Large orders and summability of eigenvalue perturbation theory: A mathematical overview

The study of large orders of perturbation theory in various problems is reviewed: the anharmonic oscillator, the Zeeman and Stark problems, double wells, and the like. Pade and Borel summability and path integral ideas are discussed. The rigorous results on the subject are summarized.

[1]  B. Simon,et al.  Schrödinger operators with magnetic fields , 1981 .

[2]  J. Avron.  Bender-Wu formulas for the Zeeman effect in hydrogen , 1981 .

[3]  B. Simon,et al.  The mathematical theory of resonances whose widths are exponentially small , 1980 .

[4]  V. Privman.  New method of perturbation-theory calculation of the Stark effect for the ground state of hydrogen , 1980 .

[5]  I. Herbst.  Exponential decay in the stark effect , 1980 .

[6]  J. Cizek,et al.  Asymptotic behavior of the ground-state-energy expansion for H/sub 2/ /sup +/ in terms of internuclear separation , 1980 .

[7]  E. Caliceti,et al.  Perturbation theory of odd anharmonic oscillators , 1980 .

[8]  J. Rosen,et al.  ASYMPTOTIC EXPANSIONS OF GAUSSIAN INTEGRALS , 1980 .

[9]  B. G. Adams,et al.  Bender-Wu formulas for degenerate eigenvalues , 1980 .

[10]  T. Spencer.  The Lipatov argument , 1980 .

[11]  J. D. Morgan,et al.  Behavior of molecular potential energy curves for large nuclear separations , 1980 .

[12]  L Benassi,et al.  Resonances in the Stark effect and strongly asymptotic approximants , 1980 .

[13]  Alan D. Sokal,et al.  An improvement of Watson’s theorem on Borel summability , 1980 .

[14]  B. G. Adams,et al.  Stark Effect in Hydrogen: Dispersion Relation, Asymptotic Formulas, and Calculation of the Ionization Rate via High-Order Perturbation Theory , 1979 .

[15]  W. E. Caswell.  Accurate energy levels for the anharmonic oscillator and a summable series for the double-well potential in perturbation theory , 1979 .

[16]  Jean-Pierre Eckmann,et al.  Borel summability of the mass and theS-matrix in ϕ4 models , 1979 .

[17]  I. Herbst.  Dilation analyticity in constant electric field , 1979 .

[18]  B. G. Adams,et al.  Bender-Wu Formula, the SO(4,2) Dynamical Group, and the Zeeman Effect in Hydrogen , 1979 .

[19]  J. Zinn-Justin,et al.  Summation of divergent series by order dependent mappings: Application to the anharmonic oscillator and critical exponents in field theory , 1979 .

[20]  C. Bender,et al.  Tight upper and lower bounds on perturbation coefficients for anharmonic oscillators , 1979 .

[21]  J. Eckmann,et al.  Time-ordered products and Schwinger functions , 1979 .

[22]  B. Simon,et al.  Bender-Wu Formula and the Stark Effect in Hydrogen , 1979 .

[23]  B. Simon,et al.  Schrödinger operators with magnetic fields. I. general interactions , 1978 .

[24]  Carl M. Bender,et al.  Perturbation theory in large order , 1978 .

[25]  H. Silverstone.  Perturbation theory of the Stark effect in hydrogen to arbitrarily high order , 1978 .

[26]  B. Simon,et al.  Some remarkable examples in eigenvalue perturbation theory , 1978 .

[27]  S. Graffi,et al.  Resonances in Stark effect and perturbation theory , 1978 .

[28]  B. Simon,et al.  Stark Effect Revisited , 1978 .

[29]  V. V. Kolosov,et al.  An asymptotic approach to the Stark effect for the hydrogen atom , 1978 .

[30]  S. Graffi,et al.  Borel summability and indeterminacy of the Stieltjes moment problem: Application to the anharmonic oscillators , 1978 .

[31]  D. Soper,et al.  Large Order Expansion in Perturbation Theory , 1978 .

[32]  E. Harrell.  On the rate of asymptotic eigenvalue degeneracy , 1978 .

[33]  Barry Simon,et al.  Analytic properties of band functions , 1978 .

[34]  E. Bogomolny,et al.  Large order calculations in gauge theories , 1977 .

[35]  R. Sénéor,et al.  Phase space cell expansion and borel summability for the Euclidean φ34 theory , 1977 .

[36]  B. Simon,et al.  The Zeeman effect revisited , 1977 .

[37]  G. Parisi,et al.  Perturbation theory at large orders for a potential with degenerate minima , 1977 .

[38]  É. Brézin,et al.  Perturbation theory at large order. II. Role of the vacuum instability , 1977 .

[39]  The weakly coupled Yukawa₂ field theory: cluster expansion and Wightman axioms , 1977 .

[40]  Jean Zinn-Justin,et al.  Perturbation theory at large order. I. The phi/sup 2//sup N/ interaction , 1977 .

[41]  J. Magnen,et al.  The wightman axioms for the weakly coupled Yukawa model in two dimensions , 1976 .

[42]  C. Bender,et al.  Statistical Analysis of Feynman Diagrams , 1976 .

[43]  R. Sénéor,et al.  Decay properties and borel summability for the Schwinger functions inP(Φ)2 theories , 1975 .

[44]  S. Varadhan,et al.  Asymptotic evaluation of certain Markov process expectations for large time , 1975 .

[45]  Barry Simon,et al.  The P(φ) 2 Euclidean Quantum Field Theory as Classical Statistical Mechanics , 1975 .

[46]  J. Feldman.  The λϕ34 field theory in a finite volume , 1974 .

[47]  J. Dimock.  Asymptotic perturbation expansion in theP(φ)2 quantum field theory , 1974 .

[48]  Tai Tsun Wu,et al.  Coupled Anharmonic Oscillators. II. Unequal-Mass Case , 1973 .

[49]  Tai Tsun Wu,et al.  Anharmonic Oscillator. II. A Study of Perturbation Theory in Large Order , 1973 .

[50]  James Glimm,et al.  Positivity of the φ 34 Hamiltonian , 1973 .

[51]  N. Fröman,et al.  On the quantal treatment of the double-well potential problem by means of certain phase-integral approximations , 1972 .

[52]  B. Simon,et al.  The (φ 2n ) 2 Field Hamiltonian for Complex Coupling Constant , 1972 .

[53]  L. Rosen.  RENORMALIZATION OF THE HILBERT SPACE IN THE MASS SHIFT MODEL. , 1972 .

[54]  C. Bender,et al.  Anharmonic Oscillator with Polynomial Self‐Interaction , 1972 .

[55]  B. Simon,et al.  The (²ⁿ)₂ field Hamiltonian for complex coupling constant , 1972 .

[56]  Tai Tsun Wu,et al.  Large order behavior of Perturbation theory , 1971 .

[57]  R. Damburg,et al.  On One‐Dimensional Model for Exchange Forces , 1971 .

[58]  B. Simon.  Borel Summability of the Ground-State Energy in Spatially Cutoff (ϕ^4)_2 , 1970 .

[59]  B. Simon,et al.  Borel summability: Application to the anharmonic oscillator , 1970 .

[60]  Barry Simon,et al.  Coupling constant analyticity for the anharmonic oscillator , 1970 .

[61]  Alain J. Martin,et al.  Pade approximants and the anharmonic oscillator , 1969 .

[62]  Phase Transition and Eigenvalue Degeneracy of a One Dimensional Anharmonic Oscillator , 1969 .

[63]  B. Simon.  Convergence of regularized, renormalized perturbation series for super-renormalizable field theories , 1969 .

[64]  Franz Rellich,et al.  Perturbation Theory of Eigenvalue Problems , 1969 .

[65]  Tai Tsun Wu,et al.  Analytic Structure of Energy Levels in a Field-Theory Model , 1968 .

[66]  R J Damburg,et al.  On asymptotic expansions of electronic terms of the molecular ion H2 , 1968 .

[67]  Martin Pincus,et al.  Gaussian processes and Hammerstein integral equations , 1968 .

[68]  Charles E. Reid,et al.  Transformation of perturbation series into continued fractions, with application to an anharmonic oscillator , 1967 .

[69]  Kurt Friedrichs,et al.  Perturbation of Spectra in Hilbert Space , 1967 .

[70]  M. Schilder,et al.  Some asymptotic formulas for Wiener integrals , 1966 .

[71]  A. Jaffe.  Divergence of perturbation theory for bosons , 1965 .

[72]  H. Hasegawa,et al.  Optical absorption spectrum of hydrogenic atoms in a strong magnetic field , 1961 .

[73]  Walter Kohn,et al.  Analytic Properties of Bloch Waves and Wannier Functions , 1959 .

[74]  E. Caianiello.  Number of feynman graphs and convergence , 1956 .

[75]  F. Dyson.  Divergence of perturbation theory in quantum electrodynamics , 1952 .

[76]  P. S. Epstein,et al.  The Stark effect from the point of view of Schroedinger's quantum theory , 1926 .