Convergence without resummation: an iterative approach to perturbative eigenvalue problems
Matteo Smerlak · arXiv (Cornell University) · 2021
We compute the eigenvectors of perturbed linear operators through fixed-point iteration instead of power series expansions. The method is elementary, explicit, and convergent under more general conditions than conventional Rayleigh-Schrodinger (which arises as a particular limiting case). We illustrate this iterative perturbation theory (IPT) with several challenging ground state computations, including even anharmonic oscillators, the hydrogenic Zeeman problem, and the Herbst-Simon Hamiltonian with finite ground state energy but vanishing Rayleigh-Schrodinger expansion. In all cases, we find that, with a suitable partitioning of the Hamiltonian, IPT converges to the correct eigenvector (hence eigenvalue) without restrictions on the coupling constant and without the need for any resummation procedure.