A global method for the analysis of all natural modes in quantum mechanics is presented. In the framework of this method the global analysis of the pole function k=k(l)(g), which gives the S-matrix poles as a function of the potential strength g, is done. The method involves the construction of the Riemann surface Rg(l) over the g plane, on which the function k=k(l)(g) is single-valued and analytic. To each natural mode of the quantum system a sheet of the Riemann surface Rg(l) is associated. A new quantum number with topological meaning is introduced in order to label a natural mode. The method is applied to various local potentials, including a coupled-channel potential, as well as a non-local potential. A new class of resonant states (exotic resonant states) was identified. The wave function of an exotic resonant state is mostly confined to the region of the potential barrier. The di-nuclear parent quasimolecular states represent a particular case of exotic resonant states for a potential well with Coulomb barrier. The method allows not only studying each state of a quantum system, but also to understand the transitions from a quantum state to another state as a result of the potential strength variation. The problems are taken from quantum physics, but the method can be applied in any field of science involving the natural modes. The described method can be applied in electromagnetism, atomic and molecular physics, nuclear physics, particle physics, solid-state physics with application in nanoscience and electronic devices, chemical physics, and optics. The purpose of this book is to provide a reference for researchers and postgraduates. The reader is assumed to be familiar with the elements of the quantum theory of scattering and the elements of the analytic functions theory.