General upper bound for the Krylov break time

Determine whether the lower bound $t_\ast(m;\epsilon)\geq \tau_m[1-o(1)]$, with $\tau_m=\sum_{j<m}1/b_j$ for the return-amplitude error of an $m$-dimensional truncation of a Jacobi chain, is also an asymptotic upper bound, so that $t_\ast(m)\simeq\tau_m$ in general.

Background

The paper proves that causal propagation on the inhomogeneous Jacobi chain guarantees a reliable time window of at least the transport time $\tau_m$. An asymptotically matching upper bound would establish that the truncation error necessarily exceeds a prescribed tolerance on this same timescale. The authors explain that such saturation depends on whether the relevant spectral component transports, information that the Lieb–Robinson bound alone does not provide.

The paper later proves saturation for selected classes, including the homogeneous chain, trace-class perturbations of the homogeneous chain, and broadband absolutely continuous chains under additional assumptions. Thus, the general question of whether the lower bound is also an upper bound remains distinct from these special cases.