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)


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].


Essential Spectrum Stark Effect Perturbation Series Ground State Binding Energy Borel Summability 
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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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