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

• URL: http://arxiv.org/abs/2105.04972v6
• Date: Thu, 27 Oct 2022 10:15:47 GMT
• Title: Perturbation theory without power series: iterative construction of non-analytic operator spectra
• Authors: Matteo Smerlak
• Abstract summary: 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.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• 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.

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