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arXiv · 2105.04972

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

Matteo Smerlak
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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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Matteo Smerlak. 2021-05-11. Perturbation theory without power series: iterative construction of non-analytic operator spectra. https://doi.org/10.1209/0295-5075%2Fac9e72

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