Quantum Dynamics of Transition Processes in the Interacting Systems

The dynamical inverse problem of representation theory (the Wigner problem) 1] is solved for a certain class of interactions of Hamiltonian systems. The constructed quantum dynamics describes transition processes and stationary regimes on an equal footing. Among other numerous applications the results may be useful for the description of the simultaneous quantum dynamics of the open and closed string elds in the uniied string eld theory with their noncanonical coupling. The aim of this short note is to describe the quantum picture for transition processes in a certain class of pairs of interacting systems. Theorem 1. Let (V 1 ; V 2) be an arbitrary isotopic pair 2], Der(V 1 ; V 2) be the Lie algebra of its derivations, and L be an arbitrary polynomial mapping from V 1 V 2 into Der(V 1 ; V 2) (a nonlinear sectional operator { cf..3]), then the equations (2)) 2 Der(V 1 ; V 2), preserves the isocommutation relations. Corollary 1. Let (V 1 ; V 2) be an arbitrary isotopic pair, T be its representation in the space U 2], i.e. a homomorphism of the isotopic pair (V 1 ; V 2) into the operator isotopic pair (End(U); End(U)). Let L be an arbitrary polynomial mapping from End(U) End(U) into Der(End(U); End(U)), then the equations (_ T t (A) =L(T t (X); T t (A)) (1) T t (A) _ T t (X) =L(T t (X); T t (A)) (2) T t (X) where X 2 V 1 , A 2 V 2 , deene a family T t (T 0 = T) of representations of the isotopic pair (V 1 ; V 2) in the space U. It means that the operators ^ X t = T t (X) and ^ A t = T t (A) obey the initial isocommutation relations.