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...