Tree expansion in time-dependent perturbation theory

The computational complexity of time-dependent perturbation theory is well known to be largely combinatorial whatever the chosen expansion method and family of parameters (combinatorial sequences, Goldstone and other Feynman-type diagrams, etc.). We show that a very efficient perturbative expansion, both for theoretical and numerical purposes, can be obtained through an original parametrization by trees and generalized iterated integrals. We emphasize above all the simplicity and naturality of the new approach that links perturbation theory with classical and recent results in enumerative and algebraic combinatorics. These tools are applied to the adiabatic approximation and the effective Hamiltonian. We prove perturbatively and nonperturbatively the convergence of Morita's generalization of the Gell-Mann and Low wave function. We show that summing all the terms associated with the same tree leads to an utter simplification where the sum is simpler than any of its terms. Finally, we recover the Rayleigh-Schrodinger time-independent equation for the wave operator and we give an explicit nonrecursive expression for the term corresponding to an arbitrary tree. (C) 2010 American Institute of Physics. [doi:10.1063/1.3447733]