Polynomial Initial-State Jumps and Christoffel Transforms in Krylov Complexity

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, $\ket{ψ_Q}=Q(H)\ket{K_0}/\sqrt{N_Q}$ with $$N_Q=\langle K_0|Q(H)^\dagger Q(H)|K_0\rangle$$. The filtered spectral measure is the positive polynomial modification $|Q(E)|^2\mathrm dμ(E)/N_Q$, 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.