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Fifty years of eigenvalue perturbation theory. (English) Zbl 0739.47006

The author gives an review of eigenvalue perturbation theory starting from famous F. Rellich’s works. It turns out, that many of the examples of interest in quantum physics do not fit into the scheme of regular perturbation theory.
The following models are discussed in detail: isoelectric atoms, autoionizing states, the anharmonic oscillator, double wells, Zeeman and Stark effects and Berry’s phase.
It must be noted that many questions of the regular perturbation theory may be completely solved by the Newton diagram method of branching theory [see M. M. Vainberg and V. A. Trenogin, “The theory of branching of solutions of nonlinear equations” (1974; Zbl 0274.47033)].

MSC:

47A55 Perturbation theory of linear operators
47N50 Applications of operator theory in the physical sciences
47A75 Eigenvalue problems for linear operators

Citations:

Zbl 0274.47033
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References:

[1] J. Aguilar and J. M. Combes, A class of analytic perturbations for one-body Schrödinger Hamiltonians, Comm. Math. Phys. 22 (1971), 269 – 279. · Zbl 0219.47011
[2] Joseph E. Avron, Bender-Wu formulas for the Zeeman effect in hydrogen, Ann. Physics 131 (1981), no. 1, 73 – 94.
[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. · Zbl 0351.47007
[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. · Zbl 0464.35086
[6] J. Avron, I. Herbst, and B. Simon, The Zeeman effect revisited, Phys. Lett. 62 A (1977), 214-216.
[7] 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. · Zbl 0830.57020
[8] 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.
[9] 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. · Zbl 0626.58033
[10] E. Balslev and J. M. Combes, Spectral properties of many-body Schrödinger operators with dilatation-analytic interactions, Comm. Math. Phys. 22 (1971), 280 – 294. · Zbl 0219.47005
[11] 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.
[12] C. Bender and T. Wu, Analytic structure of energy levels in a field-theory model, Phys. Rev. Lett. 21 (1968), 406.
[13] Carl M. Bender and Tai Tsun Wu, Anharmonic oscillator, Phys. Rev. (2) 184 (1969), 1231 – 1260.
[14] C. Bender and T. Wu, Anharmonic oscillator, II. A study of perturbation theory in large order, Phys. Rev. D7 (1973), 1620.
[15] M. V. Berry, Quantal phase factors accompanying adiabatic changes, Proc. Roy. Soc. London Ser. A 392 (1984), no. 1802, 45 – 57. · Zbl 1113.81306
[16] 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.
[17] J.-P. Eckmann and H. Epstein, Borel summability of the mass and the \?-matrix in \?\(^{4}\) models, Comm. Math. Phys. 68 (1979), no. 3, 245 – 258.
[18] J.-P. Eckmann, J. Magnen, and R. Sénéor, Decay properties and Borel summability for the Schwinger functions in P(\varphi )2 theories, Comm. Math. Phys. 39 (1975), 251-271.
[19] S. Graffi, V. Grecchi, E. M. Harrell II, and H. J. Silverstone, The 1/\? expansion for \?\(^{+}\)\(_{2}\): analyticity, summability, and asymptotics, Ann. Physics 165 (1985), no. 2, 441 – 483. · Zbl 0614.46068
[20] 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.
[21] S. Graffi, V. Grecchi, and B. Simon, Borel summability: application to the anharmonic oscillator, Phys. Lett. 32B (1970), 631 – 634.
[22] G. H. Hardy, Divergent Series, Oxford, at the Clarendon Press, 1949. · Zbl 0032.05801
[23] J. H. Hannay, Angle variable holonomy in adiabatic excursion of an integrable Hamiltonian, J. Phys. A 18 (1985), no. 2, 221 – 230.
[24] Evans M. Harrell, On the rate of asymptotic eigenvalue degeneracy, Comm. Math. Phys. 60 (1978), no. 1, 73 – 95. · Zbl 0395.34023
[25] Evans M. Harrell, Double wells, Comm. Math. Phys. 75 (1980), no. 3, 239 – 261. · Zbl 0445.35036
[26] E. Harrell and B. Simon, The mathematical theory of resonances whose widths are exponentially small, Duke Math. J. 47 (1980), no. 4, 845 – 902. · Zbl 0455.35091
[27] B. Helffer and J. Sjöstrand, Multiple wells in the semiclassical limit. I, Comm. Partial Differential Equations 9 (1984), no. 4, 337 – 408. · Zbl 0546.35053
[28] 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. · Zbl 0607.35026
[29] Ira W. Herbst, Dilation analyticity in constant electric field. I. The two body problem, Comm. Math. Phys. 64 (1979), no. 3, 279 – 298. · Zbl 0447.47028
[30] I. Herbst and B. Simon, Some remarkable examples in eigenvalue perturbation theory, Phys. Lett. B78 (1978), 304-306.
[31] I. W. Herbst and B. Simon, Stark effect revisited, Phys. Rev. Lett. 41 (1978), no. 2, 67 – 69.
[32] Ira W. Herbst and B. Simon, Dilation analyticity in constant electric field. II. \?-body problem, Borel summability, Comm. Math. Phys. 80 (1981), no. 2, 181 – 216. · Zbl 0473.47038
[33] V. Jaksic and J. Segert, Exponential approach to the adiabatic limit, Cal. Tech., preprint.
[34] T. Kato, On the adiabatic theorem of quantum mechanics, J. Phys. Soc. Japan 5 (1950), 435-439.
[35] Tosio Kato, Perturbation theory for linear operators, Die Grundlehren der mathematischen Wissenschaften, Band 132, Springer-Verlag New York, Inc., New York, 1966. · Zbl 0148.12601
[36] J. Loeffel, A. Martin, A. Wightman, and B. Simon, Padé approximants and the anharmonic oscillator, Phys. Lett. B30 (1969), 656-658.
[37] C. Lanczos, Zur Theorie des Starkeffektes in hohen Feldern, Z. für Physik 62 (1931), 204-232. · Zbl 0001.17801
[38] 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.
[39] J. Magnen and R. Sénéor, Phase space cell expansion and Borel summability for the Euclidean \?\(_{3}\)\(^{4}\) theory, Comm. Math. Phys. 56 (1977), no. 3, 237 – 276.
[40] Jonathan D. Baker, David E. Freund, Robert Nyden Hill, and John D. Morgan III, Radius of convergence and analytic behavior of the 1/\? expansion, Phys. Rev. A (3) 41 (1990), no. 3, 1247 – 1273.
[41] J. Oppenheimer, Three notes on the quantum theory of aperiodic effects, Phys. Rev. 31 (1928), 66-81. · JFM 54.0970.02
[42] 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. · Zbl 0459.46001
[43] Franz Rellich, Störungstheorie der Spektralzerlegung, Math. Ann. 113 (1937), no. 1, 600 – 619 (German). · Zbl 0016.06201
[44] Franz Rellich, Störungstheorie der Spektralzerlegung, Math. Ann. 113 (1937), no. 1, 677 – 685 (German). · Zbl 0016.06301
[45] Franz Rellich, Störungstheorie der Spektralzerlegung, Math. Ann. 116 (1939), no. 1, 555 – 570 (German). · Zbl 0020.30601
[46] Franz Rellich, Störungstheorie der Spektralzerlegung. IV, Math. Ann. 117 (1940), 356 – 382 (German). · Zbl 0023.13503
[47] Franz Rellich, Störungstheorie der Spektralzerlegung. V, Math. Ann. 118 (1942), 462 – 484 (German). · Zbl 0027.22702
[48] 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. · Zbl 0181.42002
[49] E. Schrödinger, Quantisierung als Eigenwertproblem. III, Ann. der Physik 80 (1926), 457-490. · JFM 52.0966.02
[50] Barry Simon, Coupling constant analyticity for the anharmonic oscillator. (With appendix), Ann. Physics 58 (1970), 76 – 136.
[51] Barry Simon, Quadratic form techniques and the Balslev-Combes theorem, Comm. Math. Phys. 27 (1972), 1 – 9. · Zbl 0237.35025
[52] Barry Simon, Resonances in \?-body quantum systems with dilatation analytic potentials and the foundations of time-dependent perturbation theory, Ann. of Math. (2) 97 (1973), 247 – 274. · Zbl 0252.47009
[53] Barry Simon, Semiclassical analysis of low lying eigenvalues. II. Tunneling, Ann. of Math. (2) 120 (1984), no. 1, 89 – 118. , https://doi.org/10.2307/2007072 Barry Simon, Semiclassical analysis of low lying eigenvalues. III. Width of the ground state band in strongly coupled solids, Ann. Physics 158 (1984), no. 2, 415 – 420. , https://doi.org/10.1016/0003-4916(84)90125-8 Barry Simon, Semiclassical analysis of low lying eigenvalues. IV. The flea on the elephant, J. Funct. Anal. 63 (1985), no. 1, 123 – 136. · Zbl 0652.35090
[54] Barry Simon, Holonomy, the quantum adiabatic theorem, and Berry’s phase, Phys. Rev. Lett. 51 (1983), no. 24, 2167 – 2170.
[55] 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. · Zbl 0495.65002
[56] Alan D. Sokal, An improvement of Watson’s theorem on Borel summability, J. Math. Phys. 21 (1980), no. 2, 261 – 263. · Zbl 0441.40012
[57] Clasine van Winter, Fredholm equations on a Hilbert space of analytic functions, Trans. Amer. Math. Soc. 162 (1971), 103 – 109 (1972). · Zbl 0226.45015
[58] Clasine van Winter, Complex dynamical variables for multiparticle systems with analytic interactions. I, J. Math. Anal. Appl. 47 (1974), 633 – 670. · Zbl 0321.47004
[59] Clasine van Winter, Complex dynamical variables for multiparticle systems with analytic interactions. II, J. Math. Anal. Appl. 48 (1974), 368 – 399. · Zbl 0321.47005
[60] E. Vock and W. Hunziker, Stability of Schrödinger eigenvalue problems, Comm. Math. Phys. 83 (1982), no. 2, 281 – 302. · Zbl 0528.35023
[61] Edward Witten, Supersymmetry and Morse theory, J. Differential Geom. 17 (1982), no. 4, 661 – 692 (1983). · Zbl 0499.53056
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