Abstract
The key problem in quantum scattering theory is the probability conservation, i.e., the unitarity of the S-matrix, which connects the initial with the final state of evolution of the considered physical system. This problem is not possible to solve if the scattering states neglect the so-called asymptotic convergence problem, which require that the bound and free dynamics coincide with each other at infinite distances between the colliding particles. The usual approach to scattering theory is overwhelmed by heuristic formulae, with a stereotypic explanation that a rigorous mathematical formalism would merely obscure the physical arguments. This is not the case as is documented in the present work, relying upon the basic theorems of strung topology of vector spaces and spectral operator analysis, from which all the standard synonyms of collision theory directly follow, such as the Lippmann-Schwinger integral equations, behaviours of the scattering states, probability transition from the initial to the final state, differential as well as total cross sections, etc. Furthermore, rigour in mathematical treatment is not only in absolute compatibility with physical argumentation and intuition, but also is established in this review in a simple and plausible manner. These fundamental aspects are not only relevant to the foundation of a complete quantum-mechanical scattering theory, whose essential principles are outlined in the present work, but they are also of primary significance for introducing the most adequate practical methods for versatile applications.