. Math. Phys. 2, 335–341 (1978)Google Scholar
  8. [28]
    Graffi S.; Grecchi V.: Commun.math.Phys. 62, 83–96 (1978)Google Scholar
  9. [29]
    Graffi S.; Grecchi V.: J. Phys. B 12, L 265–267 (1979)Google Scholar
  10. [30]
    Graffi S.; Grecchi V.; Simon B.: “Complete Separability of the Stark Problem in Hydrogen”; to appear in J. Phys. BGoogle Scholar
  11. [31]
    Harrell E.; Simon B.: “The Mathematical Theory of Resonances Whose Widths are Exponentially Small”; manuscript 1979Google Scholar
  12. [32]
    Hehenberger M.; Mc Intosh H.V.; Brändas E.: Phys. Rev. A 10, 1494–1506 (1974)Google Scholar
  13. [33]
    Herbst I.W.: Math. Zeitschrift 155, 55–71 (1977)Google Scholar
  14. [34]
    Herbst I.W.: Commun.math. Phys. 64, 279–298 (1979)Google Scholar
  15. [35]
    Herbst I.W.; Simon B.: Phys. Rev. Lett. 41, 67–69 (1978)Google Scholar
  16. [36]
    Herbst I.W.; Simon B.: “Dilation Analyticity in Constant Electric Field II: N-Body Problem, Borel Summability”; manuscript 1979, to be submitted to Commun.math.Phys.Google Scholar
  17. [37]
    Lanczos C.: Z. Physik 68, 204–232 (1931)Google Scholar
  18. [38]
    Oppenheimer J.R.: Phys. Rev. 31, 66–81 (1928)Google Scholar
  19. [39]
    Reinhardt W.P.: Int. J. Quant. Chem. Symp. 10, 359–367 (1976)Google Scholar
  20. [40]
    Rejto P.A.: Helv. Phys. Acta 43, 652–667 (1970)Google Scholar
  21. [41]
    Riddel R.C.: Pacific J. Math. 23, 377–401 (1967)Google Scholar
  22. [42]
    Silverstone H.J.: Phys. Rev. A 18, 1853–1864 (1978)Google Scholar
  23. [43]
    Silverstone H.J.: “Asymptotic relations between the energy shift and ionization rate in the Stark effect in hydrogen”; preprint 1979Google Scholar
  24. [44]
    Silverstone H.J.; Adams B.G.; Čížek J.; Otto P.: “Asymptotic formula for the perturbed energy coefficients and calculations of the ionization rate by means of high order perturbation theory for hydrogen in the Stark effect”; preprint 1979Google Scholar
  25. [45]
    Silverstone H.J.; Koch P.M.: “Calculation of Stark effect energy shifts by Padé approximants to Rayleigh-Schrödinger perturbation theory”; preprint 1979Google Scholar
  26. [46]
    Titchmarsh E.C.: “Eigenfunction Expansions Associated with Second Order Differential Equations”, Part II. Oxford University Press 1958Google Scholar
  27. [47]
    Veselič K.; Weidmann J.: Math. Zeitschrift 156, 93 (1977)Google Scholar
  28. [48]
    Yamabe T.; Tachibana A.; Silverstone H.J.: Phys. Rev. A 16, 877–890 (1977)Google Scholar
  29. [49]
    Yajima K.: “Spectral and scattering theory for Schrödinger operators with Stark-effect”; to appear in J. Fac. Sci. Univ. TokyoGoogle Scholar
  30. [50]
    Yajima K.: “Spectral and scattering theory for Schrödinger operators with Stark-effect, II”; preprint 1978Google Scholar

Other References

  1. [51]
    Aguilar J.; Combes J.M.: Commun.math.Phys. 22, 269–279 (1971)Google Scholar
  2. [52]
    Dodds P.G., Fremlin D.H.: “Compact operators in Banach lattices”; submitted to J. Functional AnalysisGoogle Scholar
  3. [53]
    Enss V.: Commun. math. Phys. 52, 233–238 (1977)Google Scholar
  4. [54]
    Enss V.: Commun. math. Phys. 61, 285–291 (1978)Google Scholar
  5. [55]
    Lieb E.H.: Rev. Mod. Phys. 48, 553–569 (1976)Google Scholar
  6. [56]
    Pitt L.: “A compactness condition for linear operators in function spaces”; submitted to J. Operator TheoryGoogle Scholar
  7. [57]
    Reed M., Simon B.: “Methods of Modern Mathematical Physics”. Vol. I–IV; Academic Press, New YorkGoogle Scholar
  8. [58]
    Ruelle D.: Nuovo Cimento 61 A, 655–662 (1969)Google Scholar
  9. [59]
    Schechter M.: “Spectra of Partial Differential Operators”; North Holland, Amsterdam 1971Google Scholar
  10. [60]
    Simon B.: Duke Math. J. 46, 119–168 (1979)Google Scholar
  11. [61]
    Simon B.: “Functional Integration and Quantum Physics”; Academic Press, New York 1979Google Scholar
  12. [62]
    Thirring W.: “Lehrbuch der Mathematischen Physik. 3. Quanten-mechanik von Atomen und Molekülen”; Springer Verlag, Wien, New York, 1979Google Scholar

Copyright information

© Springer-Verlag 1980

Authors and Affiliations

  • W. Hunziker
    • 1
  1. 1.Institut für Theoretische Physik ETH HönggerbergZürichSwitzerland

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