Polynomial Initial-State Jumps and Christoffel Transforms in Krylov Complexity

[2607.05294] Polynomial Initial-State Jumps and Christoffel Transforms in Krylov Complexity Skip to main content Search Submit Donate Log in High Energy Physics - Theory arXiv:2607.05294 (hep-th) [Submitted on 6 Jul 2026 ( v1 ), last revised 27 Aug 2026 (this version, v3)] Title: Polynomial Initial-State Jumps and Christoffel Transforms in Krylov Complexity Authors: Abhishek Chowdhury , Ajit Prasad Mahapatra View PDF HTML (experimental) 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 $\lvert\psi_Q\rangle=N_Q^{-1/2}Q(H)\lvert K_0\rangle$, 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. Comments: 147 pages, 2 tables. v3 adds double-scaled SYK chord dynamics, transition tomography, and all-root Nevai-class invariance with quantized subleading Jacobi asymptotics Subjects: High Energy Physics - Theory (hep-th) ; Mathematical Physics (math-ph); Commutative Algebra (math.AC); Spectral Theory (math.SP); Quantum Physics (quant-ph) MSC classes: 42C05, 33C45, 33D45, 47B36, 44A60, 81Q10 Cite as: arXiv:2607.05294 [hep-th] (or arXiv:2607.05294v3 [hep-th] for this version) https://doi.org/10.48550/arXiv.2607.05294 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Abhishek Chowdhury [ view email ] [v1] Mon, 6 Jul 2026 16:37:39 UTC (67 KB) [v2] Sun, 19 Jul 2026 16:08:57 UTC (110 KB) [v3] Thu, 27 Aug 2026 12:58:48 UTC (165 KB) Full-text links: Access Paper: View PDF HTML (experimental) TeX Source view license Current browse context: hep-th < prev | next > new | recent | 2026-07 Change to browse by: math math-ph math.AC math.MP math.SP quant-ph References & Citations INSPIRE HEP SciX Google Scholar Semantic Scholar export BibTeX citation Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer ( What is the Explorer? ) Connected Papers Toggle Connected Papers ( What is Connected Papers? ) Litmaps Toggle Litmaps ( What is Litmaps? ) scite.ai Toggle scite Smart Citations ( What are Smart Citations? ) Code, Data, Media Code, Data and Media Associated with this Article alphaXiv Toggle alphaXiv ( What is alphaXiv? ) Links to Code Toggle CatalyzeX Code Finder for Papers ( What is CatalyzeX? ) DagsHub Toggle DagsHub ( What is DagsHub? ) GotitPub Toggle Gotit.pub ( What is GotitPub? ) Huggingface Toggle Hugging Face ( What is Huggingface? ) ScienceCast Toggle ScienceCast ( What is ScienceCast? ) Demos Demos Replicate Toggle Replicate ( What is Replicate? ) Spaces Toggle Hugging Face Spaces ( What is Spaces? ) Spaces Toggle TXYZ.AI ( What is TXYZ.AI? ) Related Papers Recommenders and Search Tools Link to Influence Flower Influence Flower ( What are Influence Flowers? ) Core recommender toggle CORE Recommender ( What is CORE? ) IArxiv recommender toggle IArxiv Recommender ( What is IArxiv? ) Author Venue Institution Topic About arXivLabs arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. 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