Polynomial Initial-State Jumps and Christoffel Transforms in Krylov Complexity

Abstract

State Krylov, or spread, complexity belongs to the cyclic pair $(H,|K_0⟩)$, so changing the initial state at fixed-$H$ reorganizes the Lanczos chain within the reference cyclic subspace. For normalized polynomial descendants $|\ddot{I}{ˆ}_Q⟩\propto Q(H)|K_0⟩$, this reorganization is the positive Christoffel reweighting of the reference measure by $|Q{|}^2$. Orthogonality gives a finite-band transfer from reference Fourier-orthogonal-polynomial moments to shifted amplitudes, while a finite-rank projection of its Christoffel-Darboux kernel yields cumulative probabilities and spread complexity. Complex superpositions, confluent roots, deletion of spectral atoms and terminal closure enter the same construction. In the Heisenberg-Weyl/Charlier oscillator, root-free remainder recurrences govern arbitrary number-state jumps. Their large-index behavior distinguishes generic shifts from resonant deletion of Poisson atoms, determines the Jacobi asymptotics, and proves the finiteness of the complexity at finite time for every fixed jump. In finite $SU(2)$/Krawtchouk and tight-binding/Chebyshev chains, product identities and product-Gram factorizations in the terminal quotient determine all weight-state and localized-site connectors through the terminal edge, while Weyl reflection pairs opposite spin weights. The first-jump Charlier-Hermite scaling carries it to continuous spectral support. Finite seed families admit a matrix-valued parent measure, and the relative calculus extends to polynomial operator descendants whenever the Liouvillian has a self-adjoint realization for the chosen inner product. A solved cyclic problem determines a family of fixed-$H$ dynamics and separates preparation dependence from changes of the generator or Hilbert-space dimension.

103 pages, 2 tables. v2 substantially extends the analysis to arbitrary Charlier jumps, terminal finite chains, mixed states, and Liouville space