Polynomial Initial-State Jumps and Christoffel Transforms in Krylov Complexity
Abstract
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 lvertψQ⟩=NQ−1/2Q(H)lvert K0⟩, 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 Qrvert2 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 Kr(t)≥ K0(t), with strict inequality for r≥1 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 DrH and Drch(t) for the mean absolute Fock and chord displacements, respectively, the bounds are Kr≥ DrH and Kr(t)≥ Drch(t)≥lvertn̅r(t)−rrvert. Thus rebuilt spread bounds physical displacement and signed chord drift. At fixed 0≤ q<1 and time, Kr(t)−Drch(t)→0 as r→∞, 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.
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BibTeX
@misc{chowdhury2026polynomial,
title = {{Polynomial Initial-State Jumps and Christoffel Transforms in Krylov Complexity}},
author = {Abhishek Chowdhury and Ajit Prasad Mahapatra},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2607.05294},
eprint = {2607.05294},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.05294},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Polynomial Initial-State Jumps and Christoffel Transforms in Krylov Complexity
AU - Abhishek Chowdhury
AU - Ajit Prasad Mahapatra
PY - 2026
DA - 2026/08/27
PB - arXiv
DO - 10.48550/arxiv.2607.05294
UR - https://arxiv.org/abs/2607.05294
AB - 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⟩=N_Q^−1/2Q(H)lvert K_0⟩, 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 Qrvert^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)≥ K_0(t), with strict inequality for r≥1 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^H and D_r^ch(t) for the mean absolute Fock and chord displacements, respectively, the bounds are K_r≥ D_r^H and K_r(t)≥ D_r^ch(t)≥lvertn̅_r(t)−rrvert. Thus rebuilt spread bounds physical displacement and signed chord drift. At fixed 0≤ q<1 and time, K_r(t)−D_r^ch(t)→0 as r→∞, 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.
ER -