Remote Access Bulletin of the American Mathematical Society

Bulletin of the American Mathematical Society

ISSN 1088-9485(online) ISSN 0273-0979(print)

 

 

Fifty years of eigenvalue perturbation theory


Author: Barry Simon
Journal: Bull. Amer. Math. Soc. 24 (1991), 303-319
MSC (1985): Primary 81C12, 81G45, 46C99, 46N05
MathSciNet review: 1066108
Full-text PDF

References | Similar Articles | Additional Information

References [Enhancements On Off] (What's this?)

  • 1. J. Aguilar and J. M. Combes, A class of analytic perturbations for one-body Schrödinger Hamiltonians, Comm. Math. Phys. 22 (1971), 269–279. MR 0345551
  • 2. Joseph E. Avron, Bender-Wu formulas for the Zeeman effect in hydrogen, Ann. Physics 131 (1981), no. 1, 73–94. MR 608087, 10.1016/0003-4916(81)90184-6
  • 3. J. Avron, B. Adams, J. Cizek, M. Clay, M. Glasser, P. Otto, J. Paldus, and E. Vrscay, The Bender-Wu formula, SO(4, 2) dynamical group and the Zeeman effect in hydrogen, Phys. Rev. Lett. 43 (1979), 691-693.
  • 4. J. E. Avron and I. W. Herbst, Spectral and scattering theory of Schrödinger operators related to the Stark effect, Comm. Math. Phys. 52 (1977), no. 3, 239–254. MR 0468862
  • 5. J. E. Avron, I. W. Herbst, and B. Simon, Schrödinger operators with magnetic fields. III. Atoms in homogeneous magnetic field, Comm. Math. Phys. 79 (1981), no. 4, 529–572. MR 623966
  • 5. J. Avron, I. Herbst, and B. Simon, The Zeeman effect revisited, Phys. Lett. 62 A (1977), 214-216.
  • 6. J. E. Avron, L. Sadun, J. Segert, and B. Simon, Chern numbers, quaternions, and Berry’s phases in Fermi systems, Comm. Math. Phys. 124 (1989), no. 4, 595–627. MR 1014116
  • 6. J. E. Avron, L. Sadun, J. Segert, and B. Simon, Topological invariants in Fermi systems with time-reversal invariance, Phys. Rev. Lett. 61 (1988), no. 12, 1329–1332. MR 957875, 10.1103/PhysRevLett.61.1329
  • 7. J. E. Avron, R. Seiler, and L. G. Yaffe, Adiabatic theorems and applications to the quantum Hall effect, Comm. Math. Phys. 110 (1987), no. 1, 33–49. MR 885569
  • 8. E. Balslev and J. M. Combes, Spectral properties of many-body Schrödinger operators with dilatation-analytic interactions, Comm. Math. Phys. 22 (1971), 280–294. MR 0345552
  • 9. L. Benassi, V. Grecchi, E. Harrell, and B. Simon, The Bender-Wu formula and the Stark effect in hydrogen, Phys. Rev. Lett. 42 (1979), 704-707.
  • 10. C. Bender and T. Wu, Analytic structure of energy levels in a field-theory model, Phys. Rev. Lett. 21 (1968), 406.
  • 10. Carl M. Bender and Tai Tsun Wu, Anharmonic oscillator, Phys. Rev. (2) 184 (1969), 1231–1260. MR 0260323
  • 11. C. Bender and T. Wu, Anharmonic oscillator, II. A study of perturbation theory in large order, Phys. Rev. D7 (1973), 1620.
  • 12. M. V. Berry, Quantal phase factors accompanying adiabatic changes, Proc. Roy. Soc. London Ser. A 392 (1984), no. 1802, 45–57. MR 738926
  • 13. R. Damburg, R. Propin, S. Graffi, V. Grecchi, E. Harrell, J. Cizek, J. Palus, and H. Silverstone, 1/R expansion for H2+ : analyticity, summability, asymptotics, and calculation of exponentially small terms, Phys. Rev. Lett. 52(1984), 1112-1115.
  • 14. J.-P. Eckmann and H. Epstein, Borel summability of the mass and the 𝑆-matrix in 𝜙⁴ models, Comm. Math. Phys. 68 (1979), no. 3, 245–258. MR 544880
  • 15. J.-P. Eckmann, J. Magnen, and R. Sénéor, Decay properties and Borel summability for the Schwinger functions in P(φ)2 theories, Comm. Math. Phys. 39 (1975), 251-271.
  • 16. S. Graffi, V. Grecchi, E. M. Harrell II, and H. J. Silverstone, The 1/𝑅 expansion for 𝐻⁺₂: analyticity, summability, and asymptotics, Ann. Physics 165 (1985), no. 2, 441–483. MR 816800, 10.1016/0003-4916(85)90305-7
  • 17. S. Graffi, V. Grecchi, E. Harrell, and H. Silverstone, The 1/R expansion for H2+ : calculation of exponentially small terms and asymptotics, Phys. Rev. A33 (1986), 12-54.
  • 18. S. Graffi, V. Grecchi, and B. Simon, Borel summability: application to the anharmonic oscillator, Phys. Lett. 32B (1970), 631–634. MR 0332068
  • 19. G. H. Hardy, Divergent Series, Oxford, at the Clarendon Press, 1949. MR 0030620
  • 20. J. H. Hannay, Angle variable holonomy in adiabatic excursion of an integrable Hamiltonian, J. Phys. A 18 (1985), no. 2, 221–230. MR 784910
  • 21. Evans M. Harrell, On the rate of asymptotic eigenvalue degeneracy, Comm. Math. Phys. 60 (1978), no. 1, 73–95. MR 0486764
  • 22. Evans M. Harrell, Double wells, Comm. Math. Phys. 75 (1980), no. 3, 239–261. MR 581948
  • 23. E. Harrell and B. Simon, The mathematical theory of resonances whose widths are exponentially small, Duke Math. J. 47 (1980), no. 4, 845–902. MR 596118
  • 24. B. Helffer and J. Sjöstrand, Multiple wells in the semiclassical limit. I, Comm. Partial Differential Equations 9 (1984), no. 4, 337–408. MR 740094, 10.1080/03605308408820335
  • 25. B. Helffer and J. Sjöstrand, Effet tunnel pour l'opérateur de Schrödinger semi-classique. II. résonances, Proc. Nat. Inst. on Micoloral Analysis at "Il Ciocco", Reidel Publ. Co., 1985.
  • 26. Ira W. Herbst, Dilation analyticity in constant electric field. I. The two body problem, Comm. Math. Phys. 64 (1979), no. 3, 279–298. MR 520094
  • 27. I. Herbst and B. Simon, Some remarkable examples in eigenvalue perturbation theory, Phys. Lett. B78 (1978), 304-306.
  • 28. I. W. Herbst and B. Simon, Stark effect revisited, Phys. Rev. Lett. 41 (1978), no. 2, 67–69. MR 0495908
  • 29. Ira W. Herbst and B. Simon, Dilation analyticity in constant electric field. II. 𝑁-body problem, Borel summability, Comm. Math. Phys. 80 (1981), no. 2, 181–216. MR 623157
  • 30. V. Jaksic and J. Segert, Exponential approach to the adiabatic limit, Cal. Tech., preprint.
  • 31. T. Kato, On the adiabatic theorem of quantum mechanics, J. Phys. Soc. Japan 5 (1950), 435-439.
  • 32. Tosio Kato, Perturbation theory for linear operators, Die Grundlehren der mathematischen Wissenschaften, Band 132, Springer-Verlag New York, Inc., New York, 1966. MR 0203473
  • 33. J. Loeffel, A. Martin, A. Wightman, and B. Simon, Padé approximants and the anharmonic oscillator, Phys. Lett. B30 (1969), 656-658.
  • 34. C. Lanczos, Zur Theorie des Starkeffektes in hohen Feldern, Z. für Physik 62 (1931), 204-232.
  • 35. J. Le Guillou and J. Zinn-Justin, The hydrogen atom in strong magnetic fields: summation of the weak field series expansion, Ann. Phys. 147 (1983), 57.
  • 36. J. Magnen and R. Sénéor, Phase space cell expansion and Borel summability for the Euclidean 𝜑₃⁴ theory, Comm. Math. Phys. 56 (1977), no. 3, 237–276. MR 0523030
  • 37. Jonathan D. Baker, David E. Freund, Robert Nyden Hill, and John D. Morgan III, Radius of convergence and analytic behavior of the 1/𝑍 expansion, Phys. Rev. A (3) 41 (1990), no. 3, 1247–1273. MR 1048862, 10.1103/PhysRevA.41.1247
  • 38. J. Oppenheimer, Three notes on the quantum theory of aperiodic effects, Phys. Rev. 31 (1928), 66-81.
  • 39. Michael Reed and Barry Simon, Methods of modern mathematical physics. I, 2nd ed., Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York, 1980. Functional analysis. MR 751959
  • 40. Franz Rellich, Störungstheorie der Spektralzerlegung, Math. Ann. 113 (1937), no. 1, 600–619 (German). MR 1513109, 10.1007/BF01571652
  • 40. Franz Rellich, Störungstheorie der Spektralzerlegung, Math. Ann. 113 (1937), no. 1, 677–685 (German). MR 1513115, 10.1007/BF01571658
  • 40. Franz Rellich, Störungstheorie der Spektralzerlegung, Math. Ann. 116 (1939), no. 1, 555–570 (German). MR 1513244, 10.1007/BF01597374
  • 40. Franz Rellich, Störungstheorie der Spektralzerlegung. IV, Math. Ann. 117 (1940), 356–382 (German). MR 0002715
  • 40. Franz Rellich, Störungstheorie der Spektralzerlegung. V, Math. Ann. 118 (1942), 462–484 (German). MR 0010791
  • 41. Franz Rellich, Perturbation theory of eigenvalue problems, Assisted by J. Berkowitz. With a preface by Jacob T. Schwartz, Gordon and Breach Science Publishers, New York-London-Paris, 1969. MR 0240668
  • 42. E. Schrödinger, Quantisierung als Eigenwertproblem. III, Ann. der Physik 80 (1926), 457-490.
  • 43. Barry Simon, Coupling constant analyticity for the anharmonic oscillator. (With appendix), Ann. Physics 58 (1970), 76–136. MR 0416322
  • 44. Barry Simon, Quadratic form techniques and the Balslev-Combes theorem, Comm. Math. Phys. 27 (1972), 1–9. MR 0321456
  • 45. Barry Simon, Resonances in 𝑛-body quantum systems with dilatation analytic potentials and the foundations of time-dependent perturbation theory, Ann. of Math. (2) 97 (1973), 247–274. MR 0353896
  • 46. Barry Simon, Semiclassical analysis of low lying eigenvalues. II. Tunneling, Ann. of Math. (2) 120 (1984), no. 1, 89–118. MR 750717, 10.2307/2007072
  • 47. Barry Simon, Holonomy, the quantum adiabatic theorem, and Berry’s phase, Phys. Rev. Lett. 51 (1983), no. 24, 2167–2170. MR 726866, 10.1103/PhysRevLett.51.2167
  • 48. R. Seznec and J. Zinn-Justin, Summation of divergent series by order dependent mappings: application to the anharmonic oscillator and critical exponents in field theory, J. Math. Phys. 20 (1979), no. 7, 1398–1408. MR 538715, 10.1063/1.524247
  • 49. Alan D. Sokal, An improvement of Watson’s theorem on Borel summability, J. Math. Phys. 21 (1980), no. 2, 261–263. MR 558468, 10.1063/1.524408
  • 50. Clasine van Winter, Fredholm equations on a Hilbert space of analytic functions, Trans. Amer. Math. Soc. 162 (1971), 103–109 (1972). MR 0417849, 10.1090/S0002-9947-1971-0417849-8
  • 50. Clasine van Winter, Complex dynamical variables for multiparticle systems with analytic interactions. I, J. Math. Anal. Appl. 47 (1974), 633–670. MR 0368663
  • 50. Clasine van Winter, Complex dynamical variables for multiparticle systems with analytic interactions. II, J. Math. Anal. Appl. 48 (1974), 368–399. MR 0371308
  • 51. E. Vock and W. Hunziker, Stability of Schrödinger eigenvalue problems, Comm. Math. Phys. 83 (1982), no. 2, 281–302. MR 649163
  • 52. Edward Witten, Supersymmetry and Morse theory, J. Differential Geom. 17 (1982), no. 4, 661–692 (1983). MR 683171

Similar Articles

Retrieve articles in Bulletin of the American Mathematical Society with MSC (1985): 81C12, 81G45, 46C99, 46N05

Retrieve articles in all journals with MSC (1985): 81C12, 81G45, 46C99, 46N05


Additional Information

DOI: https://doi.org/10.1090/S0273-0979-1991-16020-9