The reviewed record of science sign in
Pith

arxiv: 2607.05294 · v1 · pith:VFR4UYJY · submitted 2026-07-06 · hep-th · quant-ph

Polynomial Initial-State Jumps and Christoffel Transforms in Krylov Complexity

Abhishek Chowdhury , Ajit Prasad Mahapatra This is my paper

Reviewed by Pith2026-07-07 19:00 UTCglm-5.2pith:VFR4UYJYopen to challenge →

classification hep-th quant-ph
keywords krylovcomplexitypolynomialfiniteinitial-statemeasureamplitudescharlier
0
0 comments X

The pith

Polynomial seed jumps solved without re-running Lanczos

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper solves a specific problem in Krylov complexity theory: when you change the initial state of a quantum system while keeping the Hamiltonian fixed, you normally have to re-run the entire Lanczos algorithm to get the new Krylov basis, amplitudes, and complexity. The authors show that for a broad class of initial-state changes—those obtained by applying a polynomial Q(H) to the original reference state—this re-run is unnecessary. The key mechanism is a classical result from orthogonal polynomial theory called a Christoffel transform. Changing the seed by Q(H) multiplies the spectral measure by |Q(E)|^2, and orthogonality forces this reweighted measure's polynomials to live inside a finite band of the original polynomial family. Specifically, the product W_Q(E) R_n(E) = sum of at most 2*deg(Q)+1 adjacent reference polynomials P_m(E). This finite-band identity means each shifted Krylov amplitude is a finite linear combination of already-computed reference amplitudes. The authors derive a complementary formula: cumulative probabilities and spread complexity come from projecting the reference Christoffel-Darboux kernel away from the roots of Q, giving a finite-rank correction. Together, these two mechanisms—an amplitude connector and a projected kernel—constitute an exact relative calculus: one solved reference problem generates an entire family of polynomially related initial-state dynamics through finite algebra alone.

Core claim

The central object is the Christoffel connector, a finite-band identity W_Q(E) R_n^Q(E) = sum_{m=n}^{n+2deg Q} Gamma_{n,m}^Q P_m(E) that relates the orthogonal polynomials of the shifted measure to those of the reference measure. This identity, combined with a projected-kernel formula for cumulative probabilities, means that all shifted Krylov data—amplitudes, probabilities, spread complexity, and even the new Lanczos coefficients—can be read off from the reference problem's already-computed Fourier-orthogonal-polynomial moments I_m(t) without re-running Lanczos in the original Hilbert space. The bandwidth is fixed by the degree of the seed polynomial, not by the system size.

What carries the argument

Christoffel transform of spectral measures; finite-band connector identity; projected Christoffel-Darboux kernel; terminal quotient algebra for finite-dimensional chains; matrix-valued parent measure for seed families

If this is right

  • In numerical many-body physics, one reference Krylov computation could generate complexity data for an entire family of polynomially related initial states, saving substantial computational cost.
  • The framework extends to operator Krylov complexity by replacing H with the Liouvillian and the seed state with an operator, making nested-commutator descendants Q(L)O tractable without re-running operator-space Lanczos.
  • The tight-binding example shows that localized-site jumps at fixed Hamiltonian are Chebyshev Christoffel transforms, connecting this abstract spectral framework to concrete lattice dynamics.
  • Support loss—where the seed polynomial vanishes on spectral atoms—provides a precise algebraic mechanism for dimension reduction in finite Krylov chains, with arithmetic conditions governing when it occurs.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The restriction to polynomial seeds means the framework captures spectral (energy-filtering) deformations but not spatially local operator insertions, unless those insertions happen to lie in the reference cyclic subspace. Extending to non-polynomial seeds such as thermal filters e^{-beta H} would produce infinite-band connectors, suggesting a natural hierarchy from polynomial to rational to analy
  • The relative-filter cocycle of Proposition 2.1 suggests that composing polynomial seed changes should be associative and path-independent when support compatibility holds, which could enable efficient exploration of the seed-parameter landscape by composing simple jumps rather than recomputing from scratch.
  • The matrix-valued parent measure construction could serve as a natural geometric object for studying how Krylov complexity varies continuously over the projective space of seed directions, even though each direction defines its own nonlinear orthogonal-polynomial family.

Load-bearing premise

The new initial state must lie in the cyclic subspace of the reference pair (H, |K_0>) and be expressible as Q(H)|K_0> for a fixed-degree polynomial Q. If the seed change involves components outside this cyclic subspace, or if the polynomial degree grows with system size, the finite-band structure and the no-re-run advantage do not apply.

What would settle it

If one constructed a polynomial seed jump Q(H)|K_0> where Q has fixed degree, and the shifted amplitudes could not be recovered from the reference moments I_m(t) via a finite-band connector with bandwidth 2*deg(Q), the central claim would fail. Alternatively, if the projected-kernel formula for cumulative probabilities did not match direct Lanczos computation of the shifted problem, the construction would be invalid.

read the original abstract

State Krylov, or spread, complexity is a property of a pair $(H,\ket{K_0})$ rather than of the Hamiltonian alone. Thus, changing the initial state at fixed $H$ generally changes the Lanczos coefficients and the ordered Krylov basis. We solve this relative initial-state problem for normalized polynomial filters, $\ket{\psi_Q}=Q(H)\ket{K_0}/\sqrt{N_Q}$ with $N_Q=\langle K_0|Q(H)^\dagger Q(H)|K_0\rangle$. The filtered spectral measure is the positive polynomial modification $|Q(E)|^2\mathrm d\mu(E)/N_Q$, and orthogonality turns this measure change into a finite-band transfer from reference Fourier-orthogonal-polynomial moments to shifted Krylov amplitudes. We derive exact finite sums for individual amplitudes and projected Christoffel-Darboux kernels for cumulative probabilities and spread complexity. The formulae cover confluent roots, complex seed coefficients, support loss, and terminal quotients in finite dimensions. We evaluate the construction in three canonical Jacobi families, the Heisenberg--Weyl/Charlier oscillator, the compact $SU(2)$/Krawtchouk spin, and the constant-coefficient tight-binding/Chebyshev chain, with a Hermite central-limit scaling of Charlier as a continuous-spectrum check of this Christoffel jump machinery. Finite seed families are organized by a matrix-valued parent measure whose scalar compressions recover the individual shifted problems. The fixed-inner-product construction carries over to operator Krylov complexity after the replacement \(H\mapsto\mathcal L\) and \(\ket{K_0}\mapsto O\); polynomial seeds then become nested-commutator descendants \(Q(\mathcal L)O\). The result is an exact relative calculus in which a solved cyclic problem generates a family of polynomially related initial-state dynamics without repeating Lanczos in the original Hilbert space.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 7 minor

Summary. This paper solves the relative initial-state problem for Krylov (spread) complexity when the new seed is a polynomial filter of the reference seed, $|ψ_Q⟩ ∝ Q(H)|K_0⟩$. The key observation is that the spectral measure of the shifted seed is a positive polynomial (Christoffel-type) modification of the reference measure. The authors prove that this measure change induces a finite-band transfer (Proposition 3.1) from reference Fourier–orthogonal-polynomial moments to shifted Krylov amplitudes, with bandwidth $L = 2 deg Q$. They also derive a finite-rank projected Christoffel–Darboux kernel (Proposition 4.1) for cumulative probabilities and spread complexity. The construction is verified in three canonical Jacobi families (Charlier, Krawtchouk, Chebyshev) plus a Hermite scaling limit, and extended to operator Krylov complexity via Liouvillian spectral filters.

Significance. The paper provides a parameter-free, exact algebraic calculus for a broad class of initial-state changes at fixed Hamiltonian. The central results—finite-band amplitude transfer and finite-rank projected kernels—are proved from standard orthogonal polynomial theory and are self-contained. The three solvable examples provide genuine independent cross-checks: Charlier recurrence coefficients (Eq. 5.8) verified against direct Lanczos, a finite SU(2) N=3 case (Eqs. E.5–E.11) checked via shifted weights, quotient reduction, and kernel projection, and tight-binding amplitudes (Eq. 5.46) matched against sine-transform lattice dynamics. The extension to operator Krylov complexity (Appendix H) and matrix-valued parent measures (Section 6) broadens the scope. The practical claim that one avoids re-running Lanczos in the original Hilbert space is well-supported and potentially useful for numerical many-body surveys of seed deformations.

minor comments (7)
  1. §5.1, Eq. (5.8): The shifted recurrence coefficients $˜a^{(1)}_0 = 1$, $(˜b^{(1)}_1)^2 = 3λ$, $˜a^{(1)}_1 = 4/3$, $(˜b^{(1)}_2)^2 = 2λ + 8/9$ are stated without showing the intermediate algebra. While Appendix D provides the derivation, a one-line pointer in the main text (e.g., 'see Appendix D for the confluent determinant derivation') would improve readability.
  2. §4, Eq. (4.27): The first-jump complexity formula $K_1(τ)$ involves the quantities $A_L = P_L(0)$ and $D_L = h_L K^μ_L(0,0)$. The notation $D_L$ is also used for the shifted norm $˜h^{(1)}_n$ in nearby equations (e.g., Eq. 5.17 uses $D_n$ for the same object). This is consistent but could confuse readers who encounter both $D_n$ and $˜h^{(1)}_n$ in the same subsection.
  3. §5.3, Eq. (5.49): The closed-form reference complexity $K_0(t)$ is stated using Rayleigh–Lommel sums (Eq. 5.48). The reference [30] is cited as the DLMF, but the specific identities used are not directly tabulated there in this form. A more precise citation or a brief derivation sketch would help readers verify this resummation.
  4. Appendix D, Eq. (D.34): The upper bound $K_1(τ) ≤ (3 + 1/λ)κ(τ)$ is derived. It would be useful to state whether this bound is tight in any limit (e.g., small $τ$ or large $λ$), or whether it is primarily a convergence tool.
  5. §6, Proposition 6.1: The matrix-valued parent measure $dM(E)$ is introduced for a finite seed block $V_0 = (|K_0⟩, ..., |K_R⟩)$. The statement that 'scalar Krylov complexity is obtained after choosing a direction $c$' is clear, but the relationship between the matrix-valued OP theory (refs [33, 34]) and the scalar compression could be made more explicit—specifically, whether block Lanczos on $V_0$ would produce the same matrix moments $M_k$.
  6. Typographical: In §2, Eq. (2.26), the polynomial division identity for $m_Q(z)$ uses $μ_j$ for reference moments, while Eq. (2.25) defines $μ_j = ∫ E^j dμ(E)$. The same symbol $μ$ is used for the measure and the moments; while context disambiguates, a distinct symbol (e.g., $m_j$) for moments would reduce notational friction.
  7. Appendix H, Eq. (H.11): The finite-temperature inner product $(A|B)^g_β$ is written schematically with a function $g(λ)$. The conditions on $g$ for positivity and Liouvillian self-adjointness are stated qualitatively but not specified. A reference or a brief statement of the standard conditions would be helpful.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for a careful and constructive reading of our manuscript. The referee's summary accurately captures the main results (finite-band amplitude transfer, projected Christoffel-Darboux kernels, three canonical Jacobi families, operator extension) and the recommendation is minor revision. The report contains no numbered major comments; we address the substantive points raised in the summary and significance sections below.

read point-by-point responses

  1. Referee: The referee's report contains no numbered major comments. We interpret the substantive content of the summary and significance sections as the points to address.

    Authors: We agree with the referee's characterization of the paper. The summary correctly identifies the central results: Proposition 3.1 (finite-band transfer with bandwidth L = 2 deg Q), Proposition 4.1 (finite-rank projected CD kernel), the three solvable families (Charlier, Krawtchouk, Chebyshev) with the Hermite scaling limit, and the operator Krylov extension via Liouvillian spectral filters. We confirm that the cross-checks mentioned in the significance section are all present in the manuscript: the Charlier recurrence coefficients (Eq. 5.8) verified against direct Lanczos, the finite SU(2) N=3 case (Eqs. E.5-E.11) checked via shifted weights, quotient reduction, and kernel projection, and the tight-binding amplitudes (Eq. 5.46) matched against sine-transform lattice dynamics. No revision is needed to the scientific content. We will make minor presentational improvements in the revised version: (1) adding a brief roadmap sentence at the end of the introduction clarifying which appendices contain which cross-checks, (2) ensuring consistent notation between the main text and appendices for the Moore-Penrose inverse symbol, and (3) adding one sentence in Section 5.3 explicitly noting that the tight-binding amplitudes of Eq. 5.46 reproduce the localized-state results of Ref. [4] in the appropriate limit, to make this independent check more visible. revision: partial

Circularity Check

0 steps flagged

No circularity found. The derivation is parameter-free and self-contained.

full rationale

The paper's central derivation chain is mathematically self-contained and does not reduce to its inputs by construction. Proposition 3.1 (finite-band transfer) is proved from orthogonality: ∫ P_m W_Q R_n^[Q] dμ = ∫ P_m R_n^[Q] dν_Q = 0 for m < n (eq. 3.4), which is a direct consequence of the shifted measure definition (eq. 2.10) and the spectral theorem — not a fitted input renamed as prediction. The connector coefficients Γ are determined by linear systems from root constraints (eq. 3.14) or Gram matrices (eq. 3.16), not by fitting to target amplitudes. Proposition 3.2 (connector dictionary) follows from coefficient comparison of two three-term recurrences. Proposition 4.1 (projected kernel) is a standard reproducing-kernel result under measure modification, proved via isometry of multiplication by Q̂. The three solvable examples provide genuine independent cross-checks: Charlier shifted recurrence coefficients (eq. 5.8) are verified against direct Lanczos recursion; the finite SU(2) N=3 case (eqs. E.5–E.11) is checked via shifted weights, quotient reduction, and kernel projection independently; tight-binding amplitudes (eq. 5.46) are checked against sine-transform lattice dynamics from ref. [4] (external authors). The only self-citation is [40] (Chowdhury & Mahapatra), which appears solely in the outlook section (section 7) regarding time-dependent extensions and is not load-bearing for any proposition or result in the paper. The mathematical foundations rest on standard external references for orthogonal polynomial theory ([16–19, 21–22, 11]). No step in the derivation chain reduces to its own inputs by definition, fit, or self-citation.

Axiom & Free-Parameter Ledger

0 free parameters · 4 axioms · 0 invented entities

No free parameters are fitted or introduced ad hoc. No new physical entities, particles, or forces are postulated. The framework is a parameter-free algebraic identity in the reference spectral data. The axioms are standard mathematical conditions (Carleman's criterion, moment finiteness) or domain-scoping assumptions (polynomial seed, self-adjoint Liouvillian) that are clearly stated.

axioms (4)
  • domain assumption The new initial state lies in the cyclic subspace of (H, |K_0⟩) and is expressible as Q(H)|K_0⟩ for a polynomial Q of fixed degree.
    Stated in eq. (1.1) and section 2. This is the scope-defining assumption; without it the finite-band transfer does not apply.
  • standard math The reference spectral measure μ is determinate (polynomials dense in L²(μ)), ensured by Carleman's criterion for noncompact support.
    Invoked in section 2 to identify the Krylov closure with the full spectral cyclic subspace. Standard in Jacobi operator theory.
  • domain assumption For operator Krylov extension, the Liouvillian L = [H, ·] is self-adjoint with respect to the chosen operator inner product.
    Stated in appendix H. Required for the scalar Christoffel construction to carry over to Liouville space.
  • standard math The first probability moment Σ n|φ̃_n|² is finite for the complexity tail identity to hold in infinite dimension.
    Stated after eq. (4.4). Verified for Charlier/Hermite examples but flagged as an additional condition in general.

pith-pipeline@v1.1.0-glm · 52799 in / 2751 out tokens · 197631 ms · 2026-07-07T19:00:10.370475+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

50 extracted references · 50 canonical work pages · 22 internal anchors

  1. [1]

    Quantum chaos and the complexity of spread of states

    V. Balasubramanian, P. Caputa, J. M. Magan and Q. Wu, “Quantum chaos and the complexity of spread of states,” Phys. Rev. D106(2022) 046007, arXiv:2202.06957

    work page internal anchor Pith review Pith/arXiv arXiv 2022

  2. [2]

    State Dependence of Krylov Complexity in $2d$ CFTs

    A. Kundu, V. Malvimat and R. Sinha, “State dependence of Krylov complexity in2dCFTs,” JHEP09(2023) 011, arXiv:2303.03426

    work page internal anchor Pith review Pith/arXiv arXiv 2023

  3. [3]

    Dependence of Krylov complexity on the initial operator and state

    P. G. Sreeram, J. Bharathi Kannan, R. Modak and S. Aravinda, “Dependence of Krylov complexity on the initial operator and state,” arXiv:2503.03400

    work page internal anchor Pith review Pith/arXiv arXiv

  4. [4]

    Variations on a theme of Krylov,

    V. Balasubramanian, P. Caputa and J. Simón, “Variations on a theme of Krylov,” arXiv:2511.03775

    work page arXiv

  5. [5]

    Lanczos Meets Orthogonal Polynomials,

    L.-C. Qu, “Lanczos Meets Orthogonal Polynomials,” arXiv:2512.15857

    work page arXiv

  6. [6]

    Krylov Distribution,

    M. Alishahiha and M. J. Vasli, “Krylov Distribution,” arXiv:2602.06150

    work page arXiv

  7. [7]

    Krylov complexity and orthogonal polynomials

    W. Mück and Y. Yang, “Krylov complexity and orthogonal polynomials,” Nucl. Phys. B984 (2022) 115948, arXiv:2205.12815

    work page internal anchor Pith review Pith/arXiv arXiv 2022

  8. [8]

    A Universal Operator Growth Hypothesis

    D. E. Parker, X. Cao, A. Avdoshkin, T. Scaffidi and E. Altman, “A universal operator growth hypothesis,” Phys. Rev. X9(2019) 041017, arXiv:1812.08657

    work page internal anchor Pith review Pith/arXiv arXiv 2019

  9. [9]

    Quantum Dynamics in Krylov Space: Methods and Applications

    P. Nandy, A. S. Matsoukas-Roubeas, P. Martínez-Azcona, A. Dymarsky and A. del Campo, “Quantum dynamics in Krylov space: methods and applications,” Phys. Rept.1125–1128 (2025) 1–82, arXiv:2405.09628

    work page internal anchor Pith review Pith/arXiv arXiv 2025

  10. [10]

    Krylov Complexity

    E. Rabinovici, A. Sánchez-Garrido, R. Shir and J. Sonner, “Krylov complexity,” arXiv:2507.06286

    work page internal anchor Pith review Pith/arXiv arXiv

  11. [11]

    The Christoffel-Darboux Kernel

    B. Simon, “The Christoffel–Darboux kernel,” inPerspectives in Partial Differential Equations, Harmonic Analysis and Applications, Proc. Sympos. Pure Math.79, AMS, 2008, pp. 295–335, arXiv:0806.1528

    work page internal anchor Pith review Pith/arXiv arXiv 2008

  12. [12]

    The uncertainty relation between energy and time in non-relativistic quantum mechanics,

    L. Mandelstam and I. Tamm, “The uncertainty relation between energy and time in non-relativistic quantum mechanics,” J. Phys. (USSR)9(1945) 249–254. – 58 –

    work page 1945

  13. [13]

    Quantum speed limits: from Heisenberg's uncertainty principle to optimal quantum control

    S. Deffner and S. Campbell, “Quantum speed limits: from Heisenberg’s uncertainty principle to optimal quantum control,” J. Phys. A50(2017) 453001, arXiv:1705.08023

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  14. [14]

    Ultimate Speed Limits to the Growth of Operator Complexity

    N. Hörnedal, N. Carabba, A. S. Matsoukas-Roubeas and A. del Campo, “Ultimate speed limits to the growth of operator complexity,” Commun. Phys.5(2022) 207, arXiv:2202.05006

    work page internal anchor Pith review Pith/arXiv arXiv 2022

  15. [15]

    Speed Limits and Scrambling in Krylov Space

    A. Gill and T. Sarkar, “Speed limits and scrambling in Krylov space,” Phys. Rev. B111 (2025) 184307, arXiv:2408.06855

    work page internal anchor Pith review Pith/arXiv arXiv 2025

  16. [16]

    T. S. Chihara,An Introduction to Orthogonal Polynomials, Gordon and Breach, 1978

    work page 1978

  17. [17]

    Szegő,Orthogonal Polynomials, AMS Colloquium Publications, Vol

    G. Szegő,Orthogonal Polynomials, AMS Colloquium Publications, Vol. 23, 4th ed., 1975

    work page 1975

  18. [18]

    Matrices, moments and quadrature,

    G. H. Golub and G. Meurant, “Matrices, moments and quadrature,” inNumerical Analysis 1993, D. F. Griffiths and G. A. Watson eds., Pitman Research Notes in Mathematics Series 303, Longman, 1994, pp. 105–156

    work page 1993

  19. [19]

    Simon,Orthogonal Polynomials on the Real Line, Part 1: Classical Theory, AMS Colloquium Publications, 2005

    B. Simon,Orthogonal Polynomials on the Real Line, Part 1: Classical Theory, AMS Colloquium Publications, 2005

    work page 2005

  20. [20]

    G. B. Folland,Real Analysis: Modern Techniques and Their Applications, 2nd ed., Wiley, New York, 1999

    work page 1999

  21. [21]

    Darboux transformation and perturbation of linear functionals,

    M. I. Bueno and F. Marcellan, “Darboux transformation and perturbation of linear functionals,” Linear Algebra Appl.384(2004) 215–242

    work page 2004

  22. [22]

    m-functions and inverse spectral analysis for finite and semi-infinite Jacobi matrices,

    F. Gesztesy and B. Simon, “m-functions and inverse spectral analysis for finite and semi-infinite Jacobi matrices,” J. Analyse Math.73(1997) 267–297

    work page 1997

  23. [23]

    Gautschi,Orthogonal Polynomials: Computation and Approximation, Oxford University Press, 2004

    W. Gautschi,Orthogonal Polynomials: Computation and Approximation, Oxford University Press, 2004

    work page 2004

  24. [24]

    B. N. Parlett,The Symmetric Eigenvalue Problem, SIAM, 1998

    work page 1998

  25. [25]

    Ben-Israel and T

    A. Ben-Israel and T. N. E. Greville,Generalized Inverses: Theory and Applications, 2nd ed., Springer, 2003

    work page 2003

  26. [26]

    A. M. Perelomov,Generalized Coherent States and Their Applications, Springer-Verlag, Berlin, 1986

    work page 1986

  27. [27]

    Second quantization representation for classical many-particle system,

    M. Doi, “Second quantization representation for classical many-particle system,” J. Phys. A9 (1976) 1465–1477

    work page 1976

  28. [28]

    Path integral approach to birth-death processes on a lattice,

    L. Peliti, “Path integral approach to birth-death processes on a lattice,” J. Physique46(1985) 1469–1483

    work page 1985

  29. [29]

    One-parameter extension of the Doi-Peliti formalism and relation with orthogonal polynomials

    J. Ohkubo, “One-parameter extension of the Doi–Peliti formalism and its relation with orthogonal polynomials,” Phys. Rev. E86(2012) 042102, arXiv:1206.0062

    work page internal anchor Pith review Pith/arXiv arXiv 2012

  30. [30]

    NIST Digital Library of Mathematical Functions, F. W. J. Olver, A. B. Olde Daalhuis, D. W. Lozier, B. I. Schneider, R. F. Boisvert, C. W. Clark, B. V. Saunders, H. S. Cohl and M. A. McClain, eds., DLMF

    work page

  31. [31]

    The differential equations of birth-and-death processes, and the Stieltjes moment problem,

    S. Karlin and J. McGregor, “The differential equations of birth-and-death processes, and the Stieltjes moment problem,” Trans. Amer. Math. Soc.85(1957) 489–546

    work page 1957

  32. [32]

    M. D. de la Iglesia,Orthogonal Polynomials in the Spectral Analysis of Markov Processes, Cambridge University Press, 2021

    work page 2021

  33. [33]

    The Analytic Theory of Matrix Orthogonal Polynomials

    D. Damanik, A. Pushnitski and B. Simon, “The analytic theory of matrix orthogonal polynomials,” Surv. Approx. Theory4(2008) 1–85, arXiv:0711.2703. – 59 –

    work page internal anchor Pith review Pith/arXiv arXiv 2008

  34. [34]

    Christoffel transformations for matrix orthogonal polynomials in the real line and the non-Abelian 2D Toda lattice hierarchy

    C. Álvarez-Fernández, G. Ariznabarreta, J. C. García-Ardila, M. Mañas and F. Marcellán, “Christoffel transformations for matrix orthogonal polynomials in the real line and the non-Abelian 2D Toda lattice hierarchy,” Int. Math. Res. Notices2017(2017) 1285–1341, arXiv:1511.04771

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  35. [35]

    Multiseed Krylov complexity

    B. Craps, O. Evnin and G. Pascuzzi, “Multiseed Krylov complexity,” Phys. Rev. Lett.134 (2025) 050402, arXiv:2409.15666

    work page internal anchor Pith review Pith/arXiv arXiv 2025

  36. [36]

    Loop groups and equations of KdV type,

    G. Segal and G. Wilson, “Loop groups and equations of KdV type,” Publ. Math. IHES61 (1985) 5–65

    work page 1985

  37. [37]

    Toda-Darboux maps and vertex operators

    M. Adler and P. van Moerbeke, “Toda–Darboux maps and vertex operators,” Int. Math. Res. Notices1998(1998) 489–511, arXiv:solv-int/9712016

    work page internal anchor Pith review Pith/arXiv arXiv 1998

  38. [38]

    Darboux transforms on Band Matrices, Weights and associated Polynomials

    M. Adler and P. van Moerbeke, “Darboux transforms on band matrices, weights and associated polynomials,” Int. Math. Res. Notices2001(2001) 935–984, arXiv:nlin/0010048

    work page internal anchor Pith review Pith/arXiv arXiv 2001

  39. [39]

    Rational spectral transformations and orthogonal polynomials,

    A. Zhedanov, “Rational spectral transformations and orthogonal polynomials,” J. Comput. Appl. Math.85(1997) 67–86

    work page 1997

  40. [40]

    Krylov Complexity of Optical Hamiltonians

    A. Chowdhury and A. Mahapatra, “Krylov Complexity of Optical Hamiltonians,” arXiv:2409.04156

    work page internal anchor Pith review Pith/arXiv arXiv

  41. [41]

    Krylov Dynamics and Operator Growth in Time-Dependent Systems via Lie Algebras

    A. Grabarits, E. Medina-Guerra and A. del Campo, “Krylov dynamics and operator growth in time-dependent systems via Lie algebras,” arXiv:2605.05290

    work page internal anchor Pith review Pith/arXiv arXiv

  42. [42]

    An iteration method for the solution of the eigenvalue problem of linear differential and integral operators,

    C. Lanczos, “An iteration method for the solution of the eigenvalue problem of linear differential and integral operators,” J. Res. Natl. Bur. Stand.45(1950) 255–282

    work page 1950

  43. [43]

    Some functions that generalize the Krall–Laguerre polynomials,

    F. A. Grünbaum, L. Haine and E. Horozov, “Some functions that generalize the Krall–Laguerre polynomials,” J. Comput. Appl. Math.106(1999) 271–297

    work page 1999

  44. [44]

    Orthogonal polynomials satisfying higher-order difference equations,

    A. J. Durán, “Orthogonal polynomials satisfying higher-order difference equations,” Constr. Approx.36(2012) 459–486

    work page 2012

  45. [45]

    The Lanczos algorithm with selective orthogonalization,

    B. N. Parlett and D. S. Scott, “The Lanczos algorithm with selective orthogonalization,” Math. Comp.33(1979) 217–238

    work page 1979

  46. [46]

    Krylov complexity of purification,

    R. N. Das and T. Mori, “Krylov complexity of purification,” arXiv:2408.00826

    work page arXiv

  47. [47]

    Bratteli and D

    O. Bratteli and D. W. Robinson,Operator Algebras and Quantum Statistical Mechanics 1:C ∗- andW ∗-Algebras, Symmetry Groups, Decomposition of States, 2nd ed., Springer, 1987

    work page 1987

  48. [48]

    Krylov complexity of density matrix operators

    P. Caputa, H.-S. Jeong, S. Liu, J. F. Pedraza and L.-C. Qu, “Krylov complexity of density matrix operators,” JHEP05(2024) 337, arXiv:2402.09522

    work page internal anchor Pith review Pith/arXiv arXiv 2024

  49. [49]

    Scaling Relations of Spectrum Form Factor and Krylov Complexity at Finite Temperature

    C. Tan, Z. Wei and R. Zhang, “Scaling relations of spectrum form factor and Krylov complexity at finite temperature,” arXiv:2401.10499

    work page internal anchor Pith review Pith/arXiv arXiv

  50. [50]

    On Krylov complexity in open systems: an approach via bi-Lanczos algorithm

    A. Bhattacharya, P. Nandy, P. P. Nath and H. Sahu, “On Krylov complexity in open systems: an approach via bi-Lanczos algorithm,” JHEP12(2023) 066, arXiv:2303.04175. – 60 –

    work page internal anchor Pith review Pith/arXiv arXiv 2023