The algebra of effective Hamiltonians and operators: Truncated operators and computational aspects

Abstract

We extend to finite orders of perturbation theory our previous analysis of effective Hamiltonians h and effective operators a which produce, respectively, exact energies and matrix elements of a time-independent operator A for a finite number of eigenstates of a time-independent Hamiltonian H. The validity of various properties is examined here for perturbatively truncated h and a, particularly, the preservation upon transformation to effective operators of commutation relations involving H and/or constants of the motion, of symmetries, and of the equivalence between dipole length and velocity transition moments. We compare formal and computational features of all a definitions and of the more limited Hellmann–Feynman theorem based ‘‘effective operators,’’ which provide only diagonal matrix elements of A in special cases. Norm-preserving transformations to effective operators are found to yield a simpler effective operator formalism from both formal and computational viewpoints.