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.- II. The Semiclassical Approximation and the JWKB Method.- III. Rayleigh-Schrödinger Perturbation Theory (RSPT).- IV. Divergence of the Perturbation Series.- V. Perturbation Series Summation Techniques.- VI. Foundations of the Variational Functional Method (VFM).- VII. Application of the VFM to One-Dimensional Systems with Trivial Boundary Conditions.- VIII Application of the VFM to One-Dimensional Systems with Boundary Conditions for Finite Values of the Coordinates.- IX Multidimensional Systems: The Problem of the Zeeman Effect in Hydrogen.- X Application of the VFM to the Zeeman Effect in Hydrogen.- XI Combination of VFM with RSPT: Application to Anharmonic Oscillators.- XII Geometrical Connection between the VFM and the JWKB Method.- B.- XIII Generalization of the Functional Method as a Summation Technique of Perturbation Series.- XIV Properties of the FM: Series with Non-Zero Convergence Radii.- XV Properties of the FM: Series with Zero Convergence Radii.- XVI Appication of the FM to the Anharmonic Oscillator.- XVII Application of the FM to Models with Confining Potentials.- XVIII Application of the FM to the Zeeman Effect in Hydrogen.- XIX Application of the FM to the Stark Effect in Hydrogen.- XX FM and Vibrational Potentials of Diatomic Molecules.- 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 ZeemanEffect in the Hydrogen Atom.- Appendix L RKR Method to Obtain Vibrational Potentials of Diatomic Molecules.- References Appendices A–L.