Perturbation theory without power series: iterative construction of non-analytic operator spectra

TL;DR

This paper introduces an iterative method to construct convergent, non-analytic operator spectra in quantum mechanics, avoiding divergence issues of traditional perturbation series, and applies it to various challenging systems at strong coupling.

Contribution

It presents a simple fixed-point iteration technique that yields convergent solutions for complex quantum systems, challenging the idea that non-analytic functions are inherently non-perturbative.

Findings

Successfully applied to anharmonic oscillators at strong coupling

Derived convergent expressions for the hydrogenic Zeeman problem

Addressed the Herbst-Simon Hamiltonian with finite energy

Abstract

It is well known that quantum-mechanical perturbation theory often give rise to divergent series that require proper resummation. Here I discuss simple ways in which these divergences can be avoided in the first place. Using the elementary technique of relaxed fixed-point iteration, I obtain convergent expressions for various challenging ground states wavefunctions, including quartic, sextic and octic anharmonic oscillators, the hydrogenic Zeeman problem, and the Herbst-Simon Hamiltonian (with finite energy but vanishing Rayleigh-Schr\"odinger coefficients), all at arbitarily strong coupling. These results challenge the notion that non-analytic functions of coupling constants are intrinsically "non-perturbative". A possible application to exact diagonalization is briefly discussed.