arXiv:2611.13633v1 Announce Type: new Abstract: Spectral estimation seeks to recover unknown frequencies from $T$ snapshots, each consisting of $\Omega+1$ noisy Fourier measurements. Super-resolution refers to the recovery of frequencies whose minimum separation $\Delta$ is smaller than $1/\Omega$. We establish state-of-the-art stability bounds for single- and multi-snapshot ESPRIT when the frequencies form well-separated clumps, with the largest clump containing $n^\star$ frequencies. In particular, for single-snapshot ESPRIT, our error bound scales as $$ \mathcal{O}(\Omega^{-1}\mathrm{SRF}^{2n^\star-2}\cdot\mathrm{Noise}), $$ where $\mathrm{SRF}:=(\Omega\Delta)^{-1}$ denotes the super-resolution factor, provided that the noise level satisfies $\mathrm{Noise}\lesssim(\Omega\Delta)^{2n^\star-2}$. In comparison with the minimax bounds [arXiv:1904.09186] and the computational resolution limits [arXiv:2003.02917, arXiv:2211.15208], our result shows that single-snapshot ESPRIT achieves the optimal stability scaling in the super-resolution regime. For multi-snapshot ESPRIT, we establish error bounds under both deterministic noise and Gaussian white noise, attaining the best-known SRF dependence under weaker conditions. Numerical experiments corroborate the theoretical error bounds.