Tree expansion in time-dependent perturbation theory

TL;DR

This paper introduces a novel tree-based parametrization for time-dependent perturbation theory, simplifying calculations and linking the method to combinatorics, with applications to adiabatic approximation and wavefunction convergence.

Contribution

The paper presents an original tree-based expansion for perturbation theory that simplifies computations and connects to combinatorial mathematics, improving both theoretical understanding and numerical efficiency.

Findings

Tree expansion simplifies perturbation calculations.

Sum over trees yields simpler expressions than individual terms.

Proves convergence of generalized wavefunction perturbation series.

Abstract

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...). 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 non-perturbatively the convergence of Morita's generalization of the Gell-Mann and Low wavefunction. We show that summing all the terms associated to the same tree leads to an utter simplification where the sum is simpler than any of its terms. Finally, we recover the time-independent equation for the wave operator and we give an explicit non-recursive expression for the term corresponding to an arbitrary tree.