Basic Notions and Main Observables in Scattering Problems Observables and elementary processes Energy as the most important physical property Classification of collisions The role of wave packets Adiabatic switching of interaction potentials Collimation of beams of projectiles General waves and quantum mechanical waves Probability character of quantum collisions Requirements of the Theory for the Experiment Elementary events versus multiple scatterings Average probabilities Total cross sections Differential cross sections Total probabilities Transmission phenomena Quantum mechanical currents and cross sections Continuous Spectrum and Eigen-Problems of Resolvents Completeness and separability of Hilbert spaces The key realizations of abstract vector spaces Isomorphism of vector spaces Eigen-problems for continuous spectra Normal and Hermitean operators Strong and weak topology Compact operators for mapping of weak to strong limits Strong differentiability and strong analyticity Linear and Bilinear Functionals Linear functionals for mapping between vector spaces and scalar fields The Ries-Freshe theorem Bilinear functionals Definition of a Quantum Scattering Event Hamiltonian operators and boundedness Evolution operators and Møller wave operators The Cauchy strong limit in non-stationary scattering theory Three criteria for a quantum collisional system The Adiabatic Theorem and the Abel Strong Limit Adiabatic theorem for scattering states Adiabatic theorem and existence of wave operators The Abel strong limit in stationary scattering theory Exponential screening of potentials and adiabatic theorem Adiabatic theorem and Green operators Adiabatic theorem and Lippmann-Schwinger equations Non-Stationary and Stationary Scattering via Strong Limits The Abel limit and Lippmann-Schwinger equations The Abel limit and Fourier integrals Scattering Matrix and Transition Matrix Abel limit and scattering operators Matrix elements of scattering operators Transition operators Spectral Analysis of Operators The Abel limit with no recourse to the Cauchy limit The spectral theorem Unitary operators and strong topology The Abel limit for Møller wave operators The link between Møller operators and Green resolvents The Existence and Completeness of Møller Wave Operators Linearity and isometry of wave operators Boundedness of wave operators in the whole Hilbert space The Schur lemma on invariant subspaces for evolution operators Intertwining relations for evolution and wave operators The role of spectral projection operators Completeness of Møller wave operators Scattering operator derived from intertwining wave operators Four-Body Theories for Fast Ion-Atom Collisions Main features of interactive four-body dynamics Notation and basic formulae The entrance channel The exit channels Perturbation Series with the Correct Boundary Conditions Lippmann-Schwinger equations Born expansions with the correct boundary conditions for four-body collisions The Dodd-Greider Series for Four-Body Collisions Derivation of the distorted waves for the initial states Double Electron Capture The CDW-4B method The SE-4B method The CDW-EIS-4B method The CDW-EFS-4B method The BDW-4B method The BCIS-4B method The CB1-4B method Comparison between theories and experiments Simultaneous Transfer and Ionization The CDW-4B method Comparison between theories and experiments Single Electron Detachment The MCB-4B method Comparison between theories and experiments Single Electron Capture The CDW-4B method The CDW-BFS (prior BDW-4B) and CDW-BIS (post BDW-4B method) Simultaneous Transfer and Excitation The CDW-4B method for the TE process The TEX mode for radiative decays of asymmetric systems The CDW-4B method for TEX modes The CDW-4B method for the TE process in asymmetric collisions Target charge ZTand the interference between RTEX and NTEX modes The TEA mode for nearly symmetrical systems: the Auger decay The CDW-4B method for TEA modes Description of the final state Cross sections for TEA modes The CDW-4B method in the Feshbach resonance formalism Comparison between theories and experiments for electron spectra near Auger peaks Concluding Remarks and Outlooks List of acronyms in the main text and bibliography References Index rt spaces The key realizations of abstract vector spaces Isomorphism of vector spaces Eigen-problems for continuous spectra Normal and Hermitean operators Strong and weak topology Compact operators for mapping of weak to strong limits Strong differentiability and strong analyticity Linear and Bilinear Functionals Linear functionals for mapping between vector spaces and scalar fields The Ries-Freshe theorem Bilinear functionals Definition of a Quantum Scattering Event Hamiltonian operators and boundedness Evolution operators and Møller wave operators The Cauchy strong limit in non-stationary scattering theory Three criteria for a quantum collisional system The Adiabatic Theorem and the Abel Strong Limit Adiabatic theorem for scattering states Adiabatic theorem and existence of wave operators The Abel strong limit in stationary scattering theory Exponential screening of potentials and adiabatic theorem Adiabatic theorem and Green operators Adiabatic theorem and Lippmann-Schwinger equations Non-Stationary and Stationary Scattering via Strong Limits The Abel limit and Lippmann-Schwinger equations The Abel limit and Fourier integrals Scattering Matrix and Transition Matrix Abel limit and scattering operators Matrix elements of scattering operators Transition operators Spectral Analysis of Operators The Abel limit with no recourse to the Cauchy limit The spectral theorem Unitary operators and strong topology The Abel limit for Møller wave operators The link between Møller operators and Green resolvents The Existence and Completeness of Møller Wave Operators Linearity and isometry of wave operators Boundedness of wave operators in the whole Hilbert space The Schur lemma on invariant subspaces for evolution operators Intertwining relations for evolution and wave operators The role of spectral projection operators Completeness of Møller wave operators Scattering operator derived from intertwining wave operators Four-Body Theories for Fast Ion-Atom Collisions Main features of interactive four-body dynamics Notation and basic formulae The entrance channel The exit channels Perturbation Series with the Correct Boundary Conditions Lippmann-Schwinger equations Born expansions with the correct boundary conditions for four-body collisions The Dodd-Greider Series for Four-Body Collisions Derivation of the distorted waves for the initial states Double Electron Capture The CDW-4B method The SE-4B method The CDW-EIS-4B method The CDW-EFS-4B method The BDW-4B method The BCIS-4B method The CB1-4B method Comparison between theories and experiments Simultaneous Transfer and Ionization The CDW-4B method Comparison between theories and experiments Single Electron Detachment The MCB-4B method Comparison between theories and experiments Single Electron Capture The CDW-4B method The CDW-BFS (prior BDW-4B) and CDW-BIS (post BDW-4B method) Simultaneous Transfer and Excitation The CDW-4B method for the TE process The TEX mode for radiative decays of asymmetric systems The CDW-4B method for TEX modes The CDW-4B method for the TE process in asymmetric collisions Target charge ZTand the interference between RTEX and NTEX modes The TEA mode for nearly symmetrical systems: the Auger decay The CDW-4B method for TEA modes Description of the final state Cross sections for TEA modes The CDW-4B method in the Feshbach resonance formalism Comparison between theories and experiments for electron spectra near Auger peaks Concluding Remarks and Outlooks List of acronyms in the main text and bibliography References Index atic theorem for scattering states Adiabatic theorem and existence of wave operators The Abel strong limit in stationary scattering theory Exponential screening of potentials and adiabatic theorem Adiabatic theorem and Green operators Adiabatic theorem and Lippmann-Schwinger equations Non-Stationary and Stationary Scattering via Strong Limits The Abel limit and Lippmann-Schwinger equations The Abel limit and Fourier integrals Scattering Matrix and Transition Matrix Abel limit and scattering operators Matrix elements of scattering operators Transition operators Spectral Analysis of Operators The Abel limit with no recourse to the Cauchy limit The spectral theorem<.