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It is well known that quantum-mechanical perturbation theory often give riseto divergent series that require proper resummation. Here I discuss simple waysin which these divergences can be avoided in the first place. Using theelementary technique of relaxed fixed-point iteration, I obtain convergentexpressions for various challenging ground states wavefunctions, includingquartic, sextic and octic anharmonic oscillators, the hydrogenic Zeemanproblem, and the Herbst-Simon Hamiltonian (with finite energy but vanishingRayleigh-Schr\"odinger coefficients), all at arbitarily strong coupling. Theseresults challenge the notion that non-analytic functions of coupling constantsare intrinsically "non-perturbative". A possible application to exactdiagonalization is briefly discussed.

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