A probabilistic evolution approach trilogy, part 2: spectral issues for block triangular evolution matrix, singularities, space extension

Abstract

This is the second part of the trilogy on the probabilistic evolution approach and related to the quantum dynamical systems as the first part is. In this sense this work extends the content of the first part to the perhaps secondary but very important details. The spectral investigation of the evolution matrix reveals important issues first and brings the importance of the zero eigenvalues to the surface. The asymptotic convergence possibility and difficulties arising from there can be softened by redefining the state vector. Beside the redefinition, the dimensional extension by adding new elements to the state vector may facilitate the utilization of evolution matrix by bringing conicality or at least multinomiality. The space extension may also help us to deal with singular Hamiltonian systems. All these issues are focused on rather phenomenologically. Illustrative or not, no comprehensive implementation is given since the main purpose is just conceptuality.