New results for Heisenberg dynamics for nonself-adjoint Hamiltonians

In a previous paper, we began our analysis on the role of non-self-adjoint Hamiltonians in connection with the Heisenberg dynamics for quantum systems. Here, motivated by the growing interest on this topic and on some recent results on dynamical systems, we continue this analysis focusing on what we believe is an unexplored (or, at least, not so explored! aspect of Heisenberg dynamics, related to the need for using vectors which are brute-force normalized. Our main interest is on conserved quantities and on conditions which guarantee that some observables of the system, or their mean values, do not evolve in time.