Some Aspects on the Hamiltonian and Liouvillian Formalism, the Special Propagator Methods, and the Equation of Motion Approach

Abstract

Publisher Summary This chapter investigates the Liouvillian eigenvalue problem LC = vC and its solution through the special propagator methods, the equation-of-motion approach, and their connection in terms of the commutator binary product and are briefly reviewed. The solution of the eigenvalue problem LC = vC in the Hilbert-Schmidt operator space is also studied in great detail. The refined solutions are expressed in terms of a new basis B (1), which spans an operator space closed under adjunction and multiplication. The refinement procedure is of particular value when separating an eigenelement associated with a degenerate eigenvalue v into components, which are excitation operators referring to specific initial and final states. The connection between the general theory and the special methods is then discussed also using the connection between the commutator binary product and the Hilbert-Schmidt binary product. In conclusion, the possibilities for further generalization of the propagator concept, as well as approximations by means of “inner projections” are also briefly discussed.