A matrix Kato–Bloch perturbation method for Hamiltonian systems

Vincent Hurtubise

We derive perturbation expansions for the mapping operators (k,l) that transform a full Hilbert space time-independent Hamiltonian H and operator A into, respectively, a finite (multidimensional) space effective Hamiltonian h and effective operator a. The eigenvalues of h are identical to some of those of H, and a produces exact matrix elements of A for the corresponding states. Our derivations are substantially both more general and simpler than most literature ones and yield simple linear recursive expressions for k and l. Both these mapping solutions and new identities involving h, a, k, and l straightforwardly produce new recursive relations for h and the first known recursive a expressions. We apply our results to the Bloch, Kato, and all norm-preserving formalisms, including the canonical one. The new h and a identities are also shown to be suitable for iterative and multireference coupled cluster approaches.

Francisco Soto, Braulio Misael Villegas-Martínez, H. M. Moya-Cessa

B. H. Brandow

Andrey Nikolaev

This work establishes a connection between canonical perturbation series in quantum mechanics and a Kato expansion for the resolvent of the Liouville superoperator. Our approach leads to an explicit expression for a generator of a block-diagonalizing Dyson’s ordered exponential in arbitrary perturbation order. Unitary intertwining of perturbed and unperturbed averaging superprojectors allows for a description of ambiguities in the generator and block-diagonalized Hamiltonian. We compare the efficiency of the corresponding computational algorithm with the efficiencies of the Van Vleck and Magnus methods for high perturbative orders.

J. čada

Daniel R. Bes, G.G. Dussel, Hugo M. Sofia

Simple expressions are given for the normalization coefficients in Rayleigh–Schroedinger, Brillouin–Wigner, and Bloch–Horowitz perturbation methods. These expressions are derivatives of the energy, or of the matrix elements of the effective Hamiltonian.