AI 中文总结
研究固定哈密顿量下改变初态时的 Krylov 复杂度问题，通过归一化多项式滤波器求解，推导了相关精确有限和及投影 Christoffel - Darboux 核公式，并在多个模型中评估，还拓展到算子 Krylov 复杂度。
AI 中文摘要
态 Krylov 或展布复杂度是一对\((H,\ket{K_0})\)的性质，而非仅哈密顿量的。固定\(H\)改变初态会改变 Lanczos 系数和有序 Krylov 基。本文针对归一化多项式滤波器解决了此相对初态问题，推导了精确有限和等公式，在多模型中评估，还拓展到算子 Krylov 复杂度。
英文摘要
At fixed Hamiltonian $H$, changing the initial state changes the cyclic pair and hence generally the Lanczos basis and spread. For every normalizable polynomial preparation $\lvertψ_Q\rangle=N_Q^{-1/2}Q(H)\lvert K_0\rangle$, we reconstruct its state-Krylov problem exactly from reference cyclic data. The transfer assumes no integrability and applies on finite or infinite cyclic support. Reweighting the reference spectral measure by $\lvert Q\rvert^2$ gives a finite-band connector for every prepared amplitude and a finite-rank Christoffel-Darboux projection for cumulative probabilities and spread, without rerunning ambient-space Lanczos. Fixed-degree seeds preserve the limiting Jacobi coefficients of asymptotically constant chains. Every Charlier number-state jump obeys $K_r(t)\ge K_0(t)$, with strict inequality for $r\ge1$ away from revivals. At every fixed finite $r$, the Hermite endpoint has exact all-level amplitudes, whereas the continuous-$q$-Hermite chord chain of double-scaled SYK has an all-level root-free re-Lanczos reconstruction. Writing $D_r^{\rm H}$ and $D_r^{\mathrm{ch}}(t)$ for the mean absolute Fock and chord displacements, respectively, the bounds are $K_r\ge D_r^{\rm H}$ and $K_r(t)\ge D_r^{\mathrm{ch}}(t)\ge\lvert\overline n_r(t)-r\rvert$. Thus rebuilt spread bounds physical displacement and signed chord drift. At fixed $0\le q<1$ and time, $K_r(t)-D_r^{\mathrm{ch}}(t)\to0$ as $r\to\infty$, with both approaching the same folded-Bessel limit. In Liouville space, the exact unnormalized gap measure family on an open inverse-temperature interval determines the positive transition-resolved measure, assuming the thermal kernel is known and strictly positive and the requisite exponential moments exist. One cyclic solution can therefore be reused across polynomial seeds while separating physical propagation, basis response, and rebuilt spread.
Comments147 pages, 2 tables. v3 adds double-scaled SYK chord dynamics, transition tomography, and all-root Nevai-class invariance with quantized subleading Jacobi asymptotics
