Resolution limits of analytic continuation from imaginary-time data
In plain words
Most electron simulations work in imaginary time, and converting their output into measurable spectra is mathematically unstable: tiny noise allows wildly different answers. The best possible resolution for a given noise level, and a way to certify an answer, are not established.
Precise statement
Given $G(\tau) = \operatorname{integral} \mathrm{d}w\, K(\tau, w) A(w)$ with $K = \operatorname{exp}(-\tau w)/(1 + \operatorname{exp}(-\beta w))$ on $0 \le \tau \le \beta$ and Gaussian noise of size $\sigma$, determine the optimal resolution of the spectral function $A(w)$ (minimal resolvable width and amplitude of a feature at frequency $w$) as a function of $\sigma, \beta \text{ and } w$, and construct estimators (maximum entropy, stochastic, Nevanlinna, Bayesian) with certified error bounds that reach it. An answer is proven bounds plus an estimator attaining them on benchmarks.
What would settle it
A theorem bounding achievable resolution from below and an estimator whose certified intervals match that bound on test spectra.
Status in the literature
Nevanlinna interpolation (Fei, Yeh and Gull 2021) enforces causality exactly but gives no certified bounds for noisy data.