Towards verifications of Krylov complexity

Krylov complexity is considered to provide a measure of the growth of operators evolving under Hamiltonian dynamics. The main strategy is the analysis of the structure of Krylov subspace KM (H, ) spanned by the multiple applications of the Liouville operator L defined by the commutator in terms of a Hamiltonian H, L: =H, acting on an operator, KM (H, ) =span\, L, , L^{M-1\}. For a given inner product (, ) of the operators, the orthonormal basis \Oₙ\ is constructed from O₀=/ (, ) by Lanczos algorithm. The moments ₘ= (O₀, LᵐO₀) are closely related to the important data \bₙ\ called Lanczos coefficients. I present the exact and explicit expressions of the moments \ₘ\ for 16 quantum mechanical systems which are exactly solvable both in the Schr\"odinger and Heisenberg pictures. The operator is the variable of the eigenpolynomials. Among them six systems show a clear sign of `non-complexity' as vanishing higher Lanczos coefficients bₘ=0, m3.