Large Order Perturbation Theory and Summation Methods in Quantum Mechanics

The book provides a general, broad approach to aspects of perturbation theory. The aim has been to cover all topics of interest, from construction, analysis, and summation of perturbation series to applications. Emphasis is placed on simple methods, as well as clear, intuitive ideas stemming from the physics of systems of interest.

A.- I. General Properties of the Eigenvalue Spectrum.- 1. Some Fundamental Properties.- 2. The Hellmann-Feynman Theorem.- 3. Hypervirial Relations and General Boundary Conditions.- References Chapter I.- II. The Semiclassical Approximation and the JWKB Method.- 4. Adiabatic Invariants.- 5. Bohr-Sommerfeld Quantization Condition and JWKB Method.- 6. Applications of the JWKB Method.- References Chapter II.- III. Rayleigh-Schrödinger Perturbation Theory (RSPT).- 7. The Rayleigh-Schrödinger Perturbation Theory.- 8. Hypervirial Method to Generate the Perturbation Expansion.- 9. Other Methods to Generate the Perturbation Expansion.- References Chapter III.- IV. Divergence of the Perturbation Series.- 10. Divergence of the perturbation series.- 11. Mathematical Methods to study the Asymptotic Behaviour of the RS coefficients.- References of Chapter IV.- V. Perturbation Series Summation Techniques.- 12. Introduction to the summability of divergent or slowly convergent series.- 13. Padé Approximants.- 14. Borel transform and Borel-Padé summation method.- 15. Euler Summation Method.- 16. Perturbation series renormalization techniques.- 17. Wick ordering and perturbation series summation.- 18. Summation of perturbation series through orderdependent mappings.- References Chapter V.- VI. Foundations of the Variational Functional Method (VFM).- 19. Energy of parameter-dependent systems.- 20. Semiclassical functional expressions for the energy.- 21. Scaling Variational Method.- 22. VFM from Heisenberg inequalities.- References Chapter VI.- VII. Application of the VFM to One-Dimensional Systems with Trivial Boundary Conditions.- 23. Anharmonic oscillators and variational functional: general properties.- 24. Translation of Coordinates and Variational Functional.- 25. Central Field Systems.- 26. Application of the variational functional to systems with confining potential.- References Chapter VII.- VIII Application of the VFM to One-Dimensional Systems with Boundary Conditions for Finite Values of the Coordinates.- 28. Functionals for Systems with Dirichlet Boundary conditions.- 29. Bounded harmonic oscillator: Approximation of its eigenvalues with the VFM.- References Chapter VIII.- IX Multidimensional Systems: The Problem of the Zeeman Effect in Hydrogen.- 30. Importance of the problem and applications of the model.- 31. Application of non-perturbative methods.- 32. Application of perturbation methods.- References Chapter IX.- X Application of the VFM to the Zeeman Effect in Hydrogen.- 33. Derivation of the variational functional.- 34. Results for several functions of physical interest.- 35. Scaling laws and semiclassical behavior of the Variational functional.- References Chapter X.- XI Combination of VFM with RSPT: Application to Anharmonic Oscillators.- 36. An elementary extension of the VFM for anharmonic oscillators.- 37. Application of the VFM to the theory of anharmonicity regimes.- 33. Another extension of the VFM for anharmonic oscillators.- References Chapter XI.- XII Geometrical Connection between the VFM and the JWKB Method.- 39. VFM and JWCB integrals for 1D systems with even potentials.- 40. VFM and JWKB integrals for 1D systems with potentials without defined parity and central field systems.- 41. Generalization of geometrical relations and RSPT.- References Chapter XII.- B.- XIII Generalization of the Functional Method as a Summation Technique of Perturbation Series.- 42. Generalization of the FM: Connection between semiclassical relations and renormalized series.- 43. Connection between the Fri and other summation techniques.- 44. Formulation of the FM from scaling laws (dilatation relationships).- References Chapter XIII 33.- XIV Properties of the FM: Series with Non-Zero Convergence Radii.- 45. Simple eigenvalue problems with branch-point singularities.- 46. Numerical Results for Simple Examples.- 47. Geometrical Series and FM.- 48. Further comments on series with non-zero con-vergence radii.- References Chapter XIV.- XV Properties of the FM: Series with Zero Convergence Radii.- 49. FM and asymptotic properties of Taylor coefficients of a series with zero convergence radius.- 50. Application of the FM to integrals of interest in field theory and statistical mechanics.- 51. Convergence conditions for the FM: Discussion of integrals with factorial divergence.- References Chapter XV.- XVI Appication of the FM to the Anharmonic Oscillator.- 52. Renormalization of the RS perturbation series with the FM: convergence to the ground state of the purely quartic oscillator.- 53. Further results for the eigenvalues of quartic anharmonic oscillators.- References Chapter XVI.- XVII Application of the FM to Models with Confining Potentials.- 54. Convergence of renormalized series in the strong coupling limit.- 55. Further results for eigenvalues of confining potential models.- References Chapter XVII.- XVIII Application of the FM to the Zeeman Effect in Hydrogen.- 56. Convergence of renormalized serie for the Landau regime.- 57. Further results for the Zeeman eigenvalues.- 58. FM approximation to the binding energy.- References Chapter XVIII.- XIX Application of the FM to the Stark Effect in Hydrogen.- 59. Approximation to Stark resonances.- 60. Upper and lower bound to the real part of the Stark resonances.- References Chapter XIX.- XX FM and Vibrational Potentials of Diatomic Molecules.- 61. Vibrational potentials for diatomic molecules.- 62. Kratzer-Fues potential and FM.- 63. Dunham series for ionic molecules.- 64. Dunham series for covalent molecules.- References Chapter XX.- Appendix A Scaling Laws of Schrödinger Operators.- Appendix B Applications of the Anharmonic Oscillator Model.- Appendix D Calculation of Integrals by the Saddle-Point Method.- Appendix E Construction of Padé Approximants.- Appendix F Normal Ordering of Operators.- Appendix G Applications of Models with Confining Potentials.- Appendix H Hamiltonian of an Hydrogen Atom in a Magnetic Field.- Appendix I Asymptotic Behavior of the Binding Energy for the Zeeman Effect in the Hydrogen Atom.- Appendix L RKR Method to Obtain Vibrational Potentials of Diatomic Molecules.- References Appendices A–L.