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We consider an autocorrelation function of a quantum mechanical systemthrough the lens of the so-called recursive method, by iteratively evaluatingLanczos coefficients, or solving a system of coupled differential equations inthe Mori formalism. We first show that both methods are mathematicallyequivalent, each offering certain practical advantages. We then propose anapproximation scheme to evaluate the autocorrelation function, and use it toestimate the equilibration time $\tau$ for the observable in question. Withonly a handful of Lanczos coefficients as the input, this scheme yields anaccurate order of magnitude estimate of $\tau$, matching state-of-the-artnumerical approaches. We develop a simple numerical scheme to estimate theprecision of our method. We test our approach using several numerical examplesexhibiting different relaxation dynamics. Our findings provide a new practicalway to quantify the equilibration time of isolated quantum systems, a questionwhich is both crucial and notoriously difficult.

Estimation of equilibration time scales from nested fraction approximations