Heisenberg Dynamics for Non Self-Adjoint Hamiltonians: Symmetries and Derivations

In some recent literature the role of non self-adjoint Hamiltonians, H not equal H-dagger, is often considered in connection with gain-loss systems. The dynamics for these systems is, most of the times, given in terms of a Schrodinger equation. In this paper we rather focus on the Heisenberg-like picture of quantum mechanics, stressing the (few) similarities and the (many) differences with respected to the standard Heisenberg picture for systems driven by self-adjoint Hamiltonians. In particular, the role of the symmetries, *-derivations and integrals of motion is discussed.