Underlying Geometric Flow in Hamiltonian Evolution

In this paper an underlying perturbed Ricci flow construction is made, within the metric operator space, originated from the Heisenberg dynamical equations, to formulate a method which appears to provide a new geometric approach for the geometric formulation of the quantum mechanical dynamics. A quantum mechanical notion of stability and local instability is introduced within the quantum mechanical theory, based on the quantum mechanical dynamical equations governing the evolution of the tensor metric operator. The stability analysis is carried in the topology of little H¨older spaces of metrics which the tensor metric operator acts on. Finally, a conjecture is introduced in attempt to characterize the stability properties of the quantum mechanical system such that it brings in the quantum mechanical dynamics into the analysis.